Qualitative study of the solutions of the damped nonlinear wave equation

We study the asymptotic behaviour of the solutions of the partial differential equation : u(tt) = (sigma(u(x)))(x) + u(xxt), x is an element of (0, 1), t > 0 (*) which describes the one-dimensional motion of a nonlinear viscoelastic material. When a is not monotone, the equilibrium solutions of(*) may not be continuous in u(x). In the first part, we use a method introduced by R. Pego to study (*), that is to transform (*) into a system of two PDEs for which we prove local existence, stability of equilibrium solutions and the persistence of discontinuities. In the second-part we discretize (*) in x to obtain a differential system. For this we study the equilibrium solutions and their stability.