A note on the initial value problem for a higher‐order Camassa–Holm equation. Issue 9 (26th August 2022) - Wang, Haiquan; Guo, Yu
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 H s ( R ) $H^{s}(R)$ 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 B 2, r s ( T ) ( s > 9 / 2, 1 ≤ r ≤ ∞ ) $B^{s}_{2, r}(T)\, (s>9/2, 1\le r\le ∞ )$ and B 2, 1 9 / 2 ( T ) $B^{9/2}_{2, 1}(T)$ are proved. Finally, the persistence property of solutions for this problem is studied.