Global well-posedness and energy decay for a one dimensional porous-elastic system subject to a neutral delay

Abstract

In this article, we consider a one-dimensional porous-elastic system with porous-viscosity and a distributed delay of neutral type. First, we prove the global existence and uniqueness of the solution by using the Faedo–Galerkin approximations along with some energy estimates. Then, based on the energy method and by constructing a suitable Lyapunov functional as well as under an appropriate assumptions on the kernel of neutral delay term, we show that despite of the destructive nature of delays in general, the damping mechanism considered provockes an exponential decay of the solution for the case of equal speed of wave propagation. In the case of lack of exponential stability, we show that the solution decays polynomially.