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Stability Analysis of Random Attractors for Stochastic Modified Swift–Hohenberg Equations with Delays

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Abstract

A new type of random attractors is introduced to study dynamics of a stochastic modified Swift–Hohenberg equation with a general delay. A compact, pullback attracting and dividedly invariant set is called a backward attractor, while the criteria for its existence are established in terms of increasing dissipation and backward asymptotic compactness of a cocycle. If the delay term in the equation is Lipschitz continuous such that the Lipschitz bound and the external force are backward limitable, then we prove the existence of a backward attractor, which further leads to the longtime stability as well as the existence of a pullback attractor, where the pullback attractor and the backward attractor are shown to be random and dividedly random, respectively. Two examples of the delay term are provided to illustrate variable and distributed delays without restricting the upper bound of Lipschitz bounds.

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Acknowledgements

This work was supported by Shandong Provincial Natural Science Foundation (Grant No. ZR2022QA054), Doctoral Foundation of Heze University (Grant No. XY22BS29), the Spanish Ministerio de Ciencia e Innovación (MCI), Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PID2021-122991NB-C21, the China Postdoctoral Science Foundation (Grant No. 2023M741266)

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Authors and Affiliations

  1. School of Mathematics and Statistics, Heze University, Heze, 274015, People’s Republic of China

    Qiangheng Zhang

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

    Tomás Caraballo & Shuang Yang

  3. Department of Mathematics, Wenzhou University, Wenzhou, 325035, Zhejiang Province, People’s Republic of China

    Tomás Caraballo

  4. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Shuang Yang

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  1. Qiangheng Zhang
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  2. Tomás Caraballo
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The three authors contributed equaly in all the task concerning the preparation of this paper.

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Correspondence to Tomás Caraballo.

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Zhang, Q., Caraballo, T. & Yang, S. Stability Analysis of Random Attractors for Stochastic Modified Swift–Hohenberg Equations with Delays. J Dyn Diff Equat (2024). https://doi.org/10.1007/s10884-024-10348-9

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  • DOI: https://doi.org/10.1007/s10884-024-10348-9

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