Stabilisation et existence globale de quelques problèmes d’évolution avec un terme de dissipation

Abstract

The purpose of this graduation thesis is to establish the stabilization and the global existence of some evolution problems with a dissipation term. Being consisted of four chapters, this work is devoted to the study of the existence, stability and blow-up of some evolution equations with nonlinear dissipative terms, viscoelastic term, delay term and logarithmic nonlinear source terms. In the first chapter is introductory where we recall some notions related to the theory of stochastic partial differential equations such as Brownian motion and Ito formula. In the second chapter, we study a non-degenerate Kirchhoff equation with general nonlinear dissipation term and time varying delay term. This chapter deals with uniform stability by using Lyapunov functional. The third chapter we consider the initial value problem for a nonlinear equation in a bounded domain with dispersion, nonlinear damping and logarithmic source terms. Under some suitable conditions on the given parameters and by using Faedo-Galarkin method, we study the existence of solutions and we show the blow up of solutions when the energy is initially negative. In chapter four states some theorems on the existence of the solution for a stochastic hyperbolic equation and eventually the explosion of the solution. We consider an initial boundary value problem of stochastic viscoelastic wave equation with nonlinear damping and logarithmic nonlinear source terms. We proved a blow-up result for the solution with decreasing kernel.