Global and exponential attractors for a nonlinear porous elastic system with delay term

Abstract

This paper is concerned with the study on the existence of attractors for a nonlinear porous elastic system subjected to a delay-type damping in the volume fraction equation. The study will be performed, from the point of view of quasi-stability for infinite dimensional dynamical systems and from then on we will have the result of the existence of global and exponential attractors. 1. Introduction. In recent years the study of continuous models of deformable bodies has intensified, in particular we have the elastic solids with voids. Due to its great applicability as for example in soil mechanics, petroleum industry, materials sciences and biomechanics, porous solids now play a prominent role in scientific research (cf. [27]). Among the various theories dealing with porous material, we can find a linear theory proposed by Cowin and Nunziato [14, 42] which is a generalization of the elastic theory for materials with voids, considering besides the material elasticity property, the volume fraction of the voids in the material. In this theory, the bulk density ρ = ρ(x, t) is given by the product of matrix density of the material γ = γ(x, t) and the volume fraction ν = ν(x, t) ρ(x, t) = γ(x, t)ν(x, t).

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

“…Consequently, we conclude that A is a maximal dissipative operator. Now, we prove that the operator Γ defined in (22)…”

“…The result in [10] for system (1) was improved by Apalara to exponential stability in [18,19]. For more papers related to our paper, please see [20][21][22][23].…”

Exponential Stabilization of a Swelling Porous-Elastic System with Microtemperature Effect and Distributed Delay

“…Also, there are many works that have studied this type of problems, of which [11,[15][16][17][18][19][20][21][22][23][24].…”

“…Therefore, the use of Poincare's inequality for ω is justified. In addition, simple substitution shows that ðu, ϕ, θ, ω, YÞ satisfies system (23). Henceforth, we work with ω instead of ω but write ω for simplicity of notation.…”

Three types of weak pullback attractors for lattice pseudo-parabolic equations driven by locally Lipschitz noise

“…The large time dynamics in terms of random attractors of the stochastic classical diffusion equation was investigated in [12,15]. The reader is referred to [5,6] for the study of attractors of other interesting models.…”