On the stability of the swelling porous elastic soils with fluid saturation and Gurtin–Pipkin thermal law

The present paper is devoted to studying the well-posedness and exponential stability of the one-dimensional system in the linear isothermal theory of swelling porous elastic soils with fluid saturation and Gurtin–Pipkin thermal law. For the well-posedness, we apply the well-known Hille–Yosida theorem of semigroup theory. To prove exponential stability without assuming that the wave speeds are the same, we use the energy method which consists of constructing a Lyapunov functional equivalent to the system’s total energy.