Nonlinear Stability of Travelling Waves for One-Dimensional Viscoelastic Materials with Non-Convex Nonlinearity

Abstract

The aim of this paper is to study the stability of travelling wave solutions with shock profiles for one-dimensional viscoelastic materials with the non-degenerate and the degenerate shock conditions by means of an elementary weighted energy method. The stress function is not necessarily assumed to be convex or concave, and the third derivative of this stress function is also not necessarily assumed to be non-negative or non-positive. The travelling waves are proved to be stable for suitably small initial disturbance and shock strength, which improves recent stability results. The key points of our proofs are to choose the suitable weight function and weighted Sobolev spaces of the solutions.