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Large time behavior of 3D functional Brinkman–Forchheimer equations with delay term

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Abstract

The relationship is studied here between the 3D incompressible Brinkman–Forchheimer problem with delay and its generalized steady state. First, with some restrictive condition on the delay term, the global well-posedness of 3D Brinkman–Forchheimer problem and its steady state problem are obtained by compactness method and Brouwer fixed point method respectively. Then the global \(\textbf{L}^{p}~ (2\le p<\infty )\) decay estimates are established for weak solution of non-autonomous Brinkman–Forchheimer equations with delay by using a retarded integral inequality. The global decay estimates can be proved for strong solution as well. Finally, the exponential stability property is investigated for weak solution of the 3D non-autonomous Brinkman–Forchheimer problem by a direct approach and also for the autonomous system by using a retarded integral inequality. Furthermore, the Razumikhin approach is utilized to achieve the asymptotic stability for strong solution of autonomous system under a relaxed restriction.

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Acknowledgements

The first author was partially supported by the Science and Technology Project of Beijing Municipal Education Commission (No. KM202210005011). The second author was partially supported by Key project of Henan Education Department (No. 22A110011). The last author was partially supported by NSFC (12171087).

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Authors and Affiliations

  1. Faculty of Science, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Rong Yang

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

    Xin-Guang Yang & Lu-Bin Cui

  3. Department of Computer Science and Technology, Dongguan University of Technology, Dongguan, People’s Republic of China

    Jinyun Yuan

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  1. Rong Yang
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  2. Xin-Guang Yang
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  3. Lu-Bin Cui
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Correspondence to Jinyun Yuan.

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Yang, R., Yang, XG., Cui, LB. et al. Large time behavior of 3D functional Brinkman–Forchheimer equations with delay term. Comp. Appl. Math. 43, 370 (2024). https://doi.org/10.1007/s40314-024-02893-2

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  • DOI: https://doi.org/10.1007/s40314-024-02893-2

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