Nonlinear stability of travelling wave solutions for viscoelastic materials with fading memory

In this paper, we shall discuss the stability of smooth monotone travelling wave solutions for viscoelastic materials with memory. It is known that a smooth monotone travelling wave solution exists for (1.1) if the end states are close and satisfy the Rankine-Hugoniot condition. For such a travelling wave, we shall show that if the initial data are close to a travelling wave solution, the solutions to (1.1) will approach the travelling wave solution in sup norm as the time goes to infinity. For the constitutive relations, we shall discuss two cases: convex and nonconvex.