Some transmission problems of waves and viscoelastic wave equations with delay and an evolutionary problem

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.