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Instability of \(H^1\)-stable Periodic Peakons for the \(\mu \)-Camassa-Holm Equation

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Abstract

By applying the method of characteristics, we prove that the periodic peakons of the \(\mu \)-Camassa-Holm (\(\mu \)CH) equation are unstable under \(W^{1,\infty }\)-perturbations. Also, we show that small initial \(W^{1,\infty }\)-perturbations of the above periodic peakons can lead to the finite time blow-up in the nonlinear evolution of the \(\mu \)CH equation.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their helpful comments and suggestions. This research was partially supported by the Natural Science Foundation of Hunan Province (No.2018JJ2272, No.2021JJ30166), by the Scientific Research Fund of Hunan Provincial Education Department (No.21A0414) and the National Natural Science Foundation of China (No. 11971163).

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Authors and Affiliations

  1. Department of Mathematics and Physics, Hunan University of Arts and Science, Changde, 415000, People’s Republic of China

    Xijun Deng

  2. Department of Mathematics, Hunan First Normal University, Changsha, 410205, People’s Republic of China

    Aiyong Chen

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  1. Xijun Deng
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  2. Aiyong Chen
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Correspondence to Xijun Deng.

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Deng, X., Chen, A. Instability of \(H^1\)-stable Periodic Peakons for the \(\mu \)-Camassa-Holm Equation. J Dyn Diff Equat 36, 515–534 (2024). https://doi.org/10.1007/s10884-022-10165-y

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