Existence and asymptotic behavior for some hyperbolic systems

Abstract

This thesis is devoted to study the Existence and asymptotic behavior for some hyperbolic systems . The first part of the thesis is composed of two chapters 2 and 3. We studied a one-dimensional linear thermoelastic system of Timoshenko type, where the heat flux is given by Cattaneo’s law, noting that in the chapter 3 we have introduced a delay term in the feedback and forcing term. We established several exponential decay results for classical and weak solutions in one-dimensional. Our technics of proof is based on the construction of the appropriate Lyapunov function equivalent to the energy of the considered solution, and which satisfies a di�erential inequality leading to the desired decay. In chapter 4, we consider a system of nonlinear wave equation with degenerate damping and strong nonlinear source terms. We prove that the solution blows up in time.