Etude D’existence, Unicité Et Stabilité De Quelques Problèmes D’évolution À Retard

Résumé: In this thesis, we study some time-delay evolution systems with the presence of different mechanisms of dissipation. We begin this thesis by presenting a brief summary about dynamical systems theory where we will introduce a historical overview of its origins. Then, we recall some reminders on functional spaces, and semigroups theory. The monograph is composed of two parts, the first one is divided to three chapters, the first chapter is devoted to study a thermo-viscoelastic system of Timoshenko-type with nonlinear damping and a distributed delay acting on transverse displacement. We use the energy method and some properties of convex functions to prove a general decay estimate. In chapter 2, we concern with a one-dimensional Timoshenko system of ther moelasticity of type III with infinite memory damped by weakly nonlinear feedbacks. We obtain a general stability estimates using the multiplier method without assuming equal or nonequal speeds of propagation of waves. The third chapter is devoted to study a thermo-viscoelastic Bresse system with second sound and delay terms. We obtain results regardless of the speeds of wave propagation and the stable number which is introduced in some works before. The second part is divided to two chapters, we study non-uniform flexible structures systems, first one with second sound and a distributed delay term, and the second in thermoelasticity with micro-temperature effect. We prove the well-posed of each system as well its stability results under suitable assumptions.

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