Theoretical and numerical studies for a delayed swelling porous thermoelastic soils model

In this paper, we consider a one-dimensional swelling problem in porous elastic soils with second sound and a constant internal delay, where the heat conduction is given by Cattaneo's law. We show that the system is well-posed using the semigroup approach. Then, based on the energy method as well as by constructing a suitable Lyapunov functional, we prove that the unique dissipation given only by the second sound is strong enough to provoke an exponential decay of the solution without any relationship between the system parameters. For the numerical study, we discretize the continuous problem by performing a temporal discretization using Euler scheme and the classical finite difference method for spatial discretization. To solve the discretized problem, we propose to introduce a fixed point algorithm and derive the condition for which the proposed algorithm converges. Finally, we present some numerical test to illustrate the theoretical results by taking different delay weights and show that the studied system is highly reactive to small delays.