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arXiv · 2303.00887

Ill-posedness for a generalized Camassa-Holm equation with higher-order nonlinearity in the critical Besov space

Wei Deng·Min Li·Xing Wu·Weipeng Zhu
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Abstract

In this paper, we prove that the Cauchy problem for a generalized Camassa-Holm equation with higher-order nonlinearity is ill-posed in the critical Besov space $B^1_{\infty,1}(\R)$. It is shown in (J. Differ. Equ., 327:127-144,2022) that the Camassa-Holm equation is ill-posed in $B^1_{\infty,1}(\R)$, here we turn our attention to a higher-order nonlinear generalization of Camassa-Holm equation proposed by Hakkaev and Kirchev (Commun Partial Differ Equ 30:761-781,2005). With newly constructed initial data, we get the norm inflation in the critical space $B^1_{\infty,1}(\R)$ which leads to ill-posedness.

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Wei Deng, Min Li, Xing Wu, Weipeng Zhu. 2023-02-06. Ill-posedness for a generalized Camassa-Holm equation with higher-order nonlinearity in the critical Besov space. https://arxiv.org/abs/2303.00887

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