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

Abstract

IN this work, 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. The system’s well-posedness is established using the semigroup approach, ensuring the existence and uniqueness of the solution. Furthermore, through energy analysis and the construction of a suitable Lyapunov functional, it is proven that the dissipation provided by the second sound mechanism leads to exponential stability, regardless of the system parameters. Finally, we give some numerical tests to illustrate the theoretical results by carrying out an Euler scheme for time discretization and finite difference method for spatial discretization. Then, we introduce a fixed point algorithm to solve the discretized problem.