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Higher-Order Continuity of Pullback Random Attractors for Random Quasilinear Equations with Nonlinear Colored Noise

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Abstract

For a nonautonomous random dynamical system, we introduce a concept of a pullback random bi-spatial attractor (PRBA). We prove an existence theorem of a PRBA, which includes its measurability, compactness and attraction in the regular space. We then establish the residual dense continuity of a family of PRBAs from a parameter space into the space of all compact subsets of the regular space equipped by Hausdorff metric. The abstract results are illustrated in the nonautonomous random quasilinear equation driven by nonlinear colored noise, where the size of noise belongs to \((0,\infty ]\) and the infinite size corresponds to the deterministic equation. The application results are the existence and residual dense continuity of PRBAs on \((0,\infty ]\) in both square and p-order Lebesgue spaces, where \(p>2\). The lower semi-continuity of attractors in the regular space seems to be a new subject even for an autonomous deterministic system.

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Funding

Y. Li was supported by National Natural Science Foundation of China grant 12271444. F. Wang was supported by China Scholarship Council (CSC No. 202106990035) and Postgraduate Research and Innovation Project of Chongqing grant CYB22109. T. Caraballo was supported by the Spanish Ministerio de Ciencia e Innovación (AEI) and FEDER under project PID2021-122991NB-C21, and Junta de Andalucia (Spain) under project P18-FR-4509.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Yangrong Li & Fengling Wang

  2. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, C/Tarfia s/n, 41012, Sevilla, Spain

    Tomás Caraballo

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  1. Yangrong Li
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  2. Fengling Wang
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Li, Y., Wang, F. & Caraballo, T. Higher-Order Continuity of Pullback Random Attractors for Random Quasilinear Equations with Nonlinear Colored Noise. J Dyn Diff Equat (2024). https://doi.org/10.1007/s10884-023-10333-8

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  • DOI: https://doi.org/10.1007/s10884-023-10333-8

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