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Sharp ill-posedness for the generalized Camassa–Holm equation in Besov spaces

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Abstract

In this paper, we consider the Cauchy problem for the generalized Camassa–Holm equation that containing, as its members, three integrable equations: the Camassa–Holm equation, the Degasperis–Procesi equation and the Novikov equation. We present a new and unified method to prove the sharp ill-posedness for the generalized Camassa–Holm equation in \(B^s_{p,\infty }\) with \(s>\max \{1+1/p, 3/2\}\) and \(1\le p\le \infty \) in the sense that the solution map to this equation starting from \(u_0\) is discontinuous at \(t = 0\) in the metric of \(B^s_{p,\infty }\). Our result covers and improves the previous work given in Li et al. (J Differ Equ 306:403–417, 2022), solving an open problem left in Li et al. (2022).

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Acknowledgements

J. Li is supported by the National Natural Science Foundation of China (11801090 and 12161004) and Jiangxi Provincial Natural Science Foundation (20212BAB211004). Y. Yu is supported by the National Natural Science Foundation of China (12101011) and Natural Science Foundation of Anhui Province (1908085QA05). W. Zhu is supported by the Guangdong Basic and Applied Research Foundation (2021A1515111018).

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Authors and Affiliations

  1. School of Mathematics and Computer Sciences, Gannan Normal University, Ganzhou, 341000, China

    **lu Li

  2. School of Mathematics and Statistics, Anhui Normal University, Wuhu, 241002, China

    Yanghai Yu

  3. School of Mathematics and Big Data, Foshan University, Foshan, 528000, Guangdong, China

    Weipeng Zhu

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  1. **lu Li
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  2. Yanghai Yu
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Correspondence to Yanghai Yu.

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Li, J., Yu, Y. & Zhu, W. Sharp ill-posedness for the generalized Camassa–Holm equation in Besov spaces. J. Evol. Equ. 22, 29 (2022). https://doi.org/10.1007/s00028-022-00792-9

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  • DOI: https://doi.org/10.1007/s00028-022-00792-9

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