  In this paper we consider a one dimensional elastic system with double
porosity structure and with frictional damping in both porous equations. We
introduce two stability numbers $\chi_{0}$ and $\chi_{1}$ and prove that the
solution of the system decays exponentially provided that $\chi_{0}=0$ and
$\chi_{1}\neq0.$ Otherwise, we prove the lack of exponential decay. Our results
improve the results of \cite{Bazarra} and \cite{Nemsi}.
