  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 $\chi_g\ne 0$, 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.
