Dynamical behaviors of stochastic local Swift-Hohenberg equation on unbounded domain

Abstract

Abstract In this paper, we first study the deterministic Swift-Hohenberg equation on a bounded domain. After obtaining some a priori estimates by the uniform Gronwall inequality, we prove the existence of an attractor by the Sobolev compact embeddings. Then, we consider the stochastic Swift-Hohenberg equation driven by additive noise on an unbounded domain and prove that the random dynamical system is asymptotically compact by uniform a priori estimates for the far-field values of the solution, which implies the existence of a random attractor for the random dynamical system associated with the stochastic Swift-Hohenberg equation.

Keywords

• stochastic Swift-Hohenberg equation
• random attractor
• additive noise
• asymptotically compact