Pullback measure attractors for non-autonomous stochastic FitzHugh-Nagumo system with distribution dependence on unbounded domains

TL;DR

The paper tackles the asymptotic dynamics of a non-autonomous, distribution-dependent stochastic FitzHugh–Nagumo system on the unbounded domain ${\mathbb{R}}^{n}$. It combines Banach-fixed-point well-posedness with a decomposition-based strategy for the $v$-component, along with tail-estimates and Vitali’s theorem to handle mean-field (McKean–Vlasov) effects and the lack of compact Sobolev embeddings. By constructing uniform energy and tail estimates and establishing ${\mathcal{D}}$-pullback asymptotic compactness in ${\mathcal{P}}_{4}( {\mathbb{L}}^{2}({\mathbb{R}}^{n}) )}$, it proves the existence and uniqueness of a ${\mathcal{D}}$-pullback measure attractor, described as the omega-limit of a pullback absorbing family. This work extends pullback attractor theory to distribution-dependent SPDEs on unbounded domains and provides a rigorous framework for the long-time behavior of mean-field FitzHugh–Nagumo dynamics under stochastic and non-autonomous forcing.

Abstract

This paper is primarily focused on the asymptotic dynamics of a non-autonomous stochastic FitzHugh-Nagumo system with distribution dependence, specifically on unbounded domains $\mathbb{R}^{n}$. Initially, we establish the well-posedness of solutions for the FitzHugh-Nagumo system with distribution dependence by utilizing the Banach fixed-point theorem. Subsequently, we demonstrate the existence and uniqueness of pullback measure attractors for this system through the application of splitting techniques, tail-end estimates and Vitali's theorem.

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