On the time decay of solutions in porous-thermo-elasticity of type II

In this paper we investigate the asymptotic behaviour of the solutions of the linear theory of thermo-porous-elasticity. That is, we consider the theory of elastic materials with voids when the heat conduction is of type II. We assume that the only dissipation mechanism is the porous dissipation. First we prove that, generically, the solutions are exponentially stable on time or, in other words, the decay of solutions can be controlled by a negative exponential for a generic class of materials. The reason lies in the fact that the temperature is strongly coupled with both the microscopic and macroscopic structures of the materials and plays the role of a "driving belt" between the dissipation at the microscopic structure and the macroscopic one. Later we note that the decay of solutions cannot be fast enough to make the solutions be zero in a finite period of time. Finally, we show that when the coupling term between the microscopic (or macroscopic) structure and the thermal variable vanishes, the solutions do not decay exponentially (generically).