Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise

Yangrong Li; Shuang Yang; Guangqing Long

doi:10.3934/dcdsb.2021303

Article Contents

2022, Volume 27, Issue 10: 5977-6008. Doi: 10.3934/dcdsb.2021303

This issue Previous Article Global dynamics of a Huanglongbing model with a periodic latent period Next Article Blowup results for the fractional Schrödinger equation without gauge invariance

Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise

1.
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
2.
Department of Mathematics, Nanning Normal University, Nanning 530001, China

^* Corresponding author: liyr@swu.edu.cn (Yangrong Li)
^* Corresponding author: liyr@swu.edu.cn (Yangrong Li)

Received: June 30, 2021

Revised: October 31, 2021

Early access: December 2021

Published: October 2022

This work was supported by the Natural Science Foundation of China Grant 11571283

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Abstract

We study the continuity of a family of random attractors parameterized in a topological space (perhaps non-metrizable). Under suitable conditions, we prove that there is a residual dense subset $ \Lambda^* $ of the parameterized space such that the binary map $ (\lambda, s)\mapsto A_\lambda(\theta_s \omega) $ is continuous at all points of $ \Lambda^*\times \mathbb{R} $ with respect to the Hausdorff metric. The proofs are based on the generalizations of Baire residual Theorem (by Hoang et al. PAMS, 2015), Baire density Theorem and a convergence theorem of random dynamical systems from a complete metric space to the general topological space, and thus the abstract result, even restricted in the deterministic case, is stronger than those in literature. Finally, we establish the residual dense continuity and full upper semi-continuity of random attractors for the random fractional delayed FitzHugh-Nagumo equation driven by nonlinear Wong-Zakai noise, where the size of noise belongs to the parameterized space $ (0, \infty] $ and the infinity of noise means that the equation is deterministic.

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Mathematics Subject Classification: Primary: 37L55; Secondary: 35B40, 60H15.

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