SPATIAL ESTIMATES IN THE DYNAMIC THEORY OF LINEAR ELASTIC-MATERIALS WITH MEMORY

We establish inequalities which provide meaningful bounds at all times for energy and we investigate spatial decay estimates for solutions o f problems such as transient processes in a viscoelastic solid subject to non-zero boundary data only, on a plane end. For a fixed time, we prove that for distances from the loaded end greater than a certain va lue, the whole activity in the body vanishes. Our method of proof uses the dissipative effects induced by memory terms in order to bound the magnitude of the stress tensor by means of an appropriate lower bound for the least work in linear viscoelasticity. On this basis we establ ish a first-order partial differential inequality for the measure of t he solution defined by the energy contained in that part of the body w hose minimum distance from the loaded end is z. The study of the parti al differential inequality gives the results.