Exponential stability of a coupling thermoelastic swelling porous system with Coleman–Gurtin heat flux

Abstract

In this paper, we consider a thermoelastic swelling soils system in which the heat flux q is given by Coleman–Gurtin’s law. Precisely, the heat flux q is defined by

$$\begin{aligned} \tau q(t)+(1-\alpha )\theta _{x}+\alpha \int _{0}^{\infty } \Lambda (s)\theta _{x}(x, t-s)ds=0,\qquad \alpha \in (0, 1), \end{aligned}$$

where $\theta$ is the temperature and $\Lambda$ is the thermal memory. In fact, the Fourier’s, Maxwell–Cattaneo’s and the Gurtin–Pipkin’s laws are special cases from the Coleman–Gurtin’s law. By constructing a suitable Lyapunov functional, we establish an exponential decay result for the considered system avoiding the imposition of any stability number. Unlike many authors who have investigated different systems with Fourier, Maxwell–Cattaneo’s and the Gurtin–Pipkin’s laws and their exponential stability obtained for those systems depends on some stability numbers. Our result extends and improves some earlier results in the literature such as the one by Apalara et al. (J King Saud Univ Sci 35(1):102460, 2023) where they investigated two thermoelastic swelling porous systems with the Fourier’s law $(\alpha =0)$ and the one by Tijani and Almutairi (Mathematics 10(23):4498, 2022) where they established an exponential stability of a thermoelastic swelling porous system with the Gurtin–Pipkin thermal effect $(\alpha =1)$.