Asymptotic behaviour for linear and nonlinear elastic waves in materials with memory

In this review, we study the Cauchy problem associated to the equation of linear and nonlinear viscoelasticity with memory. Our first point is the study of dispersive properties of the solution to the linear equation of viscoelasticity with memory. The decay estimates obtained in this first part are important to treat the corresponding nonlinear Cauchy problem. The key novelty is the fact that we admit algebraic singularities and decay at infinity for the time dependent functions in the memory kernel. This fact enables one to include models different from the classical viscoelasticity problem, where this kernel is smooth and exponentially decaying in time.