On the time behavior of a porous thermoelastic system with only thermal dissipation given by Gurtin-Pipkin law

In the present paper we consider a porous thermoelastic system with only one dissipative mechanism generated by the heat conductivity modelled by the Gurtin-Pipkin thermal law. By the use of a semigroup approach and the Lumer-Phillips theorem we prove the existence of a unique solution. We introduce a stability number ${\chi }_g$ depends on the coefficients of the system, and establish an exponential stability result provided that ${\chi }_g=0$. Otherwise, if ${\chi }_g\mathrm{\ne }0$, we prove the lack of exponential decay. Our result improves and generalizes the previous results in the literature obtained for Fourier’s and Cattaneo’s laws of thermal conductivity.