Well-posedness and asymptotic bihavior for some hyperbolic systems

This book is devoted to the study of the existence and asymptotic behavior of certain hyperbolic systems. The first part of the book is composed of two chapters 2 and 3. Both are devoted first to the study of the linear thermoelastic system in dimension one of the Timoshenko type, in which the heat flow is given by the Cattanéo law. Note here that the introduction of the lag term and the external force term is only in chapter 3, and we show the existence and the exponential stability of the solution. The proof that we have established is based on the construction of an appropriate Lyapunov function equivalent to the energy of the considered solution. This function verifies a differential inequation leading to the desired decay result. Then, and in chapter 4, we consider a system of wave equations with dissipative terms in order to prove that the solution of this system explodes in finite time.