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Invariant Manifolds for a PDE-ODE Coupled System

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Abstract

The aim of this paper is to construct invariant manifolds for a coupled system, consisting of a parabolic equation and a second-order ordinary differential equation, set on \(\mathbb {T}^3\) and subject to periodic boundary conditions. Notably, the “spectral gap condition" does not hold for the system under consideration, leading to the use of the spatial averaging principle, together with the graph transform method. This approach facilitates the construction of the relevant invariant manifold, characterized by attributes such as Lipschitz continuity, local invariance, infinite dimensionality, and exponential tracking, thus mirroring the properties traditionally associated with a classical global manifold.

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Acknowledgements

The authors wish to thank the referees who significantly contributed to improve the initial version of the paper.

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Authors and Affiliations

  1. Department of Mathematics, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Xingjie Yan

  2. Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Kun Yin

  3. Department of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, People’s Republic of China

    Xin-Guang Yang & Alain Miranville

  4. Laboratoire de Mathématiques et Applications, Université de Poitiers, Site du Futuroscope, Téléport 2, 11 Boulevard Marie et Pierre Curie, Bâtiment H3, TSA 61125, 86073, Poitiers Cedex 9, France

    Alain Miranville

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This work was partially supported by the National Nature Science Foundation of China grant (11501560) and was partially supported by the Cultivation Fund of Henan Normal University (No. 2020PL17), Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003) and Key project of Henan Education Department (No. 22A110011).

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Yan, X., Yin, K., Yang, XG. et al. Invariant Manifolds for a PDE-ODE Coupled System. J Dyn Diff Equat (2024). https://doi.org/10.1007/s10884-024-10353-y

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