Stochastic dynamic for an extensible beam equation with localized nonlinear damping and linear memory

Abdelmajid Ali Dafallah,Mohamed Y A Bakhet,Eshag Mohamed Ahmed,Fadlallah Mustafa Mosa

doi:10.30538/oms2020.0130

Abstract

In this paper, we concerned to prove the existence of a random attractor for the stochastic dynamical system generated by the extensible beam equation with localized non-linear damping and linear memory defined on bounded domain. First we investigate the existence and uniqueness of solutions, bounded absorbing set, then the asymptotic compactness. Longtime behavior of solutions is analyzed. In particular, in the non-autonomous case, the existence of a random attractor attractors for solutions is achieved.

Highlights

W e consider the following extensible beam equation with localized non-linear damping and linear memory on a bounded domain: ∞ m utt + ∆2u − k(0)(1 + |∇u|2dx)∆u −k (s)∆u(t − s)ds + a(x)g(ut) + f (u) = q(x, t) + κ ∑ hjWj(t), Ω j=1u = ∂u = 0, x ∈ ∂Γ, t ∈ R, ∂Γ u(τ, x) =u0(τ, x), ut(τ, x) = u1(τ, x), x ∈ Γ, τ ∈ R, (1)
In this paper, we concerned to prove the existence of a random attractor for the stochastic dynamical system generated by the extensible beam equation with localized non-linear damping and linear memory defined on bounded domain
In the deterministic case; that is, κ = 0 in (1), the asymptotic behavior of the solution for global attractors an extensible beam equation with localized nonlinear damping with memory has been studied in [5,17–19]

Summary

Introduction

W e consider the following extensible beam equation with localized non-linear damping and linear memory on a bounded domain:. (b) We need the following condition on q(x, t) ∈ Ł2loc(R, L2(Γ)), there exists a positive constant σ satisfy that (Q1) : −τ∞ eσr q(·, r) 2dr < ∞, ∀ r ∈ R,. In the deterministic case; that is, κ = 0 in (1), the asymptotic behavior of the solution for global attractors an extensible beam equation with localized nonlinear damping with memory has been studied in [5,17–19]. In [20], for the case of μ = 0 in (1), the authors investigated the existence of random attractor for the stochastic an extensible beam equation with localized nonlinear damping without memory.

Preliminaries

Existence and uniqueness of solution

Random absorbing set