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Some Transmission Problems of Waves and Viscoelastic Wave Equations With Delay and an Evolutionary Problem / Aissa Benseghir
Titre : Some Transmission Problems of Waves and Viscoelastic Wave Equations With Delay and an Evolutionary Problem Type de document : texte imprimé Auteurs : Aissa Benseghir, Auteur ; HAMID Bensridi, Directeur de thèse Editeur : Setif:UFA Année de publication : 2017 Importance : 1 vol (86 f.) Format : 29 cm Langues : Anglais (eng) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Transmission
Problems of Waves
Viscoelastic Wave Equations
Delay
Evolutionary ProblemIndex. décimale : 510 Mathématique Résumé : Abstract
This thesis is devoted to the study of stability and decay rates of solutions for
some wave transmission problems and viscoelastic wave equations with delay, in some
cases the delay is a time function. And the existence, uniqueness of weak solution
to a nonlinear history-dependent boundary value problem, and the same goal for an
evolution of a viscoelastic plate in frictional contact with foundation.
The first part of this thesis is composed of three chapters. In Chapter 2, we consider a transmission system with a delay. We show the well-posedness as well as the
exponential stability of the solution depending on the weight of linear damping and
the weight of the delay term. In Chapter 3, we proved the well-possedness of a system
with delay and memory. In Chapter 4, we prove a decay of a transmission problem
with memory and delay, but in this case the delay is considered as a time-varying
function.
The second part of this thesis is devoted to the study of mathematical models
of contact. More precisely, in chapter 5, we introduce a mathematical model that
describes the evolution of a viscoelastic plate in frictional contact with foundation,
we derive the variational inequality for the displacement field, then we establishe the
existence of a unique weak solution to the model.
Note de contenu : Contents
Introduction i
1 Preliminaries of Functional Analysis 15
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Definitions and Elementary Properties. . . . . . . . . . . . . . . . . . . 15
1.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.2 Useful results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3 The Theorem of Stampacchia and Lax-Milgram . . . . . . . . . . . . . 18
1.4 Element of Nonlinear Analysis . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.1 Monotone operators . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.2 Convex lower semicontinuous functions . . . . . . . . . . . . . . 23
1.5 Elliptic Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . 25
1.6 History-dependent Variational Inequalities . . . . . . . . . . . . . . . . 26
1.6.1 Spaces of vector-valued functions . . . . . . . . . . . . . . . . . 26
1.6.2 History-dependent quasivariational inequalities. . . . . . . . . . 28
1.7 Semi-groupes of linear operators . . . . . . . . . . . . . . . . . . . . . . 34
1.7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.7.2 Infinitesimal generator . . . . . . . . . . . . . . . . . . . . . . . 35
1.7.3 The Hille-Yosida and Lumer-Philips theorem . . . . . . . . . . . 35
1.8 The Hille-Yosida Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.8.1 Definition and Elementary properties . . . . . . . . . . . . . . . 36
I Transmission Problems 39
2 Decay for a transmission wave equations with delay 40
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2 Well-posedness of the problem . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.1 First Method : Galerkin Method . . . . . . . . . . . . . . . . . 42
2.2.2 Second Method : The semi-group theory . . . . . . . . . . . . . 48
2.3 Exponential decay of the solution . . . . . . . . . . . . . . . . . . . . . 52
3 Well-posedness of a transmission problem with viscoelastic term and delay 59
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Well-posedness of the problem . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Decay of the solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 Decay for a transmission problem with memory and time-varying delay 71
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Preliminaries and main results . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 General decay of the solution . . . . . . . . . . . . . . . . . . . . . . . 77
II Boundary Value Problem 85
5 An Evolutionary Boundary Value Problem 86
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4 A continuous dependence result . . . . . . . . . . . . . . . . . . . . . . 104
5.5 Dual variational formulation . . . . . . . . . . . . . . . . . . . . . . . . 107
Bibliography 115Côte titre : DM/0125 En ligne : https://drive.google.com/file/d/1V5DcMAjJr-G81MWUdxbC_jk2WljVYkJZ/view?usp=shari [...] Format de la ressource électronique : Some Transmission Problems of Waves and Viscoelastic Wave Equations With Delay and an Evolutionary Problem [texte imprimé] / Aissa Benseghir, Auteur ; HAMID Bensridi, Directeur de thèse . - [S.l.] : Setif:UFA, 2017 . - 1 vol (86 f.) ; 29 cm.
Langues : Anglais (eng)
Catégories : Thèses & Mémoires:Mathématique Mots-clés : Transmission
Problems of Waves
Viscoelastic Wave Equations
Delay
Evolutionary ProblemIndex. décimale : 510 Mathématique Résumé : Abstract
This thesis is devoted to the study of stability and decay rates of solutions for
some wave transmission problems and viscoelastic wave equations with delay, in some
cases the delay is a time function. And the existence, uniqueness of weak solution
to a nonlinear history-dependent boundary value problem, and the same goal for an
evolution of a viscoelastic plate in frictional contact with foundation.
The first part of this thesis is composed of three chapters. In Chapter 2, we consider a transmission system with a delay. We show the well-posedness as well as the
exponential stability of the solution depending on the weight of linear damping and
the weight of the delay term. In Chapter 3, we proved the well-possedness of a system
with delay and memory. In Chapter 4, we prove a decay of a transmission problem
with memory and delay, but in this case the delay is considered as a time-varying
function.
The second part of this thesis is devoted to the study of mathematical models
of contact. More precisely, in chapter 5, we introduce a mathematical model that
describes the evolution of a viscoelastic plate in frictional contact with foundation,
we derive the variational inequality for the displacement field, then we establishe the
existence of a unique weak solution to the model.
Note de contenu : Contents
Introduction i
1 Preliminaries of Functional Analysis 15
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Definitions and Elementary Properties. . . . . . . . . . . . . . . . . . . 15
1.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.2 Useful results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3 The Theorem of Stampacchia and Lax-Milgram . . . . . . . . . . . . . 18
1.4 Element of Nonlinear Analysis . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.1 Monotone operators . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.2 Convex lower semicontinuous functions . . . . . . . . . . . . . . 23
1.5 Elliptic Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . 25
1.6 History-dependent Variational Inequalities . . . . . . . . . . . . . . . . 26
1.6.1 Spaces of vector-valued functions . . . . . . . . . . . . . . . . . 26
1.6.2 History-dependent quasivariational inequalities. . . . . . . . . . 28
1.7 Semi-groupes of linear operators . . . . . . . . . . . . . . . . . . . . . . 34
1.7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.7.2 Infinitesimal generator . . . . . . . . . . . . . . . . . . . . . . . 35
1.7.3 The Hille-Yosida and Lumer-Philips theorem . . . . . . . . . . . 35
1.8 The Hille-Yosida Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.8.1 Definition and Elementary properties . . . . . . . . . . . . . . . 36
I Transmission Problems 39
2 Decay for a transmission wave equations with delay 40
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2 Well-posedness of the problem . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.1 First Method : Galerkin Method . . . . . . . . . . . . . . . . . 42
2.2.2 Second Method : The semi-group theory . . . . . . . . . . . . . 48
2.3 Exponential decay of the solution . . . . . . . . . . . . . . . . . . . . . 52
3 Well-posedness of a transmission problem with viscoelastic term and delay 59
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Well-posedness of the problem . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Decay of the solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 Decay for a transmission problem with memory and time-varying delay 71
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Preliminaries and main results . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 General decay of the solution . . . . . . . . . . . . . . . . . . . . . . . 77
II Boundary Value Problem 85
5 An Evolutionary Boundary Value Problem 86
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4 A continuous dependence result . . . . . . . . . . . . . . . . . . . . . . 104
5.5 Dual variational formulation . . . . . . . . . . . . . . . . . . . . . . . . 107
Bibliography 115Côte titre : DM/0125 En ligne : https://drive.google.com/file/d/1V5DcMAjJr-G81MWUdxbC_jk2WljVYkJZ/view?usp=shari [...] Format de la ressource électronique : Exemplaires (1)
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