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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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BADJI MOKHTAR UNIVERSITY

OF ANNABA UNIVERSITÉ BADJI MOKHTAR

DE ANNABA

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

Systemes 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

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BADJI MOKHTAR UNIVERSITY

OF ANNABA UNIVERSITÉ BADJI MOKHTAR

DE ANNABA

Sciences Faculty
Department of Mathematics

THESIS
Submitted for the obtention of

MAGISTER DIPLOMA IN MATHEMATICS
DOCTORAL SCHOOL IN MATHEMATICS

By
Khaled ZENNIR

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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

Tha1es

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mu arm that/Lizs eo to ir.P..,S.411-1-1-0W1-121, ho supervi,sed this thesis

1,th great pati,et/Lce avid Izi,t/Ldt/Less. f-te as aloft to vtkoti,vate each step of 1/14.b

orn lob reLevat/Lt revt/tarizs avid Let vte abbe to 00N/o/Lae 1,1/L vt/tu research, beei,t/LI/Li,vLe bu the D.ES Stv.0(U I txt/Li,versiku of sizi,lzda. I that/Liz hi,vt/t, verb si,t/LcereLb for hi,s availabiliku eve at a di,stat/Lce, ith hi,vt/t. I have a Lot of ft/tvL to orn.

I ish to that/Liz Prof P.. ICI-I-Opy.4, ho vtkalzes vte the hot/tour of chai,ri,vLe the jt/tru of vtku defet/Lse.

~ avw verb crateful, to z)r. L. MSS& avi,of 1)r. y LASICRI to have agreed to resci,t/Ld this thesis avid to be part of vtku defet/Lse pateL.

of course, this thesis 1,s the cyLvtki,vLati,ovL of vtkat/Lb bears of sttxdu. 1 avA. 1,o,debted to aLL those ho e[doed vt/te ot/L the road of vtkathevtkati,cs.

~ that/Liz from. vtku heart, vtku teachers hose et/Lcokraeed vtke dt/tri,vLe 1/14.b studies avid have di,stracted vt/te the vtkathevtkati,cs.

I thi,vLlz for vtku paret/Lts, hose the orn could t/Lot succeed ithout their et/LdLess support avid et/LcokraeevtkevLt. Theo -Rost i,t/L the reaLizati,ovL of this

ork, the cyLvtki,vLati,ovL of theix efforts at/Ld express vtku vtkost affecti,ovote eratikkde.

Contents

1

Preliminary

1

 

1.1

Banach Spaces - Definition 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 (l) spaces

6

 
 

1.2.2 The Sobolev space Wm'p(1)

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

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