Existence globale et stabilisation de certains problèmes d’évolution linéaires et non linéaires dans des domaines non bornés

Abstract

The present thesis is devoted to the study of global existence and asymptotic behavior in time of solution of Timoshenko system and coupled system .

This work consists of five chapters, will be devoted to the study of the global existence and asymptotic behaviour of some evolution equation with linear, nonlinear dissipative terms and viscoelastic equation. In chapter 1, we recall of some fundamental inequalities. In chapter 2, we consider the Cauchy problem for a coupled system of wave equation, we prove polynomial decay of solution. In chapter 3, we study the Cauchy problem for a coupled system of a viscoelastic wave equation, we prove exponential stability of the solution. In chapter 4, we study the Cauchy problem for a coupled system of a nonlinear weak viscoelastic wave equations, we prove existence and uniqueness of global solution and prove exponential stability of the solution. In this work, the proof an existence and uniqueness for global solution is based on stable set for small data combined with Faedo-Galerkin. The proof an decay estimate is based on multiplier method, Lyapunov functional for some perturbed energy. In chapter 5, we consider a system of viscoelastic wave equations of Petrowsky-Petrowsky type, we use a spaces weighted by density function to establish a very general decay rate of solution.