Nicolas Burq

Jean-Marc Delort

Carlos Kenig

Daniel Tataru

Tatiana Sandrine

Benoît Je Remercie Joseph

Agnieszka Bojanowska-Jackowska

Marcin Kuczma

Stanis Law Kwapień

Krzysz- Tof Oleszkiewicz

Waldemar Pa Luba

Henryk Toruńczyk

Jaros Law Wiśniewski

Davide Barilari

Jean-Yves Chemin

Patrick Gérard

Radu Ignat

Constantin Pankrashkin

Geneviève Raugel

Filippo Santambrogio

Passionnantes, En Particulier Micha

Julianna Pawe

Piotr K Kasia

Robert Et L'

n'ai pas encore mentionnés, pour les bons moments passés ensemble et les conversations

Keywords: Équation des ondes, non linéarité énergie-critique, explosion, multi-soliton Wave equation, energy-critical nonlinearity, blow-up, multisoliton

i Sur la dynamique d'équations des ondes avec non-linéarité énergie critique focalisante Résumé. Cette thèse est consacrée à l'étude du comportement global des solutions de l'équation des ondes énergie-critique focalisante. On s'intéresse tout spécialement à la description de la dynamique du système dans l'espace d'énergie. Nous développons une variante de la méthode d'énergie qui permet de construire des solutions explosives de type II, instables. Ensuite, par une démarche similaire, nous donnons le premier exemple d'une solution radiale de l'équation des ondes énergie-critique qui converge dans l'espace d'énergie vers une superposition de deux états stationnaires (bulles). En appliquant notre méthode au cas de l'équation des ondes des applications harmoniques (wave map), nous obtenons des solutions de type bulle-antibulle, en toute classe d'équivariance k > 2. Pour l'équation des ondes énergie-critique radiale, nous étudions également le lien entre la vitesse de l'explosion de type II et la limite faible de la solution au moment de l'explosion. Finalement, nous montrons qu'il est impossible qu'une solution radiale converge vers une superposition de deux bulles ayant les signes opposés.

Chapitre I Introduction (version française)

1 Généralités sur des équations des ondes L'objet de ce mémoire est d'étudier certains phénomènes non linéaires dans le comportement dynamique de solutions d'équations des ondes. On considère des solutions radiales de l'équation des ondes avec la non-linéarité focalisante énergie critique en dimension N ≥ 3 :

∂ 2 t u(t, x) = ∆u(t, x) + |u(t, x)| 4 N -2 u(t, x), (t, x) ∈ R × R N , (NLW)
et des solutions k-équivariantes de l'équation des ondes des applications harmoniques (wave map) de R 1+2 dans la sphère S 2 :

∂ 2 t u(t, r) = ∂ 2 r u(t, r) + 1 r ∂ r u(t, r) - k 2
2r 2 sin(2u(t, r)), (t, r) ∈ R × (0, +∞).

(WM) Avant de commenter les principaux résultats de la thèse, plaçons brièvement le sujet de la théorie globale des équations des ondes dans une perspective historique.

Préliminaires

Les équations des ondes non linéaires les plus simples s'écrivent sous la forme

∂ 2 t u(t, x) = ∆u(t, x) + f (x, u(t, x)), (t, x) ∈ R × R N , u(t 0 , x) = u 0 (x), ∂ t u(t 0 , x) = u0 (x), x ∈ R N , (1.1)
où ∆ est le laplacien en N variables spatiales et f est une fonction scalaire suffisamment régulière de 2 variables réelles telle que f (x, 0) = 0. Cette équation possède une structure hamiltonienne naturelle. Pour le voir, posons F (x, u) := ´u 0 f (x, v) dv et définissons la fonctionnelle d'énergie :

E : C ∞ 0 (R N ) × C ∞ 0 (R N ) u = (u, u) → E(u) ∈ R, E(u) := ˆRN 1 2 | u(x)| 2 + 1 2 |∇u(x)| 2 -F (x, u(x)) dx.
L'équation (1.1) peut être écrite de manière équivalente sous la forme

∂ t u = J • DE(u), u(t 0 ) = u 0 , (1.2)
où D est la dérivée de Fréchet, J := 0 Id -Id 0 représente la forme symplectique naturelle sur C ∞ 0 × C ∞ 0 et u = (u, u) est un élément de l'espace des phases C ∞ 0 × C ∞ 0 .

On note u, v := ´RN u • v + u • v dx. Soit u(t) une solution classique de l'équation (1.2) sur l'intervalle (t 1 , t 2 ). D'après la règle de dérivation d'une fonction composée on a d dt E(u(t)) = DE(u(t)), ∂ t u(t) = DE(u(t)), J • DE(u(t)) = 0, (1.3) donc E(u(t)) est une loi de conservation.

Remarques historiques

L'histoire de la résolution locale en temps du problème de Cauchy remonte au moins jusqu'au théorème de Cauchy-Kowalevski, qui garantit, dans le cas d'une non-linéarité analytique, l'existence d'une unique solution analytique pour toute donnée initiale analytique. Cependant, la résolution globale en temps est essentielle du point de vue physique, et le premier résultat dans cette direction a été obtenu par Jörgens [START_REF] Jörgens | Das Anfangswertproblem im Großen für eine Klasse nichtlinearer Wellengleichungen[END_REF], suivi par des versions plus abstraites de Browder [START_REF] Browder | On non-linear wave equations[END_REF] et Segal [START_REF] Segal | Non-linear semi-groups[END_REF]. Énonçons le résultat dans le cas le plus simple de l'équation de Klein-Gordon cubique, défocalisante, en dimension 1 + 3, autrement dit l'équation (1.1) avec f (x, u) = -u -u 3 :

∂ 2 t u(t, x) = ∆u(t, x) -u(t, x) -u(t, x) 3 , (t, x) ∈ R × R 3 .
(1.4)

Théorème 1.1 (Jörgens, Browder, Segal). Pour toute donnée initiale u(t 0 ) = u 0 ∈ H 1 (R 3 )× L 2 (R 3 ), il existe une unique solution globale u(t) de (1.4). De plus, si

u 0 ∈ H k+1 × H k avec k ∈ N, alors u ∈ C(R; H k+1 × H k ).
Remarque 1.2. Plus loin dans cette section nous clarifions la notion de solution dans le cadre non lisse.

La partie essentielle de la preuve consiste à établir le résultat suivant d'existence locale :

Proposition 1.3. Pour toute donnée initiale u(t 0 ) = u 0 ∈ H 1 (R 3 ) × L 2 (R 3 ), il existe une unique solution maximale u(t) : (T -, T + ) → H 1 ×L 2 de (1.4). De plus, si u 0 ∈ H k+1 ×H k avec k ∈ N, alors u ∈ C((T -, T + ); H k+1 × H k ). Si T + < +∞, alors lim t→T + u(t) H 1 ×L 2 = +∞ (de même pour T -> -∞). Enfin, sur les intervalles de temps compacts, la solution dépend continûment de la donnée initiale dans la topologie H k+1 × H k pour tout k ∈ N.

Par (1.3) et l'argument habituel d'approximation, on voit que si u(t) est la solution donnée par la Proposition 1.3, alors l'énergie

E(u(t)) = ˆRN 1 2 | u(t, x)| 2 + 1 2 |∇u(t, x)| 2 + 1 2 |u(t, x)| 2 + 1 4 |u(t, x)| 4 dx
a une valeur constante. Mais cela implique que la norme u(t) H 1 ×L 2 reste bornée, ce qui démontre le Théorème 1.1.

Comme la non-linéarité est défocalisante, on peut conjecturer que les solutions doivent décroître ponctuellement au moins comme les solutions de l'équation linéaire, rendant ainsi les effets non linéaires négligeables. Par conséquent, le comportement asymptotique de toute solution serait celui d'une solution de l'équation de Klein-Gordon libre. Ce problème s'est avéré difficile et a été résolu dans les années 80 par Brenner [START_REF] Brenner | On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations[END_REF], avec des contributions importantes de Morawetz et Strauss [START_REF] Morawetz | Decay and scattering of solutions of a nonlinear relativistic wave equation[END_REF].

Théorème 1.4 (Brenner, Morawetz, Strauss). Soit u(t) une solution de (1.4) et soit U 0 (t) le propagateur de Klein-Gordon linéaire. Alors U 0 (-t)u(t) a une limite forte dans l'espace H 1 × L 2 quand t → ±∞. Remarque 1.5. L'exsitence de la limite faible est relativement élémentaire et est connue depuis les années 60.

Pour ainsi dire, le comportement dynamique des solutions de l'équation (1.4) n'est pas très différent de la dynamique de l'équation linéaire, au moins dans l'espace d'énergie. Dans la justification de ce résultat, le caractère défocalisant de la non-linéarité ainsi que sa croissance en ±∞ (u 3 ) jouent un rôle décisif. Il est naturel d'examiner le cas d'une non-linéarité focalisante ou avec une croissance plus rapide en ±∞ (par exemple u 5 ), et c'est précisément ce qui va nous occuper dans les deux prochains paragraphes.

Effets de la non-linéarité focalisante

Considérons l'équation (1.1) avec f (x, u) = -u + u 3 (dite l'équation de Klein-Gordon cubique focalisante), en dimension N = 3 :

∂ 2 t u(t, x) = ∆u(t, x) -u(t, x) + u(t, x) 3 , (t, x) ∈ R × R 3 . (1.5)
Quant à le théorie de Cauchy locale, rien ne change et la Proposition 1.3 reste vraie. Pourtant, la fonctionnelle d'énergie dans le cas focalisant s'écrit :

E(u) = ˆR3 1 2 | u| 2 + 1 2 |∇u| 2 + 1 2 |u| 2 - 1 4
|u| 4 dx, donc, à cause du "mauvais" signe devant le terme |u| 4 , elle ne permet plus de contrôler pour tout temps la norme H 1 ×L 2 . Keller [START_REF] Keller | On solutions of nonlinear wave equations[END_REF] a été le premier à observer que le Théorème 1.1 n'est plus valable dans le cas focalisant. Une manière simple de s'en convaincre est d'analyser les solutions constantes en espace, donc les solutions de l'équation ordinaire u (t) = -u(t)+u(t) 3 . On voit que la solution associée à une donnée initiale constante en espace et suffisamment grande devient +∞ en temps fini. En utilisant la propriété de la vitesse finie de propagation, on peut construire une donnée initiale dans C ∞ 0 , pour laquelle la solution forme une singularité en temps fini.

Une autre conséquence de la non-linéarité focalisante est l'existence d'états stationnaires W , qui sont les solutions du problème elliptique ∆W (x) -W (x) + W (x) 3 = 0,

x ∈ R 3 .

(1.6)

La donnée initiale u 0 := W := (W, 0) conduit à une solution de (1.5) constante en temps. Parmi toutes les solutions stationnaires, l'état fondamental est particulièrement important. Il peut être caractérisé comme l'unique (à une translation près) solution positive de (1.6), ou comme la solution qui minimise l'énergie potentielle E(u) := ´R3 1 2 |∇u| 2 + 1 2 |u| 2 -1 4 |u| 4 dx. Dans le langage du calcul des variations, E(u) crée un puits de potentiel (appelé aussi cuvette) pour u suffisamment petit dans l'espace d'énergie, et W est le point col. Payne et Sattinger [START_REF] Payne | Saddle points and instability of nonlinear hyperbolic equations[END_REF] ont montré que cette structure variationnelle est liée au rôle de E(W ) comme seuil d'énergie pour les différents types du comportement dynamique : si une solution d'énergie E(u 0 ) < E(W ) est à l'intérieur de la cuvette ( u 0 H 1 ≤ W H 1 ), elle y reste pour toujours, et si elle est à l'extérieur ( u 0 H 1 > W H 1 ), elle subit une explosion en temps fini.

Puisque les solutions dans le puits de potentiel sont globales, il est naturel d'étudier si elles ont le comportement linéaire quand t → +∞, comme dans le cas défocalisant. Une approche générale de ce problème, appelé Conjecture de l'état fondamental ou Conjecture du seuil, a été développée par Kenig et Merle [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energycritical, focusing, non-linear schrödinger equation in the radial case[END_REF][START_REF] Kenig | Global well-posedness, scattering and blow-up for the energycritical focusing non-linear wave equation[END_REF] pour l'équation des ondes énergie critique et pour l'équation de Schrödinger énergie critique radiale. Dans le cas de l'équation de Klein-Gordon cubique, le preuve est due à Ibrahim, Masmoudi et Nakanishi [START_REF] Ibrahim | Scattering threshold for the focusing nonlinear Klein-Gordon equation[END_REF].

Théorème 1.6 (Ibrahim, Masmoudi, Nakanishi). Soit u(t) une solution de (1.5) telle que E(u 0 ) < E(W ) et u 0 H 1 ×L 2 ≤ W H 1 . Alors U 0 (-t)u(t) a une limite forte dans l'espace H 1 × L 2 quand t → ±∞.

Après avoir classifié les solutions en-dessous du seuil d'énergie E(W ), l'objectif suivant devrait être de comprendre le comportement dynamique légèrement au-dessus de ce seuil. Une solution particulièrement élégante à ce problème, exhibant un lien avec la théorie des variétés invariantes, a été proposée par Nakanishi et Schlag [START_REF] Nakanishi | Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation[END_REF][START_REF] Nakanishi | Global dynamics above the ground state for the nonlinear Klein-Gordon equation without a radial assumption[END_REF].

Non-linéarités énergie critiques

Revenons pour un instant à l'équation générale (1.1). On a vu que la structure du problème permet de contrôler, peut-être de manière indirecte comme dans le cas focalisant, la norme homogène d'énergie

u(t) 2 Ḣ1 ×L 2 := ˆRN |∂ t u(t, x)| 2 + |∇u(t, x)| 2 dx,
ce qui nous donne une certaine information sur la régularité de la solution (dans le cas de Klein-Gordon on peut contrôler également u L 2 , mais c'est moins intéressant du point de vue de la régularité). En fonction de la croissance de |f (x, u)| quand |u| → +∞, cette information peut être ou ne pas être suffisante pour exclure l'explosion de la solution. On va étudier maintenant de manière heuristique la possibilité de formation d'une singularité telle que la norme d'énergie reste bornée. Une singularité de ce type est appelée explosion de type II ou explosion géométrique, le terme explosion de type I désignant quant à lui la situation où la norme d'énergie tend vers +∞ en temps fini. Pour u(x) = (u(x), u(x)) et λ > 0, on définit le changement d'échelle énergie critique Si p(N -2)

2 -N -2 2 < 2 ⇔ p < N +2 N -2
, on voit que l'effet de la non-linéarité est négligeable par rapport au laplacien quand λ → 0. Intuitivement, la concentration de la norme d'énergie rend négligeables les effets non linéaires, en particulier ne peut pas conduire à la formation d'une singularité. Par conséquent, la seule manière de former une singularité est l'explosion de la norme d'énergie, ce qui fait partie de l'énoncé de la Proposition 1.3. On appelle ce type de non-linéarité énergie sous-critique. En revanche, si p > N +2 N -2 , alors la non-linéarité constitue la partie dominante quand λ → 0. Cela signifie qu'une concentration d'une quantité arbitrairement petite de la norme énergétique pourrait potentiellement rendre les effets non linéaires décisifs, conduisant ainsi à une formation d'une singularité. On appelle une telle non-linéarité énergie sur-critique. Le cas limite, p = N +2 N -2 , est dit énergie critique. Dans cette dernière situation, la non-linéarité agit avec la même force que le laplacien, a priori formant une singularité de type II en temps fini. L'équation des ondes énergie critique la plus simple est probablement l'équation semilinéaire avec la non-linéarité défocalisante de type puissance :

∂ 2 t u(t, x) = ∆u(t, x) -|u(t, x)| 4 N -2 u(t, x), (t, x) ∈ R × R N . (1.7)
Le caractère bien posé dans l'espace d'énergie Ḣ1 × L 2 a été démontré par Shatah et Struwe [START_REF] Shatah | Well-posedness in the energy space for semilinear wave equations with critical growth[END_REF], avec des contributions importantes de Kapitanski [START_REF] Kapitanski | Global and unique weak solutions of nonlinear wave equations[END_REF], et Ginibre, Soffer et Velo [START_REF] Ginibre | The global Cauchy problem for the critical nonlinear wave equation[END_REF].

Observons que pour tout u = (u, u) ∈ Ḣ1 × L 2 , l'énergie E(u) = ´RN

1 2 | u| 2 + 1 2 |∇u| 2 + N -2 2N |u| 2N
N -2 dx est bien définie grâce à l'injection de Sobolev critique. De plus, on vérifie facilement que E(u λ ) = E(u). Enfin, l'équation est invariante par le changement d'échelle énergie critique : si (u(t, x), u(t, x)) est une solution de (1.7) sur un intervalle I t 0 , alors pour tout λ > 0 on peut former une autre solution u t 0 + t λ , x λ , définie sur un intervalle de temps qui contient t = 0.

Il s'avère que pour les solutions de (1.7), la norme d'énergie ne peut pas se concentrer, et qu'un équivalent du Théorème 1.1 est vrai, comme l'ont démontré Struwe [START_REF] Struwe | Globally regular solutions to the u 5 klein-gordon equation[END_REF] dans le cas radial et Grillakis [START_REF] Grillakis | Regularity and asymptotic behavior of the wave equation with a critical nonlinearity[END_REF] sans l'hypothèse de symétrie. En outre, les solutions convergent vers des ondes linéaires quand t → ±∞. Dans le livre [START_REF] Tao | Nonlinear Dispersive Equations: Local and Global Analysis[END_REF]Chapter 5] on peut trouver les détails historiques concernant la résolution de ce problème.

Dans cette thèse nous nous concentrerons principalement sur l'équation (NLW), l'analogue focalisant de (1.7). L'énergie est donnée par

E(u) = ˆRN 1 2 | u| 2 + 1 2 |∇u| 2 - N -2 2N |u| 2N N -2 dx.
Le caractère bien posé se démontre en modifiant les preuves pour le cas défocalisant, cf. [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energycritical, focusing, non-linear schrödinger equation in the radial case[END_REF]. On obtient le résultat suivant.

Proposition 1.7. Pour toute donnée initiale u 0 ∈ Ḣ1 (R N ) × L 2 (R N ), il existe une unique solution maximale u(t) : (T -, T + ) → Ḣ1 ×L 2 de (NLW). Si u 0 Ḣ1 ×L 2 est suffisamment petit, alors T -= -∞, T + = +∞ et u(t) converge vers une onde linéaire dans les deux directions du temps. La solution est continue par rapport à la donnée initiale dans la topologie Ḣ1 × L 2 .

Il faut noter une différence majeure par rapport au cas sous-critique traité ci-dessus. Si T + < +∞, il n'est plus garanti que lim t→T + u Ḣ1 ×L 2 = +∞, ce dont la raison heuristique a été présentée au début de ce paragraphe. On a tout de même le critère abstrait d'explosion : si T + < +∞ et K est un sous-ensemble compact de Ḣ1 × L 2 , alors il existe τ > 0 tel que u(t) / ∈ K pour t ∈ [T + -τ, T + ). Le problème elliptique

∆W (x) + |W (x)| 4 N -2 W (x) = 0, x ∈ R N (1.8)
(appelé l'équation de Yamabe) possède la solution positive explicite suivante :

W (x) = 1 + |x| 2 N (N -2) -N -2 2 .
En raison de l'invariance de l'équation par rapport au changement d'échelle énergie-critique, pour tout λ > 0, la fonction W λ (x) est également une solution de (1.8). Il est connu que ce sont toutes les solutions à symétrie radiale d'énergie finie et, à une translation près, toutes les solutions positives de (1.8). Cependant, le problème non radial est difficile, cf. [START_REF] Del Pino | Large energy entire solutions for the Yamabe equation[END_REF]. On obtient les solutions stationnaires à symétrie radiale de (NLW) W λ = (W λ , 0). Le rôle de W λ comme le point col pour l'énergie potentielle E(u) = ´1 2 |∇u| 2 -N -2 2N |u| 2N N -2 dx résulte des travaux d'Aubin [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF] et de Talenti [START_REF] Talenti | Best constant in Sobolev inequality[END_REF].

La Conjecture de l'état fondamental pour l'équation des ondes focalisante énergie-critique a été démontrée dans les travaux de Kenig et Merle [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energycritical, focusing, non-linear schrödinger equation in the radial case[END_REF] pour N ∈ {3, 4, 5}. Pour N = 3, Krieger, Schlag et Tataru [START_REF] Krieger | Slow blow-up solutions for the H 1 (R 3 ) critical focusing semilinear wave equation[END_REF] ont construit des solutions qui développent une singularité en temps fini, et qui restent pendant tout le temps d'existence dans un voisinage arbitrairement petit (dans l'espace d'énergie) de la famille des états fondamentaux W λ . En particulier, cette construction a donné le premier exemple d'une explosion de type II. Duyckaerts, Kenig et Merle [START_REF] Duyckaerts | Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation[END_REF][START_REF] Duyckaerts | Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case[END_REF] ont montré que toute solution de (NLW) qui explose dans un voisinage de l'ensemble {W λ }, se décompose asymptotiquement dans l'espace d'énergie en une somme de l'état fondamental, après un changement d'échelle et une transformée de Lorentz, et d'un profil asymptotique. Plus précisément, dans le cas d'une symétrie radiale, si T + < +∞, alors il existe une fonction positive λ(t)

T

+ -t et u * 0 ∈ Ḣ1 × L 2 tels que lim t→T + u(t) -u * 0 -W λ(t) Ḣ1 ×L 2 = 0.
En dimension N = 3, pour les solutions à symétrie radiale, les mêmes auteurs [START_REF] Duyckaerts | Classification of the radial solutions of the focusing, energy-critical wave equation[END_REF] ont donné une classification complète du comportement dynamique des solutions de (NLW) dans l'espace d'énergie. Ils ont montré que si T + = +∞, alors il existe J ∈ N, des fonctions strictement positives λ 1 (t) λ 2 (t) . . . λ J (t) t et une solution u * (t) de l'équation des ondes linéaire tels que lim t→+∞ u(t) -u * (t) -J j=1 ±W λ j (t) Ḣ1 ×L 2 = 0.

(1.9)

Si T + < +∞, alors soit lim t→T + u(t) Ḣ1 ×L 2 = +∞, soit il existe J ∈ N, des fonctions strictement positives λ 1 (t) λ 2 (t) . . . λ J (t)

T + -t et u * 0 ∈ Ḣ × L 2 tels que lim t→T + u(t) -u * 0 - J j=1
±W λ j (t) Ḣ1 ×L 2 = 0.

(1.10) Krieger, Nakanishi et Schlag [START_REF] Krieger | Center-stable manifold of the ground state in the energy space for the critical wave equation[END_REF] ont étudié la dynamique dans un voisinage des états fondamentaux d'un point de vue différent, en relation avec la théorie des variétés invariantes. En particulier, ils ont construit la variété centre-stable, qui est une hypersurface de classe C 1 qui contient {W λ }, et ils ont montré que cette hypersurface sépare l'ensemble des solutions ayant le comportement asymptotiquement linéaire pour les temps positifs de l'ensemble des solutions qui développent une singularité de type I. Les solutions sur la variété sont celles qui restent dans un voisinage de {W λ } jusqu'à la fin de leur temps d'existence.

Remarque 1.8. Une théorie globale des équations des ondes super-critiques pour de grandes données initiales, même dans le cas défocalisant, semble actuellement inaccessible. À la connaissance de l'auteur, on dispose uniquement de résultats conditionnels sur la dynamique des solutions (aussi bien dans le cas focalisant que défocalisant), cf. Kenig et Merle [START_REF] Kenig | Nondispersive radial solutions to energy supercritical nonlinear wave equations, with applications[END_REF]. Dans le cas où la non-linéarité défocalisante est une puissance, Krieger et Schlag [START_REF] Krieger | Large global solutions for energy supercritical nonlinear wave equations on R 3+1[END_REF] ont montré l'existence de grandes données initiales telles que la solution existe pour tout temps.

Autres modèles critiques

Historiquement, le modèle énergie critique le plus étudié est le flot de la chaleur des applications harmoniques entre deux surfaces, surtout entre deux sphères de dimension 2. Pour l'état de l'art dans ce domaine, le lecteur peut consulter Topping [START_REF] Topping | Repulsion and quantization in almost-harmonic maps, and asymptotics of the harmonic map flow[END_REF], ainsi que les références citées dans cet article. En particulier, c'est dans ce cas-là que les premiers résultats de classification du comportement à l'explosion (bubbling) ont été obtenus.

Les termes "une explosion de type I" et "une explosion de type II" proviennent des travaux sur l'équation de la chaleur sur-critique ∂ t u = ∆u + |u| p u, cf. [START_REF] Matano | Classification of type I and type II behaviors for a supercritical nonlinear heat equation[END_REF].

L'équation de Schrödinger masse critique (ou L 2 critique) a été étudiée dans de nombreux travaux, aussi en vue des applications physiques. Sous certains aspects, elle ressemble à l'équation de Korteweg-de Vries généralisée L 2 critique. L'article de revue [START_REF] Martel | Near soliton dynamics and singularity formation for L 2 critical problems[END_REF] présente les avancées majeures récentes dans la compréhension de ces deux modèles.

Parmi les équations hamiltoniennes énergie critique, notons l'équation de Schrödinger semi-linéaire : i∂ t u + ∆u + |u| 4 N -2 u = 0 sur R N , et l'équation de Schrödinger des applications harmoniques (Schrödinger maps) de R 2 dans la sphère S 2 ⊂ R 3 : ∂ t u = u ∧ ∆u, où ∧ est le produit vectoriel dans R 3 . Par la suite, on évoquera parfois ces deux modèles, par souci de comparaison avec les équations des ondes.

Remarque 1.9. Les solutions des équations énergie critiques ont souvent un comportement dynamique compliqué, mais d'un autre côté elles possèdent une structure supplémentaire d'invariance par rapport au changement d'échelle énergie critique, ce qui est essentiel pour beaucoup de résultats de classification.

Éléments de la théorie de Cauchy

Dans ce paragraphe on donne quelques commentaires sur la notion de solution et de caractère bien posé. Il sera commode de noter X s := Ḣs+1 ∩ Ḣ1 pour s ≥ 0. On appelle u(t) := (u(t), u(t)) la solution du problème

Définition 1.10. Soit t 0 ∈ (t 1 , t 2 ) ⊂ R, u 0 = (u 0 , u0 ) ∈ X s × H s et h ∈ L 1 ((t
∂ 2 t u = ∆u + h, (u, ∂ t u) t=t 0 = (u 0 , u0 ).
(1.11)

Notons qu'il s'ensuit directement que u ∈ C((t 1 , t 2 ), X s × H s ) (on dit souvent que "l'équation des ondes fait gagner une dérivée"). Une version quantitative de ce fait est donnée par l'inégalité fondamentale suivante :

u(t) X s ×H s ≤ u 0 X s ×H s + ˆt t 0 h(s) H s ds , (1.12)
que l'on appelle l'estimée d'énergie. L'existence et l'unicité des solutions faibles du problème linéaire (1.11), ainsi que l'estimée d'énergie (dans un cadre plus général des systèmes symétriques) ont été démontrées par Friedrichs [START_REF] Friedrichs | Symmetric hyperbolic linear differential equations[END_REF].

Définition 1.11. Supposons que ∂ u f (x, 0) = 0. On dit que l'équation (1.1) est localement bien posée dans X s × H s si

• pour tout u 0 ∈ X s × H s et t 0 ∈ R il existe un intervalle du temps (t 1 , t 2 ) t 0 et une unique fonction u(t) = (u(t), u(t)) telle que f (x, u(t, x)) ∈ L 1 ((t 1 , t 2 ); H s ) (1.13) et u(t) est une solution de (1.11) avec h(t, x) = f (x, u(t, x)),

• u(t) dépend de u 0 de façon continue dans la topologie X s × H s .

Remarque 1.12. Pour des fonctions lisses, cette définition coïncide avec la définition habituelle d'une solution classique d'une équation différentielle.

Remarque 1.13. En général, au lieu de -∆, on devrait considérer l'opérateur T := -∆ -∂ u f (x, 0). Sous des hypothèses naturelles sur T , on peut définir la notion de solution de manière similaire, en remplaçant partout H s par (1 + T s 2 ) -1 L 2 et X s par (T

1 2 + T s+1 2 ) -1 L 2 .
Remarque 1.14. Dans des situations typiques, la condition (1.13) est équivalente au fait que u(t) appartient à un espace fonctionnel naturel, par exemple un espace de Lebesgue sur l'espace-temps. La Définition 1.11 formule un problème de point fixe. La partie principale de sa résolution consiste à trouver des bornes de f (x, u(t, x)) L 1 ((t 1 ,t 2 );H s ) en termes de h(t, x) L 1 ((t 1 ,t 2 );H s ) , où u(t, x) est la solution de (1.11). Pour cela, on utilise des inégalités de type Sobolev ou Strichartz.

Remarque 1.15. Supposons que (1.1) est localement bien posée dans X s ×H s et que u(t) est une solution avec temps maximal d'existence T + < +∞. Alors f (x, u(t, x)) L 1 ([t 0 ,T + );H s ) = +∞. Si ce n'était pas le cas, alors (1.12) impliquerait que la solution u(t) est pré-compacte dans X s × H s quand t → T + , ce qui est impossible.

Dans la même veine, si T + = +∞ et f (x, u(t, x)) L 1 ([t 0 ,+∞);H s ) < +∞, alors u(t) a le comportement asymptotiquement linéaire dans X s × H s quand t → +∞.

Remarque 1. [START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: I[END_REF]. Dans certains cas "pathologiques", comme par exemple (NLW) en grande dimension, cette définition générale n'est pas nécessairement la bonne, et il faut remplacer la condition f (x, u(t, x)) ∈ L 1 ((t 1 , t 2 ); H s ) par une autre restriction garantissant que la solution est unique et qu'elle dépend d'une manière continue de la donnée initiale.

Résultats

Dans cette thèse on considère l'équation (NLW) pour les données initiales à symétrie radiale, sauf dans le Chapitre 2, où on traite également de l'équation (WM).

Dans le prolongement des travaux cités ci-dessus, on étudie le système dynamique défini par l'équation (NLW), au voisinage de l'ensemble {W λ } dans l'espace d'énergie, c'est-à-dire les solutions u(t) de (NLW) telles que inf λ>0 u(t) -W λ Ḣ1 ×L 2 ≤ η, ∀t, où η > 0 est une petite constante. Les Chapitres 1 et 3 sont consacrés au phénomène de l'explosion de type II. Dans les Chapitres 2 et 4, on étudie le comportement local au voisinage d'une superposition de deux bulles à des échelles différentes, c'est-à-dire les solutions u(t) de (NLW) telles que inf

0<λ≤α * µ u(t) -(W µ ± W λ ) Ḣ1 ×L 2 ≤ η, ∀t,
où η > 0 est une petite constante. La motivation de ce travail vient bien évidemment des résultats de classification de [START_REF] Duyckaerts | Classification of the radial solutions of the focusing, energy-critical wave equation[END_REF] mentionnés plus haut, cf. (1.9) et (1.10). Cette thèse est composée des quatre articles suivants :

• Chapitre 1 -Jendrej, J. Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5. Prépublication, arXiv:1503.05024, 2015.

• Chapitre 2 -Jendrej, J. Construction of two-bubble solutions for energy-critical wave equations. Prépublication, arXiv:1602.06524, 2016.

• Chapitre 3 -Jendrej, J. Bounds on the speed of type II blow-up for the energy critical wave equation in the radial case. Int. Math. Res. Not., doi : 10.1093/imrn/rnv365, 2015.

• Chapitre 4 -Jendrej, J. Nonexistence of radial two-bubbles with opposite signs for the energy-critical wave equation. Prépublication, arXiv:1510.03965, 2015.

Construction de solutions explosives de type II

Dans cet article nous développons une nouvelle approche de la construction de solutions explosives de type II pour l'équation des ondes énergie critique. De telles solutions ont été d'abord construites pour l'équation (NLW) en dimension N = 3 et pour l'équation (WM) en classe d'équivariance k = 1 par Krieger, Schlag et Tataru [START_REF] Krieger | Renormalization and blow up for charge one equivariant critical wave maps[END_REF][START_REF] Krieger | Slow blow-up solutions for the H 1 (R 3 ) critical focusing semilinear wave equation[END_REF]. Ici, on considère l'équation (NLW) en dimension N = 5. On démontre les résultats suivants.

Théorème 2.1. Soit u * 0 = (u * 0 , u * 0 ) ∈ ( Ḣ5 ∩ Ḣ1 ) × H 4 une paire de fonctions à symétrie radiale avec u * 0 (0) > 0. Il existe une solution u(t) de (NLW), définie sur l'intervalle du temps (0, T 0 ), telle que lim t→0 u(t) -u * 0 -W λ(t) Ḣ1 ×L 2 = 0, (2.1) où λ(t) = 32 315π 2 u * (0, 0) 2 t 4 .

Théorème 2.2. Soit ν > 8. Il existe une solution u(t) de (NLW), définie sur l'intervalle du temps (0, T 0 ), telle que lim t→0 u(t) -u * 0 -W λ(t) Ḣ1 ×L 2 = 0, où λ(t) = t ν+1 et u * 0 est explicite. Le fait d'obtenir un continuum de vitesses d'explosion possibles indique que le comportement dynamique de nos solutions est fortement instable par rapport aux variations de la donnée initiale. Un peu plus précisément, on peut s'attendre à ce que l'ensemble des solutions ayant les mêmes caractéristiques dynamiques (par exemple la même vitesse d'explosion) soit de codimension infinie. On pourrait dire que l'on obtient des solutions de type Krieger-Schlag-Tataru, par opposition aux solutions stables construites par Hillairet et Raphaël [START_REF] Hillairet | Smooth type II blow up solutions to the four dimensional energy critical wave equation[END_REF] en dimension N = 4, qui peuvent exploser seulement avec des vitesses bien spécifiques. De telles solutions ont été aussi construites pour l'équation (WM) par Rodnianski et Sterbenz [START_REF] Rodnianski | On the formation of singularities in the critical O(3) σ-model[END_REF], et par Raphaël et Rodnianski [START_REF] Rodnianski | Stable blow up dynamics for critical corotational wave maps and the equivariant Yang Mills problem[END_REF].

Remarque 2.3. Dans ce deuxième cas, la possibilité d'une formation de singularités avait été suggérée par des expériences numériques, voir par exemple [START_REF] Bizoń | Formation of singularities for equivariant 2 + 1 dimensional wave maps into the 2-sphere[END_REF]. C'est l'explosion stable que l'on observe numériquement. Les auteurs de [START_REF] Bizoń | Formation of singularities for equivariant 2 + 1 dimensional wave maps into the 2-sphere[END_REF] suggèrent que l'ensemble des données initiales donnant lieu à une explosion est énorme. Cependant, l'existence d'un sous-ensemble ouvert de l'espace d'énergie, tel que toute donnée initiale dans ce sous-ensemble développe une singularité, reste inconnu.

Présentons maintenant les idées de la preuve du Théorème 2.1, sans nous préoccuper de tous les détails techniques. Étape 1. Supposons que u(t) = (u(t), u(t)) vérifie (2.1). Soit u * (t) = (u * (t), u * (t)) la solution de (NLW) pour la donnée initiale u * (0) = u * 0 . Notons ΛW := -∂ ∂λ W λ | λ=1 . Par la règle de dérivation d'une fonction composée on a ∂ t W λ(t) = -λ (t) λ(t) ΛW λ(t) , donc on s'attend à ce que (u(t), u(t))

u * (t) + W λ(t) , u * (t) -λ (t) λ(t) ΛW λ(t) .

(2.2)

Il semble naturel de travailler dans l'espace d'énergie. Par un changement de variable, on obtient

λ (t)
λ(t) ΛW λ(t) L 2 ∼ λ (t) (la solution explose en t = 0, d'où λ (t) > 0). Si l'on introduit le petit paramètre b(t) λ (t), on peut voir (2.2) comme le début d'un développement asymptotique de u(t) dans l'espace d'énergie en puissances de b(t).

Il s'avère que si

b (t) = 128 105π u * (t) λ(t), (2.3)
alors on peut calculer le terme suivant de ce développement et définir ainsi une solution approchée (que l'on appelle aussi ansatz )

ϕ(t) := u * (t) + U (0) λ(t) + b(t)U (1)
λ(t) + b 2 (t)U (2)
λ(t) + b 3 (t)U (3)
λ(t) , où U (0) = (W, 0) et U (1) = (0, -ΛW ). Les définitions précises de U (2) et de U (3) ne sont pas essentielles. Observons que l'équation (2.3) avec λ = b donne la valeur λ app (t) = 32 315π 2 u * (0, 0) 2 t 4 , qui est celle de l'énoncé du théorème.

Remarque 2.4. Dans nos définitions de U (2) et de U (3) , la décroissance en |x| → +∞ de W (x) joue un rôle important, et c'est la raison pour laquelle on doit se restreindre aux dimensions N ≥ 5. En grande dimension la non-linéarité |u| 4 N -2 u devient assez singulière en u = 0, ce qui introduit des difficultés techniques supplémentaires. On a choisi le cas le plus simple N = 5.

Étape 2. On considère une suite de solutions u n (t) de (NLW) ayant comme donnée initiale

u n (t n ) = u * (t n ) + W λapp(tn) , u * (t n ) - λ app (t n ) λ app (t n ) ΛW (t n ) ,
où t n > 0 et lim n→+∞ t n = 0 (un ajustement est à faire à cause de l'instabilité exponentielle au voisinage de W , mais ce n'est pas une grave difficulté). En utilisant une condition d'orthogonalité adaptée, on décompose u n (t) = ϕ(t) + g n (t), le but étant de contrôler la taille de g n (t), uniformément en n, sur un intervalle de temps (t n , t 0 ], t 0 > 0.

Pour cela, on introduit une fonctionnelle mixte énergie-viriel H n (t), qui est une petite perturbation de la fonctionnelle d'énergie E(ϕ(t) + g n (t)) -E(ϕ(t)) -DE(ϕ(t)), g n (t) . Cette fonctionnelle a la propriété de coercitivité suivante :

g n (t) 2 Ḣ1 ×L 2 H n (t)
(modulo les modes instables).

De plus, en utilisant le fait que ϕ(t) est un ansatz raffiné, on peut montrer qu'il existe une grande constante C 0 telle que

g n (t) Ḣ1 ×L 2 ≤ C 0 t 9 2 ⇒ H n (t) ≤ c 0 • C 2 0 t 8 ,
avec une petite constante c 0 . Un argument de continuité classique implique la borne uniforme g Ḣ1 ×L 2 ≤ C 0 t 9 2 . De cette estimation, on déduit par des techniques d'analyse des équations ordinaires que le paramètre de modulation λ n (t) est proche de λ app (t).

La conclusion de la deuxième étape est que

u n (t) -(u * (t) + W λapp (t), u * (t)) Ḣ1 ×L 2 t 3 ,
uniformément par rapport à n.

Étape 3. En utilisant la décomposition en profils de Bahouri et Gérard [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF], on démontre une version de la continuité faible séquentielle du flot, ce qui permet d'obtenir notre solution u(t) comme un point d'adhérence faible de la suite u n (t) dans l'espace d'énergie.

L'idée de construire une suite de solutions contrôlée uniformément, convergeant vers une solution singulière, a été introduite par Merle [START_REF] Merle | Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity[END_REF]. Fusionner cette technique avec la méthode d'énergie est une idée de Martel [START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF]. Raphaël et Szeftel [START_REF] Raphaël | Existence and uniqueness of minimal mass blow up solutions to an inhomogeneous L 2 -critical NLS[END_REF] ont utilisé une correction par un viriel de la fonctionnelle d'énergie dans un contexte similaire, dans leur étude de solutions explosives de masse minimale pour l'équation de Schrödinger non linéaire. La première étape de notre preuve est aussi inspirée par les travaux de Martel, Merle et Raphaël [START_REF] Martel | Blow up for the critical gKdV equation III: exotic regimes[END_REF] sur l'explosion exotique pour l'équation de Korteweg-de Vries L 2 -critique. Ils ont observé que la vitesse d'explosion est reliée à la taille de l'interaction de la bulle avec le "fond", ce qui est à la base de notre construction de la solution approchée.

En réalité, la taille de cette interaction apparaît explicitement dans (2.3), ce qui sera expliqué en détail dans le Chapitre 1. Nous trouvons que le fait de mettre en lumière un lien direct entre le comportement asymptotique de u * (t) en x = 0 et l'asymptotique de la fonction λ(t) est un avantage majeur de notre méthode. Par exemple, dans le Théorème 2.2 on a u *

0 ≡ 0 et u * 0 (x) ∼ |x| ν-3 2
dans un voisinage de x = 0. Un point délicat, que l'on n'aborde pas dans notre travail, serait de mieux comprendre la régularité des solutions construites. On ne dispose d'aucune information sur la régularité de la solution outre le fait qu'elle appartient à l'espace d'énergie. En même temps, on s'attend généralement à ce que ces solutions aient des singularités, et travailler au niveau de la régularité Ḣ1 × L 2 permet d'éviter de les traiter directement. Observons également que, pour construire des objets aussi instables que les nôtres, il semble naturel d'utiliser des estimées d'énergie en inversant la direction du temps, à savoir dans le sens de la défocalisation de la solution. La méthode d'énergie "dans le sens de l'explosion" implique typiquement une sorte de stabilité de codimension finie.

Dans des travaux à venir, nous espérons réaliser une construction similaire pour l'équation (WM), ainsi que dans le cas de u * 0 singulier. Remarque 2.5. À cause d'un lien avec le Chapitre 3, on signale que l'on peut aussi déterminer formellement la vitesse de l'explosion en résolvant pour λ (t) l'équation

E u * (t) + W λ(t) , u * (t) - λ (t) λ(t) ΛW λ(t) = E(W ) + E(u * 0 ).
Remarque 2.6. Certaines notations du Chapitre 1 sont différentes de celles employées ici. Nous avons choisi d'utiliser des notations cohérentes dans toutes les sections de ce chapitre introductif.

Construction de 2-bulles

Dans le Chapitre 2 nous construisons une solution à symétrie radiale de (NLW) qui existe globalement pour les temps négatifs et qui se décompose en deux bulles d'énergie dans le cône de lumière passé. À la connaissance de l'auteur, c'est le premier exemple d'une solution de ce genre.

Théorème 2.7. Il existe une solution u : (-∞,

T 0 ] → Ḣ1 × L 2 de (NLW) en dimension N = 6 telle que lim t→-∞ u(t) -(W + W 1 κ e -κ|t| ) Ḣ1 ×L 2 = 0, avec κ := 5 4 .
Il est à noter que u(t) est un exemple d'une solution de (NLW) autre que l'état fondamental qui est globale dans une direction du temps et qui ne contient pas de terme de radiation dans l'espace d'énergie, plus précisément

∀A > 0, lim sup t→-∞ ˆ|x|≥|t|-A | u(t, x)| 2 + |∇u(t, x)| 2 dx = 0.
Pour souligner le fait que l'énergie de nos solutions est exactement égale au double de l'énergie de W , sans énergie diffusée comme une onde libre, on dit qu'elles sont des 2-bulles pures. Donnons quelques commentaires sur la preuve. On observe que dans Théorème 2.1 on peut prendre u * 0 = (W, 0), ce qui produit une solution qui explose en temps fini avec (W, 0) comme profil asymptotique. C'est "presque" ce que l'on désire, sauf que la bulle se concentre et explose en temps fini au lieu d'exister et de se concentrer pour tout temps. Une manière naturelle d'obtenir un tel comportement est d'augmenter la dimension de l'espace. En dimension N = 6 l'interaction entre les deux bulles est plus faible, ce qui produit l'effet voulu. Pour N ≥ 7 on devrait arriver à une conclusion similaire, avec une vitesse de concentration de la bulle λ ∼ |t| -4 N -6 . La principale difficulté technique par rapport au Théorème 2.1 vient du fait que W a une taille fixe dans l'espace d'énergie, alors qu'auparavant, grâce à la vitesse finie de propagation, on a pu supposer que u * 0 Ḣ1 ×L 2 était petit. Afin d'obtenir des bornes sur la norme d'énergie, il est maintenant nécessaire d'étudier la coercitivité de la fonctionnelle d'énergie au voisinage d'une somme de deux bulles.

On modifie aussi notre construction pour couvrir le cas de l'équation (WM) avec k ≥ 3. On obtient le résultat suivant.

Théorème 2.8. Soit k > 2. Il existe une solution u : (-∞, T 0 ] → E de (WM) telle que lim t→-∞ u(t) -(-W + W k-2 κ (κ|t|) -2 k-2 ) E = 0, avec κ := k -2 2 8k π sin π k 1 k .
Ici, W désigne l'état fondamental de degré topologique k et E est l'espace de l'énergie. Enfin, on démontre un résultat similaire pour l'équation de Yang-Mills critique :

∂ 2 t u(t, r) = ∂ 2 r u(t, r) + 1 r ∂ r u(t, r) - 4 2r 2 u(t, r)(1 -u(t, r)) 1 - 1 2 u(t, r) .
Des résultats de classification montrant la validité de décompositions de type (1.10) ou (1.9) pour une suite de temps t n → T + ont été obtenus par Côte [START_REF] Côte | On the soliton resolution for equivariant wave maps to the sphere[END_REF] pour l'équation (WM) avec k = 1, et par Jia et Kenig [START_REF] Jia | Asymptotic decomposition for semilinear wave and equivariant wave map equations[END_REF] en plus grande généralité, y compris dans tous les cas considérés ici.

Bornes sur la vitesse d'explosion de type II

Les résultats du Chapitre 1 incitent à se pencher davantage sur la relation entre la dynamique de l'explosion et les propriétés du profil asymptotique u * 0 . Dans le Chapitre 3, on démontre deux résultats qui vont dans cette direction. Théorème 2.9.

Soit N ∈ {3, 4, 5} et s > N -2 2 , s ≥ 1. Soit u * 0 = (u * 0 , u * 0 ) ∈ H s+1 × H s à symétrie radiale. Supposons que u(t) est une solution radiale de (NLW) telle que lim t→T + u(t) -W λ(t) -u * 0 Ḣ1 ×L 2 = 0, lim t→T + λ(t) = 0, T + < +∞. (2.4)
Il existe une constante C > 0, qui dépend de u * 0 , telle que :

• si N ∈ {4, 5}, alors pour T + -t suffisamment petit λ(t) ≤ C(T + -t) 4 6-N .
• si N = 3, alors il existe une suite t n → T + telle que

λ(t n ) ≤ C(T + -t n ) 4 6-N . Théorème 2.10. Soit N ∈ {3, 4, 5}. Soit u * 0 = (u * 0 , u * 0 ) ∈ H 3 × H 2 à symétrie radiale avec u * 0 (0) < 0.
Il n'existe pas de solution u(t) de (NLW) telle que

lim t→T + u(t) -W λ(t) -u * 0 Ḣ1 ×L 2 = 0, lim t→T + λ(t) = 0, T + < +∞. ( 2.5)
L'idée principale des preuves, esquissées dans le Paragraphe 1.4 du Chapitre 3, est de borner le terme d'erreur par l'interaction (énergétique) de la bulle avec le reste de la solution, le principe auquel on a fait allusion dans la Remarque 2.5.

L'image générale qui émerge des Théorèmes 2.1, 2.2 et 2.9 est que la vitesse d'explosion des solutions de type Kriger-Schlag-Tataru est reliée à la régularité du profil asymptotique, plutôt qu'à la régularité de la solution elle-même (même si des solutions qui explosent plus vite ont tendance à être plus régulières, comme indiqué dans [START_REF] Krieger | Renormalization and blow up for charge one equivariant critical wave maps[END_REF][START_REF] Krieger | Slow blow-up solutions for the H 1 (R 3 ) critical focusing semilinear wave equation[END_REF]). Il faut noter que, même si pour des raisons techniques on demande une régularité supplémentaire de u * 0 , dans (2.4) et (2.5) la convergence a lieu seulement dans l'espace d'énergie.

Il est raisonnable de se demander si le profil asymptotique détermine de façon unique la solution qui explose dans un voisinage de {W λ }. Si c'était le cas, alors en associant à toute solution explosive son profil asymptotique, et réciproquement, on obtiendrait une classification naturelle des solutions explosives au voisinage de {W λ } dans l'espace d'énergie. H. Koch, D. Tataru et le rapporteur anonyme de l'article constituant le Chapitre 3 ont indiqué à l'auteur que ce schéma général ressemble au problème classique de la diffusion (scattering). Évidemment on devrait identifier les éléments de l'espace d'énergie qui sont les profils asymptotiques d'une certaine solution explosive. Le contenu du Théorème 2.10 est de montrer des exemples de profils qui doivent être exclus.

Même s'il serait judicieux de regarder le programme décrit ci-dessus avec une certaine réserve, il permet néanmoins de formuler des questions qui semblent plus accessibles. Par exemple, le profil asymptotique détermine-t-il la vitesse de l'explosion ?

Non existence de 2-bulles de signes opposés

Le Chapitre 4 est consacré à une preuve du résultat suivant : Théorème 2.11. Soit N ≥ 3. Il n'existe pas de solution radiale u : [t 0 , T + ) → Ḣ1 × L 2 de (NLW) telle que lim

t→T + u(t) -W λ(t) + W µ(t) Ḣ1 ×L 2 = 0 et • dans le cas T + < +∞, λ(t) µ(t) T + -t quand t → T + ,
• dans le cas

T + = +∞, λ(t) µ(t) t quand t → +∞.
Comme dans le contexte du Théorème 2.7, on travaille ici au niveau énergétique 2E(W ), qui est le seuil d'énergie pour une formation d'une multi-bulle. La motivation principale provient de l'équation (WM), où les 2-bulles peuvent soit avoir les orientations contraires, auquel cas le degré topologique de la solution vaut 0, soit avoir la même orientation, ce qui impliquerait que le degré topologique de la solution vaut ±2k. La première situation est l'objet du Théorème 2.8, et l'on sait qu'une deux-bulle peut se former, au moins dans le cas k ≥ 3. D'un autre côté, par des arguments variationnels bien connus, l'énergie potentielle de toute application k-équivariante de degré topologique 2k excède 2E(W ), donc la conservation de l'énergie implique qu'il n'y a pas de 2-bulles pures dans ce dernier cas.

Notre preuve du Théorème 2.11 est de nature variationnelle, tout comme la preuve dans le cas des applications harmoniques décrite ci-dessus. Le contenu dynamique y est réduit à l'étude de la dynamique hyperbolique induite par la présence d'une direction stable et d'une direction instable du flot au voisinage de W , ce qui constitue une différence majeure par rapport à l'équation (WM). La partie la plus difficile est d'exclure l'existence de solutions globales qui se comporteraient asymptotiquement comme une superposition d'une bulle positive à l'échelle 1 et d'une bulle négative qui se concentre. À cette fin, en utilisant la variété stable construite par Duyckaerts et Merle [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF], nous définissons des directions d'instabilité raffinées (non linéaires), qui permettent d'obtenir des propriétés de coercitivité de la fonctionnelle d'énergie suffisamment précises et robustes. Ce schéma de preuve est influencé par les résultats de Krieger, Nakanishi et Schlag [START_REF] Krieger | Center-stable manifold of the ground state in the energy space for the critical wave equation[END_REF], que l'on a déjà mentionnés dans le Paragraphe 1.4.

Chapter II

Introduction (english version) 1 Generalities on nonlinear wave equations

The subject of this thesis is to study some nonlinear phenomena in the dynamical behavior of radial solutions of the focusing wave equation with the energy-critical power nonlinearity in dimension N ≥ 3:

∂ 2 t u(t, x) = ∆u(t, x) + |u(t, x)| 4 N -2 u(t, x), (t, x) ∈ R × R N , (NLW)
and of k-equivariant solutions of the wave map equation from R 1+2 to the two-dimensional sphere S 2 :

∂ 2 t u(t, r) = ∂ 2 r u(t, r) + 1 r ∂ r u(t, r) - k 2 2r 2 sin(2u(t, r)), (t, r) ∈ R × (0, +∞). (WM)
Before stating the main results, let us briefly present the subject of the global theory of nonlinear wave equations from a historical perspective.

Preliminaries

The simplest nonlinear wave equations are of the form

∂ 2 t u(t, x) = ∆u(t, x) + f (x, u(t, x)), (t, x) ∈ R × R N , u(t 0 , x) = u 0 (x), ∂ t u(t 0 , x) = u0 (x), x ∈ R N , (1.1)
where ∆ is the Laplacian in the N space variables and f is a (smooth enough) function of 2 real variables such that f (x, 0) = 0. This equation has a natural Hamiltonian structure. Indeed, let F (x, u) := ´u 0 f (x, v) dv and define the energy functional

E : C ∞ 0 (R N ) × C ∞ 0 (R N ) u = (u, u) → E(u) ∈ R, E(u) := ˆRN 1 2 | u(x)| 2 + 1 2 |∇u(x)| 2 -F (x, u(x)) dx.
Equation (1.1) can be written in the form

∂ t u = J • DE(u), u(t 0 ) = u 0 , (1.2)
where D is the Fréchet derivative, J := 0 Id -Id 0 defines the natural symplectic form on

C ∞ 0 × C ∞ 0 and u = (u, u) is an element of the phase space C ∞ 0 × C ∞ 0 . CHAPTER II. INTRODUCTION (ENGLISH VERSION) Denote u, v := ´RN u • v + u • v dx. Let u(t) be a classical solution of (1.2) on a time interval (t 1 , t 2 ). The chain rule yields d dt E(u(t)) = DE(u(t)), ∂ t u(t) = DE(u(t)), J • DE(u(t)) = 0, (1.3)
hence E(u(t)) is a conserved quantity.

A few historical remarks

The history of the local Cauchy problem for the wave equations goes back at least to the Cauchy-Kowalevski theorem, which, for analytic nonlinearities, yields local existence and uniqueness of an analytic solution for analytic initial data. However, from the physical point of view the question of global existence is highly relevant, and the first result in this direction is due to Jörgens [START_REF] Jörgens | Das Anfangswertproblem im Großen für eine Klasse nichtlinearer Wellengleichungen[END_REF], followed by more abstract works of Browder [START_REF] Browder | On non-linear wave equations[END_REF] and Segal [START_REF] Segal | Non-linear semi-groups[END_REF]. Let us state the result in the simplest case of the cubic defocusing Klein-Gordon equation in

R × R 3 , that is equation (1.1) with f (x, u) = -u -u 3 : ∂ 2 t u(t, x) = ∆u(t, x) -u(t, x) -u(t, x) 3 , (t, x) ∈ R × R 3 . (1.4)
Theorem 1.1 (Jörgens, Browder, Segal). For any initial data

u(t 0 ) = u 0 ∈ H 1 (R 3 )×L 2 (R 3 ), there exists a unique global solution u(t) of (1.4). Moreover, if u 0 ∈ H k+1 × H k with k ∈ N, then u ∈ C(R; H k+1 × H k ).
Remark 1.2. Later in this section we will clarify the notion of a solution in the non-smooth setting.

The main ingredient of the proof is the following local existence result.

Proposition 1.3. For any initial data u(t 0 ) = u 0 ∈ H 1 (R 3 ) × L 2 (R 3 ), there exists a unique maximal solution u(t)

: (T -, T + ) → H 1 × L 2 of (1.4). Moreover, if u 0 ∈ H k+1 × H k with k ∈ N, then u ∈ C((T -, T + ); H k+1 × H k ). If T + < +∞, then lim t→T + u(t) H 1 ×L 2 = +∞ (analogously if T -> -∞).
Finally, on compact time intervals the solution is continuous with respect to the initial data in the topology H k+1 × H k for all k ∈ N.

By (1.3) and a standard approximation procedure one obtains that if u(t) is a solution given by Proposition 1.3, then the energy

E(u(t)) = ˆRN 1 2 | u(t, x)| 2 + 1 2 |∇u(t, x)| 2 + 1 2 |u(t, x)| 2 + 1 4 |u(t, x)| 4 dx is constant. But this implies that u(t) H 1 ×L 2 is bounded, thus proving Theorem 1.1.
Since the nonlinearity is defocusing, it was conjectured that the solution has to decay as t → ±∞ at least like the free solutions, causing the nonlinear effects to become negligible. Thus, any solution should behave asymptotically like a solution of a free Klein-Gordon equation (it scatters), in other words one expects asymptotic completeness in an appropriate space. This problem turned out to be quite challenging and was solved in the '80 by Brenner [START_REF] Brenner | On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations[END_REF], with essential earlier contributions of Morawetz and Strauss [START_REF] Morawetz | Decay and scattering of solutions of a nonlinear relativistic wave equation[END_REF].

Theorem 1.4 (Brenner, Morawetz, Strauss). Let u(t) be a solution of (1.4) and let U 0 (t) be the linear Klein-Gordon propagator. Then U 0 (-t)u(t) has a strong limit in H 1 × L 2 as t → ±∞.

Remark 1.5. Existence of a weak limit is relatively elementary and was already known in the '60.

Oversimplifying, one could say that the dynamical behavior of the solutions of (1.4) does not differ much from the free dynamics, at least in the energy space. In establishing these results, the sign of the nonlinearity (repulsive) and its growth at ±∞ (u 3 ) are decisive. It is natural to examine the case of a focusing nonlinearity or of a faster growth of the nonlinearity (for example u 5 ), which will be briefly discussed in the next two paragraphs.

Effects of a focusing nonlinearity

Consider equation (1.1) with f (x, u) = -u + u 3 (the focusing cubic Klein-Gordon equation), in dimension N = 3:

∂ 2 t u(t, x) = ∆u(t, x) -u(t, x) + u(t, x) 3 , (t, x) ∈ R × R 3 . (1.5)
Nothing changes as far as the local Cauchy theory is concerned (Proposition 1.3 remains valid). However, the energy functional corresponding to the focusing case is

E(u) = ˆR3 1 2 | u| 2 + 1 2 |∇u| 2 + 1 2 |u| 2 - 1 4 |u| 4 dx,
hence it no longer allows to control the H 1 × L 2 norm for large times. The fact that Theorem 1.1 fails in the focusing case was first observed by Keller [START_REF] Keller | On solutions of nonlinear wave equations[END_REF]. We see that sufficiently large constant in space initial data lead to solutions tending to +∞ in finite time. Using the finite speed of propagation, this produces initial data in C ∞ 0 which lead to a blow-up in finite time.

Another consequence of the focusing sign is the existence of stationary solutions W , that is solutions of the elliptic problem

∆W (x) -W (x) + W (x) 3 = 0, x ∈ R 3 . (1.6)
The initial data u 0 := W := (W, 0) lead to a constant in time solution of (1.5). Among the stationary solutions, the ground state plays a distinguished role. It can be characterized as the unique (up to translations) positive solution of (1.6), or as the solution minimizing the potential energy E(u) := ´R3

1 2 |∇u| 2 + 1 2 |u| 2 -1 4 |u| 4 dx.
The variational picture is that for u small in the energy norm E(u) creates a potential well and W acts as the mountain pass. Payne and Sattinger [START_REF] Payne | Saddle points and instability of nonlinear hyperbolic equations[END_REF] exhibited the role of E(W ) as the threshold energy. Namely, if a solution of energy E(u 0 ) < E(W ) is inside the potential well ( u 0 H 1 ≤ W H 1 ), it stays there forever, and if it is outside ( u 0 H 1 > W H 1 ), it blows up in finite time.

Since the solutions in the potential well are global, it is natural to study if they converge to linear solutions in the energy space, as in the defocusing case. A general approach to this problem, called the Ground State Conjecture or the Threshold Conjecture, was developed by Kenig and Merle [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energycritical, focusing, non-linear schrödinger equation in the radial case[END_REF][START_REF] Kenig | Global well-posedness, scattering and blow-up for the energycritical focusing non-linear wave equation[END_REF] for the energy-critical wave equation and the energy-critial radial Schrödinger equation. In the case of the cubic Klein-Gordon equation, the proof was carried out by Ibrahim, Masmoudi and Nakanishi [START_REF] Ibrahim | Scattering threshold for the focusing nonlinear Klein-Gordon equation[END_REF].

Theorem 1.6 (Ibrahim, Masmoudi, Nakanishi). Let u(t) be a solution of (1.5)

such that E(u 0 ) < E(W ) and u 0 H 1 ×L 2 ≤ W H 1 . Then U 0 (-t)u(t) has a strong limit in H 1 × L 2 as t → ±∞.
The next crucial issue is to understand the dynamical behavior slightly above the ground state energy. A very elegant solution of this problem, showing a link with the theory of invariant manifolds, was given by Nakanishi and Schlag [START_REF] Nakanishi | Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation[END_REF][START_REF] Nakanishi | Global dynamics above the ground state for the nonlinear Klein-Gordon equation without a radial assumption[END_REF].

Energy-critical nonlinearities

Let us return for a moment to the general equation (1.1). We have seen that the structure of the problem often allows to control directly or indirectly the homogeneous energy norm

u(t) 2 Ḣ1 ×L 2 := ˆRN |∂ t u(t, x)| 2 + |∇u(t, x)| 2 dx,
giving some information on the regularity of a solution (in the Klein-Gordon case we can control also u L 2 , but this is less interesting from the point of view of regularity). Whether this is enough to preclude a breakdown of a solution will depend on the nonlinearity, and more specifically on the growth of |f (x, u)| as |u| → +∞. We will study heuristically the possibility of a singularity formation such that the energy norm remains bounded. This type of singularity is referred to as a type II blow-up or a geometric blow-up (in opposition to a type I blow-up, referring to a situation where the energy norm tends to +∞ in finite time). For u(x) = (u(x), u(x)) and λ > 0 we define the energy-critical scale change

u λ (x) := 1 λ N -2 2 u x λ , u λ (x) := 1 λ N -2 2 u x λ , 1 λ N 2 u x λ
A straightforward change of variables shows that u λ Ḣ1 ×L 2 = u Ḣ1 ×L 2 . At the same time, the energy norm concentrates at the origin as λ → 0. Suppose to simplify that f (x, u) = f (u) does not depend on x and that N ≥ 3. If we assume that |f (u)| |u| p as |u| → +∞, then

|f (u λ )| 1 λ p(N -2) 2 -N -2 2
|u| p λ , for small λ,

whereas ∆u λ = 1 λ 2 ∆u λ . If p(N -2) 2 -N -2 2 < 2 ⇔ p < N +2 N -2
, then the effect of the nonlinearity is negligible as compared to the laplacian as λ → 0. Intuitively, concentration of the energy norm causes the nonlinear effects to become negligible, in particular cannot result in a breakdown of the solution. The only way of forming a singularity is thus the growth of the energy norm, which is a part of Proposition 1.3. This type of nonlinearity is called energy-subcritical. If, on the contrary, p > N +2 N -2 , then the nonlinearity is the dominant part as λ → 0. This means that a concentration of an arbitrarily small amount of energy could potentially cause the nonlinear effects to prevail, resulting in a singularity. Such a nonlinearity is called energy-supercritical. The borderline case, p = N +2 N -2 , is called energy-critical. In this last situation, as the data concentrates, the nonlinearity acts with the same force as the laplacian, a priori leading to a finite-energy singularity.

Probably the simplest energy-critical wave equation is the semilinear defocusing wave equation with the energy-critical power nonlinearity:

∂ 2 t u(t, x) = ∆u(t, x) -|u(t, x)| 4 N -2 u(t, x), (t, x) ∈ R × R N . (1.7)
The local well-posedness in the energy space Ḣ × L 2 is due to Shatah and Struwe [START_REF] Shatah | Well-posedness in the energy space for semilinear wave equations with critical growth[END_REF], with important contributions of Kapitanski [START_REF] Kapitanski | Global and unique weak solutions of nonlinear wave equations[END_REF] and Ginibre, Soffer and Velo [START_REF] Ginibre | The global Cauchy problem for the critical nonlinear wave equation[END_REF]. Note that for

u = (u, u) ∈ Ḣ1 × L 2 the energy E(u) = ´RN 1 2 | u| 2 + 1 2 |∇u| 2 + N -2 2N |u| 2N N -2
dx is well defined due to the critical Sobolev embedding. We also have E(u λ ) = E(u), because the nonlinearity has the same scaling as the linear part. Finally, the equation is invariant by the energy-critical change of scale: if (u(t, x), u(t, x)) is a solution of (1.7) on an interval I t 0 , then for any λ > 0 we can build another solution u t 0 + t λ , x λ , defined on some time interval containing t = 0.

It turns out that, for the solutions of (1.7), the energy norm cannot concentrate, and that an analog of Theorem 1.1 holds, which was proved by Struwe [START_REF] Struwe | Globally regular solutions to the u 5 klein-gordon equation[END_REF] in the radial case and by Grillakis [START_REF] Grillakis | Regularity and asymptotic behavior of the wave equation with a critical nonlinearity[END_REF] without the symmetry assumption. Also, the solutions scatter, i.e. asymptotically approach linear waves in the energy space as t → ±∞. In [START_REF] Tao | Nonlinear Dispersive Equations: Local and Global Analysis[END_REF]Chapter 5] one can find some details about the history of the resolution of this problem.

Most of this thesis is concerned with equation (NLW), which is the focusing counterpart of (1.7). The energy is given by

E(u) = ˆRN 1 2 | u| 2 + 1 2 |∇u| 2 - N -2 2N |u| 2N N -2 dx.
The local well-posedness can be established by modifying the proofs in the defocusing case, see [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energycritical, focusing, non-linear schrödinger equation in the radial case[END_REF]. One obtains the following result.

Proposition 1.7. For any initial data u 0 ∈ Ḣ1 (R N )×L 2 (R N ), there exists a unique maximal solution u(t) : (T -, T + ) → Ḣ1 × L 2 of (NLW). If u 0 Ḣ1 ×L 2 is sufficiently small, then T -= -∞, T + = +∞ and u(t) scatters in both time directions. The solution is continuous with respect to the initial data in the topology Ḣ1 × L 2 .

Note an important difference with respect to the sub-critical case discussed above. If T + < +∞, it is no longer guaranteed that lim t→T + u Ḣ1 ×L 2 = +∞. The reason for this was heuristically described at the beginning of this paragraph. We have however the standard abstract blow-up criterion: if

T + < +∞ and K is a compact subset of Ḣ1 × L 2 , then there exists τ > 0 such that u(t) / ∈ K for t ∈ [T + -τ, T + ). The elliptic problem ∆W (x) + |W (x)| 4 N -2 W (x) = 0, x ∈ R N (1.8)
(called the Yamabe equation) has an explicit positive solution

W (x) = 1 + |x| 2 N (N -2) -N -2 2 .
Because of the energy-critical scaling invariance, for all λ > 0, W λ (x) is also a solution of (1.8). It is known that these are the only finite-energy radially symmetric and, up to translations, the only positive solutions of (1.8). However, the nonradial problem is complicated, see [START_REF] Del Pino | Large energy entire solutions for the Yamabe equation[END_REF]. We obtain the radially symmetric stationary solutions of (NLW) W λ = (W λ , 0). The role of W λ as the mountain passes for the potential energy

E(u) = ´1 2 |∇u| 2 -N -2 2N |u| 2N
N -2 dx follows from the works of Aubin [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF] and Talenti [START_REF] Talenti | Best constant in Sobolev inequality[END_REF].

The Ground State Conjecture for the energy-critical focusing wave equation was proved in the work of Kenig and Merle [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energycritical, focusing, non-linear schrödinger equation in the radial case[END_REF] for N ∈ {3, 4, 5}. For N = 3, Krieger, Schlag and Tataru [START_REF] Krieger | Slow blow-up solutions for the H 1 (R 3 ) critical focusing semilinear wave equation[END_REF] constructed solutions developing a singularity in finite time, while staying in an arbitrarily small neighborhood (in the energy space) of the family of the ground states W λ . In particular, this construction gave the first example of a type II blow-up. Duyckaerts, Kenig and Merle [START_REF] Duyckaerts | Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation[END_REF][START_REF] Duyckaerts | Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case[END_REF] proved that any solution of (NLW) blowing up in a neighborhood of the set {W λ } asymptotically decomposes in the energy space as a sum of a ground state, rescaled and Lorentz-transformed, and an asymptotic profile. More precisely, in the case of radial data, if T + < +∞, then there exists a positive function λ(t) T + -t and u * 0 ∈ Ḣ1 ×L 2 such that lim

t→T + u(t) -u * 0 -W λ(t) Ḣ1 ×L 2 = 0.
In dimension N = 3 for radially symmetric data, the same authors [START_REF] Duyckaerts | Classification of the radial solutions of the focusing, energy-critical wave equation[END_REF] gave a complete classification of the dynamical behavior of solutions of (NLW) in the energy space. They proved that if T + = +∞, then there exists J ∈ N, positive functions λ 1 (t) λ 2 (t) . . . λ J (t) t and a solution u * (t) of the linear wave equation such that

lim t→+∞ u(t) -u * (t) - J j=1 ±W λ j (t) Ḣ1 ×L 2 = 0. (1.9) If T + < +∞, then either lim t→T + u(t) Ḣ1 ×L 2 = +∞, or there exists J ∈ N, positive functions λ 1 (t) λ 2 (t) . . . λ J (t) T + -t and u * 0 ∈ Ḣ × L 2 such that lim t→T + u(t) -u * 0 - J j=1 ±W λ j (t) Ḣ1 ×L 2 = 0. (1.10)
Krieger, Nakanishi and Schlag [START_REF] Krieger | Center-stable manifold of the ground state in the energy space for the critical wave equation[END_REF] studied the dynamics in a neighborhood of the ground states from a different perspective, establishing a connection with the theory of invariant manifolds. In particular, they construct a C 1 hypersurface containing {W λ }, the centerstable manifold, and show that it separates the set of solutions which scatter forward in time from the solutions which develop a type I singularity. The solutions on the manifold are the ones which stay close to {W λ } as long as they exist.

Remark 1.8. A global theory of energy-supercritical wave equations for large data, even in the defocusing case, currently seems to be out of reach. To the author's knowledge, only conditional results about the dynamics of solutions are available (both in the focusing and defocusing case), see Kenig and Merle [START_REF] Kenig | Nondispersive radial solutions to energy supercritical nonlinear wave equations, with applications[END_REF]. In the case of a defocusing power nonlinearity, Krieger and Schlag [START_REF] Krieger | Large global solutions for energy supercritical nonlinear wave equations on R 3+1[END_REF] proved existence of sets of large initial data leading to global, regular solutions.

Other critical models

Historically, the most intensively studied energy-critical model is the harmonic map heat flow between surfaces, especially between two-dimensional spheres. For a state of the art in this domain, see Topping [START_REF] Topping | Repulsion and quantization in almost-harmonic maps, and asymptotics of the harmonic map flow[END_REF] and the references therein. In particular, the first results classifying the blow-up behavior (bubbling) were obtained in this case.

The terminology "type I blow-up" and "type II blow-up" originates in the works on energy-supercritical heat equation ∂ t u = ∆u + |u| p u, see [START_REF] Matano | Classification of type I and type II behaviors for a supercritical nonlinear heat equation[END_REF].

An intensively studied model, relevant from the physical point of view, is the mass-critical (or L 2 -critical) focusing Schrödinger equation. In some aspects it resembles the L 2 -critical generalized Korteweg-de Vries equation. The expository article [START_REF] Martel | Near soliton dynamics and singularity formation for L 2 critical problems[END_REF] presents the recent major advances in the understanding of these two equations.

Important energy-critical Hamiltonian equations include the nonlinear Schrödinger equation: i∂ t u + ∆u + |u| 4 N -2 u = 0 on R N , and the Schrödinger map equation from R 2 to the sphere S 2 ⊂ R 3 : ∂ t u = u ∧ ∆u, where ∧ is the vector product in R 3 . In the sequel we will occasionally mention these models, for the sake of comparison with the nonlinear wave equations.

Rudiments of the Cauchy theory

For the sake of completeness, in this paragraph we comment on the notion of a solution and of well-posedness. It will be convenient to denote X s := Ḣs+1 ∩ Ḣ1 pour s ≥ 0.

Definition 1.10. Let t 0 ∈ (t 1 , t 2 ) ⊂ R, u 0 = (u 0 , u0 ) ∈ X s × H s and h ∈ L 1 ((t 1 , t 2 ); H s ). For t ∈ (t 1 , t 2 ) set u(t) := cos (t -t 0 ) √ -∆ u 0 + sin (t -t 0 ) √ -∆ √ -∆ u0 + ˆt t 0 sin (t -s) √ -∆ √ -∆ h(s) ds, u(t) := -sin (t -t 0 ) √ -∆ √ -∆u 0 + cos (t -t 0 ) √ -∆ u0 + ˆt t 0 cos (t -s) √ -∆ h(s) ds.
We call u(t) := (u(t), u(t)) the solution of the problem

∂ 2 t u = ∆u + h, (u, ∂ t u) t=t 0 = (u 0 , u0 ).
(1.11)

Note that it follows directly from the formulas that u ∈ C((t 1 , t 2 ), X s × H s ) (which is commonly expressed by saying that "the wave equation gains one derivative"). A quantitative version of this fact is the following fundamental energy estimate:

u(t) X s ×H s ≤ u 0 X s ×H s + ˆt t 0 h(s) H s ds . (1.12)
Existence and uniqueness of weak solutions of the linear problem (1.11), as well as the energy estimate (in the setting of more general symmetric systems) are due to Friedrichs [START_REF] Friedrichs | Symmetric hyperbolic linear differential equations[END_REF].

Definition 1.11. Suppose that ∂ u f (x, 0) = 0. We say that (1.1) is locally well-posed in

X s × H s if
• for any u 0 ∈ X s × H s and t 0 ∈ R there exists a time interval (t 1 , t 2 ) t 0 and a unique u(t) = (u(t), u(t)) such that

f (x, u(t, x)) ∈ L 1 ((t 1 , t 2 ); H s ) (1.13)
and u(t) is the solution of (1.11) with h(t, x) = f (x, u(t, x)),

• u(t) depends continuously in X s × H s on u 0 .

Remark 1.12. For smooth functions, this definition agrees with the usual definition of a classical solution of a differential equation.

Remark 1.13. In general one should consider the operator T := -∆ -∂ u f (x, 0) instead of -∆. Under some natural assumptions on T , we can define a solution in the same manner, replacing everywhere H s by (1 + T s 2 ) -1 L 2 and X s by (T

1 2 + T s+1 2 ) -1 L 2 .
Remark 1. Otherwise, as follows from (1.12), the solution u(t) would be precompact in X s × H s as t → T + , which is impossible.

In the same vain, if T + = +∞ and f (x, u(t, x)) L 1 ([t 0 ,+∞);H s ) < +∞, then u(t) scatters in X s × H s as t → +∞.

Remark 1. [START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: I[END_REF]. In some more "pathological" cases, such as (NLW) for large N , this general definition might not be the correct one, and it should be settled in each individual case what additional condition on u, instead of f (x, u(t, x)) ∈ L 1 ((t 1 , t 2 ); H s ), ensures that the solution is unique and continuously dependent on the initial data.

Main results

In this thesis we consider equation (NLW) for radially symmetric initial data, except in Chapter 2, where we deal also with equation (WM).

Following the works mentioned above, we undertake the study of the local behavior of the dynamical system defined by (NLW) in a small neighborhood of {W λ } in the energy space, that is the solutions u(t) of (NLW) such that

inf λ>0 u(t) -W λ Ḣ1 ×L 2 ≤ η, ∀t,
where η > 0 is a small constant. Chapters 1 and 3 are devoted to the phenomenon of the type II blow-up.

In Chapters 2 and 4 we study the local behavior in a neighborhood of a superposition of two bubbles at different scales, that is solutions u(t) of (NLW) such that inf

0<λ≤α * µ u(t) -(W µ ± W λ ) Ḣ1 ×L 2 ≤ η, ∀t,
where η > 0 is a small constant. The main motivation for this work are of course the classification results of [START_REF] Duyckaerts | Classification of the radial solutions of the focusing, energy-critical wave equation[END_REF] mentioned above, see (1.9) and (1.10). This thesis is composed of the following four articles:

• Chapter 1 -Jendrej, J. Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5. Preprint, arXiv:1503.05024, 2015.

• Chapter 2 -Jendrej, J. Construction of two-bubble solutions for energy-critical wave equations. Preprint, arXiv:1602.06524, 2016.

• Chapter 3 -Jendrej, J. Bounds on the speed of type II blow-up for the energy critical wave equation in the radial case. Int. Math. Res. Not., doi: 10.1093/imrn/rnv365, 2015.

• Chapter 4 -Jendrej, J. Nonexistence of radial two-bubbles with opposite signs for the energy-critical wave equation. Preprint, arXiv:1510.03965, 2015.

Construction of type II blow-up solutions

In this article we propose a new approach to constructing type II blow-up solutions for the energy-critical wave equation. Such solutions were first constructed for the energy-critical wave equation in dimension N = 3 and for the energy-critical wave map equation in equivariance class k = 1 by Krieger, Schlag and Tataru [START_REF] Krieger | Renormalization and blow up for charge one equivariant critical wave maps[END_REF][START_REF] Krieger | Slow blow-up solutions for the H 1 (R 3 ) critical focusing semilinear wave equation[END_REF].

Here, we consider (NLW) in dimension N = 5. We prove the following results.

Theorem 2.1. Let u * 0 = (u * 0 , u * 0 ) ∈ ( Ḣ5 ∩ Ḣ1 ) × H 4 be any pair of radial functions such that u * 0 (0) > 0. There exists a solution u(t) of (NLW) defined on a time interval (0, T 0 ) such that

lim t→0 u(t) -u * 0 -W λ(t) Ḣ1 ×L 2 = 0, (2.1)
where λ(t) = 32 315π 2 u * (0, 0)

2 t 4 .
Theorem 2.2. Let ν > 8. There exists a solution u(t) of (NLW) defined on the time interval (0, T 0 ) such that lim

t→0 u(t) -u * 0 -W λ(t) Ḣ1 ×L 2 = 0,
where λ(t) = t ν+1 and u * 0 is explicit. The fact that we obtain a continuous range of blow-up speeds leads us to believe that the dynamical behavior of our solutions is highly unstable with respect to variations of the initial data. A bit more precisely, we can expect that the set of solutions having the same dynamical characteristics (for example the same blow-up rate) has infinite codimension. We could say that we obtain solutions of Krieger-Schlag-Tataru type, in contrast with the stable solutions constructed by Hillairet and Raphaël [START_REF] Hillairet | Smooth type II blow up solutions to the four dimensional energy critical wave equation[END_REF] in dimension N = 4, which can blow up only at specific rates. Such solutions were also constructed for (WM) by Rodnianski and Sterbenz [START_REF] Rodnianski | On the formation of singularities in the critical O(3) σ-model[END_REF], and Raphaël and Rodnianski [START_REF] Rodnianski | Stable blow up dynamics for critical corotational wave maps and the equivariant Yang Mills problem[END_REF].

Remark 2.3. In this second case numerical experiments suggested the possibility of a formation of singularities, see for instance [START_REF] Bizoń | Formation of singularities for equivariant 2 + 1 dimensional wave maps into the 2-sphere[END_REF]. The blow-up observed numerically is the stable one. The authors of [START_REF] Bizoń | Formation of singularities for equivariant 2 + 1 dimensional wave maps into the 2-sphere[END_REF] suggest that the set of initial data leading to a singularity is huge. However, existence of an open subset of the energy space such that any initial data in this subset develops a singularity, remains unknown.

Let us present the main ideas of the proof of Theorem 2.1, ignoring all the technical issues.

Step 1. Suppose that u(t) = (u(t), u(t)) satisfies (2.1). Let u * (t) = (u * (t), u * (t)) be the solution of (NLW) with the initial data u * (0

) = u * 0 . Denote ΛW := -d dλ W λ | λ=1 . The chain rule yields ∂ t W λ(t) = -λ (t) λ(t) ΛW λ(t) , hence we should have (u(t), u(t)) u * (t) + W λ(t) , u * (t) - λ (t) λ(t) ΛW λ(t) . (2.2)
We find that it is a natural choice to work in the energy space. By a change of variable

λ (t) λ(t) ΛW λ(t) L 2 ∼ λ (t) (
the solution explodes at t = 0, thus λ (t) > 0). If we introduce a small parameter b(t) λ (t), (2.2) can be seen as a beginning of an asymptotic expansion of u(t) in the energy space in powers of b(t).

It turns out that if b (t) = 128 105π u * (t) λ(t), (2.3)
then we can compute the next term of the expansion and define an approximate solution (which we will also call an ansatz )

ϕ(t) := u * (t) + U (0) λ(t) + b(t)U (1)
λ(t) + b 2 (t)U (2)
λ(t) + b 3 (t)U (3)
λ(t) ,
where U (0) = (W, 0) and U (1) = (0, -ΛW ). The precise definitions of U (2) and U (3) are not essential. Note that equation (2.3) together with λ = b yields the value λ app (t) = 32 315π 2 u * (0, 0) 2 t 4 given in the formulation of the theorem.

Remark 2.4. In our definitions of U (2) and U (3) , the decay at |x| → +∞ of W (x) plays an important role, which is why we need N ≥ 5. In large dimensions the nonlinearity |u| 4 N -2 u becomes quite singular at u = 0, which causes some technical difficulties. We chose the easiest dimension N = 5.

Step 2. We consider a sequence of solutions u n (t) of (NLW) with the initial data

u n (t n ) = u * (t n ) + W λapp(tn) , u * (t n ) - λ app (t n ) λ app (t n ) ΛW (t n ) ,
where t n > 0 and lim n→+∞ t n = 0 (an adjustment has to be made because of the exponential instability near W , but this is not a major difficulty). Using a suitable orthogonality condition we decompose u n (t) = ϕ(t) + g n (t) and the goal is to control the size of g n (t), uniformly in n, on a time interval (t n , t 0 ], t 0 > 0.

To this end, we introduce a mixed energy-virial functional H n (t), which is a small perturbation of the energy functional E(ϕ(t)+g n (t))-E(ϕ(t))-DE(ϕ(t)), g n (t) . This functional has the following coercivity property:

g n (t) 2 Ḣ1 ×L 2 H n (t)
(modulo the unstable modes).

Moreover, the fact that ϕ(t) is a refined ansatz can be used to show that for some large constant C 0 we have

g n (t) Ḣ1 ×L 2 ≤ C 0 t 9 2 ⇒ H n (t) ≤ c 0 • C 2 0 t 8 ,
with a small constant c 0 . A classical continuity argument yields the uniform control g Ḣ1 ×L 2 ≤ C 0 t 9 2 . From this estimate we deduce by standard ODE techniques that the modulation parameter λ n (t) is close to λ app (t).

The conclusion of the second step is that

u n (t) -(u * (t) + W λapp (t), u * (t)) Ḣ1 ×L 2 t 3 ,
uniformly with respect to n.

Step 3. Using the profile decomposition of Bahouri and Gérard [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF], we prove a version of sequential weak continuity of the flow, which allows to obtain our solution u(t) as a weak limit of a subsequence of the sequence u n (t) in the energy space.

The idea of constructing a uniformly controlled sequence of solutions converging to a singular solution was introduced by Merle [START_REF] Merle | Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity[END_REF]. Combining this technique with energy estimates was an idea of Martel [START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF]. Raphaël and Szeftel [START_REF] Raphaël | Existence and uniqueness of minimal mass blow up solutions to an inhomogeneous L 2 -critical NLS[END_REF] used a virial correction of the energy functional in a similar context in their study of minimal mass blow-up solutions for the nonlinear Schrödinger equation. The first step of the proof is also inspired by the work of Martel, Merle and Raphaël [START_REF] Martel | Blow up for the critical gKdV equation III: exotic regimes[END_REF] on exotic blow-up for the L 2 -critical generalized Korteweg-de Vries equation. They observed that blow-up rate is directly related to the size of interaction of the bubble with the "background", which is the heart of our construction of the approximate solution.

In fact, the size of this interaction appears explicitly in (2.3), which will be explained in Chapter 1. We find that the main advantage of our method is to demonstrate a direct link between the asymptotic behavior of u * (t) at x = 0 and the asymptotics of λ(t). For example, in Theorem 2.2 we have u * 0 ≡ 0 and u * 0 (x) ∼ |x| ν-3 2 in a neighborhood of x = 0.

A delicate point, left open by our approach, is to understand better the regularity of the constructed solutions. We have no information on the regularity of the solution besides the fact that it belongs to the energy space. At the same time, it is expected that these solutions have singularities, and working at the energy level allows to avoid dealing with them directly. Note also that using "backward energy estimates" seems to be a natural method of constructing unstable objects. Energy estimates "in the direction of the blow-up" typically yield some sort of stability (modulo a finite codimension), which we do not expect here.

In later works, we hope to carry out a similar construction for equation (WM) and in the case of a singular u * 0 .

Remark 2.5. Because of a link with Chapter 3, we would like to point out that the rate of the blow-up can also be formally predicted by solving for λ (t) the equation

E u * (t) + W λ(t) , u * (t) - λ (t) λ(t) ΛW λ(t) = E(W ) + E(u * 0 ).
Remark 2.6. Some of the notation in Chapter 1 differ from the ones used above. We preferred to use coherent notation in all the sections of this introductory chapter.

Construction of two-bubble solutions

In Chapter 2 we construct a radially symmetric solution of (NLW) which exists globally for negative times and decomposes into more that one bubble of energy inside the backward light cone. To the author's knowledge, this is the first example of a solution of this kind. Note that u(t) is an example of a solution of (NLW) other than the ground state which is global in one time direction and does not contain a radiation term in the energy space, more precisely

∀A > 0, lim sup t→-∞ ˆ|x|≥|t|-A | u(t, x)| 2 + |∇u(t, x)| 2 dx = 0.
If we want to emphasize the fact that the energy of our solutions is exactly equal to twice the energy of W , with no energy radiated as a free wave, we say that they are pure two-bubbles.

Let us say a few words about the proof. Observe that in Theorem 2.1 we can take u * 0 = (W, 0), which produces a blow-up in finite time with (W, 0) as the asymptotic profile. This is "almost" what we desire, except that we need a concentration of the bubble in infinite time rather than a blow-up. A natural way to achieve this is to increase the dimension of the space. In dimension N = 6 the interaction between the two bubbles is weaker, which produces the required effect. For N ≥ 7 one should be able to obtain a similar conclusion, with the speed of concentration of the bubble λ ∼ |t| -4 N -6 . The main technical differences with respect to Theorem 2.1 come from the fact that W has a fixed size in the energy space, whereas before, using the finite speed of propagation, it could be assumed that u * 0 Ḣ1 ×L 2 is small. In order to obtain the bounds of the energy norm, we need to study the coercivity of the energy functional in a neighborhood of a sum of two bubbles.

We also extend our construction to the context of equation (WM) with k ≥ 3. We obtain the following result.

Theorem 2.8. Fix k > 2. There exists a solution u : (-∞, T 0 ] → E of (WM) such that

lim t→-∞ u(t) -(-W + W k-2 κ (κ|t|) -2 k-2 ) E = 0, with κ := k -2 2 8k π sin π k 1 k .
Here, W denotes the ground state of topological degree k and E is the natural energy space.

Finally, we prove a similar result for the critical Yang-Mills equation:

∂ 2 t u(t, r) = ∂ 2 r u(t, r) + 1 r ∂ r u(t, r) - 4 2r 2 u(t, r)(1 -u(t, r)) 1 - 1 2 u(t, r) .
Classification results proving that decompositions of type (1.10) and (1.9) hold for a sequence of times t n → T + were obtained by Côte [START_REF] Côte | On the soliton resolution for equivariant wave maps to the sphere[END_REF] for equation (WM) with k = 1, and by Jia and Kenig [START_REF] Jia | Asymptotic decomposition for semilinear wave and equivariant wave map equations[END_REF] in greater generality, including all the cases considered here.

Bounds on the speed of type II blow-up

The results of Chapter 1 encourage to further investigate the relationship between the dynamics of the blow-up and the properties of the asymptotic profile u * 0 . In Chapter 3, we prove the following two results in this direction. Theorem 2.9. Let N ∈ {3, 4, 5} and s > N -2 2 , s ≥ 1. Let u * 0 = (u * 0 , u * 0 ) ∈ H s+1 × H s be radially symmetric. Suppose that u(t) is a radial solution of (NLW) such that

lim t→T + u(t) -W λ(t) -u * 0 Ḣ1 ×L 2 = 0, lim t→T + λ(t) = 0, T + < +∞. (2.4)
There exists a constant C > 0 depending on u * 0 such that:

• if N ∈ {4, 5}, then for T + -t sufficiently small there holds

λ(t) ≤ C(T + -t) 4 6-N .
• if N = 3, then there exists a sequence t n → T + such that

λ(t n ) ≤ C(T + -t n ) 4 6-N . Theorem 2.10. Let N ∈ {3, 4, 5}. Let u * 0 = (u * 0 , u * 0 ) ∈ H 3 × H 2 be radially symmetric and u * 0 (0) < 0.
There exist no radial solutions of (NLW) such that

lim t→T + u(t) -W λ(t) -u * 0 Ḣ1 ×L 2 = 0, lim t→T + λ(t) = 0, T + < +∞. (2.5)
The main idea of the proofs, sketched in Paragraph 1.4 of Chapter 3, is to bound the error term by the (energetic) interaction of the bubble with the remainder, which was alluded to in Remark 2.5.

The general picture emerging from Theorems 2.1, 2.2 and 2.9 is that the speed of the blowup of solutions of Krieger-Schlag-Tataru type is related to the regularity of the asymptotic profile rather than the regularity of the solution itself (although the solutions which blow up faster tend to be more regular, as indicated in [START_REF] Krieger | Renormalization and blow up for charge one equivariant critical wave maps[END_REF][START_REF] Krieger | Slow blow-up solutions for the H 1 (R 3 ) critical focusing semilinear wave equation[END_REF]). Note that, even if for technical reasons we require additional regularity of u * 0 , in (2.4) and (2.5) we require convergence just in the energy space.

A natural question is whether the asymptotic profile uniquely determines the solution which blows up in a neighborhood of {W λ }. If it was the case, then assigning to each blow-up solution its asymptotic profile and vice versa would yield a classification of the blow-up solutions in a neighborhood of {W λ } in the energy space. This general scheme, as pointed out to the author by H. Koch, D. Tataru and the anonymous referee of the article constituting Chapter 3, resembles the classical scattering problem. Of course one should identify the elements of the energy space which can act as the asymptotic profile of some blow-up solution, and the content of Theorem 2.10 is to show that at least certain profiles have to be excluded.

Even if we should probably be rather dubious about the program described above, it raises questions which seem more accessible. For example, does the asymptotic profile determine the speed of the blow-up?

Nonexistence of two-bubbles with opposite signs

Chapter 4 is devoted to a proof of the following fact.

Theorem 2.11. Let N ≥ 3. There exist no radial solutions u : [t 0 , T + ) → Ḣ1 × L 2 of (NLW) such that lim

t→T + u(t) -W λ(t) + W µ(t) Ḣ1 ×L 2 = 0 and • in the case T + < +∞, λ(t) µ(t) T + -t as t → T + ,
• in the case T + = +∞, λ(t) µ(t) t as t → +∞.

An in the setting of Theorem 2.7, we work here at the energy 2E(W ), the threshold energy for a formation of a multi-bubble. The main motivation for this work comes from equation (WM), where the two bubbles can either have opposite orientations, in which case the topological degree of the solution is 0, or have the same orientation, which would imply that the topological degree of the solution is ±2k. This first situation was considered in Theorem 2.8, and we know that a two-bubble can form at least for k ≥ 3. On the other hand, by well known variational arguments, any k-equivariant map of topological degree 2k has energy > 2E(W ), hence the conservation of energy implies that there are no pure two-bubbles in this case.

Our proof of Theorem 2.11 is variational in nature, just like the proof for the wave maps described above. The dynamical content is reduced to the study of the hyperbolic dynamics induced by the presence of the linear stable and unstable modes in a neighborhood of W , which is an important difference with respect to equation (WM). The most difficult part is to exclude global solutions which behave asymptotically as a superposition of a positive bubble at scale 1 and a negative concentrating bubble. To this end, using the stable manifold constructed by Duyckaerts and Merle [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF], we introduce refined (nonlinear) instability directions, which allow to obtain sharp coercivity properties of the energy functional. The scheme of the proof is influenced by the results of Krieger, Nakanishi and Schlag [START_REF] Krieger | Center-stable manifold of the ground state in the energy space for the critical wave equation[END_REF], which we already mentioned in Paragraph 1.4.

Chapter 1

Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5

Abstract

We consider the semilinear wave equation with focusing energy-critical nonlinearity in space dimension N = 5

∂ tt u = ∆u + |u| 4/3 u, with radial data. It is known [START_REF] Duyckaerts | Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case[END_REF] that a solution (u, ∂ t u) which blows up at t = 0 in a neighborhood (in the energy norm) of the family of solitons W λ , decomposes in the energy space as (u(t), ∂ t u(t)) = (W λ(t) + u * 0 , u * 1 ) + o(1), where lim t→0 λ(t)/t = 0 and (u * 0 , u * 1 ) ∈ Ḣ1 × L 2 . We construct a blow-up solution of this type such that the asymptotic profile (u * 0 , u * 1 ) is any pair of sufficiently regular functions with u * 0 (0) > 0. For these solutions the concentration rate is λ(t) ∼ t 4 . We also provide examples of solutions with concentration rate λ(t) ∼ t ν+1 for ν > 8, related to the behaviour of the asymptotic profile near the origin.

Introduction 1.General setting

We are interested in the problem of constructing type II blow-up solutions for the energycritical wave equation in space dimension N = 5:

∂ tt u = ∆u + |u| 4/3 u, (t, x) ∈ R × R 5 .
Denote f (u) := |u| 4/3 u. It will be convenient to write the wave equation as a first-order in time system:

         ∂ ∂t u ∂ t u = ∂ t u ∆u + f (u) , u(t 0 ) ∂ t u(t 0 ) = u 0 u 1 ∈ Ḣ1 × L 2 .
(NLW)

This equation is locally well-posed in the energy space Ḣ1 × L 2 (see for example [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energycritical focusing non-linear wave equation[END_REF] and the references therein). In particular, for any initial data (u 0 , u 1 ) there exists a maximal interval of existence (T -, T + ), -∞ ≤ T -< t 0 < T + ≤ +∞, and a unique solution (u,

∂ t u) ∈ C((T -, T + ); Ḣ1 × L 2
). This solution conserves the energy:

E(u(t), ∂ t u(t)) = 1 2 ˆ|∂ t u| 2 dx + 1 2 ˆ|∇u| 2 dx -ˆF (u) dx = E(u 0 , u 1 ),
where F (u) = ´f (u) du = 3 10 |u| 10/3 (notice that ´F (u) dx is finite by the Sobolev imbedding).

For a function v : R 5 → R and λ > 0, we denote

v λ (x) := 1 λ 3/2 v x λ , v λ (x) := 1 λ 5/2 v x λ .
A change of variables shows that

E (u 0 ) λ , (u 1 ) λ = E(u 0 , u 1 ).
Equation (NLW) is invariant under the same scaling. If (u, ∂ t u) is a solution of (NLW) and λ > 0, then

t → u t -t 0 λ λ , ∂ t u t -t 0 λ λ
is also a solution with initial data (u 0 ) λ , (u 1 ) λ at time t = 0. This is why equation (NLW) is called energy-critical.

We introduce also the infinitesimal generators of scale change:

Λv := -∂λv λ λ=1 = 3 2 + x • ∇ v, Λ 0 v := -∂λv λ λ=1 = 5 2 + x • ∇ v.
A fundamental object in the study of (NLW) is the family of solutions (u, ∂ t u) = (W λ , 0), where

W (x) = 1 + |x| 2 15 -3/2 .
The functions W λ are called ground states. In this paper we are interested in radial solutions (u, ∂ t u) of (NLW) such that inf λ (u -W λ , ∂ t u) Ḣ1 ×L 2 remains small for T -< t ≤ t 0 . In the case N = 3 it was proved by Krieger, Nakanishi and Schlag [START_REF] Krieger | Center-stable manifold of the ground state in the energy space for the critical wave equation[END_REF] that such solutions form a codimension one manifold in a neighbourhood of the family {W λ }. This is expected to hold also for N = 5. The asymptotic behaviour of such (not necessarily radial) solutions as t → T -was described by Duyckaerts, Kenig and Merle in [START_REF] Duyckaerts | Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case[END_REF], both in the case T -= -∞ and T -> -∞. In the second case, which is relevant for us, they obtain the following result.

Theorem. [25, Theorem 2] Let (u, ∂ t u) be a solution of (NLW) such that T -= 0 and inf λ (u -W λ , ∂ t u) Ḣ1 ×L 2 remains small for T -< t ≤ T 0 . Then there exists a C 0 function λ(t) : (0, T 0 ) → (0, +∞), such that

lim t→0 + u(t) -W λ(t) , ∂ t u(t) = (u * 0 , u * 1 ) ∈ Ḣ1 × L 2 ,
and the convergence is strong in Ḣ1 × L 2 . In addition, λ(t) t as t → 0 + .

In this context, W λ is called the bubble of energy and (u * 0 , u * 1 ) is called the asymptotic profile.

Solutions of this type were first constructed by Krieger, Schlag and Tataru [START_REF] Krieger | Slow blow-up solutions for the H 1 (R 3 ) critical focusing semilinear wave equation[END_REF] in space dimension N = 3, where it is shown that for any ν > 1/2 there exists a solution such that the concentration speed is λ(t) ∼ t 1+ν (later Krieger and Schlag [START_REF] Krieger | Full range of blow up exponents for the quintic wave equation in three dimensions[END_REF] improved this to ν > 0). Similar results where obtained for energy-critical wave maps by the same authors [START_REF] Krieger | Renormalization and blow up for charge one equivariant critical wave maps[END_REF], for energy-critical NLS in dimension N = 3 by Ortoleva and Perelman [START_REF] Ortoleva | Nondispersive vanishing and blow up at infinity for the energy critical nonlinear Schrödinger equation in R 3[END_REF] and for energy-critical Schrödinger maps by Perelman [START_REF] Perelman | Blow up dynamics for equivariant critical Schrödinger maps[END_REF]. Using a different approach, Hillairet and Raphaël [START_REF] Hillairet | Smooth type II blow up solutions to the four dimensional energy critical wave equation[END_REF] obtained C ∞ blow-up solutions for energy-critical wave equation in dimension N = 4 with blow-up rate λ(t) = t exp -√ -log t(1 + o(1)) . Collot [START_REF] Collot | Type II blow up for the energy supercritical wave equation[END_REF] obtained a related result for supercritical wave equation in large dimension.

It follows from the classification of solutions with energy E(W ) by Duyckaerts and Merle [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF] that necessarily (u * 0 , u * 1 ) = 0. In other words, we have non-existence of minimal energy blow-up solutions. Analogous result is true also for energy-critical wave maps, energy-critical Schrödinger maps and energy-critical NLS. This is in contrast with the L 2 -critical NLS where the conformal invariance produces explicit solutions concentrating a bubble of mass and tending weakly to 0 at blow-up. Existence of blow-up solutions with a non-zero smooth asymptotic profile was first observed by Bourgain and Wang [START_REF] Bourgain | Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity[END_REF]. Blow-up solutions close to the ground state in the case of L 2 -critical NLS were extensively studied in a series of papers by Merle and Raphaël. They examined in particular the relationship between regularity of the asymptotic profile and the blow-up speed. One can consult a survey [START_REF] Martel | Near soliton dynamics and singularity formation for L 2 critical problems[END_REF] for an account of these results in a proper perspective and a presentation of recent developpements in the case of L 2 -critical gKdV.

Main results

The aim of this paper is to construct solutions which blow up by concentration of one bubble of energy in space dimension N = 5. Our approach differs substantially from [START_REF] Krieger | Slow blow-up solutions for the H 1 (R 3 ) critical focusing semilinear wave equation[END_REF] in that it produces a blow-up solution with a given asymptotic profile. This profile is seen as a source term which permits concentration of the bubble. This point of view is close to a recent construction by Martel, Merle and Raphaël [START_REF] Martel | Blow up for the critical gKdV equation III: exotic regimes[END_REF] in the case of L 2 -critical gKdV.

Denote X s := Ḣs+1 ∩ Ḣ1 . We prove the following two results.

Theorem 1. Let (u * 0 , u * 1 ) ∈ X 4 ×H 4 be any radial functions with u * 0 (0) > 0. Let (u * (t), ∂ t u * (t)) be the solution of (NLW) for the initial data (u * (0), ∂ t u * (0)) = (u * 0 , u * 1 ). There exists a solution (u, ∂ t u) of (NLW) defined on a time interval (0, T 0 ) and a C 1 function λ(t) : (0, T 0 ) → (0, +∞) such that

(u(t)-W λ(t) -u * (t), ∂ t u(t)+λ (t)(ΛW ) λ(t) -∂ t u * (t)) Ḣ1 ×L 2 = O(t 9/2 ) as t → 0 + , (1.1)
and λ(t) = 32 315π 2 u * (0, 0) 2 t 4 + o(t 4 ).

Theorem 2. Let ν > 8. There exists a solution (u, ∂ t u) of (NLW) defined on the time interval (0, T 0 ) such that

lim t→0 + u(t) -W λ(t) -u * 0 , ∂ t u(t) -u * 1 Ḣ1 ×L 2 = 0,
where λ(t) = t ν+1 , and (u * 0 , u * 1 ) is an explicit radial C 2 function.

We will refer to the situation of Theorem 1 as the non-degenerate case and to the situation of Theorem 2 as the degenerate case. Note that in Theorem 1 we allow any regular (u * 0 , u * 1 ) with u * 0 (0) > 0. Our result might be seen as a first step in a possible classification of all blow-up solutions with a non-degenerate asymptotic profile. Theorem 2 demonstrates how the asymptotic behaviour of (u * 0 , u * 1 ) at x = 0 influences the blow-up speed. The condition ν > 8 is imposed by our method. It could be improved at the cost of some technical details, but we are far from obtaining the whole range ν > 0 as in [START_REF] Krieger | Slow blow-up solutions for the H 1 (R 3 ) critical focusing semilinear wave equation[END_REF] for N = 3.

Let us mention that radiality is only a simplifying assumption. All the estimates used here are true also in the non-radial situation.

In Theorem 2, the function u * 0 is given explicitely by (4.1) and u * 1 = 0. It follows from our proof that there exists a C 1 function λ(t) : (0, T 0 ) → (0, +∞) such that λ(t) = t ν+1 + o(t ν+1 ) and the solution (u, ∂ t u) satisfies

u(t) -W λ(t) -u * (t), ∂ t u(t) + λ (t)(ΛW ) λ(t) -∂ t u * (t) Ḣ1 ×L 2 = O(t 7 6 ν- 4
3 ).

Structure of the proof

In Section 2 we present a formal computation which explains the relation between the asymptotic behaviour of (u * 0 , u * 1 ) and the blow-up speed, as well as the relevance of the condition u * 0 (0) > 0.

In Section 3 we specify an ansatz (ϕ 0 (t), ϕ 1 (t)) in the non-degenerate case and prove appropriate bounds on the error of this approximate solution.

In Section 4 we choose (u * 0 (0), u * 1 (0)) such that the same procedure leads to an approximate solution with λ(t) ∼ t 1+ν , and we prove appropriate bounds on the error in this situation.

Section 5 covers both the non-degenerate and the degenerate case. We use a well-known compactness argument introduced by Merle [START_REF] Merle | Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity[END_REF] and used by several authors starting with the work of Martel [START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] for constructions of multi-solitons. We take a decreasing sequence t n → 0 + and we define (u n , ∂ t u n ) as the solution of (NLW) such that (u n (t n ), ∂ t u n (t n )) is close to the approximate solution at time t = t n . The heart of the analysis is to obtain uniform energy bounds for this sequence. That is to say, there exists T 0 > 0 such that (u n (t), ∂ t u n (t)) stays close to (ϕ 0 (t), ϕ 1 (t)) for t n ≤ t ≤ T 0 , with bounds independent of n. Note that the exponential instability of W λ causes an additional difficulty in the argument. We use the shooting method to eliminate the unstable mode. The blow-up solution (u, ∂ t u) is obtained as a weak limit of a subsequence of (u n , ∂ t u n ). To obtain the crucial uniform energy bounds, we use a mixed energy-virial functional. This method was introduced by Raphaël and Szeftel [START_REF] Raphaël | Existence and uniqueness of minimal mass blow up solutions to an inhomogeneous L 2 -critical NLS[END_REF] for a construction of minimal mass blow-up solutions for NLS.

In Appendix A we prove sequential weak continuity of the dynamical system (NLW) under some natural (non-optimal) condition, which is an adaptation of an analogous result of Bahouri and Gérard in the defocusing case [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF]Corollary 1]. This result is required in order to extract a weak limit of the sequence (u n , ∂ t u n ).

In Appendix B we provide for reader's convenience some well-known estimates of the X 1 × H 1 norm of solutions of (NLW). The persistence of X 1 × H 1 regularity is used in Section 5. The energy estimates are used in Section 4. They are non-optimal, but sufficient for our purposes. We prove also propagation of regularity in a neighbourhood of the origin in the non-degenerate case, which is used in Section 3.

Notation

For v, w ∈ L 2 we denote v, w := ˆR5 v • w dx.
We use the same notation for the duality pairing when v ∈ Ḣ-s and w ∈ Ḣs . Linearizing -∆V -f (V ) around V = W λ we obtain a self-adjoint operator

L λ h := -∆h -f (W λ )h.
Differentiating -∆W λ -f (W λ ) = 0 with respect to λ we find

L λ (ΛW ) λ = 0.
We denote

L := L 1 = -∆ -f (W ).
We will also use the notation v(t) := (v(t), ∂ t v(t)).

We denote Z a fixed radial C ∞ 0 function such that ΛW, Z > 0. Finally, χ is a fixed standard C ∞ cut-off function (χ(r) = 1 for r ≤ 1, χ(r) = 0 for r ≥ 2, χ (r) ≤ 0).

2 Formal picture and construction of blow-up profiles

LA = κΛW + f (W ), LB = -Λ 0 ΛW. (2.1) In addition, A(r) ∼ r -1 , A (r) ∼ r -2 , A (r) ∼ r -3 and B(r) ∼ r -1 , B (r) ∼ r -2 , B (r) ∼ r -3 as r → +∞.
Proof. In the proof we will use some standard facts from the theory of Sturm-Liouville equations, see for example [START_REF] Teschl | Ordinary Differential Equations and Dynamical Systems[END_REF]Chapter 5]. Solving equation (2.1) is equivalent to solving the following ODE:

-(p(r)y ) + q(r)y = g(r), (2.2)
with r ∈ (0, +∞), p(r) = r 4 , q(r) = -r 4 f (W ) and g(r) = g A (r) = r 4 (κΛW (r) + f (W (r))) or g(r) = g B (r) = -r 4 Λ 0 ΛW (r). Notice that |g(r)| r 4 for small r.

We know that ΛW (r) is a solution of (2.2) with g(r) = 0. Let Γ(r) be a second solution normalized in such a way that

W(ΛW, Γ) = r 4 (ΛW • Γ -(ΛW ) • Γ) = 1
(W is the modified wronskian, in particular its value is independent of r).

Take r 1 < √ 15, r 2 > √ 15 (recall that r = √ 15 is the unique point where ΛW vanishes) and define

y 1 (r) := ΛW (r) • ˆr r 1 ds s 4 (ΛW (s)) 2 , for r < √ 15, y 2 (r) := ΛW (r) • ˆr r 2 ds s 4 (ΛW (s)) 2 , for r > √ 15.
It can be easily checked that y 1 and y 2 are solutions of the homogeneous equation and verify W(ΛW, y 1 ) = W(ΛW, y 2 ) = 1. Hence, we have y j = a j ΛW + Γ for some scalar coefficients a 1 , a 2 . Directly from the formulas defining y 1 and y 2 we obtain the asymptotic behaviour of y 1 as r → 0 + and of y 2 as r → ∞ :

y 1 (r) ∼ - ˆr1 r ds s 4 ∼ - 1 r 3 , r → 0 + , y 2 (r) ∼ -1 r 3 ˆr r 2 ds s 4 • s -6 ∼ -1, r → +∞.
As adding a constant multiple of ΛW does not change these asymptotics, we obtain that Γ(r) ∼ -r -3 as r → 0 + and Γ(r) ∼ -1 as r → +∞. From the relation W(ΛW, Γ) = 1 we get

Γ = r -4 + (ΛW ) • Γ ΛW ,
which immediately gives Γ (r) ∼ r -4 as r → 0 and Γ (r) ∼ ±r -1 as r → +∞ (it can be checked that the sign is " + ", but we will not use this fact). For r 0 , r ∈ (0, +∞) we define

s(r, r 0 ) := ΛW (r 0 )Γ(r) -Γ(r 0 )ΛW (r). (2.3)
We see that s(r 0 , r 0 ) = 0 and r 4 0 d dr s(r, r 0 )| r=r 0 = 1, which means that s(r, r 0 ) is the second fundamental solution of (2.2). Now using the Duhamel formula we obtain a solution of the non-homogeneous equation (2.2):

A(r) = ˆr 0 s(r, r )g A (r ) dr , B(r) = ˆr 0 s(r, r )g B (r ) dr .
Fix r > 0 and let |h| ≤ 1 2 r. In the estimates which follow, all the constants may depend on r. We have

A(r + h) -A(r) h - ˆr 0 d dr s(r, r )g A (r ) dr ≤ ˆr 0 s(r + h, r ) -s(r, r ) h - d dr s(r, r ) • |g A (r )| dr + 1 h ˆr+h r |s(r + h, r )| • |g A (r )| dr .
Formula (2.3) implies that |s( r, r 0 )| h when | r -r| ≤ h and |r -r 0 | ≤ h. Hence, the second term above converges to 0 as h → 0. For 0 ≤ r 0 ≤ r and | r -r| ≤ 1 2 r we have the bound

| d 2 dr 2 s( r, r 0 )| r -3 0 . This implies s(r + h, r ) -s(r, r ) h - d dr s(r, r ) ≤ 1 2 sup | r-r|≤h d 2 dr 2 s( r, r ) • |h| (r ) -3 • |h|,
so the first term above also converges to 0 as h → 0. This shows that A(r) (and similarly B(r)) is continuously differentiable and

A (r) = ˆr 0 d dr s(r, r )g A (r ) dr , B (r) = ˆr 0 d dr s(r, r )g B (r ) dr .
It is clear from these formulas that lim r→0 + A (r) = lim r→0 + B (r) = 0. It follows from above considerations that A and B, seen as functions on R 5 , are C 1 , so they are C ∞ by elliptic regularity. Now we consider the behaviour of A(r) and B(r) as r → +∞. From the crucial orthogonality relation ´+∞ 0 ΛW (r )g A (r ) dr = 0 we deduce that

ˆr 0 ΛW (r )g(r ) dr = ˆ+∞ r ΛW (r )g(r ) dr r -1 .
From this and the asymptotics of Γ and g A it follows that |A(r)| r We define A and B as the solutions of (2.1) satisfying the orthogonality condition

ˆR5 Z • A dx = ˆR5 Z • B dx = 0. (2.4)

Determination of blow-up speeds

Let u * (t, x) be the solution of (NLW) for initial data (u * (0), ∂ t u * (0)) = (u 0 , u 1 ). At a formal level, while computing the interaction of u * with the soliton, we will treat u * as a function constant in space and C 2 in time, u * (t, x) v * (t). (In the non-degenerate case we will take v * (t) = u * (t, 0) and in the degenerate case v * (t) = qt β , where q and β are appropriate constants.) We will construct a solution which blows up at t = 0 and is defined for small positive t. This means that in our situation the caracteristic length λ will increase in time. The usual method of performing a formal analysis of blow-up solutions in the case of the wave equation consists in defining b := λ t and searching a solution in the form of a power series in b. Following this scheme, we write

u = W λ + u * (t) + b 2 T λ + lot ∂ t u = -b(ΛW ) λ + ∂ t u * + lot.
Here, the profile T is undetermined, and we search a convenient blow up speed. Neglecting irrelevant terms and replacing λ t := d dt λ(t) by b, we compute

∂ tt u = -b t (ΛW ) λ + b 2 λ (Λ 0 ΛW ) λ + ∂ tt u * + lot.
On the other hand,

∆u + f (u) = - 1 λ b 2 (LT ) λ + f (W λ )v * + ∆u * + f (u * ) + lot.
We discover that, formally at least, we should have

LT = -Λ 0 ΛW + λ b 2 [b t ΛW + v * (t) √ λf (W )]. (2.5) Proposition 2.1 shows that if b t = κv * (t)λ 1/2 , (2.6)
then equation (2.5) has a decaying regular solution

T = B + v * (t)λ 3/2 b 2
A. We call equation (2.6) together with the equation λ t = b formal parameter equations. In the non-degenerate case v * (t) = u * (t, 0) is close to u * (0, 0), so we expect that there exists a solution of the formal parameter equations which is close to

(λ(t), b(t)) = κ 2 u * (0, 0) 2 144 t 4 , κ 2 u * (0, 0) 2 36 t 3 . (2.7)
This is indeed the case, as follows from our analysis in Section 5.

In the degenerate case we have v * (t) = qt β , and the formal parameter equations have a solution

(λ(t), b(t)) = (t 1+ν , (1 + ν)t ν ) (2.8) if we choose q = ν(1+ν) κ and β = ν-3 2 .
3 Approximate solution in the non-degenerate case 3.1 Bounds on the profile (P 0 , P 1 )

The functions A and B from the previous section do not belong to the space Ḣ1 . We will place a cut-off at the light cone, that is at distance t from the center. Given modulation parameters (λ(t), b(t)), we define:

P 0 (t) := χ • t (λ(t) 3/2 v * (t)A λ(t) + b(t) 2 B λ(t) ). (3.1)
Recall that in the non-degenerate case v * (t) = u * (t, 0) ∈ C 2 by Proposition B.6 and Schauder estimates.

Remark 3.1. Because of the finite speed of propagation, without loss of generality we can replace (u * 0 , u * 1 ) by χ

• ρ u * 0 , χ • ρ u * 1 ,
where ρ is a strictly positive constant to be chosen later. Thus, without loss of generality we can assume that the support of (u * 0 , u * 1 ) is contained in a small ball and that (u * 0 , u * 1 ) X 1 ×H 1 is small.

Remark 3.2. The fact that the profile (P 0 , P 1 ) is cut at r = t = t 1 can be considered as a coincidence. The power of t has been chosen in order to optimize the estimates. This is the only power for which we can obtain the estimate of the error term which has asymptotically the same size as the profile P 0 . Also, for this choice, P 1 L 2 (the forth term of the asymptotic expansion which will be defined in a moment) is asymptotically the same as P 0 Ḣ1 . However, the angle of the cone has no significance for us.

Remark 3.3. Notice that the orthogonality condition which we choose to define A and B has little significance due to a relatively fast decay of ΛW . We will use the same orthogonality condition as for the error term, as this choice simplifies slightly the computation. Observe that the fact that Z has compact support implies that if λ(t) t, then ´P0 (t)Z λ dx = 0 for small t.

In the error estimates which will follow, on the right hand side we will always replace λ(t) by t 4 and b(t) by t 3 , as this is the regime that we are going to consider later in the bootstrap argument. In this section, all the constants may depend on u * . Lemma 3.4. Assume that λ(t) ∼ t 4 and b(t) ∼ t 3 . Then

P 0 (t) Ḣ1 t 9/2 . (3.2) Proof. It is sufficient to show that χ( • t )A λ 2 Ḣ1
t -3 (the computation for B λ is the same). We have

χ • t A λ 2 Ḣ1 ˆ+∞ 0 χ r t A λ (r) 2 r 4 dr = ˆ+∞ 0 χ λr t A(r) 2 r 4 dr ˆ+∞ 0 χ λr t A (r) 2 r 4 dr + ˆ+∞ 0 λ t χ λr t A(r) 2 r 4 dr ˆ2t/λ 0 r 4 1 r 4 dr + λ 2 t 2 ˆ2t/λ t/λ r 4 1 r 2 dr t λ ∼ t -3 .
Lemma 3.5. Assume that λ(t) ∼ t 4 and b(t) ∼ t 3 . Then

L λ P 0 -λ 3/2 v * (t)L λ A λ -b 2 L λ B λ L 2 t 7/2 .
Proof. We will do the computation only for the terms with A. The terms with B are asymptotically the same. We need to check that

1 -χ r t f (W λ )A λ L 2 + ∆ 1 -χ r t A λ L 2 t -5/2
For the first term we have even some margin since

1 -χ r t f (W λ )A λ L 2 = 1 λ 1 -χ λr t f (W )A L 2 1 λ ˆ+∞ t/λ (r -4 r -1 ) 2 r 4 dr 1/2 ∼ 1 λ • λ t 5/2 ∼ t 7/2 .
For the second term, we have a few possibilities. Recall that ∆ = ∂ rr + 4∂r r . Either the laplacian hits directly A:

1-χ r t ∆(A λ ) L 2 = 1 λ 1-χ λr t ∆A L 2 1 λ ˆ+∞ t/λ (r -3 ) 2 r 4 dr 1/2 ∼ 1 λ • λ t ∼ t -5/2 ,
either one derivative hits χ:

1 t χ r t d dr (A λ ) L 2 = 1 t χ λr t A (r) L 2 t -1 ˆ2t/λ t/λ (r -2 ) 2 r 4 dr 1/2 ∼ t -1 • t λ ∼ t -5/2 ,
(and analogously the term

1 t χ r t 4 r (A λ ) L 2 )
, or two derivatives hit χ, and we get

1 t 2 χ r t A λ L 2 = λ t 2 χ λr t A L 2 λ t 2 ˆ2t/λ t/λ (r -1 ) 2 r 4 dr 1/2 ∼ λ t 2 • t λ 3/2 ∼ t -5/2 .
We define P 1 (t) as a formal time derivative of P 0 (t), which means that we replace λ t by b and b t by κv * (t)λ 1/2 , see (2.6), and we do not differentiate the cut-off function. Explicitely, set

P 1 (t) = χ • t v * (t) 3 2 λ 3/2 bA λ -λ 3/2 b(ΛA) λ + λ 5/2 ∂ t v * (t)A λ + 2κv * (t)λ 3/2 bB λ -b 3 (ΛB) λ . (3.3)
Notice that in the regime (2.7) the coefficient λ 5/2 is smaller than the other coefficients (all of which are, asymptotically, of the same size). However, we prefer to keep the corresponding term in the definition of P 1 .

Lemma 3.6. Assume that λ(t) ∼ t 4 and b(t) ∼ t 3 . Then

P 1 (t) L 2 t 9/2 (3.4)
Proof. All the terms except for the one mentioned above have the same asymptotics, so we will do the computation only for the first one. It is sufficient to show that χ(

• t )A λ 2 L 2 t -9 . We have χ( • t )A λ 2 L 2 ∼ χ( λr t )A(r) 2 L 2 (r 4 dr) ˆ2t/λ 0 (r -1 ) 2 r 4 dr ( t λ ) 3 ∼ t -9 .
Our ansatz ϕ(t) = (ϕ 0 (t), ϕ 1 (t)) is defined as follows:

ϕ 0 (t) = W λ(t) + P 0 (t) + u * (t), ϕ 1 (t) = -b(t)(ΛW ) λ(t) + P 1 (t) + ∂ t u * (t),
where P 0 and P 1 are given by (3.1) and (3.3). The error term ε(t) = (ε 0 (t), ε 1 (t)) is defined by the formula:

u(t) = ϕ 0 (t) + ε 0 (t), ∂ t u(t) = ϕ 1 (t) + ε 1 (t).
We shall impose the orthogonality condition ˆε0 Z λ dx = 0.

Lemma 3.7. If λ ∼ t 4 , b ∼ t 3 and t is small enough, then

|λ t -b| ≤ ε Ḣ1 ×L 2 . (3.5)
Proof. To find the formula for λ t , first we write

-b(ΛW ) λ + ∂ t u * + P 1 (t) + ε 1 (t) = ∂ t u = -λ t (ΛW ) λ + ∂ t u * + ∂ t P 0 (t) + ∂ t ε 0 ⇒ ∂ t ε 0 = (λ t -b)(ΛW ) λ + (P 1 -∂ t P 0 ) + ε 1 .
Notice that for small t and λ ∼ t 4 we have

ˆ(P 1 (t) -∂ t P 0 (t))Z λ dx = (λ t -b)[λ 3/2 v * (t) ΛA, Z L 2 + b 2 ΛB, Z L 2 ].
This follows from (2.4) and the fact that supp(Z λ ) is contained in the light cone for small t. This gives

0 = d dt ˆε0 Z λ dx = ˆ∂t ε 0 Z λ dx -λ t ˆε0 1 λ (Λ 0 Z) λ dx = ˆ(λ t -b) ΛW, Z + (λ t -b)[λ 3/2 v * (t)( ΛA, Z L 2 + b 2 ΛB, Z L 2 ] + ε 1 , Z λ L 2 -λ t ˆε0 1 λ (Λ 0 Z) λ dx,
and we obtain

(λ t -b)[ ΛW, Z + λ 3/2 v * (t) ΛA, Z + b 2 ΛB, Z ] = -ε 1 , Z λ + λ t ε 0 , 1 λ (Λ 0 Z) λ .
Rearranging the terms we get

λ t = 1 - ε 0 , 1 λ (Λ 0 Z) λ ΛW, Z L 2 + λ 3/2 v * (t) ΛA, Z + b 2 ΛB, Z -1 • • b - ε 1 , Z λ ΛW, Z + λ 3/2 v * (t) ΛA, Z + b 2 ΛB, Z . (3.6)
For t small enough, (3.5) follows.

Remark 3.8. To be precise, our rigourous argument goes the other way round -we use (3.6) and (2.6) to define the local evolution of the modulation parameters, and then by doing exactly the same computation as above, but in the opposite direction, we find that the orthogonality condition ε 0 , 1 λ Z λ L 2 = 0 is preserved if it is verified at the initial time (which will be the case). Notice also that using (2.4) we obtain

u -W λ -u * , Z λ = 0. (3.7)
Differentiating this condition we find

λ t ΛW, Z + ε 0 , 1 λ (Λ 0 Z) λ = -∂ t u -∂ t u * , Z λ . (3.8)
We need to estimate the error between the formal and the actual time derivative of P 0 : Lemma 3.9. Assume that λ(t) ∼ t 4 and b(t) ∼ t 3 . Then

∂ t P 0 -P 1 Ḣ1 √ t(t 3 + ε Ḣ1 ×L 2 ).
Proof. The error has two parts -one comes from differentiating in time the cut off function and the other one from |λ t -b|.

∂ t P 0 -P 1 = - r t 2 χ r t (λ 3/2 v * (t)A λ + b 2 B λ ) + χ r t (λ t -b)[v * (t) 3 2 λ 3/2 A λ -λ 3/2 (ΛA) λ -b 2 (ΛB) λ ].
Using Proposition 2.1, we can write:

r λ χ r t A λ Ḣ1 = rχ λr t A Ḣ1 χ λr t • 1 r L 2 + λ t rχ λr t • 1 r L 2 + rχ λr t • 1 r 2 L 2 χ λr t • 1 t L 2 + λ t χ λr t L 2 ∼ t λ 3/2 ∼ t -9/2 .
The same computation is valid also for A replaced by B. Now we have

r t 2 χ r t λ 3/2 A λ Ḣ1 λ t 2 λ 3/2 • t -9/2 ∼ t 2 t 6 t -9/2 = t 7/2 ,
and the same for the second term.

The computation for the second line is similar:

χ r t A λ Ḣ1 = χ( λr t )A Ḣ1 χ λr t • 1 r 2 L 2 + λ t χ λr t • 1 r L 2 ∼ t/λ ∼ t -3/2 .
Multiplying by √ λ(λ t -b) and using Lemma 3.7 we get the desired estimate. The last two terms are exactly the same.

Finally, the following estimate allows to stop the asymptotic expansion of the solution at P 1 . Lemma 3.10. Assume that λ(t) ∼ t 4 and b(t) ∼ t 3 . Then

∂ t P 1 L 2 √ t(t 3 + ε Ḣ1 ×L 2 ).
Proof. Consider first the terms coming from differentiating the cut-off function. Like in the proof of the previous lemma, we have

r λ χ r t A λ L 2 χ λr t L 2 ∼ t λ 5/2 , which gives r t 2 χ r t v * (t)λ 3/2 bA λ L 2 λ t 2 λ 3/2 b • t λ 5/2 ∼ t 7/2 .
The term r t 2 χ r t λ 5/2 ∂ t v * (t)A λ L 2 is even smaller. Consider now the other terms. They are of one of the following six types:

• χ r t λ t λ 1/2 bT λ , • χ r t λ t b 3 λ T λ , • χ r t b t λ 3/2 T λ , • χ r t b t b 2 T λ , • χ r t λ t λ 3/2 d t v * (t)T λ , • χ r t λ 5/2 d tt v * (t)T λ ,
where T ∈ {A, B, ΛA, ΛB, Λ 0 A, Λ 0 B, Λ 0 ΛA, Λ 0 ΛB}. In all the situations T is regular and decays like r -1 (see Proposition 2.1), so we can write

χ r t T λ L 2 ˆ2t/λ t/λ 1 r 2 r 4 dr 1/2 t λ 3/2 ∼ t -9/2 .
Using the fact that

λ ∼ t 4 , b ∼ t 3 , λ t b + ε , b t √ λ and that v * (t) is C 2 we obtain λ t λ 1/2 b + λ t b 3 λ + b t λ 3/2 + b t b 2 + λ t λ 3/2 |d t v * | + λ 5/2 |d tt v * | t 5 (t 3 + ε ),
which finishes the proof.

The last lemma shows that ϕ is "almost constant" after rescaling.

Lemma 3.11. Let c 1 > 0. If T 0 is sufficiently small, then for t ∈ (0, T 0 ] there holds

∂ t (ϕ 0 ) 1/λ Ḣ1 ≤ c 1 t .
Proof. By the definition of ϕ 0 and P 0 we get

(ϕ 0 ) 1/λ = W + χ λ • t [λ 3/2 A + b 2 B] + (u * ) 1/λ .
The terms with A and B are similar, so we only consider the first one. We observe that | λt λ | 1 t for small t, with an explicit numerical constant. Now

∂ t χ λr t λ 3/2 A = 3 2 λ t λ χ λr t λ 3/2 A - λr t 2 χ λr t λ 3/2 A.
The size of the first term is acceptable by Lemma 3.4. For the second one, it is sufficient to notice that | λr t 2 | ≤ 2 t on the support of χ. The conclusion follows again from Lemma 3.4. (Notice that we have a large margin for these two terms.)

Next, we have

∂ t (u * ) 1/λ Ḣ1 ≤ ∂ t u * Ḣ1 + λ t λ Λu * Ḣ1 .
By Proposition B.2 the first term is bounded for small t. Choosing ρ small enough (see Remark 3.1), we can guarantee that Λu * (t) Ḣ1 will stay small for small t, which is exactly what we need.

Error of the ansatz

Our next objective is to estimate the error of the approximate solution, defined as

ψ(t) = ψ 0 (t) ψ 1 (t) := ∂ t ϕ 0 (t) ∂ t ϕ 1 (t) - ϕ 1 (t) ∆ϕ 0 (t) + f (ϕ 0 (t))
.

In order to do this we first need to extract the principal terms of the nonlinear term, which is based on the following pointwise estimate:

Lemma 3.12.

|f (k + l + m) -[f (k) + f (m) + f (k)l + f (k)m]| |f (l)| + f (l)|k| + f (m)|k| + f (m)|l|. (3.9)
Proof. The inequality is homogeneous, so we can suppose that k 2 + l 2 + m 2 = 1. The right hand side vanishes only for (k, l, m) ∈ {(±1, 0, 0), (0, 0, ±1)}, so it suffices to prove the inequality in a neighborhood of these 4 points, where it is an easy consequence of the Taylor expansion of f .

Lemma 3.13. If λ(t) ∼ t 4 , b ∼ t 3 and t is small, then f (ϕ 0 (t)) -[f (W λ(t) ) + f (u * ) + f (W λ(t) )P 0 (t) + f (W λ(t) )u * (t)] L 2 t 4 .
Proof. We put in the preceding lemma k = W λ(t) , l = P 0 (t), m = u * (t), and we estimate the L 2 norm of the 4 terms on the right hand side of (3.9). When P 0 (t) appears, we split it into two parts. We sometimes forget χ, as its presence here can only help (there are no derivatives).

Term "|f (l)|":

χ r t λ 3/2 A λ 7/3 χ r t • r λ -7/3 ,
and r -14/3 is integrable near 0, so

χ r t λ 3/2 A λ 7/3 L 2 λ 7/3 χ r t r -7/3 L 2 t 4 . In a similar way, χ r t b 2 B λ 7/3 L 2 t 4 . Term "f (l)|k|": By a change of variables we get (λ 3/2 A λ ) 4/3 W λ L 2 = λ A 4/3 W L 2 ∼ t 4 (exponent of λ on the left = (3/2 -3/2) • (4/3) -3/2 = -3/2, and the L 2 scaling is -5/2).
In a similar way,

(b 2 B λ ) 4/3 W λ L 2 = λ -1 b 8/3 B 4/3 W L 2 ∼ t 4 .
Term "f (m)|k|": We use once again the L ∞ bound of u * and the fact that W λ L 2 ∼ λ.

Term "f (m)|l|": Using (3.2) and the fact that u * (t) is bounded in L 20/3 for small t (by Proposition B.2), we have

f (u * )P 0 L 2 ≤ f (u * ) L 5 • P 0 L 10/3 t 9/2 .
We can now estimate ψ(t). Proposition 3.14. Assume that λ(t) ∼ t 4 and b(t) ∼ t 3 . Then

ψ 0 (t) + (λ t -b) 1 λ (ΛW ) λ Ḣ1 √ t( ε(t) Ḣ1 ×L 2 + t 3 ), (3.10) ψ 1 (t) -(λ t -b) b λ (Λ 0 ΛW ) λ L 2 √ t( ε(t) Ḣ1 ×L 2 + t 3 ). (3.11)
Proof. The first inequality is just a reformulation of Lemma 3.9.

For the second inequality, we divide the error into several parts:

ψ 1 = ∂tϕ 1 -(∆ϕ 0 + f (ϕ 0 )) = (-b t (ΛW ) λ + bλ t λ (Λ 0 ΛW ) λ + ∂tP 1 + ∂ tt u * ) -(∆W λ + ∆P 0 + ∆u * ) -(f (W λ ) + f (u * ) + f (W λ )P 0 (t) + f (W λ )u * ),
where we have used Lemma 3.13 in order to raplace f (ϕ 0 ) by the sum of its principal terms.

Rearranging the terms and using (2.6), we can rewrite the sum above as follows:

ψ 1 = (λ t -b) b λ (Λ 0 ΛW ) λ -(∆W λ + f (W λ )) + (∂ tt u * -∆u * -f (u * )) -v * (t) √ λ(-LA + κΛW + f (W )) λ + b 2 λ (LB + Λ 0 ΛW ) λ + (-∆P 0 -f (W λ )P 0 ) -v * (t) √ λ(LA) λ - b 2 λ (LB) λ + (v * (t) -u * (t)) √ λ(f (W )) λ + ∂ t P 1 + O(t 7/2
). Now we proceed line by line.

Line 1. This is the correction that we substract in (3.11).

Line 2. Both terms equal 0.

Line 3. Both terms equal 0 by the definition of A and B.

Line 4. This error is due to the presence of the cut-off function in (3.1), and Lemma 3.5 tells us that it is acceptable.

Line 5. This error is due to the fact that we replace the interaction with u * (t) by the interaction with the constant in space function v * (t). It follows from Proposition B.6 that |v * (t) -u * (t, r)| r uniformly in time when r ≤ t and t is small. Hence,

(v * (t) -u * (t, r))f (W λ ) L 2 (r≤t) r √ λ(f (W )) λ L 2 ∼ λ 3/2 ∼ t 6 .
(We have used the fact that rf (W ) ∈ L 2 .) In the zone r ≥ t first we use the fact that v * is bounded and

f (W λ ) L 2 (r≥t) = √ λ f (W ) L 2 (r≥tλ -1 ) √ λ(λ/t) 3/2 ∼ t 13/2 .
As for u * , we know from Proposition B.2 that it is bounded in L 10 . By Hölder

u * • f (W λ ) L 2 (r≥t) ≤ u * L 10 • f (W λ ) L 5/2 (r≥t)
, and a routine computation shows that the last term is bounded by (λ/t) 2 ∼ t 6 . Line 6. This error is small by Lemma 3.10.

4 Approximate solution in the degenerate case 4.1 Bounds on the profile (P 0 , P 1 )

This section is very similar to the previous one. Formula (3.1) is still valid, but recall that in the present case we take v * (t) = qt β where q = ν(1+ν) κ and β = ν-3

2 . The function u * 0 is defined as follows:

u * 0 (x) := χ • ρ • p|x| β , p = 3q (β + 1)(β + 3) , ρ > 0 small. ( 4.1)
(by the finite speed of propagation the cut-off does not affect the behaviour at zero for small times, cf. Remark 3.1). We take u * 1 = 0. In the error estimates which will follow, on the right hand side we will always replace λ(t) by t 1+ν and b(t) by t ν , since this is the regime considered later in the bootstrap argument. Lemma 4.1. Assume that λ(t) ∼ t 1+ν and b(t) ∼ t ν . Then

P 0 (t) Ḣ1 t 3ν/2 . (4.2) Proof. Recall that v * (t) ∼ t β = t (ν-3)/2 , so λ 3/2 v * (t) ∼ b 2 ∼ t 2ν . Hence, it is sufficient to show that χ( • t )A λ 2 Ḣ1 + χ( • t )B λ 2 Ḣ1
t -ν . The computation in the proof of Lemma 3.4 gives

χ • t A λ 2 Ḣ1 t λ ∼ t -ν ,
and similarly for the second term.

Lemma 4.2. Assume that λ(t) ∼ t 1+ν and b(t) ∼ t ν . Then

L λ P 0 -λ 3/2 v * (t)L λ A λ -b 2 L λ B λ L 2 t 3ν/2-1 .
Proof. We will do the computation only for the terms with A. The terms with B are asymptotically the same. We need to check that

1 -χ r t f (W λ )A λ L 2 + ∆ 1 -χ r t A λ L 2 t -ν/2-1
The computations in the proof of Lemma 3.5 imply that the first term is bounded by

1 λ λ t 5/2 ∼ t 3ν/2-1
, and the second by

1 λ • λ t + 1 t • t λ + λ t 2 • t λ 3/2 ∼ (t • λ) -1/2 ∼ t -ν/2-1 .
In the degenerate case the profile P 1 (t) is defined by the same formula (3.3).

Lemma 4.3. Assume that λ(t) ∼ t 1+ν and b(t) ∼ t ν . Then

P 1 (t) L 2 t 3ν/2 (4.3) Proof. Notice that d dt v * (t) ∼ t β-1 ∼ t (ν-5)/2 . This implies that v * (t) • λ 3/2 b ∼ d dt v * (t) • λ 5/2 ∼ b 3 ∼ t 3ν
, so all the terms in the definition of P 1 (t) have asymptotically the same size and it suffices to show that χ(

• t )A λ 2 L 2
t -3ν (the other terms are similar). The computation in the proof of Lemma 3.6 gives

χ • t A λ 2 L 2 t λ 3 ∼ t -3ν .
Estimate (3.5) and its proof are valid in the degenerate case.

Lemma 4.4. Assume that λ(t) ∼ t 1+ν and b(t) ∼ t ν . Then

∂ t P 0 -P 1 Ḣ1 t ν/2-1 (t ν + ε Ḣ1 ×L 2 ).
Proof. As in the proof of Lemma 3.9, we write

∂ t P 0 -P 1 = - r t 2 χ ( r t )(λ 3/2 v * (t)A λ + b 2 B λ ) + χ( r t )(λ t -b)[v * (t)( 3 2 λ 3/2 A λ -λ 3/2 (ΛA) λ ) -b 2 (ΛB) λ ].
The computation in the proof of Lemma 3.9 implies

r λ χ r t A λ Ḣ1 t λ 3/2 ∼ t -3ν/2 .
Multiplying by λ t 2 λ 3/2 v * (t) ∼ t 3ν-1 we obtain the required bound on the first term. The second term of the first line is similar.

The second line is bounded exactly as in the proof of Lemma 3.9.

Lemma 4.5. Assume that λ(t) ∼ t 1+ν and b(t) ∼ t ν . Then

∂ t P 1 L 2 t ν/2-1 (t ν + ε Ḣ1 ×L 2 ).
Proof. We indicate only the modifications with respect to the proof of Lemma 3.10. The term coming from differentiating the cut-off function is estimated as before by

λ t 2 v * (t)λ 3/2 b • t λ 5/2 ∼ t 3ν/2-1 .
For the other terms, we get

χ r t T λ L 2 t λ 3/2 ∼ t -3ν/2 .

Error of the ansatz

This subsection differs from the non-degenerate case, because we work here only with X 1 regularity and some more effort is required in order to estimate the terms involving u * .

Lemma 4.6. If λ(t) ∼ t 1+ν , b ∼ t ν , ν > 8 and t is small, then f (ϕ 0 (t)) -[f (W λ(t) ) + f (u * ) + f (W λ(t) )P 0 (t) + f (W λ(t) )u * (t)] L 2 t 7 6 ν-7 3 .
Proof. As in the proof of Lemma 3.13 we use Lemma 3.12 with k = W λ(t) , l = P 0 (t) and m = u * (t). We obtain that the L 2 norm of the term "|f (l)|" is bounded by

(v * • λ) 7/3 + b 14/3 χ r t r -7/3 L 2 ,
which is better than required. For the term "f (l)|k|" we obtain the bound (v * ) 4/3 • λ + b 4/3 λ -1 ∼ t 5ν/3-1 , which is again better than required.

Term "f (m)|k|": Let (u lin * , ∂ t u lin * ) be the solution of the free wave equation for the initial data (u lin

* (0), ∂ t u lin * (0)) = (u * 0 , u * 1 ). We write f (u * )•W λ L 2 f (u lin * )•W λ L 2 (|x|≤ 1 2 t) + f (u * -u lin * )•W λ L 2 (|x|≤ 1 2 t) + f (u * )•W λ L 2 (|x|≥ 1 2 t)
and we examine separately the three terms on the right hand side. It follows from Proposition B.7 that for |x| ≤ 1 2 t we have the bound

|u lin * (t, x)| t β = t ν-3 2 , which implies f (u lin * (t)) L ∞ t 2 3 (ν-3) , hence f (u lin * ) • W λ L 2 (|x|≤ 1 2 t) t 2 3 (ν-3) W λ L 2 ∼ t 2 3 (ν-3) t ν+1 t 7 6 ν-7 3 .
From Proposition B.8 we infer

u * -u lin * L 20/3 (|x|≤ 1 2 t) t 7 6 ν-7 3 , hence f (u * -u lin * ) L 5 (|x|≤ 1 2 t) t 14 9 ν-28 9 ,
which leads to

f (u * -u lin * ) • W λ L 2 (|x|≤ 1 2 t) ≤ f (u * -u lin * ) L 5 (|x|≤ 1 2 t) • W λ L 10/3 (|x|≤ 1 2 t) t 14 9 ν-28 9 ,
which is more than sufficient for ν > 8.

For |x| ≥ 1 2 t, we know from Proposition B.2 that f (u * ) L 5 is bounded for small t. By a change of variables we obtain

W λ L 10/3 (|x|≥ 1 2 t) ˆ+∞ t/2λ (r -3 ) 10/3 r 4 dr 3 10 ∼ λ t 3/2 t 7 6 ν-7 3 .
Term "f (m)|l|": Using (4.2) we have

f (u * ) • P 0 L 2 ≤ f (u * ) L 5 • P 0 Ḣ1 t 3ν/2 t 7 6 ν-7 3 .
We can now estimate ψ(t).

Proposition 4.7. Assume that λ(t) ∼ t 1+ν and b(t) ∼ t ν . Then

ψ 0 (t) + (λ t -b) 1 λ (ΛW ) λ Ḣ1 t 7 6 ν-7 3 + t ν/2-1 ε(t) Ḣ1 ×L 2 , (4.4
)

ψ 1 (t) -(λ t -b) b λ (Λ 0 ΛW ) λ L 2 t 7 6 ν-7 3 + t ν/2-1 ε(t) Ḣ1 ×L 2 .
Proof. The first inequality follows from Lemma 4.4.

For the second inequality, as in the proof of Proposition 3.14, using Lemma 4.6 and rearranging the terms, we get:

ψ 1 = (λ t -b) b λ (Λ 0 ΛW ) λ -(∆W λ + f (W λ )) + (∂ tt u * -∆u * -f (u * )) -v * (t) √ λ(-LA + κΛW + f (W )) λ + b 2 λ (LB + Λ 0 ΛW ) λ + (-∆P 0 -f (W λ )P 0 ) -v * (t) √ λ(LA) λ - b 2 λ (LB) λ + (v * (t) -u * (t)) √ λ(f (W )) λ + ∂ t P 1 + O(t 7 6 ν- 7
3 ).

Lines 1, 2, 3, 4 and 6 are treated exactly as in the proof of Proposition 3.14, using Lemmas 4.2 and 4.5 instead of Lemmas 3.5 and 3.10. We estimate line 5 as follows:

(v * -u * )f (W λ ) L 2 (v * -u * lin )f (W λ ) L 2 (|x|≤ 1 2 t) + (u * lin -u * )f (W λ ) L 2 (|x|≤ 1 2 t) + v * • f (W λ ) L 2 (|x|≥ 1 2 t) + u * • f (W λ ) L 2 (|x|≥ 1 2 t) .
From Proposition B.7 it follows in particular that |v * (t) -u * lin (t, r)| r when r ≤ 1 2 t, hence the proof of Proposition 3.14 gives the bound

(v * -u * lin ) • f (W λ ) L 2 (|x|≤ 1 2 t) λ 3/2 t 7 6 ν-7 3 .
From Proposition B.8 and the fact that

f (W λ ) L 5/2 = f (W ) L 5/2 we get (u * -u * lin ) • f (W λ ) L 2 (|x|≤ 1 2 t) t 7 3 β+ 7 6 = t 7 6 ν-7 3 .
We have 10 and Hölder inequality we obtain the required bounds, which terminates the proof. Lemma 3.11 is still valid in the degenerate case, as well as its proof (we use Lemma 4.1 instead of Lemma 3.4).

f (W λ ) L 2 (|x|≥ 1 2 t) λ 2 |x| 4 L 2 (|x|≥ 1 2 t) ∼ λ 2 t -3/2 and f (W λ ) L 5/2 (|x|≥ 1 2 t) λ 2 |x| 4 L 5/2 (|x|≥ 1 2 t) ∼ λ 2 t -2 . Using boundedness of v * in L ∞ , boundedness of u * in L

Evolution of the error term

The evolution of the error term ε is governed by the following system of differential equations:

∂ t ε 0 ε 1 = ε 1 -ψ 0 ∆ε 0 + [f (ϕ 0 + ε 0 ) -f (ϕ 0 )] -ψ 1 . , (5.1)
coupled with the equations (3.6) and (2.6) for the modulation parameters Mod := (λ, b). We denote (T -, T + ) the maximal interval of existence of u.

We introduce the energy functional adapted to our ansatz:

I(t) := ˆ1 2 |ε 1 | 2 + 1 2 |∇ε 0 | 2 -[F (ϕ 0 + ε 0 ) -F (ϕ 0 ) -f (ϕ 0 )ε 0 ] dx.
Essentially we will perform a bootstrap argument in order to control this functional just by integrating in time its time derivative. We need a virial correction term which is defined as follows:

J(t) := b ˆε1 • 1 λ • 1 2 (∆a) λ + (∇a) λ • ∇ ε 0 dx, where a λ (r) = a( r λ ), (∇a) λ (r) = ∇a r λ , (∆a) λ (r) = ∆a r λ and a(r) :=      1 2 r 2 |r| ≤ R 15 8 Rr - 5 2 R 2 + 5 4 R 3 r -1 - 1 8 R 5 r -3 |r| ≥ R
(R is a big radius to be chosen later, see Proposition 5.2).

Lemma 5.1. The function a(r) defined above, viewed as a function on R 5 , has the following properties:

• a ∈ C 3,1 ,
• a is strictly convex,

• |a(r)| r, |a (r)| 1, |a (r)| r -1 when r → +∞ (the constant depends on R), • -1 r 3 ∆ 2 a(r) ≤ 0.
Proof. It is apparent from the formula defining a that a is regular except for r = R. A computation shows that a(r), a (r), a (r) and a (r

) are Lipschitz near r = R. For r ≥ R we have a (r) = 5 2 R r 3 -3 2 R r 5 > 0, which proves strict convexity. For r > R one can compute ∆ 2 a(r) = -15 r 3 • 1 R 2 (where ∆ = ∂ rr + 4 r ∂ r is the laplacian in dimension N = 5
). We define the mixed energy-virial functional:

H(t) = I(t) + J(t).
The proof of the following result, which will occupy most of this section, is valid both in the non-degenerate and the degenerate case. The non-degenerate case is obtained for ν = 3. We denote also:

γ := 7 2
in the non-degenerate case, We will use the notation:

∇ a,λ ε 0 2 L 2 := ˆ i,j (∂ ij a) λ ∂ i ε 0 ∂ j ε 0 dx. Proposition 5.2. Let ν = 3 or ν > 8. Suppose that λ ∼ t 1+ν , b ∼ t ν and let c > 0.
If R is chosen large enough, then there exist strictly positive constants T 0 and C 1 such that for

[T 1 , T 2 ] ⊂ (0, T 0 ] ∩ (T -, T + ) there holds H(T 2 ) ≤ H(T 1 ) + ˆT2 T 1 - b λ ∇ a,λ ε 0 2 L 2 - ˆ f (ϕ 0 + ε 0 ) -f (ϕ 0 ) ε 0 dx + c t ε 2 Ḣ1 ×L 2 + C 1 t γ • ε Ḣ1 ×L 2 dt.
(5.

2)

The proof of this result is going to be an algebraic computation which is not justified in the space Ḣ1 × L 2 . However, we do not need any uniform control of the regularity or the decay, so we can use the following density argument. We can approximate a given ε in Ḣ1 × L 2 in such a way that the initial data (u(T 1 ), ∂ t u(T 1 )) will be in X 1 × H 1 and of compact support. Then locally the evolution will have the same proprieties by Proposition B.5, and will be close to the original one in Ḣ1 × L 2 for all t ∈ [T 1 , T 2 ] by local well-posedness in Ḣ1 × L 2 . The new ε has sufficient regularity and decay to justify all the computations. Since the estimate (5.2) depends continuously (in Ḣ1 × L 2 ) on ε, we are done.

We shall split the proof of Proposition 5.2 into several Lemmas. We always work under the hypotheses of Proposition 5.2, that is λ ∼ t 1+ν , b ∼ t ν and ε Ḣ1 ×L 2 ≤ t γ+1 . Notice that γ + 1 > ν. In the non-degenerate case γ + 1 = 9 2 > 3 = ν and in the degenerate case γ + 1 = 7 6 ν -4 3 > ν because ν > 8. This means that ε Ḣ1 ×L 2 b and ε Ḣ1 ×L 2 λ t for small t. In what follows c stands for any small strictly positive constant.

We use the method introduced in [START_REF] Raphaël | Existence and uniqueness of minimal mass blow up solutions to an inhomogeneous L 2 -critical NLS[END_REF], which consists in differentiating the nonlinear term in self-similar variables. The resulting error will be corrected by the virial term J. Concretely, we have:

d dt ˆ[F (ϕ 0 + ε 0 ) -F (ϕ 0 ) -f (ϕ 0 )ε 0 ] dx = d dt ˆ[F ((ϕ 0 ) 1/λ + (ε 0 ) 1/λ ) -F ((ϕ 0 ) 1/λ ) -f ((ϕ 0 ) 1/λ )(ε 0 ) 1/λ ] dx = ˆ[f ((ϕ 0 ) 1/λ + (ε 0 ) 1/λ ) -f ((ϕ 0 ) 1/λ ) -f ((ϕ 0 ) 1/λ )(ε 0 ) 1/λ ]∂ t (ϕ 0 ) 1/λ dx + ˆ[f ((ϕ 0 ) 1/λ + (ε 0 ) 1/λ ) -f ((ϕ 0 ) 1/λ )] (ε 0t ) 1/λ + λ t λ (Λε 0 ) 1/λ dx.
The first term can be neglected, as shown by Lemma 3.11. Scaling back the second term we obtain

d dt ˆ[F (ϕ 0 + ε 0 ) -F (ϕ 0 ) -f (ϕ 0 )ε 0 ] dx ˆ[f (ϕ 0 + ε 0 ) -f (ϕ 0 )] ε 0t + λ t λ Λε 0 dx. (5.3)
Here and later the sign means that the difference of the two sides has size at most

c t ε 2 Ḣ1 ×L 2 + C 1 t γ • ε Ḣ1 ×L 2 .
Also, when we say that a term is "negligible", it always means that its absolute value is bounded by

c t ε 2 Ḣ1 ×L 2 + C 1 t γ • ε Ḣ1 ×L 2 .
Using the equations (5.1), (5.3) and integrating by parts, we obtain standard cancellations:

d dt I(t) ˆε1 ε 1t dx -ˆ[∆ε 0 + f (ϕ 0 + ε 0 ) -f (ϕ 0 )]ε 0t dx - λ t λ ˆ[f (ϕ 0 + ε 0 ) -f (ϕ 0 )]Λε 0 dx -ˆε1 ψ 1 dx + ˆ[∆ε 0 + f (ϕ 0 + ε 0 ) -f (ϕ 0 )]ψ 0 dx - λ t λ ˆ[f (ϕ 0 + ε 0 ) -f (ϕ 0 )]Λε 0 dx.
(5.4)

Consider now the virial term J(t).

Lemma 5.3.

d dt J(t) ≤ ˆε1 ψ 1 dx - b λ ∇ a,λ ε 0 2 L 2 + λ t λ ˆ[f (ϕ 0 + ε 0 ) -f (ϕ 0 )]Λ 0 ε 0 dx + c t ε 2 Ḣ1 ×L 2 + C 1 t γ • ε Ḣ1 ×L 2 .
(5.5)

Notice the cancellation of ´ε1 ψ 1 dx in (5.4) and (5.5). This is important because the bound on ψ 1 given by Proposition 3.14 and Proposition 4.7 is only 1 t ε , which is borderline but not sufficient to close the bootstrap. Moreover, Λ 0 -Λ = Id, so J eliminates the unbounded part of the operator Λ acting on ε 0 .

Proof of Lemma 5.3. We compute

d dt J(t) = b t ˆε1 • 1 λ • 1 2 (∆a) λ + (∇a) λ • ∇ ε 0 dx - bλ t λ ˆε1 • 1 λ • 1 2 (Λ 3/2 ∆a) λ + (Λ 5/2 ∇a) λ • ∇ ε 0 dx + b ˆε1t • 1 λ • 1 2 (∆a) λ + (∇a) λ • ∇ ε 0 dx + b ˆε1 • 1 λ • 1 2 (∆a) λ + (∇a) λ • ∇ ε 0t dx.
Consider the first two lines. From Lemma 5.1 and Hardy inequality it follows that

1 λ • 1 2 (∆a) λ + (∇a) λ • ∇ (5.6) and 1 λ • 1 2 (Λ 3/2 ∆a) λ -(Λ 5/2 ∇a) λ • ∇
are uniformly bounded as operators Ḣ1 → L 2 (the bound depends on R). Moreover, it is clear that |b t | + bλt λ t -1 . Hence, the first two lines are negligible.

Using again (5.1) we get

b ˆε1t • 1 λ • 1 2 (∆a) λ + (∇a) λ • ∇ ε 0 dx + b ˆε1 • 1 λ • 1 2 (∆a) λ + (∇a) λ • ∇ ε 0t dx = b ˆ(∆ε 0 + f (ϕ 0 + ε 0 ) -f (ϕ 0 )) • 1 λ • 1 2 (∆a) λ + (∇a) λ • ∇ ε 0 dx + b ˆε1 • 1 λ • 1 2 (∆a) λ + (∇a) λ • ∇ ε 1 dx -b ˆψ1 • 1 λ • 1 2 (∆a) λ + (∇a) λ • ∇ ε 0 dx -b ˆε1 • 1 λ • 1 2 (∆a) λ + (∇a) λ • ∇ ψ 0 dx.
(5.7) Proposition 3.14 and Proposition 4.7 imply that

ψ 1 L 2 1 t ε Ḣ1 ×L 2 + t γ .
Using once again uniform boundedness of the operator (5.6), we obtain that the first term of the last line is negligible. Consider now the second term. We will show that

b ˆε1 • 1 λ • 1 2 (∆a) λ + (∇a) λ • ∇ ψ 0 dx + ˆε1 ψ 1 dx ≤ c t ε 2 Ḣ1 ×L 2 + C 1 t γ • ε Ḣ1 ×L 2 . (5.8)
It follows from Proposition 3.14 and Proposition 4.7 that in (5.8) ψ 0 can be replaced by

-(λ t -b) 1 λ (ΛW ) λ and ψ 1 by (λ t -b) b λ (Λ 0 ΛW ) λ .
Hence, using (3.5), it suffices to prove that Λ 0 ΛW -1 2 ∆a + ∇a • ∇ ΛW L 2 is arbitrarily small when R is large enough. But this is clear, since 1 2 ∆a + ∇a • ∇ ΛW (r) = Λ 0 ΛW (r) for r ≤ R and | 1 2 ∆a + ∇a • ∇ ΛW (r)| r -3 for all r, with a constant independent of R.

The second line of (5.7) is 0 by integration by parts and we are left with the first line. The term with ∆ε 0 is computed via a classical Pohozaev identity:

ˆ 1 λ • 1 2 (∆a) λ + (∇a) λ • ∇ ε 0 ∆ε 0 dx = - 1 λ ∇ a,λ ε 0 2 L 2 + 1 4λ 3 ˆ(∆ 2 a) λ ε 2 0 dx.
(5.9) By Lemma 5.1, the last term is finite and ≤ 0. The nonlinear part is calculated in the following lemma.

Lemma 5.4. b ˆ 1 λ • 1 2 (∆a) λ + (∇a) λ • ∇ ε 0 • [f (ϕ 0 + ε 0 ) -f (ϕ 0 )] dx - λ t λ ˆ[f (ϕ 0 + ε 0 ) -f (ϕ 0 )]Λ 0 ε 0 dx ≤ c t ε 2 Ḣ1 ×L 2 .
We will admit for a moment that this is true and recapitulate in order to finish the proof of Lemma 5.3. Identity (5.9) implies that the term with ∆ε 0 in the first line of (5.7) is smaller thanb λ ∇ a,λ ε 0 2 L 2 . Lemma 5.4 implies that the difference between the other term of the first line of (5.7) and λt

λ ´[f (ϕ 0 + ε 0 ) -f (ϕ 0 )]Λ 0 ε 0 dx is negligible. The second line of (5.7) is 0, and the difference between the last line and ´ε1 ψ 1 dx is negligible, as follows from the computation above. This proves (5.5).

In order to prove Lemma 5.4, we need two auxiliary facts: Lemma 5.5.

|f (k + l) -f (k) -f (k)l - 1 2 f (k)l 2 | |f (l)|, |F (k + l) -F (k) -f (k)l - 1 2 f (k)l 2 | |F (l)| + |f (k)|l 3 .
Proof. For |l| ≤ 1 2 |k| this follows from the Taylor expansion and for |l| ≥ 1 2 |k| this is obvious by the triangle inequality.

Lemma 5.6. There exists a constant C 2 independent of R such that for small t,

|x| • |∇φ 0 | L 10/3 ≤ C 2 ,
(5.10)

|λ(∇a) λ | • |∇φ 0 | L 10/3 ≤ C 2 .
(5.11)

Moreover, x -λ(∇a) λ • ∇ϕ 0 L 10/3 ≤ c (5.12) if R
∇ χ x t λ 3/2 A λ (x) = ∇ χ x t A( x λ ) = 1 t (∇χ) x t A x λ + 1 λ χ x t ∇A x λ
x -λ(∇a) λ • |∇W λ | L 10/3 = (x -∇a) • |∇W | L 10/3 . By definition ∇a = x for |x| ≤ R, so (x -∇a) • |∇W | L 10/3 |x| • |∇W | L 10/3 (|x|≥R) → 0 when R → +∞. Smallness of | x -λ(∇a) λ | • |∇P 0 (t)| L 10/3 for small t follows from smallness of |x| • |∇P 0 (t)| L 10/3 .
Proof of Lemma 5.4. First, as for the linear terms, using integration by parts we transform the integral so that the unbounded operator Λ 0 (and its approximation 1 2 ∆a + ∇a • ∇) no longer acts on ε 0 :

ˆ1 λ x • ∇ε 0 f (ϕ 0 + ε 0 ) dx = ˆ1 λ x • ∇(ϕ 0 + ε 0 )f (ϕ 0 + ε 0 ) dx - ˆ1 λ x • ∇ϕ 0 f (ϕ 0 + ε 0 ) dx = -5 ˆ1 λ F (ϕ 0 + ε 0 ) dx - ˆ1 λ x • ∇ϕ 0 f (ϕ 0 + ε 0 ) dx (5.13)
and analogously

ˆ(∇a) λ • ∇ε 0 f (ϕ 0 + ε 0 ) dx = - ˆ1 λ (∆a) λ F (ϕ 0 + ε 0 ) dx -ˆ(∇a) λ • ∇ϕ 0 f (ϕ 0 + ε 0 ) dx.
Using Lemma 5.5 we see that

ˆ F (ϕ 0 + ε 0 ) -F (ϕ 0 ) + f (ϕ 0 )ε 0 + 1 2 f (ϕ 0 )ε 2 0 dx ε 3 Ḣ1 ×L 2 ≤ f ( ε Ḣ1 ×L 2 )
Similarly, from Lemma 5.5 and Lemma 5.6 we get

ˆ x • ∇ϕ 0 f (ϕ 0 + ε 0 ) -x • ∇ϕ 0 f (ϕ 0 ) + f (ϕ 0 )ε 0 + 1 2 f (ϕ 0 )ε 2 0 dx f ( ε Ḣ1 ×L 2 ). Notice that λt λ f ( ε Ḣ1 ×L 2 ) 1 t ε 2 Ḣ1 ×L 2 ,
so the above two inequalities together with (5.13) imply that

λ t λ ˆx • ∇ε 0 f (ϕ 0 + ε 0 ) dx -5 λ t λ ˆ F (ϕ 0 ) + f (ϕ 0 )ε 0 + 1 2 f (ϕ 0 )ε 2 0 dx - λ t λ ˆx • ∇ϕ 0 f (ϕ 0 ) + f (ϕ 0 )ε 0 + 1 2 f (ϕ 0 )ε 2 0 dx. (5.14)
Integrating by parts we find ˆx

• ∇ϕ 0 f (ϕ 0 ) dx = ˆx • ∇F (ϕ 0 ) dx = -5 ˆF (ϕ 0 ) dx and ˆx • ∇ϕ 0 f (ϕ 0 )ε 0 dx = ˆx • ∇f (ϕ 0 )ε 0 dx = -5 ˆf (ϕ 0 )ε 0 dx -ˆx • ∇ε 0 f (ϕ 0 ) dx.
Thus, (5.14) simplifies to

λ t λ ˆx • ∇ε 0 f (ϕ 0 + ε 0 ) -f (ϕ 0 ) dx λ t λ ˆ - 5 2 f (ϕ 0 )ε 2 0 - 1 2 x • ∇ϕ 0 f (ϕ 0 )ε 2 0 dx. (5.15)
Using just a pointwise estimate and Hölder we obtain

λ t λ ˆε0 [f (ϕ 0 + ε 0 ) -f (ϕ 0 )] dx λ t λ ˆf (ϕ 0 )ε 2 0 dx.
Combining with (5.15) we have

λ t λ ˆΛ0 ε 0 f (ϕ 0 + ε 0 ) -f (ϕ 0 ) dx - λ t 2λ ˆx • ∇ϕ 0 f (ϕ 0 )ε 2 0 dx - b 2λ ˆx • ∇ϕ 0 f (ϕ 0 )ε 2 0 dx, (5.16)
where the last almost-equality follows from the fact that

|λ t -b| ε Ḣ1 ×L 2 . Analogously, we obtain b ˆ 1 λ • 1 2 (∆a) λ + (∇a) λ • ∇ ε 0 f (ϕ 0 + ε 0 ) -f (ϕ 0 ) dx - b 2 ˆ(∇a) λ • ∇ϕ 0 f (ϕ 0 )ε 2 0 dx.
(5.17)

Comparing (5.16) and (5.17), we see that in order to finish the proof, we need to check that

ˆ x -λ(∇a) λ • ∇ϕ 0 f (ϕ 0 )ε 2 0 dx ≤ c ε 2 Ḣ1 ×L 2
when R is sufficiently large. Using Sobolev and Hölder inequalities this boils down to

(x -λ(∇a) λ ) • ∇ϕ 0 f (ϕ 0 ) L 5/2 ≤ c,
and this follows from (5.12) and boundedness of f (ϕ 0 ) in L 10 .

Proof of Proposition 5.2. From (5.4), (5.5) and the fact that Λ 0 -Λ = Id, we have

d dt H = d dt I + d dt J ≤ ˆ ∆ε 0 + f (ϕ 0 + ε 0 ) -f (ϕ 0 ) ψ 0 dx - b λ ∇ a,λ ε 0 2 L 2 + λ t λ ˆ f (ϕ 0 + ε 0 ) -f (ϕ 0 ) ε 0 dx + c t ε 2 Ḣ1 ×L 2 + C 1 t γ ε Ḣ1 ×L 2 .
Notice that

f (ϕ 0 + ε 0 ) -f (ϕ 0 ) Ḣ-1 ε 0 Ḣ1 .
This follows from the inequality |f

(k + l) -f (k)| |l| + |f (l)| and the fact that ϕ 0 is bounded in Ḣ1 . If we recall that |λt-b| λ ε Ḣ1 ×L 2 λ 1 t
, we see that in the second line we can replace λ t by b, hence to finish the proof we only have to prove that

ˆ ∆ε 0 + f (ϕ 0 + ε 0 ) -f (ϕ 0 ) ψ 0 dx ≤ c t ε 2 Ḣ1 ×L 2 + C 1 t γ ε Ḣ1 ×L 2 .
Inequalities (3.10) and (4.4) show that it is sufficient to check that

ˆ ∆ε 0 + f (ϕ 0 + ε 0 ) -f (ϕ 0 ) λ t -b λ (ΛW ) λ dx ≤ c t ε 2 Ḣ1 ×L 2 + C 1 t γ ε Ḣ1 ×L 2 ,
which in turn will follow from (3.5) and

ˆ ∆ε 0 + f (ϕ 0 + ε 0 ) -f (ϕ 0 ) (ΛW ) λ dx ≤ cλ t ε Ḣ1 ×L 2 + C 1 λt γ .
From pointwise bounds (for example the first inequality in Lemma 5.5) one deduces

f (ϕ 0 + ε 0 ) -f (ϕ 0 ) -f (ϕ 0 )ε 0 Ḣ-1 ε 2 Ḣ1 λ t ε Ḣ1 ×L 2 ,
hence it suffices to show that

ˆ ∆ε 0 + f (ϕ 0 )ε 0 )(ΛW ) λ dx = ˆε0 • [∆ + f (ϕ 0 )](ΛW ) λ dx ≤ cλ t ε Ḣ1 ×L 2 + C 1 λt γ .
Observe that [∆ + f (W λ )](ΛW ) λ = 0, so we are left with proving that ˆε0

• (f (ϕ 0 ) -f (W λ ))(ΛW ) λ dx ≤ cλ t ε Ḣ1 ×L 2 + C 1 λt γ .
By Hölder inequality it suffices to show that

f (ϕ 0 ) -f (W λ ) (ΛW ) λ L 10/7 ≤ cλ t .
(5.18)

The inequality |f

(k + l) -f (k)| |f (l)| + |f (k)| • |l| for k = W λ and l = u * (t) + P 0 (t) reduces (5.18) to checking that f (u * )(ΛW ) λ L 10/7 ≤ cλ t , (5.19
)

f (P 0 )(ΛW ) λ L 10/7 ≤ cλ t , (5.20)
u * • f ((ΛW ) λ ) L 10/7 ≤ cλ t , (5.21)
P 0 • f ((ΛW ) λ ) L 10/7 ≤ cλ t . ( 5

Construction of a uniformly controlled sequence and conclusion

In this section we will analyse finite dimensional phenomena of our dynamical systemmodulation equations and eigendirections of the linearized operator L. We will also define precisely the bootstrap assumptions and finish the proof of the main theorems. It is known that the operator L = -∆ -f (W ) has a unique simple strictly negative eigenvalue -e 2 0 (by convention e 0 > 0), with a unique positive eigenfunction Y such that Y L 2 = 1. This function Y is radial, smooth and decays exponentially. This follows from classical results of spectral theory and theory of elliptic equations, see [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF]Proposition 5.5], where it is also shown that there exists a constant c 1 > 0 such that

g ∈ Ḣ1 rad , g, Y = ∇g, ∇ΛW = 0 ⇒ g, Lg ≥ c 1 ∇g 2 L 2 . (6.1)
We need here a slight modification of this coercivity lemma.

Lemma 6.1. For any c > 0 there exist c L , C > 0 such that

g, Lg ≥ c L ∇g 2 -C g, Y 2 -c g, Z 2 . (6.2)
Proof. We first show that

g ∈ Ḣ1 rad , g, Y = g, Z = 0 ⇒ g, Lg ≥ c 2 ∇g 2 L 2 . (6.3)
To prove (6.3), decompose g = aΛW + h, h, ∆ΛW = 0. Notice that ΛW, Y = 0, thus h, Y = 0 and (6.1) implies

g, Lg = h + aΛW, Lh = h, Lh ≥ c 1 ∇h 2 L 2 .
Let ΛW be the orthogonal projection of ∆ΛW on Z ⊥ in Ḣ-1 . We have

∇h 2 L 2 = ∇g -a∇ΛW 2 L 2 = ∇g 2 L 2 -2a ∇g, ∇ΛW + a 2 ∇ΛW 2 L 2 = ∇g 2 L 2 + 2a g, ΛW + a 2 ∇ΛW 2 L 2 .
The functions ∆ΛW and Z are not perpendicular in Ḣ-1 , so ΛW Ḣ-1 < ∇ΛW L 2 , and (6.3) follows from Cauchy-Schwarz inequality.

In order to prove (6.2), we decompose

g = aY + bΛW + g, g, Y = g, Z = 0. (6.4)
Projecting (6.4) on Y and Z we have

a 2 g, Y 2 , b 2 g, Z 2 + a 2 Z, Y 2 g, Z 2 + g, Y 2 . (6.5) From (6.3) we obtain g, L f ≥ c 2 ∇ g 2 L 2 , thus g, Lg = aY + bΛW + g, -e 2 0 aY + L g = -e 2 0 a 2 + g, L g ≥ c 2 ∇ g 2 L 2 -e 2 0 a 2 .
From the inequality (xy) 2 ≥ 1 2 x 2 -y 2 we have

∇ g 2 L 2 ≥ 1 2 ∇g -b∇ΛW 2 L 2 -a 2 ∇Y 2 L 2 .
From the inequality (xy

) 2 ≥ c 1+c x 2 -cy 2 we have ∇g -b∇ΛW 2 L 2 ≥ c 1 + c ∇g 2 L 2 -cb 2 ∇ΛW 2 L 2 .
If we choose c small enough and put everything together using (6.5), we obtain (6.2).

From now on we will denote

α(g) := g, Y , α λ (g) := g, 1 λ Y λ .
We prove a version of the coercivity lemma with a localized gradient term.

Lemma 6.2. Let c > 0. If R is large enough, then there exists a constant C such that ˆ|x|≤R |∇g| 2 dx - ˆR5 f (W )g 2 dx ≥ -c ∇g 2 L 2 -C|α(g)| 2 .
In the proof we assume that g is radial, which is justified because later we use it for g = ε 0 . Notice however that the non-radial case follows by considering the radial rearrangement.

Proof. Define the projection Ψ R : Ḣ1 → Ḣ1 by the formula:

Ψ R g(r) = g(r) -g(R) if r ≤ R, 0 if r ≥ R.
By (6.2) applied to Ψ R g we have

1 + c 2 ˆ|x|≤R |∇g| 2 dx = 1 + c 2 ˆR5 |∇(Ψ R g)| 2 dx ≥ 1 + c 2 ˆf (W )|Ψ R g| 2 dx -C Ψ R g, Y 2 - c 4 Ψ R g, Z 2 .
Recall that, by the Strauss Lemma [START_REF] Strauss | Existence of solitary waves in higher dimensions[END_REF], in dimension N = 5 for a radial function g we have |g(R)| C 0 R -3 2 ∇g L 2 with a universal constant C 0 , so we have a pointwise estimate

|g| 2 ≤ C 0 1 + 2 c R -3 ∇g 2 L 2 + 1 + c 2 |Ψ R g| 2 .
Now we notice that ˆ|x|≤R f (W ) dx ∼ R, so for any δ > 0 the first term above gives a small contribution to the quadratic form for R large. Similarly,

| g -Ψ R g, Y | R -3 2 ∇g L 2 + ˆ|x|≥R |g|Y dx ≤ R -3 2 + Y L 10/7 (|x|≥R) ∇g L 2 , which is small when R is large. As Ψ R g, Y 2 ≤ 2 g, Y 2 + 2 g -Ψ R g, Y 2
, the proof is finished.

We are ready to state coercivity properties of the functional H from the previous section.

Proposition 6.3. Under the assumptions of Proposition 5.2, there exist T 0 , c H , α 0 , C 2 > 0 such that for t ∈ (0, T 0 ] ∩ (T -, T + ) there holds

|α λ (ε 0 )| ≤ α 0 ε 0 Ḣ1 ⇒ H(t) ≥ c H ε 2 Ḣ1 ×L 2 . If [T 1 , T 2 ] ⊂ (0, T 0 ] ∩ (T -, T + ) and |α λ (ε 0 )| ≤ 1 e 0 t γ+1 for all t ∈ [T 1 , T 2 ], then H(T 2 ) ≤ H(T 1 ) + c H 10 ˆT2 T 1 1 t ε 2 Ḣ1 ×L 2 dt + C 2 t 2γ+2 . (6.6)
The constants c H 10 and 1 e 0 have no special signification, but this formulation will be convenient later.

Proof. Let

I lin (t) := ˆ1 2 |ε 1 | 2 + 1 2 |∇ε 0 | 2 - 1 2 f (W λ )ε 2 0 dx.
Recall that ε 0 , Z = 0. Lemma 6.1 implies (after rescaling) that if we take α 0 small enough, then there exists a constant c > 0 such that I lin (t) ≥ c ε 2 Ḣ1 ×L 2 . We can assume that u * X 1 ×H 1 is as small as we like, so by pointwise estimates we get

|I(t) -I lin (t)| ≤ 1 3 c ε 0 2 Ḣ1 ×L 2 .
Moreover, it is clear from the definition of J that for small t we have

|J(t)| ≤ 1 3 c ε 2 Ḣ1 ×L 2 .
This proves the result with c H = 1 3 c. In order to prove (6.6), notice first that, by pointwise estimates and smallness of ϕ 0 -

W λ Ḣ1 , in (5.2) we can replace ´ f (ϕ 0 + ε 0 ) -f (ϕ 0 ) ε 0 dx by ´f (W λ )ε 2 0 dx. Convexity of a (see Lemma 5.1) implies that ∇ a,λ ε 2 L 2 ≥ ˆ|x|≤Rλ |∇ε 0 | 2 dx,
so from (5.2) and Lemma 6.2 (after rescaling) we obtain

H(T 2 ) ≤ H(T 1 ) + ˆT2 T 1 c H 20t ε 2 Ḣ1 ×L 2 + C 1 t γ • ε Ḣ1 ×L 2 + C|α(ε 0 )| 2 dt ≤ H(T 1 ) + c H 10 ˆT2 T 1 1 t ε 2 Ḣ1 ×L 2 dt + C 2 t 2γ+2 ,
where C 2 is a constant.

In order to close the bootstrap, it is necessary to control the stable and unstable directions. More precisely, it is necessary to eliminate the unstable mode. Define

α - λ (ε) := ˆYλ ε 1 - e 0 λ Y λ ε 0 dx and α + λ := ˆYλ ε 1 + e 0 λ Y λ ε 0 dx. Notice that - e 2 0
λ 2 is the unique strictly negative eigenvalue of L λ . We will define an auxiliary function l(t) which measures the distance of the modulation parameters from the approximate trajectory (2.7) or (2.8). This function has a slightly different form in the non-degenerate and degenerate cases. In the non-degenerate case we define

l(t) = 1 2 b t 3 + 2λ t 4 - κ 2 u * (0, 0) 2 24 2 + 1 2 b t 3 - 3λ t 4 - κ 2 u * (0, 0) 2 144 2 ,
and in the degenerate case

l(t) = 1 2 b t ν + ν λ t ν+1 -(ν + ν + 1) 2 + 1 2 b t ν -( ν + 1) λ t ν+1 -(ν -ν) 2 ,
where ν := -1 2 + 1 2 ν 2 + (ν + 1) 2 . We will write α + (t) and α -(t) instead of α + λ (ε) and α - λ (ε). In the next few propositions we describe the evolution of Mod(t) := (λ(t), b(t), α -(t), α + (t)) in the "modulation cylinder" defined as:

C (t) := (λ, b, α -, α + ) : l(t) ≤ t γ+1-ν and -t γ+1 ≤ α -, α + ≤ t γ+1 .
In the non-degenerate case we denote

λ app (t) := κ 2 u * (0, 0) 2 144 t 4 , b app (t) := κ 2 u * (0, 0) 2 36 t 3 ,
and in the degenerate case

λ app (t) := t ν+1 , b app (t) := (ν + 1)t ν .
Solving a 2 × 2 linear system we check easily that

l(t) ≤ t γ+1-ν ⇒ λ λ app -1 t 1 2 (γ+1-ν) , b b app -1 t 1 2 (γ+1-ν) , (6.7)
with constants which depend only on ν.

We have

α λ (ε 0 ) = 1 2e 0 (α + -α -), so Mod(t) ∈ C (t) ⇒ |α λ (ε 0 )| ≤ 1 e 0 t γ+1 . (6.8)
Remark 6.4. The formula for l is found by linearizing the parameter equations near (λ app , b app ) and diagonalizing the resulting system.

We can finally state a result on uniform in time energy bounds. Proposition 6.5. Let C 4 > 0 be a fixed constant. There exist C 0 > 0 and T 0 > 0 having the following property. Let 0 < T 1 < T 0 . Suppose that

ε(T 1 ) Ḣ1 ×L 2 ≤ C 4 T γ+1 1 , (6.9)
Mod(T 1 ) ∈ Int(C (T 1 )).

Then, either there exists a time t, T 1 ≤ t ≤ T 0 , such that Mod(t) ∈ ∂C (t), or the solution exists on [T 1 , T 0 ] and for all t ∈ [T 1 , T 0 ] there holds

ε Ḣ1 ×L 2 ≤ C 0 t γ+1 . (6.10)
Proof. Let T 0 be the time provided by Proposition 6.3. Let T + be the maximal time of existence of the solution and let T 2 := min(T 0 , T + ). Suppose that Mod(t) / ∈ ∂C (t) for T 1 ≤ t ≤ T 2 . By continuity of Mod(t) this means that Mod(t) ∈ Int(C (t)) for T 1 ≤ t ≤ T 2 . We will show first that if C 0 is large enough, then (6.10) holds for t ∈ [T 1 , T 2 ]. Argue by contradiction, assuming that there exists

T 3 < T 2 such that ε(T 3 ) Ḣ1 ×L 2 = C 0 T γ+1 3 . At t = T 3 (6.8) gives |α λ (ε 0 )| ≤ 1
e 0 t γ+1 . In particular, if C 0 is large, we will have |α λ (ε 0 )| ≤ α 0 ε 0 Ḣ1 , so by Proposition 6.3 we obtain

H(T 3 ) ≥ c H C 2 0 T 2γ+2 3 . (6.11) On the other hand, for t ∈ [T 1 , T 3 ] we have ε 2 Ḣ1 ×L 2 ≤ C 2 0 t 2γ+2 and |α λ (ε 0 )| ≤ 1 e 0 t γ+1
, so from (6.6) we deduce that

H(T 3 ) ≤ H(T 1 ) + c H C 2 0 10(2γ + 2) T 2γ+2 3 + C 2 T 2γ+2 3 . Notice that H(T 1 ) ≤ ε(T 1 ) 2 Ḣ1 ×L 2 ≤ C 2 4 T 2γ+2 1 ≤ C 2 4 T 2γ+2 3
. Returning to (6.11) we deduce

c H C 2 0 ≤ C 2 4 + c H C 2 0 10(2γ + 2) + C 2 , which is impossible if C 0 is large enough. Hence, T 3 = T 2 .
To prove that T 2 = T 0 , notice that by the Cauchy theory in the critical space there exists δ > 0 such that

(u 0 -W, u 1 ) Ḣ1 ×L 2 ≤ δ ⇒ the solution u(t) with u(0) = (u 0 , u 1 )
exists at least for t ∈ (-1, 1).

After rescaling we obtain

(u 0 -W λ , u 1 ) Ḣ1 ×L 2 ≤ δ ⇒ the solution u(t) with u(0) = (u 0 , u 1 )
exists at least for t ∈ (-λ, λ). (6.12)

If u * Ḣ1 ×L 2 is sufficiently small and T 0 is chosen sufficiently small, (3.2) and (3.4) show that our solution verifies the sufficient condition in (6.12) for any t < T 2 with λ = λ(t). Taking t close to T 2 we obtain that the solution cannot blow up at T 2 , hence T 2 = T 0 .

The crucial element of the preceding result is that the constant C 0 is independent of T 1 . From now, C 0 has a fixed value given by Proposition 6.5, and the constants which appear later are allowed to depend on C 0 . In particular, when we use the notation or O, the constant may depend on C 0 .

We examine now the evolution of the eigenvectors α -and α + . Lemma 6.6. If ε Ḣ1 ×L 2 t γ+1 , λ ∼ t ν+1 and b ∼ t ν , then

d dt α + λ - e 0 λ α + λ t γ , (6.13)
d dt α - λ + e 0 λ α - λ t γ . (6.14)
Proof. We will do the computation for (6.13), because the one for (6.14) is exactly the same.

d dt α + λ = ˆ Y λ • (-L λ ε 0 ) + e 0 λ Y λ ε 1 dx + ˆ-λ t λ (Λ 0 Y) λ ε 1 + e 0 λ (Λ -1 Y) λ ε 0 dx + ˆYλ • f (ϕ 0 + ε 0 ) -f (ϕ 0 ) -f (ϕ 0 )ε 0 dx + ˆYλ • f (ϕ 0 ) -f (W λ ) ε 0 dx + ˆ Y λ • (-ψ 1 ) + e 0 λ Y λ • (-ψ 0 ) dx.
The first line is e 0 λ α + λ and it suffices to estimate the remaining ones. For the last line we use Proposition 3.14 and L 2 -orthogonality of ΛW and Y. Using λ t ∼ t ν , λ ∼ t ν+1 and ε Ḣ1 ×L 2 ≤ Ct γ+1 the second line is seen to be bounded by Ct γ . The proof of (5.18) shows that

Y λ • f (ϕ 0 ) -f (W λ ) L 10/7 ≤ c
t , so using ε 0 L 10/3 t γ+1 we obtain the required bound for the fourth line. Finally,

f (ϕ 0 + ε 0 ) -f (ϕ 0 ) -f (ϕ 0 )ε 0 Ḣ-1 C 2 t 2γ+2 and Y λ Ḣ1 1 λ ∼ t -ν-1
, so by Cauchy-Schwarz the third line is bounded by C 2 t 2γ-ν+1 t γ .

We know from Proposition 6.5 that if we start at t = T 1 with ε small enough, then ε is controlled in Ḣ1 × L 2 unless Mod leaves the cylinder C . It turns out that it can happen only because of α + . The other parameters are trapped in the cylinder for small times: Lemma 6.7. Under the assumptions of Propositon 6.5, suppose that Mod(t

) leaves Int(C (t)) before t = T 0 . If T 2 ≤ T 0 is the first time for which Mod(T 2 ) ∈ ∂C (T 2 ), then |α + (T 2 )| = T γ+1 2 .
In addition, suppose that at time

T 3 , T 1 ≤ T 3 < T 2 , we have α + (T 3 ) > 1 2 T γ+1 3 . Then α + (T 2 ) = T γ+1 2 . Analogously, if α + (T 3 ) < -1 2 T γ+1 3 , then α + (T 2 ) = -T γ+1 2 .
Proof. Suppose, for the sake of contradiction, that for example l(T 2 ) = T γ+1-ν

2

. In particular this implies d dt l(t 1 ) ≥ 0, and we will show that it is impossible. We start with the degenerate case. Using (6.7) and

√ x = 1+x 2 + O(|1 -x| 2 ) we obtain √ λ = 1 2 t 1 2 (ν+1) + λt -1 2 (ν+1) + O(t γ+ 3 2 -1 2 ν ).
Recall that b t = (ν + 1)νt

1 2 (ν-3) √ λ, so we get b t t ν-1 = (ν + 1)ν 2 1 + λ t ν+1 + O(t γ+1-ν ). (6.15)
From Lemma 3.7 and (6.10) we have

λ t t ν = b t ν + O(t γ+1-ν ). ( 6
d dt l(t) = b t ν + ν λ t ν+1 -(ν + ν + 1) b t t ν + ν λ t t ν+1 -ν b t ν+1 -ν(ν + 1) λ t ν+2 + b t ν -( ν + 1) λ t 1+ν -(ν -ν) b t t ν -( ν + 1)
λ t t ν+1 -ν b t ν+1 -( ν + 1)(ν + 1) λ t ν+2 = 1 t b t ν + ν λ t ν+1 -(ν + ν + 1) (ν+1)ν 2 1 + λ t ν+1 + ν b t ν -ν b t ν -ν(ν + 1) λ t ν+1 + b t ν -( ν + 1) λ t 1+ν -(ν -ν) (ν+1)ν 2 1 + λ t ν+1 -( ν + 1) b t ν -ν b t ν -( ν + 1)(ν + 1) λ t ν+1 + 1 t O( l(t)t γ+1-ν ).
If we use the definition of ν, this simplifies to

d dt l(t) = -(ν -ν) b t ν + ν λ t ν+1 -(ν + ν + 1) 2 -(ν + ν + 1) b t ν -( ν + 1) λ t 1+ν -(ν -ν) 2 + 1 t O( l(t)t γ+1-ν ) = 1 t -(ν -ν)l(t) + O(t 3 2 (γ+1-ν) ) . At time t = T 2 by assumption l(T 2 ) = T γ+1-ν 2
, so for T 2 small enough the formula above implies d dt l(T 2 ) < 0, which is impossible. In the non-degenerate case the computation is similar, but we must take into account that in this case

b t = κu * (t, 0) √ λ = κu * (0, 0) √ λ(1 + O(t)),
which leads to

b t = κu * (0, 0) • 1 2 κu * (0, 0) 12 t 2 + 12 κu * (0, 0) λt -2 + O(t 3 ).
Then, the computation is the same as before:

d dt l(t) = b t 3 + 2λ t 4 - κ 2 u * (0, 0) 2 24 b t t 3 + 2λ t t 4 - 3b t 4 - 8λ t 5 + b t 3 + 2λ t 4 - κ 2 u * (0, 0) 2 24 b t t 3 + 2λ t t 4 - 3b t 4 - 8λ t 5 ≤ 1 t b t ν + 2λ t 4 - κ 2 u * (0, 0) 2 24 κ 2 u * (0, 0) 2 24 + 6λ t 4 + 2b t 3 - 3b t 3 - 8λ t 4 + 1 t b t 3 - 3λ t 4 - κ 2 u * (0, 0) 2 144 κ 2 u * (0, 0) 2 24 + 6λ t 4 - 3b t 3 - 3b t 3 + 12λ t 4 + O( l(t)) ≤ 1 t -2l(t) + O(t 7/4 ) . Since γ + 1 -ν = 3 2 < 7 4 ,
(T 3 ) > 1 2 T γ+1 3 and α + λ (T 2 ) = -T γ+1 2
. By continuity, there exists the smallest

T 4 > T 3 such that α + λ (T 4 ) = 1 2 T γ+1 4 . Necessarily d dt α + λ (T 4 ) ≤ γ+1 2 T γ 4
, which is in contradiction with (6.13).

Proposition 6.8. There exist strictly positive constants C 0 and T 0 such that for all T 1 ∈ (0, T 0 ) there exists a solution u defined on [T 1 , T 0 ] which for all t ∈ [T 1 , T 0 ] verifies

(u -W λ -u * , ∂ t u + λ t (ΛW ) λ -∂ t u * ) Ḣ1 ×L 2 ≤ C 0 t γ+1 , (6.17) λ λ app -1 ≤ C 0 t 1 2 (γ+1-ν) . (6.18)
Proof. We consider the degenerate case. The proof in the non-degenerate case is similar.

Let λ = λ app (T 1 ), b = b app (T 1 ). For a ∈ [-2 3 T γ+1 1 , 2 3 T γ+1 1 ], let ε a (T 1 ) = a 2ν (Y + λ - Y,Z
Z,Z (Z λ , 0)), and consider the corresponding evolution. Of course (6.9) is verified for a universal constant C 4 . Let C 0 be the constant provided by Proposition 6.5. We will show that there exists a parameter a for which the solution exists until t = T 0 and satisfies (6.10). Suppose this is not the case. Let

A + = {a : α + (T 2 ) = T γ+1 2 } and A -= {a : α + (T 2 ) = -T γ+1 2 }
, where T 2 is the exit time given by Lemma 6.7. By the second part of Lemma 6.7 we know that -

2 3 T γ+1 1 ∈ A -, 2 3 T γ+1 1 ∈ A +
, and that A -and A + are open sets. Indeed, let a ∈ A + . This means in particular that for

T 1 ≤ t ≤ T 2 we have α + (t) ≥ -1 2 t γ+1 and α + (T 2 ) = T γ+1 2
. By continuity of the flow, for close enough initial data we will still have α

+ (t) > -t γ+1 for T 1 ≤ t ≤ T 2 and α + (T 2 ) ≥ 1 2 T γ+1 2
. By Lemma 6.7 the corresponding solutions escape from the cylinder by positive values of α + . Thus A + ∪A -would be a partition of [-

2 3 T γ+1 1 , 2 3 T γ+1 1 ] into two disjoint open sets, which is impossible.
Using (6.10), (4.2), (4.3) and (3.5) we obtain (6.17). Estimate (6.18) follows from (6.7) and the fact that Mod(t) ∈ C (t).

Proof of Theorem 1. Let t n be a decreasing sequence such that t n > 0 and t n → 0. Let u n be the solution given by Proposition 6.8 for T 1 = t n and let λ n : [t n , T 0 ] → (0, +∞) be the corresponding modulation parameter. The sequence u n (T 0 ) is bounded in Ḣ1 × L 2 . After extracting a subsequence, it converges weakly to some function (u 0 , u 1 ). Let u(t) be the solution of (NLW) for the Cauchy data u(T 0 ) = (u 0 , u 1 ). We will show that u satisfies (1.1). Let 0 < T 1 < T 0 and T 1 ≤ t ≤ T 0 . Using (6.17), (6.18) and |λ t | t 3 we get

(u n -W λapp -u * , ∂ t u n -∂ t u * ) Ḣ1 ×L 2 ≤ C 0 t 3 4 .
This shows that if T 0 is sufficiently small, then the sequence u n satisfies the conditions of Proposition A.1 on the time interval [T 1 , T 0 ], hence

u n (T 1 ) u(T 1 ).
Weak lower semi-continuity of the norm implies that at time t = T 1 we have

(u -W λapp -u * , ∂ t u -∂ t u * ) Ḣ1 ×L 2 ≤ C 0 T 3/4 1 .
This bound holds for all T 1 such that 0 < T 1 < T 0 . In particular, the orthogonality condition:

u -W λ -u * , Z λ = 0. (6.19)
defines uniquely a continuous function λ(T 1 ) : (0, T 0 ) → (0, +∞). We will prove that λ n (T 1 ) → λ(T 1 ). Using (3.7) for the solution u n at time T 1 and passing to a limit n → ∞ we obtain that all the accumulation points of λ n (T 1 ) verify the orthogonality condition (6.19). Hence λ n (T 1 ) → λ(T 1 ). Passing to a limit in (3.8) we get d dt λ n (T 1 ) → d dt λ(T 1 ). Passing to a limit in (6.17) and (6.18) finishes the proof.

The proof of Theorem 2 follows the same lines, so we will skip it.

A Weak continuity of the flow near a fixed path

Proposition A.1. Let v : [0, 1] → Ḣ1 × L 2 be a continuous path in the energy space. There exist a constant δ > 0 with the following property. Let u n be a sequence of radial solutions of (NLW) defined on the interval [0, 1], such that

sup t∈[0,1] u n -v Ḣ1 ×L 2 ≤ δ. (A.1)
Suppose that u n (0) (u 0 , u 1 ) in Ḣ1 × L 2 and let u be the solution of (NLW) for the initial data u(0) = (u 0 , u 1 ). Then u is defined on [0, 1] and for all t ∈ [0, 1] we have

u n (t) u(t) in Ḣ1 × L 2 . (A.2)
Remark A.2. Notice that without the assumption (A.1) the result is false. More generally, existence of type II blow-up solutions in some space excludes weak continuity of the flow in this space, and existence of type II blowup solutions in our case follows from Theorems 1 and 2. One might search weaker conditions than (A.1); we have chosen a simple condition which is sufficient for our needs.

Proof.

Step 1. Suppose that u is not defined on [0, 1] and let T + ≤ 1 be its final time of existence.

In Step 2. we will prove (A.2) for all t < T + . In particular, by the lower weak semi-continuity of the norm, this shows that sup

t∈[0,T + ) u n -v Ḣ1 ×L 2 ≤ δ.
By local well-posedness in the energy space and compactness of {v(t) : t ∈ [0, 1]}, if δ > 0 is small enough, there exists τ > 0 such that the solution corresponding to the initial data u n (t) is defined at least on the interval (-τ, τ ). This means that u cannot blow up at T + , and so it is defined for t ∈ [0, 1]. If δ is chosen small enough, depending on v(1), then by the Cauchy theory the solutions u n exist on an interval (1 -t , 1 + t ) for some t > 0. By eventually choosing t smaller, we can assume that u also exists on (1 -t , 1 + t ). Hence, by repeating the same procedure we obtain weak convergence also for t = 1.

Step 2. Let t < T + . In order to prove (A.2), it is sufficient to show that any subsequence of u n (which we will still denote u n ) admits a subsequence such that the required convergence takes place. By the result of Bahouri-Gérard a subsequence of u n (0) admits a profile decomposition such that the first profile is U lin 1 (t) = S(t)(u 0 , u 1 ) (corresponding to parameters t 1,n = 0, λ 1,n = 1). By the triangle inequality u n -(u 0 , u 1 ) Ḣ1 ×L 2 ≤ 2δ, so all the other profiles are small, in particular they are global and disperse. By definition of T + the assumptions of Proposition 2.8 in [START_REF] Duyckaerts | Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation[END_REF] (which is a version of [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF]Main Theorem] for the focusing equation) are satisfied for θ n = t, in particular formula (2.22) from [START_REF] Duyckaerts | Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation[END_REF] yields:

u n (t) = u(t) + J j=2 U j n (t) + w J n (t) + r J n (t).
Here, w J n (t) = S(t)w J n (0) 0 as n → +∞ (indeed, w J n (0) 0 for J > 1 by definition of the profiles, and S(t) is a bounded linear operator). By Lemma A.3 below also U j n (t) 0 when j > 1, which finishes the proof.

Lemma A.3. Let U be a solution of equation (NLW) such that U Ḣ1 ×L 2 is small. Let t n , λ n be a sequence of parameters such that one of the following holds:

1. t n = 0 and λ n → 0, 2. t n = 0 and λ n → +∞, 3. t n /λ n → +∞, 4. t n /λ n → -∞.
Fix t ∈ R and define

U n (x) = 1 λ 3/2 n U j ( t -t n λ n , x λ n ), 1 λ 5/2 n ∂ t U j ( t -t n λ n ) . Then U n 0 in Ḣ1 × L 2 .
Proof. Again it is sufficient to show this for a subsequence of any subsequence. Thus we can assume that t-tn λn → t 0 ∈ [-∞, +∞]. Suppose first that t 0 is a finite number. Extracting again a subsequence we can assume that λ n → λ 0 ∈ [0, +∞]. If λ 0 was a strictly positive finite number, we would obtain that also t n has a finite limit, which is impossible. Thus λ n → 0 or λ n → +∞, and in both cases we get our conclusion.

In the case t-tn λn → ±∞ we have dispersion, so U n -(S(τ n )V ) λn Ḣ1 ×L 2 → 0, and it is well known that (S(τ n )V ) λn 0 when τ n → ±∞ and λ n is any sequence (in the case of space dimension N = 5 this follows for example from the strong Huyghens principle).

B Local theory in higher regularity

In this section we use the energy method to prove two results about preservation of regularity.

B.1 Energy estimates in

X 1 × H 1
Recall that we denote X s := Ḣs+1 ∩ Ḣ1 . We have classical energy estimates for the linear wave equation:

Lemma B.1. Let s ∈ N. Let I = [0, T 0 ] be a time interval, g ∈ C(I, H s ) and (u 0 , u 1 ) ∈ X s × H s . Then the Cauchy problem ∂ tt u -∆u = g, (u(0), ∂ t u(0)) = (u 0 , u 1 )
has a unique solution (u, ∂ t u) ∈ C(I, X s × H s ) and for all t ∈ I there holds

(u, ∂ t u) X s ×H s ≤ (u 0 , u 1 ) X s ×H s + ˆt 0 g(τ ) H s dτ. (B.1)
For a proof of a more general result one can consult for example [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF]Theorem 4.4]. Using finite speed of propagation and Sobolev Extension Theorem on each time slice we get a localised version of the energy estimate:

(u, ∂ t u) X s ×H s (B(0,ρ)) (u 0 , u 1 ) X s ×H s (B(0,ρ+t)) + ˆt 0 g(τ ) H s (B(0,ρ+τ ) dτ (B.2)
Now we use the case s = 1 to prove energy estimates in X 1 × H 1 for (NLW).

Proposition B.2. For all M 0 > 0 there exists T 0 = T 0 (M 0 ) > 0 such that the following is true. Let (u 0 , u 1 ) ∈ X 1 × H 1 with (u 0 , u 1 ) X 1 ×H 1 ≤ M 0 . Then the Cauchy problem:

∂ tt u -∆u = f (u), (u(0), ∂ t u(0)) = (u 0 , u 1 )
has a unique solution (u, ∂ t u) ∈ C([0, T 0 ], X 1 × H 1 ) and this solution verifies

sup t∈[0,T 0 ] (u(t), ∂ t u(t)) X 1 ×H 1 ≤ 2 (u 0 , u 1 ) X 1 ×H 1 .
Moreover, let u lin denote the solution of the free wave equation for the same initial data (u 0 , u 1 ). Then

sup t∈[0,T 0 ] (u(t), ∂ t u(t)) -(u lin (t), ∂ t u lin L(t)) X 1 ×H 1 f (u 0 , u 1 ) X 1 ×H 1 . (B.3)
This will follow easily from the following lemma.

Lemma B.3. Let u, v ∈ X 1 . Then f (u) H 1 ≤ Cf ( u X 1 ), (B.4) f (u) -f (v) H 1 ≤ C u -v X 1 • f ( u X 1 ) + f ( v X 1 ) . (B.5) Proof. We have f (u) L 2 = u 7/3 L 14/3
f ( u X 1 ) from the Sobolev imbedding. By Hölder inequality,

∇f (u) Ḣ1 = ∇u • f (u) L 2 ∇u L 10/3 • f (u) L 5 u X 1 • u 4/3 L 20/3 f ( u X 1 ),
again by Sobolev imbedding. This proves (B.4).

To prove (B.5), we write

|f (u) -f (v)| |u -v|(f (u) + f (v)), hence f (u) -f (v) L 2 u -v L 14/3 • f (u) + f (v) L 7/2 u -v L 14/3 • u 4/3 L 14/3 + v 4/3 L 14/3 u -v X 1 • f ( u X 1 ) + f ( v X 1 ) .
Finally,

|∇f (u) -∇f (v)| |∇u -∇v|(f (u) + f (v)) + |u -v|(|∇u| + |∇v|)(|f (u)| + |f (v)|),
and it suffices to notice that

|∇u -∇v|(f (u) + f (v)) L 2 ∇u -∇v L 10/3 • f (u) + f (v) L 5 u -v X 1 • f ( u X 1 ) + f ( v X 1 ) and |u -v|(|∇u| + |∇v|)(|f (u)| + |f (v)|) L 2 u -v L 10 • ( ∇u L 10/3 + ∇v L 10/3 ) • f (u) L 10 + f (v) L 10 u -v X 1 • f ( u X 1 ) + f ( v X 1 ) .
Proof of Proposition B.2. Let B denote the ball of centre 0 and radius 2 (u 0 , u 1 )

X 1 ×H 1 in the space X 1 × H 1 . Given (u, ∂ t u) ∈ C([0, T ], B), let u = Φ(u) denote the solution of the Cauchy problem ∂ tt u -∆ u = f (u), ( u(0), ∂ t u(0)) = (u 0 , u 1 )
It follows from Lemma (B.4) and (B.1) that if T ≤ M 0 Cf (2M 0 ) , then ( u, ∂ t u) ∈ C([0, T ], B). It follows from (B.5) and (B.1) that if T ≤ 1 4Cf (2M 0 ) , then Φ is a contraction, so it has a unique fixed point, which is the desired solution.

The function v := uu lin solves the Cauchy problem

∂ tt v -∆v = f (u), (v(0), ∂ t v(0)) = 0, so (B.3) follows from (B.1). B.2 Persistence of X 1 × H 1 regularity
We recall the classical Strichartz inequality:

Lemma B.4. [33] Let I = [0, T 0 ] be a time interval, g ∈ C(I, L 2 ) and (u 0 , u 1 ) ∈ Ḣ1 × L 2 .
Let u be the solution of the Cauchy problem

∂ tt u -∆u = g, (u(0), ∂ t u(0)) = (u 0 , u 1 ). Then u L 7/3 (I;L 14/3 ) (u 0 , u 1 ) Ḣ1 ×L 2 + g L 1 (I;L 2 ) ,
with a constant independent of I.

From the local theory of (NLW) in the critical space we know that if u ∈ C((T -, T + ); Ḣ1 × L 2 ) is a solution of (NLW) and

I = [T 1 , T 2 ] ⊂ (T -, T + ), then u L 7/3 (I;L 14/3 ) < +∞. (B.6) Proposition B.5. Suppose that 0 ∈ I = [T 1 , T 2 ] ⊂ (T -, T + ) and that (u 0 , u 1 ) ∈ X 1 × H 1 . Then u ∈ C(I, X 1 × H 1 ).
Proof. The proof is classical, see for example [9, Chapter 5] for more general results in the case of NLS.

We consider positive times. The proof for negative times is the same. Let T * be the maximal time of existence of u in X 1 × H 1 . Suppose that T * < T + . From Proposition B.2 it follows that lim

* -τ ), ∂ t u(T * -τ )) X 1 ×H 1 + C ∇(f (u)) L 1 (I;L 2 ) , (B.8)
with C independent of τ . From Hölder inequality we have

∇(f (u)) L 1 (I;L 2 ) ≤ ∇u L 7/3 (I;L 14/3 ) • f u L 7/3 (I;L 14/3 ) .
By (B.6), the last term is arbitrarily small when τ → 0 + , so for τ small enough the second term on the right hand side of (B.8) can be absorbed by the left hand side, which implies ∇u L 7/3 (I;L 14/3 ) < +∞ and ∇(f (u)) L 1 (I;L 2 ) < +∞. This is in contradiction with (B.7), because of the energy estimate (B.1).

B.3 Propagation of regularity around a non-degenerate point

Proposition B.6. Let (u 0 , u 1 ) ∈ X 4 × H 4 such that u 0 (0) > 0. Let (u, ∂ t u) ∈ C([0, T 0 ]; X 1 × H 1
) be the solution of the Cauchy problem:

∂ tt u -∆u = f (u), (u(0), ∂ t u(0)) = (u 0 , u 1 ),
constructed in Proposition B.2. There exists τ, ρ > 0 such that (u, ∂ t u) satisfies:

χ • ρ u, χ • ρ ∂ t u ∈ C([0, τ ]; X 4 × H 4 )
(where χ is a standard regular cut-off function).

Proof. Denote v 0 := u 0 (0) > 0 and introduce an auxiliary function

f ∈ C ∞ , f (u) = f (u) when u ≥ v 0 /2, f (u)
= 0 when u ≤ 0. Using Faà di Bruno formula one can prove an analog of Lemma B.1:

f (u) H 4 ≤ C( u X 4 ), f (u) -f (v) H 4 ≤ u -v X 4 • C( u X 4 + v X 4 ),
where C : R + → R + is a continuous function. The same procedure as in the proof of Proposition B.2 leads to the conlusion that there exists τ > 0 such that the Cauchy problem:

∂ tt u -∆ u = f ( u), ( u(0), ∂ t u(0)) = (u 0 , u 1 ) has a solution ( u, ∂ t u) ∈ C([0, τ ], X 4 × H 4 )
. By continuity and Schauder estimates, if we take τ and ρ sufficiently small, we have u(t, x) > 1 2 v 0 for |x| ≤ 4ρ and 0 ≤ t ≤ τ . We may assume that τ ≤ 2ρ. Consider v = uu. We will prove that v = 0 when 0 ≤ t ≤ τ and |x| ≤ 2ρ, which will finish the proof. The function v solves the Cauchy problem:

∂ tt v -∆v = f (u) -f ( u), (v(0), ∂ t v(0)) = 0.
We run the localized energy estimate (B.2 Let (u, ∂ t u) denote the solution of (NLW) corresponding to the intial data

) for |x| ≤ 2ρ + |t -τ |. We suppose that τ ≤ 2ρ, so |x| ≤ 4ρ, which means that f (u) -f ( u) H 1 = f (u) -f ( u) H 1 u -u X 1 (the norm is taken in the ball B(0, 2ρ + |t -τ |). From (B.
(u 0 , u 1 ) = χ • ρ p|x| β , 0 ,
where p, ρ > 0 and β > 5 2 are constants and χ is a standard cut-off function. Let (u lin , ∂ t u lin ) denote the solution of the free wave equation corresponding to the same initial data.

Proposition B.7. Let q = (β+1)(β+3) 3 p. There exist T 0 > 0 and a constant C > 0 such that for 0 ≤ t ≤ T 0 and |x| ≤ 1 2 t there holds

|u lin (t, x) -qt β | ≤ Ct β-2 |x| 2 .
Proof. Define w(y) := ∂B(0,1)

p|ω + ye 1 | β dσ(ω), - 1 √ 2 < y < 1 √ 2 ,
where B(0, 1) denote the unit ball in R 5 , dσ is the surface measure on the unit sphere and e 1 = (1, 0, 0, 0, 0). Notice that

|ω + ye 1 | β = (1 -ω 2 1 + (y + ω 1 ) 2 ) β/2 = (1 + ω 2 1 ) β/2 • 1 + y 2ω 1 1 + ω 2 1 β/2
can be developped in a power series in y which converges uniformly for -1

√ 2 < y < 1 √ 2 .
Hence, w is an analytic function. It is also symetric, so it is in fact analytic in y 2 , w(y) = w(y 2 ), w(z) analytic for |z| < 1 2 .

We have w(0) = w(0) = p.

The representation formula for solutions of the free wave equation, see for example [29, p. 77], yields

u lin (t, x) = 1 3 ∂ ∂t 1 t ∂ ∂t t 3 ∂B(x,t)
p|y| β dσ(y) .

A change of variables shows that for |x| < 1 √ 2 t and t sufficiently small we have

u lin (t, x) = 1 3 ∂ ∂t 1 t ∂ ∂t t 3 • t β w |x| 2 t 2 = t β w 1 |x| 2 t 2 ,
where w 1 (z) is analytic for |z| < 1 2 . It is easily seen that w 1 (0) = (β+1)(β+3) 3 p = q (all the terms coming from differentiating w vanish at z = 0). Hence, there exists a constant C such that | w 1 (z) -q| ≤ C|z| for |z| ≤ 1 4 , and the conclusion follows.

Proposition B.8. For t small enough there holds

u -u lin X 1 (|x|≤ 1 2 t) t 7 3 β+ 7 6 .
Proof. From (B.3) and finite speed of propagation we obtain

u -u lin X 1 (|x|≤ 1 2 t) f ( (u 0 , u 1 ) ) X 1 ×H 1 (|x|≤ 3 2 t) . We have (u 0 , u 1 ) 2 X 1 ×H 1 (|x|≤ 3 2 t) ∼ ˆ3 2 t 0 (r β-2 ) 2 r 4 dr ∼ t 2β+1 ,
and the conclusion follows.

Chapter 2

Construction of two-bubble solutions for energy-critical wave equations

Abstract

We construct pure two-bubbles for some energy-critical wave equations, that is solutions which in one time direction approach a superposition of two stationary states both centered at the origin, but asymptotically decoupled in scale. Our solution exists globally, with one bubble at a fixed scale and the other concentrating in infinite time, with an error tending to 0 in the energy space. We treat the cases of the power nonlinearity in space dimension 6, the radial Yang-Mills equation and the equivariant wave map equation with equivariance class k ≥ 3. The concentrating speed of the second bubble is exponential for the first two models and a power function in the last case.

Introduction 1.Energy critical NLW

We consider the energy critical wave equation in space dimension N = 6:

(∂ 2 t -∆)u(t, x) = |u(t, x)| • u(t, x), (t, x) ∈ R × R 6 ,
(u(t 0 , x), ∂ t u(t 0 , x)) = (u 0 (x), u0 (x)).

(1.1)

The energy functional associated with this equation is defined for

u 0 = (u 0 , u0 ) ∈ E := Ḣ1 (R 6 ) × L 2 (R 6
) by the formula

E(u 0 ) := ˆ1 2 | u0 | 2 + 1 2 |∇u 0 | 2 -F (u 0 ) dx,
where

F (u 0 ) := 1 3 |u 0 | 3 . Note that E(u 0 ) is well-defined due to the Sobolev Embedding Theorem. The differential of E is DE(u 0 ) = (-∆u 0 -f (u 0 ), u0 ), where f (u 0 ) = |u 0 | • u 0 , hence we can rewrite equation (1.1) as ∂ t u(t) = J • DE(u(t)), u(t 0 ) = u 0 ∈ E. (1.2)
Here, J := 0 Id -Id 0 is the natural symplectic structure.

Equation (1.1) is locally well-posed in the space E, see for example Ginibre, Soffer and Velo [START_REF] Ginibre | The global Cauchy problem for the critical nonlinear wave equation[END_REF], Shatah and Struwe [START_REF] Shatah | Well-posedness in the energy space for semilinear wave equations with critical growth[END_REF] (the defocusing case), as well as a complete review of the Cauchy theory in Kenig and Merle [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energycritical focusing non-linear wave equation[END_REF] (for N ∈ {3, 4, 5}) and Bulut, Czubak, Li, Pavlović and Zhang [START_REF] Bulut | Stability and unconditional uniqueness of solutions for energy critical wave equations in high dimensions[END_REF] (for N ≥ 6). By "well-posed" we mean that for any initial data u 0 ∈ E there exists τ > 0 and a unique solution in some subspace of C([t 0 -τ, t 0 + τ ]; E), and that this solution is continuous with respect to the inital data. By standard arguments, there exists a maximal time of existence (T -, T + ), -∞ ≤ T -< t 0 < T + ≤ +∞, and a unique solution u ∈ C((T -, T + ); E). If T + < +∞, then u(t) leaves every compact subset of E as t approaches T + . A crucial property of the solutions of (1.1) is that the energy E is a conservation law.

In this paper we always assume that the initial data are radially symmetric. This symmetry is preserved by the flow.

For

functions v ∈ Ḣ1 , v ∈ L 2 , v = (v, v) ∈ E and λ > 0, we denote v λ (x) := 1 λ 2 v x λ , vλ (x) := 1 λ 3 v x λ , v λ (x) := v λ , vλ .
A change of variables shows that

E (u 0 ) λ = E(u 0 ).
Equation (1.1) is invariant under the same scaling: if u(t) = (u(t), u(t)) is a solution of (1.1) and λ > 0, then t → u (tt 0 )/λ λ is also a solution with initial data (u 0 ) λ at time t = 0. This is why equation (1.1) is called energy-critical.

A fundamental object in the study of (1.1) is the family of stationary solutions u(t) ≡ ±W λ = (±W λ , 0), where

W (x) = 1 + |x| 2 24 -2
.

The functions W λ are called ground states or bubbles (of energy). They are the only radially symmetric solutions and, up to translation, the only positive solutions of the critical elliptic problem -∆u -f (u) = 0.

The ground states achieve the optimal constant in the critical Sobolev inequality, which was proved by Aubin [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF] and Talenti [START_REF] Talenti | Best constant in Sobolev inequality[END_REF]. They are the "mountain passes" for the potential energy. Kenig and Merle [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energycritical focusing non-linear wave equation[END_REF] exhibited the special role of the ground states W λ as the threshold elements for nonlinear dynamics of the solutions of (1.1) in space dimensions N = 3, 4, 5, which is believed to be a general feature of dispersive equations (the so-called Threshold Conjecture). Another major problem in the field is the Soliton Resolution Conjecture, which predicts that a bounded (in an appropriate sense) solution decomposes asymptotically into a sum of energy bubbles at different scales and a radiation term (a solution of the linear wave equation). This was proved for the radial energy-critical wave equation in dimension N = 3 by Duyckaerts, Kenig and Merle [START_REF] Duyckaerts | Classification of the radial solutions of the focusing, energy-critical wave equation[END_REF], following the earlier work of the same authors [START_REF] Duyckaerts | Profiles of bounded radial solutions of the focusing, energy-critical wave equation[END_REF], where such a decomposition was proved only for a sequence of times (this last result was generalized to any odd dimension by Rodriguez [START_REF] Rodriguez | Profiles for the radial focusing energy-critical wave equation in odd dimensions[END_REF]).

It is natural to examine the dynamics of the solutions of (1.1) in a neighborhood (in the energy space) of the family of the ground states. In dimension N = 3 the was done by Krieger, Schlag and Tataru [START_REF] Krieger | Slow blow-up solutions for the H 1 (R 3 ) critical focusing semilinear wave equation[END_REF], who showed that such solutions can blow up in finite time (by concentration of the bubble), see also [START_REF] Donninger | Nonscattering solutions and blowup at infinity for the critical wave equation[END_REF], [START_REF] Krieger | Full range of blow up exponents for the quintic wave equation in three dimensions[END_REF], [START_REF] Donninger | Exotic blowup solutions for the u 5 focusing wave equation in R 3[END_REF], [START_REF] Hillairet | Smooth type II blow up solutions to the four dimensional energy critical wave equation[END_REF], [START_REF] Jendrej | Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5[END_REF] for related results.

In view of the rich dynamics in a neighborhood of one bubble, it was expected that solutions behaving asymptotically as a superposition of many (at least two) bubbles exist, in other words that the result of [START_REF] Duyckaerts | Classification of the radial solutions of the focusing, energy-critical wave equation[END_REF] is essentially optimal. We prove that it is the case when N = 6: Theorem 1. There exists a solution u : (-∞, T 0 ] → E of (1.1) such that

lim t→-∞ u(t) -(W + W 1 κ e -κ|t| ) E = 0, with κ := 5 4 .
Remark 1.1. More precisely, we will prove that

u(t) -W + W 1 κ e -κ|t| , -e -κ|t| ΛW 1 κ e -κ|t| E ≤ C 1 • e -3 2 κ|t| ,
where ΛW := -∂ ∂λ W λ λ=1 and C 1 > 0 is a constant.

Remark 1.2. We construct here pure two-bubbles, that is the solution approaches a superposition of two stationary states, with no energy transformed into radiation. By the conservation of energy and the decoupling of the two bubbles, we necessarily have E(u(t)) = 2E(W ). Pure one-bubbles cannot concentrate and are completely classified, see [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF].

Remark 1.3. It was proved in [START_REF] Jendrej | Nonexistence of radial two-bubbles with opposite signs for the energy-critical wave equation[END_REF], in any dimension N ≥ 3, that there exist no solutions u(t)

: [t 0 , T + ) → E of (1.1) such that u(t) -(W µ(t) -W λ(t) ) E → 0 with λ(t) µ(t) as t → T + ≤ +∞.
Remark 1.4. In any dimension N > 6 one can expect an analogous result with concentration rate λ(t) ∼ |t| -4 N -6 .

Remark 1.5. In the context of the harmonic map heat flow, Topping [START_REF] Topping | An example of a nontrivial bubble tree in the harmonic map heat flow[END_REF] proved the existence of towers of bubbles for a well chosen target manifold, see also a non-existence result of van der Hout [START_REF] Van Der Hout | On the nonexistence of finite time bubble trees in symmetric harmonic map heat flows from the disk to the 2-sphere[END_REF].

Let us resume the overall strategy of the proof, which is based on the previous paper of the author [START_REF] Jendrej | Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5[END_REF].

In Section 2 we construct an appropriate approximate solution ϕ(t). We present first a formal computation which allows to predict the concentration rate and explains why the proof fails in dimension N ∈ {3, 4, 5}. It highlights also the role of the strong interaction between the two bubbles (by "strong" we mean "significantly altering the dynamics"; [START_REF] Martel | Strongly interacting blow up bubbles for the mass critical NLS[END_REF] provides an example of this phenomenon in a different context). Then we give a precise definition of the approximate solution and prove bounds on its error.

In Section 3 we build a sequence u n : [t n , T 0 ] → E of solutions of (1.1) with t n → -∞ and u n (t) close to a two-bubble solution for t ∈ [t n , T 0 ]. Taking a weak limit finishes the proof. This type of argument goes back to the works of Merle [START_REF] Merle | Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity[END_REF] and Martel [START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF]. The heart of the analysis is to obtain uniform energy bounds for the sequence u n . To this end we follow the approach of Raphaël and Szeftel [START_REF] Raphaël | Existence and uniqueness of minimal mass blow up solutions to an inhomogeneous L 2 -critical NLS[END_REF], that is we prove bootstrap estimates involving an energy functional with a virial-type correction term. This correction is designed to cancel some terms related to the concentration of the bubble W λ(t) . It has to be localized in an appropriate way, so that it does not "see" the other bubble. Finally, in order to deal with the linear instabilities of the flow, we use a classical topological ("shooting") argument.

Critical wave maps

We consider the wave map equation from the 2 + 1-dimensional Minkowski space (the energycritical case) to S 2 . We will consider solutions with k-equivariant symmetry, in which case the problem is reduced to the following scalar equation:

   ∂ 2 t u(t, r) = ∂ 2 r u(t, r) + 1 r ∂ r u(t, r) - k 2
2r 2 sin(2u(t, r)), (u(t 0 , r), ∂ t u(t 0 , r)) = (u 0 (r), u0 (r)), t, t 0 ∈ R, r ∈ (0, +∞).

(1.3)

For a presentation of the geometric content of this equation, one can consult [START_REF] Shatah | Geometric Wave Equations[END_REF]. Here we will regard (1.3) as a scalar semilinear problem. We define the space H as the completion of C ∞ 0 ((0, +∞)) for the norm

v 2 H := 2π ˆ+∞ 0 |∂ r v(r)| 2 + | k r v(r)| 2 rdr.
We will work in the energy space E := H × L 2 . Equation (1.3) can be written in the form (1.2) with the energy functional E defined for u 0 = (u 0 , u0 ) ∈ E by the formula

E(u 0 ) := π ˆ+∞ 0 ( u0 ) 2 + (∂ r u 0 ) 2 + k 2 r 2 (sin(u)) 2 rdr.
The Cauchy theory in the energy space has been established by Shatah and Tahvildar-Zadeh [START_REF] Shatah | On the Cauchy problem for equivariant wave maps[END_REF]. Note that u 0 ∈ H forces lim r→+∞ u 0 (r) = 0, but we could just as well consider states of finite energy such that lim r→+∞ u 0 (r) = π, see [START_REF] Côte | Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system[END_REF][START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: I[END_REF] for details. The stationary solutions W λ (r) := 2 arctan r λ k play a fundamental role in the study of (1.3). They are the harmonic maps of topological degree k. We will write W (r) := W 1 (r) = 2 arctan(r k ) and ΛW (r) := -∂ ∂λ W λ λ=1 = 2k r k +r -k . Note that W / ∈ H precisely because of the fact that W (r) → π as r → +∞.

The possibility of concentration of a harmonic map at the origin was first observed numerically by Bizoń, Chmaj and Tabor [START_REF] Bizoń | Formation of singularities for equivariant 2 + 1 dimensional wave maps into the 2-sphere[END_REF]. Struwe [START_REF] Struwe | Equivariant wave maps in two space dimensions[END_REF] proved that if the blow-up occurs, then W is the blow-up profile (for a sequence of times). The dynamics in a neighborhood of a harmonic map was studied by Krieger, Schlag and Tataru [START_REF] Krieger | Renormalization and blow up for charge one equivariant critical wave maps[END_REF], who constructed blow-up solutions in the energy space with the concentration rate λ(t) ∼ (T + -t) 1+ν for all ν > 1 2 . This behavior is expected to be highly unstable. Rodnianski and Sterbenz [START_REF] Rodnianski | On the formation of singularities in the critical O(3) σ-model[END_REF] constructed stable blow-up solutions, giving the first (partial) rigorous explanation of the surprising numerical results mentioned above. In the case k = 1, Côte [START_REF] Côte | On the soliton resolution for equivariant wave maps to the sphere[END_REF] proved that any solution decomposes, for a sequence of times tending to the final (finite or inifinite) time of existence, as a sum of a finite number of harmonic maps at different scales and a radiation term. A generalization of this result, including all the cases considered in this paper, was recently obtained by Jia and Kenig [START_REF] Jia | Asymptotic decomposition for semilinear wave and equivariant wave map equations[END_REF]. Motivated by these works, we prove the following result.

Theorem 2. Fix k > 2. There exists a solution u : (-∞, T 0 ] → E of (1.3) such that lim t→-∞ u(t) -(-W + W k-2 κ (κ|t|) -2 k-2 ) E = 0, with κ := k -2 2 8k π sin π k 1 k .
Remark 1.6. More precisely, we will prove that

u(t) --W + W k-2 κ (κ|t|) -2 k-2 , -(κ|t|) -k k-2 ΛW k-2 κ (κ|t|) -2 k-2 E ≤ C 1 • |t| -k+1 k-2 ,
where ΛW := -∂ ∂λ W λ λ=1 and C 1 > 0 is a constant. Remark 1.7. The constructed solution is a pure two-bubble, hence by the conservation of energy E(u(t)) = 2E(W ), and it is clear that it has the homotopy degree 0. In the case of equivariant class k = 1, Côte, Kenig, Lawrie and Schlag [START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: I[END_REF] showed that any degree 0 initial data of energy < 2E(W ) leads to dispersion (the proof is expected to generalize to all equivariance classes). Theorem 2 gives the first example of a non-dispersive solution at the threshold energy.

Note that pure two-bubbles of homotopy degree 2k (hence of type bubble-bubble and not bubble-antibubble) do not exist because the energy of such a map has to be > 2E(W ). This is similar to the case of opposite signs for (1.1), see Remark 1.3.

Remark 1.8. I believe that the proof can by adapted to deal with a more general equation

∂ 2 t u = ∂ 2 r u + 1 r ∂ r u -1 r 2 (gg )(u)
with g satisfying the assumptions of [START_REF] Côte | Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system[END_REF] and g (0) ∈ {3, 4, 5, . . .}.

Critical Yang-Mills

Finally, we consider the radial Yang-Mills equation in dimension 4 (which is the energy-critical case):

   ∂ 2 t u(t, r) = ∂ 2 r u(t, r) + 1 r ∂ r u(t, r) - 4 r 2 u(t, r)(1 -u(t, r)) 1 - 1 2 u(t, r) ,
(u(t 0 , r), ∂ t u(t 0 , r)) = (u 0 (r), u0 (r)), t, t 0 ∈ R, r ∈ (0, +∞).

(1.4)

For a derivation of this equation and further comments, see for instance [START_REF] Cazenave | Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields[END_REF]. Equation (1.4) can be written in the form (1.2) with the energy functional E defined for u 0 = (u 0 , u0 ) ∈ E by the formula

E(u 0 ) := π ˆ+∞ 0 ( u0 ) 2 + (∂ r u 0 ) 2 + 1 r 2 (u 0 (2 -u 0 )) 2 rdr.
The stationary solutions of (1.4) are W λ (r) := 2r 2 λ 2 +r 2 . We denote W (r

) := W 1 (r) = 2r 2 1+r 2
and ΛW (r

) := -∂ ∂λ W λ λ=1 = 4 (r+r -1 ) 2 .
Theorem 3. There exists a solution u : (-∞, T 0 ] → E of (1.4) such that

lim t→-∞ u(t) -(-W + W 1 κ e -κ|t| ) E = 0, with κ := 2 √ 3.
Remark 1.9. More precisely, we will prove that

u(t) --W + W 1 κ e -κ|t| , -e -κ|t| ΛW 1 κ e -κ|t| E ≤ C 1 • e -3 2 κ|t| ,
where ΛW := -∂ ∂λ W λ λ=1 and C 1 > 0 is a constant.

Remark 1.10. The case of wave maps in the equivariance class k = 2 should be very similar.

Remark 1.11. The energy 2E(W ) is the threshold energy for a non-dispersive behavior for solutions with topological degree 0, see [START_REF] Lawrie | A refined threshold theorem for (1+2)-dimensional wave maps into surfaces[END_REF].

Structure of the paper

In Sections 2 and 3 we give a detailed proof of Theorem 1. In Section 4 we treat the case of the Yang-Mills equation. We skip these parts of the proof where the arguments of Sections 2 and 3 are directly applicable. Section 5 is devoted to the wave maps equation. The main difference with respect to Section 4 is that the characteristic length of the concentrating bubble is now a power of |t| and not an exponential. Nevertheless, large parts of the previous proofs extend to this case and are skipped. It is conceivable that one could propose a unified, more general framework of the proof, encompassing all the cases under consideration. Appendix A is devoted to some elements of the local Cauchy theory needed in the proofs.

Notation

The bracket •, • denotes the distributional pairing and the scalar product in the spaces L 2 and L 2 × L 2 . For positive quantities m 1 and m 2 we write

m 1 m 2 if m 1 ≤ Cm 2 for some constant C > 0 and m 1 ∼ m 2 if m 1 m 2 m 1 .
We denote χ a standard

C ∞ cut-off function, that is χ(x) = 1 for |x| ≤ 1, χ(x) = 0 for |x| ≥ 2 and 0 ≤ χ(x) ≤ 1 for 1 ≤ |x| ≤ 2.
2 Construction of an approximate solution -the NLW case 2.1 Inverting the linearized operator

Linearizing (1.1) around W , u = W + h, one obtains ∂ t h = J • D 2 E(W )h = 0 Id -L 0 h,
where L is the Schrödinger operator

Lh := (-∆ -f (W ))h = (-∆ -2W )h.
We introduce the following notation for the generators of the Ḣ1 -critical and the L 2 -critical scale change:

Λ := 2 + x • ∇, Λ 0 := 3 + x • ∇.
This is coherent with the definition of ΛW . Notice that L(ΛW

) = d dλ λ=1 -∆W λ -f (W λ ) = 0. We fix Z ∈ C ∞ 0 such that Z, ΛW > 0, Z, Y = 0.
We will use this function to define appropriate orthogonality conditions. We denote also

κ := - ΛW, f (W ) ΛW, ΛW 1 2 = 5 4 . (2.1)
Lemma 2.1. There exist radial rational functions

P (x), Q(x) ∈ C ∞ (R 6 ) such that LP = κ 2 ΛW + f (W ), LQ = -Λ 0 ΛW, (2.2)
Z, P = Z, Q = 0, (2.3)
P (x) ∼ |x| -2 , Q(x) ∼ |x| -2 as |x| → +∞. (2.4)
Proof. By a direct computation one checks that the functions

P (x) := 1 + |x| 2 24 -3 • 1 -10 • |x| 2 24 -3 • |x| 2 24 2 , Q(x) := 1 + |x| 2 24 -3 • 1 + 11 • |x| 2 24 - 12

Formal computation

The usual method of performing a formal analysis of blow-up solutions is to search a series expansion with respect to a small scalar parameter depending on time and converging to 0 at blow-up. In our case the blow-up time is

-∞. If u(t) W + W λ(t) , then ∂ t u(t) -λ (t)ΛW λ(t) , hence u(t) (W + W λ(t) , 0) -λ (t) • (0, ΛW λ(t) ) = W + U (0) λ(t) + b(t) • U (1) λ(t) ,
with b(t) := λ (t), U (0) := (W, 0) and U (1) := (0, -ΛW ). This suggests considering b(t) = λ (t) as the small parameter with respect to which the formal expansion should be sought. Hence, we make the ansatz

u(t) = W + U (0) λ(t) + b(t) • U (1) λ(t) + b(t) 2 • U (2) λ(t) ,
and try to find the conditions under which a satisfactory candidate for U (2) = (U (2) , U (2) ) can be proposed. Neglecting irrelevant terms and replacing λ (t) by b(t), we compute

∂ 2 t u(t) = -b (t)(ΛW ) λ(t) + b(t) 2 λ(t) (Λ 0 ΛW ) λ(t) + lot.
On the other hand, using the fact that f

(W + W λ ) = f (W ) + f (W λ ) + f (W λ )W f (W ) + f (W λ ) + f (W λ ) for λ 1 and f (W λ ) = λf (W ) λ , we get ∆u(t) + f (u(t)) = - b(t) 2 λ(t) (LU (2) ) λ(t) + λ(t)f (W ) λ(t) + lot.
We discover that, formally at least, we should have

LU (2) = -Λ 0 ΛW + λ(t) b(t) 2 b (t) • ΛW + λ(t) • f (W ) . (2.5) Lemma 2.1 shows that if b (t) = κ 2 λ(t)
, then equation (2.5) has a decaying regular solution

U (2) = Q + λ(t) 2 b(t) 2 P .
The formal parameter equations

λ (t) = b(t), b (t) = κ 2 λ(t) have a solution (λ app (t), b app (t)) = 1 κ e -κ|t| , e -κ|t| , t ≤ T 0 < 0.
In any space dimension N , ignoring the problems related to slow decay of W , a similar analysis would yield b (t) = κ 2 λ(t) N -2 4 . For N < 6 this leads to a finite time blow-up, which was studied in [START_REF] Jendrej | Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5[END_REF] for N = 5. For N > 6, we obtain a global solution λ(t) ∼ |t| -4 N -6 , see Remark 1.4.

Bounds on the error of the ansatz

Let I = [T, T 0 ] be a time interval, with T ≤ T 0 < 0 and |T 0 | large. Let λ(t) and µ(t) be C 1 functions on [T, T 0 ] such that

λ(T ) = 1 κ e -κ|T | , µ(T ) = 1, (2.6) 8 9κ e -κ|t| ≤ λ(t) ≤ 9 8κ e -κ|t| , 8 9 ≤ µ(t) ≤ 9 8 . (2.7)
We define the approximate solution ϕ(t) = (ϕ(t), φ(t)) : [T, T 0 ] → E by the formula

ϕ(t) := W µ(t) + W λ(t) + S(t), φ(t) := -b(t)ΛW λ(t) , where b(t) := e -κ|T | + κ 2 ˆt T λ(τ ) µ(τ ) 2 dτ, for t ∈ [T, T 0 ], (2.8)
S(t) := χ • λ(t) 2 µ(t) 2 P λ(t) + b(t) 2 Q λ(t) , for t ∈ [T, T 0 ].
From (2.7) we get 8 (2.9)

From (2.4) we obtain

χ • P λ Ḣ1 ∂ r (χ • P λ ) L 2 (r 5 dr) P λ L 2 (0≤r≤2) + 1 λ (∂ r P ) λ L 2 (0≤r≤2) = P L 5 (0≤r≤ 2 λ ) + 1 λ ∂ r P L 2 (0≤r≤ 2 λ ) ˆ2/λ 0 (1 + r 2 ) -2 r 5 dr 1 2 + 1 λ • ˆ2/λ 0 (1 + r 3 ) -2 r 5 dr 1 2 1 λ | log λ| 1 2 , (2.10)
and analogously χ

• Q λ Ḣ1 1 λ | log λ| 1 2 , hence S(t) Ḣ1 ≤ λ 3 µ 2 χP λ Ḣ1 + λb 2 χQ λ Ḣ1 e -3κ|t| • 1 λ | log λ| 1 2 |t| • e -2κ|t| .
Thus for any c > 0 there exists T 0 such that if T < T 0 then

S(t) Ḣ1 ≤ c • e -3 2 κ|t| , for t ∈ [T, T 0 ]. (2.11) Note also that P λ L ∞ + Q λ L ∞ λ -2 , hence S(t) is bounded in L ∞ .
Since Z has compact support, for sufficiently small λ, (2.3) implies

Z λ(t) , S(t) = 0. (2.12)
We denote

ψ(t) = (ψ(t), ψ(t)) := ∂ t ϕ(t) -DE(ϕ(t)) = ∂ t ϕ(t) -φ(t), ∂ t φ(t) -(∆ϕ(t) + f (ϕ(t))) . ( 2

.13)

This function describes how much ϕ(t) fails to be an exact solution of (1.1). Before we prove bounds on ψ(t), we gather in the next elementary lemma pointwise inequalities used in various places in the text.

Lemma 2.3. Let k, l, m ∈ R. Then |f (k + l) -f (k)| ≤ f (l), (2.14)
|f (k + l) -f (k) -f (k)l| ≤ 5|f (l)|, (2.15
)

|F (k + l) -F (k) -f (k)l - 1 2 f (k)l 2 | ≤ 5F (l). ( 2
(1+l)-f (1)-f (1)l| = |(1+l) 2 -1-2l| = 2l 2 ≤ 5|f (l)| and for |l| ≥ 1 we find |f (1 + l) -f (1) -f (1)l| ≤ (1 + l) 2 + 1 + 2|l| ≤ 5|f (l)
ψ(t) + µ (t) 1 µ(t) ΛW µ(t) + (λ (t) -b(t)) 1 λ(t) ΛW λ(t) Ḣ1 e -3 2 κ|t| , (2.17)
ψ(t) - b(t) λ(t) (λ (t) -b(t))Λ 0 ΛW λ(t) L 2 e -3 2 κ|t| , (2.18)
(-∆ -f (ϕ(t)))ψ(t) Ḣ-1 e -3 2 κ|t| . (2.19)
Proof. Using the definitions of ϕ and ψ we find

ψ + µ ΛW µ + (λ -b)ΛW λ = ∂ t ϕ -φ + µ ΛW µ + (λ -b)ΛW λ = -µ ΛW µ -λ ΛW λ + ∂ t S + bΛW λ + (λ -b)ΛW λ = χ • -2µ λ 3 µ 3 P λ + 2λ λ 2 µ 2 P λ -λ λ 2 µ 2 ΛP λ + 2b bλQ λ -λ b 2 ΛQ λ .
Since ΛP and ΛQ are rational functions decaying like r -2 , we have χ

f (ϕ) -f (W µ ) -f (W λ ) -f (W λ )W µ - λ 2 µ 2 f (W λ )P λ -b 2 f (W λ )Q λ L 2 (|x|≤ √ λ)
e -3 2 κ|t| .

(2.20)

Applying (2.15) with k = W λ and l = W µ + S we get

f (ϕ) -f (W λ ) -f (W λ )(W µ + λ 2 µ -2 • P λ + b 2 • Q λ ) |f (W µ )| + |f (λ 2 µ -2 P λ + b 2 Q λ )|, which is bounded in L ∞ , hence bounded by λ 3 2 ∼ e -3 2 κ|t| in L 2 (|x| ≤ √ λ). This proves (2.20). Now we check that f (W λ )W µ - 1 µ 2 f (W λ ) L 2 e -3 2 κ|t| . (2.21) Indeed, for |x| ≤ √ λ we have |W µ (x) -1 µ 2 | = |W µ (x) -W µ (0)| |x| 2 ≤ λ ∼ e -κ|t| , hence f (W λ )W µ - 1 µ 2 f (W λ ) L 2 (|x|≤ √ λ) W µ - 1 µ 2 L ∞ (|x|≤ √ λ) • W λ L 2 e -κ|t| • λ e -3 2 κ|t| .
From (2.20) and (2.21) we obtain

f (ϕ) -f (W µ ) -f (W λ ) -µ -2 f (W λ ) -λ 2 µ -2 f (W λ )P λ -b 2 f (W λ )Q λ L 2 e -3 2 κ|t| . (2.22) Since χ = 1 in the region |x| ≤ √ λ, we have ∆ϕ = ∆(W µ )+∆(W λ )+λ 2 µ -2 ∆(P λ )+b 2 ∆(Q λ ).

From this and (2.22), using the fact that ∆(W

µ ) + f (W µ ) = ∆(W λ ) + f (W λ ) = 0, we get ∆ϕ + f (ϕ) -µ -2 λ 2 ∆(P λ ) + λ 2 f (W λ )P λ + f (W λ ) -b 2 ∆(Q λ ) + b 2 f (W λ )Q λ L 2 (|x|≤ √ λ)
e -3 2 κ|t| .

(2.23)

But formula (2.2) gives

λ 2 ∆(P λ ) + λ 2 f (W λ )P λ = (-LP ) λ = -κ 2 ΛW λ -f (W λ ) = -κ 2 λΛW λ -f (W λ ), b 2 ∆(Q λ ) + b 2 f (W λ )Q λ = b 2 λ 2 (-LQ) λ = b 2 λ Λ 0 ΛW λ ,
hence we can rewrite (2.23) as

∆ϕ + f (ϕ) + κ 2 λ µ 2 ΛW λ - b 2 λ Λ 0 ΛW λ L 2 (|x|≤ √ λ)
e -3 2 κ|t| .

(2.24)

We have To this end, we compute

∂ t φ = -b ΛW λ + λ b λ Λ 0 ΛW λ , thus -ψ + b λ (λ -b)Λ 0 ΛW λ = ∆ϕ + f (ϕ) + b ΛW λ - bλ λ Λ 0 ΛW λ + b λ (λ -b)Λ 0 ΛW λ = ∆ϕ + f (ϕ) + κ 2 λ µ 2 ΛW λ - b 2 λ Λ 0 ΛW λ , (2.25
∆(W λ ) L 2 (|x|≥ √ λ) = f (W λ ) L 2 (|x|≥ √ λ) = 1 λ f (W ) L 2 (|x|≥1/ √ λ) 1 λ ˆ+∞ 1/ √ λ r -16 r 5 dr 1 2 1 λ λ 5 2
e -3 2 κ|t| .

We need to show that ∆(χ

•P λ ) L 2 (|x|≥ √ λ) e 1 2 κ|t| and ∆(χ•Q λ ) L 2 (|x|≥ √ λ) e 1 2
κ|t| . We will prove the first bound (the second is exactly the same). Notice that

|P λ (x)| 1 λ 2 • λ 2 |x| 2 = |x| -2 and similarly |∇(P λ )(x)| |x| -3 , |∇ 2 (P λ )(x)| |x| -4 , hence we have a pointwise bound |∆(χP λ )| |∇ 2 χ|•|P λ |+|∇χ|•|∇(P λ )|+|χ|•|∇ 2 (P λ )| |∇ 2 χ|•|x| -2 +|∇χ|•|x| -3 +|χ|•|x| -4 . Of course |∇ 2 χ| • |x| -2 L 2 (|x|≥ √ λ) + |∇χ| • |x| -3 L 2 (|x|≥ √ λ)
1 e 1 2 κ|t| and we are left with the last term. We compute

|χ| • |x| -4 L 2 (|x|≥ √ λ) ˆ2 √ λ r -8 r 5 dr 1 2 λ -1 2 e 1 2 κ|t| .
This finishes the proof of (2.26). Applying (2.15) with k = W µ and l = W λ + S we get

|f (ϕ) -f (W µ )| f (W µ ) • (|W λ | + |S|) + |f (W λ )| + |f (S)| |W λ | + |S|,
where the last estimate follows from the fact that

W λ L ∞ + S L ∞ 1 for |x| ≥ √ λ.
We have χ

• P λ L 2 ´2 λ 0 (r -2 ) 2 r 5 dr 1 2 ∼ λ -1 , and similarly χ • Q λ L 2 λ -1 , which implies S L 2 e -3 2
κ|t| . There holds also

W λ L 2 (|x|≥ √ λ) = λ W L 2 (|x|≥1/ √ λ) λ ˆ∞ 1/ √ λ r -8 r 5 dr 1 2 ∼ λ 3 2 ∼ e -3 2 κ|t| , (2.27) hence f (ϕ) -f (µ) L 2 (|x|≥ √ λ)
e -3 2 κ|t| . Together with (2.26) this yields

∆ϕ + f (ϕ) L 2 (|x|≥ √ λ) e -3 2 κ|t| .
The same computation as in (2.27) gives

ΛW λ L 2 (|x|≥ √ λ) + Λ 0 ΛW λ L 2 (|x|≥ √ λ)
e -1 2 κ|t| , hence (2.25) implies that (2.18) holds also in the region |x| ≥ √ λ.

We are left with (2.19). From (2.17) it follows that it suffices to check that

(-∆ -f (ϕ))(λ -b) 1 λ ΛW λ Ḣ-1 e -3 2 κ|t| (2.28) and (-∆ -f (ϕ))µ 1 µ ΛW µ Ḣ-1 e -3 2 κ|t| . (2.29)
We start with (2.28). Since |λ -b| e -κ|t| λ, we need to show that

(-∆ -f (ϕ))ΛW λ Ḣ-1 e -3 2 κ|t| .
(2.30)

By Hölder inequality

f (W µ )ΛW λ L 3 2 ≤ f (W µ ) L 12 • ΛW λ L 12 7 λ 3 2 e -3 2 κ|t| . (2.31) Since |f (W λ + W µ ) -f (W λ )| ≤ f (W µ ), we obtain (f (W λ + W µ ) -f (W λ ))ΛW λ Ḣ-1 (f (W λ + W µ ) -f (W λ ))ΛW λ L 3 2 e -3 2 κ|t| .
As noted earlier (-∆ -f (W λ ))ΛW λ = 0, hence

(-∆ -f (W λ + W µ ))ΛW λ Ḣ-1 e -3 2 κ|t| . From (2.11) we have f (ϕ) -f (W λ + W µ ) L 3 e -3 2 κ|t|
. This implies (2.30). The proof of (2.29) is similar. It suffices to check that f (W λ )ΛW µ Ḣ-1 e -1 2 κ|t| , and in fact we even have the bound e -3 2 κ|t| , with the same proof as in (2.31).

Bootstrap control of the error term -the NLW case

In the preceding section we defined approximate solutions of (1.1). In the present section we consider exact solutions of (1.1), with some specific initial data prescribed at t = T , with T → -∞. Our goal is to control the evolution of this solution up to a time T 0 independent of T .

For technical reasons we will require the initial data to belong to the space X 1 × H 1 , where X 1 := Ḣ2 ∩ Ḣ1 . This regularity is preserved by the flow, see Proposition A.1.

Set-up of the bootstrap

It is known that L = -∆ -f (W ) has exactly one strictly negative simple eigenvalue which we denote -ν 2 (we take ν > 0). We denote the corresponding positive eigenfunction Y, normalized so that Y L 2 = 1. By elliptic regularity Y is smooth and by Agmon estimates it decays exponentially. Self-adjointness of L implies that Y, ΛW = 0.

Note that

ν < 1.

(3.1) Indeed, it is well-known that -∆ -W ≥ 0, with a one-dimensional kernel generated by W . Since 1 -W (x) > 0 almost everywhere, for any h = 0 we have

h, Lh + h, h = (-∆ -2W + 1)h, h > (-∆ -W )h, h ≥ 0.
We define

Y -:= 1 ν Y, -Y , Y + := 1 ν Y, Y , α -:= ν 2 JY + = 1 2 (νY, -Y), α + := - ν 2 JY -= 1 2 (νY, Y).
We have

J • D 2 E(W ) = 0 Id -L 0 . A short computation shows that J • D 2 E(W )Y -= -νY -, J • D 2 E(W )Y + = νY + and α -, J • D 2 E(W )h = -ν α -, h , α + , J • D 2 E(W )h = ν α + , h , ∀h ∈ E.
We will think of α -and α + as linear forms on E. Notice that α -,

Y -= α + , Y + = 1 and α -, Y + = α + , Y -= 0.
The rescaled versions of these objects are

Y - λ := 1 ν Y λ , -Y λ , Y + λ := 1 ν Y λ , Y λ , α - λ := ν 2λ JY + λ = 1 2 ν λ Y λ , -Y λ , α + λ := - ν 2λ JY - λ = 1 2 ν λ Y λ , Y λ . (3.2)
The scaling is chosen so that

α - λ , Y - λ = α + λ , Y + λ = 1. We have J • D 2 E(W λ )Y - λ = - ν λ Y - λ , J • D 2 E(W λ )Y + λ = ν λ Y + λ (3.3) and α - λ , J • D 2 E(W λ )h = - ν λ α - λ , h , α + λ , J • D 2 E(W λ )h = ν λ α + λ , h , ∀h ∈ E.
(3.4) We will need the following simple lemma in order to properly choose the initial data. Lemma 3.1. There exist universal constants η, C > 0 such that if 0 < λ < η • µ, then for all a 0 ∈ R there exists h 0 ∈ X 1 ×H 1 satisfying the orthogonality conditions Z µ , h 0 = Z λ , h 0 = 0 and such that α + µ ,

h 0 = 0, α - µ , h 0 = 0, α + λ , h 0 = a 0 , α - λ , h 0 = 0, h 0 E ≤ C|a 0 |.
Proof. We consider functions of the form:

h 0 := a + 2 Y + µ + a - 2 Y - µ + b 2 ΛW µ + a + 1 Y + λ + a - 1 Y - λ + b 1 ΛW λ , a + 2 , a - 2 , b 2 , a + 1 , a - 1 , b 1 ∈ R.
Consider the linear map Φ : R 6 → R 6 defined as follows:

Φ(a + 2 , a - 2 , b 2 , a + 1 , a - 1 , b) := α + µ , h 0 , α - µ , h 0 , 1 µ Z µ , h 0 , α + λ , h 0 , α - λ , h 0 , 1 λ Z λ , h 0
It is easy to check that the matrix of Φ is strictly diagonally dominant if η is small enough.

We consider the solution u(t) = u(a 0 ; t) : [T, T + ) → E of (1.1) with the initial data

u(T ) = W 1 κ e -κ|T | + W + h 0 , -ΛW 1 κ e -κ|T | , (3.5)
where h 0 is the function given by Lemma 3.1 with λ = 1 κ e -κ|T | , µ = 1 and some a 0 chosen later, satisfying

|a 0 | ≤ e -3 2 κ|T | .
Note that the initial data depends continuously on a 0 . For t ≥ T we define the functions λ(t) and µ(t) as the solutions of the following system of ordinary differential equations with the initial data µ(T ) = 1 and λ(T ) = 1 κ e -κ|T | :

Z λ , ΛW λ -1 λ Λ 0 Z λ , h Z λ , ΛW µ Z µ , ΛW λ Z µ , ΛW µ -1 µ Λ 0 Z µ , h • λ µ = -Z λ , ∂ t u -Z µ , ∂ t u , (3.6)
where

h = h(t) := u(t) -W µ(t) -W λ(t) , if u(t) -W µ(t) -W λ(t) Ḣ1 ≤ η, η u(t)-W µ(t) -W λ(t) Ḣ1 u(t) -W µ(t) -W λ(t) , if u(t) -W µ(t) -W λ(t) Ḣ1 ≥ η with a small constant η > 0. Notice that Z λ , ΛW λ = Z µ , ΛW µ = Z, ΛW > 0, | 1 λ Λ 0 Z λ , h | + | 1 µ Λ 0 Z µ , h | h Ḣ1 and | Z λ , ΛW µ | + | Z µ , ΛW λ | λ/µ.
For t < T 0 bounds (2.7) imply that λ/µ is small, hence equation (3.6) defines a unique solution as long as (2.7) holds. Remark 3.2. Actually the second case in the definition of h(t) will never occur in our analysis, since the bootstrap assumptions imply that h(t) Ḣ1 is small. Suppose that λ(t) and µ(t) are well defined and satisfy (2.7) for t ∈ [T, T 1 ], where T < T 1 ≤ T 0 . Suppose also that h(t) Ḣ1 < η for t ∈ [T, T 1 ], which implies that h(t) = u(t). Using (3.6) we find d dt Z µ(t) , h(t) = 0 and d dt Z λ(t) , h(t) = 0. Since Z µ(T ) , h(T ) = 0 and Z λ(T ) , h(T ) = 0, we obtain

Z µ(t) , h(t) = Z λ(t) , h(t) = 0, for t ∈ [T, T 1 ]. (3.7)
We denote h(t) := (h(t), ∂ t u(t)), so that

∂ t h = ḣ + µ ΛW µ + λ ΛW λ , ∂ t ḣ = ∆h + f (W µ + W λ + h) -f (W µ ) -f (W λ ). (3.8)
We define the function b(t) : [T, T 1 ] → R by formula (2.8) and decompose

u(t) = ϕ(t) + g(t), t ∈ [T, T 1 ].
By the definitions of g(t) and ψ(t), g(t) satisfies the differential equation

∂ t g(t) = J • DE(ϕ(t) + g(t)) -J • DE(ϕ(t)) -ψ(t). (3.9)
Finally, we denote

a + 1 (t) := α + λ(t) , g(t) , a - 1 (t) := α - λ(t) , g(t) , a + 2 (t) := α + µ(t) , g(t) , a - 2 (t) := α - µ(t) , g(t) .
The rest of this section is devoted to the proof of the following bootstrap estimate, which is the heart of the whole construction. Proposition 3.3. There exist constants C 0 > 0 and T 0 < 0 (C 0 and |T 0 | large) with the following property. Let T < T 1 < T 0 and suppose that u(t) = ϕ(t)+g(t) ∈ C([T, T 1 ]; X 1 ×H 1 ) is a solution of (1.1) with initial data (3.5) such that for t ∈ [T, T 1 ] condition (2.7) is satisfied and

g(t) E ≤ C 0 • e -3 2 κ|t| , (3.10
)

|a + 1 (t)| ≤ e -3 2 κ|t| . (3.11)
Then for t ∈ [T, T 1 ] there holds

g(t) E ≤ 1 2 C 0 e -3 2 κ|t| , (3.12
)

λ(t) - 1 κ e -κ|t| + |µ(t) -1| C 0 e -3 2 κ|t| . (3.13)
Remark 3.4. Notice that (3.12) and (3.13) are strictly stronger than (3.10) and (2.7) respectively, which will be crucial for closing the bootstrap in Subsection 3.6.

Remark 3.5. The same conclusion should be true without the assumption of X 1 × H 1 regularity, by means of a standard approximation procedure (both the assumptions and the conclusion are continuous for the topology • E ).

Modulation

Lemma 3.6. Under assumptions (2.7) and (3.10), for t ∈ [T, T 1 ] there holds

1 λ(t) Z λ(t) , g(t) = 0, 1 µ(t) Z µ(t) , g(t) c • e -3 2 κ|t| , (3.14)
|λ (t) -b(t)| + |µ (t)| g(t) E + c • e -3 2 κ|t| , (3.15)
with a constant c arbitrarily small.

Proof. We have g(t) = h(t) -S(t). Since Z has compact support, (2.12) and (3.7) yield

Z λ , g = 0. From 1 µ Z µ L ∞ 1 and χ • λ 2 P λ L 1 ≤ λ 2 P λ L 1 (|x|≤2) λ 2 ∼ e -2κ|t| (analo- gously χ • b 2 Q λ L 1 e -2κ|t| ) we obtain | 1 µ Z µ , g | e -2κ|t| . From (2.11) we have g(t) -h(t) Ḣ1 ≤ c • e -3 2 κ|t| , with a small constant c, (3.16)
Using this, (3.6) yields

Z λ , ΛW λ -1 λ Λ 0 Z λ , g Z λ , ΛW µ Z µ , ΛW λ Z µ , ΛW µ -1 µ Λ 0 Z µ , g • λ -b µ = -Z λ , ∂ t u + bΛW λ -Z µ , ∂ t u + bΛW λ + o R 2 (e -
(t) = d dt α + λ(t) , g(t) = - λ λ (Λ E * α + ) λ , g + α + λ , J • DE(ϕ + g) -J • DE(ϕ) -ψ , (3.20)
where Λ E * α + := -∂ ∂λ α + λ | λ=1 . We have

| (Λ E * α + ) λ , g | g e -1 2 κ|t| . (3.21)
Since α + λ , (ΛW λ , 0) = 0, using (2.18) we obtain

| α + λ , ψ | ψ - λ -b λ (ΛW λ , 0) E e -1 2 κ|t| . ( 3
(f (ϕ) -f (W µ ) -f (W λ ))g L 3 2 ϕ -W µ -W λ L 3 • g L 3 e -3 2 κ|t| g Ḣ1 .
Taking the sum we obtain

f (ϕ + g) -f (ϕ) -(f (W µ ) + f (W λ ))g L 3 2 g 2 Ḣ1 + e -3 2 κ|t| g Ḣ1 . (3.23) But | Y λ , f (W µ )g | Y λ L 3 2 • f (W µ ) L ∞ • g L 3 λ g Ḣ1 , hence | Y λ , f (ϕ + g) -f (ϕ) -f (W λ )g | 1 λ ( g 2 Ḣ1 + e -3 2 κ|t| g Ḣ1 ) e -
d dt a + 1 (t) = α + λ , J • D 2 E(W λ )g + o(e -1 2 κ|t| ) = ν λ a + 1 (t) + o(e -1 2 κ|t| ),
where in the last step we use (3.4). This proves (3.17 In order to prove (3.19), it is more convenient to work with h(t), which was defined right before (3.8), than with g(t). We will prove the bound for a + 2 (t). The proof for a - 2 (t) is exactly the same. Let a(t) := α + µ(t) , h(t) . We have

| Y µ , ΛW λ | ΛW λ L 1 (|x|≤1) + ΛW λ L ∞ (|x|≥1) λ 3 ˆ1 λ 0 r -4 r 5 dr + 1 λ 3 • 1 λ -4
λ.

(3.26)

Together with (2.11) this yields

|a + 2 (t) -a(t)| = | α + µ , g(t) -h(t) | b| Yµ, ΛW λ | + S Ḣ1 ≤ 1 2 e -3 2 κ|t| , hence it suffices to show that | a(t)| ≤ 1 2 e -3 2 κ|t| . (3.27)
As in the case of a + 1 (t), using (3.8) we obtain . Since c is arbitrarily small, we obtain a contradiction.

d dt a(t) = - µ µ (Λ E * α + ) µ , h + α + µ , J • D 2 E(W µ )h + α + µ , (µ ΛW µ + λ ΛW λ , f (W µ + W λ + h) -f (W µ ) -f (W λ ) -f (W µ )h) (3.28) But Y µ , ΛW µ = 0, | Y µ , ΛW λ | e -κ|t| , see (3.26), | Y µ , f (W µ + W λ + h) -f (W µ + W λ ) -f (W µ + W λ )h | h 2 Ḣ1 e -2κ|t| , | Y µ , f (W µ + W λ ) -f (W µ ) -f (W λ ) | = 2| Y µ , W µ • W λ | = 2| Y µ • W µ , W λ | e -2κ|t| , see (3.26), | Y µ , (f (W µ + W λ ) -f (W µ ))h | Y µ L 6 • W λ L 2 • h L 3 λ h Ḣ1 e -

Coercivity

Recall that we denote L := -∆ -f (W ).

Lemma 3.8. There exist constants c, C > 0 such that

• for all g ∈ Ḣ1 radially symmetric there holds

g, Lg = ˆR6 |∇g| 2 dx - ˆRN f (W )|g| 2 dx ≥ c ˆR6 |∇g| 2 dx -C Z, g 2 + Y, g 2 ,
• if r 1 > 0 is large enough, then for all g ∈ Ḣ1 rad there holds

(1 -2c) ˆ|x|≤r 1 |∇g| 2 dx + c ˆ|x|≥r 1 |∇g| 2 dx - ˆR6 f (W )|g| 2 dx ≥ -C Z, g 2 + Y, g 2 , (3.30)
• if r 2 > 0 is small enough, then for all g ∈ Ḣ1 rad there holds

(1 -2c) ˆ|x|≥r 2 |∇g| 2 dx + c ˆ|x|≤r 2 |∇g| 2 dx - ˆR6 f (W )|g| 2 dx ≥ -C Z, g 2 + Y, g 2 . (3.31)
Proof. This is exactly Lemma 2.1 in [START_REF] Jendrej | Nonexistence of radial two-bubbles with opposite signs for the energy-critical wave equation[END_REF], see also [START_REF] Martel | Construction of multi-solitons for the energy-critical wave equation in dimension 5[END_REF]Lemma 2.1].

Lemma 3.9. There exists a constant η > 0 such that if λ µ < η and U -(W µ + W λ ) E < η, then for all g ∈ E there holds

1 2 D 2 E(U )g, g +2 α - λ , g 2 + α + λ , g 2 + 1 λ Z λ , g 2 + α - µ , g 2 + α + µ , g 2 + 1 µ Z µ , g 2 g 2 E .
Proof. We will repeat with minor changes the proof of [41, Lemma 3.5].

Step 1 Without loss of generality we can assume that µ = 1. Consider the operator L λ defined by the following formula:

L λ := -∆ -f (W λ ) -f (W ) 0 0 Id . From the fact that f (U ) -f (W ) -f (W λ ) L 3 U -(W + W λ ) L 3 we obtain | D 2 E(U )g, g -L λ g, g | ≤ c g 2 E , ∀g ∈ E, (3.32)
with c > 0 small when η and λ 0 are small.

Step 2. In view of (3.32), it suffices to prove that if λ < λ 0 , then

1 2 L λ g, g + 2 α - λ 1 , g 2 + α + λ 1 , g 2 + α - λ 2 , g 2 + α + λ 2 , g 2 + 1 λ Z λ , g 2 + Z, g 2 g 2 E . Let a - 1 := α - λ , g , a + 1 := α + λ , g , a - 2 := α -, g , a + 2 := α + , g , b 1 := Z, ΛW -1 • 1 λ Z λ , g , b 2 := Z, ΛW -1 • Z, g and decompose g = a - 1 Y - λ + a + 1 Y + λ + a - 2 Y -+ a + 2 Y + + b 1 ΛW λ + b 2 ΛW + k. Using the fact that | α ± , Y ± λ | + | α ± λ , Y ± | + | 1 λ Z λ , Y | + | Z, Y λ | λ 2 , |a - 1 | + |a + 1 | + |a - 2 | + |a + 2 | + |b 1 | + |b 2 | g E , α -, Y + = α + , Y -= Z, Y = Y, ΛW = 0 we obtain α -, k 2 + α + , k 2 + α - λ , k 2 + α + λ , k 2 + Z, k 2 + 1 λ Z λ , k 2 λ 4 • g 2 E . (3.33)
Since L λ is self-adjoint, we can write

1 2 L λ g, g = 1 2 L λ k, k + L λ (a - 2 Y -+ a + 2 Y + + b 2 ΛW ), k + L λ (a - 1 Y - λ + a + 1 Y + λ + b 1 ΛW λ ), k + 1 2 L λ (a - 2 Y -+ a + 2 Y + + b 2 ΛW ), a - 2 Y -+ a + 2 Y + + b 2 ΛW + 1 2 L λ (a - 1 Y - λ + a + 1 Y + λ + b 1 ΛW λ ), a - 1 Y - λ + a + 1 Y + λ + b 1 ΛW λ + L λ (a - 2 Y -+ a + 2 Y + + b 2 ΛW ), a - 1 Y - λ + a + 1 Y + λ + b 1 ΛW λ . (3.34) It is easy to see that f (W )Y λ L 3 2 → 0, f (W )ΛW λ L 3 2 → 0, f (W λ )Y L 3 2 → 0 and f (W λ )ΛW
L λ Y -+ 2α + E * + L λ Y + + 2α - E * + L λ ΛW E * + L λ Y - λ + 2α + λ E * + L λ Y + λ + 2α - λ E * + L λ ΛW λ E * ---→ λ→0 0.
Plugging this into (3.34) and using (3.33) we obtain

1 2 L λ g, g ≥ -2a - 2 a + 2 -2a - 1 a + 1 + 1 2 L λ k, k -c g 2 E , (3.35)
where c → 0 as λ → 0. Applying (3.30) with r 1 = λ -1 2 , rescaling and using (3.33) we get, for λ small enough,

(1 -2c) ˆ|x|≤ √ λ |∇k| 2 dx + c ˆ|x|≥ √ λ |∇k| 2 dx - ˆR6 f (W λ )|k| 2 dx ≥ -c g 2 E .
(3.36)

From (3.31) with r 2 = √ λ we have (1 -2c) ˆ|x|≥ √ λ |∇k| 2 dx + c ˆ|x|≤ √ λ |∇k| 2 dx - ˆR6 f (W )|k| 2 dx ≥ -c g 2 E . (3.37)
Taking the sum of (3.36) and (3.37), and using (3.35), we obtain

1 2 L λ g, g ≥ -2a - 2 a + 2 -2a - 1 a + 1 + c k 2 E -2 c g 2 E .
The conclusion follows if we take c small enough.

Definition of the mixed energy-virial functional

Lemma 3.10. For any c > 0 and R > 0 there exists a radial function q(x) = q c,R (x) ∈ C 3,1 (R 6 ) with the following properties:

(P1) q(x) = 1 2 |x| 2 for |x| ≤ R, (P2) there exists R > 0 (depending on c and R) such that q(x) ≡ const for |x| ≥ R, (P3) |∇q(x)| |x| and |∆q(x)| 1 for all x ∈ R 6 , with constants independent of c and R,

(P4) 1≤i,j≤6 ∂ x i x j q(x) v i v j ≥ -c 6 i=1 v 2 i , for all x ∈ R 6 , v i ∈ R, (P5) ∆ 2 q(x) ≤ c • |x| -2 , for all x ∈ R 6 .
Remark 3.11. We require C 3,1 regularity in order not to worry about boundary terms in Pohozaev identities, see the proof of (3.41).

Proof. It suffices to prove the result for R = 1 since the function q R (x) := R 2 q( x R ) satisfies the listed properties if and only if q(x) does.

Let r denote the radial coordinate. Define q 0 (x) by the formula q 0 (r) :=

1 2 • r 2 r ≤ 1 8 5 • r -3 2 + 1 2 • r -2 -1 10 • r -4 r ≥ 1.
A direct computation shows that for r > 1 we have q 0 (r) = 8 5 -r -3 + 2 5 r -5 , q 0 (r) = 3r -4 -2r -6 > 0 (so q 0 (x) is convex), q 0 (r) = 12(-r -5 + r -7 ) and ∆ 2 q 0 (r) = -24r -3 . Hence q 0 satisfies all the listed properties except for (P2). We correct it as follows.

Let e j (r) := 1 j! r j • χ(r) for j ∈ {1, 2, 3} and let R 0 1. We define q(r) := q 0 (r)

r ≤ R 0 q 0 (R 0 ) + 3 j=1 q (j) 0 (R 0 ) • R j 0 • e j (-1 + R -1 0 r) r ≥ R 0 .
Note that q 0 (R 0 ) ∼ 1, q 0 (R 0 ) ∼ R -4 0 and q 0 (R 0 ) ∼ R -5 0 . It is clear that q(x) ∈ C 3,1 (R 6 ). Property (P1) holds since R 0 > 1. By the definition of the functions e j we have q(r) = q 0 (R 0 ) = const for r ≥ 3R 0 , hence (P2) holds with R = 3R 0 . From the definition of q(r) we get |∂ r q(r)| 1 and |∂ 2 r q(r)| R -4 0 for r ≥ R 0 , with a constant independent of R 0 , which implies (P3). Similarly,

|∂ x i x j q(x)| R -1 0 for |x| ≥ R 0 , which implies (P4) if R 0 is large enough. Finally |∆ 2 q(x)| R -3 0 for |x| ≥ R 0 and ∆ 2 q(x) = 0 for |x| ≥ 3R 0 . This proves (P5) if R 0 is large enough.
In the sequel q(x) always denotes a function of class C 3,1 (R 6 ) verifying (P1)-(P5) with sufficiently small c and sufficiently large R.

For λ > 0 we define the operators A(λ) and A 0 (λ) as follows:

[A(λ)h](x) := 1 3λ ∆q( x λ )h(x) + ∇q( x λ ) • ∇h(x), [A 0 (λ)h](x) := 1 2λ ∆q( x λ )h(x) + ∇q( x λ ) • ∇h(x). (3.38)
Combining these definitions with the fact that q(x) is an approximation of 1 2 |x| 2 we see that A(λ) and A 0 (λ) are approximations (in a sense not yet precised) of 1 λ Λ and 1 λ Λ 0 respectively. We will write A and A 0 instead of A(1) and A 0 (1) respectively. Note the following scalechange formulas, which follow directly from the definitions:

∀h ∈ Ḣ1 : A(λ)(h λ ) = (Ah) λ , A 0 (λ)(h λ ) = (A 0 h) λ . (3.39)
Lemma 3.12. The operators A(λ) and A 0 (λ) have the following properties:

• the families {A(λ) : λ > 0}, {A 0 (λ) : λ > 0}, {λ∂ λ A(λ) : λ > 0} and {λ∂ λ A 0 (λ) : λ > 0} are bounded in L ( Ḣ1 ; L 2 ), with the bound depending on the choice of the function q(x),

• for all h 1 , h 2 ∈ X 1 and λ > 0 there holds

A(λ)h 1 , f (h 1 + h 2 ) -f (h 1 ) -f (h 1 )h 2 = -A(λ)h 2 , f (h 1 + h 2 ) -f (h 1 ) , (3.40)
• for any c 0 > 0, if we choose c in Lemma 3.10 small enough, then for all h ∈ X 1 × H 1 there holds

A 0 (λ)h, ∆h ≤ c 0 λ h 2 Ḣ1 - 1 λ ˆ|x|≤Rλ |∇h(x)| 2 dx. (3.41)
• assuming (2.7) and (3.10), for any c 0 > 0 there holds

Λ 0 ΛW λ(t) -A 0 (λ(t))ΛW λ(t) L 2 ≤ c 0 , (3.42)
φ(t) + b(t) • A(λ(t))ϕ(t) L 3 ≤ c 0 , (3.43) ˆ1 6 ∆q x λ (f (ϕ + g) -f (ϕ))g dx - ˆf (W λ )g 2 dx ≤ c 0 C 2 0 e -3κ|t| . (3.44)
provided that the constant R in the definition of q(x) is chosen large enough.

Proof. Since ∇q(x) and ∇ 2 q(x) are continuous and of compact support, it is clear that A and A 0 are bounded operators Ḣ1 → L 2 . From the invariance (3.39) we see that A(λ) and A 0 (λ) have the same norms as A and A 0 respectively. For λ∂ λ A(λ) and λ∂ λ A 0 (λ) the proof is similar. We compute

∂ λ A(λ) = - 1 3λ 2 ∆q x λ - 1 3λ 3 x • ∇∆q x λ - 1 λ 2 x • ∇ 2 q x λ • ∇.
Since ∇q(x), ∇ 2 q(x) and ∇ 3 q(x) are continuous and of compact support, boundedness follows.

In (3.40) both sides are continuous for the X 1 topology, hence we may assume that h 1 , h 2 ∈ C ∞ 0 . We may also assume without loss of generality that λ = 1. Observe that for any h ∈ C ∞ 0 there holds h

• f (h) = 3 • F (h) and ∇h • f (h) = ∇F (h), hence Ah, f (h) = ˆ 1 3 ∆q • h + ∇q • ∇h f (h) dx = ˆ∆q • F (h) + ∇q • ∇F (h) dx = 0.
Using this for h = h 1 + h 2 and for h = h 1 , (3.40) is seen to be equivalent to

Ah 2 , f (h 1 ) + Ah 1 , f (h 1 )h 2 = 0. (3.45)
Expanding the left side using the definition of A we obtain

Ah 2 , f (h 1 ) + Ah 1 , f (h 1 )h 2 = ˆ1 3 ∆q • h 2 • f (h 1 ) + ∇q • ∇h 2 • f (h 1 ) dx + ˆ1 3 ∆q • h 1 • f (h 1 ) • h 2 + ∇q • ∇h 1 • f (h 1 ) • h 2 dx
Integrating by parts the term containing ∇h 2 and using the formulas

h 1 • f (h 1 ) = 2f (h 1 )
and ∇h 1 • f (h 1 ) = ∇f (h 1 ), this can be rewritten as

h 2 , 1 3 ∆q • f (h 1 ) -∆q • f (h 1 ) -∇q • ∇f (h 1 ) + 2 3 ∆q • f (h 1 ) + ∇q • ∇f (h 1 ) = 0,
which proves (3.45).

Inequality (3.41) follows easily from (P1), (P4) and (P5), once we check the following identity (valid in any dimension N , and used here for N = 6):

ˆ∆h(x) • 1 2λ ∆q x λ h(x) + ∇q x λ • ∇h(x) dx = - 1 4λ ˆ(∆ 2 q) x λ h(x) 2 dx - 1 λ ˆN i,j=1 ∂ ij q x λ ∂ i h(x)∂ j h(x) dx. (3.46)
Without loss of generality we can assume that λ = 1 (it suffices to replace q by its rescaled version). By a density argument, we can also assume that q, h ∈ C ∞ 0 (we use here the fact that q ∈ C 3,1 ), and (3.46) follows from integration by parts:

ˆ1 2 ∆h • ∆q • h + ∆h • ∇q • ∇h dx = ˆN i,j=1 1 2 ∂ ii h • ∂ jj q • h + ∂ ii h • ∂ j q • ∂ j h dx = ˆ-1 2 i,j ∂ i h(∂ jj q∂ i h + ∂ ijj q • h) + i 1 2 ∂ i ((∂ i h) 2 )∂ i q + i =j - 1 2 ∂ j (∂ i h) 2 ∂ j q -∂ ij q∂ i h∂ j h dx = ˆ-1 2 i,j ∂ jj q(∂ i h) 2 + 1 2 ∂ iijj q • h 2 - 1 2 i ∂ ii q(∂ i h) 2 + 1 2 i =j ∂ jj q(∂ i h) 2 - i =j ∂ ij q∂ i h∂ j h dx = ˆ-1 4 i,j ∂ iijj q • h 2 - i,j ∂ ij q∂ i h∂ j h dx.
Estimate (3.42) is invariant by rescaling, hence we can assume that λ = 1. For |x| ≤ R we have A 0 ΛW (x) = Λ 0 ΛW (x). From (P3) in Lemma 3.10 we get

|A 0 ΛW (x)| + |Λ 0 ΛW (x)| |x| -4 for |x| ≥ R, with a constant independent of R. Thus Λ 0 ΛW -A 0 ΛW L 2 ≤ c 0 if R is large enough. A similar reasoning yields ΛW λ -A(λ)W λ L 3 λ -1 as R → +∞. Since b(t) ∼ λ(t), this gives φ + bA(λ)W λ L 3 ≤ c 0 3 , if R is large enough. (3.47)
From (P2) in Lemma 3.10 it follows that supp(A(λ

)W µ ) ⊂ B(0, R•λ). Since A(λ)W µ L ∞ q 1 λ , we have bA(λ)W µ L 3 ≤ c 0 3 , if |T 0 | is large enough. (3.48)
To finish the proof, we have to check that

bA(λ) χ • λ(t) 2 P λ(t) + b(t) 2 Q λ(t) L 3 ≤ c 0 3 , if |T 0 | is large enough. (3.49)
We have the bound (χ + |∇χ|)

P λ L 3 ´2 λ 0 1 r 6 r 5 dr | log λ| (similarly with Q λ ), hence (χ + |∇χ|)(λ 2 P λ + b 2 Q λ ) L 3 (λ 2 + b 2 )| log λ| 1 as |T 0 | → +∞. We have also ∇(λ 2 P λ + b 2 Q λ ) L 3 1 λ (λ 2 +b 2 )
1. Since q is smooth and constant at infinity, we have b• 1 λ ∆q x λ + ∇q x λ 1. The constant depends on the choice of the function q, but this is not a concern here. We obtain

bA(λ) χ • λ(t) 2 P λ(t) + b(t) 2 Q λ(t) L 3 b • 1 λ ∆q x λ + ∇q x λ • • (χ + |∇χ|)(λ 2 P λ + b 2 Q λ ) L 3 + ∇(λ 2 P λ + b 2 Q λ ) L 3 1, hence (3
(f (ϕ + g) -f (ϕ))g dx - ˆ1 6 ∆q x λ (f (W µ ) + f (W λ ))g 2 dx e -3κ|t| .
Since ∇q is of compact support, we have ´1 6 ∆q x λ W µ g 2 dx g 2 Ḣ1 . As in the proof of (3.43), on can show that 1 6 ∆q x λ f (W λ ) -f (W λ ) L 3 → 0 as R → +∞. This finishes the proof.

For t ∈ [T, T 0 ] we define:

• the nonlinear energy functional

I(t) := ˆ1 2 | ġ(t)| 2 + 1 2 |∇g(t)| 2 -F (ϕ(t) + g(t)) -F (ϕ(t)) -f (ϕ(t))g(t) dx = E(ϕ(t) + g(t)) -E(ϕ(t)) -DE(ϕ(t), g(t)) ,
• the localized virial functional

J(t) := ˆġ(t) • A 0 (λ(t))g(t) dx,
• the mixed energy-virial functional

H(t) := I(t) + b(t)J(t).
From (2.16) we have

I(t) - 1 2 D 2 E(ϕ(t))g(t), g(t) g(t) 3 E .
Note that H(t) depends on the function q(x) used in the definition of A 0 (λ), see (3.38). From the first statement in Lemma 3.12 we deduce that

|J(t)| q g(t) 2 E ,
where the constant in the inequality depends on the choice of the function q(x). Thus (2.9) and (3.10) imply that for t ≤ T 0 with |T 0 | large enough there holds

H(t) - 1 2 D 2 E(ϕ(t))g(t), g(t) ≤ c g(t) 2 E , (3.50)
with c > 0 as small as we wish.

Energy estimates via the mixed energy-virial functional

Lemma 3.13. Let c 1 > 0. If C 0 is sufficiently large, then there exists a function q(x) and T 0 < 0 with the following property. If T 1 < T 0 and (2.7), (3.10), (3.11) hold for t ∈ [T, T 1 ], then for t ∈ [T, T 1 ] there holds

H (t) ≤ c 1 • C 2 0 • e -3κ|t| . (3.51)
This lemma is the key step in proving Proposition 3.3. We will postpone its slightly technical proof.

Proof of Proposition 3.3 assuming Lemma 3.13. We first show (3.13). From (3.15) and (3.10) we obtain (cC 2 0 + 1)e -3κ|t| , Eventually enlarging C 0 , we obtain (3.12), if c in (3.54) is taken sufficiently small.

|µ(t) -1| = |µ(t) -µ(T )| ˆt -∞ C 0 • e -3 2 κ 0 |τ | dτ C 0 • e -
b(t) 2 -κ 2 • λ(t) 2 µ(t) 2 C 0 • e -5 2 κ|t| . Since b(T ) = κ • λ(T ) and µ(T ) = 1, this yields b(t) 2 -κ 2 • λ(t) 2 µ(t) 2 C 0 • e -5 2 κ|t| . But b(t) + κ • λ(t) µ(t) ∼ e -κ|t| ,
Proof of Lemma 3.13. In this proof we say that a term is negligible if its contribution is

≤ c • C 2 0 • e -3κ|t| . We write Value 1 Value 2 if |Value 1 -Value 2 | ≤ c • C 2 0 • e -3κ|t|
. The order of choosing the parameters is the following: we will first choose q(x) independently of C 0 , then C 0 , which may depend on q(x), and finally |T 0 |.

Step 1 (Derivative of the energy functional) Using the definition of I(t), the conservation of energy, formulas (2.13), (3.9) and self-adjointness of D 2 E(ϕ) we compute:

I (t) = 0 -DE(ϕ), ∂ t ϕ -D 2 E(ϕ)∂ t ϕ, g -DE(ϕ), ∂ t g = -DE(ϕ), J • DE(ϕ) + ψ -J • DE(ϕ) + ψ, D 2 E(ϕ)g -DE(ϕ), J • (DE(ϕ + g) -DE(ϕ)) -ψ = -D 2 E(ϕ)ψ, g -DE(ϕ), J • (DE(ϕ + g) -DE(ϕ) -D 2 E(ϕ)g) = (∆ + f (ϕ))ψ, g -ψ, ġ -φ, f (ϕ + g) -f (ϕ) -f (ϕ)g .

The first term is

C 0 e -3κ|t| , due to (2.19) and (3.10), hence it is negligible (by enlarging C 0 if necessary). Inequality (2.18) implies that the second term can be replaced byb λ (λb) Λ 0 ΛW λ , ġ , which in turn can be replaced by -b(λ -b) A 0 (λ)ΛW λ , ġ , thanks to (3.42). From (3.43) we infer that the third term can be replaced by b• A(λ)ϕ, f (ϕ+g)-f (ϕ)-f (ϕ)g . Using formula (3.40) with h 1 = ϕ and h 2 = g we obtain

I (t) -b(λ -b) • A 0 (λ)ΛW λ , ġ -b • A(λ)g, f (ϕ + g) -f (ϕ) .
(3.55)

Step 2 (Derivative of the virial functional) We compute:

(bJ) (t) = b ˆġ • A 0 (λ)g dx + bλ ˆġ • ∂ λ A 0 (λ)g dx + b ˆġ • A 0 (λ)( ġ -ψ) dx + b ˆ(∆g + f (ϕ + g) -f (ϕ) -ψ) • A 0 (λ)g dx. (3.56)
The first two terms are negligible thanks to Lemma 3.12. Consider the third term on the right in (3.56). An integration by parts yields ´ġ • A 0 (λ) ġ dx = 0. Since A 0 (λ) : Ḣ1 → L 2 is a bounded operator, from (2.17) we see that

b ˆġ • A 0 (λ)ψ dx -b(λ -b) • A 0 (λ)ΛW λ , ġ .
Consider the forth term on the right in (3.56). The term b ´ψ • A 0 (λ)g dx is negligible. Using (3.41) and the fact that A 0 (λ) = A(λ)

+ 1 6λ ∆q( • λ ) we get b ˆ(∆g + f (ϕ + g) -f (ϕ)) • A 0 (λ)g dx ≤ b • A(λ)g, f (ϕ + g) -f (ϕ) - b λ ˆ|x|≤Rλ |∇g| 2 dx + b λ ˆ1 6 ∆q • λ (f (ϕ + g) -f (ϕ))g dx + cC 2 0 e -3κ|t| ,
with a small constant c. Putting everything back together and using (3.44) we get

(bJ) (t) ≤ -b(λ -b) • A 0 (λ)ΛW λ , ġ + b • A(λ)g, f (ϕ + g) -f (ϕ) + b λ - ˆ|x|≤Rλ |∇g| 2 dx + ˆf (W λ )g 2 dx + cC 2 0 e -3κ|t| .
Step 3 (Localized coercivity) Taking the sum of (3.55) and (3.56) we obtain

H (t) ≤ b λ - ˆ|x|≤Rλ |∇g| 2 dx + ˆf (W λ )g 2 dx + cC 2 0 e -3κ|t| .
Recall that

Shooting argument and passing to a limit

We are ready to give a proof of the main result of the paper, following a well-known scheme introduced in [START_REF] Merle | Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity[END_REF] and [START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF].

Proof of Theorem 1.

Step 1 Let t n be a decreasing sequence converging to -∞. For n large and

a 0 ∈ A := - 1 2 e -3 2 κ|Tn| , 1 2 e -3 2 κ|Tn| ,
let u a 0 n (t) : [t n , T + ) → E denote the solution of (1.1) with initial data (3.5). We will prove that there exists a 0 such that T + > T 0 and for u = u a 0 n inequalities (3.12), (3.13) hold for t ∈ [t n , T 0 ]. Suppose that this is not the case. For each a 0 ∈ A, let T 1 = T 1 (a 0 ) be the last time such that (3.12) and (3.13) hold for t ∈ [t n , T 1 ). Since {ϕ(t) : t ∈ [t n , T 1 ]} is a compact set, Corollary A.3 implies that T + > T 1 . Suppose that |a + 1 (T 1 )| < e -3 2 κ|T 1 | . Then Proposition 3.3 would imply that (3.12) and (3.13) hold on some neighborhood of T 1 , contradicting its definition. Thus a + 1 (T 1 ) = e -3 2 κ|T 1 | or a + 1 (T 1 ) = -e -3 2 κ|T 1 | . Let A + ⊂ A be the set of a 0 ∈ A which lead to a + 1 (T 1 ) = e -3 2 κ|T 1 | and let A -⊂ A be the set of a 0 ∈ A which lead to a + 1 (T 1 ) = -e -3 2 κ|T 1 | . We have proved that A = A + ∪ A -. We will show that A + and A -are open sets, which will lead to a contradiction since A is connected.

Let a 0 ∈ A + . This implies that there exists the first T 2 such that a + 1 (T 2 ) > 3 4 e -3 2 κ|T 2 | . Hence for a solution u(t) corresponding to a 0 close to a 0 we will have (by continuity of the flow) a + 1 (T 2 ) > Step 2 Call u n the solution found in Step 1. From (3.12), (3.13) and (2.11) we deduce that there exists a constant C 1 > 0 independent of n such that

u n (t) -(W 1 κ e -κ|t| -W, -e -κ|t| ΛW 1 κ e -κ|t| ) E ≤ C 1 • e -3 2 κ|t| , for t ∈ [t n , T 0 ]. (3.57)
The sequence u n (T 0 ) is bounded in E, hence its subsequence (which we still denote u n ) has a weak limit u 0 . Let u(t) we the solution of (1.1) with the initial data u(T 0 ) = u 0 . Let T < T 0 . In view of (3.57), for large n the sequence u n satisfies the assumptions of Corollary A.4 on the time interval [T, T 0 ], hence u n (T ) u(T ). Passing to the weak limit in (3.57) finishes the proof. Remark 3.14. Note that only the instability component a + 1 (t) is treated via a topological argument, whereas a + 2 (t) is estimated directly. This depends heavily on the (incidental) fact that the bootstrap bound e -3 2 κ|t| is asymptotically smaller than e -ν|t| . Were it not the case, we would have to use a topological argument based on the Brouwer fixed point theorem, as in the work of Côte, Martel and Merle [START_REF] Côte | Construction of multi-soliton solutions for the L 2supercritical gKdV and NLS equations[END_REF].

4 Bubble-antibubble for the radial critical Yang-Mills equation

Notation

In this section we denote v 2 L 2 (r 1 ≤r≤r 2 ) := 2π ´r2 r 1 |v(r)| 2 rdr. If r 1 or r 2 is not precised, then it should be understood that r 1 = 0, resp. r 2 = +∞. The corresponding scalar product is denoted v, w := 2π

´+∞ 0 v(r) • w(r) rdr. Recall that v 2 H := 2π ´+∞ 0 |∂ r v(r)| 2 + | 2 r v(r)| 2 rdr. A change of variables shows that v(r) ∈ H ⇔ v(e x ) ∈ H 1 (R), in particular v L ∞
v H (this change of variables, very helpful in proving coercivity lemmas, can be found in [START_REF] Gustafson | Schrödinger flow near harmonic maps[END_REF]). Another useful way of understanding the space H is to consider the transformation v(e iθ r) := e 2iθ v(r), which is an isometric embedding of H in Ḣ1 (R 2 ; R 2 ), whose image is given by 2-equivariant functions in Ḣ1 (R 2 ; R 2 ). Let H * be the dual space of H for the pairing •, • . The embedding just described identifies H * with the 2-equivariant distributions in Ḣ-1 (R 2 ; R 2 ).

We denote X 0 := L 2 ∩ H and X 1 := {v ∈ H : ∂ r v ∈ H and 1 r v ∈ H}. The generators of the H-critical and the L 2 -critical scale change will be denoted respectively Λ := r∂ r and Λ 0 := 1 + r∂ r .

Linearized equation and formal computation

Linearizing -∂ 2 r u -1 r ∂ r u + 4 r 2 u(1 -u)(1 -1 2 u
) around u = W we obtain the operator

L := -∂ 2 r - 1 r ∂ r - 1 r 2 (2 -6(W (r) -1) 2 ) = -∂ 2 r - 1 r ∂ r + 1 r 2 (4 -6ΛW ).
We study solutions behaving like u(t) -W + W λ(t) with λ(t) → 0 as t → -∞. As in Subsection 2.2, we expand

u(t) = -W + U (0) λ(t) + b(t) • U (1) λ(t) + b(t) 2 • U (2) λ(t) ,
with b(t) = λ (t), U (0) := (W, 0) and U (1) := (0, -ΛW ). This gives

∂ 2 t u(t) = -b (t)(ΛW ) λ(t) + b(t) 2 λ(t) (Λ 0 ΛW ) λ(t) + lot. (4.1)
Let us restrict our attention to the region r ≤ λ(t). We will see that the region r ≥ λ(t) will not have much influence on the dynamics of our system. For r ≤

√ λ we have W W λ , hence 4u(1 -u)(1 - 1 2 u) 4W λ (1 -W λ )(1 - 1 2 W λ ) + (-4 + 6ΛW λ )W + lot. Since ∂ 2 r W + 1 r ∂ r W 4 r 2 W + lot for r ≤ √ λ, we get ∂ 2 r u + 1 r ∂ r u - 4 r 2 u(1 -u)(1 - 1 2 u) = - b 2 λ (LU (2) ) λ - 1 r 2 6W λ • W + lot.
We can further simplify this using the fact that W (r) 2r 2 :

∂ 2 r u + 1 r ∂ r u - 4 r 2 u(1 -u)(1 - 1 2 u) = - b 2 λ (LU (2) ) λ -12ΛW λ + lot, thus (4.1) yields LU (2) = -Λ 0 ΛW + λ b 2 (b -12λ)ΛW.
As in Subsection 2.2, we find that the best choice of the formal parameter equations is

λ (t) = b(t), b (t) = κ 2 λ(t), with κ := 2 √ 3.
Remark 4.1. The main term of the interaction is exactly cancelled by the term -b ΛW λ for our choice of the parameters. We have seen that it is not the case for the power nonlinearity and in the next section we will see that it is not the case either for the critical equivariant wave map equation.

Bounds on the error of the ansatz

Fix Z ∈ C ∞ 0 ((0, +∞)) such that ˆ+∞ 0 Z(r) • ΛW (r) dr r > 0. (4.2)
By a direct computation we find L r 4

(1+r 2 ) 2 = r 2 (3-r 2 ) (1+r 2 ) 3 = -Λ 0 ΛW . Adding a suitable multiple of ΛW (r), we obtain a rational function Q(r) such that

LQ = -Λ 0 ΛW, ˆZ(r) • Q(r) dr r = 0, Q(r) ∼ r 2 as r → 0, Q(r) ∼ 1 as r → +∞. (4.
3) For λ, µ satisfying (2.6) and (2.7) (naturally with κ = 2 √ 3) we define the approximate solution by the formula ϕ(t) := -W µ(t) + W λ(t) + S(t),

φ(t) := -b(t)ΛW λ(t) ,
where

b(t) := e -κ|T | + κ 2 ˆt T λ(τ ) µ(τ ) 2 dτ, for t ∈ [T, T 0 ], S(t) := χ • b(t) 2 Q λ(t) , for t ∈ [T, T 0 ].
From (4.3) we obtain

χ • Q λ H ˆ2/λ 0 1 • rdr 1 2 + 1 λ • ˆ2/λ 0 ((1 + r) -1 ) 2 rdr 1 2 |t| • e κ|t| , which implies that S(t) H e -3 2 κ|t| .
Since Z has compact support, (2.3) implies, for sufficiently small λ, ˆZλ • S(t) dr r = 0.

We denote f (u) := -4u(1 -u)(1 -1 2 u) and

ψ(t) = (ψ(t), ψ(t)) := ∂ t ϕ(t) -DE(ϕ(t)) = ∂ t ϕ(t) -φ(t), ∂ t φ(t) -(∂ 2 r ϕ(t) + 1 r ∂ r ϕ(t) + 1 r 2 f (ϕ(t))) . We have f (u) = 2 -6(u -1) 2 .
Remark 4.2. By a direct computation, f (W ) = -4 + 6ΛW . Thus the potential term of the linearized operator contains a non-localized part -4 and a localized part 6ΛW . These terms need to be treated in different ways. This is a known issue coming from the fact that f (u) is not really the nonlinearity, as it "hides" the linear part near the stable equilibria:

f (u) -4u near u = 0 and f (u) 4(2 -u) near u = 2.
Sometimes it is convenient to subtract the linear part from f , but here we work simultaneously near u = 0 and near u = 2, so probably it will be simpler to keep f as it is. A similar remark could be made in the case of the equivariant wave map equation.

ψ(t) -µ (t) 1 µ(t) ΛW µ(t) + (λ (t) -b(t)) 1 λ(t) ΛW λ(t) H e -3 2 κ|t| , (4.4) ψ(t) - b(t) λ(t) (λ (t) -b(t))Λ 0 ΛW λ(t) L 2 e -3 2 κ|t| , (4.5) (-∂ 2 r - 1 r ∂ r - 1 r 2 f (ϕ(t)))ψ(t) H * e -3 2 κ|t| . (4.6)
Proof. The proof of (4.4) is the same as the proof of (2.17). In order to prove (4.5), we treat separately the regions r ≤ √ λ and r ≥ √ λ. We will show that

1 r 2 f (ϕ) -f (W λ ) + 2 r 2 µ 2 f (W λ ) -b 2 f (W λ )Q λ L 2 (r≤ √ λ) e -3 2 κ|t| . (4.7)
Since f is a polynomial of degree 3, we have

f (ϕ) = f (W λ ) + f (W λ )(-W µ + S) + 1 2 f (W λ )(-W µ + S) 2 + 1 6 f (W λ )(-W µ + S) 3 .
We treat all the terms one by one. We have

W µ (r) -2 r 2 µ 2 r 4 . Since |f (W λ )| 1, this implies 1 r 2 f (W λ )(W µ -2r 2 ) L 2 ˆ√λ 0 r 4 • rdr 1 2 e -3 2 κ|t| .
Since r is small, we have S(r) = b 2 Q λ (r), and the corresponding term is subtracted in (4.7).

Next, notice that

|W µ | + |S| r 2 . Since |f (W λ )| 1, this implies 1 r 2 f (W λ )(-W µ + S) 2 L 2 ˆ√λ 0 1 r 2 • r 4 2 rdr 1 2 ∼ λ 3 2 e -3 2 κ|t| .
The last term is estimated in a similar way. This finishes the proof of (4.7). By a direct computation

∂ 2 r + 1 r ∂ r W µ -8 µ 2 r 2 , hence ∂ 2 r + 1 r ∂ r W µ - 8 µ 2 L 2 (r≤ √ λ) e -3 2 κ|t| . (4.8)
From (4.7), (4.8) and the definition of ϕ(t) we have

(∂ 2 r + 1 r ∂ r )ϕ + 1 r 2 f (ϕ) -- 2 µ 2 f (W λ ) - b 2 λ (LQ) λ - 8 µ 2 L 2 (r≤ √ λ)
e -3 2 κ|t| .

Using (4.3) and the relation 4 + f (W ) = 6ΛW we can rewrite this as

(∂ 2 r + 1 r ∂ r )ϕ + 1 r 2 f (ϕ) -- 12 µ 2 ΛW λ + b 2 λ Λ 0 ΛW λ L 2 (r≤ √ λ) e -3 2 κ|t| ,
which is equivalent to (4.5), restricted to the region r ≤ √ λ. Consider the region r ≥ √ λ. Developping f at 2 -W µ we get

f (ϕ) = f (2-W µ )+f (2-W µ )(W λ -2+S)+ 1 2 f (2-W µ )(W λ -2+S) 2 + 1 6 f (2-W µ )(W λ -2+S) 3 .
From this and the relations f

(2-W µ ) = -f (W µ ), f (2-W µ ) = f (W µ ), we obtain a pointwise bound |f (ϕ) + f (W µ ) + f (W µ )(2 -W λ )| |S| + |2 -W λ | 2 . (4.9) Since |S(r)| b 2 , we have 1 r 2 S L 2 (r≥ √ λ) b 2 ˆ+∞ √ λ r -4 rdr 1 2
e -3 2 κ|t| . (4.10)

There holds |2 -W λ (r)| 2 λ 4 r 4 , hence 1 r 2 |2 -W λ (r)| 2 L 2 (r≥ √ λ) λ 4 ˆ+∞ √ λ r -12 rdr 1 2 e -3 2 κ|t| . (4.11) Since |W µ -2| r 2 , there holds |f (W µ ) + 4| r 2 . We also have |2 -W λ | λ 2 r 2 and |(2 -W λ ) -2λ 2 r 2 | λ 4 r 4 , hence 1 r 2 f (W µ )(2 -W λ ) + 8λ 2 r 4 L 2 (r≥ √ λ) λ 2 r 2 + λ 4 r 6 L 2 (r≥ √ λ) λ 2 ˆ+∞ √ λ r -4 rdr 1 2 + λ 4 ˆ+∞ √ λ r -12 rdr 1 2
e -3 2 |κ|t .

(4.12)

Inserting (4.10), (4.11) and (4.12) into (4.9) we obtain

f (ϕ) + f (W µ ) - 8λ 2 r 4 L 2 (r≥ √ λ) e -3 2 κ|t| . (4.13) A direct computation shows that (∂ 2 r + 1 r ∂ r )W λ (r) = -8λ 2 r 4 + O( λ 4 r 6 + λ 8 r 10 ), hence (∂ 2 r + 1 r ∂ r )W λ + 8λ 2 r 4 L 2 (r≥ √ λ) e -3 2 κ|t| . (4.14) We have ∂ r S = χ b 2 • Q λ + χb 2 • (Q λ ) and ∂ 2 r S = χ b 2 • Q λ + 2χ b 2 • (Q λ ) + χb 2 • (Q λ ) . There holds |Q| 1, |Q | 1 r and |Q | 1 r 2 , which implies |Q λ | 1, |(Q λ ) | 1 r and |(Q λ ) | 1 r 2 . This gives χb 2 • (Q λ ) L 2 (r≥ √ λ) + 1 r χb 2 • (Q λ ) L 2 (r≥ √ λ) b 2 ˆ+∞ √ λ r -4 rdr 1 2 b 2 √ λ e -3 2 κ|t| , χ b 2 • (Q λ ) L 2 (r≥ √ λ) + χ b 2 • Q λ L 2 (r≥ √ λ) + 1 r χ b 2 • Q λ L 2 (r≥ √ λ) b 2 e -3 2 κ|t| , (4.15) hence (∂ 2 r + 1 r ∂ r )S L 2 (r≥ √ λ)
e -3 2 κ|t| . Together with (4.13) and (4.14) this proves that

(∂ 2 r + 1 r ∂ r )ϕ + 1 r 2 f (ϕ) L 2 (r≥ √ λ) e -3 2 κ|t| . Since ΛW λ L 2 (r≥ √ λ) + Λ 0 ΛW λ L 2 (r≥ √ λ) ( ´+∞ 1/ √ λ 1 r 4 rdr) 1 2
√ λ, the other terms appearing in (4.5) are e -3 2 κ|t| . This finishes the proof of (4.5). The proof of (4.6) is very similar to the proof of (2.19), hence we will just indicate the differences. Since

W λ L ∞ + W µ L ∞ 1, we have |f (-W µ + W λ ) -f (W λ )| W µ and |f (-W µ + W λ ) -f (W µ )| = |f (W µ + (2 -W λ )| -f (W µ )| 2 -W λ . Next, we check that 1 r 2 W µ • ΛW λ L 1 e -3 2 κ|t| (recall that H ⊂ L ∞ , hence L 1 (R 2 ) ⊂ H * ).
To do this, we consider separately r ≤ 1 and r ≥ 1:

1 r 2 W µ • ΛW λ L 1 (r≤1) ΛW λ L 1 (r≤1) = λ 2 ΛW L 1 (r≤1/λ) λ 2 | log λ| e -3 2 κ|t| , 1 r 2 W µ • ΛW λ L 1 (r≥1) ΛW λ L ∞ (r≥1) = ΛW 1 λ ∼ λ 2 e -3 2 κ|t| .
Finally, we check that 1 r 2 (2 -W λ ) • ΛW µ L 1 e -1 2 κ|t| , again dividing into r ≤ 1 and r ≥ 1:

2 -W λ L 1 (r≤1) = 1 1 + (r/λ) 2 L 1 (r≤1) = λ 2 1 1 + r 2 L 1 (r≤1/λ) ∼ λ 2 | log λ| e -3 2 κ|t| , 2 -W λ L 1 (r≥1) = 2λ 2 1 + λ 2 e -3 2 κ|t| .
This allows to conclude, since

f (ϕ) -f (-W µ + W λ ) L ∞ S L ∞ e -3 2 κ|t| .

Modulation

Having defined the approximate solution ϕ(t), we will now analyse exact solutions close to ϕ(t). The initial data are

u(T ) = -W + W 1 κ e -κ|T | , -e -κ|T | ΛW 1 κ e -κ|T |
(there is no linear instability in the case of the Yang-Mills equation).

Similarly as in Subsection 3.2, we choose the modulation parameters λ(t) and µ(t) which verify ˆZµ

• u(t) -(-W µ(t) + W λ(t) ) dr r = 0, ˆZλ • u(t) -(-W µ(t) + W λ(t) ) dr r = 0.
We define g(t) by u(t) = ϕ(t) + g(t).

It satisfies, cf. (3.14),

1 λ(t) Z λ(t) , g(t) = 0, 1 µ(t) Z µ(t) , g(t) ≤ c • e -3 2 κ|t| .
The functions λ(t) and µ(t) are C 1 and

|λ (t) -b(t)| + |µ (t)| g(t) 2 E + c • e -3 2 κ|t|
with c > 0 arbitrarily small, cf. (3.15).

Coercivity

Recall that f (W ) = -4 + 6ΛW . In the next lemma, it is useful to separate these two terms, see Remark 4.2.

Lemma 4.4. There exist constants c, C > 0 such that

• for all g ∈ H there holds ˆ+∞

0 (g ) 2 + 4 r 2 g 2 rdr - ˆ+∞ 0 6 r 2 ΛW g 2 rdr ≥ c ˆ+∞ 0 (g ) 2 + 4 r 2 g 2 rdr -C ˆ+∞ 0 Z • g dr r 2 , (4.16)
• if r 1 > 0 is large enough, then for all g ∈ H there holds

(1 -2c) ˆr1 0 (g ) 2 + 4 r 2 g 2 rdr + c ˆ+∞ r 1 (g ) 2 + 4 r 2 g 2 rdr - ˆ+∞ 0 6 r 2 ΛW g 2 rdr ≥ -C ˆ+∞ 0 Z • g dr r 2 , (4.17)
• if r 2 > 0 is small enough, then for all g ∈ H there holds

(1 -2c) ˆ+∞ r 2 (g ) 2 + 4 r 2 g 2 rdr + c ˆr2 0 (g ) 2 + 4 r 2 g 2 rdr - ˆ+∞ 0 6 r 2 ΛW g 2 rdr ≥ -C ˆ+∞ 0 Z • g dr r 2 , (4.18)
Proof. Let g(x) := g(e x ) and Z(x) := Z(e x ). One computes that f (W (e x )) = -4+6 sech 2 (x), hence (4.16) is equivalent to

ˆR( g ) 2 + (4 -6 sech 2 ) g 2 dx ≥ c ˆR ( g ) 2 + g 2 dx -C ˆR Z • g dx 2 . (4.19)
This quadratic form corresponds to the classical operatord 2 dx 2 +(4-6 sech 2 ), for which 0 is a simple discrete eigenvalue, with the eigenspace spanned by sech 2 . Decompose g = a sech 2 +g 1 , with ´sech 2 •g 1 dx = 0. From the Sturm-Liouville theory we obtain ˆR( g

) 2 + (4 -6 sech 2 ) g 2 dx = ˆR(g 1 ) 2 + (4 -6 sech 2 )g 2 1 dx ˆg2 1 rdr. Let sech 2 = b Z + (sech 2 ) ⊥ with ´ Z • (sech 2 ) ⊥ dx = 0. Since ´ Z • sech 2 dx > 0, see (4.2), we have ´R (sech 2 ) ⊥ 2 dx < ´R(sech 2 ) 2 dx, hence ˆg2
1 dx = a 2 ˆ(sech 2 ) 2 dx -2a ˆsech 2 • g dx + ˆ g 2 dx = a 2 ˆ(sech 2 ) 2 dx -2ab ˆ Z • g dx -2a ˆ(sech 2 ) ⊥ • g dx + ˆ g 2 dx ≥ c ˆ g 2 dx -C ˆR Z • g dx 2 ,
which implies (4.19).

With the same change of variable, (4.17) and (4.18) will follow once we prove that

(1 -2c) ˆ|x|≤R (g ) 2 + 4g 2 dx + c ˆ|x|≥R (g ) 2 + 4g 2 dx -ˆR 6 sech 2 g 2 dx ≥ -C ˆR Z • g dx 2 , (4.20)
provided that R is large enough. To this end, take χ(x) := χ 2x R and h := χ • g. Since Z has compact support and R is large, we have ´R Z • h dx = ´R Z • g dx. By a standard integration by parts we get ´R( h

) 2 dx = ´R χ 2 ( g ) 2 dx + ´R 1 2 ( χ ) 2 -χ χ g 2 dx. We notice that |( χ ) 2 -χ χ | R -2
is small, in particular for any c > 0 there holds ´R χ 2 ( g

) 2 dx ≥ ´R( h ) 2 dx -c 2 ´ ( g ) 2 + 4 g 2 x dx, if R is large enough. Applying (4.19
) with h instead of g and 3c instead of c we obtain

(1 -3c) ˆ|x|≤R ( g ) 2 + 4 g 2 dx -ˆR 6 sech 2 χ 2 g 2 dx ≥ (1 -3c) ˆR χ 2 ( g ) 2 + 4 g 2 dx -ˆR 6 sech 2 χ 2 g 2 dx ≥ (1 -3c) ˆR ( h ) 2 + 4 h 2 dx - c 2
ˆR ( g ) 2 + 4 g 2 dx -ˆR 6 sech 2 h 2 dx ≥ - c 2
ˆR ( g ) 2 + 4 g 2 dx -C ˆ Z • h dx 2 ≥ - c 2
ˆR ( g ) 2 + 4 g 2 dx -C ˆR Z • g dx 2 .
But 6 sech 2 ≤ 6 sech 2 χ 2 + 2c if R is large enough, and (4.20) follows.

Lemma 4.5. There exists a constant η > 0 such that if λ µ < η and U -(-W µ +W λ ) E < η, then for all g ∈ E there holds

D 2 E(U )g, g + ˆ1 λ Z λ • g dr r 2 + ˆ1 µ Z µ • g dr r 2 g 2 E .
The proof is a modification of the proof of Lemma 3.9 and will be skipped.

Definition of the mixed energy-virial functional

Lemma 4.6. For any c > 0 and R > 0 there exists a function q(r) = q c,R (r) ∈ C 3,1 ((0, +∞)) with the following properties:

(P1) q(r) = 1 2 r 2 for r ≤ R, (P2) there exists R > 0 (depending on c and R) such that q(r) ≡ const for r ≥ R, (P3) |q (r)| r and |q (r)| 1 for all r > 0, with constants independent of c and R, (P4) q (r) ≥ -c and 1 r q (r) ≥ -c, for all r > 0,

(P5) ( d 2 dr 2 + 1 r d dr ) 2 q(r) ≤ c • r -2
, for all r > 0, (P6) r q (r) r ≤ c, for all r > 0.

Proof. It suffices to prove the result for R = 1 since the function q R (r) := R 2 q( r R ) satisfies the listed properties if and only if q(r) does.

First we define q 0 (r) by the formula

q 0 (r) := 1 2 • r 2 r ≤ 1 1 2 r 2 + c 1 1 2 (r -1) 2 -log(r) r 2 -1 4 r ≥ 1,
with c 1 small. A direct computation shows that for r > 1 we have

q 0 (r) = r 1 - c 1 log r 2 + c 1 3 4 r -1 + 1 4r , q 0 (r) = 1 - c 1 log r 2 + c 1 1 4 - 1 4r 2 , q 0 (r) = -c 1 r 2 -1 2r 3 , d 2 dr 2 + 1 r d dr 2 q 0 (r) = - c 1 r 3 . (4.21)
In particular q 0 (1) = 1 2 , q 0 (1) = 1, q 0 (1) = 1 and q 0 (1) = 0, hence q 0 ∈ C 3,1 . Let R 0 := e 2/c 1 . From (4.21) it follows that q 0 (r) verifies all the listed properties except for (P2) for r ≤ R 0 . Let e j (r) := 1 j! r j • χ(r) for j ∈ {1, 2, 3}, where χ is a standard cut-off function. We define

q(r) := q 0 (r) r ≤ R 0 q 0 (R 0 ) + 3 j=1 q (j) 0 (R 0 ) • R j 0 • e j (-1 + R -1 0 r) r ≥ R 0 .
We will show that q(r) has all the required properties if c 1 is small enough. It is clear that q(x) ∈ C 3,1 (R 6 ). Indeed, it follows from (4.21) that |q

(j) 0 (R 0 )| c 1 R 2-j 0
for j ∈ {1, 2, 3}. For r > R 0 and k ∈ {1, 2, 3, 4} we have

q (k) (r) = 3 j=1 q (j) 0 (R 0 ) • R j-k 0 e (k) j (-1 + R -1 0 r) ⇒ |q (k) (r)| c 1 R 2-k 0 .
Since q(r) ≡ const for r ≥ 3R 0 , we obtain (P1)-(P6).

We define the operators A(λ) and A 0 (λ) as follows:

[A(λ)h](r) := q r λ • h (r), [A 0 (λ)h](r) := 1 2λ q r λ + 1 2r q r λ h(r) + q r λ • h (r).
Lemma 4.7. The operators A(λ) and A 0 (λ) have the following properties:

• the families {A(λ) : λ > 0}, {A 0 (λ) : λ > 0}, {λ∂ λ A(λ) : λ > 0} and {λ∂ λ A 0 (λ) : λ > 0} are bounded in L (H; L 2 ), with the bound depending on the choice of the function q(r),

• for all λ > 0 and h 1 , h 2 ∈ X 1 there holds

A(λ)h 1 , 1 r 2 f (h 1 + h 2 ) -f (h 1 ) -f (h 1 )h 2 + A(λ)h 2 , 1 r 2 f (h 1 + h 2 ) -f (h 1 ) + 4h 2 ≤ c 0 λ ( h 1 2 H + 1) h 2 2 H + h 2 4
H , (4.22) with a constant c 0 arbitrarily small,

• for all h ∈ X 1 there holds

A 0 (λ)h, ∂ 2 r + 1 r ∂ r - 4 r 2 h ≤ c 0 λ h 2 H - 2π λ ˆRλ 0 (∂ r h) 2 + 4 r 2 h 2 rdr, (4.23)
• assuming (2.7), for any c 0 > 0 there holds

Λ 0 ΛW λ(t) -A 0 (λ(t))ΛW λ(t) L 2 ≤ c 0 , (4.24) φ(t) + b(t) • A(λ(t))ϕ(t) L ∞ ≤ c 0 , (4.25) ˆ+∞ 0 1 2 q r λ + λ r q r λ 1 r 2 f (ϕ + g) -f (ϕ) + 4g g rdr - ˆ+∞ 0 1 r 2 f (W λ ) + 4 g 2 rdr ≤ c 0 C 2 0 e -3κ|t| , (4.26)
provided that the constant R in the definition of q(r) is chosen large enough.

Remark 4.8. The condition ∂ r h 1 , 1 r h 1 , ∂ r h 2 , 1 r h 2 ∈ H is required only to ensure that the left hand side of (3.40) is well defined, but it does not appear on the right hand side of the estimate. Note also that in (4.22), (4.23) and (4.26) we extract the linear part of f , see Remark 4.2.

Proof. The proof of the first point is the same as in Lemma 3.12.

In (4.22), without loss of generality we may assume that h 1 , h 2 ∈ C ∞ 0 ((0, +∞)) and that λ = 1. From the definition of A(λ) we have

A(λ)h 1 , 1 r 2 f (h 1 + h 2 ) -f (h 1 ) -f (h 1 )h 2 + A(λ)h 2 , 1 r 2 f (h 1 + h 2 ) -f (h 1 ) + 4h 2 = ˆ+∞ 0 q h 1 • 1 r 2 f (h 1 + h 2 ) -f (h 1 ) -f (h 1 ) + q h 2 • 1 r 2 f (h 1 + h 2 ) -f (h 1 ) + 4h 2 rdr = ˆ+∞ 0 q r d dr F (h 1 + h 2 ) - d dr F (h 1 ) -h 2 • d dr f (h 1 ) -f (h 1 ) • d dr h 2 + 2 d dr (h 2 2 ) dr = ˆ+∞ 0 r q r • 1 r 2 F (h 1 + h 2 ) -F (h 1 ) -f (h 1 ) • h 2 + 2h 2 2 rdr.
Using (P6) in Lemma 4.6 and the elementary inequality

|F (h 1 + h 2 ) -F (h 1 ) -f (h 1 )h 2 | |h 1 | 2 |h 2 | 2 + |h 1 | 4 we get (4.22).
Note that as a part of this computation, we obtain

A(λ)h, 1 r 2 h ≤ c 0 λ ˆ+∞ 0 1 r 2 h 2 rdr. (4.27)
For r ≤ Rλ there holds 1 2λ q r λ + 1 2r q r λ = 1 λ and for all r, thanks to (P4), there holds

1 2λ q r λ + 1 2r q r λ ≥ -c 0 λ , hence 1 2λ q r λ + 1 2r q r λ h, 1 r 2 h ≥ 2π λ ˆRλ 0 1 r 2 h 2 rdr - c 0 λ ˆ+∞ 0 1 r 2 h 2 rdr. (4.28)
Taking the sum of (4.27) and (4.28) we obtain

A 0 (λ)h, 1 r 2 h ≥ - c 0 λ ˆ+∞ 0 1 r 2 h 2 rdr + 2π λ ˆRλ 0 1 r 2 h 2 rdr (4.29)
(c 0 has changed, but is still small).

Using identity (3.46) with N = 2 we obtain, cf. the proof of (3.41),

A 0 (λ)h, ∂ 2 r + 1 r ∂ r h ≤ c 0 λ ˆ+∞ 0 (∂ r h) 2 rdr - 2π λ ˆr≤Rλ (∂ r h) 2 rdr. (4.30)
Taking the difference of (4.30) and (4.29) we obtain (3.41). The proofs of (4.24) and (4.25) are similar to the proofs of (3.42) and (3.43) respectively. Instead of (3.48) we prove that bA(λ)W µ L ∞ ≤ c 0 3 , which follows from (P2) and (P3). We skip the details.

The proof of (4.26) is close to the proof of (3.44). Note that it is crucial that f (W λ ) + 4 vanishes at infinity. For t ∈ [T, T 0 ] we define:

• the nonlinear energy functional

I(t) := ˆ1 2 | ġ(t)| 2 + 1 2 |∇g(t)| 2 - 1 r 2 F (ϕ(t) + g(t)) -F (ϕ(t)) -f (ϕ(t))g(t) dx = E(ϕ(t) + g(t)) -E(ϕ(t)) -DE(ϕ(t), g(t)) ,
• the localized virial functional

J(t) := ˆġ(t) • A 0 (λ(t))g(t) dx,
• the mixed energy-virial functional H(t) := I(t) + b(t)J(t).

Energy estimates via the mixed energy-virial functional

The remaining part of the proof is almost identical to Subsection 3.5. We will indicate the few differences.

Instead of (3.55), we obtain now

I (t) -b(λ -b) • A 0 (λ)ΛW λ , ġ -b • A(λ)g, 1 r 2 f (ϕ + g) -f (ϕ) + 4g .
As in the proof of Lemma 3.13, we have

(bJ) (t) b ˆġ • A 0 (λ)( ġ -ψ) rdr + b ˆ ∂ 2 r + 1 r ∂ r g + 1 r 2 f (ϕ + g) -f (ϕ) -ψ • A 0 (λ)g rdr = b ˆġ • A 0 (λ)( ġ -ψ) rdr + b ˆ ∂ 2 r + 1 r ∂ r - 4 r 2 g + 1 r 2 f (ϕ + g) -f (ϕ) + 4g -ψ • A 0 (λ)
g rdr, where we recognize the terms appearing in (4.23) and (4.26). The rest of the proof applies without change. Theorem 3 follows from the argument given in Subsection 3.6.

5 Bubble-antibubble for the equivariant critical wave map equation

Notation

We use similar notation as in Section 4, with a slight modification in the definition of the norm H:

v 2 H := 2π ˆ+∞ 0 |∂ r v(r)| 2 + | k r v(r)| 2 rdr.
The transformation v(e iθ r) := e kiθ v(r) is an isometric embedding of H in Ḣ1 (R 2 ; R 2 ), whose image is given by k-equivariant functions in Ḣ1 (R 2 ; R 2 ).

Linearized equation and formal computation

Linearizing -∂ 2 r u -1 r ∂ r u + k 2 2r 2 sin(2u) around u = W we obtain the operator

L := -∂ 2 r - 1 r ∂ r + k 2 r 2 cos(4 arctan(r k )) = -∂ 2 r - 1 r ∂ r + k 2 r 2 1 - 8 (r k + r -k ) 2 .
It has a one-dimensional kernel spanned by ΛW . We fix Z ∈ C ∞ 0 ((0, +∞)) such that ˆ+∞

|U (r)| r k , |∂ r U (r)| r k-1 , |∂ 2 r U (r)| r k-2
for r small, (5.2)

|U (r)| r -k , |∂ r U (r)| r -k-1 , |∂ 2 r U (r)| r -k-2 for r large, (5.3) ˆZ(r) • U (r) dr r = 0. (5.4)
Proof. It is easy to check that the operator L factorizes as follows:

L = -∂ 2 r - 1 r ∂ r + k 2 r 2 1 - 8 (r k + r -k ) 2 = -∂ r - 1 r - ΛW (r) ΛW (r) ∂ r - ΛW (r) ΛW (r) , (5.5)
hence we can invert it explicitely using twice the variation of constants formula. Define

U 1 ∈ C ∞ ((0, +∞)) by U 1 (r) := 1 rΛW (r) ˆr 0 V (ρ)ΛW (ρ)ρ dρ.
It solves the equation

-∂ r -1 r -ΛW (r) ΛW (r) U 1 (r) = V (r).
Since |V (r)| r k and ΛW (r) ∼ r k for small r, we have

|U 1 (r)| r -1-k+k+k+1+1 = r k+1 , small r. (5.6)
From the crucial assumption ´+∞ 0 V (ρ)ΛW (ρ)ρ dρ = 0 we get

|U 1 (r)| = 1 rΛW (r) ˆ+∞ r V (ρ)ΛW (ρ)ρ dρ r -k+1 , large r. ( 5
(r)| r k , |∂ r ΛW (r)| r k-1 , |∂ 2 r ΛW (r)| r k-2 for small r and |ΛW (r)| r -k , |∂ r ΛW (r)| r -k-1 , |∂ 2 r ΛW (r)| r -k-2
for large r, (5.2) and (5.3) still hold.

We study solutions behaving like u(t) -W + W λ(t) with λ(t) → 0 as t → -∞. We expand

u(t) = -W + U (0) λ(t) + b(t) • U (1) λ(t) + b(t) 2 • U (2) λ(t) ,
with b(t) = λ (t), U (0) := (W, 0) and U (1) := (0, -ΛW ). As in Subsection 4.2, in the region r ≤ √ λ we arrive at

∂ 2 r u + 1 r ∂ r u - k 2 2r 2 sin(2u) = - b 2 λ (LU (2) ) λ - 1 r 2 • 8k 2 ((r/λ) k + (r/λ) -k ) 2 W + lot.
Using the fact that W (r) ∼ 2r k for small r we obtain

∂ 2 r u + 1 r ∂ r u - k 2 2r 2 sin(2u) = - b 2 λ (LU (2) ) λ - 16k 2 r k-2 ((r/λ) k + (r/λ) -k ) 2 + lot, thus, after rescaling, LU (2) = -Λ 0 ΛW + λ b 2 b ΛW -λ k-1 • 16k 2 r k-2 (r k + r -k ) 2 .
It is not difficult to check (using for example the residue theorem) that

ˆ+∞ 0 ΛW (r) • 16k 2 r k-2 (r k + r -k ) 2 rdr = 4k 2 π sin π k • ˆΛW (r) 2 rdr,
hence the correct choice (that is, such that Lemma 5.1 allows to invert L) of the formal parameter equations is

λ (t) = b(t), b (t) = 4k 2 π sin π k λ(t) k-1 = k 2 2κ k -2 k λ(t) k-1 , where κ := k-2 2 • 8k π sin π k 1 k
. This system has a solution

(λ app (t), b app (t)) = k -2 2κ (κ|t|) -2 k-2 , (κ|t|) -k k-2 , t ≤ T 0 < 0.

Bounds on the error of the ansatz

Let I = [T, T 0 ] be the time interval, T ≤ T 0 < 0, |T 0 | large. Let λ(t) and µ(t) be C 1 functions on [T, T 0 ] such that λ(T ) = k -2 κ (κ|T |) -2 k-2 , µ(T ) = 1, k -2 2κ (κ|t|) -2 k-2 ≤ λ(t) ≤ 2(k -2) κ (κ|t|) -2 k-2 , 1 2 ≤ µ(t) ≤ 2. ( 5.8)
Let P (r) and Q(r) by the functions obtained in Lemma 5.1 for

V (r) = k 2 2κ k-2 k ΛW (r) - 16k 2 r k-2
(r k +r -k ) 2 and V (r) = -Λ 0 ΛW (r) respectively. We define the approximate solution by the formula ϕ(t

) := -W µ(t) + W λ(t) + S(t), φ(t) := -b(t)ΛW λ(t) .
where

b(t) := (κ|T |) -k k-2 + 2κ k -2 k ˆt T λ(τ ) k-1 µ(τ ) k dτ, for t ∈ [T, T 0 ], (5.9)
S(t) := λ(t) k µ(t) k P λ(t) + b(t) 2 Q λ(t) , for t ∈ [T, T 0 ].
The definition of b(t) and (5.8) yield

b(t) ∼ |t| -k k-2 ,
with a constant depending only on k. Since P, Q ∈ H, there holds

S(t) H |t| -2k k-2 .
(5.10) Note also that ˆZλ • S(t) dr r = 0.

We denote f (u) := -k 2 2 sin(2u) (hence f (u) = -k 2 cos(2u)) and

ψ(t) = (ψ(t), ψ(t)) := ∂ t ϕ(t) -DE(ϕ(t)) = ∂ t ϕ(t) -φ(t), ∂ t φ(t) -(∂ 2 r ϕ(t) + 1 r ∂ r ϕ(t) + 1 r 2 f (ϕ(t))
) . We point out that usually, in the context of equivariant wave maps, f (u) has the opposite sign. We chose the sign which is more coherent with the traditional notation for (1.1).

Lemma 5.2. Suppose that for t ∈ [T, T 0 ] there holds |λ (t)| |t| -k k-2 and |µ (t)| |t| -k k-2 . Then ψ(t) -µ (t) 1 µ(t) ΛW µ(t) + (λ (t) -b(t)) 1 λ(t) ΛW λ(t) H |t| -2k-1 k-2 , ψ(t) - b(t) λ(t) (λ (t) -b(t))Λ 0 ΛW λ(t) L 2 |t| -2k-1 k-2 , (5.11)
(-∂ 2 r - 1 r ∂ r - 1 r 2 f (ϕ(t)))ψ(t) H * |t| -2k-1 k-2 . (5.12)
Proof. From the definition of ψ we get

ψ + µ ΛW µ + (λ -b)ΛW λ = -k λ k µ k+1 µ P λ + k λ k µ k λ P λ - λ k µ k λ ΛP λ + 2λbb Q λ -b 2 λ ΛQ λ .
The first term has size |t|

-3k k-2 |t| -2k-1 k-2 in H. The other terms have size |t| -3k-2 k-2 |t| -2k-1 k-2 in H.
In order to prove (5.11), we treat separately the regions r ≤ √ λ and r ≥ √ λ. First we will show that

1 r 2 f (ϕ) -f (W λ ) + 2 r k µ k f (W λ ) - λ k µ k f (W λ )P λ -b 2 f (W λ )Q λ L 2 (r≤ √ λ) |t| -2k-1 k-2 . (5.13)
We have an elementary pointwise inequality

|f (ϕ) -f (W λ ) -f (W λ )(-W µ + S)| |-W µ + S| 2 , (5.14)
with a constant depending only on k. We have

|W µ | r k and |S| (b 2 + λ k ) • r λ k r k , hence 1 r 2 |-W µ + S| 2 L 2 (r≤ √ λ) ˆ√λ 0 1 r 2 • r 2k 2 rdr 1 2 ∼ λ 2k-1 2 |t| -2k-1 k-2 ,
which means that the right hand side in (5.14) is negligible. From the well-known fact that | arctan(z) -z| |z| 3 for small z we get

|W µ (r) -2(r/µ) k | r 3k . This implies 1 r 2 f (W λ ) W µ -2 r k µ k L 2 ˆ√λ 0 r 6k-4 rdr 1 2 λ 3k-1 2 |t| -3k-1 k-2 |t| -2k-1 k-2 .
This proves (5.13) (see the definition of S).

We have

W µ (r) -2kr k-1 µ k r 3k-1 and W µ (r) -2(k 2 -k)r k-2 µ k r 3k-2 for small r, which implies ∂ 2 r + 1 r ∂ r W µ (r) -2k 2 r k-2 µ k r 3k-2 , hence ∂ 2 r + 1 r ∂ r W µ - 2k 2 r k-2 µ k L 2 (r≤ √ λ) ˆ√λ 0 r 6k-4 rdr 1 2 |t| -2k-1 k-2 . (5.15)
Since W λ (r) + 1 r W λ (r) + 1 r 2 f (W λ (r)) = 0, from (5.13), (5.15) and the definition of ϕ(t) we have

(∂ 2 r + 1 r ∂ r )ϕ + 1 r 2 f (ϕ) -- 2k 2 r k-2 µ k - 2r k-2 µ k f (W λ ) - λ k-1 µ k (LP ) λ - b 2 λ (LQ) λ L 2 (r≤ √ λ)
e -3 2 κ|t| .

(5.16)

By the definition of P there holds

LP = -16k 2 r k-2 (r k +r -k ) 2 + k 2 2κ k-2 k ΛW = -2k 2 r k-2 -2r k-2 f (W )+ k 2 2κ k-2
k ΛW and by the definition of Q there holds LQ = -Λ 0 ΛW , hence we can rewrite (5.16) as

(∂ 2 r + 1 r ∂ r )ϕ + 1 r 2 f (ϕ) -- k 2 2κ k -2 k λ k-1 µ k ΛW λ + b 2 λ Λ 0 ΛW λ L 2 (r≤ √ λ) e -3 2 κ|t| , which is precisely (5.11) restricted to r ≤ √ λ, cf. (2.25). Consider the region r ≥ √ λ. We have ϕ = (π -W µ ) + (W λ -π) + S, hence elementary pointwise inequalities yield |f (ϕ) -f (π -W µ ) -f (π -W µ )(W λ -π + S)| |W λ -π + S| 2 .
From this and the relations f (π

-W µ ) = -f (W µ ), f (π -W µ ) = f (W µ ), we obtain a pointwise bound |f (ϕ) + f (W µ ) + f (W µ )(π -W λ )| |S| + |π -W λ | 2 .
(5.17)

Since |S(r)| b 2 + λ k µ k ∼ |t| -2k k-2 , we have 1 r 2 S L 2 (r≥ √ λ) |t| -2k k-2 ˆ+∞ √ λ r -4 rdr 1 2 |t| -2k k-2 √ λ |t| -2k-1 k-2 . (5.18)
There holds |π -

W λ (r)| 2 = |π -2 arctan((r/λ) k )| 2 = |2 arctan((λ/r) k )| 2 λ 2k r 2k , hence 1 r 2 |π -W λ (r)| 2 L 2 (r≥ √ λ) λ 2k ˆ+∞ √ λ r -4k-4 rdr 1 2 ∼ λ 2k λ -2k+1 2 ∼ |t| -2k-1 k-2 . (5.19) Recall that f (W µ ) = -k 2 (1 -8((r/µ) k + (r/µ) -k ) -2 ), hence |f (W µ ) + k 2 | r k . We also have (by a standard asymptotic expansion of arctan) |π -W λ | λ k r k and |π -W λ -2 λ k r k | λ 3k r 3k , hence 1 r 2 f (W µ )(π -W λ ) + 2k 2 λ k r k+2 L 2 (r≥ √ λ) λ k r 2 + λ 3k r 3k+2 L 2 (r≥ √ λ) |t| -2k-1 k-2 .
(5.20) Inserting (5.18), (5.19) and (5.20) into (5.17) we obtain

1 r 2 f (ϕ) + 1 r 2 f (W µ ) - 2k 2 λ k r k+2 L 2 (r≥ √ λ) |t| -2k-1 k-2 . (5.21)
A direct computation shows that for r ≥ √ λ there holds (∂

2 r + 1 r ∂ r )W λ (r) = -2k 2 λ k r k+2 +O λ 3k r 3k+2 , hence (∂ 2 r + 1 r ∂ r )W λ + 2k 2 λ k r k+2 L 2 (r≥ √ λ) λ 3k-1 2 |t| -2k-1 k-2 .
(

The same computation as in (4.15) yields (

∂ 2 r + 1 r ∂ r )S L 2 (r≥ √ λ) b 2 √ λ + λ k √ λ |t| -2k-1 k-2 .
Together with (5.21) and (5.22) this proves that

(∂ 2 r + 1 r ∂ r )ϕ + 1 r 2 f (ϕ) L 2 (r≥ √ λ) |t| -2k-1 k-2 . Since ΛW λ L 2 (r≥ √ λ) + Λ 0 ΛW λ L 2 (r≥ √ λ) ( ´+∞ 1/ √ λ 1 r 2k rdr) 1 2 λ k-1
2 , the other terms appearing in (5.11)

are |t| -3k-3 k-2 |t| -2k-1 k-2
. This finishes the proof of (5.11). The proof of (5.12) is very similar to the proof of (2.19), so we will not give all the details. We have

λ -b λ |t| -1 and µ µ |t| -1 , hence it suffices to check that 1 r 2 (f (ϕ) - f (W λ ))ΛW λ H * |t| -k+1 k-2 and 1 r 2 (f (ϕ) -f (W µ ))ΛW µ H * |t| -k+1 k-2 , which boils down to 1 r 2 W µ • ΛW λ L 1 |t| -k+1 k-2 and 1 r 2 (π -W λ ) • ΛW µ L 1 |t| -k+1 k-2
, see the proof of (4.6). In both cases we treat separately r ≤ 1 and r ≥ 1:

1 r 2 W µ • ΛW λ L 1 (r≤1) r k-2 ΛW λ L 1 (r≤1) = λ k ΛW L 1 (r≤1/λ) λ k | log λ| |t| -k+1 k-2 , 1 r 2 W µ • ΛW λ L 1 (r≥1) ΛW L ∞ (r≥1) λ k |t| -k+1 k-2 , 1 r 2 (π -W λ ) • ΛW µ L 1 (r≤1) r k-2 arctan (λ/r) k L 1 (r≤1) = λ k r k-2 arctan(r -k ) L 1 (r≤1/λ) λ k | log λ| |t| -k+1 k-2 , 1 r 2 (π -W λ ) • ΛW µ L 1 (r≥1) π -W λ L ∞ (r≥1) λ k |t| -k+1 k-2 .

Modulation

As in Subsection 3.2, we define g(t) := u(t) -ϕ(t) with modulation parameters λ(t) and µ(t) which satisfy

1 λ(t) Z λ(t) , g(t) = 0, 1 µ(t) Z µ(t) , g(t) c|t| -k+1 k-2 , |λ (t) -b(t)| + |µ (t)| g(t) E + c|t| -k+1 k-2 , (5.23)
with a constant c arbitrarily small. The initial data are

u(T ) = -W + W k-2 2κ (κ|t|) -2 k-2 , -(κ|t|) -k k-2 ΛW k-2 2κ (κ|t|) -2 k-2
.

(5.24)

The equivalent of Proposition 3.3 can by formulated as follows.

Proposition 5.3. There exist constants C 0 > 0 and T 0 < 0 with the following property Let T < T 1 < T 0 and suppose that u

(t) = ϕ(t) + g(t) ∈ C([T, T 1 ]; X 1 × X 0 ) is a solution of (1.
3) with initial data (5.24) such that for t ∈ [T, T 1 ] condition (5.8) is satisfied and

g(t) E ≤ C 0 |t| -k+1 k-2 .
(5.25)

Then for t ∈ [T, T 1 ] there holds g(t) E ≤ 1 2 C 0 |t| -k+1 k-2 , (5.26)
λ(t) - k -2 2κ (κ|t|) -2 k-2 + |µ(t) -1| C 0 |t| -3 k-2 .
(5.27)

The rest of this section is devoted to the proof of this bootstrap estimate. Proposition 5.3 allows to construct a uniformly controlled sequence of solutions close to the ansatz and pass to a weak limit, thereby proving Theorem 2, see Subsection 3.6.

Coercivity

Recall that f (W ) = -k 2 1 - 8 (r k +r -k ) 2 .
Lemma 5.4. There exist constants c, C > 0 such that • for all g ∈ H there holds ˆ+∞

0 (g ) 2 + k 2 r 2 g 2 rdr - ˆ+∞ 0 k 2 r 2 • 8 (r k + r -k ) 2 g 2 rdr ≥ c ˆ+∞ 0 (g ) 2 + k 2 r 2 g 2 rdr -C ˆ+∞ 0 Z • g dr r 2 ,
• if r 1 > 0 is large enough, then for all g ∈ H there holds

(1 -2c) ˆr1 0 (g ) 2 + k 2 r 2 g 2 rdr + c ˆ+∞ r 1 (g ) 2 + k 2 r 2 g 2 rdr - ˆ+∞ 0 k 2 r 2 • 8 (r k + r -k ) 2 g 2 rdr ≥ -C ˆ+∞ 0 Z • g dr r 2 , (5.28)
• if r 2 > 0 is small enough, then for all g ∈ H there holds

(1 -2c) ˆ+∞ r 2 (g ) 2 + k 2 r 2 g 2 rdr + c ˆr2 0 (g ) 2 + k 2 r 2 g 2 rdr - ˆ+∞ 0 k 2 r 2 • 8 (r k + r -k ) 2 g 2 rdr ≥ -C ˆ+∞ 0 Z • g dr r 2 ,
Proof. A change of variable g(x) := g(e x/k ), Z(x) := Z(e x/k ) reduces the problem to the study of the quadratic form associated with the classical operatord 2 dx 2 + (1 -2 sech 2 ) and it suffices to repeat the proof of Lemma 4.4. Lemma 4.5 applies verbatim to the case under consideration.

Bootstrap

We use the operators A(λ) and A 0 (λ) from Subsection 4.6. We define I(t), J(t) and H(t) by the same formulas as in Subsection 4.6.

Lemma 5.5. The operators A(λ) and A 0 (λ) have the following properties:

• for all λ > 0 and h 1 , h 2 ∈ X 1 there holds

A(λ)h 1 , 1 r 2 f (h 1 + h 2 ) -f (h 1 ) -f (h 1 )h 2 + A(λ)h 2 , 1 r 2 f (h 1 + h 2 ) -f (h 1 ) + k 2 h 2 ≤ c 0 λ • h 2 2 H , (5.29)
with a constant c 0 arbitrarily small,

• for all h ∈ X 1 there holds

A 0 (λ)h, ∂ 2 r + 1 r ∂ r - k 2 r 2 h ≤ c 0 λ h 2 H - 2π λ ˆRλ 0 (∂ r h) 2 + k 2 r 2 h 2 rdr, (5.30)
• assuming (2.7), for any c 0 > 0 there holds

Λ 0 ΛW λ(t) -A 0 (λ(t))ΛW λ(t) L 2 ≤ c 0 , (5.31)
φ(t) + b(t) • A(λ(t))ϕ(t) L ∞ ≤ c 0 |t| -1 , (5.32) ˆ+∞ 0 1 2 q r λ + λ r q r λ 1 r 2 f (ϕ + g) -f (ϕ) + k 2 g g rdr - ˆ+∞ 0 1 r 2 f (W λ ) + k 2 g 2 rdr ≤ c 0 C 2 0 |t| -2k+2 k-2 , (5.33)
provided that the constant R in the definition of q(r) is chosen large enough.

The proof is almost identical to the proof of Lemma 4.7 and we will skip it.

Lemma 5.6. Let c 1 > 0. If C 0 is sufficiently large, then there exists a function q(x) and T 0 < 0 with the following property. If T 1 < T 0 and (5.8), (5.25) hold for t ∈ [T, T 1 ], then for t ∈ [T, T 1 ] there holds

H (t) ≤ c 1 • C 2 0 |t| -3k k-2 . (5.34)
Proof. We follow the lines of the proof of Lemma 3.13. We have

I (t) = ∂ 2 r + 1 r ∂ r + 1 r 2 f (ϕ) ψ, g -ψ, g -φ, 1 r 2 (f (ϕ + g) -f (ϕ) -f (ϕ)g) .
The first term is

C 0 |t| -3k k-2
, hence negligible (by enlarging C 0 if necessary). Inequality (5.11) implies that the second term can be replaced byb λ (λb) Λ 0 ΛW λ , ġ , which in turn can be replaced by -b(λ -b) A 0 (λ)ΛW λ , ġ , thanks to (5.31). From (5.32) we infer that the third term can be replaced by b

• A(λ)ϕ, 1 r 2 (f (ϕ + g) -f (ϕ) -f (ϕ)g) (indeed, 1 r 2 (f (ϕ+g)-f (ϕ)-f (ϕ)g) L 1 ´+∞ 0 1 r 2 g 2 rdr g 2 H )
. Using formula (5.29) with h 1 = ϕ and h 2 = g we obtain

I (t) -b(λ -b) • A 0 (λ)ΛW λ , ġ -b • A(λ)g, 1 r 2 (f (ϕ + g) -f (ϕ) + k 2 g) .
As in the proof of Lemma 3.13, we obtain

(bJ) (t) b(λ -b) • A 0 (λ)ΛW λ , ġ + b ˆ ∂ 2 r + 1 r ∂ r - k 2 r 2 g + 1 r 2 f (ϕ + g) -f (ϕ) + k 2 g • A 0 (λ)g rdr, hence H (t) b ˆ( ∂ 2 r + 1 r ∂ r - k 2 r 2 g • A 0 (λ)g rdr - b λ ˆ+∞ 0 1 2 q r λ + λ r q r λ 1 r 2 f (ϕ + g) -f (ϕ) + k 2 g g rdr.
and the conclusion follows from (5.30), (5.33) and (5.28).

Proof of Proposition 5.3. We first show (5.27). From (5.23) and (5.25) we obtain

|µ(t) -1| = |µ(t) -µ(T )| ˆt -∞ C 0 |t| -k+1 k-2 dt C 0 |t| -3 k-2 . (5.35)
Again from (5.23) and (5.25) we have

|λ (t) -b(t)| C 0 |t| -k+1 k-2 . Multiplying by b (t) = k 2 • 2κ k-2 k • λ(t) k-1 µ(t) k ∼ |t| -2k-2 k-2 , cf.
(5.9) and (5.8), we obtain

d dt b(t) 2 -2κ k-2 k λ(t) k µ(t) k C 0 |t| -3k-1 k-2 . Since b(T ) = 2κ k-2 k 2 • λ(T ) and µ(T ) = 1, this yields b(t) 2 -2κ k-2 k λ(t) k µ(t) k C 0 |t| -2k+1 k-2 . But b(t) + 2κ k-2 k 2 λ(t) k 2 µ(t) k 2 ∼ |t| k k-2 , see (2.7) and (2.9), hence b(t) - 2κ k -2 k 2 λ(t) k 2 µ(t) k 2 C 0 |t| -k+1 k-2 . (5.36) Bound (5.35) implies that λ(t) k 2 µ(t) k 2 -λ(t)
k 2 |t| -k+1 k-2 , thus (5.36) yields |λ (t)-2κ k-2 k 2 λ(t) k 2 | C 0 |t| -k+1 k-2 .
Integrating this differential inequality is standard. Dividing by λ(t)

k 2 ∼ |t| -k k-2 we get (λ -k-2 2 ) + k -2 2 2κ k -2 k 2 = (λ -k-2 2 ) + 2κ k -2 k-2 2 κ C 0 |t| -1 k-2 . Using λ(T ) = k-2 2κ (κ|T |) -2 k-2 we obtain λ(t) -k-2 2 -2κ k-2 k-2 2 κ|t| C 0 |t| k-3 k-2
, from which (5.27) follows.

We turn to the proof of (5.26). From (5.10) the initial data at t = T satisfy g(T

) E |t| -2k k-2 |t| -k+1 k-2 , thus H(T ) |t| -2k+2 k-2 . If C 0 is large enough, then integrating (5.34) we get H(t) ≤ c • C 2 0 |t| -2k+2 k-2
, with a small constant c. Eventually C 0 if necessary and using the coercivity of H, we obtain (5.26).

A Cauchy theory A.1 Persistence of regularity

Proposition A.1. Let u : (T -, T + ) → E be the solution of (1.1) with the initial condition u(t 0 ) = u 0 . If u 0 ∈ X 1 × H 1 , then u ∈ C((T -, T + ); X 1 × H 1 ) ∩ C 1 ((T -, T + ); E).
The proof is classical, see [START_REF] Cazenave | Semilinear Schrödinger Equations[END_REF]Chapter 5] for more general results in the case of the nonlinear Schrödinger equation. Analogous results hold in the case of equations (1.4) and (1.3).

A.2 Profile decomposition and consequences

For details about the nonlinear profile decomposition for the critical wave equation we refer to [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF] (the defocusing case), [START_REF] Duyckaerts | Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation[END_REF] (dimension N = 3) and [START_REF] Rodriguez | Profiles for the radial focusing energy-critical wave equation in odd dimensions[END_REF] (any dimension). For the reader's convenience we recall the following result [START_REF] Rodriguez | Profiles for the radial focusing energy-critical wave equation in odd dimensions[END_REF]Proposition 2.3]. We denote S(I) the Strichartz norm on the interval I.

Remark A.7. Corollaries A.3 and A.4 are crucial ingredients of the arguments in Subsection 3.6. Using the nonlinear profile decomposition of Bahouri and Gérard [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF] is nowadays a well-established method of attacking this type of questions in critical spaces. Note that [START_REF] Bahouri | High frequency approximation of solutions to critical nonlinear wave equations[END_REF] gave the first proof of the sequential weak continuity of the flow for the defocusing energycritical wave equation.

Introduction 1.Setting of the problem

Let N ∈ {3, 4, 5} be the dimension of the space. For u 0 = (u 0 , u0 ) ∈ E := Ḣ1 (R N ) × L 2 (R N ), define the energy functional

E(u 0 ) = ˆ1 2 | u0 | 2 + 1 2 |∇u 0 | 2 -F (u 0 ) dx,
where

F (u 0 ) := N -2 2N |u 0 | 2N N -2 . Note that E(u 0 ) is well-defined due to the Sobolev Embedding Theorem. The differential of E is DE(u 0 ) = (-∆u 0 -f (u 0 ), u0 ), where f (u 0 ) = |u 0 | 4 N -2 u 0 .
We consider the Cauchy problem for the energy critical wave equation:

∂ t u(t) = J • DE(u(t)), u(t 0 ) = u 0 ∈ E. (NLW)
Here, J := 0 Id -Id 0 is the natural symplectic structure. This equation is often written in the form

∂ tt u = ∆u + f (u).
Equation (NLW) is locally well-posed in the space E, see for example [START_REF] Ginibre | The global Cauchy problem for the critical nonlinear wave equation[END_REF] and [START_REF] Shatah | Well-posedness in the energy space for semilinear wave equations with critical growth[END_REF] (the defocusing case), as well as a complete review of the Cauchy theory in [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energycritical focusing non-linear wave equation[END_REF]. In particular, for any initial data u 0 ∈ E there exists a maximal time of existence (T -, T + ), -∞ ≤ T -< t 0 < T + ≤ +∞, and a unique solution u ∈ C((T -, T + ); E). In addition, the energy E is a conservation law. In this paper we always assume that the initial data is radially symmetric. This symmetry is preserved by the flow.

For functions v ∈ Ḣ1 , v ∈ L 2 , v = (v, v) ∈ E and λ > 0, we denote

v λ (x) := 1 λ (N -2)/2 v x λ , vλ (x) := 1 λ N/2 v x λ , v λ (x) := v λ , vλ .
A change of variables shows that

E (u 0 ) λ = E(u 0 ).
Equation (NLW) is invariant under the same scaling. If u = (u, u) is a solution of (NLW) and λ > 0, then t → u (tt 0 )/λ λ is also a solution with initial data (u 0 ) λ at time t = 0. This is why equation (NLW) is called energy-critical.

A fundamental object in the study of (NLW) is the family of stationary solutions (u, ∂ t u) = ±W λ = (±W λ , 0), where

W (x) = 1 + |x| 2 N (N -2) -(N -2)/2 .
The functions W λ are called ground states.

In general the energy E does not control the norm • E , and indeed this norm can tend to +∞ in finite time, which is referred to as type I blow-up. In odd space dimensions and for superconformal nonlinearities (which includes the energy-critical case) Donninger and Schörkhuber [START_REF] Donninger | Stable blow up dynamics for energy supercritical wave equations[END_REF], [START_REF] Donninger | Stable blowup for wave equations in odd space dimensions[END_REF] described large sets of initial data leading to this kind of blow-up.

It can also happen that in finite time the solution leaves every compact set of E, the norm • E staying bounded, which is referred to as type II blow-up. In dimension N = 3 in the radial case one of the consequences of the classification result of Duyckaerts, Kenig and Merle [START_REF] Duyckaerts | Classification of the radial solutions of the focusing, energy-critical wave equation[END_REF] is that any blow-up solution is either of type I or of type II. This is unknown in other cases.

A particular type of type II blow-up occurs when the solution u(t) stays close to the family of ground states W λ and λ → 0. In this situation we call W λ the bubble of energy and we say that u(t) blows up by concentration of one bubble of energy. We have the following fundamental result proved first by Duyckaerts, Kenig and Merle [START_REF] Duyckaerts | Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation[END_REF] for N = 3, by the same authors [START_REF] Duyckaerts | Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case[END_REF] for N = 5 and by Côte, Kenig, Lawrie and Schlag [START_REF] Côte | Profiles for the radial focusing 4d energycritical wave equation[END_REF] for N = 4:

Theorem ([23], [START_REF] Duyckaerts | Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case[END_REF], [START_REF] Côte | Profiles for the radial focusing 4d energycritical wave equation[END_REF]). Let u(t) be a radial solution of (NLW) which blows up at t = T + by concentration of one bubble of energy. Then there exist u * 0 ∈ E and λ ∈ C([t 0 , T + ), (0, +∞)) such that lim

t→T + u(t) -W λ(t) -u * 0 E = 0, lim t→T + (T + -t) -1 λ(t) = 0. (1.1)
In this context the function u * 0 is called the asymptotic profile. Note that in [START_REF] Duyckaerts | Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case[END_REF] a more general, non-radial version of the above theorem was proved for N ∈ {3, 5}.

Solutions verifying (1.1) were first constructed in dimension N = 3 by Krieger, Schlag and Tataru [START_REF] Krieger | Slow blow-up solutions for the H 1 (R 3 ) critical focusing semilinear wave equation[END_REF], who obtained all possible polynomial blow-up rates λ(t) ∼ (T + -t) 1+ν , ν > 0. For N = 4 smooth solutions blowing up at a particular rate were constructed by Hillairet and Raphaël [START_REF] Hillairet | Smooth type II blow up solutions to the four dimensional energy critical wave equation[END_REF]. For N = 5 the author proved in [START_REF] Jendrej | Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5[END_REF] that for any radially symmetric asymptotic profile u * 0 ∈ H 4 × H 3 such that u * 0 (0) > 0, there exists a solution u(t) such that (1.1) holds. For these solutions the concentration speed of the bubble is

λ(t) ∼ u * 0 (0) 2 (T + -t) 4 . (1.2)
In the same article, solutions with blow-up rate (T + -t) 1+ν for ν > 8 were constructed, with ν explicitely related to the asymptotic behaviour of u * 0 at x = 0.

Statement of the results

In the present paper we continue the investigation of the relationship between the behaviour of u * 0 at x = 0 and possible blow-up speeds, still in the special case when the asymptotic profile u * 0 is sufficiently regular. We prove the following result.

Theorem 1. Let N ∈ {3, 4, 5} and s > N -2 2 , s ≥ 1. Let u * 0 = (u * 0 , u * 0 ) ∈ H s+1 × H s be a radial function. Suppose that u is a radial solution of (NLW) such that lim t→T + u(t) -W λ(t) -u * 0 E = 0, lim t→T + λ(t) = 0, T + < +∞. (1.3)
There exists a constant C > 0 depending on u * 0 such that:

• if N ∈ {4, 5}, then for T + -t sufficiently small there holds

λ(t) ≤ C(T + -t) 4 6-N . (1.4)
• if N = 3, then there exists a sequence t n → T + such that

λ(t n ) ≤ C(T + -t n ) 4 6-N . (1.5)
Remark 1.1. Let u * = (u * , u * ) be the solution of (NLW) such that u * (T + ) = u * 0 and suppose that 0 ∈ supp u * 0 . We will prove that there exists a universal constant C 0 such that in the above theorem one can take

C = C 0 u * 2 6-N L ∞ ((T + -ρ,T + )×B(0,ρ)) ,
where ρ > 0 is arbitrary and B(0, ρ) is the ball of centre 0 and radius ρ in R N . Notice that u * ∈ L ∞ ((T + -ρ, T + ) × R N ) by Appendix A and the Sobolev Embedding Theorem.

If 0 / ∈ supp u * 0 , then blow-up cannot occur, as follows from the classification of solutions of (NLW) at energy level E(W ) by Duyckaerts and Merle [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF].

Remark 1.2. In the case N = 3 we will prove that for T + -t small enough there holds

ˆT+ t dτ λ(τ ) ≥ 3 √ C (T + -t) 1 3 , (1.6)
which immediately implies (1.5).

If we assume that u * ∈ H 3 ×H 2 , then (1.4) holds also in the case N = 3, see Remark 2.13. I believe that the proof of (1.5) given here could be adapted to cover the case 1 > s > 1 2 .

Remark 1.3. In dimension N = 5 the bound (1.4) is optimal, see (1.2). It is not clear if the bounds are optimal for N ∈ {3, 4}, due to slow decay of the bubble.

Remark 1.4. A natural problem is to determine sharp bounds for the blow-up speed in the case of less regular u * 0 . The method used in this paper allows to obtain some bounds for example in the case 1 ≤ s < 3 2 in dimension N = 5, but they are not optimal and I will not pursue this direction here.

In the case u * 0 (0) = 0 one could obtain various bounds depending on the asymptotics of u * 0 at x = 0, but this will not be considered in the present paper. Along the same line, one can ask if the sign of u * 0 (0) is relevant in the case when u * 0 (0) = 0. It turns out that it is, but unfortunately our method requires the additional assumption

u * 0 ∈ H 3 × H 2 : Theorem 2. Let N ∈ {3, 4, 5}. Let u * 0 = (u * 0 , u * 0 ) ∈ H 3 × H 2 be a radial function such that u * 0 (0) < 0.
There exist no radial solutions of (NLW) such that

lim t→T + u(t) -W λ(t) -u * 0 E = 0, lim t→T + λ(t) = 0, T + < +∞.
Remark 1.5. I expect that Theorems 1 and 2 could be proved by similar methods without the assumption of u * 0 being radial.

Related results

The problem of existence of an asymptotic profile at blow-up might be seen as a version of the classical question of asymptotic stability of solitons in the case when finite-time blow-up occurs. Decompositions of type (1.1) in suitable topologies are believed to hold for many models, but establishing this rigourously is a challenging problem. Historically, the study of finite type blow-up in the Hamiltonian setting received the most attention probably in the case of nonlinear Schrödinger equations (NLS). For the mass-critical NLS the conformal invariance leads to explicit blow-up solutions S(t) with the asymptotic profile u * ≡ 0. Bourgain and Wang [START_REF] Bourgain | Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity[END_REF] constructed examples of blow-up solutions with u * regular and non-zero, the speed of blow-up however being the same as for S(t). This is not a coincidence, as shown by a classification result of Merle and Raphaël [START_REF] Merle | Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation[END_REF].

For the critical gKdV equation Martel, Merle and Raphaël [START_REF] Martel | Blow up for the critical gKdV equation I: dynamics near the soliton[END_REF] proved that if the initial data decays sufficiently fast, then there is only one possible blow-up speed, given by the minimal mass blow-up solution. However, without the decay assumption other blow-up speeds are possible, as shown by the same authors in [START_REF] Martel | Blow up for the critical gKdV equation III: exotic regimes[END_REF].

These are the main two examples of the heuristic principle that the size of the interaction of the bubble with the rest of the solution influences or even determines the speed of blowup. In the present paper we try to investigate this phenomenon in the energy-critical setting. The same problem could be considered for other energy-critical models for which bubble concentration phenomenon has been observed (see results of Krieger, Schlag and Tataru [START_REF] Krieger | Renormalization and blow up for charge one equivariant critical wave maps[END_REF], Ortoleva and Perelman [START_REF] Ortoleva | Nondispersive vanishing and blow up at infinity for the energy critical nonlinear Schrödinger equation in R 3[END_REF], Perelman [START_REF] Perelman | Blow up dynamics for equivariant critical Schrödinger maps[END_REF], Merle, Rodnianski and Raphaël [START_REF] Merle | Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem[END_REF], Schweyer [START_REF] Schweyer | Type II blow-up for the four dimensional energy critical semi linear heat equation[END_REF]). It seems that the question of relationship between the asymptotic profile and the speed of blow-up has not been addressed.

Finally, let us mention that the problem of understanding the possible blow-up speeds is not limited to type II blow-up for critical equations, see for instance Merle and Zaag [START_REF] Merle | Determination of the blow-up rate for the semilinear wave equation[END_REF], [START_REF] Merle | Determination of the blow-up rate for a critical semilinear wave equation[END_REF] for the subconformal and conformal NLW, and Giga and Kohn [START_REF] Giga | Nondegeneracy of blowup for semilinear heat equations[END_REF], Mizoguchi [START_REF] Mizoguchi | Rate of Type II blowup for a semilinear heat equation[END_REF], Matano and Merle [START_REF] Matano | Classification of type I and type II behaviors for a supercritical nonlinear heat equation[END_REF] for the semilinear heat equation.

Outline of the proof

Our proofs of Theorems 1 and 2 are based on the following computation that we present here formally.

Let u : [t 0 ; T + ) → E be a solution of (NLW) which satisfies (1.1). At blow-up time, the energy of the bubble is completely decoupled from the energy of the asymptotic profile, hence

E(u) = E(u * 0 ) + E(W λ ) = E(u * 0 ) + E(W ). (1.7)
Let u * be the solution of (NLW) with the initial data u * (T + ) = u * 0 . Decompose u(t) = W λ(t) + u * (t) + g(t) (in fact we use a suitable localisation of W λ ; we will ignore here this technical point). The modulation parameter λ is determined by a suitable orthogonality condition, and a standard procedure shows that |λ (t)| g(t) E . From the Taylor formula we obtain

E(u) = E(u * + W λ ) + DE(u * + W λ ), g + 1 2 D 2 E(u * + W λ )g, g + O( g 3 E ).
Step 1. An explicit key computation shows that

E(u * + W λ ) -E(u * ) -E(W ) -u * 0 (0)λ N -2 2 .
It is clear that the sign of u * 0 (0) is decisive.

Step 2. Near blow-up time u * weakly interacts with W λ and DE(W λ ) = 0. This allows to replace DE(u * + W λ ), g by DE(u * ), g . Using the Hamiltonian structure it is seen that this quantity is, at first order in g, a conservation law. Estimating some error terms we conclude that this term can be neglected.

Step 3. Let us suppose for a moment that D 2 E(W ) is a coercive functional in the sense that D 2 E(u * + W λ )g, g g 2 E . Using (1.7) and the two preceding steps we find

|λ | 2 g 2 E u * 0 (0)λ N -2
2 . In the case u * 0 (0) < 0 this is contradictory, and in the case u * 0 (0) > 0 the conclusion follows by integrating the differential inequality for λ.

Strictly speaking, D 2 E(W ) is not a coercive functional, and much of the proof is devoted to controlling the negative directions, which are related to the eigendirections of the flow linearized around W . Clarifying the second step above is another major technical difficulty of this paper.

Notation

We introduce the inifinitesimal generators of scale change

Λ s := N 2 -s + x • ∇.
For s = 1 we omit the subscript and write Λ = Λ 1 . We denote Λ E , Λ F and Λ E * the inifinitesimal generators of the scaling which is critical for a given norm, that is

Λ E = (Λ, Λ 0 ), Λ E * = (Λ -1 , Λ 0 ).
The dimension of the space will be denoted N . The domain of the function spaces is always R N . We introduce the following notation for some frequently used function spaces:

X s := Ḣs+1 ∩ Ḣ1 for s ≥ 0, E := Ḣ1 × L 2 , F := L 2 × Ḣ-1 . The bracket •,
• denotes the distributional pairing and the scalar product in the spaces L 2 , L 2 × L 2 . Notice that E * Ḣ-1 × L 2 through the natural isomorphism induced by •, • .

For a function space A, O A (m) denotes any a ∈ A such that a A ≤ Cm for some constant C > 0. For positive quantities m 1 and m 2 we write m 1 m 2 for m 1 = O(m 2 ) and m 1 ∼ m 2 for m 1 m 2 m 1 . We denote B A (x 0 , δ) the open ball of center x 0 and radius δ in the space A. If A is not specified, it means that A = R.

The proofs 2.1 Properties of the linearized operator

Linearizing -∆u -f (u) around W , u = W + g, we obtain a Schrödinger operator

Lg = (-∆ -f (W ))g. Notice that L(ΛW ) = d dλ λ=1 -∆W λ -f (W λ ) = 0.
It is known that L has exactly one strictly negative simple eigenvalue which we denote -ν 2 (we take ν > 0). We denote the corresponding positive eigenfunction Y, normalized so that Y L 2 = 1. By elliptic regularity Y is smooth and by Agmon estimates it decays exponentially. Self-adjointness of L implies that Y, ΛW = 0. (

We define

Y -:= 1 ν Y, -Y , Y + := 1 ν Y, Y , α -:= 1 2 (νY, -Y), α + := 1 2 (νY, Y). We have J • D 2 E(W ) = 0 Id -L 0 . A short computation shows that J • D 2 E(W )Y -= -νY -, J • D 2 E(W )Y + = νY + and α -, J • D 2 E(W )g = -ν α -, g , α + , J • D 2 E(W )g = ν α + , g , ∀g ∈ E. (2.2) Notice that α -, Y -= α + , Y + = 1 and α -, Y + = α + , Y -= 0.
The rescaled versions of these objects are

Y - λ := 1 ν Y λ , -Y λ , Y + λ := 1 ν Y λ , Y λ , α - λ := 1 2 ν λ Y λ , -Y λ , α + λ := 1 2 ν λ Y λ , Y λ .
The scaling is chosen so that

α - λ , Y - λ = α + λ , Y + λ = 1. We have J • D 2 E(W λ )Y - λ = - ν λ Y - λ , J • D 2 E(W λ )Y + λ = ν λ Y + λ and α - λ , J • D 2 E(W λ )g = - ν λ α - λ , g , α + λ , J • D 2 E(W λ )g = ν λ α + λ , g , ∀g ∈ E. Let Z be a C ∞ 0 function such that Z, ΛW > 0, Z, Y = 0
(the first condition is the essential one and the second allows to simplify some computations). We recall the following result.

> 0 such that v ∈ Ḣ1 radial, Y, v = Z, v = 0 ⇒ 1 2 v, Lv ≥ c L v 2 Ḣ1 .
Lemma 2.2. There exists a constant c > 0 such that if V -W λ E < c, then for all g ∈ E such that Z λ , g = 0 there holds

1 2 D 2 E(V )g, g + 2 α - λ , g 2 + α + λ , g 2 g 2 E .
Proof. We have

D 2 E(V )g, g = D 2 E(W λ )g, g + ˆ f (V ) -f (W λ ) |g| 2 dx.
By Hölder, the last integral is c g 2 E , hence it suffices to prove the lemma with V = W λ . Without loss of generality we can assume that λ = 1. We will show the following stronger inequality:

1 2 D 2 E(W )g, g + 2 α -, g • α + , g ≥ c L g -α -, g Y --α + , g Y + 2 E . (2.3)
Let a -= α -, g , a + = α + , g and decompose g = a -Y -+ a + Y + + k, so that α -, k = α + , k = 0. From Z, Y = 0 we deduce Z, k = 0. We have g = a -+a + ν Y + k and ġ = (-a -+ a + )Y + k, hence

1 2 D 2 E(W )g, g = 1 2 a -+ a + ν Y + k, -(a -+ a + )νY + Lk + 1 2 (-a -+ a + )Y + k, (-a -+ a + )Y + k = - 1 2 (a -+ a + ) 2 Y, Y -(a -+ a + )ν Y, k + 1 2 k, Lk + 1 2 (-a -+ a + ) 2 Y, Y + (-a -+ a + ) Y, k + 1 2 k, k = -2a -a + Y, Y -2a -α + , k -2a + α -, k + 1 2 k, Lk + k, k = -2a -a + + 1 2 D 2 E(W )k, k .
Invoking Proposition 2.1 finishes the proof of (2.3).

Modulation

Recall that X s := Ḣs+1 ∩ Ḣ1 . Let u * 0 ∈ X s ×H s , T + ∈ R and let u * be the solution of (NLW) with initial data u * (T + ) = u * 0 . Without loss of generality we will assume that N -2 2 < s ≤ 2. For fixed ρ > 0 we denote c * := u * L ∞ ((T + -ρ,T + )×B(0,ρ)) . We can assume that c * > 0 (otherwise there is no blow-up, cf. Remark 1.1). Note that because of finite speed of propagation, we can also assume that u * (t) E is smaller than any fixed strictly positive constant and that u * (t) L ∞ ≤ 2c * for t close to T + .

Because of a slow decay of W , we will introduce compactly supported approximations of

W λ . Let R := (c 0 • c * ) 1 -N +2 , (2.4)
where c 0 > 0 is a small universal constant to be chosen later. We denote

V (λ)(x) := W λ (x) -ζ(λ) for |x| ≤ R √ λ, 0 for |x| ≥ R √ λ,
where

ζ(λ) := W λ (R √ λ) = 1 λ N -2 2 1 + R 2 N (N -2)λ -N -2 2 = λ + R 2 N (N -2) -N -2 2 .
We will also denote V (λ) := (V (λ), 0) ∈ E.

Notice that

∂ λ V (λ)(x) = -(ΛW ) λ (x) -ζ (λ) for |x| < R √ λ, 0 for |x| > R √ λ. Lemma 2.3. Let s > N -2 2
and s ≥ 1. The following estimates are true with universal constants:

V (λ) -W λ Ḣ1 R -N +2 2 λ N -2 4 , (2.5)
V (λ) -W λ L ∞ R -N +2 , (2.6)
∂ λ V (λ) + ΛW λ L ∞ (|x|<R √ λ) R -N , (2.7)
∂ λ V (λ) L 2N N +2 R 6-N 2 λ N -2 4 , (2.8)
∂ λ V (λ) H 1-s λ N -4 2
as λ → 0.

(2.9)

Proof. To prove (2.5), we write

V (λ) -W λ 2 Ḣ1 = ˆ|x|≥R √ λ |∇W λ | 2 dx = ˆ|x|≥R/ √ λ ∇W 2 dx ˆ+∞ R/ √ λ r -2N +2 • r N -1 dr ∼ (R/ √ λ) -N +2 .
We see that ζ(λ) ∼ R -(N -2) and ζ (λ) ∼ R -N when λ is small, which proves (2.6) and (2.7).

On the support of ∂ λ V (λ) there holds

|∂ λ V (λ)(x)| λ N -4 2 |x| -N +2 , hence ∂ λ V (λ) 2N N +2 L 2N N +2 ˆR√ λ 0 λ N -4 2 • 2N N +2 r (-N +2) 2N N +2 r N -1 dr = λ N 2 -4N N +2 ˆR√ λ 0 r -N 2 +5N -2 N +2 dr = R N (6-N ) N +2 λ N (N -2) 2(N +2) .
This proves (2.8). We will check (2.9) separately in each dimension. For N = 3 we have

|∂ λ V (λ)(x)| λ -1 2 |x| -1 and |x| -1
1. For N = 4 we have |∂ λ V (λ)(x)| |x| -2 . We suppose s > 1, hence there exists q ∈ (1, 2) such that L q ⊂ H 1-s and it is easy to check that |x| -2

L q (|x|≤R √ λ)
1. Finally for N = 5 we have |∂ λ V (λ)(x)| λ 1 2 |x| -3 . There exists q ∈ 1, 5 3 such that L q ⊂ H 1-s and it is easy to check that |x| -3

L q (|x|≤R √ λ)
1.

Note that ζ(λ) ∼ c 0 c * , which means that the cut-off is made at a radius r = R √ λ such that W λ (r) ∼ c 0 u * (t, r).

For the next lemma we will need the following version of the Implicit Function Theorem. It is obtained directly from standard proofs of the usual version, see for example [START_REF] Chow | Methods of Bifurcation Theory[END_REF]Section 2.2].

Lemma 2.4. Suppose that X, Y and Z are Banach spaces, x 0 ∈ X, y 0 ∈ Y , ρ, η > 0 and Φ : B(x 0 , ρ) × B(y 0 , η) → Z is continuous in x and continuously differentiable in y, Φ(x 0 , y 0 ) = 0 and D y Φ(x 0 , y 0 ) =: L 0 has a bounded inverse. Suppose that

L 0 -D y Φ(x, y) Z ≤ 1 3 L -1 0 -1 L (Z,Y ) for x -x 0 X < ρ, y -y 0 Y < η, Φ(x, y 0 ) Z ≤ η 3 L -1 0 -1 L (Z,Y ) for x -x 0 X < ρ.
Then there exists y ∈ C(B(x 0 , ρ), B(y 0 , η)) such that for x ∈ B(x 0 , ρ), y(x) is the unique solution of the equation Φ(x, y(x)) = 0 in B(y 0 , η).

Lemma 2.5. There exists δ 0 > 0 and λ 0 > 0 such that for any 0 ≤ δ ≤ δ 0 and t 1 < t 2 , if u : (t 1 , t 2 ) → E is a solution of (NLW) satisfying for all t ∈ (t 1 , t 2 ):

u(t) -u * (t) -W λ(t) E ≤ δ, 0 < λ(t) < λ 0 , (2.10)
then there exists a unique function λ(t) ∈ C 1 ((t 1 , t 2 ), (0, +∞)) such that

g(t) := u(t) -u * (t) -V (λ(t)) (2.11)
satisfies for all t ∈ (t 1 , t 2 ):

Z λ(t) , g(t) = 0, (2.12)

g(t) E δ + λ(t) N -2 4 , (2.13
)

|λ(t)/ λ(t) -1| δ, (2.14
)

|λ (t)| g(t) E . (2.15)
Proof. We will first show that for t 0 ∈ (t 1 , t 2 ) fixed there exists a unique λ(t 0 ) such that (2.12), (2.13) and (2.14) hold at t = t 0 . The proof is standard, see for example [START_REF] Martel | Instability of solitons for the critical generalized Korteweg-de Vries equation[END_REF]Proposition 1]. Denote v 0 := u(t 0 ) -u * (t 0 ) and l 0 := log( λ(t 0 )) (it will be convenient to consider λ(t 0 ) and λ(t 0 ) in the logarithmic scale). We define the following functional:

Φ : E × R → R, Φ(v; l) := e -l Z e l , v -V (e l ) .
We have

∂ l Φ(v; l) = -Z e l , ∂ λ V (e l ) -e -l Λ -1 Z e l , v -V (e l ) .
We apply Lemma 2.4 with x 0 = V ( λ(t 0 )) and y 0 = l 0 . It is easily checked that the assumptions hold if δ is small and η = Cδ, with a large constant C. Take λ(t 0 ) = e l 0 , where l 0 is the solution of Φ(v 0 ; l 0 ) = 0 given by Lemma 2.4. Directly from the definition of Φ we obtain (2.12). The inequality |l 0 -l 0 | ≤ η = Cδ is equivalent to (2.14), which in turn implies W λ(t 0 ) -W λ(t 0 ) Ḣ1 δ.

(2.16)

From the definition of g and (2.10) we have

g ≤ δ + W λ(t 0 ) -W λ(t 0 ) Ḣ1 + W λ(t 0 ) -V (λ(t 0 )) Ḣ1 ,
so (2.13) follows from (2.16) and (2.5).

For each t 0 ∈ (t 1 , t 2 ) we have defined λ(t 0 ). It remains to show that λ(t) is a C 1 function and that (2.15) holds. One way is to use a regularization procedure as in [START_REF] Martel | Instability of solitons for the critical generalized Korteweg-de Vries equation[END_REF]. Here we give a different argument, which might be simpler in some cases.

Take t 0 ∈ (t 1 , t 2 ) and let l 0 := log(λ(t 0 )). Denote v(t) := u(t) -u * (t) and define l : (t 0 -ε, t 0 + ε) → R as the solution of the differential equation

l (t) = -(∂ l Φ) -1 (D v Φ)∂ t v(t)
with the initial condition l(t 0 ) = l 0 . Notice that D v Φ is a continuous functional on F, so we can apply it to ∂ t v(t).

Using the chain rule we get d dt Φ(v(t); l(t)) = 0 for t ∈ (t 0 -ε, t 0 + ε). By continuity, |l(t) -l 0 | < η = Cδ in some neighbourhood of t = t 0 . Hence, by the uniqueness part of Lemma 2.4, we get l(t) = log λ(t) in some neighbourhood of t = t 0 . In particular, λ(t) is of class C 1 in some neighbourhood of t 0 .

From (2.11) we obtain the following differential equation for the error term g:

∂ t g = J • (DE(V (λ) + u * + g) -DE(u * )) -λ ∂ λ V (λ), (2.17)
which can also be written in the expanded form

∂ t g = ġ -λ ∂ λ V (λ), ∂ t ġ = ∆g + f (u * + V (λ) + g) -f (u * ) -f (V (λ)) + ∆V (λ) + f (V (λ)) . (2.18)
Differentiating (2.12) and using the first equation in (2.18) we get

0 = d dt Z λ , g = - λ λ Λ 0 Z λ , g + Z λ , ġ -λ ∂ λ V (λ) = λ Z, ΛW -Z λ , ΛW λ + ∂ λ V (λ) - 1 λ Λ 0 Z λ , g + Z λ , ġ .
We assumed that Z, ΛW > 0. When g E and λ are small enough, then

Z λ , ΛW λ + ∂ λ V (λ) + 1 λ Λ 0 Z λ , g ≤ 1 2
Z, ΛW (we use (2.7) in order to estimate the first term). This proves (2.15).

If u(t) is a solution of (NLW) satisfying (1.3), then there exists t 0 such that (2.10) holds for t ∈ [t 0 , T + ). It follows from (2.14) that, while proving Theorem 1, without loss of generality we can assume that λ(t) is the function given by Lemma 2.5. From (2.13) we obtain that g E → 0 as t → T + , which is the only information about g used in the sequel. The precise form of the right hand side of (2.13) has no importance. We will prove that (1.4) holds on some interval [t 0 , T + ) with t 0 < T + , with no information about the length of this interval. Each time we state something for t ∈ [t 0 , T + ) it should be understood that t 0 is sufficiently close to T + .

In the rest of this paper λ(t) always stands for the modulation parameter obtained in Lemma 2.5 and g(t) is the function defined by (2.11). We introduce the following notation for the joint size of the error and the interaction:

n(g, λ) := g 2 E + c * λ N -2
2 .

We will now analyze the stable and unstable directions of the linearized flow. The stable coefficient a -(t) and the unstable coefficient a + (t) are defined as follows:

a -(t) := α - λ(t
∂ t g = J • D 2 E(W λ )g + h,
where

h = h ḣ = -λ ∂ λ V (λ), f (u * + V (λ) + g) -f (u * ) -f (V (λ)) -f (W λ )g + ∆V (λ) + f (V (λ))
.

Using (2.2) we get d dt a -(t) + ν λ a -(t) = d dt α - λ , g + ν λ α - λ , g = - λ λ Λ E * α - λ , g + α - λ , J • D 2 E(W λ )g + ν λ α - λ , g + α - λ , h = - λ λ Λ E * α - λ , g + α - λ , h .
The first term is negligible due to (2.15). In order to bound the second term it suffices to check the following inequalities:

Y λ , ∂ λ V (λ) n(g, λ) 2 , Y λ , ∆V (λ) + f (V (λ)) n(g, λ) 2 , (2.20)
Y λ , f (u * + V (λ) + g) -f (u * ) -f (V (λ)) -f (W λ )g n(g, λ) 2 .
(2.21)

The first inequality follows from (2.7) and (2.1), since the region |x| ≥ R √ λ is negligible due to exponential decay of Y.

Notice that |f

(W λ ) -f (V (λ))| f (W λ )| • |W λ -V (λ)| f (W λ )c 0 c * ,
where the last inequality follows from (2.5) and (2.4). Together with the fact that ∆(W λ ) + f (W λ ) = 0 this implies

Y λ , ∆V (λ) + f (V (λ)) Y λ , ∆ W λ -V (λ) + Y λ , f (W λ ) -f (V (λ)) ∆Y λ L 1 + f (W λ )Y λ L 1 c 0 c * c * λ N -2 2 ,
which proves (2.20). We will check (2.21) in three small steps. As before, we do not have to worry about the region |x| ≥ R √ λ thanks to the fast decay of Y. First, we have a pointwise bound

|f (u * + V (λ)) -f (u * ) -f (V (λ))| f (W λ ) • c * + f (c * ), (2.22)
which implies

Y λ , f (u * + V (λ)) -f (u * ) -f (V (λ)) n(g, λ) 2 .
(2.23)

Next, we have

|f (u * + V (λ) + g) -f (u * + V (λ)) -f (u * + V (λ))g| |f (u * + V (λ))| • |g| 2 + f (|g|), (2.24) which implies Y λ , f (u * + V (λ) + g) -f (u * + V (λ)) -f (u * + V (λ))g n(g, λ) 2 . (2.25) Finally, |f (V (λ) + u * ) -f (W λ )| (|f (W λ )| + |f (V (λ) + u * -W λ )|) • |V (λ) + u * -W λ | |f (W λ )|c * .
Using Hölder and the fact that

Y λ • f (W λ ) L 2N N +2 λ N -2 2 this implies Y λ , f (u * + V (λ)) -f (W λ ) g n(g, λ) 2 . ( 2

Coercivity

By the conservation of energy, for all t ∈ [t 0 , T + ) there holds

E(V (λ) + u * + g) = E(W ) + E(u * ). (2.27)
On the other hand, using the pointwise inequality

|F (k + l) -F (k) -f (k)l - 1 2 f (k)l 2 | |f (k)||l 3 | + |F (l)|, ∀k, l ∈ R we deduce that E(V (λ) + u * + g) = E(V (λ) + u * ) + DE(V (λ) + u * ), g + 1 2 D 2 E(V (λ) + u * )g, g + O( g 3 E ).
Using (2.27) we obtain

E(V (λ) + u * ) -E(W ) -E(u * ) + DE(V (λ) + u * ), g + 1 2 D 2 E(V (λ) + u * )g, g = O( g 3 E ).
(2.28)

We start by computing the size of the first term on the left hand side.

Lemma 2.7. For T + -t small there holds

|E(V (λ) + u * ) -E(W ) -E(u * )| c * λ N -2 2 .
In addition, if u * (0) < 0, then

E(V (λ) + u * ) -E(W ) -E(u * ) c * λ N -2 2 .
(2.29)

Proof. Integrating by parts we obtain

ˆ∇V (λ) • ∇u * dx = ˆB(0,R √ λ) ∇(W λ ) • ∇u * dx = - ˆB(0,R √ λ) ∆(W λ ) • u * dx + ˆS(0,R √ λ) ∂ r (W λ ) • u * dσ = ˆB(0,R √ λ) f (W λ ) • u * dx + ˆS(0,R √ λ) ∂ r (W λ ) • u * dσ.
Developping the energy gives

E(V (λ) + u * ) -E(W ) -E(u * ) = ˆ∇V (λ) • ∇u * dx + 1 2 ˆ|∇V (λ)| 2 -|∇(W λ )| 2 dx -ˆF (V (λ) + u * ) -F (W λ ) -F (u * ) dx = ˆS(0,R √ λ) ∂ r (W λ ) • u * dσ + 1 2 ˆ|∇V (λ)| 2 -|∇W λ | 2 dx -ˆF (V (λ) + u * ) -F (W λ ) -F (u * ) -f (V (λ)) • u * dx + ˆB(0,R √ λ) f (W λ ) -f (V (λ)) • u * dx.
(2.30)

We will show that all the terms on the right hand side except for the first one are c 0 c * λ N -2 2 , where c 0 is the small constant in (2.4).

The fact that ´|∇V (λ

)| 2 -|∇W λ | 2 dx c 0 c * λ N -2 2 = R -N +2 λ N -2 2
follows directly from the proof of (2.5).

We will now show that

ˆ|F (V (λ) + u * ) -F (V (λ)) -F (u * ) -f (V (λ))u * | dx λ N -2 2 .
To this end, notice first that the integrand equals 0 for |x| ≥ R √ λ. In the region |x| ≤ R √ λ we use the pointwise estimate

|F (V (λ) + u * ) -F (V (λ)) -F (u * ) -f (V (λ))u * | f (V (λ))|u * | 2 + F (u * ).
The term F (u * ) can be neglected (it is bounded in L ∞ , so its contribution is at most λ

N 2 λ N -2
2 ). As for the first term, it is easily checked that ˆ|x|≤R

√ λ f (W λ ) dx = λ N -2 ˆ|x|≤R/ √ λ f (W ) dx λ N -2 2 .
(2.31)

Next, we show that if R is large enough, then

ˆ|F (W λ ) -F (V (λ))| dx c 0 c * λ N -2 2 .
In the region |x| ≥ R √ λ from (2.5) and Sobolev embedding we obtain that the contribution is at most λ

|F (W λ ) -F (V (λ))| ζ(λ) • |f (W λ )| + F (ζ(λ)).
The second term is in L ∞ , so its integral is at most O(λ

(V (λ))-f (W λ )| |ζ(λ)f (W λ )|+|f (ζ(λ))| it follows that ˆB(0,R √ λ) |f (V (λ)) -f (W λ )| • |u * | dx λ N -2 2 .
Now consider the first term on the right hand side of (2.30). We have

∂ r (W λ )(R √ λ) ∼ -λ N -2 2 (R √ λ) -N +1 and |u * | ≤ c * near the origin, so we get ˆS(0,R √ λ) ∂ r (W λ ) • u * dσ c * λ N -2 2 .
In the case u * 0 (0) < 0, by continuity if in the definition of c * we choose ρ small enough, then u * (t, x) ≤ -1 2 c * for (t, x) ∈ [t 0 , T + ) × B(0, ρ). In particular, ˆS(0,R

√ λ) ∂ r (W λ ) • u * dσ c * λ N -2 2 ,
where the constant in this estimate is independent of c 0 . The conclusion follows from (2.30) if c 0 is chosen small enough.

We will focus at present on the second term on the left hand side of (2.28). In Lemma 2.8 we treat the simpler case u * 0 ∈ X 2 × H 2 and in Lemma 2.9 we prove a weaker estimate in the case u * ∈ X s × H s , s > N -2 2 , s ≥ 1.

Lemma 2.8. Suppose that u * 0 ∈ X 2 × H 2 . Then for t ∈ [t 0 , T + ) there holds

| DE(V (λ(t)) + u * (t)), g(t) | √ c 0 • sup t≤τ <T + n(g(τ ), λ(τ )) 2 ,
where c 0 is the small constant in (2.4).

Proof. The proof has two steps. First we will show that

| DE(V (λ(t)) + u * (t)) -DE(u * (t)), g(t) | √ c 0 • n(g(t), λ(t)) 2 (2.32)
and then we will check that

d dt DE(u * (t)), g R n(g(t), λ(t)) 2 .
(2.33)

Clearly, integrating (2.33) and using (2.32), we obtain the conclusion for t 0 sufficiently close to T + . Note that the constant in (2.33) is allowed to depend on R (because T + -t 0 can also be chosen depending on R).

In order to prove (2.32), we begin by verifying that

DE(V (λ) + u * ), g -DE(V (λ)), g -DE(u * ), g n(g, λ) 2 . (2.34) This is equivalent to ˆ|f (V (λ) + u * ) -f (V (λ)) -f (u * )| • |g| dx n(g, λ) 2 .
By Hölder and Sobolev inequalities, it suffices to show that

f (V (λ) + u * ) -f (V (λ)) -f (u * ) L 2N N +2 λ N -2 4 .
Using (2.22) we obtain easily that the left hand side is λ

N -2 2 . Recall that R -N +2 = c 0 c * , hence (2.5) gives W λ -V (λ) Ḣ1 √ c 0 c * . Using ∆W λ + f (W λ ) = 0 and the pointwise bound |f (W λ ) -f (V (λ))| f (W λ ) • |W λ -V (λ)| one gets ∆V (λ) + f (V (λ)) Ḣ-1 ∆(W λ -V (λ)) Ḣ-1 + f (W λ ) -f (V (λ)) L 2N N +2 √ c 0 c * , hence | DE(V (λ)), g | √ c 0 • n(g, λ) 2 . (2.35)
Estimate (2.32) follows from (2.34) and (2.35). Notice that until now the assumption u * 0 ∈ X 2 × H 2 has not been used, thus (2.32) holds also in the case u * 0 ∈ X s × H s , s > N -2 2 . We move on to the proof of (2.33). Until the end of this proof all the constants are allowed to depend on R. From (2.17) we get

d dt DE(u * ), g = D 2 E(u * )∂ t u * , g + DE(u * ), J • DE(V (λ) + u * + g) -DE(u * ) -λ ∂ λ V (λ) . Notice that D 2 E(u * )∂ t u * , g = -DE(u * ), J • D 2 E(u * )g , hence it suffices to verify that DE(u * ), J • DE(V (λ) + u * + g) -DE(u * ) -D 2 E(u * )g -λ ∂ λ V (λ) n(g, λ) 2 .
Considering separately the first and the second component, cf. (2.18), we obtain that it is sufficient to verify the following bounds:

| ∆u * + f (u * ), λ ∂ λ V (λ) | n(g, λ) 2 , (2.36) | u * , f (V (λ) + u * + g) -f (V λ ) -f (u * ) -f (u * )g | n(g, λ) 2 , (2.37)
| u * , ∆V (λ) + f (V (λ)) | n(g, λ) 2 .
(2.38)

We know from Appendix A that u * (t) is bounded in X 2 , hence ∆u * + f (u * ) is bounded in L 2N N -2 by the Sobolev embedding. From (2.8) and Hölder inequality it follows that

| ∆u * + f (u * ), ∂ λ V (λ) | λ N -2
f (V (λ) + u * + g) -f (V (λ)) -f (u * ) -f (V (λ) + u * )g L 2N N +2 n(g, λ) 2 (2.39) and u * • (f (V (λ) + u * ) -f (u * )) L 2N N +2 λ N -2 4 .
(2.40)

We first prove (2.40). For |x| ≥ R √ λ the integrand equals 0, and in the region

|x| ≤ R √ λ there holds |f (V (λ))| + |f (u * )| f (W λ ). • For N = 3 u * ∈ H 2 ⊂ L ∞ and f (W λ ) L 6 5 λ 1 2 .
• For N = 4 u * ∈ H 2 ⊂ L 12 and f (W λ ) In all three cases (2.40) follows from Hölder inequality.

By a pointwise bound we have

f (V (λ) + u * ) -f (V (λ)) -f (u * ) L 2N N +2 u * • f (V (λ)) L 2N N +2 + f (u * ) • V (λ) L 2N N +2
.

It is easy to check that

f (V (λ)) L 2N N +2 ≤ f (W λ ) L 2N N +2 λ N -2 2 .
Together with (2.8) this yields

f (V (λ) + u * ) -f (V (λ)) -f (u * ) L 2N N +2 λ N -2 2 ,
and (2.39) follows from (2.24) and the Hölder inequality.

In order to prove (2.38), we write:

| u * , ∆V (λ) + f (V (λ)) | ≤ | u * , ∆(V (λ) -W λ ) | + | u * , f (V (λ) -f (W λ ) |. (2.41)
Consider the first term of (2.41). Integrating twice by parts we find

ˆu * • ∆(V (λ) -W λ ) dx = ˆ|x|≥R √ λ ∇ u * • ∇(W λ ) dx = ˆS(0,R √ λ) u * • ∂ r (W λ ) dσ - ˆ|x|≥R √ λ u * • ∆(W λ ) dx.
As for the first term, recall that

|∂ r (W λ (R √ λ))| λ N -2
2 , so it suffices to notice that by the Trace Theorem ´| u * | dσ 1 for λ 1. In order to bound the second term, we compute

f (W λ ) L 2N N +2 (|x|≥R √ λ) = f (W ) L 2N N +2 (|x|≥R/ √ λ) ∼ λ N +2 4 λ N -2 2 ,
and use Hölder.

Consider the second term of (2.41). From (2.6) we have |f

(V (λ)) -f (W λ )| f (W λ ), hence: ˆu * • f (V (λ)) -f (W λ ) dx ˆ| u * | • f (W λ ) dx,
and the required bound follows from Hölder and the fact that f (W λ )

L 2N N +2 λ N -2 2 .
Lemma 2.9. Suppose that u * 0 ∈ X s × H s , s > N -2 2 and s ≥ 1. There exists a decomposition

DE(V (λ(t)) + u * (t)), g(t) = b 1 (t) + b 2 (t)
such that for t ∈ [t 0 , T + ) there holds:

|b 1 (t)| λ(t) N -4 2 g E , (2.42
) g ". We know that ∆u * is bounded in H s-1 , hence from (2.9) we obtain

|b 2 (t)| √ c 0 • sup t≤τ <T + n(g(τ ), λ(τ )) 2 . ( 2
| ∆u * , ∂ λ V (λ) | λ N -4 2 . Since f (u * ) L 2N N -2 is bounded and N -2 4 > N -4 2 , from (2.8) we get | f (u * ), ∂ λ V (λ) | λ N -4
2 . Using (2.15), it follows that

| ∆u * + f (u * ), λ ∂ λ V (λ) | λ N -4 2
g .

The proof of (2.37) applies almost without changes, but instead of (2.40) we need to check that u *

• (f (V (λ) + u * ) -f (u * )) L 2N N +2 λ N -4
2 , which will follow from

u * • f (W λ ) L 2N N +2 λ N -4 2 .
(2.44)

We check (2.44) separately for N = 3, 4, 5. Recall that u * is bounded in H s . If N = 3, then u * L 6 and f (W λ ) L 3 2 are bounded, hence (2.44) follows from Hölder. If N = 4, then (by Sobolev) there exists q > 4 such that u * L q is bounded. It can be checked that for 1 < p < 2, f (W λ ) L p 1, hence (2.44) follows. If N = 5, then there exists q > 5 such that u * L q is bounded. It can be checked that for 5 4 < p < 2, f (W λ ) L p √ λ, hence (2.44) follows. In the proof of (2.38) we have only used the boundedness of u * in H 1 , hence it remains valid and gives the bound

| u * , ∆V (λ) + f (V (λ)) | λ N -2 2 λ N -2 4 .
Remark 2.10. It is not excluded that Lemma 2.8 holds under the assumption u * 0 ∈ X s ×H s , s > N -2

2 , but I was unable to prove it because of possible oscillations of λ(t). Note also that Lemma 2.9 could be proved for less regular u * 0 if we had some control of g(t) in suitable (for example Strichartz) norms.

Lemma 2.8 implies that if u * 0 ∈ X 2 × H 2 , then Lemma 2.9 holds with b 1 (t) = 0. For t 0 ≤ t < T + we define

ϕ(t) := C I c * λ(t) N -2 2 -b 1 (t) + 2 a -(t) 2 + a + (t) 2 (2.45)
(C I is a constant to be chosen shortly). From (2.28) we have

ϕ(t) := C I c * λ(t) N -2 2 + E(V (λ) + u * ) -E(W ) -E(u * ) + 1 2 D 2 E(V (λ) + u * )g, g + 2 a -(t) 2 + a + (t) 2 + b 2 (t) + O( g 3 E ).
(2.46)

We will consider the maximal function:

ϕ M (t) := sup t≤τ <T + ϕ(τ ). Note that ϕ M : [t 0 , T + ) → R is decreasing, lim t→T + ϕ M (t) = 0 and 0 ≥ ϕ M (t) ≥ min(0, ϕ (t)
V (λ) + u * -W λ E is small we obtain 1 2 D 2 E(V (λ(t 2 )) + u * (t 2 ))g(t 2 ), g(t 2 ) + 2 a -(t 2 ) 2 + a + (t 2 ) 2 g(t 2 ) 2 E . (2.48)
From Lemma 2.7, if we choose C I large enough, then

C I c * λ N -2 2 + E(V (λ) + u * ) -E(W ) - E(u * ) c * λ N -2
2 , hence (2.46) and (2.48) yield

ϕ(t 2 ) -b 2 (t 2 ) n(g(t 2 ), λ(t 2 )) 2 .
From Lemma 2.9 we have |b

2 (t 2 )| ≤ √ c 0 • sup t 2 ≤τ <T + n(g(τ ), λ(τ )) 2 = √ c 0 • n(g(t 2 ), λ(t 2 )) 2 ,
hence we obtain

ϕ M (t) ≥ ϕ(t 2 ) n(g(t 2 ), λ(t 2 )) 2 = sup t≤τ <T + n(g(τ ), λ(τ )) 2 ,
provided that c 0 is small enough.

Differential inequalities and conclusion

Lemma 2.12. There exists a constant C a such that for T + -t small enough there holds

|a + (t)| ≤ C a • sup t≤τ <T + n(g(τ ), λ(τ )) 2 , |a -(t)| ≤ C a • sup t≤τ <T + n(g(τ ), λ(τ )) 2 .
Proof. |a

It
-(t 1 )| ≥ d dt |a -(t)| t=t 1 ≥ C 3 • ϕ M (t 1 )
In order to prove (A.6) it suffices to check that

|∇| s (f n (u) -f n (v)) L 2 ≤ C n u -v X s • 1 + f ( u X s ) + f ( v X s ) (the estimate of f n (u) -f n (v) L 2 is a part of (A. 3
)). We write

f n (u) -f n (v) = -(v -u) ˆ1 0 f n ((1 -t)u + tv) dt,
hence by the triangle inequality

|∇| s (f n (u) -f n (v)) L 2 ≤ ˆ1 0 |∇| s (u -v)f n ((1 -t)u + tv) L 2 dt.
We will estimate the integrand for fixed t ∈ [0, 1]. We have

|∇| s (u -v)f n ((1 -t)u + tv) L 2 = |∇| s-1 ∇ (u -v)f n ((1 -t)u + tv) L 2 = |∇| s-1 ∇(u -v) • f n ((1 -t)u + tv) L 2 + |∇| s-1 (u -v) • ((1 -t)∇u + t∇v) • f n ((1 -t)u + tv) L 2 .
The first term is estimated exactly as in (A.9), so we will only consider the second one. From the Leibniz Rule we obtain

|∇| s-1 (u -v) • ((1 -t)∇u + t∇v) • f n ((1 -t)u + tv) L 2 |∇| s-1 (u -v) L 10 • (1 -t)u + tv L 10 3 • f n ((1 -t)u + tv) L 10 + u -v L 10 • |∇| s-1 ((1 -t)∇u + t∇v) L 10 3 • f n ((1 -t)u + tv) L 10 + u -v L p 1 • (1 -t)∇u + t∇v L p 2 • |∇| s-1 f n ((1 -t)u + tv) L p 3 ,
where the exponents p 1 , p 2 , p 3 ∈ (1, +∞) are chosen such that p 1 > 10, p 2 > 10 3 , p 3 < 10, X s ⊂ L p 1 ∩ W 1,p 2 and 1 2 = 1 p 1 + 1 p 2 + 1 p 3 . Estimating the first two lines is straightforward and for the last line we use the Chain Rule together with (A.8).

Proposition A.5. Let N = 5, 1 ≤ s ≤ 2 and u 0 ∈ X s × H s . There exist t 1 < t 0 < t 2 such that the solution u(t) of (NLW) satisfies

u ∈ C([t 1 , t 2 ], X s × H s ).
Proof. Using (A.4) for s = 1 and (A.2) one obtains by a standard procedure that there exists a unique maximal solution

u ∈ C([T 1 , T 2 ], X 1 × H 1 ), T 1 < t 0 < T 2 and T 1 > -∞ ⇒ lim t→T 1 u n X 1 ×H 1 = +∞, T 2 < +∞ ⇒ lim t→T 2 u n X 1 ×H 1 = +∞,
see [START_REF] Jendrej | Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5[END_REF]Proposition B.2] for details. Consider the regularized problem for n ∈ {1, 2, 3, . . .}:

(∂ tt -∆)u n = f n (u n ), (u n (t 0 ), ∂ t u n (t 0 )) = u 0 .
1 Introduction

Setting of the problem and the main result

Let N ≥ 3 be the dimension of the space. For u 0 = (u 0 , u0 ) ∈ E := Ḣ1 (R N ) × L 2 (R N ), define the energy functional

E(u 0 ) = ˆ1 2 | u0 | 2 + 1 2 |∇u 0 | 2 -F (u 0 ) dx,
where

F (u 0 ) := N -2 2N |u 0 | 2N N -2 .
Note that E(u 0 ) is well-defined due to the Sobolev Embedding Theorem. The differential of E is DE(u 0 ) = (-∆u 0 -f (u 0 ), u0 ), where f

(u 0 ) = |u 0 | 4 N -2 u 0 .
We consider the Cauchy problem for the energy critical wave equation:

∂ t u(t) = J • DE(u(t)), u(t 0 ) = u 0 ∈ E. (NLW)
Here, J := 0 Id -Id 0 is the natural symplectic structure. This equation is often written in the form

∂ tt u = ∆u + f (u).
Equation (NLW) is locally well-posed in the space E, see for example Ginibre, Soffer and Velo [START_REF] Ginibre | The global Cauchy problem for the critical nonlinear wave equation[END_REF], Shatah and Struwe [START_REF] Shatah | Well-posedness in the energy space for semilinear wave equations with critical growth[END_REF] (the defocusing case), as well as a complete review of the Cauchy theory in Kenig and Merle [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energycritical focusing non-linear wave equation[END_REF] (for N ∈ {3, 4, 5}), as well as Bulut, Czubak, Li, Pavlović and Zhang [START_REF] Bulut | Stability and unconditional uniqueness of solutions for energy critical wave equations in high dimensions[END_REF] (for N ≥ 6). In particular, for any initial data u 0 ∈ E there exists a maximal time of existence (T -, T + ), -∞ ≤ T -< t 0 < T + ≤ +∞, and a unique solution u ∈ C((T -, T + ); E). In addition, the energy E is a conservation law. In this paper we always assume that the initial data is radially symmetric. This symmetry is preserved by the flow.

For functions v ∈ Ḣ1 , v ∈ L 2 , v = (v, v) ∈ E and λ > 0, we denote

v λ (x) := 1 λ (N -2)/2 v x λ , vλ (x) := 1 λ N/2 v x λ , v λ (x) := v λ , vλ .
A change of variables shows that

E (u 0 ) λ = E(u 0 ).
Equation (NLW) is invariant under the same scaling: if u(t) = (u(t), u(t)) is a solution of (NLW) and λ > 0, then t → u (tt 0 )/λ λ is also a solution with initial data (u 0 ) λ at time t = 0. This is why equation (NLW) is called energy-critical.

A fundamental object in the study of (NLW) is the family of stationary solutions u(t) ≡ ±W λ = (±W λ , 0), where

W (x) = 1 + |x| 2 N (N -2) -(N -2)/2 .
The functions W λ are called ground states. They are the only radially symmetric solutions and, up to translation, the only positive solutions of the critical elliptic problem -∆u -f (u) = 0.

(1.1)

Note however that classification of nonradial solutions of (1. 1) is an open problem (see for example [START_REF] Del Pino | Large energy entire solutions for the Yamabe equation[END_REF] for details).

Recall that the Soliton Resolution Conjecture predicts that a generic bounded (in a suitable sense) solution of a hamiltonian system asymptotically decomposes as a sum of decoupled solitons and a dispersion. This belief is based mainly on the analysis of completely integrable systems, for instance Eckhaus and Schuur [START_REF] Eckhaus | The emergence of solitons of the Korteweg-de Vries equation from arbitrary initial conditions[END_REF]. The only complete classification of the dynamical behaviour of a non-integrable hamiltonian system is the result of Duyckaerts, Kenig and Merle [START_REF] Duyckaerts | Classification of the radial solutions of the focusing, energy-critical wave equation[END_REF], which we recall here for the reader's convenience:

Theorem ( [START_REF] Duyckaerts | Classification of the radial solutions of the focusing, energy-critical wave equation[END_REF]). Let N = 3 and let u(t) : [t 0 , T + ) → E be a radial solution of (NLW). Then one of the following holds:

• Type I blow-up: T + < ∞ and lim t→T + u(t) E = +∞.
• Type II blow-up: T + < ∞ and there exist v 0 ∈ E, an integer n ∈ N \ {0}, and for all j ∈ {1, . . . , n}, a sign ι j ∈ {±1}, and a positive function λ j (t) defined for t close to T + such that

λ 1 (t) λ 2 (t) . . . λ n (t) T + -t as t → T + lim t→T + u(t) -v 0 + n j=1 ι j W λ j (t) E = 0. (1.2)
• Global solution: T + = +∞ and there exist a solution v lin of the linear wave equation, an integer n ∈ N, and for all j ∈ {1, . . . , n}, a sign ι j ∈ {±1}, and a positive function λ j (t) defined for large t such that

λ 1 (t) λ 2 (t) . . . λ n (t) t as t → +∞ lim t→+∞ u(t) -v lin (t) + n j=1 ι j W λ j (t) E = 0. (1.3)
Of special interest are the solutions which are bounded in E and which exhibit no dispersion (that is, v 0 = 0 or v lin = 0) in one or both time directions. One of the consequences of the energy channel estimates in [START_REF] Duyckaerts | Classification of the radial solutions of the focusing, energy-critical wave equation[END_REF] is that in the case N = 3 the only solutions without any dispersion in both time directions are the stationary states W λ . This is in contrast with the case of completely integrable systems.

In the present paper we are interested in solutions with no dispersion in one time direction, say for positive times. According to Theorem 1.1, for N = 3 such a solution has to behave asymptotically as a decoupled superposition of stationary states. Such solutions are called (pure) multi-bubbles (or n-bubbles, where n is the number of bubbles). By conservation of energy, if u(t) is an n-bubble, then E(u(t)) = nE(W ).

The case n = 1 in dimension N ∈ {3, 4, 5} was treated by Duyckaerts and Merle [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF], who obtained a complete classification of solutions of (NLW) at energy level E(u(t)) = E(W ). In particular, the only 1-bubbles are W λ , W - λ and W + λ , where W -and W + are some special solutions converging exponentially to W . The authors solve also the reconnection problem by showing that for negative times W -scatters and W + blows up in norm E in finite time.

Solutions of (NLW) satisfying (1.2) or (1.3) with v 0 = 0 or v lin = 0 can exhibit non-trivial dynamical behaviour, see the results of Krieger, Schlag and Tataru [START_REF] Krieger | Slow blow-up solutions for the H 1 (R 3 ) critical focusing semilinear wave equation[END_REF], Hillairet and Raphaël [START_REF] Hillairet | Smooth type II blow up solutions to the four dimensional energy critical wave equation[END_REF], Donninger and Krieger [START_REF] Donninger | Nonscattering solutions and blowup at infinity for the critical wave equation[END_REF], Donninger, Huang, Krieger and Schlag [START_REF] Donninger | Exotic blowup solutions for the u 5 focusing wave equation in R 3[END_REF] and the author [START_REF] Jendrej | Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5[END_REF].

In the present paper we address the case n = 2, and more specifically the situation when the two bubbles have opposite signs.

Theorem 1. Let N ≥ 3. There exists no radial solutions u : [t 0 , T + ) → E of (NLW) such that lim

t→T + u(t) -W λ 1 (t) + W λ 2 (t) E = 0 (1.4)
and

• in the case T + < +∞, λ 1 (t) λ 2 (t) T + -t as t → T + ,
• in the case T + = +∞, λ 1 (t) λ 2 (t) t as t → +∞.

Remark 1.1. There exist no examples of solutions of (NLW) such that expansion (1.2) or (1.3) holds with n > 1 (with or without dispersion). Note however that spatially decoupled non-radial multi-bubbles were recently constructed by Martel and Merle [START_REF] Martel | Construction of multi-solitons for the energy-critical wave equation in dimension 5[END_REF] using the Lorentz transform. In their setting, the choice of signs seems to have little importance. On the other side, Theorem 1 is, to my knowledge, the only result proving non-existence of solutions of type multi-bubble for (NLW) in some specific cases. Existence of pure twobubbles with the same sign is an open problem. Remark 1.2. In the case of corotational wave maps existence of pure two-bubbles with the same orientation is easily excluded for variational reasons. Our proof might be seen as an adaptation of this argument to the case where the energy functional is not coercive.

Note that for corotational wave maps existence of pure two-bubbles with opposite orientations is an open problem, related to the threshold conjecture for degree 0 equivariant wave maps, see Côte, Kenig, Lawrie and Schlag [START_REF] Côte | Characterization of large energy solutions of the equivariant wave map problem: I[END_REF].

Remark 1.3. For the corresponding slightly sub-critical elliptic problem positive multibubbles cannot form, whereas multi-bubbles with alternating signs exist, see Li [START_REF] Li | Prescribing scalar curvature on S n and related problems, I[END_REF], Pistoia and Weth [START_REF] Pistoia | Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem[END_REF].

Outline of the proof

Step 1. The linearization of (NLW) around W λ has a stable direction Y - λ . We construct stable manifolds U a λ which are forward invariant sets tangent to Y - λ at W λ . They have good regularity and decay properties. They allow to define the refined unstable mode β a λ ∈ E * with the following crucial property.

Decompose any initial data close to the family of stationary states as u 0 = U a λ + g, with g satisfying natural orthogonality conditions by an appropriate choice of λ and a. We have the alternative:

• (Coercivity) | β a λ , g | g 2 E , which implies E(u 0 ) -E(W ) g 2 E , • (Destabilization) | β a λ , g | g 2 E , which implies the exponential growth of | β a λ , g |.
In other words, β a λ provides an explicit way of controlling how solutions which violate the coercivity of energy leave a neighbourhood of the stationary states for positive times.

Step 2. Let u(t) : [t 0 ; T + ) → E be a solution of (NLW) which satisfies (1.4). As already mentioned, this implies that E(u) = 2E(W ).

(1.5)

We decompose for any t ∈ [t 0 , T + ):

u(t) = U a 2 (t) λ 2 (t) -W λ 1 (t) + g(t), λ 1 (t) λ 2 (t),
with g(t) satisfying natural orthogonality conditions (in fact we use a suitable localization of W λ 1 (t) ). From the Taylor formula we obtain

E(u) = E(U a 2 λ 2 -W λ 1 ) + DE(U a 2 λ 2 -W λ 1 , g + 1 2 D 2 E(U a 2 λ 2 -W λ 1 )g, g + o( g 2 E ). (1.6)
An explicit key computation shows that

E(U a 2 λ 2 -W λ 1 ) -2E(W ) (λ 1 /λ 2 ) N -2 2
.

It is at this point that the sign condition is decisive.

Step 3. We prove that the assumption that u(t) stays close to a 2-bubble implies that

| β a 2 λ 2 , g | g 2 E + (λ 1 /λ 2 ) N -2 2
. This allows to show that the second term in the expansion (1.6) is

g 2 E + (λ 1 /λ 2 ) N -2 2
. Finally, by an elementary analysis of the linear stable and unstable modes we can prove that, at least along a sequence of times, the third term of the expansion (1.6) is coercive, that is g 2 E . Inserted in (1.6), this leads to E(u) > 2E(W ), contradicting (1.5).

Notation

We introduce the infinitesimal generators of the scale change

Λ s := N 2 -s + x • ∇.
For s = 1 we omit the subscript and write Λ = Λ 1 . We denote Λ E , Λ F and Λ E * the infinitesimal generators of the scaling which is critical for a given norm, that is

Λ E = (Λ, Λ 0 ), Λ F = (Λ 0 , Λ -1 ), Λ E * = (Λ -1 , Λ 0 ).
We use the subscript • λ to denote rescaling with characteristic length λ, critical for a norm which will be known from the context. We introduce the following notation for some frequently used function spaces:

X s := Ḣs+1 rad ∩ Ḣ1 rad for s ≥ 0, Y k := H k (1 + |x| k ) for k ∈ N, E := Ḣ1 rad × L 2 rad , F := L 2 rad × Ḣ-1 rad . Notice that E * Ḣ-1 rad × L 2
rad through the natural isomorphism given by the distributional pairing. In the sequel we will omit the subscript and write Ḣ1 for Ḣ1 rad etc. We denote J := 0 Id -Id 0 ; note that JE * = F.

For a function space A, O A (m) denotes any a ∈ A such that a A ≤ Cm for some constant C > 0. We denote B A (x 0 , η) an open ball of center x 0 and radius η in the space A. If A is not specified, it means that A = R.

For a radial function g : R N → R and r ≥ 0 we denote g(r) the value of g(x) for |x| = r.

2 Sharp coercivity properties near W λ

Properties of the linearized operator

Linearizing (NLW) around W , u = W + g, one obtains

∂ t g = J • D 2 E(W )g = 0 Id -L 0 g,
where L is the Schrödinger operator

Lg := (-∆ -f (W ))g. Notice that L(ΛW ) = d dλ λ=1 -∆W λ -f (W λ ) = 0.
It is known that L has exactly one strictly negative simple eigenvalue which we denote -ν 2 (we take ν > 0). We denote the corresponding positive eigenfunction Y, normalized so that Y L 2 = 1. By elliptic regularity Y is smooth and by Agmon estimates it decays exponentially. Self-adjointness of L implies that Y, ΛW = 0.

Fix Z ∈ C ∞ 0 such that Z, ΛW > 0, Z, Y = 0.
We have the following linear (localized) coercivity result, similar to [58, Lemma 2.1].

Lemma 2.1. There exist constants c, C > 0 such that

• for all g ∈ Ḣ1 radially symmetric there holds

g, Lg = ˆRN |∇g| 2 dx - ˆRN f (W )|g| 2 dx ≥ c ˆRN |∇g| 2 dx -C Z, g 2 + Y, g 2 , (2.1)
• if r 1 > 0 is large enough, then for all g ∈ Ḣ1 rad there holds

(1 -2c) ˆ|x|≤r 1 |∇g| 2 dx + c ˆ|x|≥r 1 |∇g| 2 dx - ˆRN f (W )|g| 2 dx ≥ -C Z, g 2 + Y, g 2 , (2.2)
• if r 2 > 0 is small enough, then for all g ∈ Ḣ1 rad there holds

(1 -2c) ˆ|x|≥r 2 |∇g| 2 dx + c ˆ|x|≤r 2 |∇g| 2 dx - ˆRN f (W )|g| 2 dx ≥ -C Z, g 2 + Y, g 2 .
(2.3)

Proof. We will prove (2.2) and (2.3). For a proof of (2.1) we refer to [40, Lemma 6.1], see also [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF]Proposition 5.5] for a different formulation. We define the projections Π r , Ψ r : Ḣ1 → Ḣ1 :

(Π r g)(x) := g(r) if |x| ≤ r, g(x) if |x| ≥ r, (Ψ r g)(x) := g(x) -g(r) if |x| ≤ r, 0 if |x| ≥ r (thus Π r + Ψ r = Id).
Applying (2.1) to Ψ r 1 g with c replaced by 3c and C replaced by C 2 we get

(1 -2c) ˆ|x|≤r 1 |∇g| 2 dx = (1 -2c) ˆRN |∇(Ψ r 1 g)| 2 dx ≥ (1 + c) ˆRN f (W )|Ψ r 1 g| 2 dx - C 2 Z, Ψ r 1 g 2 + Y, Ψ r 1 g 2 .
(2.4) By Sobolev and Hölder inequalities we have

ˆ|x|≥r 1 f (W )|g| 2 dx = ˆ|x|≥r 1 f (W )|Π r 1 g| 2 dx f (W ) L N 2 (|x|≥r 1 ) • Π r 1 g 2 Ḣ1 ≤ c 4 ˆ|x|≥r 1 |∇g| 2 dx (2.5)
if r 1 is large enough.

In the region |x| ≤ r 1 we apply the pointwise inequality

|g(x)| 2 ≤ (1 + c)|(Ψ r 1 g)(x)| 2 + (1 + c -1 )|g(r 1 )| 2 , |x| ≤ r 1 . (2.6)
Recall that by the Strauss Lemma [START_REF] Strauss | Existence of solitary waves in higher dimensions[END_REF], for a radial function g there holds

|g(r 1 )| Π r 1 g Ḣ1 • r -N -2 2 1
.

Since f (W (r)) ∼ r -4 as r → +∞, we have

ˆ|x|≤r 1 f (W ) dx r N -2 1 , as r 1 → +∞, hence ˆ|x|≤r 1 f (W ) • (1 + c -1 )|g(r 1 )| 2 dx ≤ c 4
f (W )|g| 2 dx ≤ (1 + c) ˆRN f (W )|(Ψ r 1 g)(x)| 2 dx + c 2 ˆ|x|≥r 1 |∇g| 2 dx. (2.8) Using the fact that Y ∈ L 1 ∩ L 2N N +2 we obtain | Y, Π r 1 g | Π r 1 g Ḣ1 • r -N -2 2 1 + ˆ|x|≥r 1 Y|g| dx (r -N -2 2 1 + Y L 2N N +2 (|x|≥r 1 ) ) Π r 1 g Ḣ1 , hence C 2 Y, Ψ r 1 g 2 ≤ C Y, g 2 + C Y, Π r 1 g 2 ≤ C Y, g 2 + c 4 ˆ|x|≥r 1 |∇g| 2 dx, (2.9)
provided that r 1 is chosen large enough. Similarly,

C 2 Z, Ψ r 1 g 2 ≤ C Z, g 2 + C Z, Π r 1 g 2 ≤ C Z, g 2 + c 4
|∇g| 2 dx = (1 -3c) ˆRN |∇(Π r 2 g)| 2 dx ≥ ˆRN f (W )|Π r 2 g| 2 dx - C 2 Z, Π r 2 g 2 + Y, Π r 2 g 2 .
(

|∇g| 2 dx ≥ ˆRN f (W )|g| 2 dx- C 2 Z, Π r 2 g 2 + Y, Π r 2 g 2 . (2.13) Using the fact that Y ∈ L 2N N +2 we obtain | Y, Ψ r 2 g | ˆ|x|≤r 2 Y|g| dx Y L 2N N +2 (|x|≤r 2 ) Ψ r 2 g Ḣ1 , hence C 2 Y, Π r 2 g 2 ≤ C Y, g 2 + C Y, Ψ r 2 g 2 ≤ C Y, g 2 + c 4 ˆ|x|≤r 2 |∇g| 2 dx, (2.14)
provided that r 2 is chosen small enough. Similarly, We define

C 2 Z, Π r 2 g 2 ≤ C Z, g 2 + C Z, Ψ r 2 g 2 ≤ C Z, g 2 + c 4
Y -:= 1 ν Y, -Y , Y + := 1 ν Y, Y , α -:= ν 2 JY + = 1 2 (νY, -Y), α + := - ν 2 JY -= 1 2 (νY, Y). We have J • D 2 E(W ) = 0 Id -L 0 . A short computation shows that J • D 2 E(W )Y -= -νY -, J • D 2 E(W )Y + = νY + and α -, J • D 2 E(W )g = -ν α -, g , α + , J • D 2 E(W )g = ν α + , g , ∀g ∈ E. (2.16)
We will think of α -and α + as linear forms on E. Notice that α -, Y -= α + , Y + = 1 and α -, Y + = α + , Y -= 0. The rescaled versions of these objects are

Y - λ := 1 ν Y λ , -Y λ , Y + λ := 1 ν Y λ , Y λ , α - λ := ν 2λ JY + λ = 1 2 ν λ Y λ , -Y λ , α + λ := - ν 2λ JY - λ = 1 2 ν λ Y λ , Y λ .
(2.17)

The scaling is chosen so that

α - λ , Y - λ = α + λ , Y + λ = 1. We have J • D 2 E(W λ )Y - λ = - ν λ Y - λ , J • D 2 E(W λ )Y + λ = ν λ Y + λ (2.18) and α - λ , J • D 2 E(W λ )g = - ν λ α - λ , g , α + λ , J • D 2 E(W λ )g = ν λ α + λ , g , ∀g ∈ E.
(2.19) As a standard consequence of (2.1), we obtain the following: Lemma 2.2. There exists a constant η > 0 such that if V -W λ E < η, then for all g ∈ E such that Z λ , g = 0 there holds

1 2 D 2 E(V )g, g + 2 α - λ , g 2 + α + λ , g 2 g 2 E .
Proof. For N ∈ {3, 4, 5} see [39, Lemma 2.2]. For N ≥ 6 the same proof is valid, once we notice that f

(V ) -f (W λ ) L N 2 ≤ f ( V -W λ Ḣ1 ).
We now turn to the proofs of various energy estimates for the linear group generated by

A := J • D 2 E(W ) = 0 Id -L 0 .
on its invariant subspaces, which will be needed in Subsection 2.2. This is much in the spirit of [START_REF] Bourgain | Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity[END_REF]Section 2]. It follows from (2.16) that the centre-stable subspace X cs := ker α + , the centre-unstable subspace X cu := ker α -and the centre subspace X c := X cs ∩ X cu are invariant subspaces of the operator A. Notice that α -, Y

-= α + , Y + = 1, E = X cs ⊕ {aY + } = X cu ⊕ {aY -}, X cs = X c ⊕ {aY -}, X cu = X c ⊕ {aY + }.
We define

X cc := {v = (v, v) ∈ X c | Z, v = 0}. Lemma 2.3. Let k ∈ N.
There exist constants 1 = a 0 > a 1 > . . . > a k > 0 such that the norm • A,k defined by the following formula:

v 2 A,k := k j=0 a j v, L j+1 v + v, L j v satisfies v X k ×H k ∼ v A,k for all v = (v, v) ∈ (X k × H k ) ∩ X cc .
Proof. We proceed by induction. For k = 0 we have

v A,0 = v, Lv + v, v = D 2 E(W )v, v . By Lemma 2.2, this norm is equivalent to • E on E ∩ X cc .
To check the induction step, one should show that for any k > 0 there exist a 1 , a 2 > 0 such that v

2 1 := v 2 X k-1 + a 1 v, L k+1 v v 2 X k (2.20) and v 2 2 := v 2 H k-1 + a 2 v, L k v v 2 H k . (2.21)
To prove (2.21) notice that L k = (-∆) k + (terms with at most 2k -2 derivatives).

Integrating by parts all the terms except for the first one we arrive at expressions of the form We will denote •, • A,k the scalar product associated with the norm • A,k . We define the projections:

´V • ∂ i v • ∂ j v dx
π s v := α -, v Y -, π cu := Id -π s .
We denote π cc the projection of X c on X cc in the direction Λ E W . These projections are continuous linear operators on E as well as on X k × H k for k > 0.

Proposition 2.4. The operator A generates a strongly continuous group on X k ×H k denoted e tA . Moreover, the following bounds are true for t ≥ 0:

v 0 ∈ (X k × H k ) ∩ X s ⇒ e tA v 0 X k ×H k e -νt v 0 X k ×H k , (2.22) v 0 ∈ (X k × H k ) ∩ X cu ⇒ e -tA v 0 X k ×H k (1 + t) v 0 X k ×H k , (2.23) v 0 ∈ X k × H k ⇒ e -tA v 0 X k ×H k e νt v 0 X k ×H k . ( 2

.24)

Proof. It suffices to analyse the restriction to the invariant subspace X c . Take v 0 ∈ X c and decompose

v 0 = l 0 Λ E W + w 0 , w 0 ∈ X cc (notice that Λ E W ∈ X k × H k ).
It can be checked that the operator B := π cc •A is skew-adjoint for the scalar product •, • A,k , hence it generates a unitary group w(t) = e tB w 0 by the Stone theorem. Let l(t) be defined by the formula

l(t) = l 0 + ˆt 0 Z, ẇ(t) Z, ΛW dt. (2.25) Set v(t) = w(t) + l(t)Λ E W .
This defines a linear group and lim

t→0 1 t (v(t) -v 0 ) = Bw 0 + l (0)Λ E W = Bv 0 + Z, ẇ0 Z, ΛW Λ E W = Av 0 , hence v(t) = e tA v 0 .
Estimate (2.22) follows immediately from the fact that Y -is an eigenfunction of A with eigenvalue -ν. Analogously, in (2.23) we can assume that v 0 ∈ X c (the unstable mode decreases exponentially for negative times). By the equivalence of norms and the fact that the group generated by B is unitary for the norm It is possible to finish the construction for example in the space X 1 × H 1 . However, later we will need some information on the spatial decay of the constructed functions, which forces us to use weighted spaces. We define

• A,k , w(t) X k ×H k v 0 X k ×H k for all t, (2.26
v Y k := (1 + |x| k )v H k .
One may check by induction on j = 0, 1, . . . , k that

(1 + |x| k )v 2 H j ∼ ˆ(1 + |x| 2k )(|v| 2 + |∇ j v| 2 ) dx, in particular v 2 Y k ∼ ˆ(1 + |x| 2k )(|v| 2 + |∇ k v| 2 ) dx. Lemma 2.6. Let k ∈ Z, k ≥ 0.
The following bounds are true for t ≥ 0:

v 0 ∈ (Y k+1 × Y k ) ∩ X s ⇒ e tA v 0 Y k+1 ×Y k e -νt v 0 Y k+1 ×Y k , (2.27)
v 0 ∈ (Y k+1 × Y k ) ∩ X cu ⇒ e -tA v 0 Y k+1 ×Y k (1 + t k(k+1) 2
+1 ) v 0 Y k+1 ×Y k , (2.28)
v 0 ∈ Y k+1 × Y k ⇒ e -tA v 0 Y k+1 ×Y k e νt v 0 Y k+1 ×Y k . (2.29)
Proof. The proof of (2.27) and (2.29) is the same as in Proposition 2.4, once we recall that Y -∈ Y k+1 × Y k . In order to prove (2.28), write v(t) = e -tA v 0 , so that

∂ t v = -Av = (-v, -∆v -f (W )v), hence 1 2 d dt ˆ(1 + |x| 2k )(| v| 2 + |∇v| 2 ) dx = -ˆ(1 + |x| 2k ) (∆v + f (W )v) • v + ∇ v • ∇v dx = ˆ∇(|x| 2k ) • ∇v • v + (1 + |x| 2k )(f (W )v) • v dx
(W )v L 2 v L 2N N -2 ∇v L 2 , thus d dt ˆ(1 + |x| 2k )(| v| 2 + |∇v| 2 ) dx ˆ(1 + |x| 2k-1 )(| v| 2 + |∇v| 2 ) dx. (2.30) Analogously, from 1 2 d dt ˆ(1 + |x| 2k )(|∇ k v| 2 + |∇ k+1 v| 2 ) dx = -ˆ(1 + |x| 2k ) ∇ k (∆v + f (W )v) • ∇ k v + ∇ k+1 v • ∇ k+1 v dx = ˆ∇(|x| 2k ) • ∇ k+1 v • ∇ k v + (1 + |x| 2k )∇ k (f (W )v) • ∇ k v dx we deduce d dt ˆ(1 + |x| 2k )(|∇ k v| 2 + |∇ k+1 v| 2 ) dx ˆ(1 + |x| 2k-1 )(| v| 2 + |∇ k v| 2 + |∇v| 2 + |∇ k+1 v| 2 ) dx.
(2.31) Using (2.30), (2.31) and Hölder we obtain

d dt ˆ(1 + |x| 2k )(| v| 2 + |∇ k v| 2 + |∇v| 2 + |∇ k+1 v| 2 ) dx ˆ(1 + |x| 2k )(| v| 2 + |∇ k v| 2 + |∇v| 2 + |∇ k+1 v| 2 ) dx 2k-1 2k • v 1 k X k ×H k ,
which gives, using (2.23) and integrating,

ˆ(1 + |x| 2k )(| v| 2 + |∇ k v| 2 + |∇v| 2 + |∇ k+1 v| 2 ) dx (1 + t k(k+1) ) v 0 2 Y k+1 ×Y k .
Now we can easily bound the L 2 term by Schwarz inequality:

d dt ˆ(1 + |x| 2k )|v| 2 dx ˆ(1 + |x| 2k )|v| 2 dx 1 2 • ˆ(1 + |x| 2k )| v| 2 dx 1 2 , which leads to ˆ(1 + |x| 2k )|v| 2 dx (1 + t k(k+1)+2 ) v 0 2 Y k+1 ×Y k .
We fix k ∈ N large enough. For ν > 0 the space BC ν is defined as the space of continuous functions

v : [0, +∞) → Y k+1 × Y k with the norm v BC ν := sup t∈[0,+∞) e νt v(t) Y k+1 ×Y k . Lemma 2.7. If ν ∈ (0, ν), then for any w ∈ BC ν the equation ∂ t v(t) = Av(t) + w(t) (2.32)
has a unique solution v = Kw ∈ BC ν such that α -, v(0) = 0.

In addition, K is a bounded linear operator on BC ν .

Proof. Suppose that v ∈ BC ν verifies (2.32). Denote v 0 = v(0). From the Duhamel formula we obtain

v(t) = e tA v 0 + ˆt 0 e (t-τ )A w(τ ) dτ ⇒ e -tA π cu v(t) = π cu v 0 + ˆt 0 e -τ A π cu w(τ ) dτ. (2.33)
By assumption, v(t) Y k+1 ×Y k e -ν , hence from (2.23) we infer e -tA π cu v(t) (1 + t κ )e -ν , κ := 1 2 k(k + 1) + 1. Passing to the limit t → +∞ yields

π cu v 0 = - ˆ+∞ 0 e -τ A π cu w(τ ) dτ.
If we require α -, v 0 = 0, this determines uniquely v 0 = π cu v 0 , hence, using (2.33),

v(t) = Kw(t) = - ˆ+∞ t e (t-τ )A π cu w(τ ) dτ + ˆt 0 e (t-τ )A π s w(τ ) dτ.
From (2.22) and (2.23) we obtain

Kw(t) Y k+1 ×Y k w BC ν • ˆ+∞ t (1 + (τ -t) κ )e -ντ dτ + ˆt 0 e (τ -t)ν e -ντ dτ w BC ν • e -νt ,
so K is a bounded operator.

Remark 2.8. By linearity the unique solution of (2.32) such that α -, v(0)) = a is v(t) = (Kw)(t) + e -νt aY -.

Construction of U a λ

As noted earlier, the functions U a λ were constructed in [START_REF] Duyckaerts | Dynamics of threshold solutions for energy-critical wave equation[END_REF]Section 6]. However, the construction given there does not give the additional regularity or decay, which is required in the present paper. For this reason, we provide here a different construction, which is an adaptation of a classical ODE proof, see for instance [START_REF] Chow | Methods of Bifurcation Theory[END_REF]Chapter 3.6].

We denote

R(v) := f (W + v) -f (W ) -f (W )v.
Lemma 2.9. Let ν ∈ (0, ν). There exist η > 0 such that for every b ∈ R, |b| < η there is a unique solution v = v b ∈ BC ν of the equation

∂ t v(t) = Av(t) + R(v(t)) (2.34) such that α -, v(0) = b and v BC ν < η. Moreover, v b is analytic with respect to b.
Proof. Let T : BC ν × R → BC ν be defined by the formula

T (v, b) := e -νt bY -+ K(R(v)),
where K is the operator from Lemma 2. Proposition 2.10. For any k ∈ N there exists η > 0 and an analytic function

(-η, η) a → U a -W ∈ Y k+1 × Y k such that U 0 = W , (2.35)
∂ a U a | a=0 = Y -, (2.36)
-νa∂ a U a = J • DE(U a ).

(2.37)

Proof. Evaluation at t = 0 is a bounded linear operator from

BC ν to Y k+1 × Y k , hence {v b (0) : b ∈ (-η, η)} defines an analytic curve in Y k+1 × Y k . We have v b BC ν |b|, so R(v b ) BC ν |b| 2 . By construction, v b satisfies the equation v b = be -νt Y -+ K(R(v b )), hence v b -be -νt Y - BC ν |b| 2 , in particular v b (0) -bY - Y k+1 ×Y k |b| 2 .
Because of uniqueness in Lemma 2.9, the set {v b (0

) : b ∈ (-η, η)} is forward invariant if η is small enough, hence for all b ∈ (-η, η) there exists a function b(t) such that v b (t) = v b(t) (0). The value of b(t) is determined by the condition α -, v b (t) = b(t).
Differentiating in time this condition we find

b (t) = d dt α -, v b (t) = α -, J • DE(W + v b (t)) = α -, J • DE(W + v b(t) (0)) = ψ(b(t)),
where ψ is an analytic function, ψ(0) = 0 and ψ (0) = -ν. By Lemma A.1, there exists an analytic change of variable a = a(b) which transforms the equation b (t) = ψ(b(t)) into a (t) = -νa(t) and such that a(0) = 0, a (0) = 1. We define

U a := W + v b(a) (0).
We will denote U a λ := (U a ) λ . Rescaling (2.35), (2.36) and (2.37) we obtain is a solution of (NLW) for large t. These are precisely the solutions W ± λ mentioned in the Introduction.

U 0 λ = W λ , ∂ a U a λ | a=0 = Y - λ , ∂ a U a λ = - λ νa J • DE(U a λ ). ( 2

Modulation near the stable manifold

The results of this subsection will not be directly used in the proof of Theorem 1. We include them in the paper for their own interest and because the proofs introduce in a simple setting the main technical ideas required in Section 3.

It is well known since the work of Payne and Sattinger [START_REF] Payne | Saddle points and instability of nonlinear hyperbolic equations[END_REF] that solutions of energy < E(W ) leave a neighbourhood of the family of stationary states. The aim of this subsection is to describe an explicit local mechanism leading to this phenomenon, which is robust enough not to be significantly altered by the presence of the second bubble (as will be the case in Section 3).

Note that nothing specific to the wave equation has been used so far, hence one might expect that all the proofs of Section 2 should apply to similar (not necessarily critical) models in the presence of one instability direction near a stationary state. Lemma 2.12. Let δ 0 > 0 be sufficiently small. For any 0 ≤ δ ≤ δ 0 there exists 0

≤ η = η(δ) ---→ δ→0 0 such that if u : (t 1 , t 2 ) → E is a solution of (NLW) satisfying for all t ∈ (t 1 , t 2 ) u(t) -W λ(t) E ≤ δ, λ(t) > 0,
then there exist unique functions λ(t) ∈ C 1 ((t 1 , t 2 ), (0, +∞)) and a(t

) ∈ C 1 ((t 1 , t 2 ), R) such that for g(t) := u(t) -U a(t) λ(t) (2.39)
the following holds for all t ∈ (t 1 , t 2 ): Proof. We follow a standard procedure, see for instance [57, Proposition 1].

Z λ(t) , g(t) = α - λ(t) , g(t) = 0, ( 2
Step 1. We will first show that for fixed t 0 ∈ (t 1 , t 2 ) there exist unique λ(t 0 ) and a(t 0 ) such that (2.40), (2.41) and (2.42) hold for t = t 0 . Without loss of generality we can assume that λ(t 0 ) = 1 (it suffices to rescale everything). We consider Φ : E × R 2 → R 2 defined as Φ(u 0 ; l 0 , a 0 ) = Φ 1 (u 0 ; l 0 , a 0 ), Φ 2 (u 0 ; l 0 , a 0 ) := e -l 0 Z e l 0 , u 0 -U a 0 e l 0 , α - e l 0 , u 0 -U a 0 e l 0 . One easily computes:

∂ l Φ 1 (W ; 0, 0) = Z, ΛW > 0, ∂ l Φ 2 (W ; 0, 0) = 0, ∂ a Φ 1 (W ; 0, 0) = 0, ∂ a Φ 2 (W ; 0, 0) = -α -, Y -= -1.
Applying the Implicit Function Theorem with u 0 := u(t 0 ) we obtain existence of parameters a 0 =: a(t 0 ) and λ 0 = e l 0 =: λ(t 0 ).

Step 2. We will show that λ(t) (equivalently, l(t) := log(λ(t))) and a(t) are C 1 functions of t.

Take t 0 ∈ (t 1 , t 2 ) and let a 0 := a(t 0 ), l 0 := log(λ(t 0 )). Define ( l, a)

: (t 0 -ε, t 0 + ε) → R 2 as the solution of the differential equation d dt ( l(t), a(t)) = -(∂ l,a Φ) -1 (D u Φ)∂ t u(t)
with the initial condition l(t 0 ) = l 0 , a(t 0 ) = a 0 . Notice that D v Φ is a continuous functional on F, so we can apply it to ∂ t u(t).

Using the chain rule we get d dt Φ(u(t); l(t), a(t)) = 0 for t ∈ (t 0 -ε, t 0 + ε). By continuity, | l(t) -l 0 | < η in some neighbourhood of t = t 0 . Hence, by the uniqueness part of the Implicit Function Theorem, we get l(t) = log λ(t) and a(t) = a(t) in some neighbourhood of t = t 0 . In particular, λ(t) and a(t) are of class C 1 in some neighbourhood of t 0 .

Step 3. From (2.39) we obtain the following differential equation of the error term g:

∂ t g = ∂ t (u -U a λ ) = J • DE(u) -DE(U a λ ) -∂ t U a λ -J • DE(U a λ ) .
We have

∂ t U a λ = λ ∂ λ U a λ + a ∂ a U a λ = -λ • 1 λ Λ E U a λ + a ∂ a U a λ , (2.45)
so using (2.38) we get

∂ t g = J • (DE(U a λ + g) -DE(U a λ )) + λ • 1 λ Λ E U a λ -a + νa λ ∂ a U a λ . (2.46)
The first component reads:

∂ t g = ġ + λ ΛU a λ + 1 + λa νa U a λ ,
hence differentiating in time the first orthogonality relation 1 λ Z λ , g = 0 we obtain

0 = d dt 1 λ Z λ , g = - λ λ 2 Λ 0 Z λ , g + 1 λ Z λ , ġ + λ λ 2 Z λ , ΛU a λ + 1 λ + a νa Z λ , U a λ . (2.47)
Differentiating the second orthogonality relation α - λ , g = 0 and using (2.46) we obtain

0 = d dt α - λ , g = - λ λ Λ E * α - λ , g + α - λ , J • (DE(U a λ + g) -DE(U a λ )) + λ λ α - λ , Λ E U a λ -a + νa λ α - λ , ∂ a U a λ .
Together with (2.47) this yields the following linear system for λ and λ a + νa λ :

M 11 M 12 M 21 M 22 λ λ a + νa λ = -Z λ , ḣ -λ α - λ , J • (DE(U a λ + g) -DE(U a λ ))
,

where

M 11 = 1 λ Z λ , ΛU a λ - 1 λ Λ -1 Z λ , g , M 12 = 1 νa Z λ , U a λ , M 21 = -Λ E * α - λ , g + α - λ , Λ E U a λ , M 22 = -α - λ , ∂ a U a λ .
(2.48)

Since 1 λ Z λ , ΛW λ 1, α - λ , Λ E W = 0, α - λ , Y - λ = 1, Λ E U a λ -Λ E W λ E |a| and ∂ a U a λ - Y - λ E |a|, we see that |M 11 | ∼ 1, |M 12 | 1, |M 21 | g E + |a|, |M 22 | ∼ 1.
Hence, det M 11 M 12 M 21 M 22 1 and we obtain In the rest of this section λ(t) and a(t) denote the modulation parameters obtained in Lemma 2.12 and g(t) is the function defined by (2.39).

|λ | |M 22 | • | Z λ , ġ | + |M 12 | • |λ α - λ , J • (DE(U a λ + g) -DE(U a λ )) |, a + λa ν |M 21 | • | Z λ , ġ | + |M 11 | • |λ α - λ , J • (DE(U a λ + g) -DE(U a λ )) |. (2.49) Since α - λ , J • D 2 E(W λ )g = -ν λ α - λ , g = 0, Lemma A.4 implies that | α - λ , J • (DE(U a λ + g) -DE(U a λ )) | 1 λ g E • (|a| + g E ). ( 2
For given modulation parameters λ and a we define:

β a λ := - ν 2λ J∂ a U a λ . (2.51)
We see that Proof.

β 0 λ = -ν 2λ JY - λ = α + λ ,
Step 1. We check that Similarly one obtains As an application of the preceding proposition, we now show that the stable manifold U a λ is the only source of the lack of coercivity of the energy functional restricted to the trajectories staying close to the family of stationary states. Given u 0 ∈ E, let u(t) : [0, T + ) → E denote the maximal solution of (NLW) with initial data u(0) = u 0 . For η > 0 sufficiently small we define the centre-stable set M cs as M cs := u 0 : sup

| β a λ -α + λ , g | |a| • g E , (2.52) | β a λ -α + λ , ∂ t g | 1 λ |a| • g E , (2.53)
| ∂ a β a λ , g | + | λ∂ λ β a λ , g | g E . ( 2
| β a λ -α + λ , J • (DE(U a λ + g) -DE(U a λ )) | 1 λ |a| • g E , (2.55)
| β a λ -α + λ , Λ E U a λ | + | β a λ -α + λ , ∂ a U a λ | |a|, ( 2
0≤t<T + inf λ>0 u(t) -W λ E ≤ η .
Remark 2.14. In the case N = 3 it was proved by Krieger, Nakanishi and Schlag [START_REF] Krieger | Center-stable manifold of the ground state in the energy space for the critical wave equation[END_REF] that M cs is a local C 1 manifold tangent at u 0 = W to X cs . Remark 2.15. It is not difficult to see that if M cs is a regular hypersurface, then necessarily its tangent space at U a λ is given by U a λ + ker β a λ = {U a λ + g : β a λ , g = 0}.

Hence b(t) is a natural candidate to measure how a trajectory moves away from M cs .

Corollary 2.16. If η > 0 is small enough, then there exists a constant C E > 0 such that

u 0 ∈ M cs ⇒ inf λ>0,a∈R u 0 -U a λ 2 E ≤ C E E(u 0 ) -E(W ) .
Proof.

Step 1 -Coercivity. We will prove that if g E is small enough and Z λ , g = α - λ , g = 0, then E(U a λ + g) -E(W ) + 2a β a λ , g + 2| β a λ , g | 2 ∼ g 2 E .

(2.63)

We have 2a β a λ , g = DE(U a λ ), g , hence Lemma A. As a result, |b(0)| ≤ 2C 2 E(u 0 ) -E(W ) . Since |a(0)| and g(0) E may be assumed as small as we wish, the conclusion follows from (2.67)

u 0 ∈ M c ⇒ inf λ>0 u 0 -W λ 2 E ≤ C E E(u 0 ) -E(W ) ,
where M c := M cs ∩ M cu = u 0 : sup

T -<t<T + inf λ>0 u(t) -W λ E ≤ η .
3 Nonexistence of pure two-bubbles with opposite signs

Modulation near the sum of two bubbles

Because of a slow decay of W , we will introduce compactly supported approximations of W λ . Let R > 0 be a large constant to be chosen later. We denote

V R (λ 1 , λ 2 )(x) := W λ 1 (x) -ζ(λ 1 , λ 2 ) when |x| ≤ R √ λ 1 λ 2 , 0 when |x| ≥ R √ λ 1 λ 2 , (3.1)
where

ζ(λ 1 , λ 2 ) := W λ 1 (R λ 1 λ 2 ) = 1 λ 1 N -2 2 1 + R 2 λ 2 N (N -2)λ 1 -N -2 2 = λ 1 + R 2 λ 2 N (N -2) -N -2 2 .
We have

ζ(λ 1 , λ 2 ) ∼ R -(N -2) λ -N -2 2 2 , ∂ λ 1 ζ(λ 1 , λ 2 ) ∼ R -N λ -N 2 2 and ∂ λ 2 ζ(λ 1 , λ 2 ) ∼ R -N λ -N 2 2 .
We will also denote V R (λ 1 , λ 2 ) := (V R (λ 1 , λ 2 ), 0) ∈ E.

It is straightforward to check that V R (λ 1 , λ 2 ) has weak derivatives ∂ λ 1 V R (λ 1 , λ 2 ) and ∂ λ 2 V R (λ 1 , λ 2 ), which are given by the formulas:

∂ λ 1 V R (λ 1 , λ 2 )(x) = -(ΛW ) λ 1 (x) -∂ λ 1 ζ(λ 1 , λ 2 ) when |x| < R √ λ 1 λ 2 , 0 when |x| > R √ λ 1 λ 2 , ∂ λ 2 V R (λ 1 , λ 2 )(x) = -∂ λ 2 ζ(λ 1 , λ 2 ) when |x| < R √ λ 1 λ 2 , 0 when |x| > R √ λ 1 λ 2 . (3.2)
Notice that ∂ λ j V R (λ 1 , λ 2 ) ∈ L 2 and ∂ λ j V R (λ 1 , λ 2 ) / ∈ Ḣ1 .

In the whole section we will denote λ := λ 1 λ 2 and N (g, λ) :=

g 2 E + λ N -2 2 .
Lemma 3.1. For R 1 and λ 1 the following estimates are true with constants depending only on the dimension: • |x|

V R (λ 1 , λ 2 ) -W λ 1 Ḣ1 R -N -2 2 λ N -2 4 , (3.3)
R -N λ -N 2 2 , (3.5) V R (λ 1 , λ 2 ) L 1 R 2 λ N +2 2 2 λ N 2 , (3.6)
∂ λ 1 V R (λ 1 , λ 2 ) L 1 R 2 λ N 2 2 λ N -2 2 . ( 3
λ 1 -N +2 , |∂ λ 1 V R (x)| λ -N 2 1 • |x| λ 1 -N +2
and supp V (x) = supp ∂ λ 1 V (x) = B(0, R √ λ 1 λ 2 ).

We will omit the subscript and write V (λ 1 , λ 2 ) instead of V R (λ 1 , λ 2 ). The approximate solution we will consider is defined as follows: for xx 0 X < ρ.

U (λ 1 , λ 2 , a 2 ) := U a 2 λ 2 -V (λ 1 , λ 2 ). Observe that ∂ λ 1 U (λ 1 , λ 2 , a 2 ) = -∂ λ 1 V (λ 1 , λ 2 ), (3.8)
∂ λ 2 U (λ 1 , λ 2 , a 2 ) = - 1 λ 2 Λ E U a 2 λ 2 -∂ λ 2 V (λ 1 , λ 2 ), (3.9)
∂ a 2 U (λ 1 , λ 2 , a 2 ) = ∂ a U a 2 λ 2 = - λ 2 νa 2 J • DE(U a 2
(3.12)

Then there exists y ∈ C(B(x 0 , ρ), B(y 0 , η)) such that for x ∈ B(x 0 , ρ), y = y(x) is the unique solution of the equation Φ(x, y(x)) = 0 in B(y 0 , η).

The proof is the same as standard proofs of IFT, see for instance [11, Section 2.2].

Lemma 3.3. Let δ 0 > 0 and λ 0 > 0 be sufficiently small. For any 0 ≤ δ ≤ δ 0 and 0 < λ ≤ λ 0 there exists 0 ≤ η = η(δ, λ) ----→ with constants which may depend on R.

Proof. We will follow the same scheme as in the proof of Lemma 2.12. One additional difficulty is that we cannot reduce by rescaling to modulation near one specific function as we did before.

Step 1. We consider Φ : E × R 3 → R 3 defined as Φ(u 0 ; l 1 , l 2 , a 2 ) = Φ 1 (u 0 ; l 1 , l 2 , a 2 ), Φ 2 (u 0 ; l 1 , l 2 , a 2 ), Φ 3 (u 0 ; l 1 , l 2 , a 2 )

:= 1 λ 1 Z λ 1 , u 0 -U (λ 1 , λ 2 , a 2 ) , 1 λ 2 Z λ 2 , u 0 -U (λ 1 , λ 2 , a 2 ) , α - λ 2 , u 0 -U (λ 1 , λ 2 , a 2 ) ,
where we have already written λ j instead of e l j in order to simplify the notation. We will verify that the assumptions (3.11) and (3.12) are satisfied for x 0 = U ( λ 1 , λ 2 , 0), y 0 = ( l 1 , l 2 , 0) (where l j := log λ j ), ρ small and η = Cρ with C a universal constant. We define: Using (3.8), (3.9), (3.10) and the fact that ∂ l j = λ j ∂ λ j we see that M jk (u 0 -U (λ 1 , λ 2 , a 2 ); λ 1 , λ 2 , a 2 ) = ∂ l k Φ j (u 0 ; l 1 , l 2 , a 2 ), j ∈ {1, 2, 3}, k ∈ {1, 2}, M j3 (u 0 -U (λ 1 , λ 2 , a 2 ); λ 1 , λ 2 , a 2 ) = ∂ a 2 Φ j (u 0 ; l 1 , l 2 , a 2 ), j ∈ {1, 2, 3}, hence (3.16) implies that the jacobian matrix of Φ with respect to the modulation parameters is uniformly non-degenerate in a neighbourhood of U (λ 1 , λ 2 , a 2 ). This yields parameters λ 1 (t 0 ), λ 2 (t 0 ) and a 2 (t 0 ), see Remark 3.2.

M 11 (g; λ 1 , λ 2 , a 2 ) := 1 λ 1 Z λ 1 , λ 1 ∂ λ 1 V (λ 1 , λ 2 ) - 1 λ 1 Λ -1 Z λ 1 ,
Step 2. The argument from the proof of Lemma 2.12 shows that λ 1 (t), λ 2 (t) and a 2 (t) are C 1 functions of t ∈ (t 1 , t 2 ).

Step 3. From (3.13) we obtain the following differential equation of the error term g:

∂ t g = ∂ t (u -U (λ 1 , λ 2 , a 2 )) = J • DE(U (λ 1 , λ 2 , a 2 )) -∂ t U a 2 λ 2 .
Using (3.8), (3.9) and (3.10) this can rewritten as

∂ t g = J • (DE(U (λ 1 , λ 2 , a 2 ) + g) -DE(U a 2 λ 2 )) + λ 1 ∂ λ 1 V (λ 1 , λ 2 ) + λ 2 • 1 λ 2 Λ E U a 2 λ 2 + ∂ λ 2 V (λ 1 , λ 2 ) -a 2 + ν λ 2 a 2 ∂ a U a 2 λ 2 .
(3.17)

The first component reads:

∂ t g = ġ + λ 1 ∂ λ 1 V (λ 1 , λ 2 ) + λ 2 ΛU a 2 λ 2 + ∂ λ 2 V (λ 1 , λ 2 ) -a 2 + ν λ 2 a 2 ∂ a U a 2 λ 2 ,
hence differentiating in time the first orthogonality relation 1 λ 1 Z λ 1 , g = 0 we obtain

0 = d dt Z λ 1 , g = - λ 1 λ 2 1 Λ -1 Z λ 1 , g + 1 λ 1 Z λ 1 , ġ + λ 1 λ 1 Z λ 1 , ∂ λ 1 V (λ 1 , λ 2 ) + λ 2 λ 1 Z λ 1 , ΛU a 2 λ 2 + ∂ λ 2 V (λ 1 , λ 2 ) - 1 λ 1 a 2 + ν λ 2 a 2 Z λ 1 , ∂ a U a 2 λ 2 ,
which can also be written as

M 11 • λ 1 + λM 12 • λ 2 + λM 13 • λ 2 a 2 + ν λ 2 a 2 = -Z λ 1 , ġ , (3.18)
where for simplicity we write M jk instead of M jk (g; λ 1 , λ 2 , a 2 ). Similarly, differentiating the second orthogonality relation 1 λ 2 Z λ 2 , g = 0 we obtain

0 = d dt Z λ 2 , g = - λ 2 λ 2 2 Λ -1 Z λ 2 , g + 1 λ 2 Z λ 2 , ġ + λ 1 λ 2 Z λ 2 , ∂ λ 1 V (λ 1 , λ 2 ) + λ 2 λ 2 Z λ 2 , ΛU a 2 λ 2 + ∂ λ 2 V (λ 1 , λ 2 ) - 1 λ 2 a 2 + ν λ 2 a 2 Z λ 2 , ∂ a U a 2 λ 2 ,
which can also be written as Without loss of generality we can assume that λ 2 = 1 and λ 1 = λ, hence (3. • V (λ, 1)

L N N -2 g 4 N -2 E • λ N -2 2 | log λ| N (g, λ) 2 . Finally, | Y, f (V (λ, 1)) | f (W λ ) L 1 λ N -2
2 . This finishes the proof of (3.23), hence we have shown (3.21).

Consider the inverse matrix with constants eventually depending on R.

Proof.

Step 1. Directly from the definition of a - 1 (t) we obtain

d dt a - 1 (t) = - λ 1 (t) λ 1 (t) Λ E * α - λ 1 (t) , g(t) + α - λ 1 (t) , ∂ t g(t)
.

The first term is negligible due to (3.14). We compute the second term using (3.17). We begin by treating the terms in the second line of (3.17 (of course Lemma 3.3 provides better estimates, but we do not need it here), it suffices to check that

| α - λ 1 , ∂ λ 1 V (λ 1 , λ 2 ) | + | α - λ 1 , ∂ λ 2 V (λ 1 , λ 2 ) | +| α - λ 1 , 1 λ 2 Λ E U a 2 λ 2 | + | α - λ 1 , 1 λ 2 ∂ a 2 U a 2 λ 2 | 1 λ 1 • λ 1 λ 2 N -2 2 .
The estimate is invariant by rescaling both λ 1 and λ 2 , hence we can assume that λ 2 = 1 and λ 1 = λ. For the first term we use (3.5) and rapid decay of Y. Estimating the other terms is straightforward. Now consider the first line of (3.17). It follows from (2.19) that it suffices to show that

α - λ 1 , J • DE(U (λ 1 , λ 2 , a 2 ) + g) -DE(U a 2 λ 2 ) -D 2 E(W λ 1 )g 1 λ 1 N (g, λ) 2 ,
which is equivalent to

| Y λ 1 , f (U (λ 1 , λ 2 , a 2 ) + g) -f (U a 2 λ 2 ) -∆V (λ 1 , λ 2 ) -f (W λ 1 )g | N (g, λ) 2 .
We can assume that λ Y λ , ∆ W λ -V (λ, 1) + Y λ , f (W λ ) -f (V (λ, 1))

∆Y λ L 1 + f (W λ )Y λ L 1 λ N -2 2 ,
which proves (3.27).

To fix ideas, notice that while proving the remaining inequalities we can restrict our attention to the region |x| ≤ c √ λ where c > 0 is a small constant (the region |x| ≥ c √ λ is negligible thanks to the rapid decay of Y). In this region we have W λ ≥ V (λ, 1 even if f is a negative power. This yields (3.28).

This finishes the proof of (3.24) and the proof of (3.25) is almost the same.

Step 2. The proof of (3.26) is close to the proof of Proposition 2.13, but there will be more error terms to estimate. First we need to show that

| β a 2 λ 2 -α + λ 2 , ∂ t g | 1 λ 2 |a 2 | • N (g, λ). ( 3
ν λ 2 α + λ 2 , g = | α + λ 2 , ∂ t g -J • D 2 E(W λ 2 )g | 1 λ 2 N (g, λ) 2 .
We develop ∂ t g using (3.17). Consider first the terms in the second line of (3.17). From (3.7) and (3.14) we have

| α + λ 2 , λ 1 ∂ λ 1 V (λ 1 , λ 2 ) | 1 λ 2 N (g, λ) 2 . Since |∂ λ 2 V (λ 1 , λ 2 )| λ -N 2
2 , see (3.2), using (3.14) we get

| α + λ 2 , λ 2 ∂ λ 1 V (λ 1 , λ 2 ) | 1 λ 2 N (g, λ) 2 .
The other two terms have already appeared in the proof of Proposition 2.13, see (2.59). Consider now the first line of (3.17 We can assume that λ 2 = 1 and λ 1 = λ, hence we have to prove that

| U a 2 , f (U (λ, 1, a 2 ) + g) -f (U a 1 λ 2 + g) | a 2 N (g, λ) 2 . (3.35)
In the region |x| > R √ λ the integrand equals 0. In the region |x| ≤ R √ λ we have a pointwise bound

|f (U (λ, 1, a 2 ) + g) -f (U a 2 + g)| f (U a 2 + g)W λ + f (W λ ) (f (U a 2 ) + f (g))W λ + f (W λ ).
Recall that U a 2 L ∞ |a 2 | and U a 2 L ∞ 1. Thus

| | U a 2 |, f (U a 2 )W λ | |a 2 | • W λ L 1 (|x|≤R √ λ) ∼ |a 2 |λ N -2 2 , | | U a 2 |, f (g)W λ | |a 2 | • f (g) L N 2 • W λ L N N -2 (|x|≤R √ λ) |a 2 | • g 4 N -2 Ḣ1 • λ N -2 2 | log λ| |a 2 |N (g, λ) 2 , | | U a 2 |, f (W λ ) | |a 2 | • f (W λ ) L 1 ∼ |a 2 |λ N -2 2 .
This proves (3.35).

In order to finish the proof, it suffices to check that

| D 2 E(U a 2 λ 2 )∂ t U a 2 λ 2 , g + DE(U a 2 λ 2 ), J • (DE(U a 2 λ 2 + g) -DE(U a 2 λ 2 )) | a 2 λ 2 N (g, λ) 2 ,
which is achieved exactly as in the last part of the proof of Proposition 2.13.

Coercivity near the sum of two bubbles

We have the following analogue of Lemma 2.2:

Lemma 3.5. There exist constants λ 0 , η > 0 such that if λ = λ 1 λ 2 < λ 0 and U -(W λ 2 -W λ 1 ) E < η, then for all g ∈ E such that Z λ 1 , g = Z λ 2 , g = 0 there holds

1 2 D 2 E(U )g, g + 2 α - λ 1 , g 2 + α + λ 1 , g 2 + α - λ 2 , g 2 + α + λ 2 , g 2 g 2 E .
Proof.

Step 1. Without loss of generality we can assume that λ 2 = 1 and λ 1 = λ. Consider the operator H λ defined by the following formula:

H λ := -∆ -f (W λ ) -f (W ) 0 0 Id .
We will show that for any c > 0 there holds

| D 2 E(U )g, g -H λ g, g | ≤ c g 2 E , ∀g ∈ E, (3.36)
provided that η and λ 0 are small enough. By Hölder and Sobolev, it suffices (eventually changing c) to check that

f (U ) -f (W λ ) -f (W ) L N 2 ≤ c.
Since (by pointwise estimates)

f (U ) -f (W -W λ ) L N 2 max(η, f (η)),
this will in turn follow from ∼ λ.

f (W -W λ ) -f (W λ ) -f (W ) L N 2 ≤ c. ( 3
In the region |x| ≥ √ λ we have W λ W . If N ≥ 6, then

f (W -W λ ) -f (W λ ) -f (W ) f (W λ ).
It is easy to check that f (W λ )

L N 2 (|x|≥ √ λ)
∼ λ. If N ∈ {3, 4, 5}, we obtain

f (W -W λ ) -f (W λ ) -f (W ) |f (W )| • |W λ |, hence f (W -W λ ) -f (W λ ) -f (W ) L N 2 (|x|≥ √ λ) f (W ) L 2N 6-N • W λ L 2N N -2 (|x|≥ √ λ) ∼ λ N -2 4 .
This finishes the proof of (3.37).

Step 2. In view of (3.36), it suffices to prove that if λ < λ 0 and Z, g = Z λ , g = 0, then 1 2 H λ g, g + 2 α - λ 1 , g 2 + α + λ 1 , g 2 + α - λ 2 , g 2 + α + λ 2 , g 2 g 2 E .

Let a - 1 := α - λ , g , a + 1 := α + λ , g , a - 2 := α -, g , a + 2 := α + , g and decompose

g = a - 1 Y - λ + a + 1 Y + λ + a - 2 Y - λ + a + 2 Y + + k.
Using the fact that

| α ± , Y ± λ | + | α ± λ , Y ± | + | 1 λ Z λ , Y | + | Z, Y λ | λ N -2 2 , |a - 1 | + |a + 1 | + |a - 2 | + |a + 2 | g E , α -, Y + = α + , Y -= Z, Y = 0 we obtain α -, k 2 + α + , k 2 + α - λ , k 2 + α + λ , k 2 + Z, k 2 + 1 λ Z λ , k 2 λ N -2 g 2 E .
(3.39)

Since H λ is self-adjoint, we can write → 0 as λ → 0. This and (2.17), (2.18) imply

1 2 H λ g, g = 1 2 H λ k, k + H λ (a - 2 Y -+ a + 2 Y + ), k + H λ (a - 1 Y - λ + a + 1 Y + λ ), k + 1 2 H λ (a - 2 Y -+ a + 2 Y + ), a - 2 Y -+ a + 2 Y + + 1 2 H λ (a - 1 Y - λ + a + 1 Y + λ ), a - 1 Y - λ + a + 1 Y + λ + H λ (a - 2 Y -+ a + 2 Y + ), a - 1 Y - λ + a + 1 Y + λ . (3.
H λ Y -+ 2α + E * + H λ Y + + 2α - E * + H λ Y - λ + 2α + λ E * + H λ Y + λ + 2α - λ E * ---→ λ→0 0.
Plugging this into (3.40) and using (3.39) we obtain

1 2 H λ g, g ≥ -2a - 2 a + 2 -2a - 1 a + 1 + 1 2 H λ k, k -c g 2 E , (3.41)
where c → 0 as λ → 0. Applying (2.2) with r 1 = λ -1 2 , rescaling and using (3.39) we get, for λ small enough, Taking the sum of (3.42) and (3.43), and using (3.41) we obtain

1 2 H λ g, g ≥ -2a - 2 a + 2 -2a - 1 a + 1 + c k 2 E -2 c g 2 E .
The conclusion follows if we take c small enough.

Recall that R > 0 is the constant used in the definition of the localized bubble V (λ 1 , λ 2 ), see (3.1). Lemma 3.6. There exist constants λ 0 , η, R 0 , c > 0 such that if λ = λ 1 λ 2 ≤ λ 0 , |a 2 | ≤ η and R ≥ R 0 , then E(U (λ 1 , λ 2 , a 2 )) ≥ 2E(W ) + cλ

N -2 2 .
Proof. Without loss of generality we can assume that λ 2 = 1, λ 1 = λ (it suffices to rescale). The conclusion follows from [39, Lemma 2.7] applied for u * = -U a 2 (the proof given there is valid for N ≥ 3).

Remark 3.7. In Lemma 3.5 the fact that the bubbles have opposite signs has no importance, but it is crucial in Lemma 3.6.

Conclusion of the proof

Proof of Theorem 1. Suppose by contradiction that u(t) : [0, T + ) → E is a solution of (NLW) such that (1.4) holds. Formula (1.5) and Lemma A.5 imply 2E(W ) = E(U (λ 1 , λ 2 , a 2 ) + g) = E(U (λ 1 , λ 2 , a 2 )) + DE(U (λ 1 , λ 2 , a 2 )), g

+ 1 2
D 2 E(U (λ 1 , λ 2 , a 2 ))g, g + o( g 2 E ).

(3.44)

Step 1 -Coercivity. We will prove that for all t there holds 2a This is equivalent to ˆ|f (U a 2 -V (λ, 1)) + f (V (λ, 1)) -f (U a 2 )| • |g| dx N (g, λ) 2 .

By Hölder and Sobolev inequalities, it suffices to check that f (-V (λ, 1) + U a 2 ) + f (V (λ, 1)) -f (U a 2 )

L 2N N +2 λ N -2 4 ,

1 λλ 2

 12 Une intégration par changement de variables montre que u λ Ḣ1 ×L 2 = u Ḣ1 ×L 2 . Pourtant, la norme d'énergie se concentre près de x = 0 quand λ → 0. Supposons pour simplifier que f (x, u) = f (u) ne dépend pas de x et que N ≥ 3. Si |f (u)| |u| p quand |u| → +∞, on obtient |f (u λ )| ∆u λ .

Theorem 2 . 7 .

 27 There exists a solution u : (-∞, T 0 ] → Ḣ1 × L 2 of (NLW) in dimension N = 6 such that lim t→-∞ u(t) -(W + W 1 κ e -κ|t| ) Ḣ1 ×L 2 = 0, with κ := 5 4 .

2. 1 Proposition 2 . 1 .

 121 Inverting the operator L There exist radial functions A, B ∈ C ∞ (R 5 ) such that

  -1 and similarly |B(r)| r -1 . Using the asymptotics of Γ we obtain also |A (r)| r -2 and |B (r)| r -2 . The fact that |A (r)| r -3 and |B (r)| r -3 follows from the differential equation.

7 6 ν - 7 3

 77 in the degenerate case, which is the exponent of t in the error estimates in Proposition 3.14 and Proposition 4.7 respectively.

u X 1

 1 ×H 1 = +∞. (B.7) Consider the time interval I = [T * -τ, T * ]. Derivating (NLW) once and using Lemma B.4 we get ∇u L 7/3 (I;L 14/3 ) ≤ C (u(T

  2) and Gronwall inequality we deduce that u = u when |x| ≤ 2ρ + |t -τ |, in particular when |x| ≤ 2ρ. B.4 Short-time asymptotics in the case (u 0 , u 1 ) = (p|x| β , 0).

κ 9 3e 9 3 8 3 8 3e

 9988 e -κ|t| . Integrating we get the following bound for b(t), t ∈ [T, T 0 ]: 8 -κ|t| < e -κ|T | + 8 (e -κ|t| -e -κ|T | ) ≤ b(t) ≤ e -κ|T | + 9 (e -κ|t| -e -κ|T | ) < 9 -κ|t| .

  20) with (3.21), (3.22) and (3.24) we obtain

L 3 2

 3 → 0 as λ → 0. This and (3.2), (3.3) imply

Lemma 4 . 3 .

 43 Suppose that for t ∈ [T, T 0 ] there holds |λ (t)| e -κ|t| and |µ (t)| e -κ|t| . Then

Lemma 5 . 1 .

 51 For all V (r) ∈ C ∞ ((0, +∞)) such that ´+∞ 0 ΛW (r) • V (r) rdr = 0, |V (r)| r kfor small r and |V (r)| r -k for large r, then there exists a function U (r) ∈ C ∞ ((0, +∞)) such that LU = V, (5.1)

. 7 )

 7 From the differential equation we get also |∂ r U 1 (r)| r k for small r and |∂ r U 1 (r)| r -k for large r. Now we define U ∈ C ∞ ((0, +∞)) by the formulaU (r) := ΛW (r) ˆr 0 U 1 (ρ) ΛW (ρ) dρ. It solves ∂ r -ΛW (r) ΛW (r) U (r) = U 1 (r),hence (5.5) yields (5.1). Using (5.6) and (5.7), one can check that |U (r)| r k+2 for small r and |U (r)| r -k+2 for large r. The differential equations yield |∂ r U (r)| r k+1 and ∂ 2 r U (r)| r k for small r, as well as |∂ r U (r)| r -k+1 and |∂ 2 r U (r)| r -k for large r. Adding to U a suitable multiple of ΛW we obtain (5.4). Since |ΛW

Proposition 2 . 1 ([ 40 ,

 2140 Lemma 6.1], [27, Proposition 5.5]). There exists a constant c L

N 2 λ N - 2 2.

 22 In the region |x| ≤ R √ λ we use the bound

2 . 2 2

 22 As for the first term, it is easily seen that ´|f (W λ )| dx λ N -, and we get the conclusion if we recall that ζ(λ) ∼ c 0 c * . Finally, from (2.31) and the pointwise bound |f

•

  For N = 5 u * ∈ H 2 ⊂ L 10 and f (W λ )

  where V is bounded and i, j ≤ k -1. All these expressions are controlled by• 2 H k-1 . The proof of (2.20) is almost the same. The only problem are the terms of the form ´V • ∇v • v dx and ´V • |v| 2 dx. As the potential decreases at least as f (W ), by Hardy inequality these terms are controlled by v Ḣ1 .

(

  we have integrated by parts between the first and the second line). Notice that xf (W ) ∈ L N , hence by Hölder and Sobolev xf

. 7 )-

 7 Proof. The proof of (3.3),(3.4) and (3.5) is straightforward, see [39, Lemma 2.3]; (3.6) and (3.7) follow from the fact that |V R (x)| λ

Remark 3 . 2 . 3

 323 The following version of the Implicit Function Theorem has the advantage of providing a lower bound on the size of a ball where it can be applied:Suppose that X, Y and Z are Banach spaces, x 0 ∈ X, y 0 ∈ Y , ρ, η > 0 and Φ : B(x 0 , ρ) × B(y 0 , η) → Z is continuous in x and continuously differentiable in y, Φ(x 0 , y 0 ) = 0 and D y Φ(x 0 , y 0 ) =: L 0 has a bounded inverse. Suppose that L 0 -D y Φ(x, y) for xx 0 X < ρ, yy 0 Y < η, (3.11) Φ(x, y 0 ) Z ≤ η

  22) is equivalent to | Y, -∆V (λ, 1) + f (-V (λ, 1) + U a 2 + g) -f (U a 2 + g) | N (g, λ) 2 .(3.23)We have| Y, ∆V (λ, 1) | = | ∆Y, V (λ, 1) | λ N -2 2because of (3.7). For the other term we use the bound|f (-V (λ, 1) + U a 2 + g) -f (U a 2 + g)| (f (U a 2 ) + f (g))V (λ, 1) + f (V (λ, 1)). From (3.6) we obtain | Y, f (U a 2 )V (λ, 1) | V (λ, 1) L 1 N (g, λ) 2 . Using Hölder we compute | Y, f (g) • V (λ, 1) | f (g) L N 2

). Since |λ 1 | + |λ 2 | 1 and a 2 + ν λ 2 a 2 1 λ 2

 122

2 = 1

 21 and λ 1 = λ. By the triangle inequality, it suffices to check that| Y λ , ∆V (λ, 1) + f (V (λ, 1)) | N (g, λ) 2 , (3.27) | Y λ , f (U (λ, 1, a 2 ) + g) -f (U (λ, 1, a 2 )) -f (U (λ, 1, a 2 ))g | N (g, λ) 2 , (3.28) | Y λ , f (U (λ, 1, a 2 )) -f (U a 2 ) + f (V (λ, 1)) | N (g, λ) 2 ,(3.29)| Y λ , f (U (λ, 1, a 2 )) -f (W λ ) g | N (g, λ) 2 . (3.30) Notice that |f (W λ ) -f (V (λ, 1))| f (W λ )| • |W λ -V (λ)| f (W λ ), where the last inequality follows from(3.4). Together with the fact that ∆(W λ ) + f (W λ ) = 0 this implies Y λ , ∆V (λ, 1) + f (V (λ, 1))

2

  ) 1 and |U (λ, 1, a 2 ) + W λ | ≤1 2 W λ pointwise. Inequality(3.29) follows immediately from|f (U (λ, 1, a 2 )) -f (U a 2 ) + f (V (λ, 1))| = |f (U a 2 -V (λ, 1)) -f (U a 2 ) + f (V (λ, 1))| f (W λ ).We have the bound|f (U (λ, 1, a 2 )) -f (W λ )| (|f (W λ )| + |f (U (λ, 1, a 2 ))|) • |U (λ, 1, a 2 ) + W λ | |f (W λ )|(even in the case N ≥ 6 when f is a negative power). Using Hölder and the fact thatY λ • f (W λ ) , this implies (3.30)For (3.28), we consider separately the cases N ∈ {3, 4, 5} and N ≥ 6. In the first case, (3.28) follows from the pointwise bound|f (U (λ, 1, a 2 ) + g) -f (U (λ, 1, a 2 )) -f (U (λ, 1, a 2 ))g| |f (U (λ, 1, a 2 ))| • |g| 2 + f (|g|).In the case N ≥ 6 we still have |f (U (λ, 1, a 2 ) + g) -f (U (λ, 1, a 2 )) -f (U (λ, 1, a 2 ))g| |f (U (λ, 1, a 2 ))| • |g| 2 ,

. 37 )

 37 We consider separately the regions |x| ≤ √ λ and |x| ≥ √ λ. In both cases we will use the fact that|l| |k| ⇒ f (k + l) -f (k) -f (l) f (l), for N ≥ 6, |l| |k| ⇒ f (k + l) -f (k) -f (l) |f (k)| • |l|, for N ∈ {3, 4, 5}.(3.38)In the region |x| ≤ √ λ we have W W λ , hence by (3.38)f (W -W λ ) -f (W λ ) -f (

40 )

 40 It is easy to see that f (W )Y λ L

(1 -2c) ˆ|x|≤ √ λ |∇k| 2 dx + c ˆ|x|≥ √ λ |∇k| 2

 2 dx -ˆRN f (W λ )|k| 2 dx ≥ -c g 2 E .

  1 , t 2 ); H s ). Pour t ∈ (t 1 , t 2 ) posons

	u(t) := cos (t -t 0 ) √ + ˆt t 0 sin (t -s) √ -∆ u 0 + -∆ √ h(s) ds, sin (t -t 0 ) √ -∆ √ -∆ u(t) := -sin (t -t 0 ) √ -∆ √ -∆u 0 + cos (t -t 0 ) -∆ u0 √ + ˆt t 0 cos (t -s) √ -∆ h(s) ds.	-∆ u0

  14. In typical situations the condition(1.13) is equivalent to u(t) being in some natural function space, for example a Lebesgue space on a slab of the space-time. Definition 1.11 describes a fixed-point problem. The main part of its resolution is to obtain bounds of f (x, u(t, x)) L 1 ((t 1 ,t 2 );H s ) in terms of h(t, x) L 1 ((t 1 ,t 2 );H s ) , where u(t, x) is the solution of(1.11). To this end, one uses Sobolev, Strichartz and other inequalities of this type.Remark 1.15. Suppose that (1.1) is locally well-posed in X s × H s and that u(t) is a solution with maximal time of existence T + < +∞. Then f (x, u(t, x)) L 1 ([t 0 ,T + );H s ) = +∞.

  The second term of P 0 is bounded in the same way. Notice that we obtain in fact that |x| • |∇P 0 (t)| L 10/3 is small when t is small.

, and it is sufficient to use the inequalities |A(x/λ)| λ/|x| and |∇A(x/λ)| λ 2 /|x| 2 . Clearly |λ(∇a) λ | |x| uniformly in R, so (5.11) follows from (5.10). The proof of (5.12) is similar. The terms |x| • |∇u * | L 10/3 and |λ(∇a) λ | • |∇u * | L 10/3 are small when ρ is small. By rescaling we get

  .22) Again using Hölder we get P 0 • f ((ΛW ) λ ) L 10/7 ≤ P 0 L 10/3 f ((ΛW ) λ ) L 5/2 P 0 Ḣ1 . From Lemma 3.4 (or the degenerate version Lemma 4.1) we have P 0 Ḣ1 t γ+1

	λ t . This
	proves (5.22) and (5.20) is very similar.
	From Proposition B.2 we know that u

* L 10 is bounded. Hence u * • f ((ΛW ) λ ) L 10/7 u * L 10 • f ((ΛW ) λ ) L 5/3 λ. This proves (5.21) and (5.19) is similar.

  we are done. Now suppose that |α -(T 2 )| = T γ+1

	2 λ have opposite signs, which is impossible. . As t γ+1 λ ∼ t γ-ν α -Again by contradiction, suppose that α + λ	t γ , (6.14) implies that d dt α -λ and

  • ΛP λ Ḣ1 |t| • e κ|t| and χ • ΛQ λ Ḣ1 |t| • e κ|t| , see (2.10). This implies (2.17) because |λ|, |b|, |λ |, |b |, |µ | e -κ|t| .

	In order to prove (2.18), we consider separately the regions |x| ≤	√	λ and |x| ≥	√	λ. The
	first step is to treat the nonlinearity, that is to show that

  ) have opposite signs. Hence(3.18) cannot break down at t = T 2 . The contradiction shows that (3.18) holds for t ∈ [T, T 1 ].

				).
	Similarly, we have
	d dt	a -1 (t) +	ν λ(t)	a -1 (t) ≤ c • e -1 2 κ|t| .	(3.25)
	Inequality (3.16) implies that (3.18) holds for t in a neighborhood of T . Suppose that T 2 ∈
	(T, T 1 ) is the last time such that (3.18) holds for t ∈ [T, T 2 ]. But (3.25) implies that d dt a -1 (T 2 ) and a -1 (T 2

  Again from(3.15) and (3.10) we have |λ (t) -b(t)| C 0 • e -3 2 κ|t| . Multiplying by b (t) = κ 2 • λ(t) µ(t) 2 ∼ e -κ|t| , cf. (2.8) and (2.7), we obtain d dt

	3 2 κ 0 |t| .	(3.52)

  3 4 e -3 2 κ|T 2 | and | a + 1 (t)| < e -3 2 κ|t| for t ∈ [t n , T 2 ]. Suppose that a 0 ∈ A -. Hence there exists the first T 3 > T 2 such that a + | , which is a contradiction. Hence A + is open and analogously A -is open.

	a + 1 (T 3 ) e -1 2 κ|T 3 |	e -3 2 κ|T 3	1 (T 3 ) = 3 4 e -3 2 κ|T 3 | . Estimate (3.17) yields

  .43)

	Proof. We take	b 1 (t) := DE(u * (t)), g(t) ,
	b 2 (t) := DE(V (λ(t)) + u Estimate (2.43) is exactly (2.32).
	Repeating the computation in the proof of Lemma 2.8, we see that we need to check
	inequalities (2.36), (2.37) and (2.38), with " n(g, λ) 2 " replaced by "	λ	N -4 2

* (t)) -DE(u * (t)), g(t) .

  Let t ∈ [t 0 , T + ) and let t 1 ∈ [t, T + ) be such that ϕ M (t) = ϕ(t 1 ) (such t 1 exists by the definition of ϕ M ). Using (2.47) we obtain Now let t 2 ∈ [t, T + ) be such that sup t≤τ <T + n(g(τ ), λ(τ )) 2 = n(g(t 2 ), λ(t 2 )) 2 . From Lemma 2.2 and the fact that

					)
	almost everywhere.
	Corollary 2.11. Let s > N -2 2	and s ≥ 1. For t 0 ≤ t < T + there holds
		ϕ M (t) ∼ sup	n(g(τ ), λ(τ )) 2 .
			t≤τ <T +
	Proof. Lemma 2.7 and (2.28) yield | DE(V (λ) + u * ), g | n(g, λ) 2 , hence from Lemma 2.9
	we have
		|b 1 (t)|	sup	n(g(t), λ(t)) 2 .	(2.47)
			t≤τ <T +
	ϕ M (t) = ϕ(t 1 )	sup	n(g(τ ), λ(τ )) 2 ≤ sup	n(g(τ ), λ(τ )) 2 .
		t 1 ≤τ <T +		t≤τ <T +

  follows from (2.19) that there exists C 1 > 0 such that and suppose that t 1 ∈ [t, T + ) is the smallest time such that |a + (t 1 )| ≤ C 1 • sup Clearly t 1 > t. The function on the right hand side is decreasing with respect to t 1 , hence d dt |a + (t)| t=t 1 ≤ 0. This contradicts (2.49), hence for all t ∈ [t, T + ) we have |a + (t )| ≥ C 1 • n(g(t ), λ(t )) 2 .

		|a + (t)| ≥ C 1 • n(g(t), λ(t)) 2 ⇒	d dt	|a + (t)| ≥	ν 2λ(t)	|a + (t)|.	(2.49)
	Suppose that
			|a + (t)| ≥ 2C 1 • sup	n(g(τ ), λ(τ )) 2
								t≤τ <T +
									n(g(τ ), λ(τ )) 2 .
								t 1 ≤τ <T +
									(2.50)
	Observe that	ˆT+ t	1 λ(τ )	dτ	ˆT+ t	|λ (τ )| λ(τ )	dτ = +∞.	(2.51)
	From (2.50), (2.49) and (2.51) we obtain |a + (t)| → +∞ as t → T + , a contradiction.
	We will now consider a -(t), which is less straightforward. It follows from (2.19) that
	there exists C 2 > 0 such that
		|a -(t)| ≥ C 2 • n(g(t), λ(t)) 2 ⇒	d dt	|a -(t)| ≤ -	ν 2λ(t)	|a -(t)|.	(2.52)
	From Corollary 2.11 we obtain existence of a constant C 3 > 0 such that
			|a -(t)| ≥ C 3 • ϕ M (t) ⇒ |a -(t)| ≥ C 2 • n(g(t), λ(t)) 2	(2.53)
	and a constant C 4 > 0 such that
		|a -(t)| ≥ C 4 • sup	n(g(t), λ(t)) 2 ⇒ |a -(t)| ≥ 2C 3 • ϕ M (t).
			t≤τ <T +
	Suppose that t ∈ [t 0 , T + ) is such that
			|a -(t)| ≥ C 4 • sup	n(g(τ ), λ(τ )) 2	(2.54)
								t≤τ <T +
	and let t 1 ∈ [t 0 , t] be the smallest time such that for t ∈ [t 1 , t] there holds
					|a -(t )| ≥ C 3 • ϕ M (t ).	(2.55)
	Of course t 1 < t. Suppose that t 1 > t 0 . This implies
	-	C 2 ν λ(t 1 )	n(g(t 1 ), λ(t 1 )) 2 ≥ -	ν 2λ(t 1 )

  We turn to the proof of (2.3). Applying (2.1) to Π r 2 g with c replaced by 3c and C replaced by C

	2 we get
	(1 -3c)	ˆ|x|≥r 2

ˆ|x|≥r 1 |∇g| 2 dx. (2.10) Estimate (2.2) follows from (2.4), (2.8), (2.9) and (2.10).

  Remark 2.5. The factor |t| in (2.23) is necessary, for example in dimension N = 5 we have a solution v(t) = (tΛW, ΛW ).

)

hence it suffices to bound l(t). Using (2.26) and the fact that

|l 0 | v 0 X k ×H k we get from (2.25) that |l(t)| (1 + |t|) v 0 X k ×H k .

Estimate (2.24) follows easily from (2.23).

  [START_REF] Browder | On non-linear wave equations[END_REF]. Then v is a solution of (2.34) if and only ifT (v, b) = v (see Remark 2.8).It follows from Lemma A.3 that on some neighbourhood of the origin T is analytic and a uniform contraction with respect to v, hence the conclusion follows from the Uniform Contraction Principle, cf.[START_REF] Chow | Methods of Bifurcation Theory[END_REF] Theorem 2.2].

  .38)

	Remark 2.11. Note that (2.38) implies that u(t) = U	± exp(-ν λ t) λ

  To see that the contribution of the last two terms in(2.46) is negligible, it suffices to use (2.43),(2.44), | α + λ , Λ E U a λ | |a| and | α + λ , ∂ a U a λ | 1.We can assume that a = 0 (otherwise u(t) ≡ W λ and the conclusion is obvious).The idea of the proof is that the first factor grows exponentially, while the second does not change much. From (2.44) and (2.60) we obtain a (t)(the first one follows from D 2 E(W λ )Λ E W λ = 0). Using (2.45) and (2.62) together with (2.43) and (2.44) concludes the proof.

	We compute the second term using (2.46):
	d dt	DE(U a λ ), g = D 2 E(U a λ )∂ t U a λ , g
			+ DE(U a λ ), J • (DE(U a λ + g) -DE(U a λ )) + λ •	1 λ	Λ E U a λ -a +	νa λ	∂ a U a λ .
	Observe that						DE(U a λ ), Λ E U a λ ) = -∂ λ E(U a λ ) = 0, DE(U a λ ), ∂ a U a λ ) = ∂ a E(U a λ ) = 0.	(2.61)
	Since DE(U a λ ) ∈ Y k × Y k+1 by Proposition 2.10, Lemma A.4 implies that
			DE(U a λ ), J • DE(U a λ + g) -DE(U a λ ) -D 2 E(U a λ )g	a λ	g 2 E ,	.56)
	hence (2.53) follows from (2.43) and (2.44). Note that (2.52) implies in particular that | β a hence using self-adjointness of D 2 E(U a λ ) and anti-self-adjointness of J we get g E with a universal constant. λ , g | Estimates (2.54) follow from the fact that ∂ a β a λ E * + λ∂ λ β a λ E * 1. Step 2. Consider the case d dt DE(U a λ ), g = D 2 E(U a λ ) ∂ t U a λ -J • DE(U a λ ) , g + a λ O( g 2 E ).
	The following estimates hold:		|a(t)| ≤ g(t) E .	(2.57)
		We have	d dt	b(t) = β		D 2 E(U a λ )Λ E U a λ E * D 2 E(U a λ )∂ a U a λ E *	|a|, 1	a(t) λ(t) , g .	(2.58) (2.62)
		From Lemma 2.12 we know that |λ |	g E and |a |	1 λ g E . Hence from (2.54) it follows
		that the last two terms of (2.58) are negligible.
		Using (2.57), (2.52) and (2.53) we see that it is sufficient to show that
			α + λ , ∂ t g -	ν λ	α +	1 λ	g 2 .	(2.59)
		This follows easily from (2.46). Indeed, from Lemma A.4 we deduce that
			| α +					1 λ	g 2 E .
		Step 3. Now consider the case
										g(t) E ≤ |a(t)|.	(2.60)
		Using Proposition 2.10 we get
			β a λ = -	1 2a	DE(U a λ ) ⇒ b(t) = -	1 2a(t)	• DE(U	a(t) λ(t) ), g(t) .
										a(t) + ν λ(t)	1 λ(t) g(t) , hence
			d dt	b(t) = -	a (t) a(t)	b(t) -	1 2a(t)	d dt	DE(U	a(t) λ(t) ), g(t)
					=	ν λ(t)	b(t) -	1 2a(t)	d dt	DE(U	a(t) λ(t) ), g(t) +	1 λ(t)	O( g(t) 2 E ).

a(t) λ(t) , ∂ t g(t) + λ (t) ∂ λ β a(t) λ(t) , g(t) + a (t) ∂ a β λ , g = | α + λ , ∂ t g -J • D 2 E(W λ )g | λ , J • (DE(U a λ + g) -DE(U a λ ) -D 2 E(W λ )g) |

  By (2.52) we have| β a λ , g 2 -α + λ , g 2 | |a| • g 2 , hence Lemma 2.2 yields 1 2 D 2 E(U a λ )g, g + 2| β a λ , g | 2 ∼ g 2 E ,which implies (2.63).Step 2 -Differential inequalities. Let g(t), λ(t) and a(t) be given by Lemma 2.12.If | log λ(t)| is bounded, then by the Cauchy theory T + = +∞ and (2.64) follows. From Proposition 2.13 it follows that there exists a constant C 1 such that|b(t)| ≥ C 1 g(t) 2which implies (2.66) since |a(t)| and |b(t)| are small. Suppose for the sake of contradiction that b(0) = 0 and |b(0)| ≥ 2C 2 E(u 0 ) -E(W ) . Let t 1 ≤ T + be maximal such that b(t) = 0, |b(t)| ≥ C 2 E(u 0 ) -E(W ) , ∀t ∈ [0, t 1 ). (2.68) Of course t 1 > 0. Suppose that t 1 < T + . But (2.66) and (2.65) imply that d dt |b(t)| > 0 for t ∈ [0, t 1 ]. In particular, (2.68) cannot break down at t = t 1 . Thus t 1 = T + and (2.66) implies that for t ∈ [0, T + ) there holds |b(t)| ≥ C 1 g E . By (2.65) and (2.64), this would imply |β(t)| ----→

										5 implies
	E(U a λ + g) = E(U a λ ) + DE(U a λ ), g +	1 2	D 2 E(U a λ )g, g + o( g 2 E )
				= E(W ) -2a β a λ , g +	1 2	D 2 E(U a λ )g, g + o( g 2 E ).
	Observe that					ˆT+ 0	1 λ(t)	dt = +∞.	(2.64)
	Indeed, if | log λ(t)| is unbounded, then
			ˆT+ 0	1 λ(t)	dt	ˆT+ 0	g(t) E λ(t)	dt	ˆT+ 0	|λ (t)| λ(t)	dt = +∞.
					E	⇒	d dt	|b(t)| ≥	ν 2λ(t)	|b(t)|,	∀t ∈ [0, T + ).	(2.65)
	We will show that there exists a constant C 2 such that
			|b(t)| ≥ C 2 E(u 0 ) -E(W )	⇒ |b(t)| ≥ C 1 g(t) 2 E .	(2.66)
	Indeed, we can rewrite (2.63) as
				E(u 0 ) -E(W ) + 2a(t)b(t) + 2b(t) 2 ∼ g 2 E ,	(2.67)
	hence if |b(t)| ≥ C 2 , then
	|b(t)| •	1 C 2	+ 2|a(t)| + 2|b(t)| ≥ E(u 0 ) -E(W ) + 2a(t)b(t) + 2b(t) 2	g 2 E ,
	t→T

+

+∞, which is absurd.

  applied at t = 0. Remark 2.17. It follows quite easily from Lemma 2.2 thatg ∈ X c ⇒ ∃b ∈ R : g -bΛ E W 2 cs ⇒ ∃a, b ∈ R : g -bΛ E W -aY -2Corollary 2.16 provides a nonlinear version of (2.70). By similar methods (analyzing just the linear stability and instability components α + λ and α - λ ) one can prove a nonlinear analogue of (2.69), that is

	E	1 2	D 2 E(W )g, g ,	(2.69)
	g ∈ X E	1 2	D 2 E(W )g, g .	(2.70)

  such that if u : (t 1 , t 2 ) → E is a solution of (NLW) satisfying for all t ∈ (t 1 , t 2 ) u(t) -(-W λ 1 (t) + W λ 2 (t) ) E ≤ δ, then there exist unique functions λ 1 (t) ∈ C 1 ((t 1 , t 2 ), (0, +∞)), λ 2 (t) ∈ C 1 ((t 1 , t 2 ), (0, +∞)) and a 2 (t) ∈ C 1 ((t 1 , t 2 ), R) such that for E ≤ η, |λ 1 (t)/ λ 1 (t) -1| + |λ 2 (t)/ λ 2 (t) -1| + |a 2 (t)| ≤ η.

	In addition,
	|λ 1 (t)| + |λ 2 (t)| N (g(t), λ(t)),	(3.14)
	a 2 (t) +	ν λ 2 (t)	a 2 (t)	1 λ 2 (t)	|a 2 (t)| • N (g(t), λ(t)) + N (g(t), λ(t)) 2 ,	(3.15)
				δ, λ→0	0 0 <	λ 1 (t) λ 2 (t)	≤ λ,

g(t) := u(t) -U (λ 1 , λ 2 , a 2 ) (3.13)

the following holds for all t ∈ (t 1 , t 2 ):

Z λ 1 (t) , g(t) = Z λ 2 (t) , g(t) = α - λ 2 (t) , g(t) = 0, g(t)

  g , M 12 (g; λ 1 , λ 2 , a 2 ) := 1λ 1 Z λ 1 , ΛU a 2 λ 2 + λ 2 ∂ λ 2 V (λ 1 , λ 2 ) , M 13 (g; λ 1 , λ 2 , a 2 ) := -1 λ 1 Z λ 1 , ∂ a U a 2 λ 2 , M 21 (g; λ 1 , λ 2 , a 2 ) := 1 λ 2 Z λ 2 , λ 1 ∂ λ 1 V (λ 1 , λ 2 ) , M 22 (g; λ 1 , λ 2 , a 2 ) := 1 λ 2 Z λ 2 , ΛU a 2 λ 2 + λ 2 ∂ λ 2 V (λ 1 , λ 2 ) -1 λ 2 Λ -1 Z λ 2 , g , M 23 (g; λ 1 , λ 2 , a 2 ) := -1 λ 2 Z λ 2 , ∂ a U a 2 λ 2 , M 31 (g; λ 1 , λ 2 , a 2 ) := α λ 2 , λ 1 ∂ λ 1 V (λ 1 , λ 2 ) , M 32 (g; λ 1 , λ 2 , a 2 ) := -Λ E * α λ 2 , g + α λ 2 , Λ E U a 2 λ 2 + λ 2 ∂ λ 2 V (λ 1 , λ 2 ) , M 33 (g; λ 1 , λ 2 , a 2 ) := -α λ 2 , ∂ a U a 2 λ 2 ,

	A straightforward computation yields
	|M 11 | ∼		1, |M 12 |	1, |M 13 |	1,
	|M 21 |	λ	N 2 , |M 22 |	1, |M 23 |	1,
	|M 31 |	λ

N 2 , |M 32 | N (g, λ) + |a 2 |, |M 33 | ∼ 1. (

3

.16)

  1 λ M 21 • λ 1 + M 22 • λ 2 + M 23 • λ 2 a 2 + ν λ 2 a 2 = -Z λ 2 , ġ . , Λ E U a 2 λ 2 + λ 2 ∂ λ 2 V (λ 1 , λ 2 ) -a 2 + νa 2 λ 2 α - λ 2 , ∂ a U a 2 λ 2 ,which can also be written as1 λ M 31 •λ 1 +M 32 •λ 2 +M 33 •λ 2 a 2 + ν λ 2 a 2 = -λ 2 α - λ 2 , J •(DE(U (λ 1 , λ 2 , a 2 )+g)-DE(U a 2 λ 2 ) . (3.20) Equations (3.18),(3.19) and (3.20) form a linear system for λ 1 , λ 2 andλ 2 a 2 + ν λ 2 a 2 : , ġ -Z λ 2 , ġ -λ 2 α - λ 2 , J • (DE(U (λ 1 , λ 2 , a 2 ) + g) -DE(U a 2 λ 2 )) , J • (DE(U (λ 1 , λ 2 , a 2 ) + g) -DE(U a 2 λ 2 )) | 1 λ 2 N (g, λ)(|a 2 | + N (g, λ)). (3.21) By (2.50), it suffices to show that | α - λ 2 , J • (DE(U (λ 1 , λ 2 , a 2 ) + g) -DE(U a 2 λ 2 + g)) |

	(3.19) -Z λ 1  λ 2 , g = 0 we obtain Finally, differentiating the third orthogonality relation α -0 = d dt α -λ 2 , g = -λ 2 λ 2 Λ E λ 2 λ 2 α -M 11 λM 12 λM 13 1 λ M 21 M 22 M 23 1 λ M 31 M 32 M 33     λ 1 λ 2 a 2 + λ 2 a 2 ν   =    . We will check that λ 2   | α -λ 2 1 λ 2 N (g, λ) 2 . (3.22)

* α - λ 2 , g + α - λ 2 , J • (DE(U (λ 1 , λ 2 , a 2 ) + g) -DE(U a 2 λ 2 )) + λ 1 α - λ 2 , ∂ λ 1 V (λ 1 , λ 2 ) +

  P 11 P 12 P 13 P 21 P 22 P 23 P 31 P 32 P 33We finish this subsection by analyzing the stability and instability components at both scales λ 1 (t) and λ 2 (t). At the scale λ 2 (t) we use the refined component β a 2 λ 2 introduced in Section 2, see(2.51).

	Proposition 3.4. The functions
	a -1 (t) := α -λ 1 (t) , g(t) ,	a + 1 (t) := α + λ 1 (t) , g(t) ,	b 2 (t) := β	a 2 (t) λ 2 (t) , g(t)
	satisfy
	d dt	a -1 (t) +	ν λ 1 (t)	a -1 (t)	1 λ 1 (t)	N (g(t), λ(t)) 2 ,	(3.24)
	d dt	a + 1 (t) -	ν λ 1 (t)	a + 1 (t)	1 λ 1 (t)	N (g(t), λ(t)) 2 ,	(3.25)
	d dt	b 2 (t) -	ν λ 2 (t)	b 2 (t)	1 λ 2 (t)	N (g(t), λ(t)) 2 ,	(3.26)
					M 11 λM 12 λM 13		-1
			 :=		1 λ M 21 M 22 λ M 31 M 32 1	M 23 M 33		.
	From (3.16) we obtain
	|P 11 |		1, |P 12 |	1, |P 13 |	1,
	|P 21 |		1, |P 22 |	1, |P 23 |	1,
	|P 31 |

N (g, λ) + |a 2 |, |P 32 | N (g, λ) + |a 2 |, |P 33 | 1, hence (3.21) yields (3.14) and (3.15).

  .31) Since β a 2 1 -α + L ∞ ×L ∞ |a 2 |, the proof of (3.22) gives | β a 2 λ 2 -α + λ 2 , J •(DE(U (λ 1 , λ 2 , a 2 )+g)-DE(U a 2 λ 2 +g)) | -α + λ 2 , J • (DE(U (λ 1 , λ 2 , a 2 ) + g) -DE(U a 2 λ 2 )) | Similarly one obtains | β a 2 λ 2 -α + λ 2 , ∂ λ 1 V (λ 1 , λ 2 ) | + | β a 2 λ 2 -α + λ 2 , ∂ λ 2 V (λ 1 , λ 2 ) | Step 3. Suppose that |a 2 (t)| ≤ N (g(t), λ(t)). (t) , ∂ t g(t) + λ 2 (t) ∂ λ β a 2 (t) λ 2 (t) , g(t) + a 2 (t) ∂ a β a 2 (t) λ 2 (t) , g . From Lemma 3.3 we know that |λ 2 | N (g, λ) and |a 2 | 1 λ 2 N (g, λ). Hence from (2.54) it follows that the last two terms of (3.26) are negligible. Using (3.32), (2.52) and (3.31) we see that it is sufficient to show that α + λ 2 , ∂ t g -

					(3.32)
	We have
	d dt	a 2 (t) λ 2 b 2 (t) = β
		1 λ 2	|a 2 |•N (g, λ) 2	1 λ 2	|a 2 |•N (g, λ).
	Using (2.55), we obtain
	| β a 2 λ 2 1 λ 2	|a 2 | • N (g, λ).
				1 λ 2	|a 2 |,

hence

(3.31)

follows from (2.56),

(3.14)

and

(3.15)

.

  ). From Lemma A.4 we deduce that| α + λ 2 , J • (DE(U a 2 λ 2 + g) -DE(U a 2 λ 2 ) -D 2 E(W λ 2 )g) | , J • (DE(U (λ 1 , λ 2 , a 2 ) + g) -DE(U a 2 λ 2 + g)) |whose proof is the same as the proof of(3.22).We compute the second term using (3.17) and (2.61):d dt DE(U a 2 λ 2 ), g = D 2 E(U a 2 λ 2 )∂ t U a 2 λ 2 , g + DE(U a λ ), J • (DE(U (λ 1 , λ 2 , a 2 ) + g) -DE(U a 2 λ 2 )) + λ 1 ∂ λ 1 V (λ 1 , λ 2 ) + λ 2 ∂ λ 2 V (λ 1 , λ 2 ) . N (g, λ) 2 .Until the end of this proof "negligible" meansa 2 λ 2 N (g, λ) 2 . From (3.7) and (3.2) it follows that | D(U a 2 λ 2 ), ∂ λ 1 V (λ 1 , λ 2 ) | , | D(U a 2 λ 2 ), ∂ λ 2 V (λ 1 ,λ 2 ) | By (3.14) and (3.33), the contribution of the last two terms in (3.34) is negligible. Next, we will show that | D(U a 2 λ 2 ), J • (DE(U (λ 1 , λ 2 , a 2 ) + g) -DE(U a 2 λ 2 + g)) |

									(3.34)
	We have to prove that d dt DE(U a 2 λ 2 ), g	a 2 λ 2 1 λ 2 λ N -2 2 1 λ 2	λ	N 2 .
									a 2 λ 2	N (g, λ) 2 .
									1 λ 2	N (g, λ) 2 ,
	hence it suffices to check that
		| α + λ 2 1 λ 2	N (g, λ) 2 ,
	Step 4. Now we consider the case
									N (g(t), λ(t)) ≤ |a 2 (t)|,	(3.33)
	in particular a 2 = 0.
	Recall that (see Proposition 2.10)
		β a 2 λ 2 = -	1 2a 2	DE(U a 2 λ 2 ) ⇒ b 2 (t) = -	1 2a 2 (t)	• DE(U	a 2 (t) λ 2 (t) ), g(t) .
	From (3.15) and (3.33) we obtain	a 2 (t) a 2 (t) + ν λ 2 (t)	1 λ 2 (t) N (g(t), λ(t)), hence
	d dt	b 2 (t) = -	a 2 (t) a 2 (t)	b 2 (t) -	1 2a 2 (t)	d dt	DE(U	a 2 (t) λ(t) ), g(t)
		=	ν λ 2 (t)	b 2 (t) -	1 2a 2 (t)	d dt	DE(U

a 2 (t) λ 2 (t) ), g(t) + 1 λ 2 (t)

O(N (g(t), λ(t)) 2 ).

  2 (t)b 2 (t) + 2 a - 1 (t) 2 + a + 1 (t) 2 + b 2 (t) 2 N (g(t), λ(t)) 2(3.45)(the functions a + 1 , a - 1 and b 2 are defined in Proposition 3.4). From (2.52) we have |b2 (t) 2 -α + λ 2 (t) , g(t) 2 | |a 2 | • g 2 . Since α - λ 2 (t), g(t) = 0, Lemma 3.5 and Lemma 3.6 yieldE U (λ 1 (t), λ 2 (t), a 2 (t)) -2E(W ) + 1 2 D 2 E U (λ 1 (t), λ 2 (t), a 2 (t)) g(t), g(t) + 2 a - 1 (t) 2 + a + 1 (t) 2 + b 2 (t) 2 ≥ c • N (g(t), λ(t)) 2 , for R ≥ R 0 , with a constant c > 0 independent of R.Recall that 2a 2 (t)b 2 (t) = -DE(U a 2 λ 2 , g . In view of(3.44), in order to prove (3.45) it suffices to verify that| DE(U (λ 1 , λ 2 , a 2 )) -DE(U a 2 λ 2 ), g | ≤ is large enough.Without loss of generality we can assume that λ 2 = 1 and λ 1 = λ. First we show that DE(U (λ, 1, a 2 )), g + DE(V (λ, 1)), g -DE(U a 2 ), g N (g, λ) 2 .(3.47)

	c 2	• N (g, λ) 2	(3.46)
	provided that R

L 2 (|x|≤R √ λ)

Remerciements

Proposition A.2. Let u 0,n be a bounded sequence in E admitting a profile decomposition with profiles U j lin and parameters λ j,n , t j,n . Let U j be the corresponding nonlinear profiles and let θ n be a sequence of positive numbers. Assume ∀j ≥ 1, n ≥ 1, θ n -t j,n λ j,n < T + (U j ) and lim sup n→+∞ U j S -t j,n λ j,n , θn-t j,n λ j,n < +∞.

Let u n be the solution of (1.1) with the initial data u n (0) = u 0,n . Then for n sufficiently large u n is defined on [0, θ n ] and

U j n (t) + w J n (t) + r J n (t), for all J ∈ N and t ∈ [0, θ n ],

with lim J→+∞ lim sup n→+∞ sup t∈[0,θn] r J n E = 0. An analogous statement holds for θ ≤ 0.

For a corresponding result for the Yang-Mills equation and the equivariant wave maps, see [START_REF] Jia | Asymptotic decomposition for semilinear wave and equivariant wave map equations[END_REF].

Corollary A.3. There exists a constant η > 0 such that the following holds. Let u : [t 0 , T + ) → E be a maximal solution of (1.1) with T + < +∞. Then for any compact set K ⊂ E there exists τ < T + such that dist(u(t), K) > η for t ∈ [τ, T + ).

Proof. Suppose for the sake of contradiction that there exists a sequence t n → T + such that for u 0,n := u(t n ) there holds dist(u 0,n , K) ≤ η, hence u 0,n = k n + b n with k n ∈ K and b n E ≤ η. By taking a subsequence we can assume that k n → k 0 ∈ K and that the sequence b n admits a profile decomposition, in particular b n b 0 ∈ E. This implies that u 0,n admits a profile decomposition with the first profile k 0 + b 0 and all the other profiles small in the energy norm. Proposition A.2 leads to a contradiction.

Corollary A. [START_REF] Bizoń | Formation of singularities for equivariant 2 + 1 dimensional wave maps into the 2-sphere[END_REF]. There exists a constant η > 0 such that the following holds. Let K ⊂ E be a compact set and let u n : [T 1 , T 2 ] → E be a sequence of solutions of (1.1) such that

Suppose that u n (T 1 ) u 0 ∈ E. Then the solution u(t) of (1.1) with the initial condition u(T 1 ) = u 0 is defined for t ∈ [T 1 , T 2 ] and

Proof. It suffices to prove (A.1) for a subsequence of any subsequence. Hence, we may assume that u 0,n := u n (T 1 ) admits a profile decomposition. As in the previous proof, we show that all the profiles except for the weak limit of u 0,n are small in the energy norm. Proposition A.2 implies (A.1).

Remark A.5. Corollaries A.3 and A.4 hold for the Yang-Mills and wave map equations, with the same proofs.

Remark A.6. An important point of both results is that η is independent of K. Corollary A.3 states that a blow-up cannot happen at a small distance from a compact set. Corollary A.4 establishes sequential weak continuity of the flow in a neighbourhood of any compact set. Without this additional condition weak continuity is expected to fail, a counterexample being provided by type II blow-up solutions.

Chapter 3

Bounds on the speed of type II blow-up for the energy critical wave equation in the radial case

Abstract

We consider the focusing energy-critical wave equation in space dimension N ∈ {3, 4, 5} for radial data. We study type II blow-up solutions which concentrate one bubble of energy. It is known that such solutions decompose in the energy space as a sum of the bubble and an asymptotic profile. We prove bounds on the blow-up speed in the case when the asymptotic profile is sufficiently regular. These bounds are optimal in dimension N = 5. We also prove that if the asymptotic profile is sufficiently regular, then it cannot be strictly negative at the origin.

(we use respectively (2.53), (2.52) Dividing both sides by λ(t) and integrating we get a contradiction.

We have proved the lemma with C a := max(2C 1 , C 4 ).

By modifying t 0 we can assume that Lemma 2.12 holds for t ∈ [t 0 , T + ).

Proof of Theorem 1. We define

From Lemma 2.12 and Corrolary 2.11, it is clear that

(2.56)

We will consider first the case N ∈ {4, 5}. Using (2.42) and (2.56) we obtain the following differential inequality for t ∈ [t 0 , T + ):

(2.57)

Integrating this inequality we find

6-N .

To finish the proof, recall that c * λ(t)

ϕ M (t) by Corollary 2.11. Consider now the case N = 3. The problem is that N -4 < 0, hence we cannot write

M

, as we did in the previous proof. Instead, we just have

Integrating between t and T + we obtain

This is again a differential inequality. It yields (1.6).

Remark 2.13. In the case N = 3 and u * ∈ X 2 × H 2 , we can prove (1.4) for continuous time, not only for a sequence. Indeed, in this case one can take b

, hence the proof of (2.57) applies. The end of the proof is the same as in the case N ∈ {4, 5}.

Proof of Theorem 2. Let t ∈ [t 0 , T + ) be such that n(g(t), λ(t)) = sup t≤τ <T + n(g(τ ), λ(τ )). From (2.29) and Lemma 2.2 we get

But due to Lemma 2.12, the last term on the right hand side can be omitted. This is in contradiction with (2.28) and Lemma 2.8.

A Cauchy theory in higher regularity

In this section we prove some facts about propagation of regularity for (NLW), which are applied to u * (t) in the main text. As in [40, Appendix B], the proofs rely on the classical energy estimates:

Then the solution of the linear wave equation

Proof. This is a standard application of the energy estimates and the Fixed Point Theorem, using the fact that f (u) is a monomial and X s → L ∞ . We skip the details.

In the rest of this section we consider (NLW) in dimension N = 5. In this case the nonlinearity f (u) = |u| 4 3 u is not smooth. We will use the following regularization:

In the proof of the next result we will use the Fractional Leibniz Rule and the Fractional Chain Rule in the form given in [12, Propositions 3.1, 3.3]:

Remark A.3. In [START_REF] Christ | Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation[END_REF], the Leibniz Rule and the Chain Rule are proved in the case of one space dimension, and necessary changes in order to carry out a proof in arbitrary dimension are indicated. In the present paper we use this result in dimension 5, but only for radial functions, and it can be verified that the Leibniz Rule and the Chain Rule for radial functions is a consequence of the one-dimensional result.

Lemma A.4. Let N = 5 and 1 ≤ s ≤ 2. The following estimates hold (with constants which may depend on s):

where the sign means that the constant is independent of n.

Proof. A simple computation shows that

We have

which is acceptable due to (A.7).

In order to bound

f ( u

This proves (A.1). Estimate (A.2) is a part of [START_REF] Jendrej | Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5[END_REF]Lemma B.3] and the proof of (A.3) is analogous.

From the Sobolev inequality we get

3 ) f ( u X s ), hence in order to prove (A.4) and (A.5) it suffices to check that

For s ∈ {1, 2} this is an easy algebraic computation which we will skip. For 1 < s < 2 we use Proposition A.2:

The second inequality in (A.9) is proved analogously.

CHAPTER 3. BOUNDS ON THE SPEED OF TYPE II BLOW-UP

Using (A.5) and (A.6) one can show that there exists a unique maximal solution

and

From (A.5) and the energy estimate we have

with a constant independent of n. This implies that there exist T 1 < t 0 , T 2 > t 0 and a constant C 1 independent of n such that

To this end, we notice that u n -u solves the Cauchy problem:

Since u(t) X 1 ×H 1 is bounded and u n (t) X 1 ×H 1 are uniformly bounded for t ∈ [ T 1 , T 2 ], (A.1) and (A.3) imply that for t ∈ [ T 1 , T 2 ] there holds

which yields (A.11) by the energy estimate and the Gronwall inequality. From (A.10) and (A.11) we deduce

Using once again the energy estimate together with (A.4) it is easy to see that in fact

Chapter 4

Nonexistence of radial two-bubbles with opposite signs for the energy-critical wave equation

Abstract

We consider the focusing energy-critical wave equation in space dimension N ≥ 3 for radial data. We study two-bubble solutions, that is solutions which behave as a superposition of two decoupled radial ground states (called bubbles) asymptotically for large positive times. We prove that in this case these two bubbles must have the same sign. The main tool is a sharp coercivity property of the energy functional near the family of ground states.

which follows from the inequality

Next, we prove that if R is large enough, then

(3.49)

We will prove that

In the region |x| ≥ R √ λ we have V (λ, 1) = 0 and

In the region |x| ≤ R √ λ we use the pointwise bound |f Step 2 -Differential inequalities. Observe that

The proof is the same as the proof of (2.64).

For m ∈ N, m ≥ m 0 , let t = t m be the last time such that N (g(t), λ(t)) = 2 -m . By continuity, t m is well defined if m 0 is large enough.

By Proposition 3.4, there exists a constant C 1 such that

Since this holds for all m sufficiently large, we deduce that there exists t 0 < T + such that

At time t m all the terms of (3.45) except for the term 2a - 1 (t) 2 are absorbed by the right hand side, hence

Using the definition of t m-1 we obtain

By (3.14), this implies

Dividing both sides by λ 1 (t) and integrating we obtain that log λ 1 (t) converges as t → T + . Dividing both sides by λ 2 (t), using the fact that λ 2 (t) ≥ λ 1 (t) for t ≥ t 0 and integrating we obtain that log λ 2 (t) converges as t → T + . Hence log λ(t) converges, which is impossible.

Remark 3.8. An analogous proof using the linear stability and instability components α + λ 2 and α - λ 2 instead of the refined modulation and instability component β a 2 λ 2 would yield λ 2 (0) → λ 0 ∈ (0, +∞) (hence T + = +∞) and | log λ 1 (t)| t as t → +∞, but would not (at least directly) lead to a contradiction. Proof. Without loss of generality we can assume that ψ (0) = 1. We set:

A Elementary lemmas

and it suffices to verify that ϕ has the required properties.

Recall that we denote f (u) := |u|

The same conclusion holds if we replace Y k by BC ν for ν ≥ 0.

Proof. We have an isomorphism

The series converges for |z| < 1. We have a series expansion:

large enough and that the last series converges strongly in H k if η is small.

In the case of the space BC ν the proof is the same.

Lemma A.4. There exists k = k(N ) ∈ N and η = η(N ) > 0 such that if ψ ∈ Y k and |a| ≤ η, then for all g ∈ Ḣ1 such that g Ḣ1 ≤ η there holds

with a constant depending on ψ.

Proof. For N ∈ {3, 4, 5} this follows directly from the Sobolev and Hölder inequalities (even for ψ ∈ Ḣ1 ).

For N ≥ 6 we use the pointwise bound

Here, f is a negative power. Since U a has slow decay, ψ

and k is large enough. The conclusion follows from the Hölder inequlity.

The proof of (A.2) is similar.

Lemma A.5. Let γ := min 3, 2N N -2 . For any M > 0 there exists C > 0 and η > 0 such that if v E ≤ M and g E ≤ η, then

Proof. In dimension N ∈ {3, 4, 5} this follows from the pointwise inequality

whereas for N ≥ 6 from

In order to prove bounds (A.3) and (A.4), notice that they are homogeneous and invariant by changing signs of both k and l, hence it can be assumed that k = 1 (for k = 0 the inequalities are obvious). Now for |l| ≤ 1 2 the conclusion follows from the asymptotic expansion of F (1+l) and for |l| ≥ 1 2 the bounds are evident. We also study the relationship between the speed of a type II blow-up and the weak limit of the solution at the blow-up time. Finally, we prove that there are no pure radial two-bubbles with opposite signs for the energy-critical wave equation.