On the Cauchy problem for a two-component higher order Camassa–Holm system

Abstract

In this paper, we focus on the well-posedness, blow-up phenomena, and continuity of the data-to-solution map of the Cauchy problem for a two-component higher order Camassa–Holm (CH) system. The local well-posedness is established in Besov spaces $B_{p,1}^{\frac{1}{p}}\times B_{p,1}^{2+\frac{1}{p}}$ with $1\le p<\infty$, which improves the local well-posedness result proved before in Tang and Liu [Z. Angew. Math. Phys. 66 (2015), 1559–1580], Ye and Yin [arXiv preprint arXiv:2109.00948 (2021)], Zhang and Li [Nonlinear Anal. Real World Appl. 35 (2017), 414–440], and Zhou [Math. Nachr. 291 (2018), no. 10, 1595–1619]. Next, we consider the continuity of the solution-to-data map, that is, the ill-posedness is derived in Besov space $B_{p,\infty }^{s-2}\times B_{p,\infty }^s$ with $1\le p\le \infty$ and $s>\max \{2+\frac{1}{p},\frac{5}{2}\}$. Then, the nonuniform continuous and Hölder continuous dependence on initial data for this system are also presented in Besov spaces $B_{p,r}^{s-2}\times B_{p,r}^s$ with $1\le p,r<\infty$ and $s>\max \left\{2+\frac{1}{p},\frac{5}{2}\right\}$. Finally, the precise blow-up criteria for the strong solutions of the two-component higher order CH system is determined in the lowest Sobolev space $H^{s-2}\times H^s$ with $s>\frac{5}{2}$, which improves the blow-up criteria result established before in He and Yin [Discrete Contin. Dyn. Syst. 37 (2016), no. 3, 1509–1537] and Zhou [Math. Nachr. 291 (2018), no. 10, 1595–1619].