Alain Miranville

M Arnaud

M Abdallah Elhamidi

Ihab Dannawi

Ahmad Z Fino

Mokhtar Kirane

Blow-up of solutions for a Semilinear Fractional Schrödinger equation

Keywords: Schrödinger equations, Blow-up, Fractional Laplacian. MSC : 35Q55, 35B44, Chapitre Nonlinear damped wave equation, Blow-up, Subcritical potential MSC : 35L15, 35L70, 35B33, 34B44, Cauchy problem, p-Laplacian equation, Local existence, Global nonexistence. MSC : 35K65, 35A01

À la fin de ce projet de recherche, je suis convaincu que la thèse est loin d'être un travail solitaire. En effet, je n'aurais jamais pu réaliser ce travail doctoral sans le soutien d'un grand nombre de personnes dont la générosité, la bonne humeur et l'intérêt manifestés à l'égard de ma recherche m'ont permis de progresser dans cette phase délicate de "l'apprenti-chercheur". C'est avec une certaine émotion et beaucoup de sincérité que je voudrais remercier toutes les personnes ayant soutenu et apprécié mon travail.

En premier lieu, je remercie les membres du jury de ma thèse pour l'intérêt qu'ils ont porté à mon travail et pour leur disponibilité. En particulier, je remercie M. Vincenzo Vespri et M. Mohammed Guedda d'avoir accepté d'être rapporteurs de mon manuscrit et de participer au jury de ma soutenance malgré leurs nombreuses charges. Leurs avis éclairés m'ont été précieux. Je remercie également M.

Chapitre 1 Introduction

Dans cette thèse, nous nous intéressons à l'étude de quatre équations d'évolution non-locales (cf. (1.2.1), (1.3.1), (1.4.1) et (1.5.1) ci-dessous). Les solutions de ces quatre équations peuvent exploser en temps fini.

Dans la théorie des équations d'évolution non-linéaires, une solution est qualifiée de globale si elle est définie pour tout temps positif. Au contraire, si une solution existe seulement sur un intervalle de temps [0, T ) borné, elle est dite locale. Dans ce dernier cas et quand le temps maximal d'existence est relié à une alternative d'explosion, on dit aussi que la solution explose en temps fini. Cependant, pour donner un sens à la notion d'explosion en temps fini, il faut bien préciser l'espace dans lequel on travaille et avec quelle norme on "mesure" la solution.

Dans le deuxième chapitre, nous considérons l'équation de Schrödinger non-linéaire avec une puissance fractionnaire du laplacien, et nous obtenons que la solution explose en temps fini T max > 0 pour toute condition initiale positive et non-triviale dans le cas d'exposant sous-critique.

Dans le troisième chapitre, nous étudions une équation des ondes amorties avec un potentiel d'espace-temps et un terme non-linéaire et non-local en temps. Nous obtenons un résultat d'existence locale d'une solution dans l'espace d'énergie sous des conditions restrictives sur les données initiales, la dimension de l'espace et de la croissance du terme non-linéaire. De plus, nous obtenons que la solution explose en temps fini pour toute condition initiale de moyenne strictement positive.

Le quatrième chapitre est consacré à l'étude du problème de Cauchy pour l'équation d'évolution p-Laplacien avec une mémoire non linéaire. On étudie l'existence locale d'une solution ainsi qu'un résultat de non-existence de solution globale.

Finalement, dans le cinquième chapitre, nous étudions l'intervalle maximal d'existence des solutions du problème d'évolution dans un milieu poreux avec un terme non-linéaire non-local en temps. De plus, nous obtenous que la solution n'existe pas globalement sous certaines conditions sur les données initiales.

Pour mettre en évidence les points importants et afin d'éviter qu'ils ne soient cachés dans les détails techniques, nous donnons dans cette introduction générale, des énoncés simplifiés des résultats. Les énoncés complets se trouvent dans les différents chapitres présentés ci-dessous.

Préliminaires

Préliminaires

Soit AC[0, T ] l'espace de toutes les fonctions absoluement continues sur [0, T ] avec 0 < T < ∞. Alors pour f ∈ AC[0, T ], on définit les dérivées fractionnaires de Riemann-Liouville à droite et à gauche de f d'ordre α ∈ (0, 1) par

D α 0|t f (t) := ∂ t J 1-α 0|t f (t) et D α t|T f (t) := - 1 Γ(1 -α) ∂ t T t
(s -t) -α f (s) ds, t ∈ [0, T ], (1.1.1)

où ∂ t = d dt et J α 0|t g(t) := 1 Γ(α) t 0 (t -s) α-1 g(s) ds (1.1.2)
est l'intégrale fractionnaire de Riemann-Liouville pour toute fonction g ∈ L q (0, T ) et 1 ≤ q ≤ ∞.

(Voir [START_REF] Kilbas | Theory and Applications of Fractional Differential Equations[END_REF]). De plus, pour toutes f, g ∈ C([0, T ]) telles que D α 0|t f (t), D α t|T g(t) existent et sont continues, pour tout t ∈ [0, T ], 0 < α < 1, on a la formule d'intégration par partie suivante (Voir [START_REF] Samko | Fractional integrals and derivatives, Theory and Applications[END_REF])

T 0 D α 0|t f (t)g(t) dt = T 0 f (t) D α t|T g (t) dt. (1.1.3)
Notons que pour toute f ∈ AC n+1 [0, T ] et tout entier n ≥ 0, on a (voir (2.2.30) dans [START_REF] Kilbas | Theory and Applications of Fractional Differential Equations[END_REF])

(-1) n ∂ n t .D α t|T f = D n+α t|T f, (1.1.4)
où AC n+1 [0, T ] := {f : [0, T ] → R et ∂ n t f ∈ AC[0, T ]} et ∂ n t est la n-ième dérivée. De plus, pour tout 1 ≤ q ≤ ∞, la formule suivante (voir [37, Lemma 2.4 p.74]) D α 0|t J α 0|t = Id L q (0,T ) (1.1.5) est valable presque partout sur [0, T ].

Dans ce qui suit, on utilise les résultats suivants : Si w 1 (t) = (1 -t/T ) σ + , t ≥ 0, T > 0, σ ≫ 1, alors (1.1.7) Pour la preuve de ces résultats, voir [START_REF] Fino | Finite time blow-up for a wave equation with a nonlocal nonlinearity[END_REF]Preliminaries].

D α t|T w 1 (t) = CT -σ (T -t) σ-α + , D α+1 t|T w 1 (t) = CT -σ (T -t) σ-α-1 + , D α+2 t|T w 1 (t) = CT -σ (T -t) σ-α-2 + , ( 1

Explosion en temps fini pour une équation de Schrödinger non-linéaire avec un laplacien fractionnaire

Dans le deuxième chapitre, nous nous intéressons à l'équation de Schrödinger non-linéaire avec un laplacien fractionnaire :

     i∂ t u = Λ α u + λ|u| p (t, x) ∈ [0, T ) × R N , u(x, 0) = f (x) x ∈ R N , (1.2.1)
où le laplacien fractionnaire Λ α = (-∆) α/2 avec 0 < α < 2 est un opérateur pseudo-différentiel défini par la transformée de Fourier : Λ α u(ξ) = |ξ| α u(ξ). De plus, nous supposons que T > 0, 1 < p ≤ 1 + α N , u = u(x, t) est une fonction à valeur complexe, λ ∈ C\{0} et f = f (x) ∈ H α 2 (R N ) est une fonction donnée à valeur complexe. En fait, le laplacien fractionnaire est un cas particulier de l'opérateur de Lévy L qui est bien un opérateur pseudo-differentiel défini par Lv = a(ξ) v(ξ). Comme la fonction e -ta(ξ) est bien définie et positive, alors le symbole a(ξ) peut se représenter, comme dans [START_REF] Biler | Asymptotics for conservation laws involving Lévy diffusion generators[END_REF], par la formule de Lévy-Khintchine (cf. [6, Chapitre 1, Théorème 1] ou [START_REF] Jacob | Pseudo-Differential Operators and Markov Processes[END_REF]Théorème B.2])

a(ξ) = ibξ + q(ξ) + R N (1 -e -iηξ -iηξ1 {|η|<1} (η))Π( dη).
Dans la littérature de la physique mathématique (voir [START_REF] Biler | Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws[END_REF][START_REF] Droniou | Fractal first-order partial differential equations[END_REF][START_REF] Karch | Fractal Hamilton-Jacobi-KPZ equations[END_REF]), les problèmes d'évolution nonlinéaires avec un laplacien fractionnaire décrivent la diffusion anormale (the anomalous diffusion) ou ce qu'on appelle la diffusion α-stable de Lévy (α-stable Lévy diffusion). Dans les dernières années, l'étude du calcul fractionnaire et des équations intégrodifférentielles fractionnaires a évoluée vue qu'elle s'applique à de nombreux domaines notamment la physique, (voir [START_REF] Kilbas | Theory and Applications of Fractional Differential Equations[END_REF], [START_REF] Metzler | The random walk's guide to anomalous diffusion : a fractional dynamics approach[END_REF], [START_REF] Metzler | The restaurant at the random walk : recent developments in the description of anomalous transport by fractional dynamics[END_REF] et les références citées). Meltzler et Klafter ont discuté les développements récents dans la description de la diffusion anormale par l'approche de la dynamique fractionnaires dans [START_REF] Metzler | The random walk's guide to anomalous diffusion : a fractional dynamics approach[END_REF] et [START_REF] Metzler | The restaurant at the random walk : recent developments in the description of anomalous transport by fractional dynamics[END_REF] où de nombreuses équations aux dérivées partielles fractionnaires sont dérivées asymptotiquement du modèle de marche aléatoire de Lévy, qui représente une sorte de généralisation naturelle des modèles de marche Browniens. Inspiré par l'approche du chemin de Feynman à la mécanique quantique, Laskin a utilisé le chemin intégrale de Lévy pour obtenir une équation fractionnaire de Schrödinger. Ce qui généralise un résultat classique concernant l'intégrale du chemin sur des trajectoires browniens conduisant aux équations standards de Schrödinger (voir [START_REF] Laskin | Fractional quantum mechanics and Lévy integral[END_REF], [START_REF] Laskin | Fractional quantum mechanics[END_REF]). Nous trouvons quelques papiers étudiants les équations fractionnaires de Schrödinger et leurs applications, (voir [START_REF] Guo | Existence and stability of standing waves for nonlinear fractional Schrödinger equations[END_REF], [START_REF] Rozmej | On fractional Schrödinger equation[END_REF] et les références citées). Quand α = 2, il est bien connu que (1.2.1) est bien posé localement dans H 1 (R N ) si 1 < p < 1 + 4 (N -2) + (voir [START_REF] Cazenave | Introduction aux problèmes d'évolution semi-linéaires[END_REF]). En outre, il est également connu que les solutions locales peuvent être prolongées en solutions globales en considérant des petites données lorsque p est plus grand que l'exposant de Strauss p s qui est défini comme étant la racine positive de N p 2 -(N + 2)p -2 = 0 (voir [START_REF] Cazenave | Semilinear Schrödinger Equations[END_REF]). Cependant, il n'y a pas de résultats d'existence globale pour p ≤ p s . En 2013, Ikeda et Wakasugi [START_REF] Ikeda | Small data blow-up of L 2 -solution for the nonlinear Schrödinger equation without gauge invariance[END_REF] ont montré un résultat d'explosion pour la solution du problème (1.2.1) pour des petites données quand 1 < p ≤ 1 + 2 N . L'objectif principal de cette étude est de généraliser le résultat d'explosion de Ikeda et Wakasugi [START_REF] Ikeda | Small data blow-up of L 2 -solution for the nonlinear Schrödinger equation without gauge invariance[END_REF] pour des solutions douces des équations fractionnaires de Schrödinger dont la définition est donnée ci-dessous.

Définition 1.2.1 (Solution douce) Soit f ∈ H α 2 (R N ), 0 < α ≤ 2, p > 1 et T > 0. Nous disons que u ∈ C([0, T ], H α 2 (R N )) est une solution douce du problème (1.2.1) si u est solution de l'équation intégrale u(t) = S(t)f -iλ t 0 S(t -s)|u(s)| p ds, (1.2.2)
où le générateur infinitesimal A du C 0 groupe de l'opérateur unitaire S(t) est défini par Au = -i(-∆)

α 2 u pour -∞ < t < ∞, sur L 2 (R N ).
L'existance locale est basée sur le théorème de point fixe de Banach, via la théorie des semi-groupes et le théorème de Stone concernant l'opérateur fractonnaire A = -i(-∆) α 2 qui est un générateur infinitésimal du groupe continu des opérateurs unitaires sur L 2 (voir [START_REF] Cazenave | Introduction aux problèmes d'évolution semi-linéaires[END_REF]). La méthode utilisée pour démontrer le résultat d'explosion est la méthode des fonctions tests développée par Zhang [START_REF] Tsutsumi | L 2 -solutions for nonlinear Schrödinger equations and nonlinear groups[END_REF], puis par Pohozaev et Mitidieri [START_REF] Laskin | Fractional quantum mechanics and Lévy integral[END_REF]. Pour énoncer notre résultat principal, soit λ = λ 1 + iλ 2 et f = f 1 + if 2 . Nous introduisons les hypothèses suivantes :

f 1 ∈ L 1 (R N ), λ 2 R N f 1 dx > 0 ou f 2 ∈ L 1 (R N ), λ 1 R N f 2 dx < 0 (1.2.3)
Notre résultat principal est le théorème suivant :

Théorème 1.2.2 (Explosion en temps fini) Soit λ ∈ C\{0}, et f ∈ H α 2 (R N ) satisfaisant à (1.2.3). Si 1 < p ≤ 1 + α
N alors la solution douce de (1.2.1) explose en temps fini.

       u tt -∆u + a(x)b(t)u t = t 0 (t -s) -γ |u(s)| p ds, t > 0, x ∈ R n , u(0, x) = u 0 (x), u t (0, x) = u 1 (x), x ∈ R n , (1.3.1) où u est une fonction à valeurs réelles, n ≥ 1, 0 < γ < 1 et p > 1. Le coefficient du terme d'amortissement s'écrit a(x)b(t) := a 0 (1 + |x| 2 ) -α 2 (1 + t) -β , avec a 0 > 0, α, β ≥ 0 et α + β < 1.
Dans ce travail, nous supposons que les données initiales sont dans l'espace d'énergie suivant

(u 0 , u 1 ) ∈ H 1 (R n ) × L 2 (R n ). (1.3.2)
Les notations suivantes

• q et • H 1 (1 ≤ q ≤ ∞), représentent les normes des espaces L q (R n ) et H 1 (R n ), respectivement.
Le terme non local non linéaire peut être considéré comme une approximation (avec un changement de variable approprié) de la non-linéarité de l'équation des ondes amorties semi-linéaires suivantes

u tt -∆u + a(x)b(t)u t = |u(t)| p , car lim γ→1 s -γ + = Γ(1 -γ)δ(s)
p c = 1 + 2 n ,
qui est bien connu comme l'exposant critique de Fujita pour l'équation de la chaleur u t -∆u = u p (voir [START_REF] Fujita | On the blowing up of solutions of the problem for u t = ∆u + u 1+α[END_REF]). Plus précisément, ils ont démontré l'existence globale de solution pour des petites données initiales dans le cas p > 1 + 2/n. Cependant, pour le cas 1 < p < 1 + 2/n, il y a explosion en temps fini des solutions de (1.3.3). Plus tard, Zhang [START_REF] Qi | A blow up result for a nonlinear wave equation with damping : the critical case[END_REF] et indépendamment Kirane et Qafsaoui [START_REF] Kirane | Fujita's exponent for a semilinear wave equation with linear damping[END_REF] ont montré que l'exposant critique p c = 1 + 2/n fait partie de la région d'explosion. Todorova et Yordanov supposent que les données initiales ont un support compact et utilisent essentiellement cette propriété dans [START_REF] Todorova | Critical exponent for a nonlinear wave equation with damping[END_REF]. Cependant, Ikehata et Tanizawa suppriment cette hypothèse dans [START_REF] Ikehata | Global existence for solutions for semilinear damped wave equation in R N with noncompactly supported initial data[END_REF]. Ikehata et al. ont étudié dans [START_REF] Ikehata | Critical exponent for semilinear wave equations with space-dependent potential[END_REF] le cas où le coefficient dépend de l'espace

u tt -∆u + a(x)u t = |u| p , où a(x) ≡ a 0 (1 + |x| 2 ) -α/2 , |x| → ∞, symétrique radiale et 0 ≤ α < 1.
Ils démontrent que l'exposant critique de (1.3.3) est donné par

p c = 1 + 2 n -α
en utilisant une méthode de multiplicateurs raffinée. Leur méthode dépend aussi de la propriété de propagation à vitesse finie. Récemment, Nishihara [START_REF] Nishihara | Asymptotic behavior of solutions to the semilinear wave equation with timedependent damping[END_REF] et Lin et al. [START_REF] Lin | Critical exponent for the semilinear wave equation with time-dependent damping[END_REF] ont consideré une équation d'onde semi-linéaire avec un terme d'amortissemnet en temps

u tt -∆u + b(t)u t = |u| p , où b(t) = b 0 (1 + t) -β , β ∈ (-1, 1).
Ils ont montré que l'exposant critique est

p c = 1 + 2 n .
Cela montre que les coefficients dépendants du temps sous une forme spéciale dans le terme d'amortissement n'influencent pas l'exposant critique. D'autre part, l'équation (1.3.1) peut être écrite sous la forme

u tt -∆u + a(x)b(t)u t = J δ 0|t (|u| p ) (t), (1.3.4)
où δ = 1-γ, et J δ 0|t est donné par (1.1.2). Nous présentons dans ce qui suit notre résultat principal. Nous annonçons d'abord le résultat d'existence locale de la solution du problème (1.3.1).

Définition 1.3.1 (Solution douce) Soit (u 0 , u 1 ) ∈ H 1 × L 2 . Une fonction u est dite solution douce de (1.3.1) avec données initiales u(0) = u 0 et u t (0) = u 1 si u ∈ C([0, T ); H 1 ) ∩ C 1 ([0, T ); L 2 ) et si u satisfait à l'équation intégrale suivante u(t, x) = R(t, 0)(u 0 , u 1 ) + Γ(δ) t 0 S(t, s)J δ 0|t (|u| p ) ds (1.3.5) au sens H 1 (R n ), où δ = 1-γ, S(t, s)g := R(t, s)(0, g) pour toute fonction g ∈ H 1 (R N ) et R(t,
Soit α ≥ 0, β ∈ R, 1 < p ≤ n/(n -2) pour n ≥ 3, et p ∈ (1, ∞) pour n = 1, 2
u ∈ C([0, T max ), H 1 (R n )) ∩ C 1 ([0, T max ), L 2 (R n )), où 0 < T max ≤ ∞. De plus, si T max < ∞, alors u(t) H 1 + u t (t) 2 → ∞ quand t → T max . Remarque 1.3.3
On dit que u est une solution globale de (1.3.1) si T max = ∞, alors que dans le cas où T max < ∞, on dit que u explose en temps fini.

Soit p c := 1 + 3 -γ (n -α -1 + γ) + . comme (p c = n/(n -2)) ⇐⇒ (γ = (n + 2(α -2))/n),
ceci implique, dans le cas où (n + 2(α -2))/n ≤ γ, que p c ≤ n/(n -2). Notons que

p c → 1 + 2 n -α quand γ → 1.
Notre résultat principal est le suivant Théorème 1.3.4 (Résultats d'explosion en temps fini) Sous l'hypothèse αβ = 0, on introduit

p * := 1 + 3 -γ (N -α -1 + γ) + et γ * := N + 2(α -2) N . Supposons que (u 0 , u 1 ) ∈ H 1 (R N ) × L 2 (R N ) et que R n a(x)u 0 dx > 0, R n u 1 (x) dx > 0. (1.3.6)
La solution douce du problème (1.3.1) explose en temps fini si i)

       1 < p ≤ N N -2 et γ ≥ γ * N ≥ 3, p ∈ (1, ∞) N = 1, 2. et p ≤ p * ii) N ≥ 3, 1 < p ≤ N/(N -2) et γ ≤ γ * .
La méthode des fonctions tests (voir [START_REF] Fino | Finite time blow-up for a wave equation with a nonlocal nonlinearity[END_REF][START_REF] Fino | Decay of mass for nonlinear equation with fractional Laplacian[END_REF][START_REF] Fino | Qualitative Properties of Solutions to a Time-Space Fractional Evolution Equation[END_REF][START_REF] Kirane | Critical exponents of Fujita type for certain evolution equations and systems with Spatio-Temporal Fractional derivatives[END_REF][START_REF] Mitidieri | A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities[END_REF][START_REF] Mitidieri | Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on R N[END_REF][START_REF] Qi | A blow up result for a nonlinear wave equation with damping : the critical case[END_REF]

       u t -div |∇u| p-2 ∇u = t 0 (t -s) -γ |u(s)| q-1 u(s) ds, t > 0, x ∈ R n , u(0, x) = u 0 (x), x ∈ R n , (1.4.1) où p > 2, q > 1, 0 < γ < 1 et u 0 ∈ L ∞ (R n ).
Le premier résultat sur l'explosion de la solution pour l'équation de la chaleur a été obtenu par Fujita dans [START_REF] Fujita | On the blowing up of solutions of the problem for u t = ∆u + u 1+α[END_REF]. Il a étudié le problème de Cauchy suivant Dans le cas p = 1 + 2/n, il est démontré par l'auteur de [START_REF] Hayakawa | On nonexistence of global solutions of some semilinear parabolic equations[END_REF] pour n = 1, 2 et par les auteurs de [START_REF] Kobayashi | On the blowing up problem for semilinear heat equations[END_REF] pour n ≥ 1 que (1.4.2) n'a pas de solution globale u(x, t) satisfaisant u(• , t) ∞ < ∞ pour t ≥ 0. L'auteur de [START_REF] Weissler | Existence and nonexistence of global solutions for a semilinear heat equations[END_REF] démontre que si p = 1 + 2/n, (1.4.2) n'a pas de solution globale u(x, t) satisfaisant u(• , t) q < ∞ pour t > 0 et pour q ∈ [1, +∞). La valeur p c = 1 + 2/n est appelé l'exposant critique de (1.4.2). Il joue un rôle important dans le comportement de la solution de (1.4.2). Dans les dernières années, il y a eu un certain nombre d'extensions des résultats de Fujita dans de nombreuses directions. Récemment, les auteurs de [START_REF] Cazenave | An equation whose Fujita critical exponent is not given by scaling[END_REF] ont étendu les résultats de Fujita pour l'équation de la chaleur avec un terme de mémoire non-linéaire

     u t -∆u = u p , t > 0, x ∈ R n , u(0, x) = u 0 (x), x ∈ R n , (1.4.2
       u t -∆u = t 0 (t -s) -γ u(s) p ds, t > 0, x ∈ R n , u(0, x) = u 0 (x), x ∈ R n , (1.4.3) où p > 1, 0 < γ < 1, et u 0 ∈ C 0 (R n ).
    u t -div |∇u| p-2 ∇u = u q , t > 0, x ∈ R n , u(0, x) = u 0 (x), x ∈ R n , (1.4.4)
où p > 2 et q > 1, l'auteur de [START_REF] Zhao | On the Cauchy problem and initial traces for the evolution P-Laplacian equations with strongly nonlinear sources[END_REF] a étudié le problème de Cauchy ; et il a obtenu l'exposant critique q c = p -1 + p/n à condition que p > 2n/(n + 1). En outre, l'auteur démontre que si max{1, p -1} < q < q c , alors le problème de Cauchy n'a pas de solution globale ; tandis que si q > q c et u 0 (x) est assez petite, alors le problème de Cauchy admet une solution globale positive. Les auteurs de [START_REF] Andreucci | A Fujita type result for a degenerate Neumann problem with non compact boundary[END_REF][START_REF] Liu | The critical exponent of doubly singular parabolic equations[END_REF]

u ∈ C([0, T ] : L 1 (Ω)) ∩ L p loc (0, T : W 1 p (Ω)) ∩ L ∞ loc (S T ), et Ω u(x, t)φ(x, t) dx + t t 0 Ω [-uφ t + |∇u| p-2 ∇u • ∇φ] dx dτ = Γ(δ) t t 0 Ω J δ 0|τ (|u| q-1 u)ϕ dxdτ + Ω u(x, t 0 )φ(x, t 0 ) dx (1.4.6) pour tout 0 ≤ t 0 < t ≤ T et φ ∈ C 1 (Ω × [0, T ]), φ(•, T ) = 0 au voisinage de ∂Ω, où δ = 1 -γ. De plus, lim t →0 Br |u(x, t) -u 0 (x)| dx = 0, ∀r > 0. (1.4.7)
Nous utilisons ν(a 1 , a 2 , • • • , a n ) pour définir des constantes positives dépendants seulement de

a 1 , a 2 , • • • , a n . Soit sgn η s =        1 si s > η, s η si -η ≤ s ≤ η, -1 si s < -η. Pour f ∈ L loc (R N ), nous définissons |||f ||| h = sup x∈R N 1 |B 1 (x)| B 1 (x) |f (y)| h dy 1 h . (1.4.8) Théorème 1.4.2 (Existence locale) Soit |||u 0 ||| h < ∞ où h = 1 si 1 ≤ q < p -1 + p N et h > N p (q -p + 1) si q ≥ p -1 + p N .
(1.4.9)

Alors il existe une constante ν = ν(N, p, q) ≥ 1 et une constante positive T 0 définie par

T 0 + T 0 |||u 0 ||| p-2 h + T 1- N (q-p+1) ph 0 |||u 0 ||| q-1 h = ν -1 (1.4.10)
et une solution faible u to (1.4.1) dans S T satisfaisant, pour tout 0 < t < T 0 ,

|||u(•, t)||| h ≤ ν|||u 0 ||| h , (1.4.11) |u(x, t)| ≤ νt -N k h + γ-1 q-1 |||u 0 ||| ph k h h , (1.4.12
)

où k h = N (p -2) + ph.
Notre second résultat principal est le suivant Théorème 1.4.3 (Non-existence de solutions globales)

Pour p > 1 et 0 < γ < 1, considérons q c := p -1 + (1 -γ)(p -2) + (2 -γ)p(1 + (1 -γ)(p -2)) n -(1 -γ)p . Si p -1 + (1 -γ)(p -2) < q ≤ q c
, alors le problème (1.4.1) n'a pas de solutions faibles globales non-triviales.

La méthode des fonctions tests constitue aussi l'idée principale de la preuve de ce résultat.

Durée de vie des solutions non-négatives pour des équations d'évolution non-locales en temps

Dans cette partie, nous étudions l'intervalle maximal d'existence de solution du problème

u t -∆|u| m-1 u = t 0 (t -s) -γ |u| p u(s) ds, x ∈ R N , t > 0, (1.5.1)
où p > 0, m > 1 et 0 < γ < 1 avec une condition initiale non triviale non négative continue bornée Kuiper a trouvé que la durée de vie L(σ) est bornée par Cσ -(p+1-m) quand u σ (x, 0) = σu 0 (x), σ > 0. (voir Théorème 3.6 [START_REF] Kuiper | Life span of nonnegative solutions to certain quasilinear parabolic Cauchy problems[END_REF]). Cette partie est motivée mathématiquement par les articles récents [START_REF] Cazenave | An equation whose Fujita critical exponent is not given by scaling[END_REF][START_REF] Li | Lifespan and a new critical exponent for a nonlocal parabolic equation with slowly decay initial values[END_REF] qui traitent l'exposant critique et la durée de vie pour l'équation parabolique avec un terme non linéaire non local en temps

u(x, 0) = u 0 (x) ≡ 0, u 0 (x) ≥ 0, x ∈ R N . ( 1
u t -∆u = t 0 (t -s) α-1 |u| p u ds, x ∈ R N , t > 0, (1.5.3)
où u 0 ∈ C 0 (R N ), l'espace de toutes les fonctions continues qui tendent vers zero à l'infini. Cette équation est un cas particulier de (1.5.1) correspondant à m → 1.

Notre analyse est basée sur le fait que l'équation (1.5.1) peut être écrite sous la forme suivante :

u t -∆|u| m-1 u = J α 0|t (|u| p ).
(1.5.4) L'intégrale J α 0|t joue un rôle essentiel dans la démonstration du théorème sur la durée de vie. Notons que α = 1 -γ ∈ (0, 1) dans (1.5.4).

Notre premier résultat est le suivant

Durée de vie des solutions non-négative pour des équations d'évolution Théorème 1.5.1 (Non-existence de solution globale) La méthode utilisée pour prouver la durée de vie et les conditions nécessaires pour l'existence de solution locale et globale est la méthode de fonctions tests [START_REF] Baras | Local and global solvability of a class of semilinear parabolic equations[END_REF][START_REF] Baras | Critère d'existence de solutions positives pour des équations semilinéaires non monotones[END_REF][START_REF] Fino | Qualitative Properties of Solutions to a Time-Space Fractional Evolution Equation[END_REF][START_REF] Mitidieri | A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities[END_REF][START_REF] Qi | A blow up result for a nonlinear wave equation with damping : the critical case[END_REF]. Le principe de cette méthode est le suivant : nous supposons, par l'absurde, que la solution est globale. Ensuite, nous faisons un choix approprié de la fonction test.

Soient p > 0, m > 1, 0 < γ < 1 et α = 1 -γ. Si p ≤ p * = m -1 + N α(m -

Introduction

We study the initial-value problem for the nonlinear Schrödinger equation

     i∂ t u = Λ α u + λ|u| p (t, x) ∈ [0, T ) × R N , u(x, 0) = f (x) x ∈ R N , (2.1.1)
where the pseudo-differential operator Λ α = (-∆) α/2 with 0 < α ≤ 2 is defined via the Fourier transformation :

Λ α u(ξ) = |ξ| α u(ξ). Moreover, we assume that T > 0, 1 < p ≤ 1 + α N , u = u(x, t) is a complex-valued unknown function, λ ∈ C\{0} and f = f (x) ∈ H α 2 (R N ) is a given complex- valued function.
In recent years, the study of fractional calculus and fractional integrodifferential equations applied to physics and other areas has grown considerably, (see [START_REF] Kilbas | Theory and Applications of Fractional Differential Equations[END_REF], [START_REF] Metzler | The random walk's guide to anomalous diffusion : a fractional dynamics approach[END_REF], [START_REF] Metzler | The restaurant at the random walk : recent developments in the description of anomalous transport by fractional dynamics[END_REF]) and references therein.

Meltzler and Klafter discussed recent developments in the description of anomalous diffusion by the fractional dynamics approach in [START_REF] Metzler | The random walk's guide to anomalous diffusion : a fractional dynamics approach[END_REF] and [START_REF] Metzler | The restaurant at the random walk : recent developments in the description of anomalous transport by fractional dynamics[END_REF] where many fractional partial differential equations are derived asymptotically from Lévy random walk models, a natural generalization of the brownian walk models. Inspired by the Feynman path approach to quantum mechanics, Laskin used the path integral over Lévy -like quantum mechanical paths to obtain a fractional Schrödinger equation, which extends a classical result that the path integral over brownian trajectories leads to the standard Schrödinger equations, (see [START_REF] Laskin | Fractional quantum mechanics and Lévy integral[END_REF], [START_REF] Laskin | Fractional quantum mechanics[END_REF]). There are some papers in studying fractional Schrödinger equations and their applications, (see [START_REF] Guo | Existence and stability of standing waves for nonlinear fractional Schrödinger equations[END_REF], [START_REF] Rozmej | On fractional Schrödinger equation[END_REF])and references therein. When α = 2, it is well known that local well-posedness holds for ( 2 [START_REF] Cazenave | Introduction aux problèmes d'évolution semi-linéaires[END_REF]). Moreover, it is also known that the local solutions can be extended globally for some small data when p is larger than the Strauss exponent p s , which is the positive root of N p 2 -(N + 2)p -2 = 0 (see [START_REF] Cazenave | Semilinear Schrödinger Equations[END_REF]). However, there have been no results on global existence for p ≤ p s . In 2013 Ikeda and Wakasugi [START_REF] Córdoba | A maximum principle applied to quasi-geostrophic equations[END_REF] have proved a small-data blow-up result for (1) when

.1.1) in H 1 (R N ) if 1 < p < 1 + 4 (N -2) + (see
1 < p ≤ 1 + 2 N .
The main goal in this paper is to generalize the result of blow-up of Ikeda and Wakasugi [START_REF] Córdoba | A maximum principle applied to quasi-geostrophic equations[END_REF] to the fractional Schrödinger equations (2.1.1).

The local existence is done by the Banach fixed point theorem, using the semigroup theory and stone's theorem on the fractional operator A = -i(-∆) α 2 which is the infinitesimal generator of a C 0 group of unitary operator S(t) = e -i(-∆) α 2 on L 2 (see [START_REF] Cazenave | Introduction aux problèmes d'évolution semi-linéaires[END_REF]).

The method used to prove the blow-up result is the method of the test function initiated by Baras and Pierre [START_REF] Baras | Critère d'existence de solutions positives pour des équations semilinéaires non monotones[END_REF], then developed by Zhang [START_REF] Qi | A blow up result for a nonlinear wave equation with damping : the critical case[END_REF], and later by Mitidieri and Pohozaev [START_REF] Mitidieri | Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on R N[END_REF].

The paper is organised as follows. In Section 2, we present the local existence result of solutions for (2.1.1) with some properties. Section 3 contains the blow-up result of solutions for (2.1.1).

Local existence

This section is dedicated to proving the local existence and uniqueness of mild solutions to the problem (2.1.1). Let Au = -i(-∆)

α 2 u. Applying Stone's Theorem [61, Theorem 1.10.8], we know that A is the infinitesimal generator of a C 0 group of unitary operator S(t), -∞ < t < ∞, on L 2 (R N ). We start by giving the Definition 2.2.1 (Mild solution) Let f ∈ H α 2 (R N ), 0 < α ≤ 2, p > 1 and T > 0. We say that u ∈ C([0, T ], H α 2 (R N )) is a mild solution of the problem (2.1.1) if u satisfies the following integral equation u(t) = S(t)f -iλ t 0 S(t -s)|u(s)| p ds. (2.2.1) Theorem 2.2.2 (Local existence) Given f ∈ H α 2 (R N ), λ ∈ C\{0}, and 1 < p ≤ 1 + 2α (N -α) + , then there exist a maximal time T max > 0 and a unique mild solution u ∈ C([0, T max ), H α 2 (R N )) of the problem (2.1.1). Furthermore, either T max = ∞ or else T max < ∞ and u H α 2 (R N ) → ∞ as t → T max .
Proof. By the Banach fixed point theorem, we obtain the existence of a unique mild solution

u ∈ Π T := C([0, T ], H α 2 (R N )) to the problem (2.1.1) [12, Section 4].
Using the uniqueness of solutions, we conclude the existence of a solution on a maximal interval [0, T max ) where

T max := sup {T > 0 ; there exists a mild solution u ∈ Π T to (2.1.1)} ≤ +∞. Next we prove that u H α 2 → ∞ as t → T max . We suppose lim inf t→Tmax u H α 2 < ∞.
Then we can find a sequence {t k } k∈N ⊂ [0, T max ) and a positive constant M > 0 such that

lim k→∞ t k = T max (2.2.2)
and

sup k∈N u(t k ) H α 2 ≤ M. ( 2
+ T (M )); H α 2 (R N
)) of (2.2.1) for all k ∈ N with some T (M ) > 0. However, by (2.2.2), we can take t k satisfying t k + T (M ) > T max , which contradicts the definition of T max . Therefore, we obtain

lim inf t→Tmax u H α 2 = ∞.

Blow-up of solutions

Now, we want to derive a blow-up result for Eq. (2.1.1). Our argument uses weak solutions.

Definition 2.3.1 (Weak solution) Let f ∈ H α 2 (R N ), 0 < α ≤ 2 and T > 0. We say that u is a weak solution of the problem (2.1.1) if u ∈ C([0, T ]; L p (R N )) and verifies the equation i R N f (x)ϕ(x, 0) + λ T 0 R N |u| p ϕ(x, t) = - T 0 R N u(x, t)(-∆) α/2 ϕ(x, t) -i T 0 R N u(x, t)ϕ t (x, t), (2.3.1) for all ϕ ∈ C 2 0 ([0, T ] × R N ) such that ϕ(• , T ) = 0. Lemma 2.3.2 Consider f ∈ H α 2 (R N ) and let u ∈ C([0, T ], H α 2 (R N
)) be a mild solution of (2.1.1), then u is a weak solution of (2.1.1), for all 0 < α ≤ 2 and all T > 0.

Proof. Let T > 0, 0 < α ≤ 2, f ∈ H α 2 (R N ) and let u ∈ C([0, T ], H α 2 (R N )) be a solution of (2.2.1). Given ϕ ∈ C 1 ([0, T ], H α 2 (R N ))
with compact support such that ϕ(• , T ) = 0. Then after multiplying (2.2.1) by ϕ and integrating over R N , we obtain

R N u(x, t)ϕ(x, t) = R N S(t)f (x)ϕ(x, t) -iλ R N t 0 S(t -s)|u(s)| p dsϕ(x, t).
We differentiate to obtain

d dt R N u(x, t)ϕ(x, t) = R N d dt (S(t)f (x)ϕ(x, t)) -iλ R N d dt t 0 S(t -s)|u(s)| p dsϕ(x, t). (2.3.2)
Now, using that (-∆) α/2 is a self-adjoint operator with D((-∆) α/2 ) = H α (R N ) and a property of the group S(t) ([9, Chapter 3]), we have :

R N d dt (S(t)f (x)ϕ(x, t)) = R N A (S(t)f (x)) ϕ(x, t) + R N S(t)f (x)ϕ t (x, t) = R N S(t)f (x)Aϕ(x, t) + R N S(t)f (x)ϕ t (x, t), (2.3.3)
and iλ

R N d dt t 0 S(t -s)|u(s)| p dsϕ(x, t) = iλ R N |u(s)| p ϕ(x, t) + iλ R N t 0 A (S(t -s)|u(s)| p ) dsϕ(x, t) + iλ R N t 0 S(t -s)|u(s)| p dsϕ t (x, t) = iλ R N |u(s)| p ϕ(x, t) + iλ R N t 0 S(t -s)|u(s)| p dsAϕ(x, t) + iλ R N t 0 S(t -s)|u(s)| p dsϕ t (x, t). ( 2
d dt R N u(x, t)ϕ(x, t) = R N u(x, t)Aϕ(x, t) + R N u(x, t)ϕ t (x, t) -iλ R N |u(s)| p ϕ(x, t).
We conclude by integrating in time over [0, T ] and using the fact that ϕ(• , T ) = 0.

To state our result, we put λ = λ 1 +iλ 2 and f = f 1 +if 2 . We introduce the following assumption on the data :

f 1 ∈ L 1 (R N ), λ 2 R N f 1 dx > 0, or f 2 ∈ L 1 (R N ), λ 1 R N f 2 dx < 0. (2.3.5) Theorem 2.3.3 Let λ ∈ C\{0}, and f ∈ H α 2 (R N ) satisfying (2.3.5). If 1 < p ≤ 1 + α
N then the mild solution of (2.1.1) blows-up in a finite time.

Proof. Let on the contrary u be a global weak solution to (2.1.1), then u

∈ L p ((0, R α ), L p (B 2ρ )) for R ∈ R *
+ and ρ > 0, where B 2ρ stands for the closed ball of center 0 and radius 2ρ. Let us define the function ϕ(x, t)

:= ϕ 1 (x/BR) (ϕ 2 (t)) ℓ , where ℓ = 2p-1 p-1 , R, B > 0 and 0 ≤ ϕ 1 ∈ D(∆ α/2 D ) is the first eigenfunction of the fractional Laplacian operator ∆ α/2 D in B 2 ,
(t) = ψ t R α , where ψ is a smooth non-increasing function on [0, ∞) such that ψ(r) = 1 if 0 ≤ r ≤ 1, 0 if r ≥ 2.
The constant B > 0 in the definition of ϕ 1 is fixed and will be chosen later. In fact, it plays a role in the critical case p = 1 + α/N . Only while in the subcritical case p < 1 + α/N , we simply put B = 1.

In the following, Ω 1 and Ω 2 denote the supports of ϕ 1 and ϕ 2 , respectively :

Ω 1 = x ∈ R N : |x| ≤ 2BR , Ω 2 = {t ∈ [0, ∞) : t ≤ 2R α } .
As u is a weak solution, we have

λ Ω 2 Ω 1 |u| p (x, t)ϕ(x, t) dxdt + i Ω 1 f (x)ϕ(x, 0) dx (2.3.6) = -i Ω 2 Ω 1 u(x, t)ϕ 1 (x/BR)∂ t ϕ ℓ 2 (t) dxdt + Ω 2 Ω 1 u(x, t)ϕ ℓ 2 (t)Λ α (ϕ 1 (x/BR)) dxdt.
We consider four cases : Case I : If λ 1 > 0, then R N f 2 dx < 0, therefore taking (Re) of both sides of (2.3.6), we get :

λ 1 Ω 2 Ω 1 |u| p (x, t)ϕ(x, t) dxdt - Ω 1 f 2 (x)ϕ(x, 0) dx = Re -i Ω 2 Ω 1 u(x, t)ϕ 1 (x/BR)∂ t ϕ ℓ 2 (t) dxdt + Re Ω 2 Ω 1 u(x, t)ϕ ℓ 2 (t)Λ α (ϕ 1 (x/BR)) dxdt .
The left hand side of the above equation is nonnegative.

Case II : If λ 1 < 0, then R N f 2 dx > 0, therefore taking (-Re) part of both sides of (2.3.6), we get :

-λ 1 Ω 2 Ω 1 |u| p (x, t)ϕ(x, t) dxdt + Ω 1 f 2 (x)ϕ(x, 0) dx = Re i Ω 2 Ω 1 u(x, t)ϕ 1 (x/BR)∂ t ϕ ℓ 2 (t) dxdt + (-Re) Ω 2 Ω 1 u(x, t)ϕ ℓ 2 (t)Λ α (ϕ 1 (x/BR)) dxdt .
The left hand side of the above equation is nonnegative.

Case III : If λ 2 > 0, then R N f 1 dx > 0, therefore taking (Im) part of both sides of (2.3.6), we get :

λ 2 Ω 2 Ω 1 |u| p (x, t)ϕ(x, t) dxdt + Ω 1 f 1 (x)ϕ(x, 0) dx = Im -i Ω 2 Ω 1 u(x, t)ϕ 1 (x/BR)∂ t ϕ ℓ 2 (t) dxdt + Im Ω 2 Ω 1 u(x, t)ϕ ℓ 2 (t)Λ α (ϕ 1 (x/BR)) dxdt .
The left hand side of the above equation is nonnegative.

Case IV : If λ 2 < 0, then R N f 1 dx < 0, therefore taking (-Im) part of both sides of (2.3.6), we get :

-λ 2 Ω 2 Ω 1 |u| p (x, t)ϕ(x, t) dxdt - Ω 1 f 1 (x)ϕ(x, 0) dx = Im i Ω 2 Ω 1 u(x, t)ϕ 1 (x/BR)∂ t ϕ ℓ 2 (t) dxdt + Im Ω 2 Ω 1 u(x, t)ϕ ℓ 2 (t)Λ α (ϕ 1 (x/BR)) dxdt .
The left hand side of the above equation is nonnegative. Then we only consider the Case I, since the others can be treated in the same way. In this case we may assume f 2 ∈ L 1 and

R N f 2 dx < 0.
Thus we have :

λ 1 Ω 2 Ω 1 |u| p (x, t)ϕ(x, t) dxdt ≤ ℓκB -α Ω 2 Ω 1 u(x, t)ϕ ℓ 2 (t)R -α ϕ 1 (x/BR) dxdt.
(2.3.7)

+ Im ℓ Ω 2 Ω 1 u(x, t)ϕ 1 (x/BR)ϕ ℓ-1 2 (t)∂ t ϕ 2 (t) dxdt := I 2 + I 1 .
In (2.3.7), we have used the fact that

∆ α/2 D ϕ 1 (x/BR) = R -α B -α κϕ 1 (x/R).
Hence, by the ε-Young inequality ab ≤ εa p + C(ε)b ℓ-1 (note that 1/p + 1/(ℓ -1) = 1) with ε > 0, a > 0 and b > 0, we deduce :

I 1 = Im ℓ Ω 2 Ω 1 u(x, t)ϕ 1 p ϕ -1 p ϕ 1 (x/BR)ϕ ℓ-1 2 (t)∂ t ϕ 2 (t) dxdt ≤ λ 1 4 Ω 2 Ω 1 |u| p (x, t)ϕ(x, t) dxdt + C Ω 2 Ω 1 ϕ -ℓ-1 p ϕ (ℓ-1) 1 (x/BR)ϕ (ℓ-1) 2 2 (t)|∂ t ϕ 2 (t)| ℓ-1 dxdt ≤ λ 1 4 Ω 2 Ω 1 |u| p (x, t)ϕ(x, t) dxdt + C Ω 2 Ω 1 ϕ 1 (x/BR)ϕ 2 (t)|∂ t ϕ 2 (t)| ℓ-1 dxdt,
and

I 2 = ℓλB -α Ω 2 Ω 1 u(x, t)ϕ 1 p ϕ -1 p ϕ ℓ 2 (t)R -α ϕ 1 (x/BR) dxdt ≤ λ 1 4 Ω 2 Ω 1 |u| p (x, t)ϕ(x, t) dxdt + C Ω 2 Ω 1 ϕ -ℓ-1 p ϕ ℓ(ℓ-1) 2 (t)R -α(ℓ-1) ϕ (ℓ-1) 1 (x/BR) dxdt ≤ λ 1 4 Ω 2 Ω 1 |u| p (x, t)ϕ(x, t) dxdt + C Ω 2 Ω 1 ϕ ℓ-1 2 (t)R -α(ℓ-1) ϕ 1 (x/BR) dxdt.
whereupon

λ 1 2 Ω 2 Ω 1 |u| p (x, t)ϕ(x, t) dxdt (2.3.8) ≤ C Ω 2 Ω 1 ϕ 1 (x/BR)ϕ 2 (t)|∂ t ϕ 2 (t)| ℓ-1 dxdt + C Ω 2 Ω 1 ϕ 1 (x/BR)ϕ ℓ-1 2 (t)R -α(ℓ-1) dxdt.
Recall now that ϕ 1 and ϕ 2 depend on R > 0. Hence changing the variables ξ = (BR) -1 x and τ = R -α t, we obtain from (2.3.8)

λ 1 2 Ω 2 Ω 1 |u| p (x, t)ϕ(x, t) dxdt ≤ C 2 0 |ξ|≤2 ϕ 1 (ξ)ϕ 2 (τ )R -α(ℓ-1) |∂ t ϕ 2 (τ )| ℓ-1 R N R α dξdτ + C 2 0 |ξ|≤2 ϕ 1 (ξ)ϕ ℓ-1 2 (τ )R -α(ℓ-1) B N R N R α dξdτ.
Therefore, we easily get :

λ 1 2 Ω 2 Ω 1 |u| p (x, t)ϕ(x, t) dxdt ≤ CR N +α-α(ℓ-1) , (2.3.9)
where the constant C in the right hand side of (2.3.9) is independent of R. Note that N + αα(ℓ -1) ≤ 0 if and only if p ≤ 1 + α/N. Now, we consider two cases. For p < 1 + α/N, we have N + α -α(ℓ -1) < 0. Hence, taking the limit R → ∞ in (2.3.9) and using the Lebesgue dominated convergence theorem, we obtain

λ 1 2 ∞ 0 R N |u| p (x, t)ϕ 1 (0) dxdt = lim R→∞ R 0 R N |u| p (x, t)ϕ(x, t) dxdt ≤ 0
Then u(x, t) = 0 for all x and t.

In the critical case p = 1 + α/N , we estimate the first term in the right hand side of inequality (2.3.7) using again the ε-Young inequality, and estimate the second term by the Hölder inequality (with p = p/(p -1) = ℓ -1) as follows

λ 1 Ω 2 Ω 1 |u| p ϕ(x, t) dxdt ≤ λ 1 2 Ω 2 Ω 1 |u| p ϕ(x, t) dxdt +C Ω 2 Ω 1 ϕ -p p ϕ ℓp 2 (t)ϕ p 1 (x/BR)(RB) -αp dxdt (2.3.10) +ℓ Ω 3 Ω 1 |u| p (x, t) dxdt 1/p × Ω 2 Ω 1 ϕ p 1 (x/BR)ϕ (ℓ-1)p 2 (t) |∂ t ϕ 2 (t)| p dxdt 1/p .
Here,

Ω 3 = {t ∈ [0, ∞) : R α ≤ t ≤ 2R α } is the support of ∂ t ϕ 2 . Note that lim R→∞ Ω 1 Ω 3 |u| p (x, t) dx dt = lim R→∞ Ω 1 |t|≤2R α |u| p (x, t) dx dt -lim R→∞ Ω 1 |t|≤R α |u| p (x, t) dx dt = ∞ 0 R N |u| p (x, t) dx dt - ∞ 0 R N |u| p (x, t) dx dt = 0 (2.3.11) because u ∈ L p (R N × [0, ∞)) (cf.
(2.3.9)). Now, introducing the new variables ξ = (BR) -1 x, τ = R -α t and recalling that p = 1 + α/N , we rewrite (2.3.11) as follows

λ 1 2 Ω 2 Ω 1 |u| p ϕ(x, t) dxdt ≤ C 2 0 |ξ|≤2B ϕ ℓ 2 (τ )ϕ 1 (ξ)B -α dξdτ + ℓ Ω 3 Ω 1 |u| p (x, t) dxdt 1/p × 2 0 |ξ|≤2B ϕ ℓ 2 (τ )ϕ 1 (ξ)B N |∂ t ϕ 2 (τ )| p dξdτ 1 p ≤ C 2 B -α + C 1 B N/p Ω 3 Ω 1 |u| p (x, t) dxdt 1/p , (2.3.12)
where the constants C 1 , C 2 are independent of R and B. Passing in (2.3.12) to the limit as R → +∞, using (2.3.11) and the Lebesgue dominated convergence theorem we get

λ 1 2 ∞ 0 R N |u| p ϕ(x, t) dxdt ≤ C 2 B -α . (2.3.13)
Note that we choose 1 ≤ B < R large enough such that when R → ∞ we don't have B → ∞ at the same time.

Finally, computing the limit as B → ∞ in (2.3.13), we infer that u(x, t) = 0.

Introduction

This paper concerns the Cauchy problem for the following semilinear damped wave equation

       u tt -∆u + a(x)b(t)u t = t 0 (t -s) -γ |u(s)| p ds, t > 0, x ∈ R n , u(0, x) = u 0 (x), u t (0, x) = u 1 (x), x ∈ R n , (3.1.1)
where the unknown function u is real-valued, n ≥ 1, 0 < γ < 1, p > 1. The coefficient of the damping term is given by

a(x)b(t) := a 0 (1 + |x| 2 ) -α 2 (1 + t) -β ,
with a 0 > 0, α, β ≥ 0, α + β < 1. Throughout this paper, we assume that the initial data is in the energy space

(u 0 , u 1 ) ∈ H 1 (R n ) × L 2 (R n ). (3.1.2)
Here after, • q and

• H 1 (1 ≤ q ≤ ∞) stand for the usual L q (R n )-norm and H 1 (R n )-norm, respectively.
The nonlinear nonlocal term can be considered as an approximation (with suitable change of variables) of the nonlinearity of the following semilinear damped wave equation

u tt -∆u + a(x)b(t)u t = |u(t)| p since the limit lim γ→1 s -γ + = Γ(1 -γ)δ(s)
exists in the distributional sense, where Γ is the Euler gamma function. For the equation

u tt -∆u + u t = |u(t)| p (3.1.3)
Li and Zhou [START_REF] Li | Breakdown of solutions to u + u t = |u| 1+α[END_REF] proved that if n ≥ 2, 1 < p ≤ 1 + 2/n and the data are positive on average, then the local solution of (3.1.3) must blow-up in a finite time. Todorova and Yordanov [START_REF] Todorova | Critical exponent for a nonlinear wave equation with damping[END_REF] developed a weighted energy method and determined that the critical exponent of (3.1.3) is

p c = 1 + 2 n ,
which is well known as Fujita's critical exponent for the heat equation u t -∆u = u p , u > 0 (see [START_REF] Fujita | On the blowing up of solutions of the problem for u t = ∆u + u 1+α[END_REF]). More precisely, they proved small data global existence in the case p > 1 + 2/n and blow-up for all solutions of (3.1.3) with positive in average data in the case 1 < p < 1+2/n. Later on Zhang [START_REF] Qi | A blow up result for a nonlinear wave equation with damping : the critical case[END_REF] showed that the critical exponent p = 1 + 2/n belongs to the blow-up region. We mention that Todorova and Yordanov [START_REF] Todorova | Critical exponent for a nonlinear wave equation with damping[END_REF] assumed the data to have compact support and essentially used this property. However, Ikehata and Tanizawa [START_REF] Ikehata | Global existence for solutions for semilinear damped wave equation in R N with noncompactly supported initial data[END_REF] removed this assumption. Ikehata et al. [START_REF] Ikehata | Critical exponent for semilinear wave equations with space-dependent potential[END_REF] investigated the space-dependent equation

u tt -∆u + a(x)u t = |u| p , where a(x) ≡ a 0 (1 + |x| 2 ) -α/2 , |x| → ∞, and 0 ≤ α < 1;
they proved that the critical exponent of (3.1.3) is given by

p c = 1 + 2 n -α
by using a refined multiplier method. Their method also depends on the finite propagation speed property. Recently, Nishihara [START_REF] Nishihara | Asymptotic behavior of solutions to the semilinear wave equation with timedependent damping[END_REF] and Lin et al. [START_REF] Lin | Critical exponent for the semilinear wave equation with time-dependent damping[END_REF] considered the semilinear wave equation with time-dependent damping

u tt -∆u + b(t)u t = |u| p , where b(t) = b 0 (1 + t) -β , β ∈ (-1, 1).
They proved that the critical exponent is

p c = 1 + 2 n .
This shows that, time-dependent coefficients decreasing to zero at infinity of damping term do not influence the critical exponent.

On the other hand, Equation (3.1.1) can be written as

u tt -∆u + a(x)b(t)u t = J δ 0|t (|u| p ) (t). (3.1.4)
Let us present our main results.

We first announce the following local well-posedness result.

Proposition 3.1.1 Let α ≥ 0, β ∈ R, 1 < p ≤ n/(n -2) for n ≥ 3, and p ∈ (1, ∞) for n = 1, 2.
Under the assumption (3.1.2) and γ ∈ (0, 1), the problem (3.1.1) admits a unique maximal mild solution such that

u ∈ C([0, T max ), H 1 (R n )) ∩ C 1 ([0, T max ), L 2 (R n )),
where

0 < T max ≤ ∞. Moreover, if T max < ∞, then u(t) H 1 + u t (t) 2 → ∞ as t → T max . Remark 3.1.2
We say that u is a global solution of (3.1.1) if T max = ∞, while in the case T max < ∞, we say that u blows up in a finite time.

Now, we set

p c := 1 + 3 -γ (n -α -1 + γ) + . As (p c = n/(n -2)) ⇐⇒ (γ = (n + 2(α -2))/n),
this imply, in the case when (n + 2(α -2))/n ≤ γ, that p c ≤ n/(n -2). We note that

p c → 1 + 2 n -α as γ → 1,
recovering the case studied by Ikehata and Tanizawa [START_REF] Ikehata | Global existence for solutions for semilinear damped wave equation in R N with noncompactly supported initial data[END_REF].

Our main result is Theorem 3.1.3 (Blow-up result) Under the assumption αβ = 0, we introduce the following exponents

p * := 1 + 3 -γ (N -α -1 + γ) + and γ * := N + 2(α -2) N . Suppose that (u 0 , u 1 ) ∈ H 1 (R N ) × L 2 (R N ) and R n a(x)u 0 dx > 0, R n u 1 (x) dx > 0. (3.1.5)
Then the mild solution of the problem (3.1.1) blows up in a finite time if i)

       1 < p ≤ N N -2 and γ ≥ γ * N ≥ 3, p ∈ (1, ∞) N = 1, 2. and p ≤ p * ii) N ≥ 3, 1 < p ≤ N/(N -2) and γ ≤ γ * .
The test function method (see [START_REF] Fino | Finite time blow-up for a wave equation with a nonlocal nonlinearity[END_REF][START_REF] Fino | Decay of mass for nonlinear equation with fractional Laplacian[END_REF][START_REF] Fino | Qualitative Properties of Solutions to a Time-Space Fractional Evolution Equation[END_REF][START_REF] Kirane | Critical exponents of Fujita type for certain evolution equations and systems with Spatio-Temporal Fractional derivatives[END_REF][START_REF] Mitidieri | A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities[END_REF][START_REF] Mitidieri | Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on R N[END_REF][START_REF] Qi | A blow up result for a nonlinear wave equation with damping : the critical case[END_REF] and the references therein) is the key to prove the blow-up result. This paper is organized as follows : in Section 3.2, we present some definitions and properties concerning the local existence, and the fractional integrals and derivatives. Section 3.3 contains the proof of the local existence result (Proposition 3.1.1). Finally, we prove the existence of blowingup solution (Theorem 3.1.3) in Section 3.4. Throughout this paper, C will denote positive constants that may change from line to line.

Preliminaries

In this section, we give some preliminary properties that will be used in the proof of Proposition 3.1.1 and Theorem 3.1.3. First, we start by

The linear homogeneous case

We consider the linear homogeneous equation

     u tt -∆u + a(x)b(t)u t = 0, t > 0, x ∈ R n , u(0, x) = u 0 (x), u t (0, x) = u 1 (x), x ∈ R n . (3.2.1)
It is well known that for any (u 0 , u 1 ) ∈ H 2 × H 1 (R n ), there exists a unique strong solution u of (3.2.1) (see [START_REF] Ikawa | Hyperbolic partial differential equations and wave phenomena[END_REF]Theorem 2.27]). Let us denote by R(t, s) the operator which maps the initial data (u(s), u t (s)) ∈ H 2 × H 1 given at the time s ≥ 0 to the solution u(t) ∈ H 2 at the time t ≥ s, i.e. the solution u of (3.2.1) is defined by u(t) = R(t, 0)(u 0 , u 1 ). The operator R(t, s) can be extended uniquely such that R(t, s) : [START_REF] Wakasugi | On the diffusive structure for the damped wave equation with variable coefficients[END_REF]Appendix]). Moreover, for any T > 0, the following estimation

H 1 (R n ) × L 2 (R n ) -→ C([s, ∞), H 1 (R n )) ∩ C 1 ([s, ∞), H 1 (R n )) (see
R(t, s)(u 0 , u 1 ) H 1 + ∂ t (R(t, s)(u 0 , u 1 )) L 2 ≤ C(1 + T ) (u 0 , u 1 ) H 1 ×L 2 (3.2.2)
holds for s ≤ t ≤ s + T .

The linear inhomogeneous case

Let us consider the linear inhomogeneous equation

     u tt -∆u + a(x)b(t)u t = F (t, x), t > 0, x ∈ R n , u(0, x) = u 0 (x), u t (0, x) = u 1 (x), x ∈ R n . (3.2.3) Definition 3.2.1 (Stong solution) Let (u 0 , u 1 ) ∈ H 2 × H 1 and F ∈ C([0, ∞); L 2 ). We say that a function u is a strong solution of (3.2.3) if u ∈ 2 j=0 C 2-j ([0, ∞); H j ), u(0) = u 0 , u t (0) = u 1 ,
Let (u 0 , u 1 ) ∈ H 1 × L 2 and F ∈ C([0, ∞); L 2 ). We say that a function u is a mild solution of (3.2.3) if u ∈ C([0, ∞); H 1 ) ∩ C 1 ([0, ∞); L 2
) and u has the initial data u(0) = u 0 , u t (0) = u 1 and satisfies the integral equation

u(t, x) = R(t, 0)(u 0 , u 1 ) + t 0 S(t, s)F (s, x) ds (3.2.4)
in the sense of H 1 (R n ), where S(t, s)g := R(t, s)(0, g) for a function g ∈ H 1 .

Definition 3.2.3 (Weak solution) Let (u 0 , u 1 ) ∈ H 1 × L 2 and F ∈ C([0, ∞); L 2 ). We say that a function u is a weak solution of (3.2.3) if u ∈ C((0, ∞); H 1 ) ∩ C 1 ([0, ∞); L 2
) and u has the initial data u(0) = u 0 , u t (0) = u 1 and satisfies the identity

∞ 0 R n u (ϕ tt -∆ϕ -(a(x)b(t)ϕ) t ) dx dt = R n ((a(x)u 0 (x) + u 1 (x)) ϕ(0, x) -u 0 (x)ϕ t (0, x)) dx + ∞ 0 R n F (t, x)ϕ dx dt (3.2.5) for any ϕ ∈ C ∞ 0 ([0, ∞) × R n ). Proposition 3.2.4 [69, Proposition 9.14] Let (u 0 , u 1 ) ∈ H 2 × H 1 , F ∈ C([0, ∞); H 1 ) ∩ C 1 ([0, ∞); L 2
) and u be the strong solution of (3.2.3). Then u is a mild solution. Moreover, u satisfies the following energy estimates

(u t , ∇u)(t) L 2 ×L 2 ≤ C (u 1 , ∇u 0 ) L 2 ×L 2 + C t 0 F (s, •) L 2 ds, (3.2.6)
and

u(t) L 2 ≤ C (u 0 ) L 2 + C t 0 (u 1 , ∇u 0 ) L 2 ×L 2 + s 0 F (τ, •) L 2 dτ ds. (3.2.7) Proposition 3.2.5 [69, Proposition 9.15] Let (u 0 , u 1 ) ∈ H 1 × L 2 , F ∈ C([0, ∞); L 2 ).
Then there exists a unique mild solution u of (3.2.3). Moreover, the mild solution u satisfies the estimates (3.2.6) and (3.2.7).

Local existence

We start by introducing the definitions of the mild and weak solution of (3.1.1). For nonlinear equations, it is not always true that there exist global-in-time solutions. Therefore, we consider solution defined on an interval [0, T ) for T > 0. We call such a solution local-in-time solution (or local solution) and if we can take T = ∞, then we call it global-in-time solution (or global solution).

Definition 3.3.1 (Mild solution) Let (u 0 , u 1 ) ∈ H 1 × L 2 .
We say that a function u is a mild solution of (3.1.1) if

u ∈ C([0, T ); H 1 ) ∩ C 1 ([0, T ); L 2 ), u(0) = u 0 , u t (0) = u 1
, and satisfies the integral equation

u(t, x) = R(t, 0)(u 0 , u 1 ) + Γ(δ) t 0 S(t, s)J δ 0|t (|u| p ) ds (3.3.1)
in the sense of H 1 (R n ), where δ = 1 -γ.

Definition 3.3.2 (Weak solution) Let T > 0, α ∈ [0, 1), γ ∈ (0, 1) and u 0 , u 1 ∈ L 1 loc (R n ).
We say that u is a weak solution of (3.1.1) if u ∈ C([0, T ], L p loc (R n )) and satisfies

Γ(δ) T 0 R n J δ 0|t (|u| p )ϕ dx dt + R n u 1 (x)ϕ(0, x) dx + R n u 0 (x)(a(x)ϕ(0, x) -ϕ t (0, x)) dx = T 0 R n uϕ tt dx dt - T 0 R n u(a(x)b(t)ϕ) t dx dt - T 0 R n u∆ϕ dx dt, (3.3.2)
for all compactly supported function ϕ

∈ C 2 ([0, T ] × R n ) such that ϕ(• , T ) = 0 and ϕ t (• , T ) = 0, where δ = 1 -γ.
The following lemma is useful for the proof of Theorem 3.1.3.

Lemma 3.3.3 (Mild → Weak) Let T > 0, α ∈ [0, 1) and γ ∈ (0, 1). Suppose that 1 < p ≤ n/(n -2), if n ≥ 3, and p ∈ (1, ∞), if n = 1, 2. If u ∈ C([0, T ], H 1 (R n )) ∩ C 1 ([0, T ], L 2 (R n ))
is the mild solution of (3.1.1), then u is a weak solution of (3.1.1).

Proof. Let u be a mild solution of (3.1.1) and let ϕ ∈ C 2 ([0, T ] × R n ) be a compactly supported function such that ϕ(• , T ) = 0 and ϕ t (• , T ) = 0. Under the assumption that 1 < p ≤ n/(n -2), if n ≥ 3, and p ∈ (1, ∞), if n = 1, 2, we note that by the Gagliardo-Nirenberg inequality [17, p.3], we have the estimate

f (t, u) 2 = t 0 (t -s) -γ |u(s)| p ds 2 ≤ t 0 (t -s) -γ u p 2 ds = t 0 (t -s) -γ u 2p 2 ds ≤ C t 0 (t -s) -γ ∇u σp 2 u (1-σ)p 2 ds ≤ C t 0 (t -s) -γ u p H 1 ds ≤ C u L ∞ ((0,T );H 1 ) t 0 (t -s) -γ ds ≤ C T 1-γ u L ∞ ((0,T );H 1 ) .
This inequality shows that f (u) ∈ C([0, T ]; L 2 ). We take sequences (u

(j) 0 , u (j) 1 ) ∞ j=1 ⊂ H 2 × H 1 and (F (j) ∞ j=1 ⊂ C([0, T ]; H 1 ) ∩ C 1 ([0, T ]; L 2 ) such that lim j→∞ (u (j) 0 , u (j) 1 ) = (u 0 , u 1 ) in H 1 × L 2 , lim j→∞ F (j) = f (u) in C([0, T ]; L 2 ).
Let u (j) be the strong solution of the linear inhomogeneous equation (3.2.3) with the initial data (u

(j) 0 , u (j) 1
) and the inhomogenous term F (j) . Then by Proposition 3.2.4, we have

u (j) (t, x) = R(t, 0)(u (j) 0 , u (j)
1 ) + t 0 S(t, s)F (j) (s, x) ds.

Since u is a mild solution of (3.1.1), we obtain

u (j) (t, x) -u(t, x) = R(t, 0)(u (j) 0 -u 0 , u (j)
1 -u 1 ) + t 0 S(t, s)(F (j) (s, x) -f (u(s, x))) ds and hence, by using the estimation (3.2.2), we see that

(u (j) -u, ∂ t (u (j) -u))(t) H 1 ×L 2 ≤ C(1 + T ) (u (j) 0 -u 0 , u (j) 1 -u 1 ) H 1 ×L 2 + C(1 + T )T sup s∈[0,T ] F (j) (s) -f (u(s)) L 2 .
This implies that lim j→∞ u

(j) = u in C([0, T ]; H 1 ) ∩ C 1 ([0, T ]; L 2 ). Since each u (j) is a strong solution of (3.2.3), u (j) is also a weak solution of (3.2.3), that is, ∞ 0 R n u (j) (ϕ tt -∆ϕ -(a(x)b(t)ϕ) t ) dx dt = R n a(x)u (j) 0 (x) + u (j) 1 (x) ϕ(0, x) -u (j) 0 (x)ϕ t (0, x) dx + ∞ 0 R n F (j) (t, x)ϕ dx dt.
Thus, letting j → +∞, we deduce that u satisfies the identity (3.3.2). Since ϕ is an arbitrary test function ; u is a weak solution of (3.1.1).

Proof of Proposition

3.1.1. Let T > 0, R > 0, X(T ) := C([0, T ]; H 1 (R n ) ∩ C 1 ([0, T ]; L 2 (R n )) and B R (T ) = {ϕ ∈ X(T ); ϕ X(T ) ≤ 2R},
where ϕ X(T ) := ∇ϕ 2 + ϕ t 2 . By Proposition 3.2.5, we define a mapping Φ : B R (T ) → X(T ) such that u(t, x) = (Φϕ)(t, x) is the unique mild solution to the linear inhomogeneous equation

     u tt -∆u + a(x)b(t)u t = f (t, ϕ), t > 0, x ∈ R n , u(0, x) = u 0 (x), u t (0, x) = u 1 (x), x ∈ R n , moreover (u t , ∇u)(t) L 2 ×L 2 ≤ C I 0 + C t 0 f (s, ϕ) L 2 ds,
where I 0 := u 0 H 1 + u 1 2 . Using the same calculation as in the proof of Lemma 3.3.3, we have

f (t, ϕ) 2 ≤ C T 1-γ ϕ p X(T ) . Therefore u X(T ) ≤ C I 0 + C 2 p R p T 2-γ .
We take R such that R ≥ C I 0 and then choose T sufficiently small so that

C 2 p R p-1 T 2-γ ≤ 1.
It follows that u X(T ) ≤ 2R, whereupon Φ maps B R (T ) into B R (T ).

Next, we show that Φ is a contraction by taking T smaller. Let ϕ, ϕ ∈ B R (T ), u := Φ(ϕ), u := Φ(ϕ) and w := u -u. By Proposition 3.2.5, w is the unique mild solution to the linear inhomogeneous equation

     w tt -∆w + a(x)b(t)w t = f (t, ϕ) -f (t, ϕ), t > 0, x ∈ R n , w(0, x) = 0, w t (0, x) = 0, x ∈ R n , moreover (w t , ∇w)(t) L 2 ×L 2 ≤ C t 0 f (s, ϕ) -f (s, ϕ) L 2 ds ≤ C t 0 s 0 (s -σ) -γ |ϕ(σ)| p -|ϕ(σ)| p 2 dσ ds ≤ C t 0 s 0 (s -σ) -γ ϕ p-1 (σ) 2p/(p-1) + ϕ p-1 (σ) 2p/(p-1) ϕ(σ) -ϕ(σ) 2p dσ ds = C t 0 s 0 (s -σ) -γ ϕ(σ) p-1 2p + ϕ(σ) p-1 2p ϕ(σ) -ϕ(σ) 2p dσ ds ≤ C t 0 s 0 (s -σ) -γ ϕ(σ) p-1 H 1 + ϕ(σ) p-1 H 1 ϕ(σ) -ϕ(σ) H 1 dσ ds ≤ C t 0 s 0 (s -σ) -γ ϕ p-1 X(T ) + ϕ p-1 X(T ) ϕ -ϕ X(T ) dσ ds ≤ C 2 p R p-1 T 2-γ ϕ -ϕ X(T ) ≤ 1 2 ϕ -ϕ X(T ) ,
thanks to Hölder's inequality, the fact that H 1 (R n ) ֒→ L 2p (R n ) and the following inequality

||ϕ| p -|ϕ| p | ≤ C(p)|ϕ -ϕ|(|ϕ| p-1 + |ϕ| p-1 ); (3.3.3) T is chosen such that C 2 p R p-1 T 2-γ ≤ 1 2 .
This implies that Φ is a contraction. Then, by the Banach fixed point theorem, there exists a mild solution u ∈ X(T ) to problem (3.1.1). Here we have uniqueness of solution in a small time interval but in the presence of the nonlocal term we cannot continue and obtain the uniqueness in a maximal time interval. Therefore, we prove the uniqueness of the solution of (3.1.1) using Gronwall's inequality (cf. [START_REF] Cazenave | Introduction aux problèmes d'évolution semi-linéaires[END_REF]). If u and u are solutions of (3.1.1), we have

u(t) -u(t) H 1 ≤ C 2 p R p-1 t 0 s 0 (s -σ) -γ u(σ) -u(σ) 2p dσ ds = C 2 p R p-1 t 0 t σ (s -σ) -γ u(σ) -u(σ) 2p ds dσ ≤ C 2 p R p-1 t 0 (t -σ) 1-γ u(σ) -u(σ) H 1 dσ.
So the uniqueness follows from Gronwall's inequality. Moreover, using the uniqueness of solutions, we conclude the existence of a solution on a maximal interval [0, T max ) where T max := sup {T > 0 ; there exist a mild solution u ∈ X(T ) to (3.1.1)} ≤ +∞.

Finally, if the lifespan T max is finite, then the weighted energy of the solution blows up at T max :

lim t→Tmax ( u(t) H 1 + u t (t) 2 ) = ∞. Because, if lim t→Tmax ( u(t) H 1 + u t (t) 2 ) =: M < ∞,
then there exists a time sequence {t m } m≥0 tending to T max as m → ∞ and such that

sup m∈N ( u(t m ) H 1 + u t (t m ) 2 ) ≤ M + 1
The argument before shows that there exists T (M + 1) > 0 such that the solution u(t) can be extended on the interval [t m , t m + T (M + 1)] for any m. By taking m sufficiently large so that t m ≥ T max -(1/2)T (M + 1), the solution u(t) can be extended on [T max , T max + (1/2)T (M + 1)]. This contradicts the definition of T max . Thus the proof is completed.

Blow-up result

In this section, we present the proof of Theorem 3.1.3. Proof of Theorem 3.1.3. We assume, on the contrary, that u is a global mild solution of (3.1.1). Let g(t) be the solution of the ordinary differential equation

     -g ′ (t) + (1 + t) -β g(t) = 1, g(0) = g 0 ,
where g 0 = 0 is a constant. The solution g(t) can be expressed by

g(t) = e t 0 (1+s) -β ds g 0 - t 0 e -τ 0 (1+s) -β ds dτ .
By l'Hôpital rule, we have lim t→∞ b(t)g(t) = 1 and

c 1 b(t) -1 ≤ g(t) ≤ c 2 b(t) -1 due to β ≥ -1/3.
We note that g 0 = 1 and g(t) ≡ 1 in the case β = 0. By the definition of g(t), we also have

sup t>0 |g ′ (t)| ≤ sup t>0 |b(t)g(t) -1| < ∞.
As u is a mild solution of (3.1.1), from Lemma 3.3.3 we have

Γ(δ) T 0 R n J δ 0|t (|u| p )g(t)ϕ dx dt + R n u 1 (x)g(0)ϕ(0, x) dx + R n u 0 (x)(a(x)g(0)ϕ(0, x) -(gϕ) t (0, x)) dx = T 0 R n u(gϕ) tt dx dt - T 0 R n u(a(x)b(t)g(t)ϕ) t dx dt - T 0 R n u∆(gϕ) dx dt,
for all T ≫ 1 and all function ϕ ∈ C 2 ([0, T ] × R n ) compactly supported in x such that ϕ(• , T ) = 0 and ϕ t (• , T ) = 0, where δ = 1 -γ. Then

Γ(δ) T 0 R n J δ 0|t (|u| p )g(t)ϕ dx dt + g 0 R n u 1 (x)ϕ(0, x) dx + R n u 0 (x)a(x)ϕ(0, x) dx -g 0 R n u 0 (x)ϕ t (0, x) dx = T 0 R n ug(t)ϕ tt dx dt - T 0 R n u(g ′ (t) -1)a(x)ϕ t dx dt - T 0 R n ug(t)∆ϕ dx dt, (3.4.1)
where we have used the assumption αβ = 0.

Let ϕ(x, t) = D δ t|T ( φ(x, t)) := D δ t|T ϕ ℓ 1 (x)ϕ 2 (t) with ϕ 1 (x) := Φ (|x|/B) , ϕ 2 (t) := (1 -t/T ) η + , where D δ t|T is given by (1.1.1), ℓ, η ≫ 1 and Φ ∈ C ∞ (R + ) is a cut-off non-increasing function such that Φ(r) = 1, if 0 ≤ r ≤ 1, 0, if r ≥ 2, 0 ≤ Φ ≤ 1 and |Φ ′ (r)| ≤ C 1 /
r for all r > 0. The constant B > 0 in the definition of ϕ 1 is fixed and will be chosen later. In the following, we set :

Ω(B) = {x ∈ R n : |x| ≤ 2B}, ∆(B) = {x ∈ R n : B ≤ |x| ≤ 2B}.
Equality (3.4.1) actually reads

Γ(δ) T 0 Ω(B) J δ 0|t (|u| p )g(t)D δ t|T φ dx dt + g 0 Ω(B) u 1 (x)D δ t|T φ(0, x) dx + Ω(B) u 0 (x)(a(x)D δ t|T φ(0, x) -g 0 ∂ t D δ t|T φ(0, x)) dx = T 0 Ω(B) u g(t)∂ 2 t D δ t|T φ dx dt - T 0 R n u(g ′ (t) -1)a(x)∂ t (D δ t|T φ) dx dt - T 0 R n ug(t)∆D δ t|T φ dx dt. As c 1 b(t) -1 ≤ g(t) ≤ c 2 b(t) -1 and b(s) > b(t) for 0 ≤ s ≤ t, we have g(t) ≥ c 1 b(t) = c 1 c 2 c 2 b(t) ≥ c 1 g(s)b(s) c 2 b(t) ≥ c 1 c 2 g(s), then Γ(δ) T 0 Ω(B)
J δ 0|t (|u| p g)D δ t|T φ dx dt

+ g 0 Ω(B) u 1 (x)D δ t|T φ(0, x) dx + Ω(B) u 0 (x)(a(x)D δ t|T φ(0, x) -g 0 ∂ t D δ t|T φ(0, x)) dx ≤ c 2 c 1 Γ(δ) T 0 Ω(B) J δ 0|t (|u| p )g(t)D δ t|T φ dx dt + g 0 Ω(B) u 1 (x)D δ t|T φ(0, x) dx + Ω(B) u 0 (x)(a(x)D δ t|T φ(0, x) -g 0 ∂ t D δ t|T φ(0, x)) dx = T 0 Ω(B) u g(t)∂ 2 t D δ t|T φ dx dt - T 0 R n u(g ′ (t) -1)a(x)∂ t (D δ t|T φ) dx dt - T 0 R n ug(t)∆D δ t|T φ dx dt.
From (1.1.3), (1.1.4) and (1.1.7), we conclude that

T 0 Ω(B) D δ 0|t J δ 0|t (|u| p g) φ dx dt + C T -δ-1 Ω(B) [T (u 1 (x) + a(x)u 0 (x)) + u 0 (x)] ϕ ℓ 1 (x) dx ≤ C T 0 Ω(B) u g(t)D 2+δ t|T φ dx dt + C T 0 R n u(g ′ (t) -1)a(x)D 1+δ t|T φ dx dt -C T 0 R n ug(t)∆ϕ ℓ 1 (x)D δ t|T ϕ 2 (t) dx dt,
where D δ 0|t is defined in (1.1.1). Moreover, using (1.1.5) and the fact that (??) implies

C Ω(B) u 0 (x)ϕ ℓ 1 (x) dx ≥ Ω(B) a(x)u 0 (x)ϕ ℓ 1 (x) dx ≥ 0, Ω(B) u 1 (x)ϕ ℓ 1 (x) dx ≥ 0, it follows that T 0 Ω(B) |u| p g(t) φ dx dt ≤ C T 0 Ω(B) |u|g(t)ϕ ℓ 1 D 2+δ t|T ϕ 2 dx dt + C T 0 Ω(B) |u|a(x)ϕ ℓ 1 D 1+δ t|T ϕ 2 dx dt + C T 0 ∆(B) |u|g(t)ϕ ℓ-2 1 (|∆ϕ 1 | + |∇ϕ 1 | 2 )D δ t|T ϕ 2 dx dt =: I 1 + I 2 + I 3 , (3.4.2)
where we have used the formula ∆(ϕ ℓ 1 ) = ℓϕ ℓ-1

1 ∆ϕ 1 + ℓ(ℓ -1)ϕ ℓ-2 1 |∇ϕ 1 | 2
, and the fact that ϕ 1 ≤ 1 and |g ′ (t) -1| ≤ C. Next by applying Young's inequality, we obtain

I 1 = C T 0 Ω(B) |u|g 1/p (t) φ1/p φ-1/p g 1-1/p (t)ϕ ℓ 1 D 2+δ t|T ϕ 2 dx dt ≤ 1 3p T 0 Ω(B) |u| p g(t) φ dx dt + C T 0 Ω(B) g(t)ϕ ℓ 1 ϕ -1/(p-1) 2 (D 2+δ t|T ϕ 2 ) p ′ dx dt, (3.4.3)
where p ′ = p/(p -1). Similarly,

I 2 ≤ 1 3p T 0 Ω(B) |u| p g(t) φ dx dt + C T 0 Ω(B) ϕ ℓ 1 g -1/(p-1) (t)ϕ -1/(p-1) 2
(a(x)) p ′ (D 1+δ t|T ϕ 2 ) p ′ dx dt (3.4.4) and

I 3 ≤ 1 3p T 0 Ω(B) |u| p g(t) φ dx dt+C T 0 Ω(B) g(t)ϕ ℓ-2p ′ 1 ϕ -1/(p-1) 2 (|∆ϕ 1 | p ′ +|∇ϕ 1 | 2p ′ )(D δ t|T ϕ 2 ) p ′ dx dt. ( 3
|u| p g(t) φ ≤ C T 0 Ω(B) g(t)ϕ ℓ 1 ϕ -1/(p-1) 2 (D 2+δ t|T ϕ 2 ) p ′ + C T 0 Ω(B) ϕ ℓ 1 g -1/(p-1) (t)ϕ -1/(p-1) 2 (a(x)) p ′ (D 1+δ t|T ϕ 2 ) p ′ + C T 0 Ω(B) g(t)ϕ ℓ-2p ′ 1 ϕ -1/(p-1) 2 (|∆ϕ 1 | p ′ + |∇ϕ 1 | 2p ′ )(D δ t|T ϕ 2 ) p ′ . (3.4.6)
At this stage, to prove i), we have to distinguish 2 cases.

• Case 1 < p < p c : in this case, we take B = T 1 2-α . As B < T , and g -1 (t) ≤ C, we have

T 0 Ω(B) |u| p g(t) φ dx dt ≤ C T 0 Ω(T ) g(t)ϕ ℓ 1 ϕ -1/(p-1) 2 (D 2+δ t|T ϕ 2 ) p ′ dx dt +C T 0 Ω(T 1/(2-α) ) ϕ ℓ 1 ϕ -1/(p-1) 2 (a(x)) p ′ (D 1+δ t|T ϕ 2 ) p ′ dx dt + C T 0 Ω(T ) g(t)ϕ ℓ-2p ′ 1 ϕ -1/(p-1) 2 (|∆ϕ 1 | p ′ + |∇ϕ 1 | 2p ′ )(D δ t|T ϕ 2 ) p ′ dx dt =: J 1 + J 2 + J 3 . (3.4.7)
So, using the change of variables : (s = T -1 t, y = T -1 x) in J 1 and J 3 , and

(s = T -1 t, y = T -1 2-α x) in J 2 we get from (1.1.6) that J 1 ≤ C T β-(2+δ)p ′ +1+n ≤ C T 1-α-(2+δ)p ′ +1+n = C T -((2+δ)p ′ -(2-α)-n) , J 3 ≤ C T β-2p ′ -δp ′ +1+n ≤ C T 1-α-(2+δ)p ′ +1+n = C T -((2+δ)p ′ -(2-α)-n) ,
and

J 2 ≤ C      T -a 1 , if n > αp ′ , T -a 2 ln T, if n = αp ′ , T -a 2 , if n < αp ′ ,
where

a 1 := 1 2 -α ((2 + δ(2 -α))p ′ -(2 -α) -n),
and

a 2 := 1 2 -α ((2 + δ(2 -α))p ′ -2).
Therefore, we conclude from (3.4.7) that

T 0 Ω(T 1 2-α ) |u| p g(t) φ dx dt ≤ C      T -a 1 , if n > αp ′ , T -a 1 + T -a 2 ln T, if n = αp ′ , T -a 1 + T -a 2 , if n < αp ′ . (3.4.8)
Letting T → ∞ in (3.4.8), thanks to p < p c and the Lebesgue dominated convergence theorem, it yields that

∞ 0 R n |u| p g(t) dx dt = 0,
which implies u(x, t) = 0 for all t and a.e. x. This contradicts our assumption (??).

• Case p = p c : let B = R -1 2-α T 1 2-α
, where 1 ≪ R < T is such that T and R do not go simultaneously to ∞. Moreover, from the first case and the fact that p = p c , there exists a positive constant D independent of T such that

∞ 0 R n |u| p g(t) dx dt ≤ D, which implies that T 0 ∆(R -1 2-α T 1 2-α ) |u| p g(t) φ dx dt → 0 as T → ∞. (3.4.9)
On the other hand, using Hölder's inequality instead of Young's one, we estimate the integral I 3 in (3.4.2) as follows :

I 3 ≤ C T 0 ∆(R -1 2-α T 1 2-α ) |u| p g(t) φ 1/p × T 0 Ω(R -1 T ) g(t)ϕ ℓ-2p ′ 1 ϕ -1/(p-1) 2 (|∆ϕ 1 | p ′ + |∇ϕ 1 | 2p ′ )(D α t|T ϕ 2 ) p ′ dx dt
|u| p g(t) φ dx dt ≤ C T 0 Ω(R -1 T ) g(t)ϕ ℓ 1 ϕ -1/(p-1) 2 (D 2+δ t|T ϕ 2 ) p ′ dx dt + C T 0 Ω(R -1 2-α T 1 2-α ) ϕ ℓ 1 ϕ -1/(p-1) 2 (a(x)) p ′ (D 1+δ t|T ϕ 2 ) p ′ dx dt + C T 0 ∆(R -1 2-α T 1 2-α ) |u| p g(t) φ 1/p × T 0 Ω(R -1 T ) g(t)ϕ ℓ-2p ′ 1 ϕ -1/(p-1) 2 (|∆ϕ 1 | p ′ + |∇ϕ 1 | 2p ′ )(D α t|T ϕ 2 ) p ′ dx dt 1/p ′ =: K 1 + K 2 + K 3 .
Taking account of p = p c and the scaled variables (s = T -1 t, y = RT -1 x) in K 1 and K 3 , and

(s = T -1 t, y = R 1 2-α T -1 2-α x) in K 2 we conclude that T 0 Ω(R -1 2-α T 1 2-α ) |u| p g(t) φ dx dt ≤ C R -n +C R -n-αp ′ 2-α +C R 2-n p ′ T 0 ∆(R -1 2-α T 1 2-α ) |u| p g(t) φ 1/p .
Letting T → ∞, using (3.4.9), we get

∞ 0 R n |u| p g(t) dx dt ≤ C R -n + C R -n-αp ′ 2-α ,
which implies a contradiction, when R → ∞, with (??) where we have used the fact that p = p c > n n-α . This completes the proof of Theorem 3.1.3, i).

For the proof of ii), we have two possibilities :

• γ < (n + 2(α - 2
|u| p g(t) φ dx dt ≤ C      R n T -(2+δ)p ′ +1 + R n-αp ′ T -(1+δ)p ′ +1 + R n-2p ′ T -δp ′ +1 , if n > αp ′ R n T -(2+δ)p ′ +1 + ln R T -(1+δ)p ′ +1 + R n-2p ′ T -δp ′ +1 , if n = αp ′ R n T -(2+δ)p ′ +1 + T -(1+δ)p ′ +1 + R n-2p ′ T -δp ′ +1 , if n < αp ′ .
As γ < (n + 2(α -2))/n implies p ≤ n/n -2 < 1/γ, we get a contradiction with (??) by letting the following limits : first T → ∞, next R → ∞.

• γ = (n + 2(α -2))/n : we have p ≤ n/(n -2) = p c . Using i), we get the contradiction. This completes the proof of Theorem 3.1.3, ii).

Introduction

This paper is concerned with the Cauchy problem for the following evolution p-Laplacian equation

       u t -div |∇u| p-2 ∇u = t 0 (t -s) -γ |u(s)| q-1 u(s) ds, t > 0, x ∈ R n , u(0, x) = u 0 (x), x ∈ R n , (4.1.1)
where p > 2, q > 1, 0 < γ < 1, and

u 0 ∈ L ∞ (R n ).
The study of blow-up for nonlinear parabolic equations originated from Fujita [START_REF] Fujita | On the blowing up of solutions of the problem for u t = ∆u + u 1+α[END_REF]. He studied the following Cauchy problem of the semilinear heat equation

     u t -∆u = u p , u ≥ 0 t > 0, x ∈ R n , u(0, x) = u 0 (x), x ∈ R n , (4.1.2)
where p > 1, and obtains the following results : a) If p < 1 + 2/n, then every nontrivial solution of (4.1.2) blows up in finite time. b) If p > 1 + 2/n and u 0 (x) ≤ δe -|x| 2 , (0 < δ ≪ 1), then (4.1.2) admits a global solution.

In the case p = 1 + 2/n, it is shown by the author of [START_REF] Hayakawa | On nonexistence of global solutions of some semilinear parabolic equations[END_REF] for n = 1, 2 and by the authors of [START_REF] Kobayashi | On the blowing up problem for semilinear heat equations[END_REF] for n ≥ 1 that (4.1.2) has no global solution u(x, t) satisfying u(• , t) ∞ < ∞ for t ≥ 0. The author of [START_REF] Weissler | Existence and nonexistence of global solutions for a semilinear heat equations[END_REF] proves that if p = 1 + 2/n, (4.1.2) has no global solution u(x, t) satisfying u(• , t) q < ∞ for t > 0 and some q ∈ [1, +∞). The value p c = 1 + 2/n is called the critical exponent of (4.1.2). It plays an important role in the behavior of the solution to (4.1.2).

In the sixty years, there have been a number of extensions of Fujita's results in many directions. Recently, the authors of [] extended Fujita's results to the heat equation with nonlinear memory term

       u t -∆u = t 0 (t -s) -γ u(s) p ds, t > 0, x ∈ R n , u(0, x) = u 0 (x), x ∈ R n , (4.1.3)
where p > 1, 0 < γ < 1, and u 0 ∈ C 0 (R n ). In this case, the value of the critical Fujita exponent is

p c = max{ 1 γ , 1 + 2(2 -γ)/(N -2 + 2γ) + }.
Moreover, the nonlinear nonlocal term can be considered as an approximation of the nonlinear term in the semilinear heat equation (4.1.3) since the limit

lim γ→1 s -γ + = Γ(1 -γ)δ(s)
exists in distribution sense, where Γ is the Euler gamma function.

For the evolution p-Laplacian equation

     u t -div |∇u| p-2 ∇u = u q , u ≥ 0 t > 0, x ∈ R n , u(0, x) = u 0 (x), x ∈ R n , (4.1.4)
where p > 2, q > 1, the author of [START_REF] Zhao | On the Cauchy problem and initial traces for the evolution P-Laplacian equations with strongly nonlinear sources[END_REF] discussed the Cauchy problem and obtained the critical exponent q c = p -1 + p/n provided p > 2n/(n + 1). Furthermore, the author demonstrated that if max{1, p -1} < q < q c , then the Cauchy problem has no global solution ; whereas if q > q c and u 0 (x) is small enough, then the Cauchy problem admits a global positive solution. The authors of [START_REF] Andreucci | A Fujita type result for a degenerate Neumann problem with non compact boundary[END_REF][START_REF] Liu | The critical exponent of doubly singular parabolic equations[END_REF] discussed more general quasi-linear parabolic equations and the doubly singular parabolic equations respectively, and obtain similar results.

Our goal is to study the local existence of a solution of equation ( 4.1.1) as well as the nonexistence of the global solution. Our main result is Theorem 4.1.1 (Global nonexistence) If q ≤ q c , then problem (4.1.1) has no nontrivial weak solutions. This paper is organized as follows : Section 4.2 contains the local existence theorem. In Section 4.3, the global nonexistence result (Theorem 4.1.1) will be proved.

Throughout this paper, positive constants will be denoted by C and will change from line to line.

Local existence

Using the Riemann-Liouville integral operator ; therefore, (4.1.1) takes the form

u t -div |∇u| p-2 ∇u = Γ(δ)J δ 0|t (|u| p ) (t), (4.2.1)
where δ = 1 -γ. Let us present our main results. First, we give the definition of the Definition 4.2.1 (Weak solution) A measurable function u(x, t) defined in S T = R N × (0, T ] is called a weak solution of (4.1.1) if for every bounded open set Ω with smooth boundary ∂Ω,

u ∈ C([0, T ] : L 1 (Ω)) ∩ L p loc (0, T : W 1 p (Ω)) ∩ L ∞ loc (S T ),
and

Ω u(x, t)φ(x, t) dx + t t 0 Ω [-uφ t + |∇u| p-2 ∇u • ∇φ] dx dτ = Γ(δ) t t 0 Ω J δ 0|τ (|u| q-1 u)ϕ dxdτ + Ω u(x, t 0 )φ(x, t 0 ) dx (4.2.2)
for all 0 ≤ t 0 < t ≤ T and φ ∈ C 1 (Ω × [0, T ]), φ(•, T ) = 0 near ∂Ω, where δ = 1 -γ. Moreover lim t →0 Br |u(x, t) -u 0 (x)| dx = 0, ∀r > 0. (4.2.3) We use ν(a 1 , a 2 , • • • , a n ) to denote positive constants depending only on a 1 , a 2 , • • • , a n . Let sgn η s =        1 if s > η, s η if -η ≤ s ≤ η, -1 if s < -η. For f ∈ L loc (R N ), we define |||f ||| h = sup x∈R N 1 |B 1 (x)| B 1 (x) |f (y)| h dy 1 h . (4.2.4) Theorem 4.2.2 (Local existence) Let |||u 0 ||| h < ∞ where h = 1 if 1 ≤ q < p -1 + p N and h > N p (q -p + 1) if q ≥ p -1 + p N . (4.2.5)
Then there exist a constant ν = ν(N, p, q) ≥ 1 and a positive constant T 0 defined by

T 0 + T 0 |||u 0 ||| p-2 h + T 1- N (q-p+1) ph 0 |||u 0 ||| q-1 h = ν -1 (4.2.6)
such that there exists a weak solution u to (4.1.1) in the strip S T satisfying, for all 0 < t < T 0 ,

|||u(•, t)||| h ≤ ν|||u 0 ||| h , (4.2.7
)

|u(x, t)| ≤ νt -N k h + γ-1 q-1 |||u 0 ||| ph k h h , (4.2.8)
where

k h = N (p -2) + ph.
Proof. The proof of the Theorem 4.2.2 is based on the following lemmas.

Lemma 4.2.3 Let u be any locally bounded continuous weak solution of (4.1.1) in S T for some 0 < T < ∞. Then for fixed h ≥ 1 there exists a constant ν depending only on N, p, q, h such that for every ball B 2ρ (x 0 ) and for all 0 < t < T,

||u(•, τ )|| p-2 ∞,B 2ρ (x 0 ) ρ -p + τ 1-γ sup x∈B 2ρ (x 0 ) |u| q-1 ≤ τ -1 , τ ∈ (0, t). (4.2.9)
Furthermore, the following estimate holds

||u(•, t)|| ∞,Bρ(x 0 ) ≤ νt -N +p k h t 0 B 2ρ (x 0 ) |u| h dx dτ p k h , (4.2.10)
where

k h = N (p -2) + ph.
G(t) = sup 0<τ <t B 2 (x 0 ) |u(x, τ )| h dx 1 h < ∞. (4.2.11)
Then there exists a constant ν = ν(N, p, q) such that for every ball B 2 (x 0 ), 0 < t < T, u satisfies (4.2.9)

t 0 B 1 (x 0 ) |∇u| p-1 dx dτ ≤ νt h k h G(t) 1+ h(p-2) k h , k h = N (p -2) + ph. (4.2.12)
We postpone the proofs of Lemma 4.2.3 and Lemma 4.2.4 at the end of this section.

Define a sequence of functions {f n } satisfying

f n ∈ C ∞ (R N ), f n (r) = r for r ∈ [-n, n], f n (r) = n + 1 for |r| ≥ n + 1 and |f ′ | ≤ ν.
Consider the family of approximating problem

         (u n ) t = div(|∇u n | p-2 ∇u n ) + C J δ 0|t (|f n (u n )| q-1 f n (u n )) in Q n , Q n = B n × R + , B n = {|y| < n}, u n (y, t) = 0, for |y| = n, u n (y, 0) = u 0n (y), (4.2.13) where u 0n ∈ C ∞ 0 (B n ) satisfies lim n →∞ Bρ |u 0n -u 0 | dy = 0, ∀ρ > 0,
and

|||u 0n ||| h ≤ |||u 0 ||| h .
By results of [START_REF] Dibenedetto | On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients[END_REF][START_REF] Dibenedetto | Hölder estimates for nonlinear degenerate parabolic systems[END_REF] and [START_REF] Dibenedetto | Degenerate Parabolic Equation[END_REF], there exists a solution

u n ∈ C(S T ) ∩ L ∞ (S T ), ∇u n ∈ C β, β 2 (Ω × (ǫ, T )), (u n ) t ∈ L 2 (0, T : L 2 loc (R N )) to (4.2.13),
where Ω can be any bounded open set, ǫ, T > 0 and some β ∈ (0, 1). Therefore for all t ∈ R sup

0<τ <t sup y∈R N {|u n | p-2 (y, τ ) + τ 1-γ |u n | q-1 (y, τ )} ≤ C(n) (4.2.14)
for a qualitative constant C(n) depending on the Theorem 4.2.2 will follow by a standard limiting process via the compactness results of [START_REF] Dibenedetto | On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients[END_REF] and [START_REF] Dibenedetto | Hölder estimates for nonlinear degenerate parabolic systems[END_REF] whence we show estimates (4.2.7)-(4.2.8) with u and u 0 replaced by u n and u n0 with constants independent of n. To prove these estimates we will work with (4.2.13) and drop the subscript n. Let B ρ (x) denote the ball with center x and radius ρ. Let t be the largest time satisfying that for all t ∈ (0, t)

sup x∈R N ||u(•, t)|| p-2 ∞,B 2 (x) + t 1-γ sup y∈B 2 (x) |u| q-1 (y, t) ≤ t -1 . (4.2.15)
Thus by Lemma 4.2.3, there exists a constant ν = ν(N.p, q) independent of n such that

||u(•, τ )|| ∞,B 1 (x) ≤ νt -N +p k h t 0 B 2 (x) |u| h dx dτ p k h , (4.2.16)
where

k h = N (p -2) + ph, for all 0 < t < t. Set ψ(t) = sup 0<τ <t |||u(•, τ )||| h h
and observe that ψ(t) is finite.

Next we assume (4.2.5) holds. It follows from (4.2.16) that for all 0 < t < t t sup

x∈R N |u| p-2 ≤ νt 1- N (p-2) k h ψ p(p-2) k h (t), t 2-γ sup x∈R N |u| q-1 ≤ νt 1- N (q-1) k h ψ p(q-1) k h (t). (4.2.17)
Also for δ > 0 to be chosen, we define

t * = sup{t > 0; t h ψ p-2 (t) + t 1 p (k h -N (q-1)) ψ q-1 (t) ≤ δ}. (4.2.18)
Notice that k h -N (q -1) > 0. Therefore for all 0 < t < min{t, t * },

t sup x∈R ◆ |u| p-2 (x, t) + t 2-γ sup x∈R ◆ |u| q-1 (x, t) ≤ νδ p k h .
If follows that δ = δ(p, q, N ) can be chosen small enough such that t * ≤ t. Let ξ(x) be nonnegative smooth cutoff function in B 2 (x) such that ξ = 1 on B 1 (x), |∇ξ| ≤ ν. We use (u ± ) h-1 ξ p as a testing function in (4.2.13). If h > 1, we get

t 0 B 2 (x) u t u h-1 ξ p (y) dy dτ = t 0 B 2 (x)
|∇u| p-2 ∇u∇(u h-1 ξ p (y)) dy dτ

+ Γ(δ) t 0 B 2 (x) J δ 0|t (|u| q-1 u)u h-1 ξ p (y) dy dτ.
Then

B 2 (x) u h (y, t)ξ p dy + h(h -1) t 0 B 2 (x) |∇u| p u h-2 ξ p (y) dy dτ ≤ B 2 (x) u h 0 (y)ξ p dy + ph t 0 B 2 (x) |∇u| p-1 u h-1 |∇ξ|ξ p-1 (y) dy dτ +Γ(δ) t 0 B 2 (x) J δ 0|t (|u| q-1 u)u h-1 ξ p (y) dy dτ.
By the Hölder inequality, we have

t 0 B 2 (x) |∇u| p-1 u h-1 |∇ξ|ξ p-1 (y) dy dτ ≤ t 0 B 2 (x) |∇u| p-1 u h-1 dy dτ ≤ t 0 B 2 (x) u h-1-(h-2)(p-1) p |∇u| p-1 u (h-2)(p-1) p dy dτ ≤ t 0 B 2 (x) u p(h-1)-(h-2)(p-1) dy dτ + t 0 B 2 (x) |∇u| p u h-2 dy dτ = t 0 B 2 (x) u p+h-2 dy dτ + t 0 B 2 (x)
|∇u| p u h-2 dy dτ.

Hence B 2 (x) u h (y, t) dy + C t 0 B 2 (x) |∇u| p u h-2 ξ p (y) dy dτ ≤ B 2 (x) u h (y, t) dy (4.2.19) ≤ B 2 (x) u h 0 (y) dy + t 0 B 2 (x) u p+h-2 dy dτ + Γ(δ) t 0 B 2 (x) J δ 0|t (|u| q-1 u)u h-1 ξ p (y) dy dτ,
where C is a constant depending on p and h.

On another hand

Γ(δ) t 0 B 2 (x) J δ 0|t (|u| q-1 u)u h-1 ξ p dy dτ (4.2.20) ≤ Γ(δ) t 0 B 2 (x) s 0 (s -σ) -γ u q (σ) dσ u h-1 dy dτ ≤ Γ(δ) t 0 || s 0 (s -σ) -γ u q (σ) dσ|| h L h x (B 2 (x)) × ||u h-1 || h h-1 L h h-1 x (B 2 (x)) dy dτ ≤ Γ(δ) t 0 || s 0 (s -σ) -γ u q (σ) dσ|| h L h x (B 2 (x)) dy dτ 1 h × t 0 ||u h-1 || h h-1 L h h-1 x (B 2 (x)) dy dτ h-1 h ≤ Γ(δ) c t 1-γ ||u(•, t)|| q-1 ∞,B 2ρ (x) t 0 B 2 (x)
u h (y, τ ) dy dτ

1 h × t 0 B 2 (x) u h (y, τ ) dy dτ h-1 h = Γ(δ) c t 1-γ ||u(•, t)|| q-1 ∞,B 2ρ (x) t 0 B 2 (x)
u h (y.τ ) dy dτ.

Combining (4.2.17), (

B 2 (x) u h 0 (y) dy + ν sup 0<τ <t sup x∈R N B 2 (x) 4.2.19) and (4.2.20), we get B 2 (x) u h (y, t) dy ≤
u h (y, t) dy

× t 0 τ - N (p-2) k h ψ p(p-2) k h (τ ) dτ + t 0 τ - N (q-1) k h ψ p(q-1) k h (τ ) dτ ≤ ν|||u 0 ||| h h + νψ(t){t h ψ p-2 (t) + t ( 1 p )(k h -N (q-1)) ψ q-1 (t)} p k h ≤ ν|||u 0 ||| h h + ν δ p k h ψ(t). (4.2.21)
If h = 1 take sgn η uξ p as a testing function in (4.2.13). After a Steklov averaging process and standard calculations, we get

B 2 (x) u(y,t) 0 sgn η u dsξ p dy + t 0 B 2 (x)
|∇u| p sgn ′ η uξ p dy dτ

+ p t 0 B 2 (x) |∇u| p-2 ∇u • ∇ξξ p-1 dx dτ = B 2 (x) u 0 (y) 0 sgn η u dsξ p dy + t 0 B 2 (x)
J δ 0|t (|u| q-1 u)sgn η uξ p dy dτ.

Discarding the second term on the left-hand side, which is nonnegative, letting η → 0, then sgn η u → 1 and using Lemma 4.2.4, we get

B 1 (x) |u(y, t)| dy ≤ B 2 (x) |u 0 (y)| dy + ν t 0 B 2 (x) |∇u| p-1 ξ p-1 dy dτ + t 0 B 2 (x) J δ 0|t (|u| q-1 u)ξ p dy dτ ≤ B 2 (x) |u 0 (y)| dy + νt 1 N (p-2)+p sup 0<τ <t B 2 (x) |u| dy 1+ p-2 k +ν sup 0<τ <t B 2 (x) |u| dy t + t 0 B 2 (x)
J δ 0|t (|u| q-1 u)ξ p dy dτ (4.2.22)

≤ B 2 (x) |u 0 (y)| dy + νt 1 N (p-2)+p sup 0<τ <t B 2 (x) |u| dy 1+ p-2 k +ν sup 0<τ <t B 2 (x) |u| dy t + c t 1-γ ||u(•, t)|| q-1 ∞,B 2ρ (x) t 0 B 2 (x)
|u(y, τ )| dy dτ,

where k = N (p -2) + p.

We use (4.2.17) and (4.2.22) to obtain, that for all t ∈ (0, t * ),

B 1 (x) |u(y, t)| dy ≤ B 2 (x) |u 0 (y)| dy + ν sup 0<τ <t sup x∈R N B 2 (x) |u(y, τ )| dy × t 1 k ψ p-2 k + t 0 τ -N (q-1) k (τ ) dτ + t ≤ ν|||u 0 |||
T 0 + T 0 |||u 0 ||| p-2 h + T 1- N (q-p+1) ph 0 |||u 0 ||| q-1 h = ν -1 ,
for a constant γ = γ(p, q, N ) ≥ 1. substituting (4.2.24) into (4.2.17), we get (4.2.8). Thus we have completed the proof of Theorem 4.2.2.

Proof of Lemma 4.2.3. Let ρ > 0, σ ∈ (0, 1 2 ] be fixed, let k > 0 to be chosen and for n = 0, 1, 2, • • • we set

ρ n = ρ + σ 2 n ρ, t n = t 2 - σ 2 n+1 p t, k n = k - k 2 n+1 , B n = B ρn (x 0 ), Q n = B n × (t n , t), 0 < t n < t ≤ T.
Let ξ n (x, t) be a smooth cutoff function in Q n such that

ξ n = 1 on Q n+1 , 0 ≤ ∂ξ n ∂t ≤ ν 2 np σ p t , |∇ξ n | ≤ ν 2 n+1 σρ .
In Definition 4.2.2, we take the test function φ = (u

-k n+1 ) h + ξ p n = [max{0, u -k n+1 }] h ξ p n to obtain Bn(t ′ ) u(u -k n+1 ) h + ξ p n dx + t ′ tn Bn -u (u -k n+1 ) h + ξ p n t dx dτ + t ′ tn Bn |∇u| p-2 ∇u∇ (u -k n+1 ) h + ξ p n dx dτ = Γ(δ) t ′ tn Bn J δ 0|t (|u| q-1 u)(u -k n+1 ) h + ξ p n dx dτ + Bn(tn) u(x, t 0 )(u -k n+1 ) h + ξ p n dx.
Then

1 h + 1 Bn(t ′ ) (u -k n+1 ) h+1 + ξ p n dx + h t ′ tn Bn |∇u| p-2 ∇u∇ [(u -k n+1 ) + ] (u -k n+1 ) h-1 ξ p n dx dτ + p t ′ tn Bn (u -k n+1 ) h + ξ p-1 n |∇u| p-2 ∇u∇ξ n dx dτ (4.2.25) = p t ′ tn Bn (u -k n+1 ) h+1 + ξ p-1 n ξ nt dx dτ + Γ(δ) t ′ tn Bn J δ 0|t (|u| q-1 u)(u -k n+1 ) h + ξ p n dx dτ,
where t n < t ′ < t. By the Hölder inequality

p t ′ tn Bn (u -k n+1 ) h + ξ p-1 n |∇u| p-2 ∇u∇ξ n dx dτ ≤ p t ′ tn Bn (u -k n+1 ) (h-1)(p-1) p + (u -k n+1 ) h+p-1 p + ξ p-1 n |∇(u -k n+1 )| p-1 |∇ξ n | dx dτ ≤ h 2 t ′ tn Bn (u -k n+1 ) (h-1) + ξ p n |∇(u -k n+1 )| p dx dτ +ν t ′ tn Bn (u -k n+1 ) (h+p-1) + |∇ξ n | p dx dτ. (4.2.26) Notice that if u > 2k n , u 2 > k n ⇒ u - u 2 > k n ⇒ u -k n > u 2 ⇒ u < 2(u -k n ); if k n+1 ≤ u ≤ 2k n , k n+1 -k n = k 2 n+2 > k n 2 n+2 > u 2 n+3 but u 2 n+3 < k n+1 -k n < u -k n , we get u < 2 n+3 (u -k n ).
Thus by the Hölder inequality

Γ(δ) t ′ tn Bn J δ 0|t (|u| q-1 u)(u -k n+1 ) h + ξ p n dx dτ (4.2.27) ≤ Γ(δ) t ′ tn Bn s 0 (s -σ) -γ u q (σ) dσ (u -K n+1 ) h + dx dτ ≤ Γ(δ) t ′ tn || s 0 (s -σ) -γ u q (σ) dσ|| h+1 L h+1 x (Bn) • ||(u -k n+1 ) h + || h+1 h L h+1 h x (Bn) dx dτ ≤ Γ(δ) t ′ tn || s 0 (s -σ) -γ u q (σ) dσ|| h+1 L h+1 x (Bn) dx dτ 1 h+1 × t ′ tn ||(u + k n+1 ) h + || h+1 h L h+1 h x (Bn) dx dτ h h+1 ≤ Γ(δ) c 1 t 1-γ ||u(•, t)|| q-1 ∞,B 2ρ (x 0 ) t ′ tn Bn u h+1 dx dτ 1 h+1 × t ′ tn Bn (u -k n+1 ) h+1 + dx dτ h h+1 ≤ Γ(δ) c t 1-γ ||u(•, t)|| q-1 ∞,B 2ρ (x 0 ) t ′ tn Bn (u -k n ) h+1 + dx dτ 1 h+1 × t ′ tn Bn (u -k n ) h+1 + dx dτ h h+1 = Γ(δ) c t 1-γ ||u(•, t)|| q-1 ∞,B 2ρ (x 0 ) t ′ tn Bn (u -k n ) h+1 + dx dτ.
Substituting (4.2.26) and (4.2.27) into (4.2.25), we obtain

1 h + 1 Bn(t ′ ) (u -k n+1 ) h+1 + ξ p n dx + h t ′ tn Bn |∇u| p-2 ∇u∇ [(u -k n+1 ) + ] (u -k n+1 ) h-1 ξ p n dx dτ + h 2 t ′ tn Bn |∇u| p-2 ∇u∇ [(u -k n+1 ) + ] (u -k n+1 ) h-1 + ξ p n dx dτ + ν t ′ tn Bn (u -k n+1 ) h+p-1 + |∇ξ n | p dx dτ ≤ p t ′ tn Bn (u -k n+1 ) h+1 + ξ p-1 n ξ nt dx dτ + Γ(δ) c t 1-γ ||u(•, t)|| q-1 ∞,B 2ρ (x 0 ) t ′ tn Bn (u -k n ) h+1 + dx dτ. Therefore 1 h + 1 Bn(t ′ ) (u -k n+1 ) h+1 + ξ p n dx + 3h 2 t ′ tn Bn |∇u| p-2 ∇u∇ [(u -k n+1 ) + ] (u -k n+1 ) h-1 ξ p n dx dτ + ν t ′ tn Bn (u -k n+1 ) h+p-1 + |∇ξ n | p dx dτ ≤ pν2 np σ p t + c t 1-γ sup 0<τ <t ||u(•, t)|| q-1 ∞,B 2ρ (x 0 ) Qn (u -k n ) h+1 + dx dτ.
Thus, by the following estimation

Qn ∇ (u -k n+1 ) p-1+h p + ξ n p dx dτ = Qn p -1 + h p (u -k n+1 ) h-1 p + ∇ [(u -k n+1 ) + ] ξ n + (u -k n+1 ) p-1+h p + ∇ξ n p dx dτ ≤ Qn (u -k n+1 ) h-1 + |∇(u -k n+1 ) + | p ξ p n dx dτ + Qn (u -k n+1 ) p-1+h + |∇ξ n | p dx dτ ≤ Qn (u -k n+1 ) h-1 + |∇(u -k n+1 ) + | p ξ p n dx dτ + ν 2 n+1 σρ p sup 0<τ <t ||u(•, τ )|| p-2 ∞,B 2 (x 0 ) Qn (u -k n+1 ) h+1 + dx dτ, we get ess sup tn<τ <t Bn(τ ) (u -k n+1 ) h+1 + ξ p n dx + Qn ∇ (u -k n+1 ) h+p-1 p + ξ n p dx dτ ≤ ν2 np σ p t (1 + M ) Qn (u -k n ) h+1 + dx dτ, (4.2.28)
where

M = sup 0<τ <t τ ||u(•, τ )|| p-2 ∞,B 2 (x 0 ) ρ -p + c τ 1-γ sup x∈B 2 (x 0 ) |u| q-1 .
By the Gagliardo Nirenberg inequality [17, p.3], we choose v = (u -k n+1 )

h+p-1 p + ξ n , α = p q , q = pb h+p-1 and s = (h+1)p h+p-1 , we get

Qn ξ d n (u -k n+1 ) b + dx dτ (4.2.29) ≤ ν Qn ∇ (u -k n+1 ) h+p-1 p + ξ n p dx dτ × ess sup tn<τ <t Bn(τ ) (u -k n+1 ) h+1 + ξ p n dx p N
,

where b = p + h -1 + p(h+1) N and d is large enough. Set A n = {(x, t) ∈ Q n-1 : u(x, t) ≥ k n }, n = 1, 2, • • • and observe that Qn (u -k n ) h+1 + dx dτ ≥ Qn+1 (u -k n ) h+1 + χ [u>k n+1 ] dx dτ ≥ (k n+1 -k n ) h+1 |A n+1 | = k 2 n+1 h+1 |A n+1 | ≥ ν2 -(h+1)n |A n+1 |k h+1 . ( 4
Q n+1 (u -k n+1 ) h+1 + dx dτ ≤ Qn (u -k n+1 ) h+1 + ξ (h+1)d b n dx dτ ≤ |A n+1 | s × Qn ξ d n (u -k n+1 ) b + dx dτ h+1 b ≤ ν|A n+1 | s × 2 np σ p t (1 + M ) Qn (u -k n ) h+1 + dx dτ ( N +p N )( h+1 b ) (4.2.31) ≤ νk -(b-h-1) (h+1) b C 0 (σ p t) -(1+ p N )(h+1)b × (1 + M ) (1+ p N )( h+1 b ) × Qn (u -k n ) h+1 + dx dτ 1+ p(h+1) bN
, where

s = N (p -2) + (h + 1)p N (p + h -1) + p(h + 1)
and

C 0 = 2 b-h-1+p+ p 2 N ( h+1 b ) . If k is chosen to satisfy Q 0 u - k 2 h+1 + dx dτ ≤ νk N (b-h-1) p 1 + M σ p t -(1+ N p )
, then by [42, Lemma 5.6, p.95], we get

Qn (u -k n ) h+1 + dx dτ → 0 as n → ∞, i.e., ||u + || ∞,Q∞ ≤ k. Take k = ν 1 + M σ p t 1+ N p Q 0 u h+1 + dx dτ p N (b-h-1)
.

It follows from this and the Hölder inequality that

||u + || ∞,Q∞ ≤ ν 1 + M σ p t (N +p) N (b-h-1) ||u + || ∞,Q 0 Q 0 u h + dx dτ p N (b-h-1) ≤ 1 2 ||u + || ∞,Q 0 + ν 1 + M σ p t (N +p) k h Q 0 u h + dx dτ p k h .
Similar to [1, p.393] we obtain

||u + || ∞,B 1 (x 0 )×( t 2 ,t) ≤ ν 1 + M σ p t (N +p) k h Q 0 u h + dx dτ p k h . ( 4
(
x 0 ) such that ξ = 1 on B 1 (x 0 ) and |∇ξ| ≤ 2. The calculations to follow are formal in which u + (-u -) is required to be strictly positive. The calculations can be made rigorous by replacing u + (-u -) with (u + + ǫ)(-u -+ ǫ) and letting ǫ → 0. By the Hölder inequality, we have

t 0 B 1 (x 0 ) |∇u| p-1 ξ p-1 dx dτ = t 0 B 1 (x 0 ) τ β u -1 p + |∇u| p-1 τ -β u 1 p + dx dτ ≤   t 0 B 1 (x 0 ) τ βp p-1 |∇u| p u 1 p-1 + dx dτ   p-1 p × t 0 B 1 (x 0 ) τ -βp u + dx dτ 1 p . ( 4
φ = t βp p-1 u 1-1 p-1 + ξ p (x) in (4.2.2) to obtain B 1 (x 0 ) ut βp p-1 u 1-1 p-1 + ξ p (x) dx - t 0 B 1 (x 0 ) u τ βp p-1 u 1-1 p-1 + ξ p (x) τ dx dτ + t 0 B 1 (x 0 ) |∇u| p-2 ∇u∇ τ βp p-1 u 1-1 p-1 + ξ p (x) dx dτ (4.2.34) = Γ(δ) t 0 B 1 (x 0 ) J δ 0|t (|u| q-1 u)τ βp p-1 u 1-1 p-1 + ξ p (x) dx dτ + B 1 (x 0 ) u(x, t 0 )t βp p-1 0 u 1-1 p-1 + ξ p (x) dx.
We have

t 0 B 1 (x 0 ) |∇u| p-2 ∇u∇ τ βp p-1 u 1-1 p-1 + ξ p (x) dx dτ ≤ t 0 B 1 (x 0 ) τ βp p-1 |∇u| p u 1 p-1 + dx dτ + p t 0 B 1 (x 0 ) τ βp p-1 |∇u| p-1 u 1-1 p-1 + ξ p-1 |∇ξ| dx dτ ≤ t 0 B 1 (x 0 ) τ βp p-1 |∇u| p u 1 p-1 + dx dτ + p t 0 B 1 (x 0 ) |∇u| p-1 u 1 p τ β u 1 p + u 1-1 p-1 + τ β p-1 dx dτ ≤ t 0 B 1 (x 0 ) τ βp p-1 |∇u| p u 1 p-1 + dx dτ + p t 0 B 1 (x 0 ) |∇u| p u 1 p-1 + τ βp p-1 dx dτ + p t 0 B 1 (x 0 ) u p-1 p-1 + τ βp p-1 dx dτ = C t 0 B 1 (x 0 ) τ βp p-1 |∇u| p u 1 p-1 + dx dτ + ν t 0 B 1 (x 0 ) u p-1 p-1 + τ βp p-1 dx dτ.
By the Hölder inequality

Γ(δ) t 0 B 1 (x 0 ) J δ 0|t (|u| q-1 u)τ βp p-1 u 1-1 p-1 + ξ p (x) dx dτ ≤ Γ(δ) t 0 ||τ βp 2p-3 s 0 (s -σ) -γ u q (σ) dσ|| 2p-3 p-1 L 2p-3 p-1 x (B 1 (x 0 )) × ||τ βp(p-2) (2p-3)(p-1) u 1-1 p-1 + || 2p-3 p-2 L 2p-3 p-2 x (B 1 (x 0 )) dx dτ ≤ Γ(δ) t 0 ||τ βp 2p-3 s 0 (s -σ) -γ u q (σ) dσ|| 2p-3 p-1 L 2p-3 p-1 x (B 1 (x 0 )) dx dτ p-1 2p-3 × t 0 ||τ βp(p-2) (2p-3)(p-1) u 1-1 p-1 + || 2p-3 p-2 L 2p-3 p-2 x (B 1 (x 0 )) dx dτ p-2 2p-3 ≤ Γ(δ) c 1 t 1-γ ||u(•, t)|| q-1 ∞,B 2ρ (x 0 )   t 0 B 3 2 (x 0 ) τ βp p-1 u 2p-3 p-1 + dx dτ   p-1 2p-3 ×   t 0 B 3 2 (x 0 ) τ βp p-1 u 2p-3 p-1 + dx dτ   p-2 2p-3 ≤ Γ(δ) c t 1-γ ||u(•, t)|| q-1 ∞,B 2ρ (x 0 ) t 0 B 3 2 (x 0 ) τ βp p-1 u 2-1 p-1 + dx dτ. Moreover t 0 B 1 (x 0 ) u τ βp p-1 u 1-1 p-1 + ξ p (x) τ dx dτ ≤ ν t 0 B 1 (x 0 ) uτ βp p-1 -1 u 1-1 p-1 + dx dτ + ν t 0 B 1 (x 0 ) uτ βp p-1 u -1 p-1 + u t dx dτ ≤ ν t 0 B 1 (x 0 ) uτ βp p-1 -1 u 1-1 p-1 + dx dτ + ν t 0 B 1 (x 0 ) τ βp p-1 u 1-1 p-1 + div |∇u| p-2 ∇u dx dτ +Γ(δ) t 0 B 1 (x 0 ) J δ 0|t (|u| q-1 u)τ βp p-1 u 1-1 p-1 + ξ p (x) dx dτ ≤ ν t 0 B 1 (x 0 ) uτ βp p-1 -1 u 1-1 p-1 + dx dτ + ν t 0 B 1 (x 0 ) τ βp p-1 u -1 p-1 + |∇u| p dx dτ +Γ(δ) t 0 B 1 (x 0 ) J δ 0|t (|u| q-1 u)τ βp p-1 u 1-1 p-1 + ξ p (x) dx dτ + ν t 0 B 1 (x 0 ) uτ βp p-1 -1 u 1-1 p-1 + dx dτ + ν t 0 B 1 (x 0 ) τ βp p-1 |∇u| p u 1 p-1 + dx dτ +Γ(δ) t 0 B 1 (x 0 ) J δ 0|t (|u| q-1 u)τ βp p-1 u 1-1 p-1 + ξ p (x) dx dτ,
where we have used the fact that u t = div (|∇u| p-2 ∇u) + Γ(δ)J δ 0|t (|u| q-1 u). Actually, the above estimate holds for any weak solution u by the density argument. Hence (4.2.34) gives

t 0 B 1 (x 0 ) τ βp p-1 |∇u| p u 1 p-1 + dx dτ ≤ ν t 0 B 3 2 (x 0 ) τ βp p-1 u p-1 p-1 + dx dτ + ν t 0 B 3 2 (x 0 ) τ βp p-1 -1 u 2-1 p-1 + dx dτ +c t 1-γ ||u(•, t)|| q-1 ∞,B 2ρ (x 0 ) t 0 B 3 2 (x 0 ) τ βp p-1 u 2-1 p-1 + dx dτ, therefore t 0 B 1 (x 0 ) τ βp p-1 |∇u| p u 1 p-1 + dx dτ ≤ ν(1 + M ) t 0 B 3 2 (x 0 ) τ βp p-1 -1 u 2-1 p-1 + dx dτ ≤ ν(1 + M )G(t) t 0 τ βp p-1 -1 ||u + (•, τ )|| 1-1 p-1 ∞,B 3 2 (x 0 ) dτ dτ, (4.2.35)
where |∇u| p-1 ξ p-1 dx dτ

M = sup 0<τ <t τ ||u(•, τ )|| p-2 ∞,B 2 (x 0 ) ρ -p + τ 1-γ sup x∈B 2 (x 0 ) |u| q-1 , and
G(t) = sup 0<τ <t B 2 (x 0 ) |u(x, τ )| h dx 1 h . Notice that by (4.2.10), ||u(•, t)|| ∞,Bρ(x 0 ) ≤ νt -N +p k h t 0 B 2ρ (x 0 ) |u| h dx dτ p k h ≤ νt -N k h G(t) ph k h , and t 0 B 1 (x 0 ) τ -βp u + dx dτ ≤ t 0 B 2 (x 0 ) τ -βp u + dx dτ ≤ νt 1-βp G(t). ( 4
≤ ν(1 + M )G(t) t 0 τ βp p-1 -1 νt -N k h G(t) ph k h 1-1 p-1 dτ p-1 p × νt 1-βp G(t) 1 p ≤ νG(t) 1+ h(p-2) k h t βp p-1 - N (p-2) k h (p-1) p-1 p × t 1 p -β = νG(t) 1+ h(p-2) k h t k h -N (p-2) k h p = νG(t) 1+ h(p-2) k h t h k h . Remark : If h = 1, for any constant ρ ≥ 1 we can prove t 0 Bρ(x 0 ) |∇u| p-1 dx dτ ≤ νt 1 k G(t) 1+ (p-2) k ,
where k = N (p -2) + p.

Global nonexistence

In this section we present the proof of Theorem 4.1.1.

Proof of Theorem 4.1.1. Let u be a nonnegative global weak solution of (4.1.1), we have

Γ(δ) ∞ 0 Ω J δ 0|t (u q )ϕ dx dt = ∞ 0 Ω |∇u| p-2 ∇u∇ϕ dx dt - ∞ 0 Ω uϕ t dx dt - Ω u 0 (x)ϕ(x, 0) dx.
where Ω is any bounded open set in R n . Here

ϕ(x, t) = D δ t|T ( φ(x, t)) := D δ t|T ϕ ℓ 1 (x)ϕ 2 (t) with ϕ 1 (x) := Φ (|x|/B) , ϕ 2 (t) := 1 - t T η + , (4.3.1) where D δ t|T is given by (1.1.1), T > 0, ℓ, η ≫ 1 and Φ ∈ C ∞ (R + ) is a cut-off non-increasing function such that Φ(r) =      1 if 0 ≤ r ≤ 1, 0 if r ≥ 2, 0 ≤ Φ ≤ 1 and |Φ ′ (r)| ≤ C 1 /
r for all r > 0. The constant B > 0 in the definition of ϕ 1 is fixed and will be chosen later. Then, we obtain

Γ(δ) T 0 Ω(B) J δ 0|t (u q )D δ t|T φ dx dt + Ω(B) u 0 (x)D δ t|T φ(0, x) dx = T 0 Ω(B) |∇u| p-2 ∇u∇ϕ dx dt - T 0 Ω(B) u ∂ t ϕ dx dt,
where Ω(B) := {x ∈ R n ; |x| < 2B}. From (1.1.3) and (1.1.7), we conclude that

T 0 Ω(B) D δ 0|t J δ 0|t (u q ) φ dx dt + C T -δ Ω(B) u 0 (x)ϕ ℓ 1 (x) dx = C T 0 Ω(B) |∇u| p-2 ∇u∇ϕ dx dt -C T 0 Ω(B) u ∂ t ϕ dx dt,
where D δ 0|t is defined in (1.1.1). Moreover, using (1.1.5) and the nonnegativity of u 0 and u, it follows

T 0 Ω(B) u q φ dx dt ≤ C T 0 Ω(B) |∇u| p-1 |∇ϕ| dx dt + C T 0 Ω(B) u|∂ t ϕ| dx dt := I 1 + I 2 . (4.3.2)
Next we observe that by introducing the term φ1/q φ-1/q in I 2 and applying Young's inequality ab ≤ 1 8 a q + C(q)b q ′ , a ≥ 0, b ≥ 0, q ′ = q/(q -1), we obtain

I 2 ≤ 1 8 T 0 Ω(B) u q φ dx dt + C T 0 Ω(B) ϕ ℓ 1 ϕ -1/(q-1) 2 D 1+δ t|T ϕ 2 q ′ dx dt. (4.3.3)
In order to obtain a similar estimation on I 1 , let α < 0 be an auxiliary constant such that α > max{-1, 1 -p} and let u ǫ (x, t) = u(x, t) + ǫ, ǫ > 0.

As u is a weak solution, by taking ϕ ǫ (x, t) = u α ǫ (x, t)ϕ(x, t) as a test function where ϕ is given in (4.3.1), we have

Γ(δ) ∞ 0 Ω(B) J δ 0|t (u q )u α ǫ ϕ dx dt = ∞ 0 Ω(B) |∇u| p-2 ∇u∇(u α ǫ ϕ) dx dt - ∞ 0 Ω(B) u∂ t (u α ǫ ϕ) dx dt - Ω(B) u 0 (x)u α ǫ (x, 0)ϕ(x, 0) dx.
Then, using the fact that

∇(u α ǫ ϕ) = αu α-1 ǫ ∇(u)ϕ + u α ǫ ∇(ϕ) and ∂ t (u α ǫ ϕ) = αu α-1 ǫ ∂ t (u)ϕ + u α ǫ ∂ t (ϕ), we get Γ(δ) ∞ 0 Ω(B) J δ 0|t (u q )u α ǫ ϕ dx dt = α ∞ 0 Ω(B) |∇u| p u α-1 ǫ ϕ dx dt + ∞ 0 Ω(B) (|∇u| p-2 ∇u∇ϕ)u α ǫ dx dt -α ∞ 0 Ω(B) uu α-1 ǫ ∂ t (u)ϕ dx dt - ∞ 0 Ω(B) uu α ǫ ∂ t ϕ dx dt - Ω(B) u 0 (x)u α ǫ (x, 0)ϕ(x, 0) dx. (4.3.4)
Using the definition of u ǫ and integrating by parts, we obtain

J 1 := ∞ 0 Ω(B) uu α-1 ǫ ∂ t (u)ϕ dx dt = ∞ 0 Ω(B) (u ǫ -ǫ)u α-1 ǫ ∂ t (u)ϕ dx dt = ∞ 0 Ω(B) u α ǫ ∂ t (u ǫ )ϕ dx dt -ǫ ∞ 0 Ω(B) u α-1 ǫ ∂ t (u ǫ )ϕ dx dt = 1 α + 1 ∞ 0 Ω(B) ∂ t (u α+1 ǫ )ϕ dx dt - ǫ α ∞ 0 Ω(B) ∂ t (u α ǫ )ϕ dx dt = - 1 α + 1 ∞ 0 Ω(B) u α+1 ǫ ∂ t ϕ dx dt - 1 α + 1 Ω(B) u α+1 ǫ (x, 0)ϕ(x, 0) dx + ǫ α ∞ 0 Ω(B) u α ǫ ∂ t ϕ dx dt + ǫ α Ω(B) u α ǫ (x, 0)ϕ(x, 0) dx.
Similarly, u (1-α)(p-1) ϕ 1-p |∇ϕ| p dx dt =:

J 2 = ∞ 0 Ω(B)
K 1 + K 2 + K 3 .
At this stage, we use Young's inequality and the fact that ∇(ϕ ℓ 1 ) = ℓϕ ℓ-1 ∇ϕ 1 to estimate K 1 , K 2 and K 3 , precisely, we have :

K 1 = T 0 Ω(B)
(u p-1+α φ(p-1+α)/q )(C φ-(p-1+α)/q |∇ϕ| p ϕ 1-p ) dx dt

≤ 1 8 T 0 Ω(B) u q φ + C T 0 Ω(B)
ϕ ℓ(q-p+1-α)-pq q-p+1-α

1 |∇ϕ 1 | pq q-p+1-α ϕ -p-1+α q-p+1-α 2 D δ t|T ϕ 2 q q-p+1-α K 2 = T 0 Ω(B)
(u α+1 φ(α+1)/q )(C φ-(α+1)/q |∂ t ϕ|) dx dt q-(1-α)(p-1) dx dt.

Therefore, we conclude that

I 1 ≤ 3 8 T 0 Ω(B)
u q φ dx dt

+ C T 0 Ω(B) ϕ ℓ 1 ϕ -α+1 q-1-α 2 D 1+δ t|T ϕ 2 q q-1-α dx dt + C T 0 Ω(B)
ϕ ℓ(q-p+1-α)-pq q-p+1-α

1 |∇ϕ 1 | pq q-p+1-α ϕ -p-1+α
q-p+1-α 2 D δ t|T ϕ 2 q q-p+1-α dx dt At this stage, we have to distinguish 2 cases.

+ C T 0 Ω(B)
• Case of 1 < q < q c : in this case, we take B = T |u| q φ dx dt ≤ CT -(δ+1)q ′ + n θ +1 , (4.3.9)

where C is independent of T . Letting T → ∞ in (4.3.9), thanks to q < q c and the Lebesgue dominated convergence theorem, it yields that ∞ 0 R n |u| q dx dt = 0, which implies u(x, t) = 0 for all t and a.e. x.

• Case of q = q c : let

B = R -1 θ T 1 θ
, where 1 ≪ R < T is such that T and R do not go simultaneously to ∞. Moreover, from the first case and the fact that q = q c , there exist a positive constant D independent of T such that ∞ 0 R n |u| q dx dt ≤ D, which implies that

T 0 ∆(R -1 θ T 1 θ )
|u| q φ dx dt → 0 as T → ∞, (4.3.10)

where ∆(B) := {x ∈ R n ; B < |x| < 2B}. Repeating a similar calculation as in the subcritical case (q < q c ) and using Hölder's inequality instead of Young's one in K 1 and K 3 , we get

I 2 ≤ 1 3 T 0 Ω(B) u q φ dx dt + C T 0 Ω(B) ϕ ℓ 1 ϕ -1/(q-1) 2 D 1+δ t|T ϕ 2 q ′ dx dt, K 1 ≤ C T 0 ∆(B)
u q φ dx dt p-1+α q × T 0 Ω(B) ϕ ℓ(q-p+1-α)-pq q-p+1-α 1 |∇ϕ 1 | pq q-p+1-α ϕ -p-1+α q-p+1-α 2 D δ t|T ϕ 2 q q-p+1-α dx dt q-p+1-α q , K 2 ≤ 1 3

T 0 Ω(B)
u q φ dx dt + C T 0 Ω(B)

ϕ ℓ 1 ϕ -α+1
q-1-α 2 D 1+δ t|T ϕ 2 q q-1-α dx dt, and

K 3 ≤ C T 0 ∆(B)
u q φ dx dt .

We conclude that

T 0 Ω(B)
u q φ dx dt

≤ C T 0 Ω(B)
ϕ ℓ 1 ϕ

-1/(q-1) 2 D 1+δ t|T ϕ 2 q q-1 dx dt

+ C T 0 ∆(B)
u q φ dx dt p-1+α q × T 0 Ω(B) ϕ ℓ(q-p+1-α)-pq q-p+1-α 1 |∇ϕ 1 | pq q-p+1-α ϕ -p-1+α q-p+1-α 2 D δ t|T ϕ 2 q q-p+1-α dx dt q-p+1-α q (4.3.11)

+ C T 0 Ω(B) ϕ ℓ 1 ϕ -α+1
q-1-α 2 D 1+δ t|T ϕ 2 q q-1-α dx dt

+ C T 0 ∆(B)
u q φ dx dt Let Θ L denote the restriction of Θ to the interval [0, T M ). If we take σ > Λ, then L(σ) < ∞, and we have

L(σ) ≤ Θ -1
L (R -s Ψ -1 (σR αβ V )).

(5.3.6)

In the next result we use this inequality to obtain an explicit upper bound for the life span of a solution.

Theorem 5.3.6 Let u 0 be a nonnegative nontrivial continuous function in R N . Then, there exists positive constants Λ m , C and σ 1 so that the life span L(σ) corresponding to the initial condition σu 0 with σ > Λ m satisfies L(σ) ≤ Cσ -p+1-m α+1 .

(5.3.7)

Proof. Decreasing the value of T M to a value T m if needed, we may assume that the function Θ 0 , introduced above, is decreasing on (0, T M ). We can choose Λ m such that in the meantime Λ m R -αβ V ≥ Ψ(R s Θ(T m )) and Λ m R -αβ V ≥ C 1 , where C 1 is a sufficiently large constant so that whenever σ > Λ m , then

Ψ -1 (σR -αβ V ) = [(1 -w) -1 w -w 1-w ] 1-w (σR -αβ V ) p+1-m p+1 ,
with w = m/(p + 1). We write It is easily seen that this implies equation (5.3.7).

Ψ -1 (σR -αβ V ) = γ 0 R -αβ p+1-m p+1 V p+1-

1 . 1

 11 Préliminaires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Explosion en temps fini pour une équation de Schrödinger non-linéaire avec un laplacien fractionnaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Explosion en temps fini d'une solution de l'équation d'onde amortie avec un potentiel d'espace-temps et d'un terme mémoire non linéaire . . . . . . . . . . . . . . . 1.4 Sur le problème de Cauchy pour l'équation d'évolution avec le p-Laplacien et un terme mémoire non linéaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Durée de vie des solutions non-négatives pour des équations d'évolution non-locales en temps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  .1.6) pour toute α ∈ (0, 1); doncD α t|T w 1 (T ) = 0, D α t|T w 1 (0) = C T -α , D α+1 t|T w 1 (T ) = 0 et D α+1 t|T w 1 (0) = C T -α-1 .

) où p > 1 .

 1 Il a obtienu les résultats suivants : a) Si p < 1 + 2/n, alors chaque solution non-triviale de (1.4.2) explose en temps fini. b) Si p > 1 + 2/n et u 0 (x) ≤ δe -|x| 2 , (0 < δ ≪ 1), alors (1.4.2) admet une solution globale.

2 1

 2 with the homogeneous Dirichlet boundary condition, associated to the first eigenvalue κ := κ α/, and ϕ 2

Remark 3 . 1 . 4

 314 We don't know whether there exists a global solution for problem (3.1.1) when p > p c .

  .4.5) Using (3.4.3), (3.4.4) and (3.4.5), it follows from (3.4.2) that

1/p ′ ( 3 . 4 . 10 )

 3410 Similarly to the last case, substituting (3.4.3), (3.4.4) and (3.4.10) into (3.4.2), we get

  ))/n : let B = R with the same R introduced in the case p = p c . Then, taking the scaled variables s = T -1 t, y = R -1 x, it follows from (3.4.6) that T 0 Ω(R)

Lemma 4 . 2 . 4

 424 Let the assumptions of Lemma 4.2.3 hold and set

1 +ǫI 1 u

 11 0)ϕ(x, 0) dx.Using J 1 ,J 2 , and J 3 , it follows from (4.3.C(α) b p , a > 0, b > 0.(4.3.5)Therefore, using the positivity of the solution u, we conclude that|∂ t ϕ| dx dt.Using the last inequality and Young's inequality (4.3.5), we getu (1-α)(p-1) ǫ ϕ 1-p |∇ϕ| p dx dt. (4.3.6)Apply the Fatou and Lebesgue theorems, as ǫ → 0, we obtain α+1 |∂ t ϕ| dx dt + C

  α)(p-1) φ(1-α)(p-1)/q )(C φ-(1-α)(p-1)/q ϕ 1-p |∇ϕ| p ) dx dt

1 )

 1 dx dt.Using the estimations of I 1 and I 2 into (4.3.2), we getT 0 Ω(B)u q φ dx dt (4.3.7) α)(p-1) dx dt.(4.3.8)

1 θ

 1 where θ = p(q -1)/[q -p -δp + 1 + 2δ]. So, using (1.1.6) and the change of variables : s = T -1 t, y = T -1 θ x, we get from (4.3.7) that

  α)(p-1) dx dt q-(1-α)(p-1)q

,.

  where γ 0 = (p + 1)(p + 1 -m) -p+1-m p+1 m -m p+1 . SinceΘ(T ) ≤ Θ 0 (T ) ≤ α 0 T -α+1 p+1 + β 0 T p+1-m(1+α) p+1 m , on [0, T m ), it follows that for η > β 0 T p+1-m(1+α) p+1 m .Let [0, T max ) be the maximal interval of existence of u and T ∈ [0, T max ). We defineG(R, σ) := α p+1 0 [γ 0 R -s R -αβ V Whenever T < L(σ) we have T ≤ G(R, σ). Therefore L(σ) ≤ G(R, σ).(5.3.8)

  1.3. EXPLOSION EN TEMPS FINI D'UNE SOLUTION DE L'ÉQUATION D'ONDE AMORTIE AVEC UN P 1.3 Explosion en temps fini d'une solution de l'équation d'onde amortie avec un potentiel d'espace-temps et d'un terme mémoire non linéaire Ce travail concerne le problème de Cauchy pour l'équation des ondes amorties semi-linéaire suivante

  et les références citées) est l'idée principale sur laquelle est basée la preuve du résultat d'explosion.

	Remarque 1.3.5
	On ne connait pas de résultat d'existence de solution globale du problème (1.3.1) quand p > p c .
	1.4 Sur le problème de Cauchy pour l'équation d'évolution
	avec le p-Laplacien et un terme mémoire non linéaire

Cette partie est consacrée à l'étude du problème suivant de Cauchy pour l'équation d'évolution

  and satisfies the equation(3.2.3) in the sense of L 2 (R n ). It is well known that if (u 0 , u 1 ) ∈ H 2 × H 1 and F ∈ C 1 ([0, ∞); L 2 ), then there exists a unique strong solution (see[START_REF] Ikawa | Hyperbolic partial differential equations and wave phenomena[END_REF] Theorem 2.27]).

Definition 3.2.2 (Mild solution)

  is still only qualitatively known. A quantitative lower bound can be found by substituting (4.2.24) into the definition of t * in (4.2.18). It gives that (4.2.24) holds for all 0 < t < T 0 where T 0 is the smallest root of

	1 1 + ν δ	p k ψ(t) + νtψ(t).	(4.2.23)
	Let	νt * ≤	1 2	.
	By (4.2.21) and (4.2.23), we can determine δ = δ(p, q, N ) a priory depending only on the indicated
	quantities so that
	ψ(t) ≤ ν|||u 0 ||| h h , for any 0 < t < t * .	(4.2.24)
	The number t

*

  .2.32) Also, taking φ = (-u -k n+1 ) h + ξ p n , we get (4.2.32) for u -. This implies (4.2.10).

	Proof of Lemma 4.2.4. Let ξ(x) be a piecewise smooth cutoff function in B 3 2

Je voudrais exprimer toute ma reconnaissance à M. Mokhtar Kirane qui a encadré mes recherches et qui s'est toujours soucié de m'offrir, de tout point de vue, les meilleures conditions de travail possibles. Je le remercie de m'avoir fait confiance malgré les connaissances plutôt légères que j'avais en début de ma thèse, puis pour m'avoir guidé, encouragé, conseillé, pendant ces trois années tout en me laissant une grande liberté. Je le remercie en particulier pour sa grande disponibilité et la confiance qu'il m'a accordée. Merci pour son enthousiasme permanent, son incroyable soutien et surtout ses qualités humaines exceptionnelles qui font de lui le directeur de thèse dont tout le monde rêve ! M. Ahmad Fino, merci de m'avoir pris sous ton aile et de m'avoir proposé cette thèse, merci pour ton aide exceptionelle sur le plan humain, scientifique et administratif. Merci pour tes qualités d'écoute, ta présence à mes côtés et tes conseils avisés et rassurants, merci pour tes remarques pertinentes et pour ta patience même pendant les périodes les plus chargées.. ! Ton soutien moral et scientifique au quotidien n'a pas de prix. Merci de m'avoir fait profiter de ton expérience de l'enseignement. J'espère avoir été digne de la confiance que tu m'avais accordé et que ce travail est finalement à la hauteur de tes espérances. Quoi qu'il en soit, j'ai beaucoup appris à tes côtés et je suis très honoré de t'avoir eu comme encadrant. Que tu trouves ici l'expression de ma profonde reconnaissance.

Remerciements

Life Span of a solution Bibliographie

Chapitre 5 Life span of nonnegative solutions to an evolution equation with non local in time nonlinearity

Ihab DANNAWI, Ahmad Z. FINO, Mokhtar KIRANE and A. NABTI

Abstract

We consider the problem u t -∆|u| m-1 u = t 0 (t -s) -γ |u| p u(s) ds, x ∈ R N , t > 0, where p > 0, m > 1, are real numbers with nonnegative nontrivial continuous bounded initial condition u(x, 0) = u 0 (x) ≡ 0, u 0 (x) ≥ 0, x ∈ R N .

We obtain an integral inequality that can be used to find an exponent p * ≤ p c , where p c is the critical exponent, such that this problem has no global nontrivial solution when p ≤ p * . This inequality may also be used to estimate the maximal time T max > 0 such that there is a solution for 0 ≤ t ≤ T max . This is illustrated with the initial condition u σ (x, 0) = σu 0 (x), σ > 0, by obtaining an upper bound of the form T max ≤ C 0 σ -ϑ , for some ϑ > 0.

Keywords : Hyperbolic equation, mild and weak solutions, local existence, Strichartz estimate, blow-up, Riemann-Liouville fractional integrals and derivatives.

Introduction

In this article, we investigate the maximal interval of existence of the solutions of the problem

where p > 0, m > 1, are real numbers with nonnegative nontrivial continuous bounded initial condition

If we consider (5.1.1) without nonlinear term, i.e u t -∆|u| m-1 u = 0, this equation is so-called Porous Medium Equation and have been studied by so many people. Actually many interesting results are obtained so far.

There are a number of physical applications where this model appears in a natural way, mainly to describe processes involving fluid flow, heat transfer or diffusion. Maybe the best known of them is the description of the flow of an isentropic gas through a porous medium, modeled independently by Muskat [START_REF] Muskat | The Flow of Homogeneous Fluids Through Porous Media[END_REF] around 1930. Indeed, this application was at the base of the rigorous mathematical development of the theory. Other applications have been proposed in mathematical biology, spread of viscous fluids, boundary layer theory, and other fields.

When equation (5.1.1) is considered with a nonlinearity of the form |u| p u, it reads

which is a particular case of (5.1.1) ; it corresponds to γ → 0. This equation has been considered by H. J. Kuiper [START_REF] Kuiper | Life span of nonnegative solutions to certain quasilinear parabolic Cauchy problems[END_REF]. By life span for initial condition u 0 , we mean the least upper bound of all values T such that [0, T ) is a maximal interval of existence of a solution. He found that the life span L(σ) is bounded by Cσ -(p+1-m) whenever u(x, 0) = σu 0 (x), σ > 0. (see Theorem 3.6 [START_REF] Kuiper | Life span of nonnegative solutions to certain quasilinear parabolic Cauchy problems[END_REF]).

Our article is motivated mathematically by the recent papers [START_REF] Cazenave | An equation whose Fujita critical exponent is not given by scaling[END_REF][START_REF] Li | Lifespan and a new critical exponent for a nonlocal parabolic equation with slowly decay initial values[END_REF] which deal with the critical exponent and life span for the parabolic equation with nonlocal in time nonlinearity

where u 0 ∈ C 0 (R N ), the space of all continuous functions decaying to zero at infinity, which is a particular case of (5.1.1) ; it corresponds to m → 1.

The main result of this article is to find the exponent

such that if p ≤ p * , the problem (5.1.1)-(5.1.2) has no global nontrivial solution. Suppose u σ is the solution corresponding to the nontrivial, nonnegative initial condition u σ (x, 0) = σu 0 (x). Let [0, T σ ) be its maximal interval of existence. We obtain a life span of the form

We will show for equation (5.1.1) a necessary condition for global solution. More precisely, if u is a global solution with initial condition u(x, 0) = u 0 (x), then an inequality of the form

, for some S > 0 and κ > 0, must be satisfied. Here ϕ 1 is the positive eigenfunction corresponding to the principal eigenvalue of the Dirichlet problem on the unit ball B 1 , and normalized such that B 1 ϕ 1 (ξ) dξ = 1. The constants C and κ depend on N , m, p and α.

The method used to prove the life span and necessary conditions for local and global existence is the test function method [START_REF] Baras | Local and global solvability of a class of semilinear parabolic equations[END_REF][START_REF] Baras | Critère d'existence de solutions positives pour des équations semilinéaires non monotones[END_REF][START_REF] Fino | Qualitative Properties of Solutions to a Time-Space Fractional Evolution Equation[END_REF][START_REF] Mitidieri | A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities[END_REF][START_REF] Qi | A blow up result for a nonlinear wave equation with damping : the critical case[END_REF]. The principle of this method is as follows : we assume, by absurd, that the solution is global, then we use the weak formulation of the equation on [0, T ], for some T > 0 and make an appropriate choice of the test function, then we make a change of variable and finally we take the limit as T → ∞ to get the contradiction. This paper is organized as follows. In section 5.2, we find the exponent p * . Finally, we obtain an upper bound for the maximal interval of existence.

The test function method

Let t * > 0. Suppose that u is a solution of (5.1.1)-(5.1.2) on R N × [0, t * ). We assume that

Let λ R be the principal eigenvalue for the Dirichlet problem on the ball of radius R :

where λ 1 is the principal eigenvalue of the Laplacian on the unit ball B 1 . Let ϕ 1 denote the unique nonnegative eigenfunction corresponding to the principal eigenvalue λ 1 such that

We also define ϕ(x, t)

The test function method

As u is a weak solution, we have

where we have used the fact that ∆ϕ

Using (1.1.7) and letting

.

We multiply by T α and use (1.1.6) ; we obtain

, where

Making the change of variable ξ = x/R and τ = t/R β , we deduce that

where

As

we have

It is our aim to use (5.2.1) to obtain information on the relationship between the initial condition and the length of the maximum interval of existence. Proof. In the case of s < 0 (i.e p < p * ), we use (1.1.3), (1.1.5), and (5.2.1) to obtain

which implies that u ≡ 0 is the only global solution. If s = 0 (i.e p = p * ), we note that J R (T ) is uniformly bounded for all R. Therefore by taking the limit as R → ∞ and then T → ∞ on the right-hand side of (5.2.1), we obtain again (5.2.4).

Life Span of a solution

Suppressing argument and subscripts (5.2.1) becomes

We will use this to obtain an estimate for V . First we state some lemmas.

Let η be an arbitrary positive number, then

For η sufficiently large

where w = max(w 1 , w 2 ).

We will use the notation

Then, for η sufficiently large,

Theorem 5.3.3 If u is a nonnegative solution of (5.1.1)-(5.1.2) on B R * × [0, t * ) and s given by (5.2.3), then for all (R, T ) ∈ (ρ, τ ) :

In particular, if u is a global nonnegative solution then Proof. For the sake of convenience we define

From (5.3.1) we see that

where

.

Then by Lemma 5.2, we have (5.3.3). For R sufficiently large we can use equation (5.3.2) to conclude the validity of (5.3.4).

Corollary 5.3.4 Suppose that u is a nonnegative global solution. Then

where K m is a constant depending on m, θ and p.

Proof. We easily obtain

Then, from (5.3.5), we have

where

, with

.

By Lemma 5.1, we get

. Then after substituting this into equation (5.3.4), the proof is complete.

When we are dealing with the problem originally considered by Fujita (m → 1 and α → 0), then J 1 = p(p + 1) -(p+1) p and K 1 = (p + 1)p -p p+1 , and we see that the above inequality reduces to

This is precisely the result found in [START_REF] Lee | Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem[END_REF]. Note that if m > 1 and α → 0, we recover the case of Kuiper [START_REF] Kuiper | Life span of nonnegative solutions to certain quasilinear parabolic Cauchy problems[END_REF]. As it is done in that article, we can deduce the following result. The proof is complete by letting k → ∞. Now, we consider the problem (5.1.1)-(5.1.2). By the life span for initial condition u 0 , we mean the least upper bound of all values T such that [0, T ) is the maximal interval of existence of the solution to (5.1.1)-(5.1.2). Let us fix u 0 ≡ 0 and u 0 ≥ 0 for all x ∈ R N . We denote by L(σ), σ > 0, the life span corresponding to initial condition σu 0 . There exists a value Λ such that

where T M is the value of T at which Θ(T ) attains its minimum value.