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

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 $B_{p,r}^s$ with $s>1+\frac{1}{p}$ and $s=1+\frac{1}{p},r=1,p\in [1,\mathrm{\infty }).$ 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.

Keywords:

• A weakly dissipative Camassa-Holm equation
• local well-posedness
• global existence
• blow up
• Ill-posedness

Mathematics Subject Classifications:

• 35Q53
• 35B10
• 35B65
• 35C05

Acknowledgments

The authors thank the referee for valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China (Grant No. 12171493), the Macao Science and Technology Development Fund (Grant No. 0091/2018/A3), the Guangdong Special Support Program (Grant No. 8-2015).