Random Attractors for Stochastic Ginzburg-Landau Equation on Unbounded Domains

Abstract

We prove the existence of a pullback attractor in ${\mathbb{L}}^2$ $({ℝ}^n)$ for the stochastic Ginzburg-Landau equation with additive noise on the entire n-dimensional space ${ℝ}^n$. We show that the stochastic Ginzburg-Landau equation with additive noise can be recast as a random dynamical system. We demonstrate that the system possesses a unique $\mathcal{D}$-random attractor, for which the asymptotic compactness is established by the method of uniform estimates on the tails of its solutions.