Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms

TL;DR

This paper investigates the mathematical properties of a porous-elastic system with damping and source terms, establishing conditions for existence, blow-up, and exponential decay of solutions, supported by numerical validation.

Contribution

It introduces new theoretical results on existence, blow-up, and decay for porous-elastic systems with nonlinear damping and sources, using adapted analytical techniques.

Findings

Local existence of weak solutions established

Finite time blow-up for solutions with negative initial energy

Exponential decay of global solutions proved

Abstract

In this paper we consider a porous-elastic system consisting of nonlinear boundary/interior damping and nonlinear boundary/interior sources. Our interest lies in the theoretical understanding of the existence, finite time blow-up of solutions and their exponential decay using non-trivial adaptations of well-known techniques. First, we apply the conventional Faedo-Galerkin method with standard arguments of density on the regularity of initial conditions to establish two local existence theorems of weak solutions. Moreover, we detail the uniqueness result in some specific cases. In the second theme, we prove that any weak solution possessing negative initial energy has the latent blow-up in finite time. Finally, we obtain the so-called exponential decay estimates for the global solution under the construction of a suitable Lyapunov functional. In order to corroborate our theoretical decay, a numerical example is provided.