Uniform measure attractors of McKean-Vlasov stochastic reaction-diffusion equations on unbounded thin domain

TL;DR

This paper investigates the existence, uniqueness, and properties of uniform measure attractors for non-autonomous McKean-Vlasov stochastic reaction-diffusion equations on unbounded thin domains, addressing non-compactness issues.

Contribution

It establishes the existence, uniqueness, and upper semi-continuity of uniform measure attractors for these equations on unbounded thin domains, using uniform tail estimates.

Findings

Proved existence and uniqueness of uniform measure attractors.

Established asymptotic compactness via tail estimates.

Demonstrated upper semi-continuity as domains collapse.

Abstract

This article addresses the issue of uniform measure attractors for non-autonomous McKean-Vlasov stochastic reaction-diffusion equations defined on unbounded thin domains. Initially, the concept of uniform measure attractors is recalled, and thereafter, the existence and uniqueness of such attractors are demonstrated. Uniform tail estimates are employed to establish the asymptotic compactness of the processes, thereby overcoming the non-compactness issue inherent in the usual Sobolev embedding on unbounded thin domains. Finally, we demonstrate that the upper semi-continuity of uniform measure attractors defined on $(n + 1)$-dimensional unbounded thin domains collapsing into the space $\mathbb{R}^n$.