On wave propagation in linear viscoelasticity

We discuss the initial value problem in one-dimensional linear visco-elasticity with a step-jump in the initial data. If the memory kernel is sufficiently smooth on <!-- MATH $\left[ {0,\infty } \right)$ --> , the solution exhibits discontinuities propagating along characteristics and a (higher order) stationary discontinuity at the position of the original step-jump. For a singular memory kernel, the propagating waves are smoothed in a manner depending on the nature of the singularity in the kernel, but the stationary discontinuity remains. We also discuss the effects of these phenomena on the regularity of solutions with arbitrary initial data.