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
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 Problem
Index. 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 115
Côte titre :	DM/0125
En ligne :	https://drive.google.com/file/d/1V5DcMAjJr-G81MWUdxbC_jk2WljVYkJZ/view?usp=shari [...]
Format de la ressource électronique :	pdf

Some Transmission Problems of Waves and Viscoelastic Wave Equations With Delay and an Evolutionary Problem [texte imprimé] / Aissa Benseghir, Auteur . - [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 Problem
Index. 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 115
Côte titre :	DM/0125
En ligne :	https://drive.google.com/file/d/1V5DcMAjJr-G81MWUdxbC_jk2WljVYkJZ/view?usp=shari [...]
Format de la ressource électronique :	pdf