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

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 <=infinity)$B<^>{s}_{2,r}(\mathbf {T})\, (s>9/2, 1\le r\le \infty )$ 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.