In the present work, we consider a one-dimensional porous-elastic system with infinite memory and a nonlinear damping term. We establish the well-posedness of the system using semigroup theory and show the general decay for the case of nonequal speeds of wave propagation. Introducing some conditions on the kernel of the infinite memory term helps estimate the nonequal speed term even if this complementary control is not strong enough to stabilize the system exponentially. Our result is an extension of many other works in this area.

The one-dimensional full von Kármán beam is stabilized using the dissipative effect through heat conduction acting on the longitudinal component and the frictional damping acting on the transversal component of the beam.

In this article, we study a porous-elastic system with dissipation only due to microtemperatures effect. We introduce a new stability number and prove that the unique dissipation due to the microtemperatures is strong enough to drive the system to the equilibrium state in an exponential manner. This result is new and improves previous results in the literature.