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

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

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  • 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.

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


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