Spatial and temporal behavior in elastic materials with voids

The aim of this paper is to study the spatial and temporal behavior of the solution to the initial-boundary value problems associated with the linear theory of elastic materials with voids. As regards the spatial behavior, a domain of influence is established in the sense that for eacht ∈ [0,T] the whole activity is zero in the part of the body for whichr≥ct, wherer represents the distance from the support\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat D_T $$ \end{document} of the given data on [0,T] andc is a positive constant depending on the elastic properties of the material. Forr≤ct, a spatial decay estimate of Saint-Venant type is established for which the decay rate is independent of time. As regards the temporal behavior, some relations are established describing the asymptotic partition of the Cesàro mean of the total energy. When the dissipation is absent the above relations give an equipartition theorem of the Cesàro means of various energies.