On a generalized Camassa–Holm equation with the flow generated by velocity and its gradient

Abstract

In this paper, we consider a generalized Camassa–Holm equation with the flow generated by the vector field and its gradient. We first establish the local well-posedness of equation in the sense of Hadamard in both critical Besov spaces and supercritical Besov spaces. Then we gain a blow-up criterion. Under a sign condition we reach the sign-preserved property and a precise blow-up criterion. Applying this precise criterion we finally present two blow-up results and the precise blow-up rate for strong solutions to equation.

Keywords:

• A generalized Camassa–Holm equation
• Cauchy problem
• local well-posedness
• blow-up criterion
• blow-up
• blow-up rate

AMS Subject Classifications:

• 35G25
• 35L05
• 35Q53

Acknowledgements

We thank the referee for the valuable comments and suggestions.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by NNSFC [grant number 11271382]; RFDP [grant number 20120171110014]; MSTDF [grant number 098/2013/A3]; Guangdong Special Support Program [grant number 8-2015].