Theoretical and numerical study of a porous thermoelastic system

Abstract

In the present work, we consider a one dimensional porous-thermoelastic system with dissipation only due to microtemperatures effect where the heat conduction is given by Cattaneo's law. First, we give an existence and uniqueness of the solution using semiroup theory. Then, by constructing a suitable Lyapunov functional using the multipliers method and by introducing a stability number that obtained at the first in [20], we prove that the dissipation given only by the microtemperature is strong enough to give an exponential stability of the energy. Also, by constructing a suitable Lyapunov functional using the multipliers method, we establish a polynomial decay result of the solution in the case when the stability number not holds. Then, we give some numerical tests to illustrate the theoretical results by carrying out an Euler scheme for time discretization and the classical finite difference method for the spatial discretization.