On the stability of a double porous elastic system with visco-porous dampings

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 ${\chi }_1\mathrm{\ne }0.$ Otherwise, we prove the lack of exponential decay. Our results improve the results of \cite{Bazarra} and \cite{Nemsi}.