Exponential stability of a porous thermoelastic system with Gurtin–Pipkin thermal law

In this paper we consider a one-dimensional porous thermoelastic system with hereditary heat conduction. Existence and uniqueness of a solution are obtained by the use of Lumer–Phillips and Lax–Milgram theorems. Using a semigroup approach, 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 speeds. This result extends the results of Messaoudi and Fareh (Discrete Contin Dyn Syst B 20(2):599–612, 2015) and Muñoz Rivera and Quintanilla (J Math Anal Appl 338:1296–1309, 2008) to a more general case of heat conduction.