No 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.

In this paper, we consider a one-dimensional system known as Shear beam model (no rotary inertia) where the transverse displacement equation is subject to a distributed delay of neutral type. Under some assumptions on the kernel h, we first achieved the global well- posedness of the system by using the classical Faedo-Galerkin approximations along with two a priori estimates. Next, we find the energy expression and by using technique of Lyapunov functional we demonstrate, although delays are known to be of a destructive nature in the general case, that this system is exponentially stable regardless any relationship between coefficients of the system.