Existence et comportement asymptotique des solutions d’une équation de viscoélasticité non Linéaire de type hyperbolique

ﻲﻤﻠﻌﻟﺍ ﺚﺤﺒﻟﺍﻭ ﱄﺎﻌﻟﺍ ﻢﻴﻠﻌﺘﻟﺍ ﺓﺭﺍﺯﻭ

BADJI MOKHTAR UNIVERSITY

OF ANNABA

UNIVERSITÉ BADJI MOKHTAR

DE ANNABA

ﺭﺎﺘﺨﻤ ﻲﺠﺎﺒ ﺔﻌﻤﺎﺠ

ﺔﺒﺎﻨﻋ

Faculté des Sciences

Département de Mathématiques

MÉMOIRE

Présenté en vue de l’obtention du diplôme de

MAGISTER EN MATHÉMATIQUES

ÉCOLE DOCTORALE EN MATHÉMATIQUES

Par

Khaled ZENNIR

Intitulé

Existence et Comportement Asymptotique des

Solutions d’une Equation de Viscoélasticité

Non Linéaire de type Hyperbolique

Dirigé par

Prof. Hocine SISSAOUI

Option

Systèmes Dynamiques et Analyse Fonctionnelle

Devant le jury

PRÉSIDENT

_{B. KHODJA}

PROF.

U. B. M. ANNABA

RAPPORTEUR

H. SISSAOUI

PROF.

U. B. M. ANNABA

ÉXAMINATEUR

L. NISSE

M. C.

U. B. M. ANNABA

ÉXAMINATEUR

Y. LASKRI

M. C.

U. B. M. ANNABA

INVITE

B. SAID-HOUARI

Dr.

U. B. M. ANNABA

ﻲﻤﻠﻌﻟﺍ ﺚﺤﺒﻟﺍﻭ ﱄﺎﻌﻟﺍ ﻢﻴﻠﻌﺘﻟﺍ ﺓﺭﺍﺯﻭ

BADJI MOKHTAR UNIVERSITY

OF ANNABA

UNIVERSITÉ BADJI MOKHTAR

DE ANNABA

ﺭﺎﺘﺨﻤ ﻲﺠﺎﺒ ﺔﻌﻤﺎﺠ

ﺔﺒﺎﻨﻋ

Sciences Faculty

Department of Mathematics

THESIS

Submitted for the obtention of

MAGISTER DIPLOMA IN MATHEMATICS

DOCTORAL SCHOOL IN MATHEMATICS

By

Khaled ZENNIR

Existence et Comportement Asymptotique des

Solutions d’une Equation de Viscoélasticité

Non Linéaire de type Hyperbolique

Directed by

First Advisor: Prof. Hocine SISSAOUI

Second Advisor : Dr. Belkacem SAÏD-HOUARI

Option

Dynamic Systems and Functional Analysis

PRESIDENT

B. KHODJA

PROF.

U. B. M. ANNABA

SUPERVISOR

H. SISSAOUI

PROF.

U. B. M. ANNABA

EXAMINER

L. NISSE

M. C.

U. B. M. ANNABA

EXAMINER

Y. LASKRI

M. C.

U. B. M. ANNABA

2008

Existence and Asymptotic Behavior of

Solutions of a Non Linear Viscoelastic

In the first place at all, my gratitude goes to

Prof. H.

SISSAOUI

, my supervisor. He always listens, advices, encourages and

guides me, he was behind this work. I shall always be impressed by his

availability and the care with which he comes to read and correct my

manuscript.

My warm thanks go to

Dr.B.SAID-HOUARI

, who supervised this thesis

with great patience and kindness. He was able to motivate each step of my

work by relevant remarks and let me able to advance in my research,

beginning by the DES study in university of Skikda. I thank him very

sincerely for his availability even at a distance, with him I have a lot of fun

to work.

I wish to thank

Prof B. KHODJA

, who makes me the honour of chairing

the jury of my defense.

I am very grateful to

Dr. L. NISSE

and

Dr. Y. LASKRI

to have agreed to

rescind this thesis and to be part of my defense panel.

Of course, this thesis is the culmination of many years of study. I am

indebted to all those who guided me on the road of mathematics.

I thank from my heart, my teachers whose encouraged me during my

studies and have distracted me the mathematics.

I think for my parents, whose the work could not succeed without their

endless support and encouragement. They find in the realization of this

work, the culmination of their efforts and express my most affectionate

gratitude.

1 Preliminary 1

1.1 Banach Spaces - De…nition and Properties . . . 2

1.1.1 The weak and weak star topologies . . . 3

1.1.2 Hilbert spaces. . . 4

1.2 Functional Spaces . . . 6

1.2.1 The Lp_{( ) spaces . . . . 6}

1.2.2 The Sobolev space Wm;p( ) . . . 8

1.2.3 The Lp(0; T; X) spaces . . . 10

1.2.4 Some Algebraic inequalities . . . 12

1.3 Existence Methods. . . 14

1.3.1 The Contraction Mapping Theorem . . . 14

1.3.2 Gronwell’s lemma . . . 14

1.3.3 The mean value theorem . . . 15

2 Local Existence 17 2.1 Local Existence Result . . . 18

3 Global Existence and Energy Decay 39 3.1 Global Existence Result . . . 40

3.2 Decay of Solutions . . . 46

4 Exponential Growth 59 4.1 Growth result . . . 60

Abstract

Our work, in this thesis, lies in the study, under some conditions on p, m and the functional g, the existence and asymptotic behavior of solutions of a nonlinear viscoelastic hyperbolic problem

of the form ₈ > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > : utt u ! ut+ t R 0 g(t s) u(s; x)ds +a_jutj m 2 ut = bjuj p 2 u; x_{2 ; t > 0} u (0; x) = u0(x) ; x2 ut(0; x) = u1(x) ; x 2 u (t; x) = 0; x_{2 ; t > 0} ; (P)

where _{is a bounded domain in R}N _{(N 1), with smooth boundary ; a; b; w are positive}

constants, m 2; p 2;and the function g satisfying some appropriate conditions.

Our results contain and generalize some existing results in literature. To prove our results many theorems were introduced.

Keywords: Nonlinear damping, strong damping, viscoelasticity, nonlinear source, local solutions,

Résumé

Notre travail, dans ce memoire consiste à étudier l’éxistence et le comportement asymptotique des solutions d’un problème de viscoelasticité non lineaire de type hyperbolique suivant:

8 > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > : utt u ! ut+ t R 0 g(t s) u(s; x)ds +a_jutjm 2ut= bjujp 2u; x2 ; t > 0 u (0; x) = u0(x) ; x 2 ut(0; x) = u1(x) ; x₂ u (t; x) = 0; x_{2 ; t > 0} ; (P)

où, _{est un domaine borné de R}N _{(N 1), avec frontière assez regulière ; a; b; w sont des}

constantes positives, m 2; p 2;et la fonction g satisfaite quelques conditions.

Nos résultats contiennent et généralisent certains résultats d’existences dans la littérature. Pour la preuve, beaucoup théorèmes ont été présentés

Mots clés: Dissipation nonlinéaire, viscoelasticité, source nonlinéaire, solutions locale, solutions

Notations

: _{bounded domain in R}N_: :topological boundary of : x = (x1; x2; :::; xN) : generic point of RN: dx = dx1dx2:::dxN :Lebesgue measuring on : ru : gradient of u: u : Laplacien of u: f+_{; f : max(f; 0); max( f; 0):}

a:e : almost everywhere.

p0 _{:conjugate of p; i:e} 1

p +

1

p0 = 1:

D( ) : space of di¤erentiable functions with compact support in :

D0_{( ) : distribution space.}

Ck_{( ) : space of functions k times continuously di¤erentiable in :}

C0( ) : space of continuous functions null board in :

Lp_{( ) : Space of functions p th power integrated on with measure of dx:}

kfkp = R _jf(x)jp 1 p : W1;p_{( ) =}n_u 2 Lp_{( ) ;} ru 2 (Lp_{( ))}No :

kuk1;p = _kukp_p+_krukp_p

1 p :

W₀1;p( ) : the closure of D ( ) in W1;p_{( ) :}

W 1;p0( ) : the dual space of W₀1;p( ) :

H :Hilbert space. H1 0 = W 1;2 0 : If X is a Banach space Lp_{(0; T ; X) = f : (0; T )} ! X is measurable ; T R 0 kf(t)kpXdt <1 : L1_{(0; T ; X) =} (

f : (0; T )_{! X is measurable ;ess} sup

t2(0;T ) kf(t)k p X <1 ) : Ck_{([0; T ] ; X) :}

Space of functions k times continuously di¤erentiable for [0; T ] ! X:

D ([0; T ] ; X) :space of functions continuously di¤erentiable with compact support in [0; T ] :

Introduction

In this thesis we consider the following nonlinear viscoelastic hyperbolic problem 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : utt u ! ut+ t R 0 g(t s) u(s; x)ds +a_jutj m 2 ut= bjuj p 2 u; x_{2 ; t > 0} u (0; x) = u0(x) ; ut(0; x) = u1(x) ; x2 u (t; x) = 0; x_{2 ; t > 0} ; (1)

where _{is a bounded domain in R}N _{(N 1), with smooth boundary ; a; b; w are positive constants,}

and m 2; p 2:The function g(t) is assumed to be a positive nonincreasing function de…ned on

R+ _{and satis…es the following conditions:}

(G1): g : R+ ! R+ is a bounded C1-function such that

g(0) > 0; 1 1 Z 0 g(s)ds = l > 0: (G2): g(t) 0; g0(t) 0; g(t) g0(t); _{8 t} 0; > 0:

In the physical point of view, this type of problems arise usually in viscoelasticity. This type of problems have been considered …rst by Dafermos [12], in 1970, where the general decay was discussed. A related problems to (1) have attracted a great deal of attention in the last two decades, and many results have been appeared on the existence and long time behavior of solutions.

See in this directions [2; 3; 5 8; 18; 29; 33; 34; 38] and references therein.

In the absence of the strong damping ut; that is for w = 0; and when the function g vanishes

identically ( i.e: g = 0); then problem (1) reduced to the following initial boundary damped wave equation with nonlinear damping and nonlinear sources terms.

utt u + ajutj

m 2

ut = bjuj

p 2

u: (2)

Some special cases of equation (2) arise in quantum …eld theory which describe the motion of charged mesons in an electromagnetic …eld.

Equation (2) together with initial and boundary conditions of Dirichlet type, has been extensively studied and results concerning existence, blow up and asymptotic behavior of smooth, as well

as weak solutions have been established by several authors over the past three decades. Some interesting results have been summarized by Said-Houari in his master thesis [42]:

For b = 0, that is in the absence of the source term, it is well known that the damping term a_jutj

m 2

ut assures global existence and decay of the solution energy for arbitrary initial data ( see

for instance [17] and [21]):

For a = 0; the source term causes …nite-time blow-up of solutions with a large initial data ( negative initial energy). That is to say, the norm of our solution u(t; x) in the energy space reaches +1 when the time t approaches certain value T called " the blow up time", ( see [1] and [20] for more details).

The interaction between the damping term a jutjm 2ut and the source term b jujp 2u makes the

problem more interesting. This situation was …rst considered by Levine [23; 24] in the linear damping case (m = 2), where he showed that solutions with negative initial energy blow up in …nite time T . The main ingredient used in [23] and [24] is the " concavity method" where the basic idea of this method is to construct a positive function L(t) of the solution and show that for some

> 0; the function L (t) is a positive concave function of t: In order to …nd such ; it su¢ ces to

verify that:

d2L (t)

dt2 = L

2_{(t) LL}00 _{(1 + )L}02_{(t) 0; 8t 0:}

This is equivalent to prove that L(t) satis…es the di¤erential inequality

LL00 (1 + )L02(t) 0; _8t 0:

Unfortunately, this method fails in the case of nonlinear damping term (m > 2):

Georgiev and Todorova in their famous paper [14], extended Levine’s result to the nonlinear damp-ing case (m > 2): More precisely, in [14] and by combindamp-ing the Galerkin approximation with the

contraction mapping theorem, the authors showed that problem (2) in a bounded domain with

initial and boundary conditions of Dirichlet type has a unique solution in the interval [0; T ) provided that T is small enough. Also, they proved that the obtained solutions continue to exist globally in

time if m p and the initial data are small enough. Whereas for p > m the unique solution of

problem (2) blows up in …nite time provided that the initial data are large enough. ( i.e: the initial energy is su¢ ciently negative).

This later result has been pushed by Messaoudi in [35] to the situation where the initial energy

E(0) < 0: For more general result in this direction, we refer the interested reader to the works of

In the presence of the viscoelastic term (g 6= 0) and for w = 0; our problem (1) becomes 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : utt u + t R 0 g(t s) u(s; x)ds +a_jutjm 2ut= bjujp 2u; x2 ; t > 0 u (0; x) = u0(x) ; ut(0; x) = u1(x) ; x2 u (t; x) = 0; x_{2 ; t > 0} : (3)

For a = 0; problem (3) has been investigated by Berrimi and Messaoudi [3]: They established the local existence result by using the Galerkin method together with the contraction mapping theorem. Also, they showed that for a suitable initial data, then the local solution is global in time and in addition, they showed that the dissipation given by the viscoelastic integral term is strong enough to stabilize the oscillations of the solution with the same rate of decaying ( exponential or polynomial) of the kernel g: Also their result has been obtained under weaker conditions than those used by Cavalcanti et al [7]; in which a similar problem has been addressed.

Messaoudi in [29]; showed that under appropriate conditions between m; p and g a blow up and global existence result, of course his work generalizes the results by Georgiev and Todorova [14] and Messaoudi [29]:

One of the main direction of the research in this …eld seems to …nd the minimal dissipation such that the solutions of problems similar to (3) decay uniformly to zero, as time goes to in…nity. Consequently, several authors introduced di¤erent types of dissipative mechanisms to stabilize these

problems. For example, a localized frictional linear damping of the form a(x)utacting in sub-domain

w has been considered by Cavalcanti et al [7]: More precisely the authors in [6] looked into

the following problem

utt u +

t Z

0

g(t s) u(s; x)ds + a(x)ut+_{juj u = 0:} (4)

for > 0; g a positive function and a : _{! R}+ a function, which may be null on a part of the

domain :

By assuming a(x) a0 > 0 on the sub-domain w ; the authors showed a decay result of an

exponential rate, provided that the kernel g satis…es

and kgkL1_(0;1) is small enough.

This later result has been improved by Berrimi and Messaoudi [2], in which they showed that the viscoelastic dissipation alone is strong enough to stabilize the problem even with an exponential rate.

In many existing works on this …eld, the following conditions on the kernel

g0(t) gp(t); t 0; p 1; (6)

is crucial in the proof of the stability.

For a viscoelastic systems with oscillating kernels, we mention the work by Rivera et al [36]. In that work the authors proved that if the kernel satis…es g(0) > 0 and decays exponentially to zero, that is for p = 1 in (6), then the solution also decays exponentially to zero. On the other hand, if the kernel decays polynomially, i.e. (p > 1) in the inequality (6), then the solution also decays polynomially with the same rate of decay.

In the presence of the strong damping (w > 0) and in the absence of the viscoelastic term (g = 0), the problem (1) takes the following form

8 > > > > > > > < > > > > > > > : utt u ! ut+ ajutjm 2ut= bjujp 2u; x2 ; t > 0 u (0; x) = u0(x) ; ut(0; x) = u1(x) ; x 2 u (t; x) = 0; x_{2 ; t > 0} : (7)

Problem (7) represents the wave equation with a strong damping ut. When m = 2, this problem

has been studied by Gazzola and Squassina [13]: In their work, the authors proved some results on well posedness and asymptotic behavior of solutions. They showed the global existence and polynomial decay property of solutions provided that the initial data is in the potential well. The proof in [13] is based on a method used in [19]. Unfortunately their decay rate is not opti-mal, and their result has been improved by Gerbi and Said-Houari [16], by using an appropriate modi…cation of the energy method and some di¤erential and integral inequalities.

Introducing a strong damping term ut makes the problem from that considered in [42] and [14],

for this reason less results where known for the wave equation with strong damping and many problem remain unsolved. ( See [13] and the recent work by Gerbi and Said-Houari [15]):

In this thesis, we investigated problem (1), in which all the damping mechanism have been considered

form those studied in the literature, specially the blow up result / exponential growth of solutions (chapter4).

This thesis is organized as follows: Chapter1:

In this chapter we introduce some notation and prepare some material needed for our work. The

main results of this chapter such as: the Lp _{spaces, the Sobolev spaces, di¤erential and integral}

inequalities and other theorems of functional analysis, can found in the books [4] and [43]: Chapter2:

This chapter is devoted to the study of the local existence result, the main ingredient used in this chapter is the Galerkin approximations ( the compactness method) introduced in the book of Lions

[26]; together with the …x point method.

Indeed, we consider …rst for u 2 C ([0; T ] ; H1

0) given, the following problem

vtt v ! vt+

t Z 0

g(t s) v(s; x)ds + a_jvtjm 2vt= bjujp 2u; x2 ; t > 0 (8)

with the initial data

v (0; x) = u0(x) ; vt(0; x) = u1(x) ; x 2 (9)

and boundary conditions of the form

v (t; x) = 0; x_{2 ; t > 0;} (10)

and we will show that problem (8) (10) has a unique local solutions v by the Faedo-Galerkin

method, which consists in constructing approximations of the solution, then we obtain a priori estimates necessary to guarantee the convergence of these approximations. We recall here that the

presence of nonlinearity on the damping term a jvtj

m 2

vt forces us to go until the second a priori

estimate. We point out that the contraction semigroup method fails here, because of the presence of the nonlinear terms.

Once the local solution v exists, we will use the contraction mapping theorem to show the local existence of our problem (1). This will be done under the assumption that T is required to be small enough (see formula (2:77)):

Chapter 3:

Our main purpose in this chapter is tow-fold:

First, we introduce a set W de…ned in (3:5) called " the potential well " or " stable set" and we show that if we restrict our initial data in this set, then our solution obtained in chapter 2 is global in time, that is to say, the norm

in the energy space L2_{( ) H}1

0( ) of our solution is bounded by a constant independent of the

time t:

Second, We show that, if our solution is global in time, ( i.e: by assuming that the initial data

u0 2 W ) and if our function g satis…es the condition (6) ( for p = 1); then our solution decays time

asymptotically to zero. More precisely we prove that the decay rate is of the form (1 + t)2=(2 m)

if m > 2; whereas for m = 2; we obtain an exponential decay rate. (See Theorem 3.2.1). The main tool used in our proof is an inequality due to Nakao [37]; in which this inequality has been introduced in order to study the stability of the wave equation, but it is still works in our problem. Chapter 4:

In this chapter we will prove that if the initial energy E(0) of our solution is negative ( this means that our initial data are large enough), then our local solutions in bounded and

kutk2+kruk2 ! 1

as t tends to +1: In fact it will be proved that the Lp _{norm of the solution grows as an exponential}

function. An essential tool of the proof is an idea used by Gerbi and Said-Houari [15], which based on an auxiliary function ( which is a small perturbation of the total energy), in order to obtain a di¤erential inequality leads to the exponential growth result provided that the following conditions

1 Z 0 g(s)ds < p 2 p 1; holds.

Preliminary

Abstract

In this chapter we shall introduce and state some necessary materials needed in the proof of our results, and shortly the basic results which concerning the Banach spaces, the weak and weak star

topologies, the Lp space, Sobolev spaces and other theorems. The knowledge of all this notations

1.1

Banach Spaces - De…nition and Properties

We …rst review some basic facts from calculus in the most important class of linear spaces "Banach spaces".

De…nition 1.1.1 A Banach space is a complete normed linear space X: Its dual space X0 is the

linear space of all continuous linear functional f : X ! R:

Proposition 1.1.1 ([43])

X0 _{equipped with the norm k:kX}

0 de…ned by

kfkX0 = supfjf(u)j : kuk 1g ; (1.1)

is also a Banach space.

We shall denote the value of f 2 X0 _{at u 2 X by either f(u) or hf; ui}

X0; X:

Remark 1.1.1 ([43]) From X0 _{we construct the bidual or second dual X}00 _{= (X}0₎0_{: Furthermore,}

with each u 2 X we can de…ne '(u) 2 X00 _{by '(u)(f ) = f (u); f 2 X}0_{, this satis…es clearly}

k'(x)k kuk : Moreover, for each u 2 X there is an f 2 X0 with f (u) = kuk and kfk = 1, so it

follows that k'(x)k = kuk :

De…nition 1.1.2 Since ' is linear we see that

' : X _{! X}00;

is a linear isometry of X onto a closed subspace of X00_{, we denote this by}

X ,_{! X}00:

De…nition 1.1.3 If ' ( in the above de…nition) is onto X00 _{we say X is re‡exive, X u X}00:

Theorem 1.1.1 ([4]; Theorem III.16)

Let X be Banach space. Then, X is re‡exive, if and only if,

BX =fx 2 X : kxk 1g ;

is compact with the weak topology (X; X0_{) : (See the next subsection for the de…nition of (X; X}0₎₎

De…nition 1.1.4 Let X be a Banach space, and let (un)_n2N be a sequence in X. Then un converges

strongly to u in X if and only if

lim

n!1kun ukX = 0;

and this is denoted by un_{! u; or lim}

1.1.1

The weak and weak star topologies

Let X be a Banach space and f 2 X0_{: Denote by}

'_f : X _{! R}

x_{7! 'f}(x); (1.2)

when f cover X0_{, we obtain a family '}

f f2X0 of applications to X in R:

De…nition 1.1.5 The weak topology on X; denoted by (X; X0_{) ; is the weakest topology on X for}

which every '_{f f2X0} is continuous.

We will de…ne the third topology on X0_{, the weak star topology, denoted by (X}0_{; X) : For all}

x_{2 X: Denote by}

'_x : X0 _{! R}

f _{7! 'x}(f ) =_{hf; xi}X0; X;

(1.3)

when x cover X, we obtain a family ('_x)_x2X0 of applications to X0 in R:

De…nition 1.1.6 The weak star topology on X0 _{is the weakest topology on X}0 _{for which every}

('_x)_x2X0 is continuous.

Remark 1.1.2 ([4]) Since X X00_{; it is clear that, the weak star topology (X}0_{; X) is weakest}

then the topology (X0_{; X}00_{), and this later is weakest then the strong topology.}

De…nition 1.1.7 A sequence (un) in X is weakly convergent to x if and only if

lim

n!1f (un) = f (u);

for every f 2 X0; and this is denoted by un * u.

Remark 1.1.3 ([42]; Remark 1.1.1)

1. If the weak limit exist, it is unique.

2. If un _{! u 2 X (strongly), then u}n * u (weakly):

3. If dim X < +1; then the weak convergent implies the strong convergent.

Proposition 1.1.2 ([43])

On the compactness in the three topologies in the Banach space X : 1- First, the unit ball

in X is compact if and only if dim(X) < 1:

2- Second, the unit ball B0 _{in X}0 _{(The closed subspace of a product of compact spaces) is weakly}

compact in X0 _{if and only if X is re‡exive.}

3- Third, B0 is always weakly star compact in the weak star topology of X0.

Proposition 1.1.3 ([4]; proposition III.12)

Let (fn) be a sequence in X0_{. We have:}

1. hfn * f in (X0; X)

i

, [fn(x)! f(x); 8x 2 X] :

2. If fn _{! f (strongly), then f}n * f, in (X0; X00),

If fn * f in (X0; X00), then fn* f, in (X0; X) :

3. If fn * f, in (X0; X), then _kfnk is bounded and kfk lim infkfnk :

4. If fn * f, in (X0_{; X) and x}

n ! x (strongly) in X, then fn(xn)! f(x):

1.1.2

Hilbert spaces

The proper setting for the rigorous theory of partial di¤erential equation turns out to be the most important function space in modern physics and modern analysis, known as Hilbert spaces. Then, we must give some important results on these spaces here.

De…nition 1.1.8 _{A Hilbert space H is a vectorial space supplied with inner product hu; vi such that}

kuk =phu; ui is the norm which let H complete.

Theorem 1.1.2 ([42], Theorem 1.1.1)

Let (un)_n2N is a bounded sequence in the Hilbert space H, then it possess a subsequence which

converges in the weak topology of H:

Theorem 1.1.3 ([42], Theorem 1.1.2)

In the Hilbert space, all sequence which converges in the weak topology is bounded.

Theorem 1.1.4 ([42], Corollary 1.1.1)

Let (un)_n2N be a sequence which converges to u; in the weak topology and (vn)_n2N is an other sequence

which converge weakly to v; then

lim

n!1hvn; uni = hv; ui: (1.5)

Theorem 1.1.5 ([42], Theorem 1.1.3)

Let X be a normed space, then the unit ball

B0 =nl _{2 X}0 :_klk 1o; (1.6)

Proposition 1.1.4 ([42]; Proposition 1.1.1)

Let X and Y be two Hilbert spaces, let (un)_{n2N 2 X be a sequence which converges weakly to u 2 X;}

let A 2 L(X; Y ): Then, the sequence (A (un))_n2N converges to A(u) in the weak topology of Y:

Proof. _{For all u 2 X, the function}

u_{7! hA(u); vi}

is linear and continuous, because

jhA(u); vij kAk_{L(X; Y )}kukXkvkY ; _{8u 2 X; v 2 Y:}

So, according to Riesz theorem, there exists w 2 X such that hA(u); vi = hu; wi, 8u 2 X: Then,

lim

n!1hA (un) ; vi = limn!1hun; wi

= _{hu; wi = hA(u); vi:} (1.7)

1.2

Functional Spaces

1.2.1

The

L

p

( ) spaces

De…nition 1.2.1 Let 1 p _{1; and let be an open domain in R}n; n_{2 N: De…ne the standard}

Lebesgue space Lp( ); by Lp( ) = 8 < :f : ! R : f is measurable and Z jf(x)jpdx <₁ 9 = ;: (1.8)

Notation 1.2.1 _{For p 2 R and 1} p < _{1, denote by}

kfkp = 0 @Z _jf(x)jp dx 1 A 1 p : (1.9) If p = 1; we have

L1_{( ) =ff : ! R : f is measurable and there exists a constant C}

such that, jf(x)j C a.e in _g:

(1.10)

Also, we denote by

kfk1=Inf fC; jf(x)j C a.e in g : (1.11)

Notation 1.2.2 Let 1 p _{1; we denote by q the conjugate of p i.e.} 1

p +

1

q = 1:

Theorem 1.2.1 ([48])

It is well known that Lp_{( ) supplied with the norm}

k:kp is a Banach space, for all 1 p _1:

Remark 1.2.1 In particularly, when p = 2, L2( ) equipped with the inner product

hf; giL2_{( )} =

Z

f (x)g(x)dx; (1.12)

is a Hilbert space.

Theorem 1.2.2 ([43], Corollary 3.2)

Some integral inequalities

We will give here some important integral inequalities. These inequalities play an important role in applied mathematics and also, it is very useful in our next chapters.

Theorem 1.2.3 ([48], Hölder’s inequality )

Let 1 p _{1. Assume that f 2 L}p_{( ) and g}

2 Lq_{( ); then, f g}

2 L1_{( ) and}

Z

jfgj dx kfkpkgkq: (1.13)

Corollary 1.2.1 (Hölder’s inequality - general form)

Lemma 1.2.1 Let f1; f2; :::fk be k functions such that, fi 2 Lpi( ); 1 i k, and

1 p = 1 p1 + 1 p2 + ::: + 1 pk 1:

Then, the product f1f2:::fk 2 Lp( ) and kf1f2:::fkkp kf1kp₁:::kfkkp_k:

Lemma 1.2.2 ( [48], Young’s inequality)

Let f 2 Lp (R) and g 2 Lq (R) with 1 < p < 1; 1 < q < 1 and 1 r = 1 p + 1 q 1 0. Then f g _{2 L}r (R) and kf gkLr_(R) kfkLp_(R)kgkLq_(R): (1.14)

Lemma 1.2.3 ([43]; Minkowski inequality)

For 1 p _{1; we have} ku + vkLp kukLp +kvkLp: (1.15) Lemma 1.2.4 ([43]) Let 1 p r q; 1 r = p + 1 q ; and 1 1: Then

kukLr kukLpkuk

1 Lq : (1.16) Lemma 1.2.5 ([43]) If ( ) <_{1; 1} p q _{1; then L}q _, ! Lp_{; and} kukLp ( ) 1 p 1 q _kukL q:

1.2.2

The Sobolev space W

m;p

( )

Proposition 1.2.1 ([26])

Let _{be an open domain in R}N_;

Then the distribution T 2 D0_{( ) is in L}p_{( ) if there exists a}

function f 2 Lp_{( ) such that}

hT; 'i = Z

f (x)'(x)dx; for all '_{2 D( );}

where 1 p _{1; and it’s well-known that f is unique.}

De…nition 1.2.2 _{Let m 2 N and p 2 [0; 1] : The W}m;p_{( ) is the space of all f}

2 Lp_{( ), de…ned} as Wm;p_{( ) =} ff 2 Lp_{( ); such that @ f} 2 Lp_{( ) for all} 2 Nm _{such that} j j =Pnj=1 j m; where, @ = @ 1 1 @22::@nng: (1.17) Theorem 1.2.4 ([9])

Wm;p_{( ) is a Banach space with their usual norm}

kfkWm;p_{( )} =

X j j m

k@ fkLp; 1 p < 1, for all f 2 Wm;p( ): (1.18)

De…nition 1.2.3 Denote by W₀m;p( ) the closure of D( ) in Wm;p_{( ):}

De…nition 1.2.4 When p = 2, we prefer to denote by Wm;2( ) = Hm( ) and W₀m;2( ) = H₀m( )

supplied with the norm

kfkHm_{( )} = 0 @X j j m (_{k@ fkL}2) 2 1 A 1 2 ; (1.19)

which do at Hm_{( ) a real Hilbert space with their usual scalar product}

hu; viHm_{( )} = X j j m Z @ u@ vdx (1.20) Theorem 1.2.5 ([42]; Proposition 1.2.1)

1) Hm_{( ) supplied with inner product}

h:; :iHm_{( )} is a Hilbert space.

2) If m m0_{, H}m_{( ) ,}

! Hm0

( ), with continuous imbedding :

Lemma 1.2.6 ([26])

Since D( ) is dense in H0m( ) ; we identify a dual H m( ) of H0m( ) in a weak subspace on ;

and we have

D( ) ,! H0m( ) ,! L

2_{( ) ,}

Lemma 1.2.7 (Sobolev-Poincaré’s inequality) If 2 q 2n n 2; n 3 q 2; n = 1; 2; then kukq C(q; )_kruk2; (1.21) for all u 2 H1 0( ) :

The next results are fundamental in the study of partial di¤erential equations

Theorem 1.2.6 ([9] Theorem 1.3.1)

Assume that _{is an open domain in R}N _{(N 1), with smooth boundary . Then,}

(i) if 1 p n; we have W1;p _Lq_{( ); for every q}

2 [p; p ] ; where p = np

n p:

(ii) if p = n we have W1;p _Lq_{( ); for every q}

2 [p; 1) :

(iii) if p > n we have W1;p _L1_{( )\ C}0; _{( ); where =} p n

p :

Theorem 1.2.7 ([9] Theorem 1.3.2)

If is a bounded, the embedding (ii) and (iii) of theorem 1.1.4 are compacts. The embedding (i) is

compact for all q 2 [p; p ) :

Remark 1.2.2 ([26])

For all ' 2 H2_{( ); '}

2 L2_{( ) and for su¢ ciently smooth, we have}

k'(t)kH2_{( )} Ck '(t)kL2_{( )}: (1.22)

Proposition 1.2.2 ([43], Green’s formula)

For all u 2 H2_{( ); v} 2 H1_{( ) we have} Z uvdx = Z rurvdx Z @ @u @ vd ; (1.23) where @u @ is a normal derivation of u at .

1.2.3

The

L

p

(0; T; X) spaces

De…nition 1.2.5 Let X be a Banach space, denote by Lp_{(0; T; X) the space of measurable functions}

f : ]0; T [_{! X} t _{7 ! f(t)} such that 0 @ T Z 0 kf(t)kpXdt 1 A 1 p =_kfkLp_(0;T;X)<1; for 1 p < 1: (1.24) If p = 1, kfkL1(0;T;X) = sup t2]0;T [ ess_kf(t)kX: (1.25) Theorem 1.2.8 ([42])

The space Lp_{(0; T; X) is complete.}

We denote by D0_{(0; T; X) the space of distributions in ]0; T [ which take its values in X, and let us}

de…ne

D0(0; T; X) =_{L (D ]0; T [ ; X) ;}

where L ( ; ') is the space of the linear continuous applications of to ': Since u 2 D0_{(0; T; X) ;}

we de…ne the distribution derivation as @u

@t(') = u

d'

dt ; 8' 2 D (]0; T [) ; (1.26)

and since u 2 Lp_{(0; T; X) ; we have}

u(') = T Z 0

u(t)'(t)dt; _{8' 2 D (]0; T [) :} (1.27)

We will introduce some basic results on the Lp_{(0; T; X)space. These results, will be very useful in}

the other chapters of this thesis.

Lemma 1.2.8 ([26] Lemma 1.2 )

Let f 2 Lp_{(0; T; X) and} @f

@t 2 L

p_{(0; T; X); (1 p}

1) ; then, the function f is continuous from

Lemma 1.2.9 ([26])

Let ' = ]0; T [ _{an open bounded domain in R R}n_{;and let g ; g are two functions in L}q_{(]0; T [ ; L}q_{( )) ;}

1 < q < _{1 such that} kg kLq_(0;T;Lq_{( ))} C; 8 2 N (1.28) and g _{! g in ';} then g * g in Lq(') : Theorem 1.2.9 ([9]; Proposition 1.4.17)

Lp_{(0; T; X) equipped with the norm}

k:kLp_(0;T;X), 1 p 1 is a Banach space.

Proposition 1.2.3 ([14])

Let X be a re‡exive Banach space, X0 it’s dual, and 1 p <_{1; 1} q <_1; 1

p +

1

q = 1: Then the

dual of Lp_{(0; T; X) is identify algebraically and topologically with L}q_{(0; T; X}0_):

Proposition 1.2.4 ([9])

Let X; Y be to Banach space, X Y with continuous embedding, then we have Lp_{(0; T; X) L}p_(0;

T; Y ) with continuous embedding.

The following compactness criterion will be useful for nonlinear evolution problems, especially in the limit of the non linear terms.

Proposition 1.2.5 ([26]).

Let B0; B; B1 be Banach spaces with B0 B B1; assume that the embedding B0 ,! B is compact

and B ,! B1 are continuous. Let 1 < p < 1; 1 < q < 1; assume further that B0 and B1 are

re‡exive. De…ne

W _{fu 2 L}p(0; T; B0) : u0 _{2 L}q(0; T; B1)_{g :} (1.29)

1.2.4

Some Algebraic inequalities

Since our study based on some known algebraic inequalities, we want to recall few of them here.

Lemma 1.2.10 ([48];The Cauchy-Schwarz inequality)

Every inner product satis…es the Cauchy-Schwarz inequality

hx1; x2i kx1k kx2k : (1.30)

The equality sign holds if and only if x1 and x2 are dependent.

Young’s inequalities :

Lemma 1.2.11 _{For all a; b 2 R}+, we have

ab a2+ b

2

4 ; (1.31)

where is any positive constant.

Proof. Taking the well-known result

(2 a b)2 0 _{8a; b 2 R}

for all > 0, we have

4 2a2+ b2 4 ab 0: This implies 4 ab 4 2a2+ b2 consequently, ab a2+ 1 4 b 2_:

This completes the proof.

Lemma 1.2.12 ([43])

For all a; b 0; the following inequality holds

ab a p p + bq q; where, 1 p + 1 q = 1:

Proof. _{Let G = (0; 1), and f : G ! R is integrable, such that} f (x) = 8 > < > : p log a, 0 x 1 p q log b, 1 p x 1 ;

for all a; b 0and 1

p +

1

q = 1:

Since '(t) = et is convex, and using Jensen’s inequality

' 0 @ 1 (G) Z G f (x)dx 1 A 1 (G) Z G ' (f (x)) dx: (1.32) Consequently, we have 1 (G) Z G ' (f (x)) dx = 1 Z 0 ef (x)dx = 1=p Z 0 ep log adx + 1 Z 1=p eq log bdx = 1=p Z 0 apdx + 1 Z 1=p bqdx = 1 p(a p_{) + 1} 1 p b q_: (1.33) = a p p + bq q ; where, (G) = 1 and ' 0 @ 1 (G) Z G f (x)dx 1 A = e(R01f (x)dx) = e( R1=p 0 p log adx+ R1 1=pq log bdx)

= e(log a+log b) = elog ab

= ab: (1.34)

1.3

Existence Methods

1.3.1

The Contraction Mapping Theorem

Here we prove a very useful …xed point theorem called the contraction mapping theorem. We will apply this theorem to prove the existence and uniqueness of solutions of our nonlinear problem.

De…nition 1.3.1 _{Let f : X ! X be a map of a metric space to itself. A point x 2 X is called a}

…xed point of f if f (x) = x.

De…nition 1.3.2 Let (X; dX) and (Y; dY) be metric spaces. A map ' : X ! Y is called a

contrac-tion if there exists a positive number C < 1 such that

dY('(x); '(y)) CdX(x; y); (1.35)

for all x; y 2 X.

Theorem 1.3.1 (Contraction mapping theorem [45] )

Let (X; d) be a complete metric space. If ' : X ! X is a contraction, then ' has a unique …xed point.

1.3.2

Gronwell’s lemma

Theorem 1.3.2 ( In integral form)

Let T > 0; and let ' be a function such that, ' 2 L1(0; T ); ' 0; almost everywhere and be

a function such that, _{2 L}1_{(0; T ); 0; almost everywhere and '}

2 L1_{[0; T ] ; C} 1; C2 0. Suppose that (t) C1+ C2 t Z 0 '(s) (s)ds; for a.e t_{2 ]0; T [ ;} (1.36) then, (t) C1exp 0 @C2 t Z 0 '(s)ds 1 A ; for a.e t 2 ]0; T[ : (1.37) Proof. Let F (t) = C1+ C2 t Z 0 '(s) (s)ds; _{for t 2 [0; T ] ;} (1.38) we have, (t) F (t);

From (1:38) we have F0(t) = C2'(t) (t) C2'(t) (t); for a.e t 2 ]0; T [ : (1.39) Consequently, d dt 8 < :F (t) exp 0 @ t Z 0 C2'(s)ds 1 A 9 = ; 0; (1.40) then, F (t) C1exp 0 @C2 t Z 0 '(s)ds 1 A ; for a.e t 2 ]0; T[ : (1.41)

Since F; then our result holds.

In particle, if C1 = 0; we have = 0 _{for almost everywhere t 2 ]0; T [ :}

1.3.3

The mean value theorem

Theorem 1.3.3 _{Let G : [a; b] ! R be a continues function and ' : [a; b] ! R is an integral positive}

function, then there exists a number x in (a; b) such that b Z a G(t)'(t)dt = G(x) b Z a '(t)dt: (1.42)

In particular for '(t) = 1; there exists x 2 (a; b) such that b Z a G(t)dt = G(x) (b a) : (1.43) Proof. Let m = inf_{fG(x); x 2 [a; b]g} (1.44) and M = sup_{fG(x); x 2 [a; b]g} (1.45)

of course m and M exist since [a; b] is compact. Then, it follows that

m b Z a '(t)dt b Z a G(t)'(t)dt M b Z a '(t)dt: (1.46)

By monotonicity of the integral. dividing through byR_ab'(t)dt; we have that m Rb a G(t)'(t)dt Rb a '(t)dt M: (1.47)

Since G(t) is continues, the intermediate value theorem implies that there exists x 2 [a; b] such that G(x) = Rb a G(t)'(t)dt Rb a '(t)dt : (1.48)

Local Existence

Abstract

Our goal in this chapter is to study the local existence ( local well-possedness) of the problem (P );

for u in C ([0; T ] ; H1

0( )). For this purpose we consider, …rst the related problem for u …xed in

C ([0; T ] ; H₀1( )) ₈ > > > > > > > > > > > > < > > > > > > > > > > > > : vtt vt v + Rt 0 g(t s) v(s; x)ds +_jvtj m 2 vt=juj p 2 u; x_{2 ; t > 0} v (0; x) = u0(x) ; vt(0; x) = u1(x) ; x2 v (t; x) = 0; x_{2 ; t > 0} ; (2.1)

and we will prove the local existence of this problem by using the Faedo-Galerkin method. Then, by using the well-known contraction mapping theorem, we can show the local existence of (P ): Our techniques of proof follows carefully the techniques due to Georgiev and Todorova [14]; with necessary modi…cations imposed by the nature of our problem. The …rst step of our proof is the choice of the space where the local solution exists. The minimal requirement for this space is that

u(t; x)be time continuous. The weak space satisfying the above requirement is C ([0; T ] ; H), where

H = H1

2.1

Local Existence Result

In order to prove our local existence results, let us introduce the following space

YT = 8 > > < > > : u : u_{2 C ([0; T ] ; H}1 0( )) ; ut _{2 C([0; T ] ; H0}1( ))_{\ L}m([0; T ] ) 9 > > = > > ; : (2.2)

Our main result in this chapter reads as follows:

Theorem 2.1.1 Let (u0; u1)2 (H01( ))

2

be given. Suppose that m 2; p 2 be such that

max_{fm; pg} 2 (n 1)

n 2 ; n 3: (2.3)

Then, under the conditions (G1) and (G2), the problem (P ) has a unique local solution u(t; x) 2 YT,

for T small enough.

The proof of theorem 2.1.1 will be established through several lemmas. The presence of the term

jujP 2u in the right hand side of our problem (P ) ; gives us negative values of the energy. For this

purpose we …xed u 2 C ([0; T ] ; H1

0( )) in the right hand side of (P ) and we will prove that our

problem (2:1) ; admits a solution.

Lemma 2.1.1 ([14]; Theorem 2.1) Let (u0; u1)2 (H01( ))

2

, assume that m 2; p 2 and (2:3)

holds. Then, under the conditions (G1) and (G2), there exists a unique weak solution v 2 YT to the

problem (2:1); for any u 2 C ([0; T ] ; H01( )) given.

The proof of the above Lemma follows the techniques due to Lions [26], in order to deal with the

convergence of the non linear terms in our problem, we must take …rst our initial data (u0; u1) in a

high regularity (that is, u0 _{2 H}2_{( )}

\ H1

0( ); and u1 2 H01( )\ L2(m 1)( )).

Lemma 2.1.2 _{([26]; Theorem 3.1) Let u 2 C ([0; T ] ; H0}1( )). Suppose that

u0 _{2 H}2( )_{\ H0}1( ); (2.4)

u1 2 H01( )\ L

2(m 1)_{( ); (2.5)}

assume further that m 2; p 2. Then, under the conditions (G1) and (G2) there exists a unique

solution v of the problem (2:1) such that

vt2 L1 [0; T ] ; H01( ) ; (2.7)

vtt _{2 L}1 [0; T ] ; L2( ) ; (2.8)

vt2 Lm([0; T ] ( )) : (2.9)

The following technical Lemma will play an important role in the sequel.

Lemma 2.1.3 _{For any v 2 C}1(0; T; H2( )) we have

Z _Zt 0 g(t s) v(s):v0(t)dsdx = 1 2 d dt (g rv) (t) 1 2 d dt 2 4 t Z 0 g(s) Z jrv(t)j2dxds 3 5 1 2(g 0 _{rv) (t) +}1 2g(t) Z jrv(t)j2dxds: where (g u)(t) = t Z 0 g(t s) Z ju(s) u(t)_j2dxds:

The proof of this result is given in [31], for the reader’s convenience we repeat the steps here.

Proof. It’s not hard to see

Z _Zt 0 g(t s) v(s):v0(t)dsdx = t Z 0 g(t s) Z rv0(t):_rv(s)dxds = t Z 0 g(t s) Z rv0(t): [_{rv(s) rv(t)] dxds} t Z 0 g(t s) Z rv0(t):_rv(t)dxds: Consequently, Z _Zt 0 g(t s) v(s):v0(t)dsdx = 1 2 t Z 0 g(t s)d dt Z jrv(s) rv(t)j2dxds t Z 0 g(s) 0 @d dt 1 2 Z jrv(t)j2dx 1 A ds

which implies, Z _Zt 0 g(t s) v(s):v0(t)dsdx = 1 2 d dt 2 4 t Z 0 g(t s) Z jrv(s) rv(t)j2dxds 3 5 1 2 d dt 2 4 t Z 0 g(s) Z jrv(t)j2dxds 3 5 1 2 t Z 0 g0(t s) Z jrv(s) rv(t)j2dxds +1 2g(t) Z jrv(t)j2dxds:

This completes the proof. Proof of Lemma 2.1.2. Existence:

Our main tool is the Faedo-Galerkin’s method, which consist to construct approximations of the so-lutions, then we obtain a prior estimates necessary to guarantee the convergence of approximations. Our proof is organized as follows. In the …rst step, we de…ne an approach problem in bounded

dimension space Vn which having unique solution vn and in the second step we derive the various

a priori estimates. In the third step we will pass to the limit of the approximations by using the compactness of some embedding in the Sobolev spaces.

1. Approach solution:

Let V = H1

0( )\ H2( ) the separable Hilbert space. Then there exists a family of subspaces fVng

such that

i) Vn V (dimVn<1); 8n 2 N:

ii) Vn ! V; such that, there exist a dense subspace # in V and for all v 2 #; we can get sequence

fvng_n2N 2 Vn;and vn! v in V:

iii) Vn Vn+1 and [n2N Vn= H01( )\ H2( ):

For every n 1; let Vn = Span_fw1; :::; wng ; where fwig, 1 i n; is the orthogonal complete

system of eigenfunctions of _{such that kw}jk2 = 1, wj 2 H2( )\ L2(m 1)( ) for all j = 1; :::; n.

Denote by f jg the related eigenvalues, where wj are solutions of the following initial boundary

value problem ₈ > > < > > : wj = jwj wj = 0 on j = 1; :::; in : (2.10)

According to (iii) ; we can choose vn0; vn12 [w1; :::; wn] such that vn0 n X j=1 jnwj ! u0 in H01( )\ H 2_{( ), (2.11)} vn1 n X j=1 jnwj ! u1 in H01( )\ L 2(m 1)_{( ); (2.12)} where jn= R u0wjdx jn= R u1wjdx: We seek n functions 'n 1; :::; 'nn2 C2[0; T ] ; such that vn(t) = n X j=1 'n_j(t)wj(x); (2.13)

solves the problem 8 > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > : R ((v00 n(t) v0n(t) vn(t)) dx +R t R 0 g(t s) vn(s)ds +jvn0(t)j m 2 v0 n(t) dx =R _ju(t)jp 2u(t) dx vn(0) = vn0; v0n(0) = vn1 ; (2.14)

where the prime "0_{" denotes the derivative with respect to t. For every 2 V}

n and t 0: Taking

= wj; in (2:14) yields the following Cauchy problem for a ordinary di¤erential equation with

unknown 'n j: 8 > > > > > > > > > < > > > > > > > > > : '00n j (t) + j'0nj (t) + j'nj(t) + j t R 0 g(t s)'n j(s)ds + '0h j (t) m 2 '0n j (t) = j(t) 'n_j(0) =R u0wj; '0n_j (0) =R u1wj; j = 1; :::; n ; (2.15)

for all j; where

j(t) =

Z

By using the Caratheodory theorem for an ordinary di¤erential equation, we deduce that, the

above Cauchy problem yields a unique global solution 'n

j 2 H3[0; T ] ; and by using the embedding

Hm_{[0; T ] ,}

! Cm 1_{[0; T ], we deduce that the solution '}n

j 2 C2[0; T ]. In turn, this gives a unique

vn de…ned by (2:13) and satisfying (2:14):

2. The a priori estimates:

The next estimate prove that the energy of the problem (2:1) is bounded and by using a result in

[9];we conclude that; the maximal time tn of existence of (2:15) can be extended to T .

The …rst a priori estimate:

Substituting = v0 n(t) into (2:14), we obtain Z v00_n(t)v0_n(t)dx Z v_n0(t)v_n0(t)dx Z vn(t)vn0(t)dx + Z _Zt 0 g(t s) vn(s)dsvn0(t)dx + Z jvn0(t)j m 2 v_n0(t)v_n0(t)dx (2.16) = Z f (u)v0_n(t)dx:

for every n 1; _{where f (u) = ju(t)j}p 2u(t);Since the following mapping

L2( ) _{! L}2( )

u _{7! juj}p 2u

is continues, we deduce that,

jujp 2u_{2 L}1 [0; T ] ; L2( ) :

So, f 2 H1_{([0; T ] ; H}1_{( )). Consequently, by using the Lemma 2.1.3 and (G1) we get easily}

1 2 d dtkv 0 n(t)k 2 2+krvn0(t)k 2 2+ 1 2 d dtkrvn(t)k 2 2 +1 2 d dt 2 4 t Z 0 g(t s) Z jrvn(s) rvn(t)j 2 dxds 3 5 1 2 d dt 0 @krvn(t)k 2 2 t Z 0 g(s):ds 1 A 1 2 t Z 0 g0(t s) Z jrvn(s) rvn(t)j2dxds + 1 2g(t)krvn(t)k 2 2 +kv0n(t)k m m = Z f (u):v_n0(t)dx:

Therefore, we obtain d dtEn(t) Z f (u):v0_n(t)dx +_krvn0(t)_k2₂+_kv0_n(t)_km_m = 1 2(g 0 rvn)(t) 1 2g(t)krvn(t)k 2 2; where En(t) = 1 2 8 < :kv0n(t)k 2 2+ 0 @1 t Z 0 g(s)ds 1 A krvn(t)k 2 2+ (g rvn)(t) 9 = ;;

is the functional energy associated to the problem (2:1): It’s clear by using (G2), that

d

dtEn(t)

Z

f (u):v_n0(t)dx +_krv0_n(t)_k2₂+_kvn0(t)_km_m 0; _8t 0:

Which implies, by using Young’s inequality, for all > 0

d dtEn(t) +krv 0 n(t)k 2 2+kv0n(t)k m m 1 4 kf(u)k 2 2+ kvn0(t)k 2 2: (2.17)

Integrating (2:17) over [0; t] ; (t < T ), we obtain

En(t) + t Z 0 krvn0(s)k 2 2ds + t Z 0 kvn0(s)k m mds 1 4 T Z 0 kf(u)k22ds + T Z 0 kv0n(s)k 2 2ds + 1 2 ku1nk 2 2+kru0nk 2 2 : Since f 2 H1_{(0; T; H}1 0( )); we deduce En(t) + t Z 0 krvn0(s)k 2 2ds + t Z 0 kvn0(s)k m mds CT ku1nk 2 2+kru0nk 2 2 ; (2.18)

for, small enough and every n 1, where CT > 0 is positive constant independent of n:

Then, by the de…nition of En(t) and by using (2:11) ; (2:12), we get

kvn0(t)k 2 2+ 0 @1 t Z 0 g(s)ds 1 A krvn(t)k 2 2+ (g rvn)(t) KT; (2.19) and t Z 0 kvn0(s)k m mds KT; (2.20)

and t Z 0 krv0n(s)k 2 2ds KT; (2.21)

where KT = CT ku1nk2₂+kru0nk2₂ ; by (2:19), we get tn= T; 8n:

However, the insu¢ cient regularity of the nonlinear operator, jvtjm 2vt; with the presence of the

viscoelastic term and strong damping, we must prove in the next a prior estimates that, the family of

approximations vnde…ned in (2:13) is compact in the strong topology and by using compactness of

the embedding H1_{([0; T ] ; H}1_{( )) ,}

! L2_{([0; T ] ; L}2_{( )); we can extract a subsequence of vn denoted}

also by v such that v0 converges strongly in L2_{([0; T ] ; L}2_{( )): To do this, it’s su¢ ces to prove}

that v0_n is bounded in L1_{([0; T ] ; H}1

0( )) and v

00

n is bounded in L1([0; T ] ; L2( )), then by using

Aubin-Lions Lemma, our conclusion holds. The second a priori estimate:

Substituting = wj in (2:14) and taking wj = jwj, multiplying by '

0_n

j (t)and summing up the

product result with respect to j; we get by Green’s formula. Z rvn00:rv0ndx + Z v_n0: v_n0dx + Z vn: vn0dx Z _Zt 0 g(t s) vn: vn0dsdx + n X i=1 Z @ @xi jv 0 njm 2v_n0 @v 0 n @xi dx (2.22) = Z rf(u):rvn0dx: As in [26], we have n X i=1 Z @ @xi jv 0 njm 2v_n0 @v 0 n @xi dx = (m 1) n X i=1 Z _@ @xi jv 0 nj m 2 2 @v 0 n @xi 2 dx (2.23) = 4(m 1) m2 n X i=1 Z _@ @xi jv 0 nj m 2 2 _v0 n 2 dx:

Also, the fourth term in the left hand side of (2:22) can be written as follows Z _Zt 0 g(t s) vn: v0ndsdx = 1 2g(t)k vnk 2 2+ 1 2(g 0 _v n) 1 2 d dt 8 < :(g vn) k vnk 2 2 t Z 0 g(s)ds 9 = ;: (2.24) Therefore, (2:22) becomes 1 2 d dt 2 4krv0_nk2 2+ (g vn) +k vnk 2 2 0 @1 t Z 0 g(s)ds 1 A 3 5 +4(m 1) m2 n X i=1 Z _@ @xi jv 0 nj m 2 2 _v0 n 2 dx (2.25) +_{k v}0_nk2₂+ 1 2g(t)k vnk 2 2 1 2(g 0 _v n) = Z rf(u):rvn0dx:

Let us de…ne the energy term

Kn(t) = 1 2 2 4krv0_nk2 2+ (g vn) +k vnk 2 2 0 @1 t Z 0 g(s)ds 1 A 3 5 (2.26)

Then, it’s clear that (2:25) takes the form d dtKn(t) Z rf(u):rvn0dx +k vnk0 2 2 + n X i=1 Z _@ @xi jv 0 nj m 2 2 _v0 n 2 dx (2.27) = 1 2(g(t)k vnk 2 2) + 1 2(g 0 _v n): Using (G1) ; (G2) and integrating (2:27) over [0; t] ; we obtain

Kn(t) + n X i=1 t Z 0 Z @ @xi jv 0 nj m 2 2 _v0 n 2 dxds + t Z 0 k v0nk22ds T Z 0 Z rf(u):rv0ndxds + 1 2(krv1nk 2 2+k v0nk 2 2): (2.28)

Obviously, by using Young’s inequality, we get T Z 0 Z rf(u):rvn0dxds 1 4 T Z 0 krf(u)k22ds + T Z 0 krv0nk22ds:

Inserting the above estimate into (2:28); to get

Kn(t) + n X i=1 t Z 0 Z @ @xi jv 0 nj m 2 2 _v0 n 2 dxds + t Z 0 k vnk0 22ds CT(krv1nk2₂+k v0nk2₂): Thus, Kn(t) + n X i=1 t Z 0 Z @ @xi jv 0 nj m 2 2 _v0 n 2 dxds + t Z 0 k v0nk22ds CT;

for, small enough and every n 1, where CT > 0is positive constant independent of n: Therefore,

this equivalent by the de…nition of Kn(t)

krv0nk22+ (g vn) +k vnk 2 2 0 @1 t Z 0 g(s)ds 1 A CT; (2.29) and n X i=1 t Z 0 Z _@ @xi jv 0 nj m 2 2 _v0 n 2 dxds CT; (2.30) and t Z 0 k v0nk22ds CT: (2.31)

Then, from (2:29), (2:30) and (2:31) ; we conclude

v_n0 is bounded in L1([0; T ] ; H₀1( )); (2.32) vn is bounded in L1([0; T ] ; H2( )); (2.33) @ @xi jv 0 nj m 2 2 _v0 n is bounded in L 2_{([0; T ] ; L}2_{( )); i = 1; :::; n: (2.34)}

The third a priori estimate: It’s clear that

d dt 2 4 t Z 0 g(t s) vn(s)ds 3 5 = g(0) vn+ t Z 0 g0(t s) vn(s)ds: (2.35)

Performing an integration by parts in (2:35) we …nd that d dt 2 4 t Z 0 g(t s) vn(s)ds 3 5 = g(t) un0+ t Z 0 g(t s) v0_n(s)ds: (2.36)

Now, returning to (2:14), di¤erentiating throughout with respect to t, and using (2:36), we obtain Z (v_n000(t) v_n00(t) v0_n(t)) dx + Z 0 @ t Z 0 g(t s) v0_n(s)ds + g(t) un0+ (m 1)(jvn0(t)j m 2 v_n00(t)) 1 A dx = Z (f (u))0) dx; (2.37) where, (f (u))0 = @f @t. By substitution of = v 00 h(t) in (2:37); yields Z v_n000(t)v_n00(t)dx Z v_n00(t)v_n00(t)dx Z v_n0(t)v_n00(t)dx + Z 0 @ t Z 0 g(t s) v_n0(s)ds + g(t) un0 1 A v00 n(t)dx +(m 1) Z jv0n(t)j m 2 v00_n(t)v00_n(t)dx (2.38) = 0 Z (f (u))0v_n00(t)dx:

The fourth term in the left hand side of (2:38) can be analyzed as follows. It’s clear that Lemma 2.1.3 implies Z _Zt 0 g(t s) v_n0(s):v00_n(t)dsdx = Z _Zt 0 g(t s)_rv0_n(s):_rvn00(t)dsdx = 1 2 d dt (g rv 0 n) (t) 1 2 d dt t Z 0 g(s)_krv0_n(t)_k2₂ds (2.39) 1 2(g 0 _rv0 n) (t) + 1 2g(t)krv 0 n(t)k 2 2:

Also, (m 1)_hjvn0(t)_jm 2v_n00(t); v00_n(t)_{i =} 4(m 1) m2 Z @ @t jv 0 nj m 2 2 _v0 n(t) 2 dx: (2.40)

Inserting (2:39) and (2:40) into (2:38), we obtain 1 2 d dt kv 00 nk22 +krv00nk 2 2+ 1 2 d dtkrv 0 nk22+ 1 2 d dt 8 < :(g rvn0) (t) krv0nk 2 2 t Z 0 g(s)ds 9 = ; +4(m 1) m2 Z _@ @t jv 0 nj m 2 2 _v0 n(t) 2 dx + (g(t) un0) Z v00_n(t)dx (2.41) = Z (f (u))0v00_n(t)dx + 1 2(g 0 rv0n) (t) 1 2g(t)krv 0 nk22: Let us denote by n(t) = 1 2 8 < :kv00nk 2 2+ + (g rv0n) (t) + (1 t Z 0 g(s)ds)_krv0_nk2₂ 9 = ;: (2.42)

We obtain, from (2:41) ; (G2) that d dt n(t) Z (f (u))0:v_n00(t)dx +4(m 1) m2 Z _@ @t jv 0 nj m 2 2 _v0 n(t) 2 dx +_krvnk00 2₂+ (g(0) un0) Z v_n00(t)dx = 1 2(g 0 _rv0 n) (t) 1 2g(t)krv 0 n(t)k 2 2 0:

Integration the above estimate over [0; t], we conclude

n(t) + 4(m 1) m2 t Z 0 Z @ @t jv 0 nj m 2 2 _v0 n(t) 2 dxds + t Z 0 krvnk00 22ds + (g(0) un0) t Z 0 Z v_n00(s)dxds 1 2kv 00 n0k22+ 1 2krvn1k 2 2+ 1 4 T Z 0 (f (u))0 2₂ds + T Z 0 kvnk00 22ds:

Then, for small enough, we deduce n(t) + 4(m 1) m2 t Z 0 Z @ @t jv 0 nj m 2 2 _v0 n 2 dxds + t Z 0 krvnk00 22ds + (g(0) un0) t Z 0 Z v00_ndxds (2.43) CT kv00n0k 2 2+krvn1k 2 2 :

In order to estimate the term kv_n0k00 2

2, taking t = 0 in (2:14), we …nd kv00n0k22 = Z vn1:vn000 dx + Z vn0:vn000 dx Z jvn1jm 2vn1:vn000 dx + Z f (u(0)):v_n000 dx:

Thanks to Cauchy-Schwartz inequality (Lemma 1.2.10), we write

kv00n0k22 kv00n0k2 0 B B @k vn1k2+k vn0k2+kf(u(0))k2+ 0 @Z _jv1nj2(m 1)dx 1 A 1 2 1 C C A ; which implies, by using (2:11) and (2:12) ; that

kv00n0k2 k vn1k2+k vn0k2+kf(u(0))k2+ 0 @Z _jv1nj 2(m 1) dx 1 A 1 2 C: (2.44) Then, n(t) LT; (2.45) 4(m 1) m2 t Z 0 Z _@ @t jv 0 nj m 2 2 _v0 n 2 dxds LT; (2.46) and t Z 0 krv00nk22ds LT; (2.47) and t Z 0 Z v_n00dxds LT: (2.48)

where LT > 0:

From (2:11); (2:12) and (2:44) (2:48) ; we deduce

v_n00 is bounded in L1([0; T ] ; L2( )); (2.49) vn is bounded in L1([0; T ] ; H01( )); (2.50) @ @t jv 0 nj m 2 2 _v0 n is bounded in L 2_{([0; T ] ; L}2_{( )); i = 1; :::; n: (2.51)}

3.Pass to the limit:

By the …rst, the second and the third estimates, we obtain

vn is bounded in L1([0; T ] ; H2( )\ H01( )); (2.52)

v_n0 is bounded in L1([0; T ] ; H₀1( )); (2.53)

v_n00 is bounded in L1([0; T ] ; L2( )); (2.54)

v0_n is bounded in Lm((0; T ) ): (2.55)

Therefore, up to a subsequence, and by using the (Theorem 1.1.5), we observe that there exists a

subsequence v of vn and a function v that we my pass to the limit in (2:14); we obtain a weak

solution v of (2:1) with the above regularity

v * v in L1([0; T ] ; H₀1( )_{\ H}2( ));

v0 * v0in L1([0; T ] ; H₀1( )_{\ L}m( ));

v00* v00in L1([0; T ] ; L2( )):

By using the fact that

L1([0; T ] ; L2( )) ,_{! L}2([0; T ] ; L2( )); L1([0; T ] ; H₀1( )) ,_{! L}2([0; T ] ; H₀1( )): We get v_n0 is bounded in L2([0; T ] ; H₀1( )); v_n00 is bounded in L2([0; T ] ; L2( )); therefore, v_n0 is bounded in H1([0; T ] ; H1( )): (2.56)

Consequently, since the embedding

is compact, then we can extract a subsequence v0 _{such that} v0 _{! v}0 in L2([0; T ] ; L2( )). (2.57) which implies v0 _{! v}0 a.e on (0; T ) . By (2:20), we have v_n0 is bounded in Lm([0; T ] ):

and by using Theorem 1.2.2

jv0jm 2v0 * # in Lm0([0; T ] ; Lm0( ))

where m

m 1 = m

0_:

The estimates (2:34) and (2:46) imply that jv0j

m 2

2 _v0 _{* in H}1_{([0; T ] ; H}1_{( )}

by using the fact that, the mapping u 7 ! jujm 2uis continuous, (Lemma 1.2.9) and since the weak

topology is separate, we deduce

# =_jv0_jm 2v0 (2.58)

=_jv0_j

m 2

2 _v0 _(2.59)

Then, by using the uniqueness of limit, we deduce

jv0jm 2v0 *_jv0_jm 2v0 in Lm0([0; T ] ; Lm0( )) (2.60)

jv0jm22 _v0 _*jv0_j

m 2

2 _v0 _{in H}1_{([0; T ] ; H}1_{( ) (2.61)}

Now, we will pass to the limit in (2:14), by the same techniques as in [26]:

Taking = wj, n = and …xed j < ;

Z v00(t):wjdx + Z rv (t):rwjdx + Z rv0(t):_rwjdx Z _Zt 0 g(t s)_{rv (s):rw}jdsdx + Z jv0(t)_jm 2v0(t):wjdx (2.62) = Z f (u):wjdx:

We obtain, by using the property of continuous of the operator in the distributions space Z v00(t):wjdx * Z v00(t):wjdx; in D0(0; T ) Z rv (t):rwjdx * Z rv(t):rwjdx; in L1(0; T ) Z rv0(t):_rwjdx * Z rv0(t):_rwjdx; in L1(0; T ) Z _Zt 0 g(t s)_{rv (s):rw}jdsdx * Z _Zt 0 g(t s)_rv(s):rwjdsdx; in L1(0; T ) Z jv0(t)_jm 2v0(t):wjdx * Z jv0(t)_jm 2v0(t):wjdx; in L1(0; T )

We deduce from (2:62), that Z v00(t):wjdx + Z rv(t):rwjdx + Z rv0(t):_rwjdx Z _Zt 0 g(t s)_rv(s):rwjdsdx + Z jv0(t)_jm 2v0(t):wjdx (2.63) = Z f (u):wjdx:

Since, the basis wj (j = 1; :::)is dense in H1

0( )\ H2( ) ; we can generalize (2:63) ; as follows

Z v00(t):'dx + Z rv(t):r'dx + Z rv0(t):_r'dx Z _Zt 0 g(t s)_{rv(s):r'dsdx +} Z jv0(t)_jm 2v0(t):'dx = Z f (u):'dx; _{8' 2 H0}1( )_{\ H}2( ) : Then, v _{2 L}1 [0; T ] ; H2( )_{\ H0}1( ) ; vt _{2 L}1 [0; T ] ; H₀1( ) ; vtt 2 L1 [0; T ] ; L2( ) ; vt 2 Lm([0; T ] ( )) :

Uniqueness:

Let v1; v2 two solutions of (2:1), and let w = v1 v2 satisfying :

w00 w w0+

t Z

0

g(t s) wds + _jv1j0 m 2v0_{1 jv2j}0 m 2v₂0 = 0: (2.64)

Multiplying (2:64), by w0 and integrating over , we get

1 2 d dt 0 @kw0 n(t)k 2 2 + 0 @1 t Z 0 g(s)ds 1 A krwn(t)_k2₂+ (g _rwn)(t) 1 A + Z jv1j0 m 2v₁0 _jv2j0 m 2v₂0 w0dx = _krw0_n(t)_k2₂+ 1 2(g 0 _rw n)(t) 1 2g(t)krwn(t)k 2 2: Denote by J (t) =_kw0_n(t)_k2₂+ 0 @1 t Z 0 g(s)ds 1 A krwn(t)k2₂+ (g rwn)(t): (2.65)

Since the function y 7! jyjm 2y is increasing, we have

Z jv01jm 2v0_{1 jv2j}0 m 2v0₂ w0dx 0 and since 1 2(g 0 _rw n)(t) 0; we deduce d dtJ (t) 0 : (2.66)

This implies that J (t) is uniformly bounded by J (0) and is decreasing in t; since w(0) = 0; we

obtain w = 0 and v1 = v2:

Proof of Lemma 2.1.1.

As in [14]; since D( ) = H2( ), we approximate, u0; u1 by sequences (u 0) ; (u 1) in D( ), and u

by a sequence (u ) in C ([0; T ] ; D( )), for the problem (2:1). Lemma 2.1.2 guarantees the existence

of a sequence of unique solutions (v ) satisfying (2:6) (2:9) :Now, to complete the proof of Lemma

2.1.1, we proceed to show that the sequence (v ) is Cauchy in YT equipped with the norm

kuk2YT =kuk

2

H +kutk

2

where kuk2H = max_{0 t T} 8 < : Z u2_t + l_jruj2 (x; t)dx 9 = ;

Denote w = v v for ; given. Then w is a solution of the Cauchy problem:

8 > > > > > > > > > > > > < > > > > > > > > > > > > : wtt w wt+ L(w) + k(vt) k(vt) = f (u ) f (u ); x_{2 ; t > 0} w(0; x) = u 0 u 0; wt(0; x) = u 1 u 1; x2 w(t; x) = 0; x_{2 ; t > 0} ; (2.67) where, k(v_t) = _jvtjm 2v_t f (u ) =_{ju j}p 2u L(w) = t Z 0 g(t s) w(s; x)ds:

The energy equality reads as 1 2 d dt 8 < :kwtk 2 2+ 0 @1 t Z 0 g(s)ds 1 A krwk2 2+ (g rw)(t) 9 = ; + Z k(v_t) k(v_t) wtdx +krwtk2₂ (2.68) = Z f (u ) f (u ) wtdx + 1 2(g 0 _{rw) (s)ds} 1 2krw(s)k 2 2 t Z 0 g(s)ds: The term, _Z k(v_t) k(v_t) wtdx = Z jvtjm 2v_t v_t m 2 v_t v_t v_t dx is nonnegative. We need to estimate Z

ful…lled for u; u; v; v 2 H1

0( ); where C is a constant depending on ; l; p only. Then, Hölder’s

inequality yields, for 1

q + 1 n + 1 2 = 1 (q = 2n n 2); Z f (u ) f (u ) wtdx = Z ju jp 2u u p 2u v_t v_t dx C u u _Lq vt vt L2 ku k p 2 n(p 2)+ u p 2 n(p 2) : (2.69)

The Sobolev embedding Lq _,

! H1 0( ) gives u u Lq_{( )} C ru ru _L2_{( )}: Then, ku kp 2n(p 2)+ u p 2 n(p 2) C(ku k p 2 L2_{( )}+ u p 2 L2_{( )}):

The necessity to estimate ku kn(p 2) by the energy norm kukH requires a restriction on p. Namely,

we need n(p 2) 2n

n 2; then the Sobolev embedding L

q _, ! H1 0( ) gives ku kp 2n(p 2) kuk p 2 H :

Therefore, (2:69) takes the form Z f (u ) f (u ) wtdx C v_t v_t L2_{( )} ru ru L2( ) kru k p 2 L2_{( )}+ ru p 2 L2_{( )} (2.70)

under the fact that

t Z 0 (g0 _{rw) (s)ds} 0; we conclude t Z 0 (g0 _{rw) (s)ds + (g rw) (t) + krw(t)k}2₂ t Z 0 g(s)ds 0: Thus, 1 2kw(t; :)k 2 H 1 2kw(0; :)k 2 H + C t Z 0 ru ru L2_{( )}kwt(s; :)kHds:

The Gronwell Lemma and Young’s inequality guarantee that

Since

v (t; :) v (t; :) _H C v (0; :) v (0; :) _H + CT u u _{C([0;T ];H)}; (2.71)

then fv g is a Cauchy sequence in C([0; t] ; H); since fu g and fv (0; :)g are Cauchy sequences in

C([0; T ] ; H) and H, respectively.

Now, we shall prove that fvtg is a Cauchy sequence, in Lm_{([0; T ] ), to control the norm}

kvtk2Lm_{([0;T ] )}: By the following algebraic inequality

j jm 2 j jm 2 ( ) C_{j j}m; (2.72)

which holds for any real ; and c, we get

Z k(v_t) k(v_t) wtdx = Z v_{t jvtj}m 2 v_t v_t m 2 v_t v_t dx C v_t v_t 2 Lm_{([0;T ] )}:

This estimate combined with (2:68) gives

v_t v_t 2 Lm_{([0;t] )} C v (0; :) v (0; :) Lm([0;t] ) +CR t Z 0 u u _Lm_{([0;t] )} v (t; :) v (t; :) _Lm_{([0;t] )}ds:

So by using Gromwell Lemma, we obtain fvtg is a Cauchy sequence, in Lm_{([0; T ] )and hence}

fv g is a Cauchy sequence in YT:Let v its limit in YT and by Lemma 2.1.2, v is a weak solution of

(2:1).

Now, we are ready to show the local existence of the problem (P ) Proof of Theorem 2.1.1.

Let (u0; u1)2 (H01( ))

2

; and

R2 = _kru0k2₂+ku1k2₂ :

For any T > 0; consider

MT = u2 YT : u(0) = u0; ut(0) = u1 and kukY_T R :

Let

: MT ! MT

We will prove as in [13] that,

(i) (MT) MT:

(ii) is contraction in MT:

Beginning by the …rst assertion. By Lemma 2.1.1, for any u 2 MT we may de…ne v = (u);

the unique solution of problem (2:1). We claim that, for a suitable T > 0, is contractive map

satisfying

(MT) MT:

Let u 2 MT; the corresponding solution v = (u)satis…es for all t 2 [0; T ] the energy identity :

1 2 8 < :kv0(t)k 2 2+ 0 @1 t Z 0 g(s)ds 1 A krv(t)k2 2+ (g rv)(t) 9 = ; + t Z 0 krv0(s)_k2₂ds + t Z 0 kv0(s)_km_m)ds (2.73) = 1 2 kv1k 2 2+krv0k 2 2 + t Z 0 Z ju(s)jp 2u(s)v0(t)dxds: We get 1 2kv(t)k 2 YT 1 2kv(0)k 2 YT + t Z 0 Z ju(s)jp 2u(s)v0(t)dxds: (2.74)

We estimate the last term in the right-hand side in (2:74) as follows: thanks to Hölder’s, Young’s

inequalities, we have _Z

ju(s)jp 2u(s)v0(t)dx C_kukp_Y

TkvkYT; then, kv(t)k2YT kv(0)k 2 YT + CR p t Z 0 kvkYT ds;

where C depending only on T; R: Recalling that u0; u1 converge, then

kv(t)kYT kv(0)kYT + CR

p_T:

Choosing T su¢ ciently small, we getkvkYT R; which shows that

Now, we prove that is contraction in MT:Taking w1 and w2 in MT, subtracting the two equations

in (2:1); for v1 = (w1) and v2 = (w2);and setting v = v1 v2;we obtain for all 2 H01( ) and

a.e. t 2 [0; T ] Z vtt: dx + Z rvr dx + Z rvtr dx + Z _Zt 0 g(t s)_{rvr dsdx} + Z jvtjm 2vt dx = Z jw1jp 2w1 jw2jp 2w2 dx: (2.75)

Therefore, by taking = vt in (2:75) and using the same techniques as above, we obtain

kv(t; :)k2YT C t Z 0 kw1k p 2 YT +kw2k p 2 YT kw1 w2kYTkv(s; :)k 2 YT ds: (2.76)

It’s easy to see that

kv(t; :)k2Yt =k (w1) (w2)k

2

Yt kw1 w2k

2

Yt; (2.77)

for some 0 < < 1 where = 2CT RP 2:

Finally by the contraction mapping theorem together with (2:77); we obtain that there exists a

unique weak solution u of u = (u) and as (u) _{2 Y}T we have u 2 YT: So there exists a unique

weak solution u to our problem (P ) de…ned on [0; T ] ; The main statement of Theorem 2.1.1 is proved.

Remark 2.1.1 Let us mention that in our problem (P ) the existence of the term source ( f (u) =

jujp 2u ) in the right hand forces us to use the contraction mapping theorem. Since we assume a

little restriction on the initial data. To this end, let us mention again that our result holds by the well depth method, by choosing the initial data satisfying a more restrictions.

Global Existence and Energy Decay

Abstract

In this chapter, we prove that the solution obtained in the second chapter (Local solution ) is global in time: In addition, we show that the energy of solutions decays exponentially if m = 2 and polynomial if m > 2, provided that the initial data are small enough. The existence of the source

term jujp 2u forces us to use the potential well depth method in which the concept of so-called

stable set appears. We will make use of arguments in [44] with the necessary modi…cations imposed by the nature of our problem.