Stability in an unbounded domain of a coupling thermoelastic swelling porous elastic soils system with Coleman-Gurtin heat flux

The subject of the present paper is to study the stability in the whole line $ \mathbb{R} $ of a thermoelastic swelling system consisting of coupled two wave equations in one dimensional case, where the coupling terms are of two different orders with respect to the space variable, and combined with a heat conduction given by Coleman-Gurtin's law acting only on the second wave equation. We consider two types of coupling terms between the second wave equation and the heat conduction equation. The main result of this paper is showing that the thermoelatic dissipation generated by Coleman-Gurtin's law is able to stabilize the whole system at least polynomially in the sense of $ L^2 (\mathbb{R}) $-norm, where the polynomial decay in the $ L^2 (\mathbb{R}) $-norm of the solution and its higher order derivatives with respect to the space variable are specified in terms of the regularity of the initial data. The proofs are based on the energy method and Fourier analysis combined with some well chosen multiplier functions and the help of some arguments devised in [20,21] by the third author of the present paper.