Study of the global existence and stabilization of some evolutions problems

Abstract

This thesis focuses on the study of global existence and long-time behavior of the solutions of certain evolution equations. This work consists of four chapters. In the first chapter, we intro- duced some basic notions on Sobolev spaces and some main theorems in functional analysis. In the second chapter, we considered the nonlinear plate equation in the presence of dissipative terms: a viscoelastic dissipative term and a strong dissipative term of general form. By the Faedo-Galerkin method we have obtained the global existence of solutions in Sobolev spaces. Moreover, under conditions on the relaxation function the general stability is given by Lya- punov's method combined with some properties of convex functions. In the third chapter, we studied the global existence, the regularity of the solution and the internal stabilization of the nonlinear Petrovsky equation by a strong locally distributed damping. We showed that, under certain conditions on the damping term a(z) which allow it to be zero on a subdomain of 2, this problem admits a unique solution by the theory of semigroup and by the Faedo-Galerkin method we find that the solution is regular. Using a piecewise multiplier method, we proved the energy decreases exponentially and polynomially towards zero, under geometric conditions. Finally, in the fourth chapter, we considered a coupled system consisting of the Kirchhoff equa- tion and the heat equation in a bounded domain. We showed the existence and uniqueness of a global solution based on the Faedo-Galarkin approximations. And we used the method of multipliers to find a general stability.