Evolution problems in materials with fading memory

Abstract

Evolution problems in materials with memory are here considered.
Thus, linear integro-differential equations with Volterra type kernel are
investigated. Specifically, initial boundary value problems are studied;
physical properties of the material under investigation are shown to induce the choice of a suitable function space, where solutions are looked for. Then, combination with the application of Fourier transforms, allows to prove existence and uniquenes of the solution. Indeed, the original evolution problem is related to an elliptic one: existence and uniqueness results are proved for the latter and, thus, for the original problem. Two different evolution initial boundary value problems with memory which arise, in turn, in the framework of linear heat conduction and of linear viscoelasticity are compared.