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

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par Khaled ZENNIR
Université Badji Mokhtar Algérie - Magister en Mathématiques 2009
  

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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 jutjm~2 ut = b jujp~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 RN (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 jutjm~2 ut = b jujp~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 RN (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 RN.

~ : topological boundary of a.

x = (xi, x2, ...,xN) : generic point of RN.

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.

Ck (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

.

11f11p = (11 I f(x)Ip)

W1,p (a) = {u E Lp (a) , Vu E (Lp (a))N1 .

1

p .

= (11urp+ 11Vurp)

W 1;p

0(a) : the closure of D (a) in W1,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)11pX < oo .

tE(0,T)

Ck ([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.

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