The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients

Penghui Lv,Yuting Sun,Guoguang Lin

doi:10.1515/math-2022-0003

Abstract

Abstract In this paper, the stochastic asymptotic behavior of the nonautonomous stochastic higher-order Kirchhoff equation with variable coefficients is studied. By using the Galerkin method, the solution of this kind of equation is obtained, and stochastic dynamical system under this kind of equation is obtained; by using the uniform estimation, the existence of the family of D k {{\mathcal{D}}}_{k} -absorbing sets of the stochastic dynamical system Φ k {\Phi }_{k} is obtained, and the asymptotic compactness of Φ k {\Phi }_{k} is proved by the decomposition method. Finally, the D k {{\mathcal{D}}}_{k} -stochastic attractor family of the stochastic dynamical system Φ k {\Phi }_{k} in V m + k ( Ω ) × V k ( Ω ) {V}_{m+k}\left(\Omega )\times {V}_{k}\left(\Omega ) is obtained.

Highlights

Let Ω ⊂ RN be a bounded domain with smooth boundary
We study the asymptotic behavior of nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients on Ω:
⎩⎪u(x, τ) = uτ(x), ut(x, τ) = u1τ(x), x ∈ Ω, where Γ is the smooth boundary of Ω, v is the outer normal vector on the boundary Γ, m > 1, a(x) and b(x) are variable coefficient functions, f (x, t) ∈ Ll2oc(R, Vk(Ω)) is a time-dependent external force term, w is a one-dimensional bilateral standard Wiener process, h(x)∂w describes white noise, and g(x, u) is a nonlinear

Summary

Introduction

Let Ω ⊂ RN be a bounded domain with smooth boundary (i.e., the derivative of the function at the boundary exists and is continuous). We study the asymptotic behavior of nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients on Ω:. The random attractor is an important tool for studying the long-term asymptotic behavior of stochastic dynamical systems. Lin and Chen [34], Lin and Jin [35] have performed a detailed study on the long-term dynamical behavior of higher-order wave equations and proposed the concept of the family of attractors. Combined with the current research results, there are no relevant research results on the long-time dynamics of the nonautonomous stochastic higher-order Kirchhoff equation, and the asymptotic behavior of the higher-order Kirchhoff equation with variable coefficients has not been studied. This article will study the family of random attractors of nonautonomous random higher-order Kirchhoff equation with variable coefficients.

Preparatory knowledge

Uniform estimates of solutions

The existence of the family of random attractors