On the Cauchy problem for a higher-order $\mu$-Camassa-Holm equation

Abstract

In this paper, we study the Cauchy problem of a higher-order $\mu$-Camassa-Holm equation. We first establish the Green's function of $(\mu -{\partial }_x^2+{\partial }_x^4{)}^{-1}$ and local well-posedness for the equation in Sobolev spaces $H^s(\mathbb{S})$, $s>\frac{7}{2}$. Then we provide the global existence results for strong solutions and weak solutions. Moreover, we show that the solution map is non-uniformly continuous in $H^s(\mathbb{S})$, $s\ge 4$. Finally, we prove that the equation admits single peakon solutions.

27 pages. arXiv admin note: substantial text overlap with arXiv:1712.07996