The aim of this paper is to study the spatial behavior of the solutions to the final-boundary-value problems associated with the linear theory of elastic materials with voids. More precisely, the present study is devoted to porous materials with a memory effect for the intrinsic equilibrated body forces. An appropriate time-weighted volume measure is associated with the backward-in-time thermoelastic processes. Then a first-order partial differential inequality in terms of such measure is established and it is further shown how the inequality implies the spatial exponential decay of the thermoelastic process in question.