Random Attractor for Stochastic Wave Equation with Arbitrary Exponent and Additive Noise on $\mathbb{R}^n$

TL;DR

This paper proves the existence of a pullback random attractor for a stochastic wave equation with large nonlinearities and additive noise on unbounded domains, advancing understanding of its long-term behavior.

Contribution

It establishes the existence of a pullback random attractor for stochastic wave equations with arbitrary nonlinear exponent on unbounded domains, overcoming previous compactness challenges.

Findings

Existence of a pullback random attractor is proven.

Pullback asymptotic compactness is established for the cocycle.

Results apply to equations with arbitrary nonlinear exponent on $\\mathbb{R}^n$.

Abstract

Asymptotic random dynamics of weak solutions for a damped stochastic wave equation with the nonlinearity of arbitrarily large exponent and the additive noise on $\mathbb{R}^n$ is investigated. The existence of a pullback random attractor is proved in a parameter region with a breakthrough in proving the pullback asymptotic compactness of the cocycle with the quasi-trajectories defined on the integrable function space of arbitrary exponent and on the unbounded domain of arbitrary dimension.