Stabilization of the Coleman-Gurtin thermal coupling with swelling porous system: general decay rate

Abstract

This paper is concerned with a thermoelastic swelling system with Coleman-Gurtin’s law, when the heat flux q is given by

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

where \(\theta \) is the temperature supposed to be known for negative times. \(\Psi \) is the convolution thermal kernel, a nonnegative bounded convex function on \([0, + \infty )\) belongs to a broad class of relaxation functions satisfying the unitary total mass, and some additional properties that will be specified later. By using the Dafermos history framework and constructing a suitable Lyapunov functional, we established a general decay result, from which the exponential and polynomial decay rates are only special cases. The stability result in this manuscript is obtained without imposing any stability number, and extends and improves many earlier results in the literature.