Existence et comportement asymptotique des solutions d'une équation de viscoélasticité non linéaire de type hyperbolique

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

utt ~ Xu ~ W~ut + _Rt g(t - s)iu(s, x)ds

0

+a jutj^m~2 ut = b juj^p~2 u, x 2 l, t > 0

u(0,x) = u0 (x), x 2 ~ ut (0,x) = u1 (x), x 2 ~

u(t,x) = 0, x 2 [', t > 0

₈

<>>>>>>>>>>>>>>> >

>>>>>>>>>>>>>>>>:

, (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, global solutions, exponential decay, polynomial decay, growth.

Résumé

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

utt ~ Xu ~ W~ut + _Rt g(t - s)iu(s, x)ds

0

+a jutj^m~2 ut = b juj^p~2 u, x 2 l, t > 0

u(0,x) = u0 (x), x 2 ~ ut (0,x) = u1 (x), x 2 ~

u(t,x) = 0, x 2 [', t > 0

₈

<>>>>>>>>>>>>>>> >

>>>>>>>>>>>>>>>>:

, (P)

on, est un domaine borné de R^N (N ~ 1), avec frontière assez regulière [1, a, b, w sont des constantes positives, n-i ~ 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 dlés: Dissipation nonlinéaire, viscoelasticité, source nonlinéaire, solutions locale, solutions globale, décroissance exponentielle de l'énergie, décroissance polynomiale, croissement.

iv

Notations

a : bounded domain in R^N.

~ : topological boundary of a.

x = (xi, x2, ...,xN) : generic point of R^N.

dx = dx1dx2...dxN : Lebesgue measuring on a.

Vu : gradient of u.

Au : Laplacien of u.

f+, f^- : max(f, 0), max(--f,0).

a.e : almost everywhere.

+

1 p^,

= 1.

p^' : conjugate of p, i.e 1

p

D(a) : space of differentiable functions with compact support in a.

D^'(a) : distribution space.

C^k (a) : space of functions k--times continuously differentiable in a. Co (a) : space of continuous functions null board in a.

Lp (a) : Space of functions p--th power integrated on a with measure of dx.

1
p

.

11f11_p = (1₁ I f(x)Ip)

W^1,p (a) = {u E L^p (a) , Vu E (L^p (a))^N1 .

1

^p .

= (11ur_p+ 11Vur_p)

W 1;p

0(a) : the closure of D (a) in W^1,p (a). W ^~1;^p0(a) : the dual space of W0 "^p (a).

H : Hilbert space. H1 0 =W0 ^1;2 .

If X is a Banach space

T

Lp (0, T; X) = {f : (0, T) --> X is measurable ; f 11f(t)111 dt < oo} .

0

{ )

L^°° (0, T; X) = f : (0, T) --> X is measurable ;ess -- sup 11f(t)11^p_X < oo .

tE(0,T)

C^k ([0, T] ; X) : Space of functions k--times continuously differentiable for [0, T] --> X.

D ([0, T] ; X) : space of functions continuously differentiable with compact support in [0, T] . Bx = Ix E X; 11x11 < 1} : unit ball.