Exponential stability of a porous thermoelastic system with Gurtin Pipkin thermal law and distributed delay time

In this paper, we consider a one-dimensional porous thermoelastic system with herditary heat conduction and a distributed delay time acting only on the porous equation, where the heat conduction is given by Gurtin Pipkin law. Existence and uniqueness of a solution are obtained by the use of Hille-Yosida theorem. Then, based on the energy method as well as by constructing a suitable Lyapunov functional, we prove under some assumptions on the derivative of the heat-flux kernel, that the solution of the system decays exponentially without any assumption on the wave speed.