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In the present paper we consider a porous thermoelastic system with only onedissipative mechanism generated by the heat conductivity modelled by theGurtin-Pipkin thermal law.  By the use of a semigroup approach and the Lumer-Phillips theorem we provethe existence of a unique solution. We introduce a stability number $\chi_g$depends on the coefficients of the system, and establish an exponentialstability result provided that $\chi_g=0$. Otherwise, if $\chi_g\ne 0$, weprove the lack of exponential decay. Our result improves and generalizes theprevious results in the literature obtained for Fourier's and Cattaneo's lawsof thermal conductivity.

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