Well-posedness and global solutions to the higher order Camassa-Holm equations with fractional inertia operator in Besov space

Abstract

In this paper, we study well-posedness and the global solutions to the higher-order Camassa-Holm equations with fractional inertia operator in Besov space. When $a\in [\frac{1}{2},1),$ we prove the existence of the solutions in space $B_{p,1}^s(\mathbb{R})$ with $s\ge 1+\frac{1}{p}$ and $p<\frac{1}{a-\frac{1}{2}}$, the existence and uniqueness of the solutions in space $B_{p,1}^s(\mathbb{R})$ with $s\ge 1+2a-\min \{\frac{1}{p},\frac{1}{p^{\prime }}\},$ and the local well-posedness in space $B_{p,1}^s(\mathbb{R})$ with $s>1+2a-\min \{\frac{1}{p},\frac{1}{p^{\prime }}\}$. When $a>1,$ we obtain the existence of the solutions in space $B_{p,1}^s(\mathbb{R})$ with $s\ge a+\max \{\frac{1}{p},\frac{1}{2}\}$ and the local well-posedness in space $B_{p,1}^s(\mathbb{R})$ with $s\ge 1+a+\max \{\frac{1}{p},\frac{1}{2}\}$. Moreover, we obtain two results about the global solutions.

higher-order Camassa-Holm equations with fractional inertia operator; well-posedness; global solution