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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
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PRÉSIDENT
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B. KHODJA
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PROF.
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U. B. M. ANNABA
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RAPPORTEUR
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H. SISSAOUI
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PROF.
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U. B. M. ANNABA
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ÉXAMINATEUR
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L. NISSE
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M. C.
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U. B. M. ANNABA
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ÉXAMINATEUR
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Y. LASKRI
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M. C.
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U. B. M. ANNABA
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INVITE
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B. SAID-HOUARI
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Dr.
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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
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PRESIDENT
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B. KHODJA
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PROF.
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U. B. M. ANNABA
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SUPERVISOR
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H. SISSAOUI
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PROF.
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U. B. M. ANNABA
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EXAMINER
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L. NISSE
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M. C.
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U. B. M. ANNABA
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EXAMINER
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Y. LASKRI
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M. C.
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U. B. M. ANNABA
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Tha1es
lin, the first pace at aLL, vtku eratikkde does to Prof.
1--/-. S ISSA-0 k I, VIA.0 stxpervi,sor.
t--te al aus Li,stet/Ls, advi,ces, et/Lcot/traees avid evi,des vie,
he as behi,t/Ld this ork. I shall aL aus be 1,vwpressed bu hi,s availabiliku
avid the care ith hich he covwes to read avid correct 1/14.b ta1tscript.
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
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1
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Preliminary
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1
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1.1
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Banach Spaces - Definition and Properties
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2
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1.1.1 The weak and weak star topologies
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3
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1.1.2 Hilbert spaces
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4
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1.2
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Functional Spaces
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6
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1.2.1 The Lp (l) spaces
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6
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1.2.2 The Sobolev space Wm'p(1)
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8
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1.2.3 The Lp (0, T, X) spaces
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10
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1.2.4 Some Algebraic inequalities
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12
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1.3
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Existence Methods
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14
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1.3.1 The Contraction Mapping Theorem
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14
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1.3.2 Gronwell's lemma
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14
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1.3.3 The mean value theorem
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15
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2
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Local Existence
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17
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2.1 Local Existence Result
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18
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3
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Global Existence and Energy Decay
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39
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3.1
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Global Existence Result
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40
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3.2
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Decay of Solutions
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46
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4
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Exponential Growth
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59
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4.1 Growth result
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60
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