A note on the initial value problem for a higher‐order Camassa–Holm equation. Issue 9 (26th August 2022)

• Record Type: Journal Article
• Authors: Wang, Haiquan; Guo, Yu
• Is Part Of: Mathematische Nachrichten. Volume 295:Issue 9(2022)
• Page Start: 1783
• Page End: 1811
• Publication Date: 2022-08-26
• Subjects: a higher‐order Camassa–Holm equation -- Besov spaces -- non‐uniform dependence -- persistence property -- Sobolev spaces; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.202000016
• Languages: English
• ISSNs: 0025-584X

Abstract

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 ) $H^{s}(\mathbf {R})$ 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 B 2, r s ( T ) ( s > 9 / 2, 1 ≤ r ≤ ∞ ) $B^{s}_{2, r}(\mathbf {T})\, (s>9/2, 1\le r\le \infty )$ and B 2, 1 9 / 2 ( T ) $B^{9/2}_{2, 1}(\mathbf {T})$ are proved. Finally, the persistence property of solutions for this problem is studied.