On the Cauchy problem for a weakly dissipative Camassa-Holm equation in critical Besov spaces

TL;DR

This paper investigates the well-posedness, global existence, and blow-up phenomena of a weakly dissipative Camassa-Holm equation within critical Besov spaces, advancing understanding of its mathematical behavior.

Contribution

It establishes local and global well-posedness results, blow-up criteria, and ill-posedness in critical Besov spaces for the equation, filling gaps in the mathematical theory.

Findings

Local well-posedness in Besov spaces for s > 1 + 1/p

Global existence for small initial data

Blow-up criteria and ill-posedness results

Abstract

In this paper, we mainly consider the Cauchy problem of a weakly dissipative Camassa-Holm equation. We first establish the local well-posedness of equation in Besov spaces $B^{s}_{p,r}$ with $s>1+\frac 1 p$ and $s=1+\frac 1 p , r=1,p\in [1,\infty).$ Then, we prove the global existence for small data, and present two blow-up criteria. Finally, we get two blow-up results, which can be used in the proof of the ill-posedness in critical Besov spaces.