Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains

This paper is concerned with pullback attractors of the stochastic p  -Laplace equation defined on the entire space Rⁿ${\mathbb{R}}^n$. We first establish the asymptotic compactness of the equation in L²(Rⁿ)$L^2({\mathbb{R}}^n)$ and then prove the existence and uniqueness of non-autonomous random attractors. This attractor is pathwise periodic if the non-autonomous deterministic forcing is time periodic. The difficulty of non-compactness of Sobolev embeddings on Rⁿ${\mathbb{R}}^n$ is overcome by the uniform smallness of solutions outside a bounded domain.     

Regularity of random attractors for fractional stochastic reaction–diffusion equations on Rn

We investigate the regularity of random attractors for the non-autonomous non-local fractional stochastic reaction–diffusion equations in $H^s({\mathbb{R}}^n)$ with $s\in (0,1)$. We prove the existence and uniqueness of the tempered random attractor that is compact in $H^s({\mathbb{R}}^n)$ and attracts all tempered random subsets of $L^2({\mathbb{R}}^n)$ with respect to the norm of $H^s({\mathbb{R}}^n)$. The main difficulty is to show the pullback asymptotic compactness of solutions in $H^s({\mathbb{R}}^n)$ due to the noncompactness of Sobolev embeddings on unbounded domains and the almost sure nondifferentiability of the sample paths of the Wiener process. We establish such compactness by the ideas of uniform tail-estimates and the spectral decomposition of solutions in bounded domains.     

Pullback attractors for the non-autonomous FitzHugh–Nagumo system on unbounded domains

The existence of a pullback attractor is established for the singularly perturbed FitzHugh–Nagumo system defined on the entire space Rⁿ${\mathbb{R}}^n$ when external terms are unbounded in a phase space. The pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far-field values of solutions. Although the limiting system has no global attractor, we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded, and hence do not blow up as a small parameter approaches zero.