Well-Posedness and Stability Results for a Nonlinear Damped Porous–Elastic System with Infinite Memory and Distributed Delay Terms

Abstract

In the present paper, we consider an important problem from the application perspective in science and engineering, namely, one-dimensional porous–elastic systems with nonlinear damping, infinite memory and distributed delay terms. A new minimal conditions, placed on the nonlinear term and the relationship between the weights of the different damping mechanisms, are used to show the well-posedness of the solution using the semigroup theory. The solution energy has an explicit and optimal decay for the cases of equal and nonequal speeds of wave propagation.

Dynamics of a one-dimensional nonlinear poroelastic system weakly damped

“…Recently in [1], Al-Mahdi et al studied the issue of general decay for the following thermoelastic porous model with memory effect: Compared to linear ones, there are few works related to the asymptotic behavior of nonlinear poroelastic systems. In this case, it may happen that the objective of the work is to study the decay of the solutions of the system as in [4,22,25] or to study the existence of global attractors (Definition A.2) for the dynamical system associated with the solutions of the system; we will focus on the second. It is noteworthy that the global attractor, when it exists, is unique.…”