Random Attractors for Non-Autonomous Benjamin–Bona–Mahony Equations Driven by Laplace Multiplier Noise in High-Dimensional Unbounded Channel

Abstract

We investigate the long-time dynamics of a non-autonomous stochastic Benjamin–Bona–Mahony equation on a high-dimensional unbounded channel domain $\mathcal{O}\subseteq {\mathbb{R}}^d$ , driven by an operator-type Laplace-multiplier noise. We construct a non-autonomous random dynamical system on $H_0^1(\mathcal{O})$ and prove the existence of a compact pullback random attractor, which is cocycle-invariant and pullback attracts every tempered random set in $H_0^1(\mathcal{O})$ . The lack of compactness of Sobolev embeddings on unbounded domains is overcome by combining spectral decomposition on bounded truncations with uniform tail estimates outside large bounded sets.