Regularity and stability for a viscoelastic material with a singular memory kernel

Abstract

We consider a linear viscoelastic material whose relaxation function may exhibit an initial singularity. We show that the Laplace transform method is still applicable in order to study existence, uniqueness and asymptotic behaviour of the solution to the dynamic problem. In order to provide these results, we impose on the relaxation function only restrictions deriving from Thermodynamics. Moreover, by using energy estimates, we establish a stability theorem. Finally, for a class of singular kernels, we obtain a regularity result which ensures the asymptotic stability of the solution.