Keywords: Problémes paraboliques-dégénérés, estimations de gradient, explosion en un seul point, solutions de viscosité, comportement asymptotique, Théorème de type Liouville Degenerate parabolic equations, gradient estimates, single point gradient blow-up, viscosity solutions, asymptotic behavior, Liouville-type theorem

Ces 4 années de thèse ont été pour moi des années de bonheur et de questionnement. Je tiens à témoigner ma reconnaissance à tous ceux qui ont fait que cette expérience fut enrichissante et formatrice.

Tout d'abord je tiens à remercier chaleureusement mon directeur Philippe Souplet pour son soutien, sa patience, sa pédagogie et ses précieux conseils. La première fois que j'ai rencontré Philippe fut lors de son cours de M2 à Chevaleret où j'ai pu apprécier les EDP paraboliques et leurs applications. Ensuite, ce fut pour moi une très grande joie qu'il ait accepté d'encadrer mes premiers pas dans le monde de la recherche lors du stage de Master et un très grand honneur qu'il est accepté humblement de poursuivre l'aventure en encadrant ma thèse. Je le remercie pour toute la disponibilité qu'il m'a accordée (surtout lors des moments où j'avais besoin de nouvelles pistes), et pour m'avoir fait profiter de son expérience. Je le remercie de m'avoir permis de participer à des conférences dans différents endroits de ce monde. J'ai pu nouer des contacts et découvrir des cultures à travers et en parallèle de l'univers des maths. Cela a été un véritable plaisir de travailler à ses côtés.

Je tiens ensuite à remercier vivement Mr Alessio Porretta pour avoir accepté d'être l'un des rapporteurs de cette thèse, pour sa sympathie et pour avoir attiré mon attention sur la régularité des solutions de viscosité après le temps d'explosion. Mes sincères sentiments de reconnaissance vont à Mr Olivier Ley pour sa lecture attentive et ses remarques pertinentes. C'est un grand honneur pour moi que vous soyez les rapporteurs de cette thèse.

L'occasion m'est donné d'exprimer toute ma gratitude à celui qui a pris le temps de me faire découvrir sous un angle ludique et agréable la théorie des solutions de viscosité : Pr Barles. Cette collaboration a été un très grand enrichissement autant mathématique qu'humain. J'en ai appris énormément et cela a ouvert mes horizons. Merci ! ! ! Je tiens aussi à remercier Pr Imbert, Pr Rakotoson, Pr Wiessler et Pr Zaag pour avoir accepté de faire partie de mon jury de thèse. Je remercie Pr Weissler et Pr Zaag pour l'intérêt et le soutien apporté lors de ces 4 années de thèse.

Je veux aussi remercier Pr Carmen Cortazar (belle famille de mathématiciens !), Marta Huidobro ainsi que toute l'équipe de la faculté de mathématiques de l'Université PUC du Chili (Duvan Henao, Manuel Elgueta, Myriam Tsutsumi..) de l'accueil chaleureux qui m'a été fait pendant mon séjour très agréable à Santiago, en novembre 2011. Merci de l'intérêt qu'ils ont manifesté pour les problèmes que j'étudiais, et pour m'avoir fait découvrir des thématiques liées au p-Laplacien. Ce travail n'a pas encore abouti mais j'espère qu'on y avancera.

Mes remerciements s'adressent au laboratoire LAGA et à l'ensemble de ses membres, autant les chercheurs que le personnel administratif (Isabelle, Yolande, Jean Philippe Dru, Jean-Philippe Dommergue, Gilles ...) qui par leur soutien m'ont permis d'optimiser le temps consacré à la recherche mais aussi d'avoir des souvenirs agréables de la vie de labo.

L'enseignement et la transmission du savoir constituent certainement une partie enrichissante pour tout thésard. Je suis heureuse d'avoir pu bénéficier d'un monitorat lors de cette thèse. Je remercie mes différents étudiants avec qui j'ai pu partager mes connais-

Résumé

Cette thèse est consacrée à l'étude des propriétés qualitatives de solutions d'une équation d'évolution de type Hamilton-Jacobi avec une diffusion donnée par l'opérateur p-Laplacien. On s'attache principalement à l'étude de l'effet de la diffusion non-linéaire sur le phénomène d'explosion du gradient. Les principales questions qu'on étudie portent sur l'existence locale, régularité, profil spatial d'explosion et la localisation des points d'explosion. En particulier on montre un résultat d'explosion en seul point du bord. Dans le chapitre 4, on utilise une approche de solutions de viscosité pour prolonger la solution explosive au delà des singularités et on étudie son comportement en temps grands. Dans l'avant dernier chapitre on s'intéresse au caractère borné des solutions globales du problème unidimensionnel. Dans le dernier chapitre on démontre une estimation de gradient locale en espace et on l'utilise pour obtenir un résultat de type Liouville. On s'inspire et on compare nos résultats avec les résultats connus pour le cas de la diffusion linéaire.

sances. Ils ont été pour moi une bonne source de motivation pour la recherche ! Je suis à mon tour redevable à tous ces enseignants, qui du lycée au Master, m'ont permis d'acquérir un certain bagage mathématique pour pouvoir entamer cette thèse. Je suis reconnaissante à Gwenola, François, Julien, Élise et les élèves de l'association Science Ouverte pour l'enrichissement pédagogique apporté au cours de ces dernières années.

Comment ne pas remercier ceux qui, au quotidien ont rendu ma vie de thésarde joyeuse et riche : les thésards du LAGA. Pour faire simple, procédons par ordre alphabétique : Abderahmane, Alexandre (frère de thèse, merci pour les discutions mathématiques, les relaxations et ta générosité, je garde de très agréables souvenirs des différents colloques et séminaires !) Asma (merci pour la bsisa, les fous rires et tes explications sur l'équation des ondes), Amine (merci pour tes discutions sur les lois de parois, les moments de détente et l'aide apportée) Bakari (merci pour ta sympathie, bonne humeur et le repas malien !), Cécile (amie depuis le master et ça continue pour après !), David, Elisa, Eva, Julien, Kaouther (la première amie que j'ai rencontré au LAGA et toujours d'une grande aide, merci pour la bonne ambiance, les moments de partage qui ont fait du bureau A301 un lieu agréable et l'ouverture vers les probas !), Khue (longtemps mon voisin de bureau adoré, merci pour ta générosité et amabilité), Linglong (merci pour les encouragements et le sourire) Phan (grand frère de thèse toujours là pour lire mes gribouillis de maths et poser les bonnes questions, que des souvenirs joyeux), Nejib (petit frère de thèse, je garde un bon souvenir de Hammamet !) Phong (merci pour les différentes aides), Rémi, Roland, Van Tuan, Van Tien (merci pour m'avoir invité à ton mariage à la Basilique St Denis). Je n'oublie pas Maher (futur Professeur des universités et le super coach des petit thésards !). Merci pour les remarques, corrections, questions et anecdotes autour des maths et des mathématiciens ! ! ! J'ai eu la chance d'être dans un labo où différentes thématiques des maths (algèbre, analyse, probas,...) sont représentées et où un cadre de travail agréable est fourni.

C'est avec plaisir que je remercie maintenant mes amis n'ayant pas de lien direct avec la thèse mais qui ont joué un grand rôle pour son aboutissement par tous ces bols d'air frais et l'équilibre qu'ils ont su m'apporter. Je remercie en particulier tous les membres de la vélo école Vivre à Vélo en Ville. Les moments passés à vos côtés ont été un vrai bonheur où j'ai eu la joie de rencontrer des gens de tous horizons. Merci François pour m'avoir initié aux joies de la petite reine qui m'ont permis de m'évader et mieux réfléchir ! Je remercie aussi les bénévoles de Vélocipaide pour leur bienveillance et encouragements.

Pour finir, je tiens à remercier la solide équipe de soutien de toujours : ma famille. Je remercie particulièrement mes parents (Salwa et Tawfik, merci pour TOUT, il n'y aura jamais assez de mots pour remercier ses parents ! ! !), mon grand frère Wassim (et oui on cherche encore Mr Zéro, ils sont fous ces matheux ! merci pour les chats et les blagues !), tante Fouza (merci pour les encouragements depuis NY et les appels chaleureux et hebdomadaires : une vraie co-directrice de thèse !) et Nazouh (merci pour ton soutien depuis Tunis), mon défunt oncle Habib (tu nous manques !) et bien sûr je n'oublie pas l'essentielle : ma soeur Farah (à toi seule il te faudrait un chapitre de remerciements ! je te les réserve pour d'autres occasions j'espère !). dans Ω × (0, T ),

Table des matières

où ∆ p u = div (|∇u| p-2 ∇u).

Les équations (1.1), notamment dans le cas de la diffusion linéaire p = 2, sont importantes de plusieurs points de vue : a. Elles fournissent une méthode de construction des solutions de l'équation de Hamilton-Jacobi u t = |∇u| q , par régularisation parabolique évanescente (i.e. on met un paramètre devant le terme de diffusion et on regarde la limite → 0 + ). Cette dernière équation est fondamentale en théorie du contrôle optimal [START_REF] Fleming | Controlled Markov processes and viscosity solutions[END_REF]. b. Elles interviennent dans le modèle physique de Kardar-Parisi-Zhang (KPZ) qui décrit des processus de croissance d'interfaces rugueuses par déposition de particules sur une surface [START_REF] Kardar | Dynamic scaling of growing interfaces[END_REF]. L'équation KPZ est à l'origine une EDP stochastique qui décrit l'évolution au cours du temps de la hauteur h d'une interface

h t = λ∆h + λ 2 |∇h| 2 + η(x, t).
La diffusion tient compte de la relaxation, le terme en gradient vient de l'effet de croissance par déposition de nouvelles particules et η(x, t) est un bruit blanc de moyenne nulle produit par des forces stochastiques avec faible corrélation. L'équation de KPZ déterministe a ensuite été généralisée par Krug et Spohn [START_REF] Krug | Universality classes for deterministic surface growth[END_REF] afin d'étudier l'effet d'une non-linéarité plus forte (|∇u| q avec q > 2) sur le comportement de la solution. L'équation (1.1) présente aussi un très grand intérêt mathématique en elle-même. En effet la compétition entre le terme source qui dépend seulement du gradient et le terme de diffusion est à l'origine d'une grande richesse de phénomènes : structure des états stationnaires, si Ω = R N existence de solutions globales non bornées en norme C 1 , pour Ω = R N comportement en temps grand de type diffusif pour certaines valeur de q et p ou bien de type hyperbolique ou le terme hamiltonien l'emporte... Le phénomène qui nous intéresse le plus ici est celui de l'apparition d'une singularité en gradient et non en amplitude : si T max (u 0 ) < ∞ (ici T max (u 0 ) est le temps maximal d'existence de la solution classique), alors u reste bornée en norme L ∞ (ceci découle d'une application simple du principe du maximum faible) mais pour q > p et certaines données initiales, avec des conditions de Dirichlet homogènes au bord, on a lim t→Tmax(u 0 ) |∇u(t)| = ∞. Ce phénomène d'explosion dépend fortement de la donnée initiale, de la taille de la non-linéarité, du domaine et des conditions aux limites. Rappelons que quand on s'intéresse à des équations quasi-linéaires paraboliques de la forme :

u t - N i,j=1
a i,j (x, u, ∇u)

∂ 2 u ∂x i ∂x j -F (x, u, ∇u) = 0 (1.2)
avec des coefficients a i,j et F assez réguliers, les questions d'existence locale et d'unicité pour le problème de Cauchy-Dirichlet sont assez bien comprises (du moins pour le cas uniformément parabolique) [START_REF]Second order parabolic differential equations[END_REF][START_REF] Ladyzenskaja | Linear and quasi-linear equations of parabolic type[END_REF][START_REF] Friedman | Partial differential equations of parabolic type[END_REF]. Sous certaines conditions sur les termes non linéaires a i,j et F , des estimations a priori des solutions ont été obtenues. Il a été observé que si ces conditions ne sont pas satisfaites alors on a apparition de solutions "explosives". Il est connu que la condition de croissance de Bernstein-Nagumo [START_REF] Ladyzenskaja | Linear and quasi-linear equations of parabolic type[END_REF][START_REF] Ar | On the solvability of some boundary value problems for a class of quasilinear parabolic equations[END_REF] F (x, u, p) A(x, u, p) ≤ K(u)h(|p|)

avec h satisfaisant la restriction ∞ 1 s ds h(s) = ∞,
garantit une borne sur le gradient de la solution une fois que l'on dispose d'une estimation de la norme L ∞ de la solution. Si cette condition n'est pas satisfaite, alors on peut avoir des solutions bornées dont le gradient explose soit au bord du domaine soit à l'intérieur. Cette condition a été affaiblie en écrivant F comme la somme de deux fonctions qui vérifient une certaine monotonie et des conditions de croissance relaxées (voir [115,[START_REF] Ph | Gradient bounds for solutions of semilinear parabolic equations without Bernstein's quadratic condition[END_REF]). Dans cette thèse on se placera donc principalement dans le cas q > p > 2 et on s'attachera à répondre à certaines questions relatives au phénomène d'explosion du gradient (conditions suffisantes d'explosion, profil spatial, localisation des points d'explosion...) en étudiant l'influence de l'opérateur p-Laplacien sur ce phénomène. Le cas de la diffusion linéaire p = 2 pour l'équation (1.1) ayant fait l'objet d'un certain nombre d'études, on essaiera de s'en inspirer et on les comparera au cas de la diffusion non-linéaire.

Signalons que l'étude de la formation de singularités en temps fini pour les solutions de certaines EDP paraboliques semi-linéaires a fait l'objet de nombreux travaux ces dernières années. La grande majorité d'entre eux s'est concentrée sur l'explosion de la solution en amplitude, c'est à dire en norme L ∞ . Ces travaux regroupent des résultats sur les critères d'explosion, localisation des points d'explosion, vitesse et profil spatial de d'explosion, etc. En particulier pour l'équation de la chaleur semi-linéaire bien connue u t -∆u = u q (1.3) nous renvoyons le lecteur à [START_REF] Quittner | Superlinear parabolic problems : blow-up, global existence and steady states[END_REF][START_REF] Samarskii | Blow-up in quasilinear parabolic equations[END_REF] et les références qui s'y trouvent. Étant donné la dégénérescence de l'équation (1.1) aux points où ∇u = 0, on ne peut espérer obtenir en général des solutions régulières sans imposer certaines conditions sur les données initiale et au bord. Ainsi selon les problèmes qu'on tentera de résoudre, différentes notions de solutions, telles que les solutions faibles, classiques ou de viscosité doivent être considérées. Dans un premier temps on s'intéressera aux solutions faibles Lipschitziennes en espace pour avoir un bon cadre de travail sur les singularités en gradient. Dans un second temps on se placera dans un cas particulier ou la théorie classique des EDP paraboliques quasi-linéaires nous donne l'existence de solutions classiques, ce qui facilitera l'étude de la localisation des points d'explosion. Dans le chapitre 4 nous verrons une autre approche qui est basée sur la notion de solution de viscosité et qui permettra l'étude de la continuation des solutions explosives.

Etat de l'art et résultats du chapitre 2 : Théorie locale et profil spatial de l'explosion

Dans ce chapitre on s'intéresse au problème de Cauchy-Dirichlet associé à l'équation (1.1) :

   ∂ t u -div(|∇u| p-2 ∇u) = |∇u| q , x ∈ Ω, t > 0, u(x, t) = g(x), x ∈ ∂Ω, t > 0, u(x, 0) = u 0 (x),

x ∈ Ω,

où -Ω ⊂ R N est un domaine borné régulier, p > 2 et q > p -1, u 0 ∈ W 1,∞ (Ω), g est la trace d'une fonction C 2 . On suppose que u 0 satisfait la condition de compatibilité u 0 (x) = g(x) pour x ∈ ∂Ω.

(1.5) D'après les résultats de [START_REF] Dlotko | Examples of parabolic problems with blowing-up derivatives[END_REF][START_REF] Galaktionov | The problem of blow-up in nonlinear parabolic equations[END_REF][START_REF] Ph | Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions[END_REF]110] nous savons que pour p = 2 les solutions positives du problème (1.4) existent globalement lorsque 1 < q < 2 tandis qu'elles peuvent exploser en temps fini pour q > 2 (Ceci est clairement détaillé dans le livre de Quittner-Souplet et les références qui s'y trouvent [START_REF] Quittner | Superlinear parabolic problems : blow-up, global existence and steady states[END_REF]). Pour le cas p > 2 où l'équation peut être dégénérée (pour |∇u| = 0), J. Zhao [START_REF] Zhao | Existence and nonexistence of solutions for u t = div(|∇u| p-2 ∇u) + f (∇u, u, x, t)[END_REF] a étudié les problèmes d'existence et de non existence de solutions faibles de (1.4) pour q ≤ p -1. Dans ce cas il prouve l'existence de solutions (globales) pour toute donnée initiale u 0 . Pour q < p une borne L ∞ locale du gradient des solutions est prouvée dans [START_REF] Burczak | L ∞ a priori bounds for gradients of solutions to quasilinear inhomogeneous fast-growing parabolic systems[END_REF]. Pour le cas q > p -1, Chen, Nakao et Ohara [START_REF] Chen | Global existence and gradient estimates for quasilinear parabolic equations of the m-laplacian type with a strong perturbation[END_REF] ont montré l'existence de solutions faibles globales mais seulement sous une condition de petitesse de la donnée initiale et avec l'hypothèse d'une courbure moyenne positive du bord ∂Ω.

Notre première contribution vient compléter ces résultats en montrant l'existence et l'unicité d'une solution maximale en temps dans W 1,∞ (Ω) sans restriction de taille sur la donnée initiale et en mettant en évidence l'alternative d'explosion dans W 1,∞ .

Tout d'abord la notion de solution faible pour le problème (1.4) est définie de manière standard comme suit : Définition 1.2.1. Soit r = max(p, q). Une fonction u(x, t) est appelée solution faible du problème (1.4) dans Q T := Ω × (0, T ) si u ∈ C(Ω × [0, T )) ∩ L r ((0, T ); W 1,r (Ω)), u t ∈ L 2 ((0, T ); L 2 (Ω)), u(x, 0) = u 0 (x), u = g sur ∂Ω et l'égalité

Q T u t ψ + |∇u| p-2 ∇u • ∇ψ dx dt = Q T |∇u| q ψ dx dt (1.6)
est satisfaite pour tout ψ ∈ C 0 (Q T ) ∩ L p ((0, T ); W 1,p (Ω)) telle que ψ = 0 sur ∂Ω × (0, T ).

Rappelons que la notion de solution faible (ou au sens des distributions) n'est pas la seule notion dont on dispose pour résoudre le problème (1.1). En effet, la théorie des solutions de viscosité fournit un cadre plus général pour les problèmes d'existence et d'unicité, mais celui-ci est moins adapté pour l'étude des problèmes de singularités qui nous intéressent. En effet ces solutions ne sont a priori pas assez régulières pour "voir" les singularités du gradient. La notion de solution faible qui fait intervenir les espaces de Sobolev et qui demande un peu plus de régularité est certainement une meilleure alternative. Théorème 1.2.1. On suppose que q > p -1 > 1. Soient M > 0 et u 0 , g satisfaisant la condition de compatibilité (1.5) et ∇u 0 ∞ ≤ M . Alors (i) Il existe un temps T = T (M, p, q, N, g C 2 ) > 0 et une solution faible u de (1.4) dans [0, T ), qui de plus satisfait u ∈ L ∞ loc ([0, T ); W 1,∞ (Ω)). (ii) Pour tout T > 0 le problème (1.4) admet au plus une solution u telle que u ∈ L ∞ loc ([0, T ); W 1,∞ (Ω)). (iii) Il existe une (unique) solution faible, maximale de (1.4), notée u. Soit T max (u 0 ) le temps maximal d'existence, alors min

Ω u 0 ≤ u ≤ max Ω u 0 dans Ω × (0, T max (u 0 )) (1.7) et si T max (u 0 ) < ∞, alors lim t→Tmax(u 0 ) ∇u L ∞ (Ω) = ∞.
La méthode classique pour montrer l'existence locale de solutions faibles est d'introduire un problème approché uniformément parabolique permettant de construire des solutions classiques u ε . Le but est de démontrer que les suites u ε et ∇u ε sont uniformément bornées dans un espace L m . Avec ces estimations et quitte à extraire une sous-suite, on a la convergence forte des u ε par le théorème d'Ascoli mais seulement une convergence faible des ∇u ε qui est insuffisante pour passer à la limite dans le terme non-linéaire. La principale difficulté consiste alors à avoir de meilleures estimations sur ∇u ε . Pour venir à bout de cette difficulté, notre principale nouveauté par rapport à [START_REF] Chen | Global existence and gradient estimates for quasilinear parabolic equations of the m-laplacian type with a strong perturbation[END_REF] est un contrôle du gradient des solutions approchées près du bord pour un temps petit via des fonctions barrière bien choisies. L'invariance par translation en espace de l'équation (1.4) nous permet ensuite d'avoir un contrôle du gradient sur tout le domaine via un principe de comparaison. On utilise alors un résultat fort de Friedman-DiBenedetto sur la régularité Hölderienne du gradient de certaines EDP paraboliques dégénérées [START_REF] Dibenedetto | Degenerate parabolic equations[END_REF][START_REF] Dibenedetto | Hölder estimates for nonlinear degenerate parabolic sytems[END_REF].

Notre deuxième contribution porte sur un effet régularisant sur u t . En utilisant l'homogénéité du p-Laplacien et un résultat de comparaison on a pu établir le résultat suivant. On note osc(u 0 ) = max

Ω u 0 -min Ω u 0 .
Théorème 1.2.2. On suppose que q > p -1 > 1 et u 0 ∈ W 1,∞ (Ω). Soit u l'unique solution faible de (1.4) dans L ∞ loc ([0, T max (u 0 )); W 1,∞ (Ω)). Alors

u t ≤ 1 p -2 osc(u 0 ) t dans D (Ω × (0, T )). (1.8)
Remarquons que grâce à l'estimation unilatérale de u t on a une estimation de ∆ p u dans l'esprit des estimation de Bénilan et Crandall. En effet on a

∆ p u = u t -|∇u| q ≤ 1 p -2 u 0 ∞ t . (1.9)
L'estimation de semi-concavité (1.9) a été montré dans [START_REF] Esteban | Approximate solutions to first and second order quasilinear evolution equations via nonlinear viscosity[END_REF] seulement pour q = p et dans le cas du problème de Cauchy Ω = R N . Elle est aussi valide pour le problème de Cauchy pour 1 < q ≤ p (voir [START_REF]Non-Diffusive Large Time Behavior for a Degenerate Viscous Hamilton-Jacobi Equation[END_REF]). Rappelons que les estimations de ce type peuvent servir pour démontrer des inégalités de Harnack ou bien obtenir des résultats de régularité. Signalons qu'en utilisant des arguments similaires à ceux de [START_REF] Ph | Regularizing effects of homogeneous evolution equations, Contributions to analysis and geometry[END_REF], il est possible de montrer que pour q < p -1

u t ≥ -1 p -2 u 0 ∞ t .
Dans les travaux de Aronson-Bénilan on rencontre aussi des estimations unilatérales similaires. Dans le chapitre 5 cette estimation est complétée par une estimation inférieure sur u t .

Théorème 1.2.3. On suppose que q > p -1 > 1. Soit u l'unique solution faible de (1.4) dans L ∞ loc ([0, T max (u 0 )); W 1,∞ (Ω)). Fixons t 0 ∈ (0, T max ), alors

u t ≥ - q -p + 1 p -2 sup [0,t 0 ]×Ω |∇u| q - 1 p -2 osc(u 0 )
t dans D ((0, t 0 ) × Ω) . (1.10) L'estimation (1.8) et la méthode de Bernstein (qu'on détaillera un peu plus loin dans le dernière section de l'introduction) nous ont permis d'établir une estimation locale sur le gradient de la solution donnant le profil spatial de l'explosion. On note δ(x) = dist(x, Ω). Théorème 1.2.4. On suppose que q > p -1 > 1. Soient M > 0 et u 0 satisfaisant les conditions de compatibilité et ∇u 0 ∞ ≤ M . Soit u l'unique solution faible de (1.4) dans L ∞ loc ([0, T max (u 0 )); W 1,∞ (Ω)). Alors

|∇u| ≤ C 1 δ -1/(q-p+1) (x) + C 2 .
dans Ω × (0, T max (u 0 )) .

(1.11)

où C 1 = C 1 (q, p, N ) > 0 et C 2 = C 2 (q, p, Ω, M, g C 2 ) > 0.
Comme on le verra dans les chapitres 3 et 5, cette estimation est optimale. Pour terminer nous donnons une condition suffisante qui induit l'explosion du gradient en temps fini. Proposition 1.2.1 (Méthode de la fonction propre). On suppose que q > p > 2. Soit u l'unique solution faible de (1.4)

dans L ∞ loc ([0, T max (u 0 )); W 1,∞ (Ω)). Soit α ≥ 1 telle que p-1 q-p+1 < α < q -1, alors il existe une constante C = C(q, p, α, Ω, g ∞ ) > 0 telle que si Ω u 0 ϕ α 1 dx ≥ C, alors T max (u 0 ) < ∞.
Dans [START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF] un autre critère suffisant qui repose sur une autre méthode est fourni : pour

tout r ∈ [1, ∞), il existe une constante C 1 = C 1 (p, q, r, Ω) > 0 telle que, si u 0 ∈ W 1,∞ et u 0 r ≥ C 1 , alors T max (u 0 ) < ∞.

Questions ouvertes ou avec des réponses partielles

Avec quelle vitesse ?

Après avoir obtenu le profil spatial de l'explosion, il est naturel de s'intéresser à la vitesse avec laquelle ce phénomène se produit. Un première réponse a été donnée dans les travaux de [START_REF] Guo | Blowup rate estimates for the heat equation with a nonlinear gradient source term[END_REF][START_REF] Conner | Asymptotics of blowup for a convectiondiffusion equation with conservation[END_REF] pour le cas de la diffusion linéaire. Théorème 1.2.5 (J.S. Guo et B. Hu). Soit le problème (1.4) avec q > 2 = p. Si le gradient explose en temps fini T * , alors il existe une constante C 0 > telle que :

sup x∈Ω,0≤τ ≤t |∇u(x, τ )| ≥ C 0 (T * -t) -1 q-2 .
Il est intéressant de constater que pour cette équation la vitesse d'explosion ne coïncide pas avec celle suggérée par l'invariance de l'équation. En effet, soit u une solution de (1.4), la fonction u λ = λ -k u(λx, λ 2 t) ; k = q -2 q -1 résout aussi (1.4). On peut alors envisager l'existence de solutions auto-similaires de la forme

w(t, x) = (T -t) k 2 V x √ T -t , (1.12)
dans un demi-espace, avec la condition w = 0 sur le bord (ceci ne peut se produire dans l'espace entier, car on sait que toutes les solution existent globalement cf. [START_REF] Quittner | Superlinear parabolic problems : blow-up, global existence and steady states[END_REF]). S'il existe des solutions w de la forme (1.12) avec ∇V ∈ L ∞ , alors w aurait impliqué que

∇u(t, ) L ∞ ∼ (T -t) - 1
2(q-1) .

Mais ceci et incompatible avec le Théorème 1.2.5 car 1 q-2 > 1 2(q-1) . Autrement dit la vitesse d'explosion du gradient est plus rapide que la vitesse autosimilaire. L'estimation supérieure de la vitesse d'explosion est encore un problème ouvert. Néanmoins pour M assez grand et pour les solutions croissantes en temps du problème 1D :

u t -u xx = |u x | p , x ∈ (0, 1), t > 0 u(t, 0) = 0, u(t, 1) = M, t > 0. (1.13) on a c 0 (T * -t) -1 q-2 ≤ max 0≤x≤1 |u x (t, x)| ≤ c 1 (T * -t) -1 q-2 . (1.14)
Dans [START_REF] Zhang | Gradient blowup rate for a viscous hamilton-jacobi equation with degenerate diffusion[END_REF] Z. Zhang a généralisé ce résultat au cas de la diffusion non linéaire pour des solutions classiques croissantes en temps et en espace. On a

C 1 (T * -t) -1 (q-p) ≤ max 0≤x≤1 |u x (t, x)| ≤ C 2 (T * -t) -1 (q-p) . (1.15)
La preuve de ces deux derniers résultats repose sur l'application du principe du maximum à une fonctionnelle bien choisie et sur le lemme de Hopf appliqué à u t . Notons qu'on ne sait pas si cette vitesse est la seule possible si on ne suppose pas que la solution est croissante en temps.

Solution de viscosité vs solutions faibles

Le lien entre solutions de viscosité et solutions faibles (au sens des distributions) n'est pas encore très bien compris. Quand ces deux notions sont-elles équivalentes ? En utilisant l'unicité des solutions dans la classe des solutions de viscosité et dans la classe des solutions Lipschitziennes, on peut néanmoins montrer le résultat suivant Proposition 1.2.2. On suppose que q > p -1 > 1 et u 0 ∈ W 1,∞ (Ω). Notons T max (u 0 ) le temps maximal d'existence de l'unique solution faible u Lip , Lipschitzienne, maximale. Alors la solution de viscosité u V is (globale) du problème (1.4) coïncide avec la solution faible sur [0, T max (u 0 )).

En effet prenons u

0 ∈ W 1,∞ (Ω) avec ∇u 0 ∞ ≤ M et notons u Vis la solution de viscosité associée et u Lip la solution faible u Lip ∈ L ∞ loc ([0, T max (u 0 )); W 1,∞ (Ω)) associée. Fixons T ∈ (0, T max (u 0 )) et posons T = sup s > 0 tel que u Vis = u Lip dans [0, s] × Ω .
(1.16)

On sait que T ≥ t 0 = t 0 (M ) > 0. En effet, pour un temps petit t 0 (M ), nous disposons d'estimations uniformes en ε de la norme L ∞ des solutions u ε des problèmes approchés ainsi que des estimations Hölderiennes (locales) de ∇u ε (voir le chapitre 2). Ces estimations impliquent que (u ε ) est relativement compact dans C(Ω × [0, t 0 (M )]). D'une part, en utilisant les résultats de stabilité et de comparaison [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF][START_REF] Attouchi | Global continuation beyond singularities on the boundary for a degenerate diffusive hamilton-jacobi equation[END_REF][START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF], on conclut que u ε convergent uniformément vers l'unique solution de viscosité u Vis du problème (1.4) (voir chapitre 4 pour l'unicité). D'autre part les estimations (suffisamment fortes) sur le gradient des u ε sur [0, t 0 (M )] nous permettent aussi de conclure que les u ε convergent vers l'unique solution faible Lipschitzienne u Lip du problème (1.4) (voir chapitre 2). Il s'en suit que

u Vis = u Lip sur [0, t 0 (M )]. Supposons que T < T . Notons A = sup t∈[0,T ] u(t) W 1,∞ . Pour tout η ∈ (0, T )
il existe un τ (A) > 0 (indépendant de η) tel que la solution de viscosité et la solution faible associées au problème ci-dessous coïncident (pour la même raison que précédemment).

   ∂ t w -div(|∇w| p-2 ∇w) = |∇w| q , x ∈ Ω, t > 0, w(x, t) = u( T -η, x), x ∈ ∂Ω, t > 0, w(x, 0) = u( T -η, x), x ∈ Ω, (1.17) Par conséquent u Vis et u Lip coïncident sur [0, T -η + τ ]. Puisque T -η + τ > T pour η
suffisamment petit, on obtient une contradiction avec la définition de T . Signalons également que pour p = 2 et q > 1, Poretta et Zuazua [START_REF] Porretta | Null controllability of viscous Hamilton-Jacobi equations[END_REF] ont montré récemment que les solutions de viscosité (qui sont globales en temps) redeviennent Lipschitziennes (et vérifieront donc les conditions au bord au sens classique) après un certain temps. Plus précisément pour g = 0 et u 0 ∈ C(Ω) (sans condition de signe), ils ont montré qu'il existe des constantes positives λ, K, C (dépendant seulement de q et Ω) tel que les solutions de viscosité du problème (1.4) vérifient

u(t) L ∞ + ∇u(t) L ∞ ≤ Ce -λt , pour t ≥ K u 0 L ∞ .
(1.18)

La preuve repose sur des arguments de comparaison et sur l'étude du problème linéarisé. Le lecteur peut trouver une étude de l'équivalence de la notion de solution de viscosité et solution faible dans [START_REF] Juutinen | On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation[END_REF] pour le cas parabolique et dans les travaux de Ishii [START_REF]On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions[END_REF][START_REF] Julin | A new proof for the equivalence of weak and viscosity solutions for the p-Laplace equation[END_REF] pour le cas elliptique. Ces études sont basées sur d'autres méthodes (résultat de comparaison, régularisation par sup et inf-convolution et passage à la limite) et montrent l'équivalence des deux notions pour des sous ou sur-solutions.

Régularité jusqu'au bord des solutions locales

La régularité Hölderienne du gradient jusqu'au bord et pas seulement à l'intérieur du domaine Ω est une question intéressante. Pour certaines données au bord et certains domaines, cette régularité pourrait s'obtenir en prolongeant par réflexion la solution dans un domaine plus grand contenant Ω et en utilisant les résultats de régularité de Friedman et DiBenedetto. Pour un résultat dans ce sens voir l'étude du problème unidimensionnel (chapitre 5).

Etat de l'art et résultats du chapitre 3 : localisation des points d'explosion.

La localisation et la taille de l'ensemble des points d'explosion des solutions du problème (1.4) est une question assez délicate en général. Dans la littérature on trouve principalement deux types d'explosion du gradient :

1. l'explosion a lieu à l'intérieur du domaine Ω.

l'explosion a lieu au bord ∂Ω.

Le premier type est observé par exemple pour les solutions de l'équation : -L'ensemble des points d'explosion est un ensemble discret, notamment réduit à un singleton. -L'ensemble des points d'explosion est de mesure finie et positive (connu sous le nom de "regional blow-up") L'ensemble des points d'explosion peut éventuellement être un ensemble plus complexe. Précisons ici que l'ensemble des points d'explosion du gradient de u est défini par :

   u t -u xx = |u| p-1 u|u x | q , t > 0, -1 < x < 1, u(t, ±1) = A ± , t > 0, u(0, x) = u 0 (x), -1 < x < 1. (1.19) avec q > 2, p ≥ 1 u 0 ∈ C 1 et u 0 (-1) = A -≤ u 0 ≤ A + = u 0 (1
GBU S(u 0 ) := x 0 ∈ ∂Ω; ∇u est non borné dans (T max -η, T max ) × (Ω ∩ B(x 0 , η))
pour tout η > 0} .

(1.20) La première possibilité est triviale dans le cas où le domaine Ω est une boule et la donnée initiale u 0 est radiale. Pour le cas de la diffusion linéaire, Souplet et Li [START_REF] Li | Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[END_REF] se sont intéressés à la validité de la deuxième possibilité. Ils ont d'abord montré que pour des domaines bornés réguliers Ω ⊂ R N avec N ≥ 2, l'ensemble des points d'explosion peut être localisé dans un voisinage arbitraire de n'importe quel point x 0 ∈ ∂Ω. Ensuite, pour certains types de domaines bidimensionnels, ils ont pu construire des données initiales qui garantissent l'explosion du gradient en un seul point du bord. Ils ont établi le résultat suivant dans le cas Ω = B(0, 1) ⊂ R 2 . Théorème 1.3.1 (Souplet et Li). On suppose que q > 2 = p et g = 0 dans le problème (1.4). On note ũ0 (r, θ) = u 0 (r cos θ, r sin θ). On a le résultat suivant

(i) Il existe u 0 ∈ X + := u 0 ∈ C 1 (Ω); u = 0 sur ∂Ω, u 0 ≥ 0 , telle que T * (u 0 ) < ∞ et      u 0 est symétrique par rapport à la ligne y = 0, ∂ ũ0 ∂θ ≤ 0 dans B + := {(r, θ); 0 < r < 1, 0 < θ < π} , GBU S(u 0 ) = ∂Ω.
(ii) Pour une telle donnée initiale u 0 , GBU S(u 0 ) contient seulement le point (x, y) = (1, 0).

Le deuxième type de domaine qu'ils ont considéré sont les domaines symétriques par rapport à l'axe x = 0 et avec un bord contenant une portion plate centrée en l'origine (c'est à dire il existe ρ > tel que (-ρ, ρ) × {0} ⊂ ∂Ω). Dans ce cas ils ont réussi à construire des données initiales qui garantissent que le seul point d'explosion est (0, 0).

Le problème d'explosion en un seul point pour des équations paraboliques semi-linéaires a été traité pour la première fois par Weissler [START_REF] Weissler | Single point blow-up for a semilinear initial value problem[END_REF] pour l'équation de la chaleur unidimensionnelle avec un terme source u α . Ensuite Friedman et Mcleod [START_REF] Friedman | Blow-up of positive solutions of semilinear heat equations[END_REF] ont étendu ce résultat en introduisant une fonctionnelle J de la forme J = u x + c(x)u q et en lui appliquant le principe du maximum. Depuis, leur méthode est devenue une des méthode "phare" pour démontrer des résultats d'explosion en un seul point. Néanmoins cette technique est utilisée en majorité pour l'explosion en amplitude et dans le cas unidimensionnel ou pour des solutions radiales.

La méthode de Souplet et Li [START_REF] Li | Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[END_REF] est une adaptation astucieuse de la technique de Friedman-Mcleod au cas de la dimension 2 en espace et pour le phénomène nouveau de l'explosion du gradient. Dans le cas qui nous intéresse (q > p > 2) et pour des domaines localement plats, nous allons adapter la stratégie utilisée dans [START_REF] Li | Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[END_REF] aux diverses complications apportées par la diffusion non linéaire pour montrer que l'ensemble des points d'explosion du problème (1.4) est un singleton.

Tout d'abord précisons les hypothèses géométriques sur le domaine Ω. On suppose que pour L 1 , L 2 > 0, Ω ⊂ R 2 est un domaine borné régulier de classe C 2+ pour un ∈ (0, 1); (1.21) Ω est symétrique par rapport à l'axe x = 0;

(1.22)

Ω ⊂ {y > 0} et Ω contient le rectangle (-L 1 , L 1 ) × (0, 2L 2 );
(1.23) Ω est convexe dans la direction x des abscisses.

(1.24)

Dans le problème (1.4) on pose g = µy où µ > 0 est une constante et on suppose que la donnée initiale u 0 est dans V µ , où

V µ := u 0 ∈ C 1 (Ω), u 0 ≥ µy dans Ω, u 0 = µy sur ∂Ω .
Ensuite, on impose des conditions spécifiques sur la donnée initiale (profil concentré près de l'origine, symétrie et décroissance en la première variable d'espace, croissance par rapport à la deuxième variable) qui se transmettent à la solution u. Grâce à ce profil particulier, nous avons le résultat suivant. Théorème 1.3.2 (Attouchi et Souplet). On suppose que dans le problème (1.4), q > p > 2, g = µy, u 0 ∈ V µ et Ω vérifie les hypothèses (1.21)- (1.24). On note Ω + := Ω ∩ {x > 0}.

Alors

(i) Pour tout ρ ∈ (0, L 1 ), il existe µ 0 = µ 0 (p, q, Ω, ρ) > 0 tel que, pour tout µ ∈ (0, µ 0 ], il existe une donnée initiale u 0 dans V µ ∩ C 2 (Ω) pour laquelle la solution u du problème (1.4) vérifie les propriétés suivantes :

T := T max (u 0 ) < ∞ et GBU S(u 0 ) ⊂ [-ρ, ρ] × {0}, (1.25)
u(•, t) est symétrique par rapport à l'axe x = 0, pour tout t ∈ (0, T ), (1.26)

u x ≤ 0 dans Ω + × (0, T ), (1.27)
u y ≥ µ/2 dans Ω × (0, T ).

(1.28)

(ii) Pour tout µ et u 0 vérifiant les conditions de (i), on a que

GBU S(u 0 ) = {(0, 0)}.
Bien que nous ne réussissions pas à passer outre, la restriction à des solutions qui vérifient (1.28) (d'où le choix de la donnée au bord) semble être d'ordre technique. En effet, à cause du terme non linéaire |∇u| p-2 dans la partie principale du ∆ p u, les quantités qui apparaissent dans le calcul de l'équation de J := u x + c(x)d(y)F (u) ne dépendent pas seulement de u et ∇u mais aussi de D 2 u et u t . Ne disposant pas d'estimations assez fortes de D 2 u, on se sert de l'équation pour ré-écrire les termes qui font apparaître D 2 u en fonction de puissances de ∇u dont certaines sont négatives. Ainsi pour pouvoir dériver une équation sur la fonctionnelle J, l'hypothèse (1.28) nous a été cruciale et les quantités qui y apparaissent explosent quand µ tend vers 0 (voir le chapitre 3 pour plus de détail). Ce genre de difficulté n'apparaît pas pour un opérateur de diffusion linéaire.

La preuve est longue et technique, plus encore que dans [START_REF] Li | Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[END_REF]. Les outils clés de la preuve sont l'estimation locale du gradient, une construction d'une donnée initiale bien préparée qui nous permet de localiser l'ensemble des points d'explosion dans un voisinage de l'origine. Ensuite, pour obtenir les conditions aux bords et initiales pour la fonction auxiliaire J, on a recours au lemme de Hopf classique et une version parabolique du lemme du coin de Serrin dont on donnera une preuve. Bien que le lemme de Serrin soit assez connu dans le cas elliptique [START_REF] Serrin | A symmetry problem in potential theory[END_REF][START_REF] Sweers | Semilinear elliptic problems on domains with corners[END_REF] notamment pour l'étude de problèmes de symétrie, on ne trouve que de très rares versions paraboliques de ce lemme [START_REF] Rubinstein | Fast reaction, slow diffusion, and curve shortening[END_REF]. Une version parabolique du lemme de Serrin pour une équation de la chaleur dans un rectangle est prouvée dans [START_REF] Li | Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[END_REF] en utilisant des fonctions à variables séparables en espace. Notre preuve est différente et repose sur une modification de celle utilisée dans [START_REF] Serrin | A symmetry problem in potential theory[END_REF]. Notons que la propriété (1.28) garantit que la solution u est classique et par conséquent ses dérivées secondes et troisièmes en espace (D 2 u, D 3 u) sont localement bornées ce qui nous servira pour utiliser les lemmes de Hopf et la version parabolique du lemme de Serrin.

Problèmes ouverts ou avec des réponses partielles

L'étude de la localisation des singularités pour le problème (1.4) est loin d'être achevée. Elle peut être poursuivie dans diverses directions en étudiant un certain nombre de questions intéressantes. La première direction viendrait compléter ce travail sur le caractère discret de l'ensemble des points d'explosion. Peut on étendre les théorèmes 1.3.2 et 1.3.1 à des domaines de dimension N > 2 (en utilisant des données initiales symétriques) ? Peuton s'affranchir de la condition de bord localement plat ? Comment traiter le cas µ = 0 et relaxer l'hypothèse (1.28) ? Ensuite, il serait aussi intéressant de savoir s'il est possible de construire (comme c'est déjà le cas pour certaines équations semi-linéaires [START_REF] Chen | Convergence, asymptotic periodicity, and finite-point blowup in one-dimensional semilinear heat equations[END_REF]) des solutions dont l'ensemble de points d'explosion est discret fini mais non réduit à un singleton (c'est à dire qu'il contient 2, 3 ou n points) et s'il est possible de construire des solutions qui explosent en des points qu'on aura fixé [START_REF] Merle | Solution of a nonlinear heat equation with arbitrarily given blow-up points[END_REF].

D'autre part on peut se demander si on peut construire des solutions qui valident la possibilité de "régional blow-up".

Etat de l'art et résultats du chapitre 4 : prolongement des solutions au delà des singularités

Pour les solutions classiques qui explosent en temps fini il est intéressant de savoir s'il est possible de les étendre au delà du temps maximal d'existence et d'étudier le comportement asymptotique de ces solutions prolongées. Il est alors important de savoir en quel sens on peut le faire. L'apparition d'une singularité en gradient est un obstacle non négligeable pour étendre les solutions classiques. Une première tentative peut être trouvée dans les travaux [START_REF] Giga | Interior derivative blow-up for quasilinear parabolic equations[END_REF][START_REF] Fila | Derivative blow-up and beyond for quasilinear parabolic equations[END_REF]. Fila et Lieberman [START_REF] Fila | Derivative blow-up and beyond for quasilinear parabolic equations[END_REF], ont étudié l'équation (1.4) pour N = 1 et p = 2 et avec une non-linéarité F (u x ) plus générale. Pour Ω = (0, L) et g = 0, ils ont montré que pour L assez grand et pour certaines données initiales, le gradient de la solution de (1.4) n'explose qu'en x = 0. Ensuite, ils ont pu prolonger la solution explosive par une solution classique qui satisfait la condition au bord en x = L mais qui ne la satisfait pas au point où a lieu l'explosion du gradient. Plus précisément la solution prolongée vérifie U (0, t) > 0 et U x (0, t) = +∞ pour t > T max (u 0 ). Les ingrédients clés de ce résultat sont l'étude du problème stationnaire "singulier" associé à l'équation (1.4), l'équation satisfaite par une fonction auxiliaire h(u x ) et le cadre unidimensionnel. Des résultats sur le comportement en temps grand de la solution prolongée ont été obtenus montrant la convergence vers l'unique état stationnaire "singulier" Φ (i.e Φ (0) = +∞ et Φ(L) = 0).

Pour notre équation et pour pouvoir traiter le cas d'une dimension N ≥ 1, la théorie des solutions de viscosité offre un bon cadre pour la continuation des solutions au delà des singularités et l'étude des propriétés des solutions prolongées. En effet, récemment pour le cas de la diffusion linéaire (p = 2), Barles et Da Lio [START_REF]On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations[END_REF] ont introduit la notion de solutions de viscosité généralisées qui ont l'avantage d'autoriser la perte des conditions au bord. Généralement, pour ces solutions les conditions au bord sont comprises dans un sens faible comme suit : soit la solution atteint la condition au bord au sens classique, soit l'équation elle même est satisfaite au bord (au sens de viscosité).

Dans ce chapitre on s'intéresse plus particulièrement au problème suivant :

u t -div(|∇u| p-2 ∇u) + |∇u| q = f (x, t), x ∈ Ω, t > 0, u(x, t) = g(x), x ∈ ∂Ω, t > 0, (1.29) u(x, 0) = u 0 (x), x ∈ Ω.
Contrairement aux chapitres précédents, on demande moins de régularité sur les fonctions u 0 et g. On suppose que u 0 , g et f sont des fonctions continues et que u 0 (x) = g(x) pour x ∈ ∂Ω. NB : Le signe de la non-linéarité n'est pas important dans cette partie. Il suffit de changer u en -u et les résultats restent valables à condition de changer sous-solutions par sursolutions et vice versa.

Dans [START_REF]On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations[END_REF][START_REF] Barles | Remarks on the dirichlet and state-constraint problems for quasilinear parabolic equations[END_REF], on peut trouver une étude du lien entre l'explosion du gradient (due à la forte non-linéarité du terme du premier ordre) et l'éventuelle perte des conditions aux bord. Lorsque q > 1 = (p-1), cette étude est faite en utilisant le lien entre l'équation (1.29) et un problème de contrôle stochastique. En effet, dans ce cas, Barles et Da Lio proposent une expression explicite de la solution de viscosité de (1.29) comme étant la fonction-valeur d'un problème de contrôle stochastique de type temps de sortie. Pour le cas elliptique, signalons que B. Kawohl et N. Kutev [START_REF]A study on gradient blow up for viscosity solutions of fully nonlinear, uniformly elliptic equations[END_REF] ont aussi étudié cette problématique en donnant des conditions plus optimales qui garantissent l'explosion du gradient à l'intérieur ou au bord. Le problème de pertes des conditions aux limites est lié aux "couches limites" et il faudrait alors construire des "barrières" pour assurer que celles-ci sont bien satisfaites.

Dans un premier temps nous rappelons la définition d'une solution de viscosité.

Définition 1.4.1 (Sous-solutions, sur-solutions, solutions). On dit qu'une fonction u : [0, T ] × Ω → R est sous-solution de viscosité de (1.29) si u est semi-continue-supérieure (SCS) dans [0, T ] × Ω et si, pour toute fonction-test φ ∈ C 2 ([0, T ] × Ω) telle que u -φ a un maximum local en un point (t 0 , x 0 ) ∈ ]0, T [×Ω, on a

φ t (t 0 , x 0 ) -div(|∇φ| p-2 ∇φ) + |∇φ(t 0 , x 0 )| q ≤ f (t 0 , x 0 ).
La condition de Dirichlet au bord associée à (3.9.20) doit être relaxée et comprise au sens de viscosité de la manière suivante : si

(t 0 , x 0 ) ∈]0, T ] × ∂Ω alors min (φ t -div(|∇φ| p-2 ∇φ) + |∇φ| q , u -g) ≤ 0.
De même la condition initiale est comprise au sens suivant : si

(t 0 , x 0 ) ∈ {0} × Ω alors min (φ t -div(|∇φ| p-2 ∇φ) + |∇φ| q , u -u 0 ) ≤ 0.
Symétriquement, on dit qu'une fonction u :]0, T [×Ω est sur-solution de viscosité de (1.29) si u est semi-continue-inférieure (SCI) dans ]0, T [×Ω et si, pour toute fonction-test φ ∈ C 2 (]0, T [×Ω) telle que u -φ a un minimum local en un point (t 0 , x 0 ) ∈ ]0, T [×Ω, on a

φ t (t 0 , x 0 ) -∆ p (φ(t 0 , x 0 )) + |∇φ(t 0 , x 0 )| q ≥ f (t 0 , x 0 ), si (t 0 , x 0 ) ∈]0, T ] × ∂Ω on a max (φ t -div(|∇φ| p-2 ∇φ) + |∇φ| q , u -g) ≥ 0, et si (t 0 , x 0 ) ∈ {0} × Ω alors min (φ t -div(|∇φ| p-2 ∇φ) + |∇φ| q , u -u 0 ) ≤ 0.
Enfin, u : [0, T ] × Ω est solution de viscosité de (1.29) si u est sous et sur-solution de (1.29).

Le sens avec lequel on définit les conditions aux limites se justifie grâce au résultat de stabilité discontinue et la méthode des semi-limites relaxées introduits par Barles et Perthame [START_REF] Barles | Exit time problems in optimal control and vanishing viscosity method[END_REF][START_REF] Barles | Solutions de viscosité et équations elliptiques du deuxième ordre, notes de cours[END_REF] (qui requièrent seulement une borne L ∞ uniforme pour les solutions u ε des problèmes régularisés).

Dans [START_REF]On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations[END_REF] un résultat d'existence-unicité d'une solution de viscosité globale en temps du problème (1.29) a été prouvé pour le cas du laplacien (p = 2). Notre première contribution vient étendre ce résultat pour le cas de la diffusion non linéaire p > 2.

Théorème 1.4.1 (Attouchi, Barles). On suppose que q > p > 2 et u 0 , g, f sont continues. Alors il existe une unique solution de viscosité du problème (1.29), qui est définie pour tout temps t > 0.

L'argument clé pour monter le théorème 1.4.1 est un résultat de comparaison fort (i.e un résultat de comparaison pour des sur et sous-solutions discontinues). Notre approche est légèrement différente de celle de [START_REF]On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations[END_REF] puisqu'on utilise l'astuce de la sup-convolution temporelle introduite dans les travaux de [START_REF] Lasry | A remark on regularization in Hilbert spaces[END_REF] et le résultat de régularité ci-dessous pour EDP elliptiques avec une forte non linéarité du gradient. Cette astuce nous permet d'approcher toute sous-solution du problème (1.29) (qui a priori peut n'être que SCS) par une sous-solution continue et qui de ce fait satisfait automatiquement la condition de cône et de pouvoir ainsi terminer la preuve dans l'esprit de [START_REF]On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations[END_REF]. La méthode de Perron [START_REF] Ishii | Perron's method for Hamilton-Jacobi equations[END_REF] nous permettant alors de montrer l'existence. Cette méthode consiste à d'abord construire une sous-solution et une sur-solution qui se comporte bien sur le bord parabolique, on construit alors une sous-solution maximale et on montre que c'est aussi une sur-solution.

Théorème 1.4.2 (Attouchi, Barles : Régularité elliptique des sous-solutions).

Soit l'équation

-div(|∇u| p-2 ∇u)

+ |∇u| q = f (x) dans Ω, (1.30)
où q > p ≥ 2 et f est une fonction continue sur Ω à valeurs réelles. Toute sous-solution de viscosité de l'équation (1.30) est dans C 0,β (Ω) avec β = q -p q -p + 1 et la norme dans C 0,β ne dépend que de p, q, f ∞ .

Le théorème 1.4.2 est une généralisation des résultats de I. Capuzzo Dolcetta, F. Leoni et A. Porretta [START_REF] Capuzzo Dolcetta | Hölder estimates for degenerate elliptic equations with coercive Hamiltonians[END_REF] et de sa version revisitée par G. Barles [15] au cas quasi-linéaire. Notre 1.4. Etat de l'art et résultats du chapitre 4 : prolongement des solutions au delà des singularités preuve est différente de celle de [START_REF] Capuzzo Dolcetta | Hölder estimates for degenerate elliptic equations with coercive Hamiltonians[END_REF]Théorème 2.11] et s'appuie sur la méthode utilisée dans [START_REF]A short proof of the C 0,α -regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications[END_REF]. Signalons que c'est la croissance forte du terme |∇u| q qui donne la régularité. De plus on a comme conséquence directe que l'on ne peut pas résoudre le problème de Dirichlet au sens classique pour toute donnée au bord g. En effet le théorème 1.4.2 implique que g doit être au moins Hölderienne.

Une autre application du théorème 1.4.2 se trouve dans l'étude du comportement asymptotique de la solution globale de (1.29). Désormais on suppose que f (x, t) = f (x) dans (1.29). Une approche classique pour l'étude du comportement en temps grand de la solution globale réside dans l'introduction du "problème ergodique" associé à (1.29) consistant à trouver un couple (c, u ∞ ) solution de l'équation

-div |∇u ∞ | p-2 ∇u ∞ + |∇u ∞ | q -f (x) = c dans Ω, (1.31)
associée à la contrainte d'état au bord :

-div |∇u ∞ | p-2 ∇u ∞ + |∇u ∞ | q -f (x) ≥ c sur ∂Ω. (1.32)
On a le résultat suivant qui vient étendre celui de [START_REF]A short proof of the C 0,α -regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications[END_REF].

Questions ouvertes ou avec des réponses partielles

Il est naturel de se demander si on peut avoir une description plus précise du comportement en temps grand des solutions globales de l'équation (1.29). Quand le problème stationnaire admet une solution, on s'attend à ce que u converge vers l'unique état stationnaire (voir [START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF]110,[START_REF] Benachour | Decay estimates for a viscous Hamilton-Jacobi equation with homogeneous Dirichlet boundary conditions[END_REF]).

Quand le problème stationnaire n'admet pas de solutions, un résultat classique nous dit que la solution u du problème de Dirichlet (1.29) devrait se comporter comme

-ct + u ∞ (x) + o t (1)
quand t → ∞, où u ∞ est une solution du problème ergodique stationnaire qui correspond à une unique constante ergodique c. Ceci a été démontré dans les travaux de Tchamba [START_REF] Tchamba | Large time behavior of solutions of viscous Hamilton-Jacobi equations with superquadratic Hamiltonian[END_REF] pour q > 2 = p. Cette étude plus précise se base généralement sur deux ingrédients clés : la régularité des solutions et le principe de comparaison fort (qui se réduit au principe du maximum fort pour le cas de la diffusion linéaire p = 2). Le problème ergodique n'admet pas en général une unique solution même à une constante additive près rendant l'analyse du comportement asymptotique assez délicate. Mais dans le cas de solutions qui sont Lipschitziennes, l'application d'un principe de comparaison fort étendu aux solutions de viscosité permet de montrer l'unicité à une constante additive près. Pour le cas de la diffusion non-linéaire p > 2, la régularité Lipschitzienne des solutions pourrait être obtenue en adaptant les arguments de [START_REF] Capuzzo Dolcetta | Hölder estimates for degenerate elliptic equations with coercive Hamiltonians[END_REF]. Par conséquent, la principale difficulté consiste à démontrer un principe de comparaison fort (ce qui est assez délicat vu le caractère quasi-linéaire de l'équation). Signalons qu'un principe du maximum fort a été démontré pour le p-Laplacien [START_REF]Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations[END_REF]. D'autre part, signalons aussi que la résolubilité du problème stationnaire est capitale pour l'étude du comportement asymptotique des solutions globales (voir chapitre 5). Une autre difficulté liée au caractère quasi-linéaire du p-Laplacien consiste à établir un résultat de comparaison fort pour le problème stationnaire (qui ne satisfait donc pas une condition de monotonie) en utilisant seulement la donnée d'une sous-solution stricte (voir [START_REF] Tchamba | Large time behavior of solutions of viscous Hamilton-Jacobi equations with superquadratic Hamiltonian[END_REF] pour le cas p = 2). Le problème stationnaire semble n'être résoluble que dans le cas c ≤ 0 (voir [START_REF] Tchamba | Large time behavior of solutions of viscous Hamilton-Jacobi equations with superquadratic Hamiltonian[END_REF]110]. L'étude du problème ergodique vient donc apporter une précision sur le comportement asymptotique des solutions globales dans le cas de non-existence de solutions stationnaires.

Etat de l'art et résultats du chapitre 5 : classification des solution globales

Nous revenons ici à l'étude des solutions faibles Lipschitziennes maximales (c.f Théorème 1.2.1). Étant donné que dans l'étude de l'équation (1.4) on a l'alternative suivante :

-solutions globales en temps qui sont uniformément bornées en temps en norme W 1,∞ (c'est à dire non explosives) -solutions globales en temps qui explosent en temps infini (c'est à dire des solutions qui existent pour tout tout 0 < t < ∞ mais qui vérifient lim t→∞ ∇u(t) ∞ = +∞) -solutions qui explosent en temps fini, il est naturel de s'intéresser à la "classification" des solutions. Dans ce chapitre on apporte une contribution à cette problématique en étendant le résultat de Arrieta-Rodriguez Bernal-Souplet. On se place dans le cadre unidimensionnel et on considère le problème suivant :

   u t -(|u x | p-2 u x ) x = |u x | q x ∈ (0, 1), t > 0, u(t, 0) = 0, u(t, 1) = M t > 0, u(0, x) = u 0 (x)
x ∈ (0, 1).

(1.33)

1.5. Etat de l'art et résultats du chapitre 5 : classification des solution globales Théorème 1.5.1 (Classification des solutions globales). On suppose que q > p > 2 et que u 0 ∈ W 1,∞ satisfait la condition de compatibilité. On pose M c = q-p+1 q-p q-p+1 p-1

1 p-1-q .
Alors (i) Si 0 ≤ M < M c alors toutes les solutions globales de (1.33) sont bornées en norme C 1 et convergent dans C 1 ([0, 1]) vers l'unique solution stationnaire.

(ii) Si M > M c alors toutes les solutions de (1.33) explosent en temps fini.

Ici M c est la valeur critique pour l'existence d'une solution stationnaire (unique). Notons que le lien entre l'existence et les propriétés des solutions de l'équation stationnaire et le comportement asymptotique des solutions globales est crucial.

La preuve du théorème 1.5.1 s'inspire de celle de [5] et procède par contradiction. Elle se compose de trois étapes. La première étape consiste à construire une fonctionnelle de Lyapunov ayant de bonnes propriétés. Etant donné qu'on est en dimension 1, on peut utiliser la méthode de Zelenyak pour en construire une. Cette méthode s'applique principalement aux EDP uniformément paraboliques. Étant donné le manque de régularité des solutions faibles de (1.33), une des principales difficultés de la preuve du théorème 1.5.1 est d'établir des estimations assez fortes sur les fonctions approchées (u ε ) permettant d'avoir la convergence jusqu'au bord des (u ε ) x vers u x . En effet on construit en fait une fonctionnelle de Lyapunov approchée et on doit avoir assez de compacité pour pouvoir passer à la limite dans les termes non-linéaires en (u ε ) x . Cette fonctionnelle nous permet de montrer la convergence des solutions globales (éventuellement non bornées dans W 1,∞ ) vers la solution stationnaire. Enfin on utilise le profil de la solution stationnaire et des estimations sur la dérivée pour aboutir à une contradiction.

Questions ouvertes ou avec des réponses partielles

Que se passe-t-il pour M = M c ? On sait que toutes les solutions globales doivent exploser en temps fini ou infini. Mais existe-t-il des solutions globales qui explosent en temps infini ? Dans le cas de la diffusion linéaire, Souplet et Vàzquez [108] ont montré que pour M = M b , la solution globale u tend vers l'état stationnaire singulier (unique) noté v ss dés que u 0 est majorée par v ss . Une étude précise et assez technique de la formation des singularités est fournie dans [108]. Notons aussi que dans le cas de la diffusion linéaire si la non linéarité est remplacée par une non-linéarité exponentielle l'alternative d'explosion en temps infini à été montrée dans [START_REF] Zhang | Boundedness of global solutions for a heat equation with exponential gradient source[END_REF]. D'autre part l'explosion du gradient en temps infini a aussi été observé pour des équations quasi-linéaires de courbure moyenne [START_REF] Asai | On the interior derivative blow-up for the curvature evolution of capillary surfaces[END_REF].

Étant donné que la méthode de Zelenyack pour construire des fonctionnelles de Lyapunov est restreinte au cadre unidimensionnel, comment traiter le cas d'une dimension quelconque ? Par ailleurs il serait intéressant d'avoir des estimations uniformes sur les solutions.

Enfin on regroupe ici quelques résultats récents [START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF][START_REF] Stinner | Convergence to steady states in a viscous Hamilton-Jacobi equation with degenerate diffusion[END_REF] sur le comportement en temps grand de l'équation (1.4) pour d'autres valeur de p et q. On prend g = 0 dans (1.4) et on suppose que p > 2.

-On suppose que q < p -1. Si N = 1 ou bien u 0 est radiale, alors il existe une famille d'états stationnaires et l'unique solution de viscosité de (1.4) converge vers l'un d'eux quand t → ∞. -Pour p -1 < q ≤ p et des données initiales quelconques ou q > p et des données initiales suffisamment petites, l'unique solution de viscosité de (1.4) converge vers 0 à la vitesse t -1 p-2 et t -1 p-2 u(t, x) tend vers une fonction ξ solution positive de

-∆ p ξ -|∇f | p-1 - f p -2 = 0.
1.6 Etat de l'art et résultats du chapitre 6 : un théorème de type Liouville

Dans ce chapitre on s'intéresse à une autre propriété des solutions de l'équation (1.1) qui n'est pas reliée au phénomène d'explosion du gradient mais qui est une application directe d'une estimation, locale en espace du gradient des solutions localement bornées. Plus précisément, on s'intéresse à un théorème de Liouville pour les solutions anciennes (t < 0) dans l'espace entier R N de l'équation (1.1). Les estimations de gradient s'obtiennent en général via des techniques de type Bernstein [START_REF] Bernstein | Sur les équations du calcul des variations[END_REF]. La technique a été introduite par Bernstein (1910) et a été étendue par Serrin dans les années 60 pour étudier certaines propriétés d'EDP elliptiques quasi-linéaires et récemment généralisée par Barles au cadre des solutions de viscosité [START_REF]A weak Bernstein method for fully nonlinear elliptic equations[END_REF]. Cette technique consiste à appliquer le principe du maximum à l'inconnue |∇v| 2 où u = f (v). Le choix de f doit être assez judicieux. Nous renvoyons le lecteur aux travaux [START_REF] Bartier | Gradient estimates for a degenerate parabolic equation with gradient absorption and applications[END_REF]110] où différents choix de la fonction f ont permis d'obtenir des estimations de gradient de natures différentes. Signalons aussi que pour les estimations locales, on doit introduire une fonction de troncature bien adaptée. Quand on applique cette méthode aux solutions de l'équation (1.1) on obtient le résultat suivant.

Théorème 1.6.1. On suppose que q > p -

1 > 1, x 0 ∈ R N et R, T > 0. On pose Q T,R = B(x 0 , R) × (0, T ). Soit u une solution faible dans L ∞ ((0, T ); W 1,∞ (B(x 0 , R)) de ∂ t u -∆ p u = |∇u| q dans Q T,R .
On suppose que |u| ≤ M pour une certaine constante M ≥ 1. Alors,

|∇u| ≤ C(p, N, q) t -1 q + R -1 + R -1 q-p+1 M dans Q T, R 2 .
(1.34)

Le théorème 1.6.1 vient étendre les estimations obtenues dans [110] au cas de la diffusion non-linéaire (p > 2). Notons que dans [110] une borne supérieure locale sur u suffit pour avoir l'estimation de gradient. Signalons aussi que, comparativement à l'estimation (1.11), on n'utilise que la norme L ∞ locale de la solution mais ceci a un prix puisque la puissance sur R est moins bonne.

Comme application directe du théorème 1.6.1, on a le résultat de type Liouville suivant.

1.6. Etat de l'art et résultats du chapitre 6 : un théorème de type Liouville Théorème 1.6.2. On suppose que q > p -

1 > 1 et on pose σ = min 1, 1 q-p+1 . Soit u ∈ L ∞ loc ((-∞, 0); W 1,∞ loc (R N )) une solution faible de u t -∆ p u = |∇u| q , x ∈ R N , -∞ < t < 0, satisfaisant |u(x, t)| = o(|x| σ + |t| 1 q ), quand |x| σ + |t| 1 q → ∞. (1.35)
Alors u is constante.

Ce résultat peut être vu comme une version parabolique du résultat récent de Bidaut-Véron, Véron et Huidobro [START_REF] Bidaut-Veron | Local and global properties of solutions of quasilinear hamilton-jacobi equations[END_REF] où une estimation du gradient et un théorème de type Liouville ont été obtenus pour l'équation elliptique associée à (1.1).

Quand q = p, la transformation de Hopf-Cole v = e s/(p-1) -1 permet de relier l'équation (1.1) à l'équation |z| p-2 z t = ∆ p z.

Dans ce cas, pour p < 2, F. Wang [START_REF] Wang | Gradient estimates for the p-Laplace heat equation under the Ricci flow[END_REF] obtient une estimation de gradient similaire à (1.34) pour des solutions bornées supérieurement de l'équation (1.1) dans des variétés Riemanniennes avec une métrique décrite par un flot de Ricci. Pour des résultats de type Liouville pour le p-Laplacien sans le terme de gradient, nous renvoyons le lecteur à [START_REF] Dibenedetto | Liouville-type theorems for certain degenerate and singular parabolic equations[END_REF][START_REF] Teixeira | An intrinsic liouville theorem for degenerate parabolic equations[END_REF]. Dans [START_REF] Teixeira | An intrinsic liouville theorem for degenerate parabolic equations[END_REF] une nouvelle approche basée sur des normes intrinsèques, un argument de régularité et un argument de blow-up est utilisée pour démontrer un théorème de type Liouville.

Questions ouvertes ou avec des réponses partielles

La condition de croissance (1.35) est importante comme le montre l'exemple de la fonction u := x + t. Cependant on ne sait pas si elle est optimale.

Chapitre 2 Théorie locale d'existence et profil spatial de l'explosion du gradient

Dans ce chapitre nous établissons une théorie locale en temps, dans la classe naturelle des données initiales lipschitziennes, avec alternative d'explosion sur le gradient, qui fournit un bon cadre pour l'étude des singularités. Nous obtenons aussi une estimation du gradient près du bord donnant le profil spatial de l'explosion. Cette estimation sera utile pour le chapitre suivant.

Introduction and main results

This chapter is concerned with the existence and qualitative properties of weak solutions of the initial boundary value problem of the p-Laplacian with a nonlinear gradient source term

   ∂ t u -div(|∇u| p-2 ∇u) = |∇u| q , x ∈ Ω, t > 0, u(x, t) = g(x), x ∈ ∂Ω, t > 0, u(x, 0) = u 0 (x), x ∈ Ω, (2.1.1)
where Ω is a bounded domain in R N of class C 2+α for some α > 0, p > 2 and q > p -1.

Throughout this chapter we assume that the boundary data g ≥ 0 is the trace on ∂Ω of a regular function in C 2 (Ω), also denoted g, and the initial data u 0 satisfies

u 0 ∈ W 1,∞ (Ω), u 0 ≥ 0, u 0 (x) = g(x) for x ∈ ∂Ω. (2.1.2)
We note that, as far as bounded solutions are concerned, there is no loss of generality in assuming g, u 0 ≥ 0, since the partial differential equation in (2.1.1) is unchanged when adding a constant to u. When p = 2, the differential equation of (2.1.1) is the so-called viscous Hamilton-Jacobi equation and it appears in the physical theory of growth and roughening of surfaces, where it is known as the Kardar-Parisi-Zhang equation (q = 2), and has been studied by many authors (see for example [START_REF] Ben-Artzi | The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces[END_REF][START_REF] Quittner | Superlinear parabolic problems : blow-up, global existence and steady states[END_REF] and the references therein). It is known that, under certain conditions, |∇u| blows up in a finite time t = T max while, by the maximum principle, all solutions are uniformly bounded (cf. [START_REF] Ph | Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions[END_REF][START_REF] Hesaaraki | Blow-up of positive solutions for a family of nonlinear parabolic equations in general domain in R N[END_REF]110]). We shall call such phenomenon gradient blow-up (GBU). This is different from the usual blow-up in which the L ∞ norm of the solution tends to infinity as t → T max (cf. [START_REF] Quittner | Superlinear parabolic problems : blow-up, global existence and steady states[END_REF]). Sharp results on gradient blowup analysis, including blow-up rate, blow-up set, blow-up profile and continuation after blow-up have been recently obtained, see e.g. [START_REF] Li | Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[END_REF][START_REF] Guo | Blowup rate estimates for the heat equation with a nonlinear gradient source term[END_REF][START_REF] Hesaaraki | Blow-up of positive solutions for a family of nonlinear parabolic equations in general domain in R N[END_REF][START_REF] Quittner | Superlinear parabolic problems : blow-up, global existence and steady states[END_REF]5,108] and the references therein.

When p > 2, equation (2.1.1) is a degenerate parabolic equation for |∇u| = 0 and one cannot expect the existence of classical solutions. Weak solutions can be obtained by approximation with solutions of regularized problems. This was done in [START_REF] Zhao | Existence and nonexistence of solutions for u t = div(|∇u| p-2 ∇u) + f (∇u, u, x, t)[END_REF] when the right hand side in (2.1.1) is replaced with a general nonlinearity f (u, ∇u, x, t). In the case where f depends on ∇u, typically for problem (2.1.1), the results in [START_REF] Zhao | Existence and nonexistence of solutions for u t = div(|∇u| p-2 ∇u) + f (∇u, u, x, t)[END_REF] require the assumption q ≤ p -1, in which case a global solution is directly constructed for any initial data. Local-in-time existence results are also given in [START_REF] Zhao | Existence and nonexistence of solutions for u t = div(|∇u| p-2 ∇u) + f (∇u, u, x, t)[END_REF] but they require that f actually does not depend on ∇u. In [START_REF] Chen | Global existence and gradient estimates for quasilinear parabolic equations of the m-laplacian type with a strong perturbation[END_REF], the existence of a global weak solution for q > p -1 was proved for small data, under the assumption that the mean curvature of ∂Ω is nonpositive. In the articles [START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF][START_REF] Barles | Convergence to steady states for radially symmetric solutions to a quasilinear degenerate diffusive Hamilton-Jacobi equation[END_REF], problem (2.1.1) was studied in the framework of viscosity solutions, but only in situations where global existence of a W 1,∞ solution is guaranteed, namely for q ≤ p or for suitably small initial data when q > p. On the other hand, when q > p, global existence is not expected in general for large initial data. A result in this direction was given in [START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF]Theorem 5.2], where it was proved that problem (2.1.1) (with g = 0) cannot admit a global, Lipschitz continuous, weak solution for large initial data. See [START_REF] Lieberman | The first initial-boundary value problem for quasilinear second order parabolic equations[END_REF][START_REF] Dlotko | Examples of parabolic problems with blowing-up derivatives[END_REF][START_REF] Giga | Interior derivative blow-up for quasilinear parabolic equations[END_REF] and the references therein for earlier counter-examples concerning related quasilinear equations.

Our first goal will be to complete the above results by constructing a unique, maximal in time, W 1,∞ solution, without size restriction on the initial data and to establish the blow up alternative in W 1,∞ norm. This will enable us to interpret the above mentioned global nonexistence result from [START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF] appropriately as a gradient blow-up (GBU) result (see Theorem 2.1.4 and Remark 2.4.1 below), and will provide the grounds for the subsequent analysis of the asymptotic behavior of GBU solutions. For the local existence part, we will follow and suitably modify the approximation procedure used in [START_REF] Zhao | Existence and nonexistence of solutions for u t = div(|∇u| p-2 ∇u) + f (∇u, u, x, t)[END_REF].

The main difficulty is to get relevant estimates on the first order derivatives of the approximate solutions in order to pass to the limit in the nonlinear source term. To deal with this difficulty, our main new ingredient with respect to [START_REF] Zhao | Existence and nonexistence of solutions for u t = div(|∇u| p-2 ∇u) + f (∇u, u, x, t)[END_REF] is the construction of suitable barrier functions, in order to get uniform pointwise estimates on the gradients near the boundary for small time. We then use a strong result of DiBenedetto and Friedman [START_REF] Dibenedetto | Hölder estimates for nonlinear degenerate parabolic sytems[END_REF][START_REF] Dibenedetto | Degenerate parabolic equations[END_REF] on the Hölder regularity of gradients of weak solutions of degenerate parabolic equations and consequently we will use the framework of weak rather than viscosity solutions.

First, let us state the precise definition of solution. Let

Q T = Ω × (0, T ) and ∂ p Q T = {∂Ω × [0, T ]} ∪ Ω × {0} , T > 0.
Throughout this chapter, we will use the following definition of weak solution for (2.1.1).

Definition 2.1.1. Set m = max(p, q). A function u(x, t) is called a weak super-(sub-) solution of problem (2.1.1) on Q T if u ∈ C(Ω × [0, T )) ∩ L m ((0, T ); W 1,m (Ω)), u t ∈ L 2 ((0, T ); L 2 (Ω)), u(x, 0) ≥ (≤) u 0 (x), u ≥ (≤) g on ∂Ω and Q T u t ψ + |∇u| p-2 ∇u • ∇ψ dx dt ≥ (≤) Q T |∇u| q ψ dx dt (2.1.3) holds for all ψ ∈ C 0 (Q T ) ∩ L p ((0, T ); W 1,p (Ω)) such that ψ ≥ 0, ψ = 0 on ∂Ω × (0, T ). A function u is a weak solution of (2.1.1)
if it is a super-solution and a sub-solution.

Our first result concerns local existence and uniqueness of weak solutions (see also Section 2 for a comparison principle).

Theorem 2.1.1. Assume that q > p -1 > 1. Let M > 0 and let u 0 satisfy (2.1.2) and ∇u 0 ∞ ≤ M . Then (i) There exist a time T = T (M, p, q, N, g C 2 ) > 0 and a weak solution u of (2.1.1) on [0, T ), which moreover satisfies u ∈ L ∞ loc ([0, T ); W 1,∞ (Ω)). (ii) For any T > 0 the problem (2.1.1) has at most one weak solution u such that u ∈ L ∞ loc ([0, T ); W 1,∞
). (iii) There exists a (unique) maximal, weak solution of (2.1.1), still denoted by u. Let T max (u 0 ) be its existence time.

Then min

Ω u 0 ≤ u ≤ max Ω u 0 in Ω × (0, T max (u 0 )) (2.1.4) and if T max (u 0 ) < ∞, then lim t→Tmax(u 0 ) ∇u L ∞ = ∞ (gradient blow up GBU). Remark 2.1.1. Concerning Definition 2.1.1, we note that if 0 < T 1 < T 2 < ∞ and u is a weak solution on Q T 2 , then the restriction of u to Q T 1 is a weak solution on Q T 1 (this can be easily checked, taking any test function ψ on Q T 1 , by extending ψ as ψn (x, t) = ψ(x, T 1 )[1 -n(t -T 1 )] + for t ∈ (T 1 , T 2 ] and letting n → ∞).
Then, in Theorem 2.1.1(iii), by u being the maximal weak solution of (2.1.1), we mean that u is a weak solution on Q τ for any τ ∈ (0, T max (u 0 )) but cannot be extended to a weak solution on Q T for any

T > T max (u 0 ).
In what follows, the (unique) solution constructed in Theorem 2.1.1, will be called the maximal weak solution u ∈ L ∞ loc ([0, T max (u 0 )); W 1,∞ (Ω)) or, for short, the weak Lipschitz solution of (2.1.1).

We next establish a precise gradient estimate involving the distance to the boundary. Here and in the rest of the chapter we denote δ(x) = dist(x, ∂Ω).

Theorem 2.1.2. Let q > p -1 > 1. Let M > 0 and let u 0 satisfy (2.1.2) and ∇u 0 L ∞ ≤ M . Let u be the unique weak solution of (2.1.1) in L ∞ loc ([0, T max (u 0 )); W 1,∞ (Ω)). Then |∇u| ≤ C 1 δ -1/(q-p+1) (x) + C 2 .
in Ω × (0, T max (u 0 )) .

(2.1.5)

where

C 1 = C 1 (q, p, N ) > 0 and C 2 = C 2 (q, p, Ω, M, g C 2 ) > 0.
This estimate in particular implies that |∇u| remains bounded away from the boundary. Therefore, when T max (u 0 ) < ∞, the blow-up may only take place on the boundary and (2.1.5) provides information on the blow-up profile near ∂Ω. Estimate (2.1.5) is sharp in one space dimension, see [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF]. Similar results are already available for p = 2 and have been established in [110], [5]. For p > 2, only global-in-space gradient estimates were available up to now (ie for Ω = R N , see [START_REF] Bartier | Gradient estimates for a degenerate parabolic equation with gradient absorption and applications[END_REF]). The proof of estimate (2.1.5) is based on similar arguments as for the case p = 2, namely Bernstein type arguments, but they are much more technical. Moreover, the proof of (2.1.5) also relies on a regularizing effect for solutions to (2.1.1) which seems to be new and which is stated below.

Theorem 2.1.3. Assume that q > p-1 > 1 and let u be the unique weak Lipschitz solution of problem (2.1.1). Then

u t ≤ 1 p -2 u 0 L ∞ t in D (Ω) a.e. t > 0. (2.1.6)
Let us note that due to the positivity of the source term, this inequality implies the semi-concavity estimate

∆ p (u) = div |∇u| p-2 ∇u ≤ C t , (2.1.7)
which was obtained in the case Ω = R N by a different method in [START_REF] Esteban | Approximate solutions to first and second order quasilinear evolution equations via nonlinear viscosity[END_REF] and for q = p in [START_REF]Non-Diffusive Large Time Behavior for a Degenerate Viscous Hamilton-Jacobi Equation[END_REF].

Finally we give the following blow-up result, which is a variant of a global nonexistence result in [START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF], reinterpreted in terms of GBU in the light of Theorem 2.1.1. Let ϕ 1 be the first eigenfunction of -∆ with homogeneous Dirichlet boundary conditions Theorem 2.1.4. Assume that q > p > 2 and let u be the unique weak Lipschitz solution

of (2.1.1). Let α ≥ 1 satisfy p -1 q -p + 1 < α < q -1, then there exists a constant C = C(q, p, α, Ω, g ∞ ) > 0 such that if Ω u 0 ϕ α 1 dx ≥ C, then T max (u 0 ) < ∞, i.e.

gradient blow-up occurs.

For results concerning other aspects of equation (2.1.1) and the corresponding Cauchy problem, see e.g. [START_REF] Chen | On the Cauchy problem of evolution p-Laplacian equation with nonlinear gradient term[END_REF][START_REF] Shi | Self-similar singular solution of a p-laplacian evolution equation with gradient absorption term[END_REF][START_REF] Chen | Global existence and gradient estimates for quasilinear parabolic equations of the m-laplacian type with a strong perturbation[END_REF][START_REF] Zhao | Existence and nonexistence of solutions for u t = div(|∇u| p-2 ∇u) + f (∇u, u, x, t)[END_REF][START_REF] Bartier | Gradient estimates for a degenerate parabolic equation with gradient absorption and applications[END_REF] and the references therein. Asymptotic behavior of global solution is investigated in [START_REF] Stinner | Convergence to steady states in a viscous Hamilton-Jacobi equation with degenerate diffusion[END_REF][START_REF] Barles | Convergence to steady states for radially symmetric solutions to a quasilinear degenerate diffusive Hamilton-Jacobi equation[END_REF][START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF][START_REF]Non-Diffusive Large Time Behavior for a Degenerate Viscous Hamilton-Jacobi Equation[END_REF][START_REF] Ph | Localized non-diffusive asymptotic patterns for nonlinear parabolic equations with gradient absorption[END_REF]1,2] and references therein.

The rest of the chapter is organized as follows : In Section 2.2 we prove the wellposedness of (2.1.1) in W 1,∞ (Ω), as well as the regularizing effect. Section 2.4 is devoted to the proof of Theorem 2.1.2. Finally in section 4 we prove the sufficient blow-up criterion of Theorem 2.1.4.

2.2 Proof of Theorem 2.1.1 and Theorem 2.1.3

Local existence

Consider the following approximate problems for (2.1.1) :

         ∂ t u n -div |∇u n | 2 + 1 n (p-2)/2 ∇u n = |∇u n | 2 + 1 n q/2 - 1 n q/2 , x ∈ Ω, t > 0, u n (x, t) = g(x), x ∈ ∂Ω, t > 0, u n (x, 0) = u 0 (x),
x ∈ Ω.

(2.2.1) For each fixed n ∈ N, problem (2.2.1) is no longer degenerate and the regularity theory of quasilinear parabolic equations [START_REF] Ladyzenskaja | Linear and quasi-linear equations of parabolic type[END_REF] provides local-in-time solutions u n , which are smooth for t > 0 and continuous up to t = 0.

To find the limit function u(x, t) of the sequence {u n (x, t)}, we divide our proof into 5 steps. Recall that there exists η 0 > 0 small such that, for any x ∈ Ω with δ(x) ≤ η 0 , the point x := proj ∂Ω (x) (the projection of x onto the boundary) is well defined and unique. STEP 1. There exist a small time T 0 > 0, η ∈ (0, η 0 ) and M 2 > 0, all independent of n and depending on u 0 through M only, such that

u n L ∞ (Q T 0 ) ≤ M 1 := max ( u 0 L ∞ , g L ∞ ) , (2.2.2)
and

sup x∈Ω δ(x)≤η |u n (x, t) -u n ( x, t)| δ(x) ≤ M 2 , 0 < t ≤ T 0 . (2.2.3) Estimate (2.2.
2) is a direct consequence of the maximum principle since M 1 is a super solution for any n.

In order to prove estimate (2.2.3), we are going to construct a local barrier function under the exterior sphere condition satisfied by the domain Ω, i.e. for any x near ∂Ω, a supersolution in a neighborhood of x.

Let ρ > 0 be such that for all x ∈ ∂Ω, B ρ (x + ρν x ) ∩ Ω = {x}, where ν x is the unit outward normal vector on ∂Ω at x. Fix an arbitrary x 0 ∈ Ω such that δ(x 0 ) ≤ η where η ∈ (0, η 0 ) will be chosen later. Define x 1 = x 0 + ρν x 0 . Without loss of generality we may assume that x 1 = 0 and we write r = |x|. Let us denote, for s ≥ 0,

a(s) = s + 1 n (p-2)/2
, and

κ = 2a (s)s a(s) ∈ [0, p -2].
(2.2.4)

x 1 ∂Ω Γ x 0 x 0 ρ η η + ρ Figure 2.

-Local barrier function

We recall that for a function φ(x) = φ(|x|), we have :

∇φ(x) = φ (r) x r , D 2 φ(x) = φ (r) x ⊗ x r 2 + φ (r)Id r -φ (r) x ⊗ x r 3 , ∆φ(x) = φ (r) + (N -1)φ (r) r , (2.2.5)
where Id is the unit matrix and (x ⊗ x) ij = x i x j . The barrier function will have the form

v(x, t) = φ(r -ρ) + g(x),
where φ is a smooth function of one variable which is increasing and concave. First let us write

div |∇v| 2 + 1 n (p-2)/2 ∇v = a(|∇v| 2 )∆v + 2a (|∇v| 2 )(∇v) t D 2 v∇v, = a(|∇v| 2 ) ∆v + κ(|∇v| 2 ) (∇v) t D 2 v∇v |∇v| 2 . (2.2.6)
Using (2.2.5), we have

∆v + κ(|∇v| 2 ) (∇v) t D 2 v∇v |∇v| 2 = φ (r -ρ) + (N -1)φ (r -ρ) r + ∆g + κ(|∇v| 2 ) φ (r -ρ)(∇v • x) 2 r 2 |∇v| 2 + κ(|∇v| 2 ) φ (r -ρ) r -κ(|∇v| 2 ) φ (r -ρ)(∇v • x) 2 r 3 |∇v| 2 + κ(|∇v| 2 ) (∇v) t D 2 g∇v |∇v| 2 . Since φ (r -ρ) ≥ 0, r ≥ ρ, κ(|∇v| 2 ) ≥ 0 and 0 ≥ φ (r -ρ), we have -∆v + κ(|∇v| 2 ) (∇v) t D 2 v∇v |∇v| 2 ≥ -φ (r -ρ) - (N -1 + κ(|∇v| 2 )) ρ φ (r -ρ) -∆g ∞ -κ(|∇v| 2 ) D 2 g L ∞ . (2.2.7) On the other hand |∇v| = φ (r -ρ) x r + ∇g ≤ φ (r -ρ) + |∇g| ≤ 2φ (r -ρ) provided that φ (r -ρ) ≥ ∇g L ∞ . (2.2.8)
In this case we have

|∇v| 2 + 1 n (q-p+2)/2 ≤ 4(φ (r -ρ)) 2 + 1 (q-p+2)/2 . (2.2.9)
We take φ(s) = s(s + µ) -β , s ≥ 0, where β = β(q, p) ∈ (0, 1) is to be chosen later. We denote Γ := B(x 1 , ρ + η) ∩ Ω (see figure 2.1). Our aim is to show that v is a super-solution in Γ × (0, T 0 ) where T 0 , µ > 0 and η ∈ (0, η 0 ) small enough. In the rest of the proof, the constants T 0 , η, δ and C will be independent of x 0 , n and will depend on the initial data u 0 through M only (and they will depend on the other data p, q, N, Ω and g C 2 without other mention). We calculate

φ (s) = [(1 -β)s + µ] (s + µ) -β-1 , φ (s) = -β [(1 -β)s + 2µ] (s + µ) -β-2 .
We are looking for condition on β and µ such that

-div |∇v| 2 + 1 n ∇v ≥ |∇v| 2 + 1 n q/2 - 1 n q/2 . (2.2.10)
Due to (2.2.6), it suffices to have

-∆v + κ(|∇v| 2 ) (∇v) t D 2 v∇v |∇v| 2 ≥ |∇v| 2 + 1 n q-p+2 2 , ( 2
-φ (r -ρ) + (3 -N -p) ρ φ (r -ρ) ≥ 4(φ (r -ρ)) 2 + 1 (q-p+2)/2 + (p -2 + √ N ) D 2 g L ∞ .
(2.2.12) Using that ρ < r + η + ρ and (3 -N -p) < 0, then (2.2.10) holds if

(r -ρ + µ) -β-2 2βµ + (3 -N -p) (η + µ) 2 ρ ≥ 4(r -ρ + µ) -2β + 1 (q-p+2)/2 + (p -2 + √ N ) D 2 g L ∞ .
Assume that η and µ are such that

   4(r -ρ + µ) -2β ≥ 4(η + µ) -2β ≥ 1, 2βµ + (3 -N -p) ρ (η + µ) 2 ≥ βµ, (2.2.13)
then to get (2.2.10) it is sufficient to have

βµ(r -ρ + µ) -β-2 ≥ (r -ρ + µ) -β(q-p+2) 4 (q-p+3) , (2.2.14)
and βδ(r

-ρ + µ) -β-2 ≥ 4(p -2 + √ N ) D 2 g L ∞ .
(2.2.15)

Inequality (2.2.14) holds if we choose η = µ, β = 1 2(q-p+2)
, and µ satisfying 4 p-q-4 β ≥ µ (q-p+3)/(2q-2p+4) .

Inequalities (2.2.14)-(2.2.15) and (2.2.8) hold if we choose µ small enough. We have thus shown that if η = µ is small, then v is a supersolution on Γ × (0, T 0 ) for any T 0 > 0. Now we need to have a control on the parabolic boundary of Γ × (0, T 0 ) for T 0 > 0 small. For this purpose, we introduce another comparison function

ū(x, t) = (2C 2 K 2 + 2 ∇g 2 L ∞ + 1) q/2 t + C(1 -e -K(r-ρ) ) + g L ∞ .
It is easy to see that if we fix K sufficiently large K > N + p -3 ρ , then we can find

C = C(p, N, M, Ω, g C 2 ) > 0 sufficiently large so that -div |∇ū| 2 + 1 n (p-2)/2 ∇ū ≥ 0 in Ω. Indeed, since Ω is bounded there exists R(Ω) > 0 such that Ω ⊂ B(x 1 , R(Ω)) and hence r -ρ ≤ R(Ω). Now once K > 2(N +p-3) ρ is fixed using (2.2.7) it is sufficient to require that CKe -K(r-ρ) K - N + p -3 ρ ≥ (p -2 + √ N ) D 2 g L ∞ , 2.2. Proof of Theorem 2.1.1 and Theorem 2.1.3 which is satisfied if C ≥ 2e KR(Ω) (p -2 + √ N ) D 2 g L ∞ K 2 .
Thus

∂ t ū -div |∇ū| 2 + 1 n (p-2)/2 ∇ū ≥ |∇ū| 2 + 1 n q/2 - 1 n q/2
.

Next we can also choose

C > 0 large enough such that C(1 -e -K(r-ρ) ) + g L ∞ ≥ u 0 (x) in Ω. Since ū ≥ g on ∂Ω ⊂ x ∈ R N , |x| ≥ ρ
, by the maximum principle we get that for any n, u n ≤ ū in Q T . Thus

u n (x, t) ≤ (2C 2 K 2 + 2 ∇g 2 L ∞ + 1) q/2 t + C(1 -e -Kη ) + g ∞ ≤ 2 -β η 1-β + g(x) = v(x, t) on {x ∈ Ω, |x| = ρ + η} × [0, T 0 ],
provided T 0 and η = µ are small enough (depending only on M, p, q, Ω, g C 2 ). Next we can choose η = µ small enough such that

u 0 (x) ≤ g(x) + M |x -x| ≤ g(x) + M |r -ρ| ≤ g(x) + (r -ρ) (2η) -β -∇g L ∞ ≤ v(x, 0).

On the other hand u

n = g ≤ v on ∂Ω×[0, T 0 ]. We conclude that v is a super solution on Γ × (0, T 0 ). Similarly v := g -φ(r -ρ) is a sub-solution.
Applying the maximum principle we get v ≤ u n ≤ v on Γ × [0, T 0 ], and hence in particular

|u n (x 0 , t) -u n ( x0 , t)| |x 0 -x0 | ≤ sup 0≤s≤η |φ (s)| + ∇g ∞ ≤ η -β + ∇g L ∞ =: M 2 , 0 < t ≤ T 0 , which yields (2.2.

3).

STEP 2. There holds

∇u n L ∞ (Q T 0 ) ≤ M 3 := sup(M, M 2 + ∇g L ∞ ). ( 2
:= u n (x -h, t) is a classical solution of (2.2.1) in Ω h × (0, T 0 )
where

Ω h := x ∈ R N | x -h ∈ Ω . Let t ∈ [0, T 0 ] and x ∈ ∂(Ω ∩ Ω h ).
We may assume for instance x ∈ ∂Ω, the case x + h ∈ ∂Ω being similar. Then using | y -z| ≤ |y -z| and (2.2.3), we get

|u n (x, t) -u n (x + h, t)| = |u n ( x, t) -u n ( x + h, t) + u n ( x + h, t) -u n (x + h, t)| ≤ ∇g ∞ | x -x + h| + M 2 δ(x + h) ≤ ( ∇g ∞ + M 2 )|h| = M 3 |h|. In particular u n (x, t) ≤ u h n (x, t) + M 3 |h| on ∂(Ω ∩ Ω h ) × [0, T 0 ]. Applying the maximum principle, we have u n (x, t) ≤ u h n (x, t) + M 3 |h| on (Ω ∩ Ω h ) × [0, T 0 ]. By the same argument u h n (x, t) -M 3 |h| ≤ u n (x, t) on (Ω ∩ Ω h ) × [0, T 0 ], hence |u n (x, t) -u h n (x, t)| ≤ M 3 |h|. Since |h| ≤ η is arbitrary, the conclusion follows. STEP 3. Let > 0 and set Q T 0 , = {x ∈ Ω, δ(x) > } × ( , T 0 -). There exists a constant M 4 > 0 independent of n, such that |∇u n (x 1 , t 1 ) -∇u n (x 2 , t 2 )| ≤ M 4 |x 1 -x 2 | α + |t 1 -t 2 | α 2 (2.2.17)
for any pair of points (x i , t i ) ∈ Q T 0 , , where M 4 and α are positive constants depending only on T 0 , M 3 and . Indeed we know from a result of DiBenedetto and Friedman [START_REF] Dibenedetto | Hölder estimates for nonlinear degenerate parabolic sytems[END_REF][START_REF] Dibenedetto | Degenerate parabolic equations[END_REF]

that if f ∈ L r (Ω T ) for some r > pN p -1 then weak solutions of degenerate parabolic equation of the form ∂ t v -div |∇v| p-2 ∇v = f (x, t) (2.2.18) are of class C 1,α loc (Q T ) with Hölder norm depending only on f L r , ∇u L p and u L ∞ t ,L 2
x .

STEP 4. There exists a constant M 5 > 0 independent of n, such that

∂ t u n L 2 (Q T 0 ) ≤ M 5 . (2.2.19)
To see this , multiplying (2.2.1) by ∂ t u n and integrating over Q T 0 , we have

T 0 0 Ω (∂ t u n ) 2 dxdt ≤ - T 0 0 Ω |∇u n | 2 + 1 n (p-2)/2 ∇u n • ∇(∂ t u n )dxdt + T 0 0 Ω |∇u n | 2 + 1 n q/2 ∂ t u n dxdt.
By Hölder's inequality and

T 0 0 Ω |∇u n | 2 + 1 n (p-2)/2 ∇u n • ∇(∂ t u n )dxdt = 1 p Ω |∇u n (x, T 0 )| 2 + 1 n p/2 - 1 p Ω |∇u n (x, 0)| 2 + 1 n p/2
, we get

T 0 0 Ω (∂ t u n ) 2 dxdt ≤ 2 p Ω |∇u n (x, 0)| 2 + 1 n p/2 dx + 2 T 0 0 Ω |∇u n | 2 + 1 n q dxdt ≤ M .
for some M = M |Ω|, M 3 , T 0 , p, q > 0.

STEP 5. We recall that by the Arzelá-Ascoli theorem we have

W 1,∞ (Ω) c → C(Ω) → L 2 (Ω). ( 2
u n → u in C Ω × [0, T 0 ] , ∇u n → ∇u in C(Q T 0 , ), ∂ t u n → ∂ t u weakly in L 2 (Q T 0 ).    (2.2.21)
We multiply (2.2.1) by a test function and integrate. Then by the Lebesgue's dominated convergence theorem and (2.2.21) we can pass to the limit and check that u is a weak solution of (2.1.1).

The blow-up alternative

Let us temporarily assume the uniqueness result which will be proved in the next section. The construction of the weak solution as a limit of classical solutions implies the blow-up alternative.

Indeed suppose that the maximal existence time T max (u 0 ) < ∞ and that there exist M > 0 and t k → T max (u 0 ) such that for all k

∇u(t k ) L ∞ (Ω) ≤ M.
(2.2.22)

Then we can find τ = τ (M) > 0 independent of k, such that the problem

   ∂ t u -div(|∇u| p-2 ∇u) = |∇u| q , x ∈ Ω, t > 0, u(x, t) = g(x), x ∈ ∂Ω, t > 0, u(x, 0) = u(x, t k ), x ∈ Ω, (2.2.23) admits a unique weak solution v k on [0, τ ). Setting ũ(t) = u(t) for t ∈ [0, t k ) v k (t -t k ) for t ∈ [t k , t k + τ ) ,
it is easy to see that we get a weak solution defined on [0,

t k + τ ). Since for k large enough t k + τ > T max (u 0 ), this contradicts the definition of T max (u 0 ). Hence T max (u 0 ) < ∞ ⇒ lim t→Tmax(u 0 ) ∇u L ∞ (Ω) = ∞.

Uniqueness

In this section we prove the uniqueness of the weak solution. This result will be a consequence of the following comparison principle which, in turns, also guarantees (2.1.4). Proposition 2.2.1. Let u, v be respectively, sub-, super-solutions of (2.1.1). Assume that

u, v ∈ L ∞ ((0, T ); W 1,∞ (Ω)). Then u ≤ v on Ω × (0, T ).
The proof of Proposition 2.2.1 is mostly based on the following algebraic lemma from which we can show that the source term can be counterbalanced by the diffusion effect (c.f [START_REF] Bojarski | p-harmonic equation and quasiregular mappings[END_REF] and [START_REF] Lindqvist | Notes on the p-laplace equation[END_REF] for usefull inequalities on the p-Laplacian).

Lemma 2.2.1 (Monotonicity Property). Let σ > 1. For all a and b ∈ R N :

|a| σ-2 a -|b| σ-2 b, a -b ≥ 4 σ 2 |a| (σ-2)/2 a -|b| (σ-2)/2 b 2 .
Proof of Proposition 2.2.1. We set w = (u -v) + . By definition we have w = 0 on ∂Ω. By Remark 2.1.1, for any τ ∈ (0, T ), using ψ = w as test-function, we have

τ 0 Ω ww t dxdt ≤ τ 0 {w(•,t)>0} [|∇u| q -|∇v| q ] w dxdt B - τ 0 {w(•,t)>0} |∇u| p-2 ∇u -|∇v| p-2 ∇v • ∇w dxdt H .
We set a = ∇u and b = ∇v. We get by lemma 2.2.1

H ≥ c(p) τ 0 {w(•,t)>0} |∇u| (p-2)/2 ∇u -|∇v| (p-2)/2 ∇v 2 dxdt. (2.2.24)
Let's consider the term B. We put h(s) = s 2q p for s ≥ 0. Given that q ≥ p -1 ≥ p 2 , we have

h (s) = 2q p s 2q-p
p . The mean value theorem yields

|∇u| q -|∇v| q 2 ≤ Ch (θ) 2 |∇u| (p-2)/2 ∇u -|∇v| (p-2)/2 ∇v 2 , for some 0 ≤ θ ≤ max(|∇u| p 2 , |∇v| p 2 ). Since we assumed u, v ∈ L ∞ ((0, T ); W 1,∞ (Ω)), it follows that |∇u| q -|∇v| q 2 ≤ C |∇u| (p-2)/2 ∇u -|∇v| (p-2)/2 ∇v 2 .
On the other hand, the Young inequality implies

B ≤ τ 0 {w(•,t)>0} | |∇u| q -|∇v| q | 2 dxdt + C( ) τ 0 {w(•,t)>0} w 2 dxdt.
Combining these two inequalities, we arrive at

B ≤ C τ 0 {w(•,t)>0} |∇u| (p-2)/2 ∇u -|∇v| (p-2)/2 ∇v 2 dxdt + C( ) τ 0 {w(•,t)>0}
w 2 dxdt.

(2.2.25)

Choosing small enough, we get

Ω w 2 (τ ) dx ≤ Ω w 2 (0) dx + C( ) τ 0 Ω w 2 dxdt, 0 < τ < T. (2.2.26)
The Gronwall lemma implies that for any t ∈ (0, T )

Ω w 2 (x, t) dx ≤ e Ct Ω w(x, 0) 2 dx.
We conclude that w ≡ 0 almost everywhere.

Remark 2.2.1. (a) The inequality in lemma 2.2.1 for σ ∈ (1, 2) can be deduced from the inequality for σ ≥ 2 in [START_REF] Lindqvist | Notes on the p-laplace equation[END_REF] as follows :

We set a = |∇u| σ-2 ∇u and b = |∇v| σ-2 ∇v.

|∇u| σ-2 ∇u -|∇v| σ-2 ∇v, ∇u -∇v = a -b, a |a| 2-σ σ-1 -b |b| 2-σ σ-1 = a -b, a |a| m-2 -b |b| m-2 .
(2.2.27)

where m = σ σ-1 > 2. (b)
The question of uniqueness was partially open in [START_REF] Stinner | Convergence to steady states in a viscous Hamilton-Jacobi equation with degenerate diffusion[END_REF]. The preceding proof of Poposition 2.2.1 can be applied to show uniqueness in the case p -1 ≥ q ≥ p 2 with p ≥ 2.

(c) In [4] we have a weaker inequality for p ∈ (1, 2) but it is sufficient to prove uniqueness for the case q > 1 :

|a| p-2 a -|b| p-2 b, a -b ≥ (p -1)|a -b| 2 (|a| p + |b| p ) p-2 p .

Regularizing effect

We use a technique developed by Zhao for the the p-Laplace equation without source term [START_REF]A note to the regularity of solutions for the evolution p-Laplacian equations[END_REF]. The idea is to apply a Stampacchia maximum principle argument to the equation satisfied by λ γ u(x, λt) -u(x, t) and then let λ → 1 + . Let u be a weak solution of (

2.1.1) in L ∞ loc ([0, T ); W 1,∞ (Ω)). Set u λ (x, t) = λ γ u(x, λt), λ > 1, γ = 1 p -2 .
Then u λ is a weak solution of

   ∂ t u λ -div(|∇u λ | p-2 ∇u λ ) = λ -(q-p+1)γ |∇u λ | q , x ∈ Ω, t ∈ 0, T λ , u λ (x, t) = λ γ g(x), x ∈ ∂Ω, t ∈ 0, T λ , u λ (x, 0) = λ γ u 0 (x), x ∈ Ω.
We set v(x, t) = u(x, t)+k where k := (λ γ -1) u 0 L ∞ , then v satisfy the same equation as u with v(x, 0) = u 0 (x) + k and v(x, t) = g(x) + k on ∂Ω × (0, T ). Given that λ γ u 0 (x) =

u 0 (x) + (λ γ -1)u 0 (x) ≤ u 0 (x) + (λ γ -1) u 0 L ∞ and λ γ g(x) ≤ g(x) + (λ γ -1) g L ∞ , we have u λ (x, 0) ≤ v(x, 0) in Ω and u λ ≤ v on ∂Ω × 0, T λ .
Since λ > 1 and q > p -1, we have λ -(q-p+1)γ |∇u λ | q ≤ |∇u| q and hence u λ is a sub-solution of the equation. Using proposition 2.2.1, we have

u λ (x, t) ≤ v(x, t) in Ω × 0, T λ that is λ γ u(λt, x) -u(x, t) ≤ (λ γ -1) sup( u 0 L ∞ , u 0 L ∞ ). ( 2
+ tu t (x, t) ≤ γ u 0 L ∞ .
We conclude using the positivity of u.

Remark 2.2.2. The homogeneity of the operator and the boundedness of u are essential.

Gradient estimate : proof of Theorem 2.1.2

The proof of (2.1.5) relies on a modification of the Bernstein technique and the use of a suitable cut-off function. It requires the study of the partial differential equation satisfied by |∇u| 2 . We follow the ideas used in [110] and [START_REF] Bartier | Gradient estimates for a degenerate parabolic equation with gradient absorption and applications[END_REF]. Let

x 0 ∈ Ω be fixed, 0 < t 0 < T < T max (u 0 ), R > 0 such that B(x 0 , R) ⊂ Ω and write Q t 0 T,R = B(x 0 , R) × (t 0 , T ) Let α ∈ (0, 1) and set R = 3R 4 . We select a cut-off function η ∈ C 2 (B(x 0 , R )), 0 < η < 1, with η(x 0 ) = 1 and η = 0 for |x -x 0 | = R , such that |∇η| ≤ CR -1 η α |D 2 η| + η -1 |∇η| 2 ≤ CR -2 η α for |x -x 0 | < R (2.3.1)
with C = C(α) > 0 (see [110] for an example of such function). First let us state the following lemma.

Lemma 2.3.1. Let u 0 , u be as in Theorem 2.1.2. We denote w = |∇u| 2 and z = ηw.

Then at any point (x 1 , t 1 ) ∈ Q t 0 T,R such that |∇u(x 1 , t 1 )| > 0, z is smooth and satisfies the following differential inequality

Lz + Cz 2q-p+2 2 ≤ C sup( u 0 L ∞ , g L ∞ ) t 0 2q-p+2 q + CR -2q-p+2 q-p+1 ,
where

Lz = ∂ t z -Az -H • ∇z, (2.3.2) Az = |∇u| p-2 ∆z + (p -2)|∇u| p-4 (∇u) t D 2 z∇u, (2.3.3)
H is defined by (2.3.7) and C = C(p, q, N ) > 0.

Proof of lemma 2.3.1 We know that a solution u of (2.1.1) is smooth at points where |∇u| > 0 [START_REF] Bartier | Gradient estimates for a degenerate parabolic equation with gradient absorption and applications[END_REF]. More precisely, we know that ∇u ∈ C 2,1 in a neighborhood of such points and hence we can differentiate the equation. As observed in [START_REF] Bartier | Gradient estimates for a degenerate parabolic equation with gradient absorption and applications[END_REF], w = |∇u| 2 satisfies the following differential equation :

∂ t w -Aw = -2|∇u| p-2 |D 2 u| 2 + H • ∇w Indeed, for i = 1, • • • , N , put u i = ∂u ∂x i and w i = ∂w ∂x i . Differentiating (2.1.1) in x i , we have ∂ t u i -|∇u| p-2 ∆u i - p -2 2 |∇u| p-4 N j=1 ∂w i ∂x j u j - p -2 2 |∇u| p-4 N j=1 w j ∂u i ∂x j = q 2 w q-2 2 w i + p -2 2 w p- 4
2 w i ∆u + (p -2)(p -4) 4 w p-6 2 (∇u • ∇w)w i . (2.3.4)
Multiplying (2.3.4) by 2u i , summing up, and using ∆w = 2∇u

• ∇(∆u) + 2|D 2 u| 2 , we deduce that Lw = -2w p-2 2 |D 2 u| 2 , (2.3.5)
where

Lw := ∂ t w -|∇u| p-2 ∆w -(p -2)|∇u| p-4 (∇u) t D 2 w∇u -H • ∇w, (2.3.6)
H := (p -2)w p-4 2 ∆u + (p -2)(p -4) 2 w p-6 2 ∇u • ∇w + qw q-2 2 ∇u + p -2 2 w p-4 2 ∇w. (2.3.7)
Setting z = ηw, we get

Lz = ηLw + wLη -2w p-2 2 ∇η • ∇w -2(p -2)w p-4 2 (∇η • ∇u)(∇w • ∇u).
Now we shall estimate the different terms. In what follows δ i > 0 can be chosen arbitrarily small.

-Estimate of |2w p-2

2 ∇η • ∇w|. Using Young's inequality, we have

|2w p-2 2 ∇η • ∇w| ≤ w p-2 2 Cη -1 |∇η| 2 w + δ 1 η|D 2 u| 2 , (2.3.8)
where we used the fact that ∇w = 2D 2 u∇u.

-Estimate of |2(p -2)w p-4 2 (∇η • ∇u)(∇w • ∇u)|. |2(p -2)w p-4 2 (∇η • ∇u)(∇w • ∇u)| ≤ w p-2 2 Cη -1 |∇η| 2 w + δ 2 η|D 2 u| 2 . (2.3.9) -Estimate of |w H • ∇η|. |w H • ∇η| ≤ w p-2 2 Cη -1 |∇η| 2 w + δ 3 [D 2 u| 2 η (1) + w p-2 2 Cη -1 |∇η| 2 w + δ 4 [D 2 u| 2 η (2) + w p-2 2 Cη -1 |∇η| 2 w + δ 5 [D 2 u| 2 η (3) +Cw q+1 2 |∇η|. ( 2
2 ∇w • ∇η. Finally choosing δ i such that -2 + δ 1 + δ 2 + δ 3 + δ 4 + δ 5 = -1, we arrive at Lz + ηw p-2 2 |D 2 u| 2 ≤ C(p, q, N )w p 2 |D 2 η| + |∆η| + η -1 |∇η| 2 + |∇η|w q+1 2 .
Using the properties of the cut-off function η, we get

Lz + ηw p-2 2 |D 2 u| 2 ≤ C(p, q, N )R -2 η α w p 2 + C(p, q, N )R -1 η α w q+1 2 . (2.3.11)
Using the result of Theorem 2.1.3, we shall estimate |∇u| p-2 |D 2 u| 2 in terms of a power of w.

For (x 1 , t 1 ) ∈ Q t 0 T,R such that |∇u(x 1 , t 1 )| > 0, we have |∇u(x 1 , t 1 )| q = ∂ t u(x 1 , t 1 ) -div |∇u| p-2 ∇u(x 1 , t 1 ) ≤ sup( u 0 L ∞ , g L ∞ ) (p -2)t 0 + (p -2 + √ N )|∇u| p-2 |D 2 u(x 1 , t 1 )|. Hence 1 2(p -2 + √ N ) 2 |∇u(x 1 , t 1 )| 2q ≤ sup( u 0 L ∞ , g L ∞ ) (p -2)(p -2 + √ N )t 0 2 + |∇u| 2p-4 |D 2 u(x 1 , t 1 )| 2 .
There are two cases :

either 1 2(p -2 + √ N ) 2 |∇u(x 1 , t 1 )| 2q ≤ 2 sup( u 0 L ∞ , g L ∞ ) (p -2)(p -2 + √ N )t 0 2 , or 1 2(p -2 + √ N ) 2 |∇u(x 1 , t 1 )| 2q-p+2 ≤ 2|∇u| p-2 |D 2 u(x 1 , t 1 )| 2 .
In both cases we arrive at

1 C(N, p) |∇u(x 1 , t 1 )| 2q-p+2 ≤ C(p, q, N ) sup( u 0 L ∞ , g L ∞ ) t 0 2q-p+2 q + |∇u| p-2 |D 2 u(x 1 , t 1 )| 2 .
Using this inequality, it follows from (2.3.11) that, at (x 1 , t 1 ),

Lz + 1 C(N, p) η|∇u| 2q-p+2 ≤ C(p, q, N ) sup( u 0 L ∞ , g L ∞ ) t 0 2q-p+2 q + CR -2 η α w p 2 + CR -1 η α w q+1 2 .
We take α = q + 1 2q -p + 2 ∈ (0, 1) (since q > p -1). Using Young's inequality, we have

CR -1 η α w q+1 2 ≤ CR -2q-p+2 q-p+1 + 1 4C(N, p) ηw 2q-p+2 2 , CR -2 η α w p 2 ≤ CR -2q-p+2 q-p+1 + 1 4C(N, p) η q+1 p w 2q-p+2 2
.

Using that η ≤ 1, we get

Lz + 1 C(N, p) η|∇u| 2q-p+2 ≤ C(p, q, N ) sup( u 0 L ∞ , g L ∞ ) t 0 2q-p+2 q + CR -2q-p+2 q-p+1 + 1 2C(N, p) η|∇u| 2q-p+2 . Hence Lz + 1 2C(N, p) z 2q-p+2 2 ≤ C(p, q, N ) sup( u 0 L ∞ , g L ∞ ) t 0 2q-p+2 q + CR -2q-p+2 q-p+1 . (2.3.12)
Proof of theorem 2.1.2

First let us note that by the proof of the local existence there exists t 0 ∈ 0, T max (u 0 ) with t 0 = t 0 (M, p, q, N, g C 2 ), such that

sup 0≤t≤t 0 ∇u L ∞ ≤ C(p, q, Ω, M, g C 2 ).
(2.3.13)

We also know that ∇u is a locally Hölder continuous function and thus z is a continuous function on

B(x 0 , R ) × [t 0 , T ] = Q, for any T < T max(u 0 ) . Therefore, unless z ≡ 0 in Q, z must reach a positive maximum at some point (x 1 , t 1 ) ∈ B(x 0 , R ) × [t 0 , T ]. Since z = 0 on ∂B R × [t 0 , T ], we deduce that x 1 ∈ B R . Therefore ∇z(x 1 , t 1 ) = 0 and D 2 z(x 1 , t 1 ) ≤ 0. Now we have either t 1 = t 0 , or t 0 < t 1 ≤ T . If t 1 = t 0 , then z(x 1 , t 1 ) ≤ ∇u(t 0 ) 2 L ∞ ≤ C(p, q, Ω, M, g C 2 ). If t 0 < t 1 ≤ T , we have ∂ t z(x 1 , t 1
) ≥ 0 and therefore Lz ≥ 0. Using (2.3.12) we arrive at

1 2C(N, p) z(x 1 , t 1 ) 2q-p+2 2 ≤ C(p, q, N ) sup( u 0 L ∞ , g L ∞ ) t 0 2q-p+2 q + CR -2q-p+2 q-p+1 , (2.3.14) that is z(x 1 , t 1 ) ≤ C(p, q, N ) sup( u 0 L ∞ , g L ∞ ) t 0 1 q + C(p, q, N )R -1 q-p+1 . (2.3.15)
Since z(x 0 , t) ≤ z(x 1 , t 1 ) and η(x 0 ) = 1, we get

|∇u(x 0 , t)| ≤ C(p, q, N ) sup( u 0 L ∞ , g L ∞ ) t 0 1 q + C(p, q, N )R -1 q-p+1 for t ∈ [t 0 , T ].
The proof of (2.1.2) follows by taking R = δ(x 0 ), letting T → T max (u 0 ) and using (2.3.13).

2.4 Blow-up criterion : proof of Theorem 2.1.4

Assume that T max (u 0 ) = ∞, taking ϕ α 1 as test-function, we have for any τ > 0

τ 0 Ω u t ϕ α 1 dxdt = τ 0 Ω |∇u| q ϕ α 1 dxdt -α τ 0 Ω |∇u| p-2 ϕ α-1 1 ∇u • ∇ϕ 1 dxdt. (2.4.1) Set y(t) = Ω u(t) ϕ α 1 dx. Since by definition u t ∈ L 2 loc ((0, ∞); L 2 (Ω)), we have y ∈ W 1,1 loc (0, ∞) and y (t) = Ω u t ϕ α 1 dx. Differentiating (2.4.1
) with respect to τ we have, for a.e. τ > 0

y (τ ) = Ω |∇u(τ )| q ϕ α 1 dx -α Ω |∇u(τ )| p-2 ϕ α-1 1 ∇u(τ ) • ∇ϕ 1 dx. (2.4.2)
Assume that α > p-1 (q-p+1) . Since q > p > 1 and ∇ϕ 1 L ∞ ≤ C , using Hölder and Young inequalities we get :

α Ω |∇u(τ )| p-2 ϕ α-1 1 ∇u(τ ) • ∇ϕ 1 dx ≤ 1 2 Ω |∇u(τ )| q ϕ α 1 dx + C Ω ϕ α-q/(q-p+1) 1 dx ≤ 1 2 Ω |∇u(τ )| q ϕ α 1 dx + C.
Here we used the fact that Ω ϕ -l 1 dx < ∞ for l < 1. Therefore

y (τ ) ≥ 1 2 Ω |∇u(τ )| q ϕ α 1 dx -C.
Assuming that α < q -1, we get

Ω |∇u(τ )| dx = Ω |∇u(τ )| ϕ α q 1 ϕ -α q 1 dx ≤ Ω |∇u(τ )| q ϕ α 1 dx 1/q Ω ϕ -α q-1 1 dx q-1 q ≤ C Ω |∇u(τ )| q ϕ α 1 dx 1/q .
2.4. Blow-up criterion : proof of Theorem 2.1.4

On the other hand using that Ω |u(τ

)| dx ≤ C u L ∞ (∂Ω) + C Ω |∇u(τ )| dx, we have Ω u(τ ) ϕ α 1 dx ≤ ϕ α 1 L ∞ Ω u(τ ) dx ≤ C + C Ω |∇u(τ )| dx.
Combining these two inequalities we arrive at

Ω |∇u(τ )| q ϕ α 1 dx ≥ C Ω u(τ ) ϕ α 1 dx q -C.
Finally we get the blow-up inequality

y (τ ) ≥ C 1 y(τ ) q -C 2 , for a.e. τ > 0, with C 1 = C 1 (p, q, Ω) > 0 and C 2 = C 2 (p, q, α, Ω, g L ∞ ).
Remark 2.4.1. Instead of assuming that Ω u 0 φ α 1 dx is large in Theorem 2.1.4, it would be sufficient to assume that u 0 r is large for some r ∈ [1, ∞). In fact, assuming without loss of generality r ≥ (2q -p)/(q -p) and denoting y(t) = Ω u r (t)dx, the Poincaré and Hölder inequalities can be used in order to prove the blow-up inequality y ≥ C 1 y (q+r-1)/r -C 2 (see [START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF]).

Chapitre 3 Localisation des points d'explosion

Dans ce travail en collaboration avec Philippe Souplet, on s'intéresse à la localisation des point d'explosion du gradient. On a vu précédemment que l'ensemble des points d'explosion est contenu dans le bord du domaine. Il s'agit ici de montrer que pour des domaines bien choisis et des données initiales bien préparées, l'ensemble des points d'explosion se réduit à un singleton. Pour arriver à cette fin on adaptera la méthode de Souplet et Li aux différentes difficultés apportées par une diffusion non-linéaire.

Introduction

Problem and main result

This chapter is a contribution to the study of the influence of nonlinear diffusion on the qualitative properties of equations of Hamilton-Jacobi type and, in particular, on the formation of finite-time singularity. More specifically, we consider the following problem

     u t = ∆ p u + |∇u| q , (x, y) ∈ Ω, t > 0, u(x, y, t) = µy, (x, y) ∈ ∂Ω, t > 0, u(x, y, 0) = u 0 (x, y), (x, y) ∈ Ω, (3.1.1)
where ∆ p denotes the p-Laplace operator, ∆ p = ∇ • (|∇u| p-2 ∇u). Throughout this paper, we assume that µ ≥ 0 is a constant and that q > p > 2.

(3.1.2)

For reasons that will appear later, we restrict ourselves to a class of planar domains Ω which satisfy certain geometric properties. We assume that, for some L 1 , L 2 > 0, Ω ⊂ R 2 is a smooth bounded domain of class C 2+ for some ∈ (0, 1); (3.1.3) Ω is symmetric with respect to the axis x = 0;

(3.1.4) Ω ⊂ {y > 0} and Ω contains the rectangle (-L 1 , L 1 ) × (0, 2L 2 );

(3.1.5) Ω is convex in the x-direction.

(3.1.6)

In particular, by (3.1.5), ∂Ω has a flat part, centered at the origin (0, 0). Note that assumption (3.1.6) is equivalent to the fact that Ω ∩ {y = y 0 } is a line segment for each y 0 . The initial data u 0 is taken in V µ , where

V µ := u 0 ∈ C 1 (Ω), u 0 ≥ µy, ∂ y u 0 ≥ µ/2
in Ω and u 0 = µy on ∂Ω .

We shall use the following notation throughout :

Ω + := {(x, y) ∈ Ω; x > 0} . For T > 0, set Q T = Ω × (0, T ), S T = ∂Ω × (0, T ) and ∂ P Q T = S T ∪ (Ω × {0}) its parabolic boundary. Problem (3.1.1
) is well posed locally in time (see Section 2 for details), with blow-up alternative in W 1,∞ norm. For brevity, when no confusion arises, the existence time of its maximal solution u will be denoted by

T := T max (u 0 ) ≤ ∞.
It is known (see [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF][START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF]) that global nonexistence, i.e. T < ∞, occurs for suitably large initial data (more generally, for problem (3.1.1) in an n-dimensional bounded domain with Dirichlet boundary conditions). Note that the condition q > p is sharp, since the solutions are global and bounded in W 1,∞ if 1 < q ≤ p (see [START_REF] Stinner | Convergence to steady states in a viscous Hamilton-Jacobi equation with degenerate diffusion[END_REF][START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF]). Since it follows easily from the maximum principle that u itself remains uniformly bounded on Q T , global nonexistence can only occur through gradient blow-up, namely

sup Q T |u| < ∞ and lim t→T ∇u(•, t) ∞ = ∞.
This is different from the usual blow-up, in which the L ∞ norm of the solution tends to infinity as t → T max , which occurs for equations with zero-order nonlinearities, such as u t = ∆ p u + u q (see [START_REF] Fujii | Asymptotic behavior of blowup solutions of a parabolic equation with the p-Laplacian[END_REF][START_REF] Galaktionov | Single point blow-up for N -dimensional quasilinear equations with gradient diffusion and source[END_REF]). The study of -L ∞ or gradient-blow-up singularities, in particular their location, time and spatial structure is very much of interest for the understanding of the physical problems modelled by such equation, as well as for the mathematical richness that they involve. The L ∞ blow-up for the equation u t = ∆ p u + u q has been extensively studied, both in the case of linear (p = 2) and nonlinear (p > 2) diffusion ; see respectively the monographs [START_REF] Quittner | Superlinear parabolic problems : blow-up, global existence and steady states[END_REF] and [START_REF] Samarskii | Blow-up in quasilinear parabolic equations[END_REF] and the numerous references therein.

As for equation (3.1.1) with p = q = 2, it is known as the (deterministic version of the) Kardar-Parisi-Zhang (KPZ) equation, describing the profile of a growing interface in certain physical models (see [START_REF] Kardar | Dynamic scaling of growing interfaces[END_REF]), where u then represents the height of the interface profile. The case of p = 2 and q ≥ 1 is a more general model which was developed by Krug and Spohn, aiming at studying the effect of the nonlinear gradient term on the properties of solutions (see [START_REF] Krug | Universality classes for deterministic surface growth[END_REF]). Our main interest in this paper is to study the effect of a quasilinear gradient diffusivity on the localization of the singularities.

For the case of linear diffusion p = 2, various sufficient conditions for gradient blowup and global existence were provided and qualitative properties were investigated, such as : nature of the blow-up set, rate and profile of blow-up, maximum existence time and continuation after blow-up, boundedness of global solutions and convergence to a stationary state. We refer for these to the works [5,[START_REF] Guo | Blowup rate estimates for the heat equation with a nonlinear gradient source term[END_REF]108,110,[START_REF] Quittner | Superlinear parabolic problems : blow-up, global existence and steady states[END_REF][START_REF] Li | Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[END_REF] and the references therein.

The case p > 2 is far from being completely understood and fewer works deal with the nonlinear diffusion. The large time behavior of global solutions in bounded or unbounded domains has been studied in [START_REF] Ph | Localized non-diffusive asymptotic patterns for nonlinear parabolic equations with gradient absorption[END_REF][START_REF] Stinner | Convergence to steady states in a viscous Hamilton-Jacobi equation with degenerate diffusion[END_REF][START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF][START_REF] Bartier | Gradient estimates for a degenerate parabolic equation with gradient absorption and applications[END_REF][START_REF] Barles | Convergence to steady states for radially symmetric solutions to a quasilinear degenerate diffusive Hamilton-Jacobi equation[END_REF][START_REF]Non-Diffusive Large Time Behavior for a Degenerate Viscous Hamilton-Jacobi Equation[END_REF]. Concerning the asymptotic description of singularities, results on the gradient blow-up rate in one space dimension can be found in [START_REF]Boundedness of global solutions of a p-laplacian evolution equation with a nonlinear gradient term[END_REF][START_REF] Zhang | Boundedness of global solutions for a heat equation with exponential gradient source[END_REF][START_REF] Zhang | Gradient blowup rate for a viscous hamilton-jacobi equation with degenerate diffusion[END_REF]. On the other hand, in any space dimension, it is known [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF] that gradient blow-up can take place only on the boundary, i.e.

GBU S(u

0 ) ⊂ ∂Ω,
where the gradient blow-up set is defined by GBU S(u 0 ) = x 0 ∈ Ω; for any ρ > 0, sup

(Ω∩Bρ(x 0 ))×(T -ρ,T ) |∇u| = ∞ .
Moreover, the following upper bounds for the space profile of the singularity were obtained in [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF] :

|∇u| ≤ Cδ -1 q-p+1
and u ≤ Cδ

q-p q-p+1 in Q T , where δ(x) = dist(x, ∂Ω), (3.1.7)
and they are sharp in one space dimension [START_REF]Boundedness of global solutions of a p-laplacian evolution equation with a nonlinear gradient term[END_REF]. (Our nondegeneracy lemma 3.7.1 below indicates that they are also sharp in higher dimensions.) It is easy to see, by considering radially symmetric solutions with Ω being a ball, that GBU S(u 0 ) can be the whole of ∂Ω. A natural question is then :

Can one produce examples (in more than one space dimension) when GBU S(u 0 ) is a proper subset of ∂Ω, especially a single point ?

The goal of this work is to provide an affirmative answer to this question. Our main result is the following. (i) For any ρ ∈ (0, L 1 ), there exists µ 0 = µ 0 (p, q, Ω, ρ) > 0 such that, for any µ ∈ (0, µ 0 ], there exist initial data u 0 in V µ ∩ C 2 (Ω) such that the corresponding solution u of (3.1.1) enjoys the following properties :

T := T max (u 0 ) < ∞ and GBU S(u 0 ) ⊂ [-ρ, ρ] × {0}, (3.1.8)
u(•, t) is symmetric with respect to the line x = 0, for all t ∈ (0, T ), (3.1.9)

u x ≤ 0 in Ω + × (0, T ), (3.1.10) u y ≥ µ/2 in Q T . (3.1.11)
(ii) For any such µ and u 0 , we have

GBU S(u 0 ) = {(0, 0)}.
A class of initial data satisfying the requirements of Theorem 3.1.1 is provided in Lemma 3.6.1 below. We note that, in the semilinear case p = 2, a single-point boundary gradient blow-up result was obtained in [START_REF] Li | Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[END_REF]. Although we follow the same basic strategy, the proof here is considerably more complicated. We point out right away that, in view of property (3.1.11), the equation is not degenerate for the solutions under consideration. However, since the essential goal of this work is to study the effect of nonlinear diffusion on gradient blow-up, what is relevant here are the large values of the gradient in the diffusion operator (rather than the issues of loss of higher regularity that would arise from the degenerate nature of the equation near the level ∇u = 0). It is an open question whether or not single-point gradient blow-up can still be proved in the case µ = 0. Actually, the lower bound (3.1.11) on |∇u| is crucially used at various points of the proof, which is already very long and involved, due to the presence of the nonlinear -even though nondegenerate -diffusion term (see the next subsection for more details).

Section 3.2 is devoted to local in time well-posedness and regularity results. The proof of Theorem 3.1.1 will be split into several sections, namely sections 3.3-3.7 for assertion (i) and sections 3.8-3.9 for assertion (ii) (the latter uses also section 3.5). Finally, in two appendices, we provide the proofs of some regularity properties and a suitable parabolic version of Serrin's corner lemma. Since the proof is quite long and involved, for the convenience of readers, we now give an outline of the main steps of the proof.

Outline of proof

For sake of clarity, we have divided the proof into a number of intermediate steps, each of which being relatively short (one or two pages, say), except for step (f), which involves long and hard computations. For the convenience of readers, we outline the structure of the proof.

(a) Preliminary estimates (Lemmas 3.3.1-3.3.4) : The symmetry in the variable x and the decreasing property for x > 0 are basic features in order to expect single-point gradient blow-up. Besides u being bounded, we also have boundedness of u t . Moreover, for sufficiently small µ and under a suitable assumption on u 0 , we show that u y ≥ µ/2. These bounds on u t and u y seem necessary in the very long calculations of the key step (f) below. In turn, the positivity of u y guarantees that solutions are actually classical and that D2 u, D 3 u satisfy some bounds which seem also necessary to the argument, especially in view of the application of the Hopf lemma and the Serrin corner lemma in step (g).

(b) Finite time gradient blow-up for suitably concentrated initial data (Lemma 3.4.1) : by using a rescaling argument and known blow-up criteria, we show that the solution blows up in finite time provided the initial data is suitably concentrated in a small ball near the origin.

(c) Local boundary gradient control (Lemma 3.5.1) : if the gradient remains bounded on the boundary near a given boundary point, then the gradient remains also bounded near that point inside the domain, hence it is not a blow-up point. This is proved by a local Bernstein type argument.

(d) Localization of the gradient blow-up set (Lemma 3.5.3) : if an initial data is suitably concentrated near the origin, then the gradient blow-up set is contained in a small neighborhood of the origin. This is proved by constructing comparison functions which provide a control of the gradient on the boundary outside a small neighborhood of the origin, and then applying step (c). One then constructs (Lemma 3.6.1) initial data which also fulfill the assumptions in (a) and (b). This ensures the existence of "well-prepared" initial data and thereby completes the proof of assertion (i) of Theorem 3.1.1.

(e) Nondegeneracy of gradient blow-up (Lemma 3.7.1) : if the solution is only "weakly singular" in a neighborhood of a boundary point, then the singularity is removable. 1 More precisely, we show the existence of m = m(p, q) ∈ (0, 1) such that, for a given point (x 0 , 0) on the flat part of ∂Ω, if u(x, y, t) ≤ c(x)y m near (x 0 , 0) for t close to T , then (x 0 , 0) is not a gradient blow-up point. In view of step (c), it suffices to control the gradient on the boundary near the point (x 0 , 0). This is achieved by constructing special comparison functions, taking the form of "regularizing (in time) barriers".

(f) Verification of a suitable parabolic inequality for an auxiliary function J, of the form J(x, y, t) = u x + kxy -γ u α .

(Proposition 3.8.1). This is the most technical step and gives rise to very long computations. Those computations make use, among many other things, of the singular, Bernstein-type boundary gradient estimate (3.1.7), obtained in [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF]. They use the bound on u t and the lower bound on u y , obtained at step (a) (it is not clear if the latter could be relaxed 2 ).

We note that a similar function J was introduced in [START_REF] Li | Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[END_REF] to treat the semilinear case p = 2. The function in [START_REF] Li | Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[END_REF] was a 2D-modification of a one-dimensional device from [START_REF] Friedman | Blow-up of positive solutions of semilinear heat equations[END_REF], used there to show single point L ∞ blow-up for radial solutions of equations of the form u t -∆ p u = u q (for p = 2, see also [START_REF] Galaktionov | Single point blow-up for N -dimensional quasilinear equations with gradient diffusion and source[END_REF] for p > 2). Although the ideas are related, the calculations here are considerably harder than in [START_REF] Friedman | Blow-up of positive solutions of semilinear heat equations[END_REF][START_REF] Galaktionov | Single point blow-up for N -dimensional quasilinear equations with gradient diffusion and source[END_REF][START_REF] Li | Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[END_REF].

(g) Verification of initial-boundary conditions for the auxiliary function J in a small subrectangle near the origin. This requires a delicate parabolic version of Serrin's corner lemma, which we prove in Appendix 2 (see Proposition 3.11.1).

(h) Derivation of a weakly singular gradient estimate near the origin and conclusion. Steps (d) and (e) imply J ≤ 0 by the maximum principle. By integrating this inequality we obtain an inequality of the form u(x, y, t) ≤ c(x)y k as y → 0, for each small x = 0 and some k > m. In view of step (e), this shows that (0, 0) is the only gradient blow-up point.

Local well-posedness and regularity

In this section we consider the question of local existence and regularity for problem (3.1.1). Actually, we consider the slightly more general problem

u t -∆ p u = |∇v| q in Q T , (3.2
.1) u = g on S T , (3.2.2) u(•, 0) = u 0 in Ω, (3.2.3)
where the boundary data g and initial data u 0 satisfy : g ≥ 0 is the trace on ∂Ω of a regular function in C 2+γ (Ω) for some γ ∈ (0, 1) (3.2.4) and u 0 ∈ W 1,∞ (Ω), u 0 ≥ 0, u 0 = g on ∂Ω.

(3.2.5)

A function u is called a weak super-(sub-) solution of problem (3.2.1)-(3.2.3) on Q T if u(•, 0) ≥ (≤) u 0 in Ω, u ≥ (≤) g on S T , u ∈ C(Ω × [0, T )) ∩ L q (0, T ; W 1,q (Ω)), u t ∈ L 2 (0, T ; L 2 (Ω))
and the integral inequality (i) There exists a time T 0 = T 0 (p, q, Ω, M 1 , g C 2 ) > 0 and a weak solution u of (3.2.1)-(3.2.3) on [0, T 0 ), which moreover satisfies u ∈ L ∞ (0, T 0 ; W 1,∞ (Ω)). Furthermore, ∇u is locally Hölder continuous in Q T 0 .

Q T u t ψ + |∇u| p-2 ∇u • ∇ψ dx dt ≥ (≤) Q T |∇u| q ψ dx dt holds for all ψ ∈ C 0 (Q T ) ∩ L p (0, T ; W 1,p (
(ii) For any τ > 0, the problem (3.2.1)-(3.2.3) has at most one weak solution u such that u ∈ L ∞ (0, τ ; W 1,∞ (Ω)).

(iii) There exists a (unique) maximal, weak solution of (3.2.1)

-(3.2.3) in L ∞ loc ([0, T ); W 1,∞ (Ω))
, still denoted by u, with existence time denoted by T = T max (u 0 ). Then

min Ω u 0 ≤ u ≤ max Ω u 0 in Q T , (3.2
Q T (v 1 -v 2 ) ≤ sup ∂Q T (v 1 -v 2 ).
As one expects, the solution will possess additional regularity if we know that |∇u| remains bounded away from 0. This is made precise by the following result, which is a consequence of regularity theory [START_REF] Ladyzenskaja | Linear and quasi-linear equations of parabolic type[END_REF][START_REF] Lieberman | The first initial-boundary value problem for quasilinear second order parabolic equations[END_REF] for quasilinear uniformly parabolic equations. However, for completeness, we provide a proof in Appendix 1.

Theorem 3.2.2. Under the assumptions of Theorem 3.2.1, suppose also that inf Q T |∇u| > 0.

(i) Then u is a classical solution in Q T and

u ∈ C 2+α,1+α/2 loc (Ω × (0, T )). (3.2.7)
for some α ∈ (0, 1).

(ii) Moreover,

∇u ∈ C 2+β,1+β/2 loc (Ω × (0, T )). (3.2.8)
for some β ∈ (0, 1).

(iii) If the boundary conditions in (3.2.2) depend only on y, then

u x ∈ C 2+β,1+β/2 loc
(Ω × (0, T )).

(3.2.9)

for some β ∈ (0, 1).

Preliminary estimates :

x-symmetry, lower bound on u y and bound on u t Notation. Throughout the paper, we shall use the summation convention on repeated indices, in expressions of the form a ij u ij or a ij u i u j . Also, the letter C will denote positive constants which may vary from line to line, and whose dependence will be indicated if necessary.

Lemma 3.3.1. Let µ > 0 and u 0 ∈ V µ . Assume u 0 is symmetric with respect to the line x = 0, (3.3.1)

∂ x u 0 ≤ 0 in Ω + . (3.3.2)
Then we have u(x, y, t) ≥ µy in Q T .

(3.3.3)

and properties (3.1.9)-(3.1.10) are satisfied.

Proof. Property (3.1.9) is a direct consequence of (3.3.1) and the local-in-time uniqueness. Due to the assumption u 0 ∈ V µ , v = µy is a subsolution of (3.1.1). This implies (3.3.3).

To prove (3.1.10), fix h > 0 and let

u ± = u(x ± h, y, t) for (x, y) ∈ Ω h := {(x, y) ∈ Ω + ; (x + h, y) ∈ Ω} and t ∈ (0, T ).
Owing to (3.1.4) and (3.1.6), we see that (x -h, y) ∈ Ω for all (x, y) ∈ Ω h , so that u -is well defined. The functions u ± are weak solutions of (3.1.1) 1 in Ω h × (0, T ). Also u + ≤ u - at t = 0, due to (3.

u + (x, y, t) = u(x + h, y, t) = µy ≤ u -(x -h, y, t).
We deduce from the comparison principle that u + ≤ u -in Ω h × (0, T ), which implies (3.1.10).

Our next lemma provides a useful supersolution of problem (3.1.1).

Lemma 3.3.2. For 0 < ρ < L 1 , denote Σ ρ = [-ρ/2, ρ/2] × {0} and Σ ρ = ∂Ω \ ((-ρ, ρ) × {0}).
There exist µ 0 = µ 0 (p, q, Ω, ρ) > 0 and a function U ∈ C 2 (Ω), depending on p, q, ρ, with the following properties :

U > 0 on Ω ∪ Σ ρ , (3.3.4) U = 0 on Σ ρ , (3.3.5) |U y | ≤ 1/2 in Ω (3.3.6)

Preliminary estimates :

x-symmetry, lower bound on u y and bound on u t and, for all 0 < µ ≤ µ 0 , the function Ū = µ(y + U ) satisfies

-∆ p Ū ≥ |∇ Ū | q in Ω. (3.3.7)
Proof. Fix a nonnegative function φ ∈ C 3 (R 2 ) such that φ = 1 on Σ ρ and φ = 0 on Σ ρ . We shall look for U under the form U = εV , ε > 0, where V ∈ C 2 (Ω) is the classical solution of the linear elliptic problem

-V xx + (p -1)V yy = 1 in Ω, V = φ on ∂Ω. (3.3.8)
Note that V > 0 in Ω by the strong maximum principle which, along with the boundary conditions in (3.3.8), will guarantee (3.

3.4)-(3.3.5). Let Ū = µ(y + U ). Assume 0 < ε < 1/( V x ∞ + 2 V y ∞ ),
∆ p Ū = |∇ Ū | p-2 ∆ Ū + (p -2) Ūi Ūj Ūij |∇ Ū | 2 = µε|∇ Ū | p-2 V xx + V yy + (p -2) (εV x ) 2 V xx + 2(εV x )(1 + εV y )V xy + (1 + εV y ) 2 V yy (1 + εV y ) 2 + (εV x ) 2 = µε|∇ Ū | p-2 V xx + (p -1)V yy + (p -2) (εV x ) 2 (V xx -V yy ) + 2(εV x )(1 + εV y )V xy (1 + εV y ) 2 + (εV x ) 2 ≤ µε|∇ Ū | p-2 -1 + Cε 2 + Cε
in Ω, with C > 0 depending only on Ω, ρ, p (through V ). We may then choose ε depending only on Ω, ρ, p, such that -∆ p Ū ≥ µε 2 |∇ Ū | p-2 . Next, since q > p, for any µ ≤ µ 0 with µ 0 = µ 0 (p, q, Ω, ρ) > 0 sufficiently small, we have

-∆ p Ū ≥ µε 2 |∇ Ū | p-2 ≥ µε 2(2µ) q+2-p |∇ Ū | q ≥ |∇ Ū | q in Ω.
Based on Lemma 3.3.2, we construct a class of solutions such that u y satisfies a positive lower bound. Lemma 3.3.3. Let 0 < ρ < L 1 and let µ 0 , Ū be given by Lemma 3.3.2. Assume that 0 < µ ≤ µ 0 and u 0 ∈ V µ satisfy

u 0 (x, y) ≤ µ y + cχ (-ρ/2,ρ/2)×(0,L 2 ) in Ω, (3.3.9)
with c = c(p, q, Ω, ρ) > 0 sufficiently small. (i) Then u ≤ Ū in Q T .

(ii) Moreover, using the assumption that

∂ y u 0 ≥ µ/2 in Ω, (3.3.10)
we have that

∂ y u ≥ µ/2 in Q T . (3.3.11)
Proof. (i) Let Ū = µ(y + U ) be given by Lemma 3.3.2. From (3.3.4), we know that c := min

[-ρ/2,ρ/2]×[0,L 2 ]
U > 0.

Under assumption (3.3.9), we thus have u 0 ≤ Ū in Ω. Since u = µy ≤ Ū on S T , we infer from the comparison principle that u ≤ Ū in Q T .

(ii) Set δ 0 = µ/2, fix h > 0 and let Ωh = {(x, y) ∈ Ω; (x, y + h) ∈ Ω}. We observe that u 1 := u(x, y, t) + δ 0 h and u 2 := u(x, y + h, t)

are weak solutions of (3.1.1) 1 in Ωh × (0, T ). Let (x, y) ∈ ∂ Ωh . If (x, y) ∈ ∂Ω, then, by (3.3.3), we have

u 2 (x, y, t) = u(x, y + h, t) ≥ µ(y + h) = u(x, y, t) + µh ≥ u 1 (x, y, t).
Otherwise, we have (x, y) ∈ Ω and (x, y + h) ∈ ∂Ω. So there is a minimal h ∈ (0, h] such that (x, y + h) ∈ ∂Ω. By the mean-value inequality, it follows that, for some θ ∈ (0, 1),

U (x, y) = U (x, y + h) -hU y (x, y + θ h) ≤ |U y (x, y + θ h)|h ≤ h/2,
where we used (3.3.6). Therefore, Ū (x, y) ≤ µ(y + h/2), hence u(x, y + h, t) -u(x, y, t) ≥ µ(y + h) -Ū (x, y) ≥ µh 2 .

We have thus proved that u 2 ≥ u 1 on ∂ Ωh .

(3.3.12)

On the other hand, using (3.3.10) and the fact that u 0 = µy on ∂Ω, it is not difficult to show that y → u 0 (x, y) -δ 0 y is nondecreasing in Ω. (Note that in case Ω is nonconvex, this is not a mere consequence of (3.3.10) alone). It follows that u(x, y + h, 0) ≥ u(x, y, 0) + δ 0 h in Ωh .

(3.3.13)

Owing to (3.3.12)-(3.3.13), we may then apply the comparison principle to deduce that u 2 ≥ u 1 in Ωh × (0, T ). Since h is arbitrary, the desired conclusion (3.3.11) follows immediately.

Assuming that u 0 is sufficiently regular, we also get an estimate on the time derivative.

Lemma 3.3.4. Let µ ≥ 0 and assume that u 0 ∈ V µ ∩ C 2 (Ω). Then

|u t | ≤ C1 := ∆ p u 0 + |∇u 0 | q ∞ in Q T . (3.3.14)
Proof. It is easy to see that v ± (x, y, t) := u 0 (x, y) ± C1 t are respectively super-and subsolution of (3.1.1) in Q T . The comparison principle implies that

u 0 (x, y) -C1 t ≤ u(x, y, t) ≤ u 0 (x, y) + C1 t in Q T . (3.3.15)
Now fix h ∈ (0, T ) and set w ± (x, y, t) := u(x, y, t + h) ± C1 h. By (3.3.15), we have w -(x, y, 0) ≤ u 0 (x, y) ≤ w + (x, y, 0) and it follows that w ± are respectively super-and sub-solution of (3.1.1) in Q T -h . By a further application of the comparison principle, we deduce that |u(x, y, t

+ h) -u(x, y, t)| ≤ C1 h in Q T -h .
Since h is arbitrary, we conclude by dividing by h and sending h → 0.

Finite-time gradient blow-up for concentrated initial data

In this section, by a rescaling argument, we show that the solution of (3.1.1) blows up in finite time provided the initial data is suitably concentrated in a small ball near the origin. For such concentrated initial data, under some additional assumptions, we will show in section 3.6 that the gradient blow-up set is contained in a small neighborhood of the origin.

The following lemma shows in particular that gradient blow-up may occur for initial data of arbitrarily small L ∞ -norm (but the W 1,∞ norm has to be sufficiently large). We note that we do not assume (3.3.10) here, so that in the proof, we work only with weak solutions (in particular, we cannot use the continuity of ∇u up to the boundary). Lemma 3.4.1. Let κ = (q -p)/(q -p + 1). There exists

C 1 = C 1 (p, q) > 0 such that, if ε ∈ (0, min(L 1 , L 2 )), µ ≥ 0 and u 0 ∈ V µ satisfies u 0 (x, y) ≥ C 1 ε κ in B ε/3 (0, ε) ⊂ Ω, (3.4.1)
then T max (u 0 ) < ∞.

Proof. We denote

B r = B r (0, 0) ⊂ R 2 for r > 0. Fix a radially symmetric function h ∈ C ∞ 0 (B 1 ), h ≥ 0, such that supp(h) ⊂ B 1/3 and h ∞ = 1. We consider the following problem    v t -∆ p v = |∇v| q , x ∈ B 1 , t > 0 v(x, y, t) = 0, x ∈ ∂B 1 , t > 0 v(x, y, 0) = v 0 (x, y) := C 1 h(x, y), x ∈ B 1 . (3.4.2)
We know from [START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF] and chapter 2 that, there exists C 0 = C 0 (p, q) > 0 such that if v 0 L 1 ≥ C 0 , then T max (v 0 ) < ∞. Therefore, if we take C 1 = C 1 (p, q) > 0 large enough then ∇v blows up in finite time in L ∞ norm.

Next we use the scale invariance of the equation, considering the rescaled functions

v ε (x, y, t) := ε κ v x ε , y -ε ε , t ε (2q-p)/(q-p+1) .
Pick ε ∈ (0, min(L 1 , L 2 )) and denote Bε = B ε (0, ε), which is included in Ω and tangent to ∂Ω at the origin. Set T ε = ε (2q-p)/(q-p+1) T max (v 0 ) and Tε = min T max (u 0 ), T ε . We shall show that, for each τ ∈ (0, Tε ),

∇v ε L ∞ (Qτ ) ≤ max ∇v ε (•, 0) ∞ , ∇u L ∞ (Qτ ) .
(3.4.3)

Since gradient blow-up occurs in finite time T ε for v ε , this will guarantee

T max (u 0 ) ≤ T ε < ∞.
First observe that v ε solves (3.4.2) in Bε ×(0, T ε ), with initial data v ε (x, y, 0) ≤ u 0 (x, y), due to (3.4.1). It follows from the comparison principle that v ε ≤ u in Bε × (0, Tε ). In particular, for each h ∈ (0, ε) and t ∈ (0, Tε ), since u(0, 0, t) = 0, we get that

v ε (0, h, t) h ≤ u(0, h, t) h ≤ ∇u(•, t) ∞ . (3.4.4)
We shall next show that the first quantity in (3.4.4) can be suitably bounded from below in terms of the sup norm of ∇v ε . Fix h ∈ (0, ε) and let Bh

ε := B ε (0, ε -h) Since v h ε (x, y, t) := v ε (x, y + h, t
) is a solution of (3.4.2) in Bh ε , it follows from the comparison principle (see Remark 3.2.1) that, for any 0 < τ < Tε , sup

( Bε∩ Bh ε )×(0,τ ) |v ε (x, y + h, t) -v ε (x, y, t)| ≤ max sup Bε∩ Bh ε |v h ε (x, y, 0) -v ε (x, y, 0)|, sup ∂( Bε∩ Bh ε )×(0,τ ) |v h ε (x, y, t) -v ε (x, y, t)| .
(3.4.5)

We claim that, for any 0 < t < T ε , sup

∂( Bε∩ Bh ε ) |v h ε (x, y, t) -v ε (x, y, t)| ≤ v ε (0, h, t). (3.4.6)
First consider the case (x, y) ∈ ∂ Bε . Then (x, y + h) -(0, ε) ≥ (0, h) -(0, ε) , due to

x 2 + (y + h -ε) 2 -(h -ε) 2 = ε 2 -(y -ε) 2 + (y + h -ε) 2 -(h -ε) 2 = 2hy ≥ 0.
Since v ε is radially symmetric and non-increasing with respect to the point (0, ε), we deduce that ) that, for each 0 < h < ε -y and 0 < t < τ < Tε ,

|v ε (x, y + h, t) -v ε (x, y, t)| = v ε (x, y + h, t) ≤ v ε (0, h, t). Next consider the case (x, y) ∈ ∂ Bh ε that is, (x, y + h) ∈ ∂ Bε . Then (x, y) -(0, ε) ≥ (0, h) -(0, ε) , due to x 2 + (y -ε) 2 -(h -ε) 2 = ε 2 -(y + h -ε) 2 + (y -ε) 2 -(h -ε) 2 = 2h(2ε -h -y) ≥ 0. Therefore, |v ε (x, y + h, t) -v ε (x, y, t)| = v ε (x, y, t) ≤ v ε (0, h, t),
|v ε (0, y + h, t) -v ε (0, y, t)| h ≤ max ∇v ε (•, 0) ∞ , ∇u L ∞ (Qτ )
hence, letting h → 0,

|∂ y v ε (0, y, t)| ≤ max ∇v ε (•, 0) ∞ , ∇u L ∞ (Qτ ) .
For each 0 < τ < Tε , taking supremum over y ∈ [0, ε) and t ∈ (0, τ ) and using the fact that v ε is radially symmetric, we obtain (3.4.3). This concludes the proof of the lemma.

Local boundary control for the gradient and localization of the gradient blow-up set

For simplicity we shall here assume (3.3.9) and (3.3.10), so as to have the continuity of ∇u up to the boundary (although one might possibly relax this assumption at the expense of additional work). Lemma 3.5.1. Let ρ, µ, u 0 be as in Lemma 3.3.3 (ii) and let (x 0 , y 0 ) ∈ ∂Ω. If there exist M 0 , R > 0 such that

|∇u| ≤ M 0 in (B R (x 0 , y 0 ) ∩ ∂Ω) × [0, T max (u 0 )), (3.5.1)
then (x 0 , y 0 ) is not a gradient blow-up point.

The proof is based on a local Bernstein technique. For (x 0 , y 0 ) ∈ ∂Ω, R > 0 and given α ∈ (0, 1), we may select a cut-off function η ∈ C 2 (B R (x 0 , y 0 )), with 0 < η ≤ 1, such that

η = 1 on B R/2 (x 0 , y 0 ), η = 0 on ∂B R (x 0 , y 0 ) and |∇η| ≤ CR -1 η α |D 2 η| + η -1 |∇η| 2 ≤ CR -2 η α on B R (x 0 , y 0 ), (3.5.2)
where C = C(α) > 0 (see e.g. [110] for an example of such function). Also, for 0 < t 0 < τ < T = T max (u 0 ), we denote

Q t 0 τ,R = (B R (x 0 , y 0 ) ∩ Ω) × (t 0 , τ ).
For the proof of Lemma 3.5.1, we then rely on the following lemma from chapter 2 (cf. [7, Lemma 3.1] ; it was used there to derive upper estimates on |∇u| away from the boundary). Lemma 3.5.2. Let µ ≥ 0 and u 0 ∈ V µ . Let (x 0 , y 0 ) ∈ ∂Ω, R > 0, 0 < t 0 < τ < T and choose α = (q + 1)/(2q -p + 2). Denote w = |∇u| 2 and z = ηw. Then z ∈ C 2,1 (Q t 0 τ,R ) and satisfies the following differential inequality

Lz + C 2 z 2q-p+2 2 ≤ C 3 u 0 ∞ t 0 2q-p+2 q + C 3 R -2q-p+2 q-p+1 , (3.5.3)
where

C i = C i (p, q) > 0, Lz = z t -Āz -H • ∇z, (3.5.4) Āz = |∇u| p-2 ∆z + (p -2)|∇u| p-4 (∇u) t D 2 z∇u, (3.5.5)
and H is defined by

H := (p -2)w p-4 2 ∆u + (p -2)(p -4) 2 w p-6 2 ∇u • ∇w + qw q-2 2 ∇u + p -2 2 w p-4 2 ∇w. (3.5.6)
Proof of Lemma 3.5.1 Let t 0 = T /2 < τ < T and set

M 1 := sup 0≤t≤t 0 ∇u L ∞ < ∞.
By Lemma 3.3.3(ii) and Theorem 3.2.2, we know that ∇u is a continuous function on Ω × (0, T ), hence z ∈ C(Q t 0 τ,R ). Therefore, unless z ≡ 0 in Q t 0 τ,R , z must reach a positive maximum at some point (x 1 , y 1 , t 1 ) ∈ Q t 0 τ,R . Since

z = 0 on ∂B R (x 0 , y 0 ) ∩ Ω × [t 0 , τ ], (3.5.7)
we deduce that either (x 1 , y 1 ) ∈ B R (x 0 , y 0 ) ∩ Ω or (x 1 , y 1 ) ∈ B R (x 0 , y 0 ) ∩ ∂Ω.

• If t 1 = t 0 , then z(x 1 , y 1 , t 1 ) ≤ ∇u(t 0 ) 2 L ∞ ≤ M 2 1 .
(3.5.8)

• If t 0 < t 1 ≤ τ and (x 1 , y 1 ) ∈ B R (x 0 , y 0 ) ∩ ∂Ω, then, by (3.5.1), z(x 1 , y 1 , t 1 ) ≤ M 2 0 .
(3.5.9)

• Next consider the case t 0 < t 1 ≤ τ and (x 1 , y 1 ) ∈ B R (x 0 , y 0 ) ∩ Ω. Then we have ∇z(x 1 , y 1 , t 1 ) = 0, z t (x 1 , y 1 , t 1 ) ≥ 0 and D 2 z(x 1 , y 1 , t 1 ) ≤ 0, and therefore Lz ≥ 0. Using (3.5.3) we arrive at

C 2 z(x 1 , y 1 , t 1 ) 2q-p+2 2 ≤ C 3 u 0 ∞ t 0 2q-p+2 q + C 3 R -2q-p+2 q-p+1 that is, z(x 1 , y 1 , t 1 ) ≤ C u 0 ∞ t 0 1 q + CR -1 q-p+1 =: M 2 > 0. ( 3
max Q t 0 τ,R z ≤ M 2 3 , with M 3 = max {M 0 , M 1 , M 2 }. Since z = |∇u| 2 in B R/2 (x 0 , y 0 ) ∩ Ω × (t 0 , τ ) and τ ∈ (t 0 , T ) is arbitrary, we get |∇u| ≤ M 3 in B R/2 (x 0 , y 0 ) ∩ Ω × (t 0 , T ),
and we conclude that (x 0 , y 0 ) is not a gradient blow-up point.

By combining Lemmas 3.3.3 and 3.5.1, we can now easily obtain a class of initial data whose possible gradient blow-up set is contained in a small neighborhood of the origin. (3.5.11) But (3.5.11) easily follows from a comparison with the function Ū provided in Lemma 3.3.2. Indeed, under the assumptions of Lemma 3.3.3(i), we already know that u ≤ Ū in Q T . Also, u = µy = Ū on Σ ρ × (0, T ). From this, along with (3.3.3), it follows that

∂ ν Ū ≤ ∂ ν u ≤ µ ∂ ν y on Σ ρ × (0, T ).
(3.5.12) From (3.5.12) and (3.1.1) 2 , we get

|∇u| 2 ≤ µ 2 + |∂ ν u| 2 ≤ C on Σ ρ × (0, T ),
hence (3.5.11), and the lemma is proved.

Existence of well-prepared initial data : proof of Theorem 3.1.1(i)

We need to construct initial data meeting the requirements from sections 3-5. This will be achieved in the following lemma. Let us fix an even function ϕ ∈ C ∞ (R) such that sϕ (s) ≤ 0, with ϕ(s) = 1 for |s| ≤ 1/3 0 for |s| ≥ 2/3.

(3.6.1) Lemma 3.6.1. Let κ = (q -p)/(q -p + 1) and let C 1 = C 1 (p, q) > 0 be given by Lemma 3.4.1. For ε ∈ (0, min(L 1 , L 2 /2)), define

ψ ε (y) =      ϕ y -ε ε for 0 ≤ y ≤ ε ϕ y -ε L 2 for y ≥ ε (3.6.2)
and let u 0 be defined by

u 0 (x, y) = µy + C 1 ε κ ϕ x ε ψ ε (y).
Next fix 0 < ρ < L 1 and let µ 0 = µ 0 (p, q, Ω, ρ) > 0 and c = c(p, q, Ω, ρ) > 0 be given by Lemmas 3.3.2 and 3.3.3. For any µ ∈ (0, µ 0 ], there exists ε 0 = ε 0 (p, q, Ω, µ, ρ) > 0 such that, for all ε ∈ (0, ε 0 ], the function u 0 ∈ V µ and satisfies u 0 is symmetric with respect to the line x = 0, (3.6.3)

∂ x u 0 ≤ 0 in Ω + , (3.6.4) ∂ y u 0 ≥ µ/2 in Ω, (3.6.5) u 0 (x, y) ≤ µ y + cχ (-ρ/2,ρ/2)×(0,L 2 ) in Ω, (3.6.6) u 0 (x, y) ≥ C 1 ε κ in B ε/3 (0, ε) ⊂ Ω.
(3.6.7)

Proof. Assume ε ≤ min(L 1 , L 2 /12). Then ψ ε (y) = 0 for y ∈ [ ε 3 , 3L 2 4 ] (3.6.8) (indeed, y ≥ 3L 2 4 implies y-ε L 2 ≥ 3 4 -1 12 = 2
3 ) and therefore u 0 ∈ V µ . Properties (3.6.3)-(3.6.4) are clear by the choice of ϕ.

To check (3.6.5), we note that

∂ y u 0 = µ + C 1 ε κ ϕ x ε ψ ε (y).
For 0 ≤ y ≤ ε, we have ψ ε (y) ≥ 0, hence ∂ y u 0 ≥ µ. Whereas, for y ≥ ε, we have

ψ ε (y) = L -1 2 ϕ (y -ε)/L 2 ≥ -L -1 hence ∂ y u 0 ≥ µ -C 1 ε κ L -1 2 ϕ ∞ ≥ µ/2 whenever ε κ ≤ µL 2 /(2C 1 ϕ ∞ ).
As for (3.6.6), if C 1 ε κ ≤ µc and ε ≤ ρ/2, it immediately follows from ϕ, ψ ε ≤ 1 and supp(ϕ) ⊂ (-1, 1). Finally, since ϕ(x/ε) = 1 for |x| ≤ ε/3 and ψ ε (y) = 1 for |y -ε| ≤ ε/3, we have (3.6.7). The lemma is proved.

Proof of Theorem 3.1.1(i) Let µ and u 0 be as in Lemma 3.6.1.

• The fact that T max (u 0 ) < ∞ follows from Lemma 3.4.1.

• Next, we have GBU S(u 0 ) ⊂ [-ρ, ρ] × {0} as a consequence of Lemma 3.5.3.

• Properties (3.1.9)-(3.1.10) follow from Lemma 3.3.1.

• Finally, property (3.1.11) is a consequence of Lemma 3.3.3(ii). This proves the assertion.

Nondegeneracy of gradient blow-up points

In this section, we show that if u is only "weakly singular" in a neighborhood of a boundary point (x 0 , 0), then the singularity is removable. Lemma 3.7.1. Let ρ, µ, u 0 be as in Lemma 3.3.3(ii) and let x 0 ∈ (-L 1 , L 1 ). There exist c 0 = c 0 (p, q) > 0 such that, if u 0 ∈ V µ with T := T max (u 0 ) < ∞ and u(x, y) ≤ c 0 y (q-p)/(q-p+1)

in

(B R (x 0 , 0) ∩ Ω) × [t 0 , T ), (3.7.1)
for some R > 0 and t 0 ∈ (0, T ), then (x 0 , 0) is not a gradient blow-up point.

Proof. Let x 0 ∈ (-L 1 , L 1 ). Then for some constants r ∈ (0, R) and d ∈ (0, L 2 ), we have that

ω 1 := (x, y) ∈ R 2 ; |x -x 0 | < r, 0 < y < d ⊂ B R (x 0 , 0) ∩ Ω.
Setting β = 1/(q -p + 1), we define the comparison function

v = v(x, y, t) = εyV -β in Q := ω 1 × (t 0 , T ) with V = y + η r 2 -(x -x 0 ) 2 (t -t 0 ),
where η, ε > 0 are to be determined later. We compute, in Q,

v t = -εβηy(r 2 -(x -x 0 ) 2 )V -β-1 , v x = 2εβηy(x -x 0 )(t -t 0 )V -β-1 , v y = εV -β -εβyV -β-1 = εV -β 1 -β y V , v xx = 2εβηy(t -t 0 )V -β-1 -4εβη 2 y(x -x 0 ) 2 (t -t 0 ) 2 (-β -1)V -β-2 = 2εβη(t -t 0 )V -β-1 y + 2(β + 1)η(x -x 0 ) 2 (t -t 0 ) y V , v yy = -2εβV -β-1 + εβ(β + 1)yV -β-2 = εβV -β-1 -2 + (β + 1) y V , v xy = 2εβη(x -x 0 )(t -t 0 )V -β-1 -2εβ(β + 1)ηy(x -x 0 )(t -t 0 )V -β-2 = 2εβη(x -x 0 )(t -t 0 )V -β-1 1 -(β + 1) y V .
Noting that β < 1 and y V ≤ 1, we see that, in Q,

0 ≤ v xx ≤ 2εβηT d + 2(β + 1)ηr 2 T V -β-1 , |v xy | ≤ 2εβηrT V -β-1 and v yy ≤ εβ(β -1)V -β-1 < 0.
It follows that

∆ p v = |∇v| p-2 ∆v + (p -2) v i v j v ij |∇v| 2 ≤ |∇v| p-2 (p -1)v xx + v yy + (p -2)|v xy | ≤ εβ|∇v| p-2 V -β-1 2(p -1)ηT d + 2(β + 1)ηr 2 T + (β -1) + 2(p -2)ηrT .
On the other hand, we have

|∇v| ≥ |v y | ≥ ε(1 -β)V -β ≥ ε(1 -β)(d + ηT r 2 ) -β , hence v t ≥ -εβηdr 2 V -β-1 ≥ -εβ|∇v| p-2 V -β-1 ηdr 2 ((1 -β)ε) 2-p (d + ηT r 2 ) (p-2)β .
Therefore,

v t -∆ p v ≥ εβ|∇v| p-2 V -β-1 × -ηdr 2 ((1 -β)ε) 2-p (d + ηT r 2 ) (p-2)β -2(p -1)ηT d + 2(β + 1)ηr 2 T -2(p -2)ηrT + (1 -β) .
Since also

|v x | ≤ 2εβηrT V -β , |v y | ≤ εV -β ,
if we choose η = η(p, q, d, r, T, ε) > 0 small enough, we get that, in Q,

|∇v| ≤ 2εV -β and v t -∆ p v ≥ εβ(1 -β) 2 |∇v| p-2 V -β-1 , hence v t -∆ p v ≥ εβ(1 -β) 2 |∇v| p-2 (2ε) -β+1 β |∇v| β+1 β = β(1 -β) 4 (2ε) -1 β |∇v| q ,
due to β = 1/(q -p + 2). If ε = ε 0 (p, q) > 0 is small enough, we thus obtain

v t -∆ p v ≥ |∇v| q . (3.7.2)
Now we shall check the comparison on the parabolic boundary of ω 1 × (t 0 , T ). On

ω 1 × {t 0 }, choosing c 0 = 2 -β ε 0 , we have u ≤ c 0 y 1-β = 2 -β ε 0 y 1-β ≤ v. (3.7.3)
On the lateral boundary part {(x, y) ∈ R 2 ; |x -x 0 | = r, 0 ≤ y ≤ d} × (t 0 , T ), inequality (3.7.3) holds also. On the surface {(x, y) ∈ R 2 ; |x -x 0 | ≤ r, y = 0} ⊂ ∂Ω, we have for t 0 < t < T , u(., ., t) = v(., ., t) = 0

Finally, on {(x, y) ∈ R 2 ; |x -x 0 | ≤ r, y = d} × (t 0 , T ), assuming in addition that η satisfies η ≤ dT -1 r -2 , we get

u ≤ c 0 d 1-β ≤ ε 0 d(d + ηr 2 T ) -β ≤ v.
Using the comparison principle, we get that

u ≤ v in ω 1 × (t 0 , T ). (3.7.4)
This implies that

|u y | ≤ ε η(r 2 -|x -x 0 | 2 )(t -t 0 ) -β ≤ M 0 on (B r/2 ((x 0 , 0)) ∩ ∂Ω) × ((t 0 + T )/2, T )
for some constant M 0 > 0. Lemma 3.7.1 is then a direct consequence of Lemma 3.5.1.

The auxiliary function J and the proof of singlepoint gradient blow-up

In all this section, we fix ρ, x 1 with

0 < ρ < x 1 < L 1 (3.8.1)
and we assume that µ and u 0 ∈ V µ ∩ C 2 (Ω) satisfy the assumption of Theorem 3.1.1(ii) i.e., the corresponding solution of (3.1.1) fulfills properties (3.1.8)-(3.1.11). We denote as before T = T max (u 0 ). We consider the auxiliary function

J(x, y, t) := u x + c(x)d(y)F (u), with    F (u) = u α , c(x) = kx, k > 0, d(y) = y -γ and 1 < α < 1 + q -p, γ = (1 -2σ)(α -1), (3.8.2)
where

0 < σ < 1 2(q -p + 1) (3.8.3)
is fixed. Letting D = (0, x 1 ) × (0, y 1 ), our goal is to use a comparison principle to prove that

J ≤ 0 in D × (T /2, T ),
provided α > 1 is chosen close enough to 1 (hence making γ > 0 small) and y 1 ∈ (0, L 2 ) and k > 0 are chosen sufficiently small.

Parabolic inequality for the auxiliary function J

By the regularity of u (see Theorem 3.2.2), we have

J ∈ C 2,1 (Q T ).
A key step is to derive a parabolic inequality for J. To this end, we define the operator

PJ := J t -|∇u| p-2 ∆J -(p -2)|∇u| p-4 D 2 J ∇u, ∇u + H • ∇J + AJ, (3.8.4)
where the functions H = H(x, y, t) and A = A(x, y, t) are given by formulae (3.9.9)-(3.9.12) below.

Proposition 3.8.1. Assume (3.8.1), (3.8.3) and let µ, u 0 satisfy the assumption of Theorem 3.1.1(ii). There exist α, γ satisfying (3.8.2), y 1 ∈ (0, L 2 ) and k 0 > 0, all depending only on p, q, Ω, µ, u 0 C 2 , σ, such that, for any k ∈ (0, k 0 ], the function J satisfies

PJ ≤ 0 in D × (T /2, T ). (3.8.5)
Moreover,

H, A ∈ C(D × (0, T )) and A ∈ L ∞ (D × (T /2, τ )) for each τ ∈ (T /2, T ). (3.8.6)
The proof of Proposition 3.8.1 is very long and technical. In order not to disrupt the main line of argument, we postpone it to section 9 and now present the rest of the proof of Theorem 3.1.1(ii).

Boundary conditions for the auxiliary function J

The verification of the appropriate boundary and initial conditions for the function J depends on an essential way on the applicability to u x of the Hopf boundary lemma at the points (x 1 , 0) and (0, y 1 ), up to t = T . To this end, besides the nondegeneracy of problem (3.1.1), guaranteed by (3.1.11), we also need the following local regularity lemma, which ensures that D 2 u remains bounded up to t = T away from the gradient blow-up set. Lemma 3.8.1. Let ρ ∈ (0, L 1 ) and let µ, u 0 satisfy the assumption of Theorem 3.1.1(ii). Let ω ⊂ ω ⊂ Ω be such that dist(ω , Ω \ ω) > 0 and t 0 ∈ (0, T ). If

sup ω×(0,T ) |∇u| < ∞, then sup ω ×(t 0 ,T ) |D 2 u| < ∞.
Proof. Introduce an intermediate domain ω with ω ⊂ ω ⊂ ω, such that dist(ω , Ω\ω) > 0 and dist(ω , Ω \ ω ) > 0. Write the PDE in (3.1.1) as

-∇ • (|∇| p-2 ∇u) = |∇u| q -u t .
Using |∇u| ≥ ∂ y u ≥ µ/2 (cf. (3.1.11)) and the boundedness of u t in Q T (cf. Lemma 3.3.4), it follows from the elliptic estimate in [80, Theorem V.5.2] that there exists θ ∈ (0, 1)

such that ∇u(•, t) C θ (ω ) ≤ C, t 0 /2 ≤ t < T.
The boundedness of u t in Q T and the interpolation result in [80, Lemma II.3.1] then guarantee the estimate

∇u(x, y, •) C β ([t 0 /2,T )) ≤ C, (x, y) ∈ ω , with a ij (x, y, t) = |∇u| p-2 δ ij + (p -2) u i u j |∇u| 2 , B(x, y, t) = (p -2)|∇u| p-4 ∇u∆u + (p -2)(p -4)|∇u| p-6 D 2 u ∇u, ∇u ∇u + q|∇u| q-2 ∇u + 2(p -2)|∇u| p-4 (D 2 u ∇u). Fix ρ < x 3 < x 2 < x 1 . Since GBU S ⊂ [-ρ, ρ] × {0}, we have |∇u| ≤ C in Ω\ {(-x 3 , x 3 ) × (0, y 1 /3)} × (0, T ).
It follows from Lemma 3.8.1 that

|D 2 u| ≤ C in Ω\ {(-x 2 , x 2 ) × (0, y 1 /2)} × (T /4, T ), hence |B| ≤ C in Ω\ {(-x 2 , x 2 ) × (0, y 1 /2)} × (T /4, T ).
Moreover, the matrix A(x, y, t) is uniformly elliptic (cf. (3.8.7)). We may thus apply the strong maximum principle and the Hopf boundary point Lemma [98, Theorem 6 p. 174], to get

u x ≤ -c 1 y on {x 1 } × (0, y 1 ) × (T /2, T ), u x ≤ -c 1 x on (0, x 1 ) × {y 1 } × (T /2, T ).
Also, since x 1 > ρ and u(x, 0, t) = 0, we get that, for some c 2 > 0, u ≤ c 2 y on {x 1 } × (0, y 1 ) × (T /2, T ).

Consequently, using α > γ + 1 and (3.2.6), we have for 0 < k ≤ k 1 (y 1 ) sufficiently small

J(x, y 1 , t) ≤ -c 1 x + kxy -γ 1 u 0 α ∞ ≤ 0 on (0, x 1 ) × {y 1 } × (T /2, T ), J(x 1 , y, t) ≤ -c 1 y + kx 1 y α-γ c α 2 ≤ 0 on {x 1 } × (0, y 1 ) × (T /2, T ).
This, along with (3.8.10)-(3.8.11), proves (3.8.9).

Initial conditions for J

(ii)

. There exists k 2 > 0 such that, for any k ∈ (0, k 2 ], the function J satisfies

J(x, y, T /2) ≤ 0 in [0, x 1 ] × [0, L 2 ].
The proof relies on a parabolic version of the Serrin corner lemma applied to u x . This is provided by Proposition 3.11.1, which we state and prove in Appendix 2.

Proof. The function z = u x satisfies equation (3.8.12). We shall apply Proposition 3.11.1 to this equation, with

τ 1 = T /4, τ 2 = 3T /4, X 1 = x 1 , Y 1 = L 2 , X1 = L 1 , Ŷ1 = 2L 2 .
We thus need to check the assumption (3.11.2). Let us denote DT = (0, ĉbX 1 )×(0, Ŷ1 )×(T /4, 3T /4).

For x = 0 or y = 0, we have u x = 0, hence a 12 = a 21 = (p -2)|∇u| p-2 u x u y = 0. Due to the regularity of u (cf. (3.2.7)), we deduce that

a 12 + a 21 ≥ -C(x ∧ y) in DT . ( 3
B 1 = (p -2)|∇u| p-4 (∆u)u x + (p -2)(p -4)|∇u| p-6 D 2 u ∇u, ∇u u x + q|∇u| q-2 u x + 2(p -2)|∇u| p-4 u xx u x + u xy u y ≥ -Cx in DT . (3.8.14)
Next, for y = 0 and 0 < x < X1 , we have u t = 0 and u x = u xx = 0. Recalling (3.2.7), we thus have

(u y ) q = |∇u| q = u t -∆ p u = -|∇u| p-2 ∆u + (p -2) D 2 u ∇u, ∇u |∇u| 2 = -|∇u| p-2 u xx + u yy + (p -2) u xx u 2 x + 2u xy u x u y + u yy u 2 y u 2 x + u 2 y = -(p -1)(u y ) p-2 u yy .
It follows that, for 0 < x < X1 and t ∈ [T /4, 3T /4],

B 2 (x, 0, t) = (p -2)(p -4)|∇u| p-6 u xx u 2 x + 2u xy u x u y + u yy u 2 y u y + (p -2)|∇u| p-4 (u xx + u yy )u y + q|∇u| q-2 u y + 2(p -2)|∇u| p-4 u yx u x + u yy u y = (p -2)(p -4)(u y ) p-3 u yy + (p -2)(u y ) p-3 u yy + q(u y ) q-1 + 2(p -2)(u y ) p-3 u yy = (p -2)(p -1)(u y ) p-3 u yy + q(u y ) q-1 = (q + 2 -p)(u y ) q-1 ≥ (q + 2 -p)(µ/2) q-1 > 0.
Therefore, owing to (3.2.7), there exists η > 0 such that In view of (3.8.13)-(3.8.15), we may thus apply Proposition 3.11.1 to deduce

B 2 (x, y, t) ≥ 0 on (0, X1 ) × [0, η] × [T /4, 3T /4],
u x (x, y, T /2) ≤ -c 3 xy in [0, x 1 ] × [0, L 2 ].
Let C := ∇u(•, T /2) ∞ . Since α > γ + 1, we get that, for k ∈ (0, k 2 ] with k 2 > 0 small enough,

J(x, y, T /2) ≤ -c 3 xy + kxC α y α-γ ≤ [kC α L α-γ-1 2 -c 3 ]xy ≤ 0 in [0, x 1 ] × [0, L 2 ].

Proof of Theorem 3.1.1(ii)

Let α, γ, y 1 , k 0 be given by Proposition 3.8.1 and let k 1 , k 2 be given by Lemmas 3.8.2-3.8.3. We take k = min(k 0 , k 1 , k 2 ). By these results and the maximum principle, we have

J ≤ 0 in D × (T /2, T ). (3.8.16)
Integrating inequality (3.8.16) over (0, x) for 0 < x < x 1 , with fixed y, we get that

u ≤ Cx -2/(α-1) y 1-2σ in D × (T /2, T ),
where

C = C(α, k, σ) > 0. Using that 1-2σ > q -p q -p + 1
, it follows from the nondegeneracy property in Lemma 3.7.1 that no point (x 0 , 0) with 0 < |x 0 | ≤ ρ can be a gradient blow-up point. In view of (3.1.8), we conclude that GBU S(u 0 ) = {(0, 0)}.

3.9 Proof of the main parabolic inequality (Proposition 3.8.1)

The proof is quite technical. For sake of clarity, some of the intermediate calculations will be summarized in Lemma 3.9.1 and 3.9.2 below.

We first compute

J t = u xt + cdF (u)u t = (∆ p u) x + (|∇u| q ) x +cdF ∆ p u (0p) + cdF |∇u| q (0q)
. and

(∆ p u) x = |∇u| p-2 ∆(u x ) + (p -2)∆u|∇u| p-4 ∇u • ∇u x + (p -2)|∇u| p-4 D 2 u x ∇u, ∇u + (p -2)(p -4)|∇u| p-6 ∇u • ∇u x D 2 u ∇u, ∇u + 2(p -2)|∇u| p-4 D 2 u ∇u, ∇u x .
Using that u x = J -cdF (u), we write

∇u x = ∇J -cdF ∇u -F c d d c , D 2 u x = D 2 J -cdF D 2 u -cdF u 2 x u x u y u x u y u 2 y -F (u) c d c d c d d c -F (u) 2c du x cd u x + c du y cd u x + c du y 2cd u y , ∆u x = T race(D 2 u x ) = ∆J -cdF ∆u -cdF |∇u| 2 -F [c d + d c] -2F c dJ + 2F F c cd 2 -2F (u)d cu y , D 2 u x , ∇u, ∇u = D 2 J, ∇u, ∇u -cdF |∇u| 4 -cdF D 2 u, ∇u, ∇u -F c du 2 x -2F c d u x u y -F cd u 2 y -2F |∇u| 2 (cd u y + dc u x )
and

D 2 u, ∇u, ∇u x = D 2 u, ∇u, ∇J -cdF D 2 u, ∇u, ∇u -F c d∇u • ∇J + cc d 2 F F |∇u| 2 + d 2 F 2 (c ) 2 J -d 2 F 2 (c ) 2 cdF + c cd dF 2 u y -F d c(u x u xy + u y u yy ).
Therefore,

(∆ p u) x = |∇u| p-2 ∆J -cdF ∆u -cdF |∇u| 2 -2c dF J +2cc d 2 F F -2d cF u y -F (c d + d c) + (p -2)∆u|∇u| p-4 ∇u • ∇J -cdF |∇u| 2 -u y cd F -c dF J + c cd 2 F 2 + (p -2)|∇u| p-4 D 2 J∇u, ∇u -cdF D 2 u∇u, ∇u -cdF |∇u| 4 -2c dF |∇u| 2 J + 2c cd 2 F F |∇u| 2 -2cd F u y |∇u| 2 -c dF (u x ) 2 -2c d F Ju y + 2c cd dF 2 u y -d cF (u y ) 2 + (p -2)(p -4)|∇u| p-6 D 2 u∇u, ∇u [∇u • ∇J -cdF |∇u| 2 -u y cd F -c dF J + c cd 2 F 2 + 2(p -2)|∇u| p-4 D 2 u ∇u, ∇J -cdF D 2 u ∇u, ∇u -c dF ∇u • ∇J + c cd 2 F F |∇u| 2 + (c d) 2 F 2 J -(c d) 2 F 3 cd + c d cdF 2 u y -d cF u yy u y -d cF ∇J • L + d dc 2 F F u y J -d d 2 c 3 F F 2 u y + (d cF ) 2 J -(d cF ) 2 cdF ,
where L = 0 u x . This can be rewritten as

(∆ p u) x = |∇u| p-2 ∆J + (p -2)|∇u| p-4 D 2 J ∇u, ∇u + H 1 • ∇J + A 1 (x, y, t)J -F |∇u| p-2 [c d + d c] -(p -2)F |∇u| p-4 c du 2 x + d cu 2 y -2(p -2)cdF |∇u| p-4 [(c dF ) 2 + (d cF ) 2 ] -(p -1)cdF |∇u| p +4(p -2)F 2 c cd d|∇u| p-4 u y    (1) ≤ 0 +(4p -6)c cd 2 F F |∇u| p-2 -2(p -1)cd F |∇u| p-2 u y -2(p -2)d d 2 F F 2 c 3 |∇u| p-4 u y (2) ≥ 0 -(p -1)cdF ∆ p u (3) -2(p -2)d cF |∇u| p-4 u y u yy (4) -(p -2)cd F u y |∇u| p-4 ∆u + (p -4)|∇u| p-6 D 2 u∇u, ∇u (5)
+(p -2)c cd 2 F 2 |∇u| p-4 ∆u + (p -4)|∇u| p-6 D 2 u∇u, ∇u (6)

,

To deal with (4), we set

u yy = u t -|∇u| q -∇u x • M w ,
where

M := |∇u| p-2 + (p -2)|∇u| p-4 u 2 x 2(p -2)|∇u| p-4 u x u y and w = |∇u| p-2 + (p -2)|∇u| p-4 u 2 y .
Since u x = J -cdF , we get

∇u x • M = -cdF J |(p -1)∇u| p-2 + (p -2)|∇u| p-4 u 2 y -2(p -2)cd F u y |∇u| p-4 J +2(p -2)c 2 d dF 2 u y |∇u| p-4 + c 2 d 2 F F |(p -1)∇u| p-2 + (p -2)|∇u| p-4 u 2 y +∇J • M -c dF |∇u| p-2 + (p -2)|∇u| p-4 u 2 x .
It follows that

u yy = u t -|∇u| q w - ∇J • M w + cdF J (p -1)|∇u| p-2 + (p -2)|∇u| p-4 u 2 y w + 2(p -2)cd F u y |∇u| p-4 J w + c dF [|∇u| p-2 + (p -2)|∇u| p-4 u 2 x ] w - c 2 d 2 F F (p -1)|∇u| p-2 + (p -2)|∇u| p-4 u 2 y w - 2(p -2)c 2 d dF 2 u y |∇u| p-4 w .
Now, to treat the contribution of u yy in (5) and (6), we set N = u 2

x 2u x u y and rewrite

|∇u| p-4 ∆u + (p -4)|∇u| p-6 D 2 u ∇u, ∇u = ∆ p u |∇u| 2 -2|∇u| p-6 D 2 u ∇u, ∇u = u t -|∇u| q |∇u| 2 -2|∇u| p-6 ∇u x • N + u 2 y u yy .
We have

∇u x • N = ∇J • N -cdF J u 2 x + 2u 2 y + c 2 d 2 F F u 2 x + 2u 2 y -c dF u 2 x -2cd F Ju y + 2d dc 2 F 2 u y .
The expression in (4) then becomes

(4) = 2(p -2)d cF |∇u| p-4 u y M • ∇J w (4 ∇ ) - 2(p -2)c 2 d dF F |∇u| p-4 u y (p -1)|∇u| p-2 + (p -2)|∇u| p-4 u 2 y J w - 4(p -2) 2 (d cF ) 2 |∇u| p-4 u 2 y |∇u| p-4 J w      (4 J ) + 2d cF (p -2)|∇u| p-4 |∇u| q u y w + 2(p -2)c 3 d d 2 F F 2 |∇u| p-4 u y (p -1)|∇u| p-2 + (p -2)|∇u| p-4 u 2 y w      (4 -) ≤ 0 + 4(p -2) 2 (d cF ) 2 cdF |∇u| p-4 u 2 y |∇u| p-4 w - 2(p -2)d dc cF 2 u y |∇u| p-4 [|∇u| p-2 + (p -2)|∇u| p-4 u 2 x ] w      (4 + ) ≥ 0 - 2d cF (p -2)|∇u| p-4 u t u y w (4 t ).
The other two terms can be rewritten as

(5) = 2cd F (p -2)u y |∇u| p-6 N • ∇J - 2(p -2)cd F u 3 y |∇u| p-6 M • ∇J w (5 ∇ ) -2(p -2)c 2 d dF F u y |∇u| p-6 u 2 x + 2u 2 y J -4(p -2)(cd F ) 2 u 2 y |∇u| p-6 J + 2(p -2)c 2 d dF F u 2 y |∇u| p-6 u y (p -1)|∇u| p-2 + (p -2)|∇u| p-4 u 2 y J w + 4(p -2) 2 (d cF ) 2 u 4 y |∇u| 2p-10 J w            (5 J ) +(p -2)cd F u y |∇u| q-2 + 2(p -2)c 3 d d 2 F F 2 u y |∇u| p-6 u 2 x + 2u 2 y + 2(p -2)c cd dF 2 u y |∇u| p-6 u 2 y [|∇u| p-2 + (p -2)|∇u| p-4 u 2 x ] w - 4(p -2) 2 (cd F ) 2 cdF u 4 y |∇u| 2p-10 w            (5 -) ≤ 0 -2(p -2)c cd dF 2 u y |∇u| p-6 u 2 x + 4(p -2)(cd F ) 2 cdF u 2 y |∇u| p-6 - 2(p -2)cd F u y |∇u| p-4 |u 2 y |∇u| q-2 w - 2(p -2)c 3 d d 2 F F 2 u y |∇u| p-6 u 2 y (p -1)|∇u| p-2 + (p -2)|∇u| p-4 u 2 y w            (5 + ) ≥ 0 - (p -2)cd F u y u t |∇u| 2 + 2(p -2)cd F u 2 y |∇u| p-6 u t u y w (5 t )
and (noticing that ( 6) can be obtained from ( 5) by formally multiplying with -c d 2 F d uy )

(6) = -2(p -2)c cd 2 F 2 |∇u| p-6 N • ∇J + 2(p -2)c cd 2 F 2 u 2 y |∇u| p-6 M • ∇J w (6 ∇ ) +2(p -2)c c 2 d 3 F F 2 |∇u| p-6 u 2 x + 2u 2 y J + 4(p -2)c c 2 d d 2 F 3 u y |∇u| p-6 J - 2(p -2)c c 2 d 3 F F 2 u 2 y |∇u| p-6 (p -1)|∇u| p-2 + (p -2)|∇u| p-4 u 2 y J w - 4(p -2) 2 c c 2 d d 2 F 3 |∇u| p-6 u 3 y |∇u| p-4 J w          (6 J ) -(p -2)c cd 2 F 2 |∇u| q-2 -2(p -2)c c 3 d 4 F F 3 |∇u| p-6 u 2 x + 2u 2 y - 2(p -2)(c ) 2 cd 3 F 3 |∇u| p-6 u 2 y [|∇u| p-2 + (p -2)|∇u| p-4 u 2 x ] w + 4(p -2) 2 c c 3 d d 3 F 4 u 3 y |∇u| 2p-10 w          (6 -) ≤ 0 +2(p -2)(c ) 2 cd 3 F 3 |∇u| p-6 u 2 x -4(p -2)c c 3 d d 3 F 4 u y |∇u| p-6 + 2(p -2)c cd 2 F 2 |∇u| p-4 u 2 y |∇u| q-2 w + 2(p -2)c c 3 d 4 F F 3 |∇u| p-6 u 2 y (p -1)|∇u| p-2 + (p -2)|∇u| p-4 u 2 y w          (6 + ) ≥ 0 + (p -2)c cd 2 F 2 u t |∇u| 2 - 2(p -2)c cd 2 F 2 u 2 y |∇u| p-6 u t w . (6 t )
We shall now collect and relabel the numerous positive and negative terms that we just obtained, when expanding (1)-( 7) in the process of eliminating u yy . A number of positive and negative terms will then be paired together according to certain cancellations. Then, the remaining positive terms, as well as the terms involving u t , will be eventually controlled by using the negative terms.

Using that d ≤ 0 and F , F , u y ≥ 0, we first have positive terms :

(a) := -2(p -2)c 3 d d 2 F F 2 |∇u| p-4 u y (b) := - 2(p -2)cd F u 2 y |∇u| p-6 |∇u| q u y w (c) := - 2(p -2)c 3 d d 2 F F 2 u y |∇u| p-6 u 2 y (p -1)|∇u| p-2 + (p -2)|∇u| p-4 u 2 y w (d) := +2(p -2)(c ) 2 cd 3 F 3 |∇u| p-6 u 2 x (e) := + 2(p -2)c c 3 d 4 F F 3 |∇u| p-6 u 2 y (p -1)|∇u| p-2 + (p -2)|∇u| p-4 u 2 y w (f ) := - 2(p -2)d dc cF 2 u y |∇u| p-4 [|∇u| p-2 + (p -2)|∇u| p-4 u 2 x ] w (f 1 ) -2(p -2)c cd dF 2 u y |∇u| p-6 u 2 x (f 2 )
Finally observing that and using ( f ), ( ĩ), ( j), ( l) ≤ 0, we obtain the following lemma.

Lemma 3.9.2. Recalling the definition (3.8.4) of the parabolic operator P :

PJ := J t -|∇u| p-2 ∆J -(p -2)|∇u| p-4 D 2 J ∇u, ∇u -H • ∇J -A(x, y, t)J,
we have

PJ ≤ (f ) + (g) + (h) + (i) + (j) + (l) + (m) + (g) + ( h) +(3 t ) + (4 t ) + (5 t ) + (6 t ).
Completion of proof of Proposition 3.8.1 Starting from Lemma 3.9.2, we shall estimate the remaining positive and u t terms by the key negative terms (g) and ( h), after appropriate choice of the parameters. An essential tool in this step will the Bernstein-type estimates (see [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF]Theorem 1.2])

|∇u| ≤ C 0 y -1/(q-p+1) and u ≤ C 0 y (q-p)/(q-p+1) in [0, x 1 ] × (0, L 2 ] × (0, T ), (3.9.13)

where C 0 = C 0 (p, q, Ω, µ, ∇u 0 ∞ ) > 0, and we will use also the lower bound from (3.1.11) :

|∇u| ≥ u y ≥ δ 0 = µ/2 > 0.

(3.9.14)

First, using w ≥ |∇u| p-2 , we get

(f ) ≤ -2(p -2)pc cd dF 2 u y |∇u| p-4 .
Assume y 1 ≤ 1. Due to (3.9.13), we have dF ≤ C α 0 y -γ+α(q-p)/(q-p+1) ≤ C α 0 .

(3.9.15)

Here we used that γ ≤ α -1 and hence α(q -p)/(q -p + 1) -γ ≥ 1 -α/(q -p + 1) ≥ 0.

Assume k 0 = k 0 (p, q, Ω, µ, ∇u 0 ∞ ) > 0 sufficiently small so that

0 < k 0 ≤ |∇u| q-p+2 4pdF and 0 < k 3 0 ≤ |∇u| q-p+4 8x 2 F 3 d 3 , (3.9.16)
which is possible due to (3.9.14), x ≤ L 1 and (3.9.15)). We then have

(f ) + (g) + (g) ≤ p -2 2 cd F u y ≤0   |∇u| q-2 -8k 3 x 2 F 3 d 3 |∇u| p-6 + |∇u| q-2 -4pk|∇u| p-4 dF ≥0   ≤ 0.
Now, we may choose α = α(p, q, Ω, µ, u 0 C 2 , σ) > 1 close enough to 1 in such a way that γ = (α -1)(1 -2σ) is small enough to satisfy

γ q 2 2σ C q-p+1 0 + 25(p -2) 2 K 2 u 0 ∞ 2σδ q-p 0 ≤ (q -p + 1)α (3.9.28)
and

(α -1) 2σ - 1 q -p + 1 + 2(q -p) q -p + 1 ≥ 0, (3.9.29) α q -p q -p + 1 -(α -1)(1 -2σ) ≥ 0. (3.9.30)
Finally, once α is fixed (hence γ is also fixed small), recalling that y ≤ y 1 ≤ 1, x ≤ L 1 and γ ≤ 2, we take k 0 = k 0 (p, q, Ω, µ, u 0 C 2 , σ) > 0 possibly smaller, in such a way that

k 0 (q + 2(p -2))C α+q-p+2 0 δ -2 0 + α(4p -6)δ -2 0 C α+1 0 +k 2 0 4(p -2)(p -1)γ 2 L 2 1 C 2α 0 δ -2 0 + k 0 3(p -2) u 0 α ∞ K δ p-2 0 ≤ 2p -3 2 σγ, (3.9.31)
and next we take

y 1 = y 1 (p, q, Ω, µ, u 0 C 2 , σ) > 0 small enough such that (p -2)αKy 1 µδ p-2 0 ≤ 2p -3 2 σγ.
(3.9.32)

Then it follows from (3.9.27) that PJ ≤ 0 in D × (T /2, T ).

(3.9.33)

Finally, we need to check (3.8.6). The continuity statement is clear from the definition of A, H. Let us show that A is bounded in D × (T /2, τ ) for each τ < T . For this purpose, let us observe that due to |∇u| ≤ C(τ ), u ≤ C(τ )y and α -1 ≥ γ, we have for y ≤ 1 and τ ∈ (T /2, T )

|F d| = αu α-1 y -γ ≤ C α-1 (τ )y α-1-γ ≤ αC α-1 (τ ) (3.9.34) |F d | = γu α y -γ-1 ≤ γC α (τ ) (3.9.35) |F d| = u α y -γ ≤ C α (τ ).
(3.9.36)

We also have by (3.9.13) and (3.9.14) :

|∇u| r ≤ C r (τ ), if r > 0, δ r 0 , if r < 0.
(3.9.37)

The assertion then follows easily from (3.9.10), (3.9.12) and (3.2.7).

APPENDIX 1. Proof of regularity results (Theorem 3.2.2)

Proof. (i) Since, by Theorem 3.2.1(iii), ∇u is continuous in Q T , there exist λ, ρ, M 2 > 0 such that

λ ≤ |∇u| ≤ M 2 in Q ρ := B ρ (x 0 , y 0 ) × [t 0 -ρ, t 0 + ρ] ⊂ Q T . (3.10.1)
For any unit vector e and 0 < h < ρ/2, let us introduce the differential quotients

D h u = h -1 (τ h u -u), where τ h u = u (x, y) + h e, t .
We have

|∇τ h u| q -|∇u| q = d h (x, y, t) • ∇(τ h u -u) in Q ρ/2 ,
where

|d h (x, y, t)| ≤ C independent of h. Next denote b(s) = s (p-2)/2 and a i (p) = b(|p| 2 )p i where p = (p 1 , p 2 ), so that ∆ p u = ∂ i (a i (∇u))
. Following [80, p.445], we write

a i (∇τ h u) -a i (∇u) = 1 0 d ds a i (s∇τ h u + (1 -s)∇u)) ds = ãh ij ∂ j (τ h u -u),
where

ãh ij (x, y, t) = 1 0 ∂a i ∂p j (s∇τ h u + (1 -s)∇u)) ds.
Subtracting the PDE in (3.1.1) for u and for τ h u and dividing by h, we see that D h u is a local weak solution of

∂ t (D h u) -∂ i ãh ij ∂ j (D h u) = d h (x, y, t) • ∇(D h u) in Q ρ/2 . (3.10.2) Moreover, since ∂a i ∂p j ξ i ξ j = b(|p| 2 )|ξ| 2 + 2b (|p| 2 )p i p j ξ i ξ j ≥ b(|p| 2 )|ξ| 2 ≥ λ p-2 |ξ| 2 in Q ρ/2 by (3.10.1), we have ãh ij ξ i ξ j ≥ λ p-2 |ξ| 2 in Q ρ/2 . We then test (3.10.2) with ϕ 2 D h u, where ϕ ∈ C ∞ 0 (Q ρ/2
) is a cut-off function such that ϕ = 1 on Q ρ/3 . By integration by parts and some simple manipulations, it is easy to see that λ p-2

Q ρ/3 |∇D h u| 2 dxdydt ≤ Q ρ/2 ãh ij ∂ i (D h u)∂ j (D h u)ϕ 2 dxdydt ≤ C. It follows that D 2 u ∈ L 2 (Q ρ/3 ). Consequently, we obtain that u ∈ W 2,1 2 (Q ρ/3
) and is a local strong solution of equation (3.1.1) written in nondivergence form, i.e. :

u t -a ij u ij = f in Q ρ/3 , where a ij = |∇u| p-2 δ ij + (p -2) u i u j |∇u| 2 , f = |∇u| q . (3.10.3)
Since, by Theorem 3.2.1(iii), a ij , f are Hölder continuous in Q ρ/3 , it follows from interior Schauder parabolic regularity [80, Theorem III.12.2] that, for some α ∈ (0, 1),

u ∈ C 2+α,1+α/2 (Q ρ/4 ). (3.10.4)
(ii) Thanks to (3.10.4), we know that u is a classical solution of (3.10.3) in Q ρ/4 . Keeping the above notation, for 0 < h < ρ/8, we then have

(D h u) t -a ij (D h u) ij = F h := D h f + (D h a ij )(τ h u ij ) in Q ρ/8 .
Moreover, as a consequence of (3.10.4), we have, for 1 < r < ∞,

F h L r (Q ρ/8 ) ≤ C ∇f L r (Q ρ/4 ) + ∇A L r (Q ρ/4 ) D 2 u L ∞ (Q ρ/4 ) ≤ C, 0 < h < ρ/8.
It thus follows from interior parabolic L r estimates (see [80, Theorem III.12.2]) that, for

0 < h < ρ/8, D 2 D h u L r (Q ρ/16 ) + ∂ t D h L r (Q ρ/16 ) ≤ C(ρ) F h L r (Q ρ/8 ) + D h u L r (Q ρ/8 ) ≤ C.
We deduce that Du t , D 3 u ∈ L r loc (Q T ). Then differentiating (3.10.3) in space, we see that the function z

= ∂ u x ( = 1, 2) is a local strong solution of z t -a ij z ij = f in Q ρ/16 , (3.10.5)
where 16 due to (3.10.4), it follows from interior Schauder parabolic regularity [80, Theorem III.12.2] that z ∈ C 2+α,1+α/2 (Q ρ/20 ) for some α ∈ (0, 1).

f = ∂ f -u ij ∂ a ij . Since a ij , f are Hölder continuous in Q ρ/
Proof of Theorem 3.2.2 (continued) (ii) This is a direct consequence of Lemma 3.10.1.

(iii) It follows from (i)(ii) that v = u x ∈ C 2,1 (Q T ) ∩ C(Ω × (0, T ))
is a classical solution of (3.10.5) in Q T , where a ij are defined in (3.10.3). Moreover, v = g x = 0 on S T . Taking θ(t) a cut-off in time and setting w = θv, we see that w solves

w t -a ij w ij = f := θ f + θ t v in Q T , (3.10.6)
with 0 initial-boundary conditions. By [89, Theorem 4.28], since f is locally Hölder continuous in Ω×[0, T ) due to (i), there exists a solution to this problem in C 2+β,1+β/2 (Ω×[0, T )) for some β ∈ (0, 1). Since we have uniqueness in the class C 2,1 (Q T ) ∩ C(Ω × [0, T )) by the maximum principle, the conclusion (3.2.9) follows.

APPENDIX 2. A parabolic version of Serrin's corner lemma

In [88, p. 512], a Serrin corner property in a rectangle was shown for a parabolic equation involving the Laplacian. This was proved by comparison with a suitable product of functions of x, t and y, t. This result and method are no longer sufficient here and we shall establish a result for general nondivergence operators by modifying the original proof of [START_REF] Serrin | A symmetry problem in potential theory[END_REF] for the elliptic case.

Proposition 3.11.1. Let 0 < X 1 < X1 , 0 < Y 1 < Ŷ1 , D = (0, X1 ) × (0, Ŷ1 ) ⊂ R 2 , 0 < τ 1 < τ 2 , Dτ = D × (τ 1 , τ 2 ). Let the coefficients a ij = a ij (x, y, t) satisfy a ij ξ i ξ j ≥ λ|ξ| 2 in Dτ (3.11.1)
for some λ > 0 and assume that

a ij , B i ∈ C( Dτ ), a 12 + a 21 ≥ -C(x ∧ y), B 1 ≥ -Cx, B 2 ≥ -Cy in Dτ . (3.11.2) Let z ∈ C 2,1 ( Dτ ) ∩ C( Dτ ) satisfy Lz := z t -a ij z ij -B i z i ≤ 0 in Dτ , z(x, y, t) ≤ 0 in Dτ , z(0, 0, t) = 0. (3.11.3)
Then, for each t 0 ∈ (τ 1 , τ 2 ), there exists c 0 > 0 such that

z ≤ -c 0 xy in (0, X 1 ) × (0, Y 1 ) × [t 0 , τ 2 ). (3.11.4)
Proof. Let a = min(X 1 , Y 1 , t 0 -τ 1 2 ) and τ 3 = τ 1 +t 0 2 , so that τ 1 < τ 3 < t 0 < τ 2 . Fix t 1 ∈ [t 0 , τ 2 ) and let

K 1 := (x, y, t); x 2 + (a -y) 2 + (t 1 -t) 2 < a 2 , x > 0, t ≤ t 1 . Observe that K 1 ⊂ D × [τ 3 , t 1 ] and set K 2 = B (0, 0), a/2 × [τ 3 , t 1 ] and K = K 1 ∩ K 2 . Now set v(x, y, t) := e -α(x 2 +(y-a) 2 +(t-t 1 ) 2 ) -e -αa 2 , v(x, y, t) = e -α(x 2 +(y-a) 2 +(t-t 1 ) 2 ) ,
with α > 0 to be chosen later on, and define the auxiliary function h = xv. It is clear that h > 0 in K. We compute

h t = -2αx(t -t 1 )v, ∇h = v -2αx 2 v -2αx(y -a)v , D 2 h = v -6αx + 4α 2 x 3 -2α(y -a) + 4α 2 x 2 (y -a) -2α(y -a) + 4α 2 x 2 (y -a) -2αx + 4α 2 x(y -a) 2 , B(x, y, t) • ∇h = -2αxv[xB 1 + (y -a)B 2 ] + B 1 v.
Using that a ij ξ i ξ j ≥ λ|ξ| 2 , we have

a ij h ij = va 11 (-6αx + 4α 2 x 3 ) + va 22 -2αx + 4α 2 x(y -a) 2 + v (a 12 + a 21 ) -2α(y -a) + 4x 2 α 2 (y -a) = αxv 4α a 11 x 2 + (y -a)x(a 12 + a 21 ) + (y -a) 2 a 22 -6a 11 -2a 22 - 2(y -a)(a 12 + a 21 ) x ≥ αxv 4αλ(x 2 + (y -a) 2 ) -6a 11 -2a 22 + 2(a -y)(a 12 + a 21 ) x , hence Lh ≤ αxv -4αλ(x 2 + (y -a) 2 ) + 2(t 1 -t) + 6a 11 + 2a 22 - 2(a -y)(a 12 + a 21 ) x +2xB 1 + 2(y -a)B 2 - B 1 αx 1 -e α(x 2 +(y-a) 2 +(t-t 1 ) 2 -a 2 ) .
On the one hand, on K, we have y < a/2, hence x 2 + (y -a) 2 > a 2 /4. On the other hand, using part of assumptions (3.11.2) along with 0 ≤ a -y ≤ a and 0 ≤ 1 -e α(x 2 +(y-a) 2 +(t-t 1 ) 2 -a 2 ) ≤ 1 on K, it follows that for α > 1 large enough,

Lh ≤ αxv -αλa 2 + 2(t 1 -t) + 6a 11 + 2a 22 + 2Ca + 2xB 1 + 2(y -a)B 2 + C ≤ - λα 2 a 2 xv 2 < 0 in K. (3.11.5)
We now set w = z + εh where ε is a positive constant to be chosen. By (3.11.3) and (3.11.5), we have Lw < 0 in K.

Denote M = max K w ≥ 0. Since L is (uniformly) parabolic, by the usual proof of the maximum principle, it follows from (3.11.6) that w cannot attain the value M in K (observe that for each s ∈ [τ 3 , t 1 ], the section K ∩ {t = s} is an open, possibly empty, subset of R 2 ). To show M = 0 (for sufficiently small ε > 0), it thus suffices to verify that w ≤ 0 on

∂ P K = ∂K \ (K ∩ {t = t 1 }). We have ∂ P K = Γ 1 ∪ Γ 2 , where Γ 1 = ∂K 1 ∩ K 2 and Γ 2 = ∂K 2 ∩ K 1 .
On Γ 1 we have either

x 2 + (y -a) 2 + (t 1 -t) 2 = a 2 or x = 0,
so that h = 0 and z ≤ 0, hence w ≤ 0. Next observe that on Γ 2 we have

x 2 + (y -a) 2 + (t 1 -t) 2 < a 2 and x 2 + y 2 = a 2 /4,
hence τ 1 < τ 3 ≤ t ≤ t 1 and a/8 < y < a/2 (in other words, (x, y) is "far" from the corners of D). Therefore, by the Hopf boundary point lemma [START_REF] Protter | Maximum principles in differential equations[END_REF]Theorem 6 p. 174] and the strong on some time interval [0, T max (u 0 )), with the property that its gradient blows up on the boundary ∂Ω while the solution itself remains bounded. We refer the reader to [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF] [76] and [START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF] for the degenerate parabolic case and to [110] for the uniformly parabolic case. This singularity is a difficulty to extend the solution past T max (u 0 ). A natural question is then : Can we extend the weak solution past t = T max (u 0 ) and in which sense ? Let us mention here that a result in this direction where the continuation beyond gradient blow-up does not satisfy the original boundary conditions was obtained in [START_REF] Fila | Derivative blow-up and beyond for quasilinear parabolic equations[END_REF][START_REF] Fila | Rate of convergence to a singular steady state of a supercritical parabolic equation[END_REF].

Recently, for the linear diffusion case (p = 2), Barles and Da Lio [START_REF]On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations[END_REF] showed that such gradient blow-up is related to a loss of boundary condition and address the problem through a viscosity solutions approach. They proved a "Strong Comparison Result" (that is a comparison result between discontinuous viscosity sub and supersolutions) which allowed them to obtain the existence of a unique continuous, global in time viscosity solution of (4.1.1)-(4.1.3), the Dirichlet boundary condition being understood in the generalized sense of viscosity solution theory. They also provided an explicit expression of the solution of (4.1.1)-(4.1.3) in terms of a value function of some exit time control problem, which allows a simple explanation of the losses of boundary condition when it arises.

We recall that the formulation of the generalized Dirichlet boundary condition for (4.1.1)-(4.1.3) in the viscosity sense reads

min u t -div |Du| p-2 Du + |Du| q -f (x, t), u -g ≤ 0 on ∂Ω × (0, +∞), (4.1.5)
and

max u t -div |Du| p-2 Du + |Du| q -f (x, t), u -g ≥ 0 on ∂Ω × (0, +∞). (4.1.6)
Our first result mainly extends the investigation of [START_REF]On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations[END_REF] to the degenerate diffusion case p > 2.

Theorem 4.1.1. Assume that q > p ≥ 2 and that Ω is a bounded domain with a C 2boundary. For any u 0 ∈ C(Ω), f ∈ C Ω × [0, T ] and g ∈ C (∂Ω × [0, T ]) satisfying (4.1.4), there exists a unique continuous solution u of (4.1.1)-(4.1.3) which is defined globally in time.

As it is classical in viscosity solutions theory, the proof of Theorem 4.1.1 relies on a Strong Comparison Result (SCR in short), the existence of the global solution u being an almost immediate consequence of the Perron's method introduced in the context of viscosity solutions by Ishii [START_REF] Ishii | Perron's method for Hamilton-Jacobi equations[END_REF] (see also [START_REF] Lio | Comparison results for quasilinear equations in annular domains and applications[END_REF]).

The most important difficulties in the proof of Strong Comparison Results come from the formulation of the boundary condition in the viscosity sense, the discontinuity of the sub and the supersolution to be compared and the strong nonlinearity of the Hamiltonian term |Du| q . A key argument in the proof of the SCR in [START_REF]On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations[END_REF] is the "cone condition" which is useful in the treatment of boundary points. Roughly speaking the "cone condition" holds if at any point (x, t) of the boundary ∂Ω × (0, T ), an usc subsolution u satisfies u(x, t) = lim k→∞ u(x k , t k ) where {(x k , t k )} k is a sequence of points of Ω × (0, T ) with the following properties

(x k , t k ) → (x, t) and d ∂Ω (x k , t k ) ≥ b |x k -x| + |t k -t| ,
where b is a positive constant.

Our approach is slightly different : instead of directly proving the "cone condition" for any viscosity subsolution of (4.1.1)-(4.1.3) as it was done in [START_REF]On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations[END_REF], we use a combination of a C 0,β regularity result for subsolutions of stationary problems, strongly inspired by the result of Capuzzo Dolcetta, Leoni and Porretta [START_REF] Capuzzo Dolcetta | Hölder estimates for degenerate elliptic equations with coercive Hamiltonians[END_REF], together with a regularization by a sup-convolution in time. These arguments provide an approximation of the (a priori only usc) subsolution by a continuous subsolution, which automatically satisfies the "cone condition", allowing to borrow the methods of [START_REF] Barles | Remarks on the Dirichlet problem for quasilinear elliptic and parabolic equations[END_REF] to conclude.

The generalisation of the C 0,β regularity result of [START_REF] Capuzzo Dolcetta | Hölder estimates for degenerate elliptic equations with coercive Hamiltonians[END_REF] is the following.

Theorem 4.1.2. If u is a locally bounded, usc viscosity subsolution of

-div |Du| p-2 Du + |Du| q ≤ C in Ω , (4.1.7)
where Ω is an open subset of R N and C is a positive constant, and if q > p ≥ 2, then

u ∈ C 0,β loc (Ω) with β = q -p q -p + 1 .
Moreover, if Ω is a bounded domain with a C 2 -boundary, then u is bounded on Ω and it can be extended as a C 0,β -function on Ω and

|u(x) -u(y)| ≤ M |x -y| β for all x, y ∈ Ω, (4.1.8)
for some positive constant M depending only on p, q, C and ∂Ω.

The regularity result of [START_REF] Capuzzo Dolcetta | Hölder estimates for degenerate elliptic equations with coercive Hamiltonians[END_REF] was revisited in [START_REF]A short proof of the C 0,α -regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications[END_REF], where an interpretation was given in terms of state-constraint problems together with several possible applications. Our proof will rely on the arguments of [START_REF]A short proof of the C 0,α -regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications[END_REF].

A second motivation where such regularity results are useful, is the asymptotic behavior as t → +∞ of solutions of the evolution equation. For this purpose, one has first to study the ergodic (or additive eigenvalue) problem

-div |Du ∞ | p-2 Du ∞ + |Du ∞ | q -f (x) = c in Ω, (4.1.9)
associated to a state-constraint boundary condition on ∂Ω

-div |Du ∞ | p-2 Du ∞ + |Du ∞ | q -f (x) ≥ c on ∂Ω. (4.1.10)
We recall that, in this type of problems, both the solution u ∞ and the constant c (the ergodic constant) are unknown. First we have the following result.

Theorem 4.1.3. Assume that Ω is a bounded domain with a C 2 -boundary, f ∈ C(Ω) and q > p ≥ 2, then there exists a unique constant c such that the state-constraints problem (4.1.9)-(4.1.10) has a continuous viscosity solution u ∞ .

A typical result that connects the study of the ergodic problem to the large time behavior of the solution u of (4.1.1)-(4.1.3) is the following. Theorem 4.1.4. Assume that Ω is a bounded domain with a C 2 -boundary, u 0 ∈ C(Ω), g ∈ C(∂Ω) satisfying (4.1.4) and assume that f (x, t) = f (x) with f ∈ C(Ω) and q > p ≥ 2. If (c, u ∞ ) is the solution of (4.1.9)-(4.1.10) and if u is the unique viscosity solution of (4.1.1)-(4.1.3), then u + c + t is bounded, where c + = max(c, 0). In particular

lim t→∞ u(x, t) t = -c + uniformly on Ω.
The next step in the study of the asymptotic behavior would be to show that u(x, t) + ct → u ∞ (x) as t → ∞ where u ∞ solves (4.1.9)-(4.1.10). The main difficulty to prove such more precise asymptotic behavior comes from the fact that (4.1.9)-(4.1.10) does not admit a unique solution ((4.1.9)-(4.1.10) is invariant by addition of constants). Such results were obtained recently in [START_REF] Tchamba | Large time behavior of solutions of viscous Hamilton-Jacobi equations with superquadratic Hamiltonian[END_REF] for the uniformly elliptic case p = 2 through the use of the Strong Comparison Principle (i.e. a result which allows to apply the Strong Maximum Principle to the difference of solutions) and the Lipschitz regularity of u ∞ . But, for p > 2, such Strong Comparison Principle is not available since the equation is quasilinear and not semilinear. We recall that a Strong Maximum Principle is available for p > 2, see [START_REF] Bardi | On the strong maximum principle for fully nonlinear degenerate elliptic equations[END_REF]. Another difficulty comes from the proof of a strong comparison result for the steady problem in case of an operator that does not fulfill a monotonicity property, even if there exits a strict subsolution. Let us mention the works of [START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF][START_REF] Barles | Convergence to steady states for radially symmetric solutions to a quasilinear degenerate diffusive Hamilton-Jacobi equation[END_REF] for more results on the asymptotic behavior of global solutions.

Finally we point out that it was shown in [START_REF] Barles | On the large time behavior of solutions of the dirichlet problem for subquadratic viscous Hamilton-Jacobi equations[END_REF] that the expected asymptotic behavior, namely u(x, t) + ct → u ∞ (x), is not always true in the p = 2-case when the nonlinearity is sub quadratic in Du.

This chapter is organized as follows : in Section 4.2, we present the needed results on viscosity solutions for the stationary and evolution problems we consider ; in particular, we analyze the losses of boundary conditions for subsolutions. In Section 4.3 we prove the Hölder regularity result of Theorem 4.1.2. In Section 4.4 we study the ergodic problem. Section 4.5 is devoted to the proof of Theorem 4.1.1 and the asymptotic behavior of solutions of the evolution equation.

Preliminaries and Analysis of Boundary Conditions

In this section we collect some preliminary properties of viscosity subsolutions (the boundary conditions being always understood in the viscosity sense) and we also formulate SCR under different forms, some of them being only useful as a step in the proof of the complete regularity result. These results are concerned with either problem (4.1.1)-(4.1.3) or the following two nonlinear elliptic problem

-(p -1)|Du| p-2 λ i (D 2 u)>0 λ i (D 2 u) + |Du| q = C in Ω, u = g in ∂Ω. (4.2.1)
and

-div |Du| p-2 Du + |Du| q + λu -f = 0 in Ω, u = g in ∂Ω. (4.2.2)
where q > p ≥ 2, C, λ ≥ 0, f ∈ C(Ω) and g ∈ C(∂Ω).

From now on, we assume that Ω is a smooth domain with a C 2 -boundary. We define the distance from x ∈ Ω to ∂Ω by d ∂Ω (x) := dist (x, ∂Ω). For δ > 0, we denote by

Ω δ := {x ∈ Ω | d ∂Ω (x) < δ} , (4.2.3
)

Ω δ := {x ∈ Ω | d ∂Ω (x) > δ} . (4.2.4)
As a consequence of the regularity of ∂Ω, d ∂Ω is a C 2 -function in a neighborhood Ω δ of the boundary for all 0 < δ ≤ δ 0 . We denote by d a C 2 -function agreeing with d ∂Ω in Ω δ such that |Dd(x)| ≤ 1 in Ω δ . We also denote by n(x) the C 1 -function defined by n(x) = -Dd(x) in Ω δ ; if x ∈ ∂Ω, then n(x) is just the unit outward normal vector to ∂Ω at x. Our first result says that there is no loss of boundary conditions for the subsolutions, namely that the subsolutions satisfy the boundary condition in the classical sense. Proposition 4.2.1. Assume that q > 0 and p ≥ 2. We have the following i) If u is a bounded, usc subsolution of (4.1.1)-(4.1.3) on a time interval (0, T ), then

u ≤ g on ∂Ω × (0, T ). (4.2.5)
ii) If u is a bounded, usc subsolution of (4.2.1) or (4.2.2) , then

u ≤ g on ∂Ω. (4.2.6)
Proof. We only give the proof for the time dependent problem, the proof for the stationnary problems being similar. We use a result of Da Lio [START_REF] Lio | Comparison results for quasilinear equations in annular domains and applications[END_REF]Corollary 6.2]. We denote by S N the space of real symmetric N × N matrices. For x ∈ Ω, t ∈ (0, T ), ξ ∈ R N and M ∈ S N , we define the function F by

F (x, t, ξ, M ) = -|ξ| p-2 T r(M ) -(p -2)|ξ| p-4 M ξ, ξ + |ξ| q -f (x, t),
so that the equation can be written as u t + F (x, t, Du, D 2 u) = 0. From [START_REF] Lio | Comparison results for quasilinear equations in annular domains and applications[END_REF], we know that, if u(x 0 , t 0 ) > g(x 0 , t 0 ) at some point (x 0 , t 0 ) ∈ ∂Ω × (0, T ), then the following conditions hold lim inf

(y,t)→(x 0 ,t 0 ) α↓0 o(1) α + F y, t, Dd(y) + o(1) α , - Dd(y) ⊗ Dd(y) + o(1) α 2 ≤ 0 lim inf (y,t)→(x 0 ,t 0 ) α↓0 o(1) α + F y, t, Dd(y) + o(1) α , D 2 d(y) + o(1) α ≤ 0. (4.2.7)
But the first condition cannot hold since

F y, t, Dd(y) + o(1) α , - Dd(y) ⊗ Dd(y) + o(1) α 2 ≥ (p -1) α p (1 + o(1)) + 1 -o(1) α q -f (y, t),
and the right hand side is going to +∞ as α → 0 since p ≥ 2, q > 0 and all terms converge to +∞.

Let us point out that the above computation shows that there is no competition between the nonlinear Hamiltonian term and the slow diffusion operator since they both produce positive contribution which prevent any loss of boundary conditions for the subsolution.

Next, we remark that there cannot be loss of initial condition.

Proof. Fix x 0 ∈ Ω and define for ε > 0 and C ε > 0 the function φ ε (x, t) by

φ ε (x, t) = u(x, t) - |x -x 0 | ε 2 -C ε t.
This function attains a global maximum on Ω×[0, T ) at x ε , t ε . Using the boundedness of u, it is easy to see that, for any C ε > 0, (x ε , t ε ) → (x 0 , 0) as ε → 0. Arguing as in [START_REF] Barles | Remarks on the Dirichlet problem for quasilinear elliptic and parabolic equations[END_REF], choosing C ε sufficiently large depending on ε, we are left with (x ε , t ε ) ∈ (∂Ω × (0, T )) ∪ Ω × {0} and the two following possibilities

either t ε = 0 and u(x ε , 0) ≤ u 0 (x ε ), or t ε > 0, x ε ∈ ∂Ω and u(x ε , t ε ) ≤ g(x ε , t ε ).
In either case, since u(x 0 , 0) ≤ φ ε (x ε , t ε ) ≤ u(x ε , t ε ), we get the desired result for u letting ε → 0 and using the continuity of u 0 and g. The argument for v is similar. Now we claim that under some assumptions (set out below), a SCR holds for semicontinuous viscosity sub-and supersolutions of (4.1.1)-(4.1.3) or (4.2.1) or (4.2.2). The proof being somehow technical we refer the reader to the appendice for a detailed proof of the following two propostions. The stationary version of the SCR is used either in the proof of the C 0,β -regularity or for solving the ergodic problem.

Hölder Regularity of Viscosity Subsolutions for the Degenerate Elliptic Problem

In this section we are going to prove that equation of type (4.1.7) enters into the general framework described in [START_REF]A short proof of the C 0,α -regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications[END_REF] which allows us to state that, if u is a locally bounded, usc viscosity subsolution of (4.1.7), then u is Hölder continuous with exponent β = q -p q -p + 1 .

The key point is that the strong growth of the first order term balances the degeneracy of the second order term, providing a control on |Du|.

Using that -Dd(x) = µ(x)x for some µ(x) ≥ 0 and that q > p > 2, we have

|Dw 1 (x)| q-p+2 = |C 1 |x| β-2 x| + |C 2 d β-1 (x)Dd(x)| q-p+2 ≥ |C 1 |x| β-2 x| q-p+2 + |C 2 d β-1 (x)Dd(x)| q-p+2 = C q-p+2 1 |x| (β-1)(q-p+2) + C q-p+2 2 d (β-1)(q-p+2) (x)|Dd(x)| q-p+2 ,
and

|Dw 1 (x)| p-2 ≥ C p-2 1 |x| (β-1)(p-2) + C p-2 2 d (β-1)(p-2) (x)|Dd(x)| p-2 .
Using that d(x) = h(1 -|x|), with h being C2 , non-decreasing and concave, 0 < β < 1, we have

D 2 w 1 (x) ≤ C 1 |x| β-2 Id+C 2 d β-1 (x) h |x| Id -h x |x| ⊗ x |x| +(1-β)C 2 d β-2 (x)Dd(x)⊗Dd(x), and
λ i (D 2 w 1 (x)) ≤ C 1 |x| β-2 + C 2 d β-1 (x) h |x| -h + (1 -β)C 2 d β-2 (x)|Dd(x)| 2 .
At this point, it is worth noticing that because of the properties of h, the term h |x| -h is bounded.

These properties imply that, we can (almost) consider the two terms (in |x| and in d(x)) separately. Since (β -1)(q -p + 2) = (β -2), the C 1 β |x| β term yields

-(p -1)C 1 |x| β-2 + |C 1 |x| β-2 x| q-p+2 = |x| β-2 -(p -1)C 1 + C q-p+2 1 .
By choosing C 1 large enough, we can have for any

K 1 > 0 |x| β-2 -(p -1)C 1 + C q-p+2 1 ≥ K 1 |x| β-2 in B 1 (0)\ {0} .
On the other hand the

C 2 β d β (0) -d β (x) term yields -(p-1)C 2 d β-1 (x) h |x| -h +(β-1)(p-1)C 2 d β-2 (x)|Dd(x)| 2 +C q-p+2 2 |d β-1 Dd(x)| q-p+2 . (4.3.2)
We have to consider two cases : either |x| ≥ 1 2 and then h = 1, h = 0 and Dd(x) = -x |x| ;

hence the above quantity is given by

-(p -1)C 2 d β-1 (x) 1 |x| -(p -1)(1 -β)C 2 d β-2 (x) + C q-p+2
Recalling that (β -1)(q -p + 2) = (β -2), then for C 2 large enough we have for any

K 2 > 0 -(p -1)C 2 d β-1 (x) 1 |x| -(p -1)(1 -β)C 2 d β-2 (x) + C q-p+2 2 d (β-1)(q-p+2) ≥ K 2 d β-2 (x). Now for |x| ≤ 1 2
, the quantity (4.3.2) coming from the d(x)-term is bounded and can be controlled by the |x|-term. Hence, for any constant C > 0, choosing first C 2 large enough and then C 1 large enough, we have in

B 1 (0)\ {0} -(p -1)|Dw 1 (x)| p-2 λ i (D 2 w 1 (x)) + |Dw 1 (x)| q ≥ |Dw 1 (x)| p-2 K 1 |x| β-2 + K 2 d β-2 (x) ≥ K 1 |x| (β-1)(p-1)-1 ≥ C.
Next we set w r (x) := r β w 1 x r .

It is easy to check that for 0

< r ≤ 1, G(Dw r , D 2 w r ) ≥ r (β-1)(p-1)-1 C -C ≥ 0 on B r (0)\ {0}.
H3. Comparison result. Let v be any bounded usc viscosity subsolution of G r (Dv,

D 2 v) ≤ 0 in B r (0)\ {0} then v(y) ≤ v(x) + r β w 1 y -x r . (4.3.3)
We use the fact that v(0) + w r (x) is a strict super-solution up to the boundary and that it is a continuous function. It follows that the comparison is a direct consequence of Proposition 4.2.3. Since the hypotheses are satisfied, we can apply Proposition 2.1 of [START_REF]A short proof of the C 0,α -regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications[END_REF] to obtain the C 0,β regularity of subsolutions, both locally and globally with further assumptions on Ω. Remark 4.3.1. As far as the exponent β is concerned, the value is the best one can expect in the assumption of the above theorem (see [START_REF] Capuzzo Dolcetta | Hölder estimates for degenerate elliptic equations with coercive Hamiltonians[END_REF]). It is well-known that the degeneracy of the p-Laplacian is an an obstruction to the solvability of the Dirichlet problem in the classical sense. The presence of the strongly non-linear term with q > p is another source of obstruction, even in the uniformly elliptic case since examples of boundary layers can occur [START_REF]On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations[END_REF][START_REF] Lasry | Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem[END_REF]. By the previous result, we know that every continuous solution to (4.1.7) is Hölder continuous up to the boundary. Hence, a necessary condition in order that the solution can attain continuously the boundary data g is the existence of some C ≥ 0 such that |g(x) -g(y)| ≤ C[x -y| β for all x, y ∈ ∂Ω, β = q -p q -p + 1 .

For the uniformly elliptic case p = 2, a more detailed study including several gradient bounds and applications can be found in [START_REF] Lasry | Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem[END_REF].

As an application of the previous regularity result, we consider the generalized Dirichlet problem consisting in solving (4.2.2). Theorem 4.3.1. Let Ω ⊂ R N be a bounded domain with a C 2 -boundary. Assume that q > p ≥ 2, f ∈ C(Ω), g ∈ C(∂Ω) and λ > 0. Let u and v be respectively a bounded usc subsolution and a bounded lsc super-solution of (4.2.2) with u satisfying for x ∈ ∂Ω u(x) = lim sup y→x y∈Ω u(y).

Then, u ≤ v on Ω. Moreover Problem (4.2.2) has a unique viscosity solution which belongs to C 0,β (Ω).

Proof. For the comparison part, Theorem 4.1.2 implies that u is Hölder continuous, hence the comparison u ≤ v is a direct consequence of Proposition 4.2.3. Once noticed that -λ

-1 || f || L ∞ + ||g|| L ∞ and + λ -1 || f || L ∞ + ||g|| L ∞
are respectively sub and super-solution, we can apply the Perron's method with the version up to the boundary (see [START_REF] Lio | Comparison results for quasilinear equations in annular domains and applications[END_REF]). Since a solution is also a subsolution, the Hölder regularity is a direct consequence of Theorem 4.1.2. In this part we study the existence of a pair (c, u ∞ ) ∈ R × C(Ω) for which u ∞ is a viscosity solution of the state-constraints problem (4.1.9)-(4.1.10), to gather with the uniqueness of the ergodic constant c. For this purpose, we introduce a λu-term in the equation, as it is classical, with the aim to let λ tend toward 0. This key step is described by the following Lemma. q -p q -p + 1

The Ergodic Problem

. For 0 < λ < 1 and q > p, there exists a unique viscosity solution u λ ∈ C 0,β (Ω) of the state constraint problem

-div (|Du λ | p-2 Du λ ) + |Du λ | q + λu λ = f (x) in Ω, (4.4.1) -div (|Du λ | p-2 Du λ ) + |Du λ | q + λu λ ≥ f (x) on ∂Ω. (4.4.2)
Moreover there exists a constant C > 0 such that, for all 0 < λ < 1,

|λu λ | ≤ C in Ω. (4.4.3)
Proof. For R > 0, we consider the following generalized Dirichlet problem

-div (|Du R,λ | p-2 Du R,λ ) + |Du R,λ | q + λu R,λ = f (x) in Ω, u R,λ = R in ∂Ω. (4.4.4)
By Theorem 4.3.1, this problem admits a unique viscosity solution u R,λ . Moreover, u R,λ satisfies

-λ -1 f L ∞ ≤ u R,λ ≤ - M 1 β d β (x) + M 2 λ in Ω. (4.4.5)
Indeed, on the one hand, it is easy to see that -λ -1 f L ∞ is a subsolution. On the other hand, borrowing arguments from [START_REF] Tchamba | Large time behavior of solutions of viscous Hamilton-Jacobi equations with superquadratic Hamiltonian[END_REF], we claim that for some

M 1 , M 2 > 0 chosen large enough, ū(x) = - M 1 β d β (x) + M 2 λ
is a supersolution of (4.4.1)-(4.4.2). Indeed, using that

q(β -1) = (p -2)(β -1) + (β -2), we have -div (|Dū| p-2 Dū) + |Dū| q + λū -f (x) = M p-1 1 |Dd| p-2 d (p-2)(β-1) (p -1)(β -1)d β-2 |Dd| 2 + d β-1 ∆d + (p -2)d β-1 D 2 d Dd, Dd + M q 1 d q(β-1) |Dd| q -λ M 1 β d β + M 2 - f = M p-1 1 |Dd| p-2 d q(β-1) (p -1)(β -1)|Dd| 2 + d∆d + (p -2)d D 2 d Dd, Dd + M q-p+1 1 |Dd| q-p+2 -λ M 1 β d β + M 2 -f .
In Ω δ where |Dd| = 1 and 0 ≤ d ≤ δ, we have

-div (|Dū| p-2 Dū) + |Dū| q + λū -f (x) = M p-1 1 d q(β-1) (p -1)(β -1) + d∆d + (p -2)d D 2 dD d, Dd + M q-p+1 1 -λ M 2-p 1 β d β(2-p)+p + M 2 -f . Taking M 1 > 1 and M 2 > 0 such that M q-p+1 1 ≥ (p -1)(1 -β) + (p -2 + √ N )δ D 2 d L ∞ + δ β(2-p)+p β (4.4.6) and M 2 ≥ 2 f L ∞ , ( 4.4.7)
then we have -div (|Dū| p-2 Dū) + |Dū| q + λū -f (x) ≥ 0 in Ω δ . Now in Ω δ , we have |Dd| ≤ 1 and δ ≤ d(x) ≤ C(Ω). Using that 0 < λ < 1, then we have

-div (|Dū| p-2 Dū) + |Dū| q + λū -f (x) ≥ M p-1 1 (p -1)(β -1) d (β-1)(p-1)-1 L ∞ -(p -2 + √ N ) d (β-1)(p-1) L ∞ D 2 d L ∞ - M 1 β d β L ∞ + M 2 -f L ∞
.

Hence if we take M 1 as in (4.4.6) and M 2 such that

M 2 ≥ M p-1 1 (p -1)(1 -β) d (β-1)(p-1)-1 L ∞ + (p -2 + √ N ) d (β-1)(p-1) L ∞ D 2 d L ∞ + M 1 β d β L ∞ + 3 f L ∞ , ( 4
= u R,λ for R > M 2 λ . We have -max f L ∞ , M 2 ≤ λu λ ≤ max f L ∞ , M 2 .
Now we are in position to prove Theorem 4.1.3. Using that

u λ ≥ -λ -1 f L ∞ in Ω, we have -div(|Du λ | p-2 Du λ ) + |Du λ | q -f ≤ f L ∞ in Ω.
Theorem 4.1.2 implies uniform Hölder estimates with respect to λ for the functions u λ . Consequently if x 0 is an arbitrary point in Ω, we get that w λ := u λ (x) -u λ (x 0 ) is also uniformly bounded in C 0,β (Ω) (recall that Ω is connected). From (4.4.3), we also know that {-λu λ (x 0 )} λ is bounded. It follows that, by Ascoli's Theorem, we can extract a uniformly converging subsequence from {w λ } λ and we can assume that {-λu λ (x 0 )} λ converges along the same subsequence. Denoting by u ∞ and c, the limits of {w λ } λ and {-λu λ (x 0 )} λ respectively and taking into account that w λ solves

-div(|Dw λ | p-2 Dw λ ) + |Dw λ | q -f (x) + λw λ = -λu λ (x 0 ) in Ω,
we can pass into the limit λ → 0 and conclude by the stability result for viscosity solutions that, (c, u ∞ ) solves the ergodic problem. Now let (c 1 , u Once one noticed that

u 1 (x, t) = t f L ∞ + g L ∞ + u 0 L ∞ and u 2 (x, t) = -t f L ∞ - g L ∞ -u 0 L ∞
are respectively super-solution and subsolution of (4.1.1)-(4.1.3), the existence and uniqueness of a continuous global solution can be obtained by Perron's method, combining classical arguments of [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF] (see also [START_REF] Ishii | Perron's method for Hamilton-Jacobi equations[END_REF]), the version up to the boundary of Da Lio [START_REF] Lio | Comparison results for quasilinear equations in annular domains and applications[END_REF] and the Strong Comparison Result of the Proposition 4.2.2 on any time interval [0, T ].

Large Time Behavior

Let u ∞ be a bounded solution of (4.1.9)-(4.1.10). If c ≤ 0, then u is uniformly bounded. Indeed, if C > u ∞ L ∞ + u 0 L ∞ + g L ∞ , then u ∞ -C is a subsolution of (4.1.1)-(4.1.3). On the other hand, if x is a point far enough from Ω, then |x -x| 2 is a super-solution. To see this,it suffices to take x such that B(x, R)

∩ Ω = ∅ with R > max(1, ( f L ∞ + (p -1)) 1 q-p+2 ).
Hence applying the Strong Comparison Result, we have

u ∞ (x) -C ≤ u(x, t) ≤ |x -x| 2 + C on Ω × (0, +∞),
-ct + u ∞ -C ≤ u(x, t) ≤ u ∞ -ct + C on Ω × (0, +∞).
The result follows by dividing by t and then letting t → +∞.

(x 0 , t 0 ) ∈ ∂Ω × (Kα, T -Kα).

Next, using the regularity of the boundary, we can find a C 2 -function ξ : R N → R N which is equal to n = -Dd in a neighborhood of ∂Ω. Now we consider the auxiliary function

Φ ε : Ω × Ω × [Kα, T -Kα] × [Kα, T -Kα] → R defined by Φ ε (z, w, t, s) = u α µ (z, t) -v(w, s) -ω(α) -η α (t -Kα) - z -w ε -χ z + w 2 4 - |t -s| 2 ε 2
Let (z, w, t, s) be a global maximum point of

Φ ε on Ω×Ω×[Kα, T -Kα]×[Kα, T -Kα].
For notational simplicity we drop again the dependance of (z, w, t, s) on ε, µ and η. Using the inequality Φ ε (z, w, t, s) ≥ Φ ε (x 0 , x 0 , t 0 , t 0 ) and the boundedness of u α µ , v and χ, we have

z -w ε ≤ C, t - s ε ≤ C, for some constant C > 0 depending on u L ∞ , v L ∞ and α. By the compactness of Ω × [Kα, T -Kα],
we can assume that (z, t), (w, s) converge to (x, t)

∈ Ω × [Kα, T -Kα].
Moreover, using the continuity of u α , we have On the other hand, we have also

Φ ε (z, w, t, s) ≥ Φ ε (x 0 -εξ(x 0 ), x 0 , t 0 , t 0 ) = M α µ,η -o ε (1), as ε → 0,
lim sup ε→0 Φ ε (z, w, t, s) ≤ lim sup ε→0 (u α µ (z, t) -v(w, s) -η α ( t -Kα) -ω(α)) -lim inf ε→0 z -w ε -χ z + w 2 4 -lim inf ε→0 | t -s| 2 ε 2 ≤ M α µ,η . (4.5.4)
Therefore, combining (4.5.3) and (4.5.4) with classic arguments, we have

z -w ε -χ z + w 2 4 = o ε (1), | t -s| 2 ε 2 = o ε (1), (4.5.5)
u α µ (z t) -v(w, s) -η α ( t -Kα) -ω(α) → u α µ (x, t) -v(x, t) -η α ( t -Kα) -ω(α) = M α µ,η as ε → 0. (4.5.6)
is sufficiently close to 1, then we have M µ = max Ω (u µ -v) > M/2 > 0. Since u is usc and v is continuous this maximum is achieved at x 0 . We may assume that x 0 ∈ ∂Ω. We drop the dependence of x 0 on µ.

Next, using the regularity of the boundary, we can find a C 2 -function ξ : R N → R N which is equal to n in a neighborhood of ∂Ω. Now we consider the test function

Φ ε : Ω × Ω → R defined by Φ ε (z, w) = µu(z) -u(w) - z -w ε + ξ z + w 2 4 .
Let (x, y) be a global maximum point of Φ ε on Ω × Ω. For notational simplicity we drop the dependence of x and y on ε. and µ. Using the boundedness of u and v, it is clear that x -y = O(ε) and it follows that, along a subsequence, x, y → x ∈ Ω. Since u is lsc and v is continuous, we get that lim sup ε→0 Φ ε (x, y) ≤ M µ .

On the other hand we have, using the continuity of v, we have

Φ ε (x, y) ≥ Φ ε (x 0 , x 0 -εξ(x 0 )) ≥ M µ + O(ε). It follows that lim inf ε→0 Φ ε ≤ M µ . Hence we get that Φ ε (x, y) → M µ as ε → 0. (4.5.17)
Standard arguments allow us to deduce from (4.5.17) that

x -y ε + ξ x + y 2 Next we claim that, for ε small enough the viscosity inequalities hold for u and v. This is obviously the case if y ∈ Ω. Using Proposition 4.2.1 and arguing similarly as in the previous proof, we get that, if y ∈ ∂Ω than v(y) ≤ g(y) and the viscosity inequality holds also in this case. On the other hand, using (4.5.19), we get that

x = y -εξ(y) + o ε (1) (4.5.20)
which implies by the smoothness of the domain and the properties of ξ that x lies in Ω for ε small enough and hence the viscosity inequality holds for µu.

Using the same arguments as the previous proof, we get that the elements (q 1 , X) ∈ J 2,+ u µ and (q 2 , Y ) ∈ J 2,-v given by the Jensen-Ishii's Lemma satisfy the fact that the singularities may only take place near the boundary, allow us to prove the following convergence result : any global solution, even unbounded in W 1,∞ must converge in C([0, 1]) to a stationary solution W of (5.1.1) with W (0) = 0, W (1) = M (see Proposition 5.3.3). On the other hand, if u were unbounded, then our gradient estimates would imply that W x (0) = +∞ or W x (1) = -∞. But such a W does not exist if M = M b , leading to a contradiction. Although the scheme of proof follows that in [5] for p = 2, we have to face a number of additional technical difficulties, caused by the lack of regularity of solutions. In particular, we have to work at the level of regularized problems, including for the construction of Lyapunov-Zelenyak functional. This, in turn requires good convergence properties and estimates of regularized solutions. For this, we heavily rely on results from our previous work [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF] (which concerned the higher dimensional problem as well) and an extension up to the boundary of a result of DiBenedetto-Friedman on the regularity of the derivative of weak solutions of a degenerate parabolic problem (see Proposition 5.2.1). Let us mention some results concerning related equations possessing solutions with unbounded gradient. When the nonlinearity is replaced with an exponential one and p = 2, results on boundedness and existence of infinite time gradient blow-up solutions are obtained in [START_REF] Zhang | Boundedness of global solutions for a heat equation with exponential gradient source[END_REF][START_REF] Zhu | Rate of approach to the steady state for a diffusion-convection equation on annular domains[END_REF]. A phenomenon of infinite time gradient blow-up has been observed for quasilinear equations involving mean curvature type operators [START_REF] Chen | Infinite time blow-up of solutions to a nonlinear parabolic problem[END_REF]. For results on interior gradient blow-up we refer the reader to [3,[START_REF] Asai | On the interior derivative blow-up for the curvature evolution of capillary surfaces[END_REF]. Finally for other results concerning existence, asymptotic behavior of global solutions for the corresponding Cauchy problem and a viscosity solution approach see [START_REF] Chen | On the Cauchy problem of evolution p-Laplacian equation with nonlinear gradient term[END_REF][START_REF]Non-Diffusive Large Time Behavior for a Degenerate Viscous Hamilton-Jacobi Equation[END_REF][START_REF] Quittner | Superlinear parabolic problems : blow-up, global existence and steady states[END_REF] and references therein.

λ(1 + O(ε))µ p-2 u µ (x) -v(y) ≤ |q 2 | p-2 o ε (1) + o(ε)|q 1 | q-p+2 + (1 -µ p-1-q )|q 1 | q-p+2 + µ p-1 (1 + O(ε)) f (x) -f (y). ( 4
The rest of the chapter is organized as follows. Section 5.2 contains some useful preliminary material, including smoothing properties of solutions and estimate of the derivative u x . In section 5.3, we employ the technique of Zelenyak [START_REF] Zelenyak | Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable[END_REF], along with a trick used in [5], to construct an approximate Lyapunov functional for weak solutions to (5.1.1). Section 5.4 is devoted to the proof of Theorem 5.1.1.

Preliminary estimates and steady states

Space and time derivative estimates

For u 0 ∈ W 1,∞ ((0, 1)), u 0 (0) = 0, u 0 (1) = M , by a (weak) solution of (5.1.1) on [0, T ], we mean a function u ∈ C([0, T ) × [0, 1]) ∩ L q ((0, T ); W 1,q (0, 1)) such that

u t ∈ L 2 ((0, T ); L 2 (0, 1)), u(0, x) = u 0 (x), u(t, 0) = 0, u(t, 1) = M and T 0 1 0 u t ψ + |u x | p-2 u x • ψ x dx dt = T 0 1 0 |u x | q ψ dx dt, (5.2.1)
holds for all ψ ∈ C 0 ([0, T ] × [0, 1]) ∩ L p ((0, T ); W 1,p 0 ((0, 1)). It is known (see e.g., [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF]) that there exists T max = T max (u 0 ) ∈ (0, ∞] such that for each T ∈ (0, T max ), (5.1.1) admits a unique solution u such that u ∈ L ∞ (0, T ); W 1,∞ (0, 1) . In the rest of this chapter, the maximal weak solution of problem (5.1.1) will refer to this solution. Now let us state the following result (which will be very useful in the sequel) on the Hölder regularity of the derivative of solutions to a possibly degenerate parabolic problem. This result is an extension up to the boundary (in one space dimension) of an interior estimate of DiBenedetto-Friedman [START_REF] Dibenedetto | Hölder estimates for nonlinear degenerate parabolic sytems[END_REF] (see the appendix for a proof). The definition of a weak solution of (5.2.2) is the same as in (5.2.1), with |u x | q replaced by F and u ∈ L q (0, T ; W 1,q (0, 1)) replaced by v ∈ L p (0, T ; W 1,p (0, 1)). Proposition 5.2.1. Let ε ∈ [0, 1), u 0 ∈ W 1,∞ , C > 0 and F ∈ L r ((0, T ) × (0, 1)) for some r > 2 with F L r ((0,T )×(0,1)) ≤ C. Let v be a weak solution of

     v t = (|v x | 2 + ε 2 ) p-2 2 v x x + F (t, x), t > 0, x ∈ (0, 1), v(t, 0) = 0, v(t, 1) = M, t > 0, v(0, x) = u 0 (x),
x ∈ (0, 1).

(5.2.2)

Then, for each η > 0, v x ∈ C α ([η, T -η] × [0, 1]) where α > 0 and the Hölder norm of v x depend only on C, v x L p and v L ∞ t ,L 2
x . As a direct consequence of this proposition, we get that u is a C 1 -function w.r.t. the space variable in (0, T ) × [0, 1] and that its derivative u x is locally Hölder continuous.

In order to describe the asymptotic behavior, we need to collect some preliminary estimates. We first give the following theorem which is of independent interest. It gives a useful regularizing property for local solutions of (5.1.1) as well as a uniform bound on u t away from t = 0 for any space dimension.

Let Ω ⊂ R N be a bounded domain of class C 2+α for some α > 0. Consider the following problem .2.11) This implies that y < 0 on (a, b) so that necessarily a = 0. Integrating inequality (5.2.11), it follows that y(x) ≤ q-p+1 p-1 x 1-p q-p+1 on (0, b) and y(0) > 0. If y ≡ 0, then we can find c = c(t) ∈ (0, 1] such that y > 0 in (0, c) and y = 0 in [c, 1). Therefore we get y(x) ≤ q-p+1 p-1 x 1-p q-p+1 on (0, 1) and (5.2.9) is readily deduced. In the same manner, considering

y + y q p-1 ≤ (|u x | p-2 u x ) x -C 2 + |u x | q ≤ 0. ( 5
y(x) = -|u x | p-2 u x (t, 1 -x) -C 2 x
+ , we get (5.2.10).

Remark 5.2.1. Similar gradient estimates in any space dimension are already obtained in chapter 2 using a more technical Bernstein type argument.

The following corollary is a direct consequence of Lemma 5.2.1 that states that, when gradient blow-up occurs on the boundary, it can only be towards +∞ at x = 0 or towards -∞ at x = 1.

Corollary 5.2.1. Let u be a maximal weak Lipschitz solution of (5.1.1) and t 0 ∈ (0, T max ). There exists C 3 = C 3 (t 0 , p, q, osc(u 0 )) > 0 such that for all t ∈ [t 0 , T max ),

u x (t, 0) ≥ -C 3 and u x (t, 1) ≤ C 3 .
(5.2.12)

Steady states

It is a well-known fact that the large-time behavior of evolution equations is closely connected to the existence and properties of the stationary states. In this part we are looking for nonnegative stationary solutions W of (5.1.1), that is weak solution of

(|W x | p-2 W x ) x + |W x | q = 0, x ∈ (0, 1), W (0) = 0, W (1) = M ≥ 0. (5.2.13) More precisely, W ∈ C([0, 1]) ∩ C 1 (0, 1) is a weak solution of (5.2.13) if W (0) = 0, W (1) = M and W satisfies 1 0 |W x | p-2 W x φ x -|W x | q φ dx = 0 for any φ ∈ C 1 c (0, 1). ( 5

.2.14)

It is not difficult to show that any weak solution in the above sense is actually a classical C 2 solution in (0, 1) (for any x 0 ∈ (0, 1), consider separately the cases W x (x 0 ) = 0 and W x (x 0 ) = 0). For small values of M ≥ 0, problem (5.2.13) admits a unique weak solution ]). Namely, this happens for 0 ≤ M < M b , where M b is the critical value,

W M = W M (x) ∈ C 2 ([0, 1
M b = q -p + 1 q -p q -p + 1 p -1 -1/(q-p+1)
.

More precisely, for M = 0 we have W 0 = 0 and for 0 ≤ M < M b , there exists k

= k(M ) ∈ [0, ∞) such that W M = M b (x + k)
q-p q-p+1 -k q-p q-p+1 . On the other hand, there is no steady state if M > M b . In the critical case M = M b , there still exists a steady state W M b = U , given by the explicit formula U (x) = M b x q-p q-p+1 . U belongs to C([0, 1]) ∩ C 2 ((0, 1]), but it is singular in the sense that it has infinite derivative on the left-hand boundary, U x (0) = ∞.

Lyapunov functional and convergence to steady states

Since (5.1.1) is a degenerate problem, we do not have sufficient regularity properties of the trajectories to construct a good smooth Lyapunov functional (which exists for onedimensional uniformly parabolic equations). Hence we first consider a regularized problem, then the main estimate which plays a key role in the proof of the convergence to steady states will be proved by passing to the limit ε → 0 in the regularizing parameter.

Approximate problem

Let ε ∈ (0, 1/2). We consider the following approximate problems :

     (u ε ) t = (|(u ε ) x | 2 + ε 2 ) p-2 2 (u ε ) x x + B ε ((u ε ) x ) (t, x) ∈ (0, +∞) × (0, 1), u ε (t, 0) = 0, u ε (t, 1) = M, t > 0, u ε (0, x) = u 0 (x),
x ∈ (0, 1), (5.3.1) where

B ε (v) = (|v| 2 + ε 2 ) q-2 2 (|v| 2 + ε 2 p-1
). Here we collect some useful properties of the sequence {u ε } which we will use later on.

Let u ∈ L ∞ [0, T ); W 1,∞ ((0, 1)) for any T ∈ (0, T max ) be the unique, maximal weak solution of problem (5.1.1) and let u ε be the unique, maximal classical solution of (5.3.1) and T (u ε ) be its existence time. We have the following proposition. Proposition 5.3.1. Let A > 0 and assume that u 0 W 1,∞ ≤ A. Then for 0 < T < T max and ε small, we have

a) T (u ε ) > T , u ε → u in C([0, T ] × [0, 1]) and (u ε ) x → u x in C loc ((0, T ] × [0, 1]). b) |(u ε ) x (t, x)| ≤ C = C(A, T ) on [0, T ] × [0, 1] and T 0 1 0 (u ε ) 2 t dxdt ≤ C(A, T ).
Proof. For the convenience of the proof, we shall actually replace the initial data u 0 in the approximate problem (5.3.1) with a sequence u ε,0 ∈ W 1,∞ ((0, 1)), where u ε,0 → u 0 in W 1,∞ ((0, 1)), and prove that Proposition 5.3.1 remains true in this more general situation. We know from chapter 2 that there exist a small time τ = τ (A) > 0 and a

Construction of the Lyapunov functional

Now we construct a Lyapunov functional for (5.3.1) with the help of the technique developed by Zelenyak [START_REF] Zelenyak | Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable[END_REF]. Let D K = [-K, K] × R, where K is the constant in (5.3.4). We look for a pair of functions Φ ε ∈ C 1 (D K ; R) and Ψ ε ∈ C(D K ; (0, ∞)) with the following property : For any solution u ε of ( 5

.3.1) with |u ε | ≤ K, defining L ε (u (t)) = 1 0 Φ ε (u ε (t, x), (u ε ) x (t, x)) dx, it holds d dt L ε (u ε (t)) = - 1 0 Ψ ε (u ε (t, x), (u ε ) x (t, x)) (u ε ) 2 t (t, x) dx. Since (u ε ) t (t, 0) = (u ε ) t (t, 1) = 0, we have d dt 1 0 Φ ε (u ε , (u ε ) x )dx = 1 0 (u ε ) t • (Φ ε ) u (u ε , (u ε ) x ) + (u ε ) xt • (Φ ε ) v (u ε , (u ε ) x ) dx = 1 0 (u ε ) t (Φ ε ) u (u ε , (u ε ) x ) -(u ε ) x • (Φ ε ) uv (u ε , (u ε ) x ) -(u ε ) xx • (Φ ε ) vv (u ε , (u ε ) x ) dx.
So it is natural to require that

(Φ ε ) u (u ε , (u ε ) x ) -(u ε ) x • (Φ ε ) uv (u ε , (u ε ) x ) -(u ε ) xx • (Φ ε ) vv (u ε , (u ε ) x ) = -Ψ ε (u ε , (u ε ) x ) • (u ε ) t = -Ψ ε (u ε , (u ε ) x ) (p -1) |(u ε ) x | 2 + ε 2 p-4 2 |(u ε ) x | 2 + ε 2 p -1 (u ε ) xx + |(u ε ) x | 2 + ε 2 q-2 2 |(u ε ) x | 2 + ε 2 p -1 A sufficient condition is (Φ ε ) vv (u, v) = (p -1)Ψ ε (u, v) v 2 + ε 2 p-4 2 v 2 + ε 2 p -1 , (5.3.5) (Φ ε ) u (u, v) -v(Φ ε ) uv (u, v) = -Ψ ε (u, v) v 2 + ε 2 q-2 2 v 2 + ε 2 p -1 , (5.3.6)
that is Φ ε satisfies the differential equation :

(Φ ε ) u (u, v) -v(Φ ε ) uv (u, v) + (v 2 + ε 2 ) q-p+2 2 p -1 (Φ ε ) vv (u, v) = 0. (5.3.7)
We follow the method used in [5] to find such nice functions. For a given function ρ ε (u, v), let us denote

H ε = (ρ ε ) u + (v 2 + ε 2 ) q-p+2 2 p -1 (ρ ε ) vv -v(ρ ε ) uv .
Here we assume that ρ ε , (ρ ε ) u , (ρ ε ) v , (ρ ε ) uv are continuous and C 1 in v in D K , and that (ρ ε ) vv is continuous in D K and, except perhaps at v = 0, C 1 in v.

We want to have

(H ε ) v = 0, so that H ε (u, v) = H ε (u, 0) = H ε (u). We compute (H ε ) v = (v 2 + ε 2 ) q-p+2 2 p -1 (ρ ε ) vvv + q -p + 2 p -1 v v 2 + ε 2 q-p 2 (ρ ε ) vv -v(ρ ε ) uvv .
For this, it suffices that f ε = (ρ ε ) vv satisfies the following conditions :

   (f ε ) u - q -p + 2 p -1 (v 2 + ε 2 ) q-p 2 f ε - (v 2 + ε 2 ) q-p+2 2 (p -1)v (f ε ) v = 0 |u| ≤ K, v = 0, (f ε ) v (u, 0) = 0.
(5.3.8)

Now, the equation (5.3.8) can be solved by the method of characteristics. For any K > 0 such that |u| ≤ K, one finds that the function defined by

f ε (u, v) = 1 + q -p p -1 v 2 + ε 2 q-p 2 (K + 1 -u) -q-p+2 q-p > 0 is a solution of (5.3.8) on [-K, K] × R. Define ρ ε by ρ ε (u, v) = v 0 z 0 f ε (u, s) ds dz ≥ 0,
and let then

Φ ε (u, v) = ρ ε (u, v) - u 0 H ε (s, 0) ds + K + 1.
(5.3.9)

We added the constant K + 1 to ensure that Φ ε ≥ 0. In fact, given that ε ≤ 1/2, 2 < p and 0 ≤ (ρ ε ) vv ≤ 1, we get (ρ ε ) u (s, 0) = 0 and

0 ≤ H ε (s, 0) = (ρ ε ) u (s, 0) + ε q-p+2 p -1 f ε (s, 0) ≤ 1.
(5.3.10)

Consequently, using that |u| ≤ K, we get

- u 0 H ε (s, 0) ds ≥ -u ≥ -K for u ∈ [0, K], - u 0 H ε (s, 0) ds ≥ 0 for u ∈ [-K, 0].
Using the definition of H ε and the fact that H ε (u, v) = H ε (u, 0), we see that :

(Φ ε ) u -v(Φ ε ) uv (u, v) + (v 2 + ε 2 ) q-p+2 2 p -1 (Φ ε ) vv (u, v) = 0,
i.e. Φ ε satisfies (5.3.7), hence (5.3.5)-(5.3.6) with

Ψ ε (u, v) = v 2 + ε 2 p -1 -1 (v 2 + ε 2 ) 4-p 2 (ρ ε ) vv p -1 ≥ (v 2 + ε 2 ) 2-p 2 (ρ ε ) vv p -1 > 0. (5.3.11) It follows that d dt L ε (u ε (t)) = - 1 0 ((u ε ) 2 x + ε 2 ) 2-p 2 (ρ ε ) vv (p -1) (u ε ) 2 x + ε 2 p -1 (u ε ) 2 t dx = - 1 0 Ψ ε (u ε , (u ε ) x ) (u ε ) 2 t dx.
Due to q > p > 2, we remark that, ∀ε ∈ (0, 1), |u| ≤ K and v ∈ R,

Ψ ε (u, v) ≥ A(v) = (v 2 + 1) 2-p 2 p -1 1 + (q -p) (p -1) v 2 + 1 q-p 2 (2K + 1) -q-p+2 q-p . (5.3.12)
As a consequence of the existence of the approximate Lyapunov functional, we have the following estimate.

Proposition 5.3.2. Assume that q > p > 2 and let u be a global weak solution of (5.1.1).

Then for any T > 1 and δ > 0, There exists C = C( u 0 W 1,∞ , δ, p, q) > 0 such that

T 1 1-δ δ (u t ) 2 dxdt ≤ C. ( 5

.3.13)

Proof. First let us remark that Lemma 5.2.1 implies that, for any δ > 0,

|u x | ≤ C(δ) in [1, ∞) × [δ, 1 -δ]. (5.3.14)
Now we fix T > 1 and δ ∈ (0, 1/2). On the one hand, by (5.3.12), we have

T 0 1-δ δ A ((u ε ) x ) • (u ε ) 2 t (t, x) dx dt ≤ T 0 1 0 Ψ ε (u ε , (u ε ) x ) • (u ε ) 2 t (x, t) dx dt = L ε (u(0)) -L ε (u ε (T )) (5.3.15) ≤ C( u 0 W 1,∞ ).
On the other hand, by Proposition 5.3.1, there exists ε 0 (δ, T ) such that, for all ε < ε 0

, x ∈ [0, 1], t ∈ [1, T ], |(u ε ) x (t, x) -u x (t, x)| ≤ C(δ).
Then, by (5.3.14)

, |(u ε ) x | ≤ 2C(δ) for (t, x) ∈ [1, T ] × [δ, 1 -δ] so that T 0 1-δ δ A ((u ε ) x ) • (u ε ) 2 t (t, x) dx dt ≥ T 1 1-δ δ A ((u ε ) x ) • (u ε ) 2 t (t, x) dx dt ≥ θ(2C(δ)) T 1 1-δ δ (u ε ) 2 t (x, t) dx dt,
where θ(R) = inf {A(v); |v| ≤ R} > 0. Letting ε → 0 and using a lower semicontinuity argument as well as (5.3.15), we obtain

θ(2C(δ)) T 1 1-δ δ (u) 2 t (t, x) dx dt ≤ C( u 0 W 1,∞ ), (5.3.16)
The result immediately follows. Proof. Assume that u is a global weak solution of (5.1.1). Fix a sequence (t k ) k∈N , 1 ≤ t k → ∞ and set w k (t, x) = u(t + t k , x). By (5.1.3), we know that

Convergence to steady states

|u| ≤ max { u 0 ∞ , M } in [1, ∞) × [0, 1], (5.3.17)
Using lemma 5.2.1 we have

|u x | ≤ C(δ), in [1, ∞) × [δ/2, 1 -δ/2]. (5.3.18)
Thus applying a result of DiBenedetto-Friedman [START_REF] Dibenedetto | Hölder estimates for nonlinear degenerate parabolic sytems[END_REF], we have that {w k } and {(w

k ) x } are Hölder continuous in [δ, T -δ] × [δ, 1 -δ] with a Hölder norm independent of k. It follows that {w k } and {(w k ) x } are relatively compact in C ([δ, T -δ] × [δ, 1 -δ]) for any δ, T > 0.
Thus, by the Arzelà-Ascoli theorem and a diagonal procedure, there exist a subsequence (t k l ) l∈N of (t k ) and a function W ∈ C ((0, ∞) × (0, 1)), W x ∈ C ((0, ∞) × (0, 1)) such that for any δ, T > 0

w k l → W strongly in C ([δ, T -δ] × [δ, 1 -δ]) as l → ∞. (5.3.19) (w k l ) x → W x strongly in C ([δ, T -δ] × [δ, 1 -δ]) as l → ∞. (5.3.20)
and W is a distributional solution of

W t -(|W x | p-2 W x ) x = |W x | q , t > 0, x ∈ (0, 1).
z is Lipschitz in (0, 1) with a Lipschitz bound depending on t, p, q, ||u 0 || ∞ and ||u x (t)|| q ∞ . Moreover the function z satisfies almost everywhere

z + z q/(p-1) = (|u x | p-2 u x (t, x)) x 1 {ux>0} + |u x | p-2 u + x (t, x) + C p-1 q 1 q p-1 ≥ [(|u x | p-2 u x (t, x)) x + |u x | q ] 1 {ux>0} + C 1 ≥ 0. For 0 ≤ y ≤ x ≤ 1, an integration yields z(x) (p-1-q)/(p-1) ≤ z(y) (p-1-q)/(p-1) + q -p + 1 p -1 (x -y), that is (5.4.1) with C 4 = C p-1 q 1 and C 5 = q -p + 1 p -1 . It follows that for 0 ≤ y ≤ x ≤ 1, we have |u x | p-2 u + x (t, x) + C 4 p-1-q p-1 ≤ |u x | p-2 u + x (t, y) + C 4 p-1-q p-1 + C 5 (x -y).
The estimate (5.4.2) can be obtained similarly by considering z

(x) = |u x | p-2 (-u x ) + (t, 1 - x) + C p-1 q 1 .
Remark 5.4.1. Lemma 5.4.1 yields in particular a lower bound on the gradient blow-up profile, which complements the upper bounds in (5.2.9)-(5.2.10). Namely, if x = 0 is a GBU point (in finite or infinite time), i.e. if |u x | is unbounded in any neighborhood of T max and 0, then lim sup t→Tmax u x (t, x) ≥ C(p, q)x -1/(q-p+1) for all sufficiently small x > 0. The analogous estimate holds if x = 1 is a GBU point. Now let us state the following lemma which is a direct consequence of the convergence of u to the steady state. It results that max We proceed by contradiction. Assume that u is a global weak solution which is unbounded in W 1,∞ . We know that when t → ∞, u converges to W = W M in C[0, 1] and in C 1 [δ, 1 -δ] for all δ > 0. Since u x is unbounded and can only blow up to +∞ at x = 0, there exist sequences t n → ∞, x n → 0 such that u x (t n , x n ) → +∞ (5.4.6)

Taking t = t n and y = x n in (5.4.1) and sending n → ∞, we deduce that, for any x ∈ (0, 1)

|W x (x)| p-2 W x + C 4 p-1-q p-1 ≤ C 5 x.
This would imply that

|W x | p-2 W x + C 4 ≥ (C 5 x)
1-p q-p+1 .

Passing to the limit x → 0 we get a contradiction since W = W M ∈ C 1 ([0, 1]). So all the global solutions are bounded in W 1,∞ .

Proof of the convergence in

C 1 norm for M ∈ [0, M b )
This follows from the proof of Proposition 5.3.3, with (5.3.14) replaced by the boundedness of u x on [0, ∞) × [0, 1], and using Proposition 5.2.1 which is an extention of the result in [START_REF] Dibenedetto | Hölder estimates for nonlinear degenerate parabolic sytems[END_REF]. Next, since F (t, x)

L r ((0,T )×(-1,2))

≤ C F L r (0,T )×(0,1)) , using a result of DiBenedetto and Friedman (see [START_REF] Dibenedetto | Hölder estimates for nonlinear degenerate parabolic sytems[END_REF] and [47, chapter 9] for the case ε > 0) on Hölder regularity of gradient of some degenerate parabolic problems, we get that, for any η > 0, v * x ∈ C α loc ([η, Tη] × (-1, 2)) where α > 0 and the norm of v *

x depend only on F L r (0,T )×(0,1)) , v *

x L p and v * L ∞ t ,L 2

x . We get the desired result recalling that v * x = v x on [0, 1] and using that [0, 1] ⊂ (-1, 2).

Introduction

In this chapter, we are interested in qualitative properties of solutions of the non-linear degenerate parabolic equation u t -∆ p u = |∇u| q , (6.1.1)

where ∆ p u = div(|∇u| p-2 ∇u), q > p -1 > 1.

The kind of result we are going to prove are gradient estimates for local solutions in time-space, and a Liouville type theorem for ancient solutions. In the last years, gradient estimates have played a key role in geometry and PDE since at least the early work of Bernstein. Gradient a priori estimates are fundamental for elliptic and parabolic equations, leading to Harnack inequalities, Liouville theorems, and compactness theorems for both linear and nonlinear PDE. For the corresponding elliptic equation of (6.1.1), gradient estimates were first considered by Lions [START_REF] Lions | Quelques remarques sur les problèmes elliptiques quasilinéaires du second ordre[END_REF] for the linear diffusion case p = 2. These estimates were based upon the Bernstein technique. Recently for the possibly degenerate elliptic equation with q > p -1 > 0, Bidaut-Véron, Huidobro, Véron [START_REF] Bidaut-Veron | Local and global properties of solutions of quasilinear hamilton-jacobi equations[END_REF] obtained a priori universal gradient estimate for equations on a domain Ω of R and they extended their estimates to equations on complete non compact manifolds satisfying a lower bound estimate on the Ricci curvature. These estimates allowed them to derive some Liouville type theorems.

It is natural to look also for parabolic Liouville-type-theorems. In the linear diffusion case p = 2 and for q > 1, Souplet and Zhang [110] obtained local gradient estimate for locally upper bounded solution of (6.1.1) (u ≤ M ) of the form |∇u| ≤ C(p, N, q) t -1 q + R -1 + R -1 q-1 (M + 1 -u) in B(x 0 , R) × (0, T ).

Relying on this estimate they proved that, under some growth condition at infinity, ancient solutions in the whole of R N are constant. Motivated by their result, we generalize the gradient estimate and Liouville theorem to the case 1 < p -1 < q. We also require that the solution is locally lower bounded. Using a Bernstein method, we have the following gradient estimate. Theorem 6.1.1. Let q > p -1 > 1, x 0 ∈ R N and R, T > 0. We set Q T,R = B(x 0 , R) × (0, T ). Let u be a solution in L ∞ ((0, T ); W 1,∞ (B(x 0 , R))) of

∂ t u -∆ p u = |∇u| q in Q T,R .
Suppose that |u| ≤ M for some constant M ≥ 1. Then, |∇u| ≤ C(p, N, q) t -1

q + R -1 + R -1 q-p+1 M in Q T, R 2 . ( 6.1.2)
For the Cauchy-Dirichlet problem associated to (6.1.1), a gradient estimate involving the W 1,∞ norm of the initial data has been obtained in [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF]110]. In Theorem 6.1.1 we only use the local L ∞ norm of the solution but we get a weaker estimate regarding the exponent on the distance to the boundary R. Recently, for the singular diffusion case 1 < p < 2 and for q = p, F. Wang [START_REF] Wang | Gradient estimates for the p-Laplace heat equation under the Ricci flow[END_REF] established gradient estimates similar to (6.1.2) for smooth, upper bounded, local solutions to (6.1.1) on closed manifolds or on complete noncompact Riemannian manifolds evolving under a Ricci flow. These estimates are of the form :

|∇u| 1 -u (x, t) ≤ C(N, p) R -1 + t -1 p + K 2 p + K in Q T, R 2 (6.1.3)
where K > 0 is a constant related to the Ricci flow and the sectional curvature of the manifold. These estimates allowed to the author to provide some Harnack inequalities for positive solutions of the following p-Laplace heat equation The estimates (6.1.3) have been obtained by deriving an equation for w = |∇v| p , v = f -1 (-u) and f (s) = e s/(p-1) -1. For q > p > 2, we take a different auxiliary function f , adapted to the degenerate diffusion case and to the fast growing gradient non-linearity.

As an application of the gradient estimate (6.1.2), we can state the following Liouville theorem for (6.1.1). Theorem 6.1.2. Assume that q > p -1 > 1 and let σ = min 1, 1 q-p+1 . Assume that u ∈ L ∞ loc ((-∞, 0); W 1,∞ loc (R N )) is a weak solution of u t -∆ p u = |∇u| q , x ∈ R N , -∞ < t < 0, Then u is constant.

Remark 6.1.1. The growth hypothesis (6.1.5) is important (see the example of the function u(x, t) = x 1 + t). However, we do not know if the exponents are sharp.

Besides the works mentioned above, there are few other studies on gradient estimates and nonlinear Liouville theorems for a parabolic type equation on noncompact Riemannian manifolds. In this case the proof mostly relies on two types of gradient estimates or a combination of them. These estimates are known as Hamilton gradient estimate (the estimate only involves ∇u and u) [START_REF] Hamilton | A matrix Harnack estimate for the heat equation[END_REF] and Li-Yau's gradient estimate (the estimate involves ∇u, u and u t ) [START_REF] Li | On the parabolic kernel of the Schrödinger operator[END_REF]. Let us also mention that the linear heat equation on noncompact manifolds was studied by Souplet and Zhang in [109] where they obtained a local gradient estimate related to the elliptic Cheng-Yau estimate and Hamilton's estimate for the heat equation on compact manifolds. A Liouville theorem was also proved in [109]. Hamilton-type gradient estimates were also used in [START_REF] Wang | Liouville theorems for the ancient solution of heat flows[END_REF][START_REF] Ma | Gradient estimate for the degenerate parabolic equation u t = ∆F (u) + H(u) on manifolds[END_REF][START_REF] Zhu | Hamilton's gradient estimates and Liouville theorems for porous medium equations on noncompact Riemannian manifolds[END_REF]. For q = p > 1, a nonlinear analogue of Li-Yau's estimate has been established in [START_REF] Kotschwar | Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula[END_REF] for positive solutions of (6.1.1) on compact manifolds with nonnegative Ricci curvature. In [START_REF] Kotschwar | Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula[END_REF], the gradient estimate was not used to get Liouville theorems but to obtain an entropy formula. Nevertheless, Liouville theorems should be obtained as a consequence of the obtained gradient estimate.

This chapter is organized as follows : In Section 6.2, we provide the proof of the gradient estimate (6.1.2) and we prove Theorem 6.1.2. In Sections 6.3 we give the proof of a technical auxiliary lemma that appears in the proof of the gradient estimate.

Bernstein-type gradient estimate

The proof of Theorem 6.1.1 is based on the following technical lemma which is based on a Bernstein method. The most significant difficulty being the choice of the auxiliary function f and the estimates coming from the cut-off argument. Let us mention that for different suitable choice of f , gradient bounds global in space for the Cauchy problem associated to (6.1.1) have been obtained in [START_REF] Bartier | Gradient estimates for a degenerate parabolic equation with gradient absorption and applications[END_REF]. First let us make precise that by local weak solution of (6.1.1) we mean a function u ∈ C(R N × (0, T )) ∩ L ∞ loc (0, T ; W 1,∞ loc (R N )) such that the integral equality |∇u| q ψ dx dτ By Lemma 6.2.1 we get that, in a small neighbourhood Q of (x, t), z satisfies Lz ≤ -2(q -1)(f ) q-2 f w q+2 2 η + C(p, N, α)(f ) p-2 R -2 η α w p 2 + C(p, q, α)R -1 η α w p+1 2 (f ) p-3 f + w q+1 2 (f ) q-1 .

Hence

(f ) 1-q Lz ≤ -2(q -1)

f f w q+2 2 η + C(p, N, α)(f ) p-1-q R -2 η α w p 2 + C(p, q, α)R -1 η α w p+1 2 (f ) p-q-1 f f + w q+1 2
.

Since v ∈ 0, (3) ≤ 1 (6.2.6) Using (6.2.6) together with the fact that 1 ≤ M ≤ f and p -q -1 < 0, we get that (f ) 1-q Lz ≤ -2(q -1) 3(p -1)(N + 1) w .

We take α = max q+1 q+2 , p+1 q+2 . Using the Young's inequality and recalling that η ≤ 1, then -for the conjugate exponents r 1 = q+2 p , s 1 = q+2 q-p+2 we have that C(N, p, α)R -2 η α w p 2 = η p q+2 w p 2 C(N, p, q, α)η α-p/(q+2) R -2

≤ ε 1 (N, p, q)ηw q+2 2 + C(N, p, q, α)R -2(q+2) q-p+2 , -for the conjugate exponents r 2 = q+2 p+1 , s 2 = q+2 q-p+1 we have that C(N, p, q, α)R -1 η α w

p+1 2 = η p+1 q+2 w p+1 2 C(N, p, q, α)R -1 η α-p+1 q+2 ≤ ε 2 ηw q+2 2 + C(N, p, q, α)R -(q+2) q-p+1
-and finally for the conjugate exponent r 3 = q+2 q+1 , s 3 = (q + 2) we have that C(N, p, q, α)R -1 η α w

It follows that

|∇u| ≤ M γ|∇v| ≤ C(N, p, q)(A + t

-2 q ) 1/2 M in Q T, R 2
. (6.2.9)

Here we used the fact that 2 -u M γ-1 γ ≤ 1. Hence we have |∇u| ≤ C(N, p, q) R -1 + R -1

q-p+1 + t -1 q M in Q T, R 2 .
and the proof of Theorem 6.1.1 is complete.

Proof of Theorem 6.1.2 Fix x 0 ∈ R and t 0 ∈ (-∞, 0). Take R ≥ 1, T = R σq and set Q = B(0, R) × (0, T ). Now we consider the function U := u(x + x 0 , t + t 0 -T ). Using (6. 6.3 Proof of Lemma 6.2.1

Our proof consists of three steps.

Step 1 : computations

Let f be a C 3 -function to be determined. We assume that f , f > 0. We put v = f -1 (-u) and w = |∇v| 2 . By a straightforward computation, we have that v satisfies the following equation -2(q -1)(f ) q-2 f w q+2 2 η + C(p, N, α)(f ) p-2 R -2 w p 2 η α + C(p, q, α)η α R -1 w p+1 2 (f ) p-3 f + w q+1 2 (f ) q-1 .

∂ t v = (f ) p-
Step 3 : suitable choice for the function f

To get rid of the term (f ) p-3 f w

RESUME en anglais

This thesis is devoted to the study of qualitative properties of solutions of an evolution equation of Hamilton-Jacobi type with a p-Laplacian diffusion. It is mainly concerned with the study of the effect of the non-linear diffusion on the gradient blow-up phenomenon. The main issues we are studying are: local existence and uniqueness, regularity, spatial profile of gradient blow-up and localization of the singularities. We provide examples where the gradient blow-up set is reduced to a single point. In Chapter 4, a viscosity solution approach is used to extend the blowing-up solutions beyond the singularities and an ergodic problem is also analyzed in order to study their long time behavior. In the penultimate chapter, we address the question of boundedness of global solutions to the one-dimensional problem.

In the last chapter we prove a local in space, gradient estimate and we use it to obtain a Liouville-type theorem.

Discipline : Mathématique MOTS-CLES : Problémes paraboliques-dégénérés, estimations de gradient, explosion en un seul point, solutions de viscosité, comportement asymptotique, Théorème de type Liouville.

LAGA (Laboratoire d'Analyse Géomètrie et Applications), 99 Avenue Jean Baptise Clément, Villetaneuse 93430.

1 Introduction générale 1 . 1 1

 111 Motivation et présentation générale du problème . . . . . . . . . . . . . . . 1.2 Etat de l'art et résultats du chapitre 2 : Théorie locale et profil spatial de l'explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Etat de l'art et résultats du chapitre 3 : localisation des points d'explosion. 1.4 Etat de l'art et résultats du chapitre 4 : prolongement des solutions au delà des singularités . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . de type Liouville 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Bernstein-type gradient estimate . . . . . . . . . . . . . . . . . . . . . . . 6.3 Proof of Lemma 6.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Motivation et présentation générale du problème L'objectif de ce travail de thèse est d'étudier l'effet d'une diffusion non linéaire sur le comportement qualitatif des solutions des équations de type Hamilton-Jacobi. Plus précisément nous nous intéressons à l'équation suivante u t -∆ p u = |∇u| q

Theorem 3 . 1 . 1 .

 311 Assume (3.1.2)-(3.1.6).

Theorem 3 . 2 . 1 .

 321 Ω)) such that ψ ≥ 0 and ψ = 0 on S T . A function u is a weak solution of (3.2.1)-(3.2.3) if it is a super-solution and a sub-solution.The following result was established in Theorem 2.1.1 (actually in any space dimension). Assume (3.1.3) and q > p -1 > 1. Let M 1 > 0, let u 0 , g satisfy (3.2.4)-(3.2.5) and ∇u 0 ∞ ≤ M 1 . Then :

  3.1)-(3.3.2) and (3.1.6).Let (x, y) ∈ ∂Ω h . If x = 0, then u + (x, y, t) = u(h, y, t) = u(-h, y, t) = u -(x, y, t) by (3.1.9). If x > 0, then (x + h, y) ∈ ∂Ω as a consequence of (x, y) ∈ ∂Ω h and (3.1.6). So, by (3.3.3), we have

  and the claim (3.4.6) is proved. Now fix y ∈ [0, ε). It follows from (3.4.4)-(3.4.6

.5. 10 )

 10 It follows from (3.5.7)-(3.5.10) that

Lemma 3 . 5 . 3 .

 353 Let ρ, µ, u 0 be as inLemma 3.3.3 (ii). Then GBU S(u 0 ) ⊂ [-ρ, ρ] × {0}. Proof. Denote again Σ ρ = [-ρ/2, ρ/2] × {0} and Σ ρ = ∂Ω \ ([-ρ, ρ] × {0}). In view of Lemma 3.5.1, it suffices to show that sup (x,y)∈Σ ρ , t∈(0,T ) |∇u(x, y, t)| < ∞.

Lemma 3 . 8 . 3 .

 383 Assume (3.8.1)-(3.8.3) and let µ, u 0 satisfy the assumption of Theorem 3.1.1

.8. 13 )

 13 On the other hand, for x = 0 and 0 < y < Ŷ1 , we have u xy = u x = 0. Also, by (3.2.9), we have |(u xy ) x | = |(u x ) yx | ≤ C in DT . Consequently |u x | + |u xy | ≤ Cx in DT . Using (3.2.7) and (3.1.11), we deduce

which implies B 2 ≥ 64 3. 9 .

 2649 -Cy in DT . (3.8.15) Proof of the main parabolic inequality (Proposition 3.8.1)

  (a) + (ã) ≤ 0, (b) + ( b) ≤ 0, (c) + (c) ≤ 0, (d) + ( d) ≤ 0, (e) + (ẽ) ≤ 0

Lemma 4 . 2 . 1 .

 421 Assume that q > p ≥ 2, f ∈ C Ω × [0, T ] and u 0 ∈ C(Ω), g ∈ C (∂Ω × [0, T ]) satisfy (4.1.4). Let u and v be respectively a bounded usc viscosity subsolution and a bounded lsc super-solution of (4.1.1)-(4.1.3) then u(x, 0) ≤ u 0 (x) ≤ v(x, 0) on Ω.

Proposition 4 . 2 . 2 (( 4 . 2 . 9 )

 422429 Parabolic SCR). Assume that q > p ≥ 2, f ∈ C Ω × [0, T ] and u 0 ∈ C(Ω), g ∈ C (∂Ω × [0, T ]) satisfy (4.1.4). Let u and v be respectively a bounded usc viscosity subsolution and a bounded lsc super-solution of (4.1.1)-(4.1.3), then u ≤ v in Ω × [0, T ]. Moreover, if we define ũ on Ω × [0, T ] by setting ũ(x, t) :=    lim sup u(y, s) (y,s)→(x,t) (y,s)∈Ω×(0,T ) for all (x, t) ∈ ∂Ω × (0, T ] u(x, t) otherwise, then ũ remains an usc subsolution of (4.1.1)-(4.1.3) and ũ ≤ v on Ω × [0, T ]. (4.2.10)

Proposition 4 . 2 . 3 (

 423 Elliptic SCR). Assume that q > p ≥ 2, f ∈ C Ω and g ∈ C (∂Ω).

  (i) Let u and v be respectively a bounded usc viscosity subsolution and a bounded lsc super-solution of (4.2.1). If v is continuous on Ω and is a strict supersolution of (4.2.1), thenu ≤ v on Ω.(4.2.11)(ii) Let u and v be respectively a bounded usc viscosity subsolution and a bounded lsc supersolution of (4.2.2). Assume that either λ > 0 or λ = 0 and v is a strict supersolution.We define ũ on Ω by setting then ũ remains an usc subsolution of (4.1.1)-(4.1.3) and ũ ≤ v on Ω. (4.2.13)

4. 4 . 1

 41 Existence of the pair (c, u ∞ )

Lemma 4 . 4 . 1 .

 441 Let f ∈ C(Ω) and β =

  If c > 0, then u ∞ -ct+C is a supersolution of (4.1.1)-(4.1.3) with state constraint condition on ∂Ω. On the other hand, u ∞ -ct -C is a subsolution of (4.1.1)-(4.1.3) which is below u 0 at t = 0 and below g on ∂Ω. Applying the Strong Comparison Result, we have

Φ

  ε (z, w, t, s) ≥ M α µ,η .(4.5.3)

4 =

 4 o ε (1), (4.5.18) µu(x) -v(y) → µu(x) -v(x) = M µ as ε → 0. (4.5.19)

.5. 21 )

 21 Letting ε → 0 and then µ → 1, we get a contradiction. It follows that ũ ≤ v on Ω.

Remark 5 . 1 . 1 .

 511 (a) For the critical case M = M b , all solutions must blow up in either finite or infinite time. The existence of global solution which are unbounded in W 1,∞ norm (that is infinite time gradient blow-up) should occur for some suitable initial data as it is the case for the corresponding semilinear equation (namely for initial data u 0 below the singular steady state), but this still is an open problem. Moreover, we know from Proposition 5.3.3 that, even in this case the solutions will converge in C([0, 1]) ∩ C 1 loc ((0, 1]) to the unique singular steady state.(b) Since the technique of Zelenyak to obtain a Lyapunov functional is restricted to the one-dimensional setting, the large time behavior and the boundedness of global solution in W 1,∞ norm are still open problems in higher dimension.

Proposition 5 . 3 . 3 .

 533 Let u be a global weak solution of (5.1.1). Then M ≤ M b and u(t) converges in C([0, 1]) to a steady state of (5.1.1) as t → ∞. Moreover the convergence also holds in C 1 ([δ, 1 -δ]) for all δ > 0.

Lemma 5 . 4 . 2 .

 542 Let M ≥ 0 and let u be a global weak solution of (5.1.1).Then it holds lim Since u(t, 1) = M , we get max[0,1] u(t, x) ≥ M . Next, using that w → W in C([0, 1]) (see Proposition 5.3.3) and W ≤ M , it holds that ∀ε > 0, ∃t ε > 0 such that if t > t ε then u(t, x) ≤ w(x) + ε ≤ M + ε x ∈ [0, 1].

[0, 1 ]

 1 u(t, x) ≤ M + ε if t > t ε . 123Proof of the boundedness of u x for 0 < M < M b

  Proof of Theorem 5.1.1 for M > M b This is an immediate consequence of Proposition 5.3.3 and the fact that (5.2.13) admits no solution for M > M b . Further regularity for global solutions for 0 < M < M b As a consequence of the convergence of global solutions to the steady state in C 1 [(0, 1)], we have the following proposition which is of independent interest. It gives a result of further regularity of global solutions for large time. It is unknown whether or not such property is true in the case M = 0. Proposition 5.4.1. Assume that 0 < M < M b and let u be a global weak solution of (5.1.1). Then there exist T > 0 and η > 0 such that u x ≥ η on [ T , +∞) × [0, 1]. Moreover, u becomes a classical solution on [ T , +∞) × [0, 1]

|z| p- 2

 2 z t = ∆ p z.(6.1.4)

1 q ), as |x| σ + |t| 1 q

 11 satisfying |u(x, t)| = o(|x| σ + |t|

  x, t)ψ(x, t) -u(x, s)ψ(x, s)) dx + t s R N -uψ t + |∇u| p-2 ∇u • ∇ψ dx dτ = t s R N

1 γ - 1 1 γ ≤ 3

 1113 , γ, M ≥ 1, we have 1 ≤ v + 1 ≤ (3) and hence1 3(p -1)(N + 1) ≤ f f ≤ 1 (p -1)(N + 1)

q+2 2 ηη α w p 2 +η α w p+1 2 + w q+1 2

 2222 + C(N, p, α)R -2 C(p, q, α)R -1

2 C≤ ε 3 ηw q+2 2 +

 22 (N, p, q, α)R -1 η α-q+1 q+2 C(N, p, q, α)R -(q+2) .

1 q

 1 1.5), we have that |U | ≤ M R in Q, where M R := sup B(x 0 ,R)×(t 0 -T,t 0 ) |u| = o(T + R σ ) = o(R σ ), as R → ∞.Applying Theorem 6.1.1 to U in Q, we get that|∇u(x 0 , t 0 )| = |∇U (0, T )| ≤ C(N, p, q)R -σ M Rand the conclusion follows by sending R to +∞.

2 w p-2 2 ∆v + (p - 2 )f w p 2 -(f ) q-1 w q 2 =For i = 1 , 2 C 1 1 + (f ) p-2 w p-2 2 C 2 2 + (f ) p-2 w p-2 2 C 3 3 +2(p - 1 ) 2 2 w

 22221211222233122 D 2 v, ∇v, ∇v w + (p -1)(f ) p-3 ..., N , we set v i = ∂v ∂x i . In a neighbourhood Q := ω × (τ 1 , τ 2 ) of any point (x, t) ∈ Q T,R for which |∇u| = f (v)|∇v| > 0, the equation is uniformly parabolic and -Estimate of |wH • ∇η| |wH • ∇η| ≤ (f ) p-2 w p-2 (p, N, δ 1 )η -1 |∇η| 2 w + δ 1 [D 2 v| 2 η (p, N, δ 2 )η -1 |∇η| 2 w + δ 2 [D 2 v| 2 η (p, N, δ 3 )η -1 |∇η| 2 w + δ 3 [D 2 v| 2 η (f ) p-3 f w p+1 2 |∇η| + q(f ) q-1w comes from an estimate via the Young's inequality of |(p -2)w∆v∇v • ∇η|.Recalling that ∇w = (2D 2 v, ∇v), (2) comes from an estimate of (p-2) 2 w∇w • ∇η and (3) come from an estimate of (p-2)(p-4) (∇v • ∇w) (∇v • ∇η) .

3 . 2 C 4 4 .f w p+2 2 +f f 2 w p+2 2

 3244222 Estimate of 2|∇u| p-2 |∇η • ∇w|. Using the Young inequality, we have2|∇u| p-2 |∇η • ∇w| ≤ (f ) p-2 w p-2 (p, N, δ 4 )η -1 |∇η| 2 w + δ 4 |D 2 v| 2 η . Estimate of 2(p -2)(∇u • ∇η)(∇w • ∇u) |2(p -2)(∇u • ∇η)(∇w • ∇u)| ≤ (f ) 2 w C 5 (N, p, δ 5 )η -1 |∇η| 2 w + |D 2 v| 2 η .Finally recalling that ∇u = f ∇v and choosing δ i in such way that -2+δ 1 +δ 2 +δ 3 +δ 4 +δ 5 = -1 and then recalling the properties of the function η, we arrive at L(z) ≤ 2(p -1)η (f ) p-3 N (p -1)(f ) p-2

p+2 2 +f f 2 w p+2 2 ( 6 12 )

 222612 N (p -1)(f ) p-2Résumé en français Cette thèse est consacrée à l'étude des propriétés qualitatives de solutions d'une équation d'évolution de type Hamilton-Jacobi avec une diffusion donnée par l'opérateur p-Laplacien. On s'attache principalement à l'étude de l'effet de la diffusion non-linéaire sur le phénomène d'explosion du gradient. Les principales questions qu'on étudie portent sur l'existence locale, régularité, profil spatial d'explosion et la localisation des points d'explosion. En particulier on montre un résultat d'explosion en seul point du bord. Dans le chapitre 4, on utilise une approche de solutions de viscosité pour prolonger la solution explosive au delà des singularités et on étudie son comportement en temps grands. Dans l'avant dernier chapitre on s'intéresse au caractère borné des solutions globales du problème unidimensionnel. Dans le dernier chapitre on démontre une estimation de gradient locale en espace et on l'utilise pour obtenir un résultat de type Liouville. On s'inspire et on compare nos résultats avec les résultats connus pour le cas de la diffusion linéaire. TITRE en anglais : A qualitative study of a Hamilton-Jacobi equation with a nonlinear diffusion

  Théorème 1.4.3. On suppose que q > p > 2, Ω est un domaine borné de classe C 2 et f ∈ C(Ω), alors il existe une unique constante c tel que le problème (1.31)-(1.32) admet une solution de viscosité u ∞ ∈ C(Ω).

	L'existence de (c, u ∞ ) nous permet d'avoir le résultat suivant.
	Théorème 1.4.4. On suppose que q > p > 2, Ω est un domaine borné de classe C 2 ,
	u 0 ∈ C(Ω), g ∈ C(∂Ω) satisfaisant la condition de compatibilité et f ∈ C(Ω). Soit (c, u ∞ )
	une solution de (1.31)-(1.32) et u l'unique solution de viscosité de (1.29), alors u + c + t est
	bornée, avec c + = max(c, 0). En particulier on a
	lim t→∞	u(x, t) t	= -c +
	uniformément dans Ω.

  T 0 ] . By virtue of (2.2.16)-(2.2.17)-(2.2.19), the Ascoli-Arzelà theorem and the relative compactness of {u n } in C Ω × [0, T 0 ] , we can find a subsequence, still denoted by {u n } for convenience, such that, for each > 0,

	.2.20)
	Using (2.2.2)-(2.2.16)-(2.2.19)-(2.2.20) and the compactness theorem in [[105] Corollary 4],
	we have that {u

n } is relatively compact in C [0, T 0 ]; C(Ω) = C Ω × [0,

  which implies (3.3.6), as well as |1 + εV y | ≥ 1/2 and |∇ Ū | ≤ 2µ. To check (3.3.7), we compute :

  .4.8) then the function ū satisfies the supersolution inequality in Ω δ . The estimate follows by applying the SCR to -λ -1 f L ∞ , u R,λ and ū.It is worth pointing out that, if M 2 is as in (4.4.8), then

	u R,λ < R on Ω for any R >	M 2 λ	.
	It follows that u R,λ is a viscosity solution of (4.4.1)-(4.4.2) for all R >	M 2 λ	. Theorem 4.3.1
	implies that u λ

  1 ∞ ) and (c 2 , u 2 ∞ ) be two solutions of the ergodic problem. If c 1 < c 2 or c 1 > c 2 , we could use Proposition 4.2.3 to obtain either u1 ∞ ≤ u 2 ∞ or u 2 ∞ ≤ u 1 ∞ .But such comparison cannot hold since, for all k ∈ R, u i ∞ + k are solutions as well of the ergodic problem, proving the uniqueness of c.

In this context, the term "nondegeneracy" describes a property of finite-time singularities (like in, e.g., Giga and Kohn[START_REF] Giga | Nondegeneracy of blowup for semilinear heat equations[END_REF]) and should not be confused with the notion of nondegenerate diffusion mentioned after Theorem 3.1.1.

It seems that the constants in some of the key estimates there are nonuniform as µ → 0 + , which prevents us to argue by a limiting procedure from the case µ > 0.

ϕ ∞ ,

d (β-1)(q-p+2) .

Remerciements

where β = θ/(1 + θ). Therefore ∇u C β (ω ×[t 0 /2,T )) ≤ C. The conclusion now follows by applying standard Schauder parabolic estimates to the PDE in (3.1.1), rewritten under the form

Indeed, the matrix (a ij ) = (a ij (x, y, t)) is uniformly elliptic due to (3.1.11) and

and there exists ν ∈ (0, 1) such that a ij , f ∈ C ν (ω × [t 0 /2, T ]), for each T < T , with norm independent of T .

Lemma 3.8.2. Assume (3.8.1)-(3.8.3), let y 1 ∈ (0, L 2 ) and let µ, u 0 satisfy the assumption of Theorem 3.1.1(ii). Then J ∈ C(D × (0, T ))

and there exists k 1 > 0 (depending in particular on y 1 ) such that, for any k ∈ (0, k 1 ], the function J satisfies J ≤ 0 on ∂D × (T /2, T ).

(3.8.9)

Proof. Since u = 0 for y = 0 and |∇u| ≤ C(τ ) in Ω × [0, τ ] for each τ < T , we have

Due to γ < α, we may therefore extend the function c(x)d(y)F (u) continuously to be 0 for y = 0. Property (3.8.8) then follows from the regularity of u (see Theorem 3.2.2) and we have J = 0 on (0, x 1 ) × {0} × (T /2, T ).

(3.8.10) By (3.1.10), we have u x = 0 on {0} × (0, y 1 ) × (0, T ), hence J = 0 on {0} × (0, y 1 ) × (T /2, T ). (3.8.11) Next, the function w = u x is ≤ 0 in Ω + × (0, T ) (cf. (3.1.10)) and satisfies there : On the other hand, we have (|∇u| q ) x = q|∇u| q-2 ∇u • ∇u x = q|∇u| q-2 ∇u • ∇J -qc dF |∇u| q-2 J + [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF], where

:= -qcdF |∇u| q (7 -)≤0

+ qcc d 2 F 2 |∇u| q-2 -qcd F |∇u| q-2 u y (7 + )≥0

.

we have thus proved the following lemma.

Lemma 3.9.1. Define the parabolic operator :

LJ := J t -|∇u| p-2 ∆J -(p -2)|∇u| p-4 D 2 J ∇u, ∇u -H 2 • ∇J -A 2 J.

Then LJ = (0 p ) + (0 q ) + (1) + (2) + (3) + (4) + ( 5) + ( 6) + [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF].

(3.9.4)

As a significant difficulty as compared with the semilinear case p = 2, many additional terms appear in the contributions (1), ( 2), ( 4)- [START_REF] Asai | On the interior derivative blow-up for the curvature evolution of capillary surfaces[END_REF], and especially nonlinear, second order terms in (4)- [START_REF] Asai | On the interior derivative blow-up for the curvature evolution of capillary surfaces[END_REF]. To proceed further, we need to observe that, among the second derivatives of u, u yy needs a special treatment, since it is not immediately expressed in terms of ∇J unlike u xx and u xy . Namely we shall eliminate u yy by expressing it in terms of u t , ∇u, u xx and u xy by using the equation. Although this will make the computation even more involved, by producing a lot of additional terms, this seems to be the only way to control the effects of u yy . The bound on u t given by Lemma 3.3.4 will be helpful in this process. (i) := -qd cF |∇u| q-2 u y (j) := +qc cd 2 F 2 |∇u| q-2 (j 1 ) + 2(p -2)c cd 2 F 2 |∇u| p-4 u 2 y |∇u| q-2 w

First we have

They give rise to the following decompositions :

With these terms, we have the following decompositions (using c = 0) :

It follows from (3.9.4) in Lemma 3.9.1 and (3.9.5)-(3.9.7) that LJ = (0 q ) + (0 p )

Reordering the terms, we obtain

(3.9.8)

Collecting the terms with J (reps., ∇J) in (3.9.8), together with those in A 2 (resp., H 2 ) and using (3.9.3), we define

where we recall that

with L = 0 u x , and

(3.9.12)

Next, we have

and, owing to Lemma 3.3.4,

Here and in the rest of the proof, K denotes a constant depending on u 0 C 2 , p and q. Consequently

(3.9.17) Using Young's inequality, we obtain that

(3.9.18) By (3.9.13), we have also u|∇u| q-p ≤ C q-p+1 0 .

(3.9.19)

Using again Young's inequality, and (3.9.19), we have

(3.9.20)

(3.9.21)

Next, using (3.9.13) and (3.9.14), it follows that

|∇u| p-4 u 2 y y 2 , (3.9.22)

and Using the bounds u ≤ u 0 ∞ and (3.9.14) we have

(3.9.26)

Combining (3.9.17)-(3.9.26), we get that

We pick smooth functions b = b τ and F = F τ with the following properties : b(s) = s (p-2)/2 and F (s) = s q/2 for δ

By the results in [80, Chapter V] (see Remark 3.10.1 below for details), there exists a (unique

Since v is also a weak solution of (3.2.1)-(3.2.3) in Q τ , by uniqueness of weak solutions (cf. Theorem 3.2.1(ii)), it follows that u = v τ on Q τ , hence (3.2.7). Remark 3.10.1. More precisely, in the special case when u 0 ∈ C 2+α (Ω) and u 0 satisfies the second order compatibility conditions, the existence of v claimed in the above proof follows from [80, Theorem V.6.1]. In the general case u 0 ∈ W 1,∞ (Ω), with u 0 = g on ∂Ω, this follows by a standard approximation procedure of u 0 by such smooth u 0,n . Namely, if v n denotes the solution originating from u 0,n , then, by [80, Theorems V.4.1, V.1.1 and V.5.4] respectively, we get uniform a priori estimates for the sequence v n in the spaces

(Ω × (0, τ ]) for some α ∈ (0, 1). We may then pass to the limit along a subsequence and obtain a solution with the announced properties.

In the proof of Theorem 3.2.2(ii)(iii), we shall use the following local regularity lemma. We note that only statement (ii) will be used here. The global version of statement (i) was already proved in Theorem 3.2.2(i). However, we give and prove its local version for completeness, since it was mentioned without proof in [7, p. 2487]. Lemma 3.10.1. Under the assumptions of Theorem 3.2.1, let u be the (maximal) weak solution of (3.1.1) and let P 0 = (x 0 , y 0 , t 0 ) ∈ Q T . Assume |∇u(P 0 )| > 0. Then :

(i) for some α ∈ (0, 1), u is a classical C 2+α,1+α/2 -solution on a space-time neighborhood of P 0 ;

(ii) for some β ∈ (0, 1), ∇u is C 2+β,1+β/2 on a space-time neighborhood of P 0 .

maximum principle, there exists c > 0 (independent of t 1 ), such that z ≤ -cx on Γ 2 . Choosing ε ∈ (0, c), we then have w ≤ -cx + εx < 0 on Γ 2 .

We have thus proved that

Now exchanging the roles of x, y and noticing that the assumptions (3.11.2) are symmetric in x, y, the conclusion already obtained guarantees that also z(x, y, t 1 ) ≤ -aεαe -αa 2 xy for 0 < y ≤ x < ã, hence (3.11.4) in (0, ã) 2 × [t 0 , τ 2 ). The extension to the remaining part of the rectangle (0, X 1 ) × (0, Y 1 ) (away from the corner (0, 0)) follows from the Hopf boundary lemma and the strong maximum principle.

Chapitre 4

Prolongement de la solution au delà des singularités via les solutions de viscosité Dans ce travail en collaboration avec Guy Barles, nous nous intéressons à la continuation des solutions explosives au delà du temps d'explosion. Pour se faire la théorie des solutions de viscosité offre un bon cadre de travail. Le comportement asymptotique des solutions prolongées est étudié via la considération standard d'un problème ergodique.

Introduction and Main Results

In this chapter we are interested in the following generalized Dirichlet problem for second-order degenerate parabolic partial differential equations

where q > p ≥ 2, u 0 and g are continuous functions satisfying the compatibility condition

Most of works devoted to this degenerate diffusive Hamilton-Jacobi equation concerned the case where Ω = R N , providing results on well-posedness, gradient estimates and asymptotic behavior of either classical or weak solutions in the sense of distributions (see [START_REF] Ben-Artzi | The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces[END_REF]1,[START_REF] Quittner | Superlinear parabolic problems : blow-up, global existence and steady states[END_REF][START_REF] Bartier | Gradient estimates for a degenerate parabolic equation with gradient absorption and applications[END_REF] and the references therein). Some other works are concerned with the solvability of the Cauchy-Dirichlet problem. They proved that, under suitable assumptions on u 0 and g, there exists a weak solution Proof of Theorem 4.1.2. If u is a subsolution of (4.1.7), then it is a subsolution in B r (x) = y ∈ R N ; |y -x| < r of the simpler equation

Now we are going to check the required hypotheses in [START_REF]A short proof of the C 0,α -regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications[END_REF].

H2. There exists a super-solution up to the boundary

for some η r > 0.

Despite the construction of the functions w r is a rather easy adaptation of [START_REF]A short proof of the C 0,α -regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications[END_REF], we reproduce it for the sake of completeness and for the reader's convenience. In order to build w r , we first build w 1 and then use the scale invariance of the equation. To do so, we borrow arguments from [START_REF]A short proof of the C 0,α -regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications[END_REF]. For C 1 , C 2 > 0 to be chosen later on and for β = q -p q -p + 1 , we consider the function

| q can be written as

Therefore, in order to prove the claim, we are going to show that, for C 1 , C 2 > 0 large enough, the bracket is positive and bounded away from 0 and that |Dw 1 (x)| p-2 remains large.

Computing the derivatives of w 1 in B 1 (0)\ {0}, we have

Appendice : A General Strong Comparison Result

A : Properties of the Regularization by Sup-convolution of Viscosity Subsolutions

To circumvent the lack of smoothness of the viscosity subsolution u, we consider instead the more regular time sup-convolution u α . Such regularization was first introduced by Lasry and Lions [START_REF] Lasry | A remark on regularization in Hilbert spaces[END_REF], and for 0 < α ≤ 1 and u a bounded usc, viscosity subsolution is defined by

iii) u α is Hölder continuous w.r.t the space variable x on Ω uniformly w.r.t the time for t > Kα.

Proof. Since u(x) is bounded, the supremum in (4.5.1) is attained at some point s * (t) which belong to the interval (t -Kα, t + Kα).

reaches a local maximum at (x 0 , s * (t 0 )). Recalling that u is a viscosity subsolution of (4.1.1), we get by definition

Next, let h > 0 small enough, then

A first estimate of u α t (from below) follows by dividing the previous inequality by h and sending h → 0. Exchanging the role of t + h and t provides the estimate from above.

The second assertion comes from the upper semi-continuity of u and the fact that u(x, 0) ≤ u 0 (x). Indeed

with s * (Kα) → 0 as α → 0. Taking the lim sup we get lim sup

Similarly, we use the semi-continuity of u and Proposition 4.2.1 to prove that

The last assertion is a consequence of Theorem 4.1.2 where

Let us note that, using the lower semi-continuity of v and v(x, 0) ≥ u 0 (x), we have v(x, Kα) ≥ u 0 (x) -o α (1). Hence

for some ω(α) satisfying lim α→0 ω(α) = 0.

B : Proof of Proposition 4.2.2

In order to prove the SCR, we are going to show that ũα -v ≤ ω(α) in Ω×[Kα, T -Kα]. Inequality (4.2.10) follows by passing to the limit as α → 0. To do so, the continuity of u α is a key point since it allows to use the arguments of [START_REF] Barles | Remarks on the Dirichlet problem for quasilinear elliptic and parabolic equations[END_REF][START_REF]A short proof of the C 0,α -regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications[END_REF].

For the sake of simplicity of notations, we drop the on ũα . The key idea is to compare u α µ := µu α and v with 0 < µ < 1 close to 1 in order to take care of the difficulty due to the |Du| q term. We argue by contradiction assuming that M α = max

If µ is sufficiently close to 1 and if η α > 0 is a constant small enough, then we have

We denote by

The existence of (x 0 , t 0 ) is guaranteed by the upper and lower semi-continuity of u α and v respectively (we drop the dependence of (x 0 , t 0 ) on η α , α and µ for the sake of simplicity of notations). Since M α η,µ > 0, we necessarily have t 0 > Kα in view of (4.5.2). By the Maximum Principle of the "Users guide" [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF], we have It follows that u α µ (z, t) → u α µ (x, t) and v(w, s) → v(x, t). Now, recalling the properties of u α and v at t = Kα, we have t, s > Kα for ε small enough. Next we claim that, for ε small enough the viscosity inequalities hold for u α and v. This is obviously the case for v if w ∈ Ω. If on the contrary w ∈ ∂Ω, then we necessarily have v(w, s) < g(w, s). Indeed if w ∈ ∂Ω then x ∈ ∂Ω. Since there is no loss of boundary conditions for subsolution s as clearly specified in Proposition 4.2.1, we have

Using that M α µ,η > 0, we cannot have v(x, t) ≥ g(x, t) since we would then have

a contradiction by sending µ → 1.

It follows that, if x ∈ ∂Ω, then we have necessarily that v(x, t) < g(x, t) and µu α (x, t) ≤ µ(g(x, t) + ω(α)). (4.5.7)

Hence, using that v(w, s) → v(x, t) < g(x, t), we deduce that if w ∈ ∂Ω, then v(w, s) < g(w, s) for ε small enough and the viscosity inequality holds also in this case.

On the other hand, from (4.5.5) we get that

which implies by the smoothness of the domain and the properties of χ that z lies in Ω for ε small enough and hence the viscosity inequality for u α µ holds too.

Next, we notice that u α µ satisfies

and we can also re-write it as

where ξ = ξ |ξ| for ξ = 0 and ξ = 0 if ξ ≡ 0.

The Jensen-Ishii's Lemma [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF] ensures the existence of X, Y ∈ S N , a, b ∈ R, q 1 , q 2 ∈ R N such that (a, q 1 , X) ∈ P 2,1,+ u α µ (z, t), (b, q 2 , Y ) ∈ P 2,1,-v(w, s), (4.5.9)

and (4.5.10) is an easy consequence of the boundedness of Dχ.

Moreover the viscosity inequalities for u α µ and v read

In the sequel we fix η α > 2o α (1) (recall that the o α (1) comes from the sup-convolution procedure and is fixed, therefore we can choose in such a way η α ). Since we may have a singularity at q 1 = 0 or q 2 = 0, we have to consider separately three cases. First we assume that there exists a constant γ > 0 such that

In this case the matrix A(ξ) = Id + (p -2)( ξ ⊗ ξ) is positive definite, so that its matrix square root σ exists and satisfies

Combining (4.5.10) with the fact that (4.5.11) implies that X ≤ Y + o ε (1), we have

)

At this point, we recall that the Lipschitz continuity of u α implies that |a| ≤ 2µK α . On the other hand we remark that, since 1 -µ p-1-q < 0, for fixed µ the term o(ε)|q 1 | q-p+2 is controlled by the (1 -µ p-1-q )|q 1 | q-p+2 term. Now we are going to let ε → 0 : if we assume that q 1 , q 2 (which depend on ε) are bounded, we may assume that they converge (we still denote their limits as q 1 , q 2 respectively). For µ close enough to 1, we get as ε → 0

Recalling that η α > 2o α (1), we get a contradiction when µ → 1 since the last term of the right-hand side is negative. Of course, we get the same contradiction if (at least for some subsequence) q 1 or q 2 → ∞.

If q 1 , q 2 = 0 but q 1 → 0, q 2 → 0 then, noticing that

, we can pass to the limit ε → 0 in the same way and obtain

and we also get a contradiction.

If q 1 = 0 or q 2 = 0, then necessarily q 1 = q 2 = 0 and, by subtracting (4.5.13) from (4.5.12), we have

We get a contradiction when ε → 0.

In all cases fixing η α > 2o α (1) we get a contradiction for ε small enough and µ close to 1 and the conclusion follows.

C : Proof of Proposition 4.2.3

The proof of (i) and (ii) are very similar. Indeed we know by Theorem 4.1.2 that subsolutions of (4.2.2) are Hölder continuous, so we are always in a case where the subsolution or the supersolution are continous. We will only give details of the proof for the p -Laplacian operator in the case where λ > 0 and v ∈ C(Ω). The other cases are an easy adaptation (the equation (4.2.1) is even easier to study). Since v is assumed to be continuous, we follow the proof of [START_REF] Barles | Remarks on the Dirichlet problem for quasilinear elliptic and parabolic equations[END_REF] with the same trick as before in order to take care of the strong growth of the gradient term. We argue by contradiction assuming that M = max

Classification des solutions globales pour le problème unidimensionnel

Dans ce chapitre on s'intéresse à la classification des solutions globales des solutions du problème 1D. Notre objectif est d'exclure l'existence de solutions globales non bornées en norme W 1,∞ (i.e l'explosion en temps infini) pour certaines données au bord et initiales étendant ainsi les résultat connus pour le cas de la diffusion linéaire.

Introduction and main results

In this chapter we are interested in the asymptotic behavior of global solutions to the following one-dimensional degenerate diffusive Hamilton-Jacobi equation

with q > p > 2, M ≥ 0 and suitably regular initial data u 0 . Problem (5.1.1) models a variety of physical phenomena which arise for example in the study of surface growth where a stochastic version of it is known as the Kardar-Parisi-Zhang equation (p = 2, q = 2). It has also a mathematical interest through the viscosity approximation of Hamilton-Jacobi type equations from control theory.

Solutions of (5.1.1) exhibit a rich variety of qualitative behaviors, according to the values of p ≥ 2 and q ∈ (0, ∞).

If q ≤ p, it is known that all solutions are global and bounded in W 1,∞ norm [START_REF] Ladyzenskaja | Linear and quasi-linear equations of parabolic type[END_REF]. For q ∈ [p -1, p] it was proved in [START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF] that nonnegative viscosity solutions of (5.1.1) with homogeneous Dirichlet boundary condition decay to 0 and the rate of convergence was also obtained, see also [START_REF] Benachour | Decay estimates for a viscous Hamilton-Jacobi equation with homogeneous Dirichlet boundary conditions[END_REF] for the semilinear case. Concerning the large time behavior of global weak solutions to (5.1.1) with homogeneous boundary conditions and q ∈ (0, p -1), it has been shown that there exists a one parameter family of nonnegative steady states, and any solution converges uniformly to one of these stationary solutions (cf. [START_REF] Stinner | Convergence to steady states in a viscous Hamilton-Jacobi equation with degenerate diffusion[END_REF][START_REF] Barles | Convergence to steady states for radially symmetric solutions to a quasilinear degenerate diffusive Hamilton-Jacobi equation[END_REF][START_REF] Ph | Convergence to steady states for a one-dimensional viscous Hamilton-Jacobi equation with Dirichlet boundary conditions[END_REF]).

For q > p ≥ 2, the situation is quite different. It is known that for any M ≥ 0 and suitably large u 0 , there exist solutions of (5.1.1) for which the L ∞ norm of the gradient blows up in finite time (the L ∞ norm of the solution remaining bounded) [START_REF] Ph | Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions[END_REF][START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF], while there exist global and decaying solutions for u 0 sufficiently small [110]. In view of a classification of all solutions of (5.1.1), it is then a natural question to ask whether or not C 1 -unbounded global solutions may exist. The question of the boundedness of global solutions of (5.1.1) was initiated for the semilinear case p = 2 in [5] and further investigated in [108,110]. Denoting M c := (q -1) q-2 q-1 /(q -2), the result of [5] says that if 0 ≤ M < M c , then any global solution of (5.1.1) is bounded in C 1 norm for t ≥ 0, that is,

On the other hand, it is known from [108] that some unbounded global solutions do exist if M = M c and u 0 ≤ U (x) where U (x) := M c x (q-2)/(q-1) is the unique singular steady state. Moreover the precise exponential rates of the gradient blow-up in infinite time was obtained.

Motivated by the results of the papers [5,108], we modify the method used by Arrieta, Rodriguez-Bernal and Souplet and extend their results on the classification of large time behavior of global solutions to the degenerate parabolic equation case p > 2.

From now on, we assume that q > p > 2. By a solution of (5.1.1), we mean a weak solution (see Section 2 below for a precise definition and well-posedness results). We recall that weak solutions of (5.1.1) satisfy a comparison principle, hence in particular min min

(5.1.3) Our main result is then the following :

) to the unique steady state.

(ii) If M > M b , then all weak solutions of (5.1.1) exhibit gradient blow-up in finite time.

The proof of Theorem 5.1.1 proceeds by contradiction. It relies on the analysis of steady states and the existence of a Lyapunov functional which enjoys nice properties on any global trajectory of (5.1.1), even if it were unbounded in C 1 norm. The construction of such a nice Lyapunov functional which is handled through the Zelenyak technique, together with

x ∈ ∂Ω, t > 0, u(x, 0) = u 0 (x),

x ∈ Ω, (5.2.3) where the boundary data g is the trace on ∂Ω of a regular function in C 2 (Ω), also denoted g, and the initial data u 0 satisfies

for x ∈ ∂Ω.

(5.2.4)

Theorem 5.2.1. Assume that q > p -1 and let u be a maximal weak Lipschitz solution of problem (5.2.3). We have the following statements.

(5.2.5)

(5.2.6)

.2.7)

Proof. The initial data being bounded and the sought-for estimate being invariant by addition of a constant, we may replace u with u -B, where B = min Ω u 0 ≥ min ∂Ω g. Then u ≥ 0 by the maximum principle. (i) This has been proved in chapter 2.

(ii) Fix t 0 ∈ (0, T max ) and let D = sup

where

Since u ≥ 0, λ ≥ 1 and γ ≥ 0, we have w ≥ u on {0} × Ω ∪ {(0, t 0 /λ) × ∂Ω}. On the other hand, we have on (0, t 0 /λ) × Ω :

Hence, by the comparison principle, we get that w ≥ u on (0, t 0 /λ) × Ω, that is

Dividing (5.2.8) by (λ -1) and letting λ → 1 + , we get γu + t ∂ t u ≥ -(q -p + 1)γD q t in D ((0, t 0 ) × Ω) .

The estimate (5.2.6) follows.

(iii) Fix t 0 ∈ (0, T max ). By (5.2.5)-(5.2.6), for h > 0 small, we have

where

Due to the translation invariance of (5.1.1), for t > t 0 /2, u(t + h, x) is still a solution of (5.1.1). Applying a comparison principle, we obtain that

Since h is arbitrary small, we conclude that

Thanks to the upper bound of u t , we derive the following lemma giving lower and upper bounds on u x , showing that u x remains bounded away from the boundary.

Lemma 5.2.1. Let u be a maximal weak Lipschiz solution of (5.1.1). For all t 0 ∈ (0, T max ), there exists C 2 = C 2 (t 0 , p, osc(u 0 ), M ) > 0 such that for all t ∈ [t 0 , T max ) and 0 < x < 1 ,

(5.2.10)

On any interval (a, b) with 0 < a < b < 1 where y > 0, the function y satisfies in the classical sense y +y q p-1 ≤ 0. Indeed, for each x ∈ (a, b), we have |u x | p-2 u x > C 2 x > C 2 a > 0 and the function u is smooth at such points since the equation is uniformly parabolic [START_REF] Ladyzenskaja | Linear and quasi-linear equations of parabolic type[END_REF]. Using theorem 5.2.1, we get that subsequence {u εn } of {u ε } such that u εn converges in C([0, τ ] × [0, 1]) ∩ C 0,1 loc ((0, τ ) × (0, 1)) to a solution ũ of (5.1.1). This was actually proved for u ε,0 ≡ u 0 , but an inspection of the proof shows that this is true in the general case. The uniqueness of the solution of (5.1.1) implies that ũ = u and that the whole sequence converges to u. We recall that u ε is bounded in L ∞ ([0, τ ]; W 1,∞ (0, 1)) (see Step 3 of the proof of Theorem 1.1 [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF]). The boundary regularity result of Proposition 5.2.1 implies that the convergence of {(u ε ) x } to u x holds in C loc ((0, τ ) × [0, 1]) (that is up to the boundary). Now fix T ∈ (0, T max ) and let T := sup {s > 0 such that T (u ε ) > s for ε > 0 small and

We know that

For any η ∈ (0, T ), we have

Thanks to (5.3.2) and the small-time existence and convergence result mentioned at the beginning of the proof, we can find τ = τ (A 1 ) > 0 (independent of η) and ε 0 = ε 0 (η) > 0 such that the problem

admits a unique classical solution u η ε on [0, τ ]. Moreover, we have

). Since T -η + τ > T for η small enough, this contradicts the definition of T . The second assertion follows from the estimates given in [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF]Inequalities 2.16 and 2.19].

Let us also note that due to q > p > 2, we have for ε small enough (p -1)ε p cosh(εx) p-1 ≥ ε q cosh(εx) q (it suffices to take 0 < ε < cosh(1) p-1-q q-p ). Hence u 0 L ∞ +M +2-cosh(εx) is a supersolution for problem (5.3.1). It is also easy to see that -u 0 L ∞ is a subsolution. Therefore there exists K > 0 depending only on u 0 L ∞ such that, ∀ε ∈ (0, 1/2), u ε (t, x) L ∞ ≤ K.

(5.3.4) Further, using lemma 5.2.1 and q > p, we get that for some r > 1

Since (w k l ) t → W t in D ((0, ∞) × (0, 1)) and δ ∈ (0, 1) is arbitrary, it follows that W t ≡ 0. Thus W is a steady state of (5.1.1) which implies that M ≤ M b . Given that the sequence t k → ∞ is arbitrary and the steady states (for given M ) are unique, it follows that the whole solution u(t) converges to W . Thanks to (5.2.6) in Theorem 5.2.1, we shall derive the following lemma providing a lower bound on the blow up profile of u x in case GBU occurs in finite or infinite time near x = 0 or 1. Lemma 5.4.1. Let u be a global weak solution of (5.1.1) and t 0 > 0. Let C 1 > 0 be the constant given in the estimate (5.2.7) of Theorem 5.2.1. There exist C 4 = C 4 (C 1 , p, q), C 5 = C 5 (p, q) > 0 with the following property. For all t ∈ [t 0 , +∞) and 0 ≤ y ≤ x ≤ 1

and

(5.4.2)

, where C 1 is given by the estimate of |u t | in Theorem 5.2.1. Using that |u t | ≤ C 1 and |u x (t)| q is bounded, we get that, Thanks to this property we can rule out infinite time gradient blow-up towards -∞ when x → 1.

Lemma 5.4.3. Let u be a global weak solution of (5.1.1). Then inf [0,∞)×(0,1) u x > -∞.

(5.4.4)

Proof. We proceed by contradiction. Assume that the lemma is false. Then, by Lemma 5.2.1, there exist a sequence t n → +∞ and x n → 0 such that u x (t n , 1 -x n ) → -∞. Fix ε > 0, then for n ≥ n(ε) large enough, we have

Taking t = t n and y = x n in (5.4.2), we get that for n ≥ n 0 (ε) large enough, we have for

This implies that

Choosing ε = ε(C 5 , C 4 ) > 0 small enough, we get that u

Using that u(t n , 1 -x n ) → M (by Proposition 5.3.3) and recalling Lemma 5.4.2, we end up with a contradiction.

Remark 5.4.2. Thanks to lemma 5.4.3 we deduce that, for the case M = M b , if there exist global solutions with infinite time gradient blow up (we expect that this could occur for some particular initial data), then u x can only blow up at x = 0. Proof of the boundedness of u x for M = 0 Lemma (5.4.3) is sufficient to prove the main theorem in the case M = 0. Let u be a global solution of (5.1.1). For M = 0, we note that w(t, x) := u(t, 1 -x) solves (5.1.1) with u 0 (1 -x) as initial data. Lemma (5.4.3) implies that u x and w x are bounded below on [0, +∞) × (0, 1), therefore u x is bounded. See the Subsection 5.4.2 for the proof of the convergence to the steady state in the W 1,∞ (0, 1) norm.

Proof. First let us note that there exists η > 0 such that (W M ) x ≥ 2η > 0 in [0, 1]. Next, by Theorem 5.1.1, we know that u x → W x uniformly on [0, 1]. Hence, there exists T > 0 such that u x (t, x) > (W M ) x (x) -η ≥ η for all x ∈ [0, 1], t > T .

(5.4.7)

The last inequality implies that the differential equation is uniformly parabolic for (t, x) ∈ [ T , ∞] × [0, 1]. Hence, by the standard theory (see [START_REF] Ladyzenskaja | Linear and quasi-linear equations of parabolic type[END_REF]) we know that u ∈ C 1,2 (( T , ∞) × [0, 1])

Appendix

Proof of Proposition 5.2.1 on the regularity of the derivative up to the boundary

and that the partial differential equation in (5.2.2) is satisfied in the sense of equality of functions in L 2 ((0, T ) × (0, 1)).

Next, we define an extension v * of v to [-1, 2] by setting

(5.4.9)

We will prove that v * is a weak solution of the following problem

x ∈ (0, 1).

(5.4.10)

where

)). Due to (5.4.8), for a.e. t ∈ (0, T ), we have

]). By elementary distribution theory (jump formula), it readily follows that (|v *

]). For a.e. t ∈ (0, T ), we can thus write : holds for all 0 < s < t < T and for all testing function ψ ∈ C ∞ c (R N × (0, T )). Now let α ∈ (0, 1) to be chosen later on. Set R = 3R 4 . We select a cut-off function η

for some C = C(α) > 0 (see [110] for the existence of such function). Lemma 6.2.1. Assume that u is a local weak solution of (6.1.1) and that |u| ≤ M in Q T,R for some M > 1. We consider a C 3 smooth increasing function f satisfying f > 0, the following differential equation

and mapping

Then at any point where |∇u| > 0, z satisfies the following differential inequality

where L(z) := ∂ t z -Az + H • ∇z (6.2.4)

with A is given by (6.3.4) H is given by (6.3.5).

The proof of lemma 6.2.1 is postponed to the the next section.

Proof of Theorem 6.1.1

Let u ∈ L ∞ loc ((0, ∞); W 1,∞ loc (Ω)) be a local weak solution of (6.1.1). Since u and ∇u are locally bounded, using the result of Di Benedetto and Friedman [START_REF] Dibenedetto | Hölder estimates for nonlinear degenerate parabolic sytems[END_REF][START_REF] Dibenedetto | Degenerate parabolic equations[END_REF], we get that ∇u is a locally Hölder continuous function. Thus z is a continuous function on B(x 0 , R )×[0, T ] = Q, for any 0 < T . Therefore, unless z ≡ 0 in Q, z must reach a positive maximum at some point

Since z(x, t) > 0, we have that |∇u| = f (v)|∇v| > 0 and hence we can use Lemma 6.2.1. Now let us take f (s) = M (s + 1) γ -2M where γ is given by

It is easy to see that f satisfies the differential equation (6.2.2) and f , f > 0 and f maps 0, 3

Choosing ε i in such way that ε

, we get that

q-p+1 + C(N, p, q, α)R -(q+2) .

(6.2.7)

Using the fact that 1 q-p+1 ≤ 2 q-p+2 ≤ 1 for q ≥ p, 1 ≤ 2 q-p+2 ≤ 1 q-p+1 for q ≤ p, we have that (f ) 1-q Lz ≤ -(q -1) 2(p -1)(N + 1) w q+2 2 η + C(N, p, q, α) R -q+2 q-p+1 + R -(q+2) .

Setting A = A(R, p, q, N ) := C(N, p, q) R -1 q-p+1 + R -1 2 and using that (f ) q-1 ≥ M q-1 ≥ 1, it follows that Lz ≤ -(q -1) 4(p -1)(N + 1) z q+2 2

in {(x, t) ∈ Q T,R ; z(x, t) ≥ A} . (

Next for λ = λ(q, N, p) > 0 suitably chosen, the function ψ(t) = λt -2 q satisfies ψ (t) ≥ -(q -1) 4(p -1)(N + 1)

Now for t 0 ∈ (0, T ) fixed, we define z(t) := z(t + t 0 , x) -ψ(t). It is easy to see that Lz ≤ 0 in {(x, t) ∈ Q T -t 0 ,R ; z(x, t) ≥ A} .

Since z(t) ≤ 0 for t > 0 sufficiently small, we deduce from the maximum principle that z(t) ≤ A, i.e. z(x, t + t 0 ) ≤ A + ψ(t) in Q T -t 0 ,R . Finally using that z = η|∇v| 2 , letting t 0 to 0, we get that |∇v| ≤ C(N, p, q)(A + t -2 q ) 1/2 .

Using that

v + 1 = 2 -u M

Here and in all the manuscript, the variable v is omitted in the expression of f , f , f f , etc. The equalities are understood in a classical sense in Q. Multiplying (6. Here, when passing from (6.3.2) to (6.3.3), the terms have been transformed according to 2 (∇v • ∇w) ∇v -q(f ) q-1 w q-2 2 ∇v (6. -2(q -1)(f ) q-2 f w

Leading estimates

Recalling that f is increasing and that f > 0, we get the following estimates.

1. Estimate of ηN w ηN w ≤ 2(p -1)((f ) p-3 f ) w p+2 2 η -2(q -1)(f ) q-2 f w Hence we get that L(z) ≤ -2(q -1)(f ) q-2 f w + C(p, q, α)R -1 η α w p+1 2 (f ) p-3 f + w q+1 2 (f ) q-1 .