A note on the initial value problem for a higher‐order Camassa–Holm equation

Haiquan Wang and Yu Guo

Mathematische Nachrichten, 2022, vol. 295, issue 9, 1783-1811

Abstract: Considered herein is the Cauchy problem for a higher‐order Camassa–Holm equation. Based on the local well‐posedness results for this problem, the non‐uniformly continuous dependence on initial data is established in Sobolev spaces Hs(R)$H^{s}(\mathbf {R})$ with s>9/2$s>9/2$ on the line by using the method of approximate solutions. In the periodic case, the non‐uniformly continuous dependence on initial data in Besov spaces B2,rs(T)(s>9/2,1≤r≤∞)$B^{s}_{2,r}(\mathbf {T})\, (s>9/2, 1\le r\le \infty )$ and B2,19/2(T)$B^{9/2}_{2,1}(\mathbf {T})$ are proved. Finally, the persistence property of solutions for this problem is studied.

Date: 2022
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