Global conservative solution for a dissipative Camassa-Holm type equation with cubic and quartic nonlinearities

Abstract

This paper is devoted to the global conservative solutions of a dissipative Camassa-Holm type equation with cubic and quartic nonlinearities. We first transform the equation into an equivalent semilinear system by introducing a new set of variables. Using the standard ordinary differential equation theory, we then obtain the global solutions of the semilinear system. Returning to the original variables, we get the global conservative solution of the equation. Finally, we show that the peakon solutions of the equation still conserve in $H^1$.

Keywords:

• The Camassa-Holm type equation
• global conservative solutions
• the semilinear system
• Peakon solutions

Mathematics Subject Classifications:

• 35Q53
• 35B10
• 35C05

Acknowledgments

The authors thank the referees for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by National Natural Science Foundation of China (NNSFC) [grant numbers 12171493 and 11671407], FDCT [grant number 0091/2018/A3], Guangdong Special Support Program [grant number 8-2015] and the key project of NSF of Guangdong Province [grant number 2016A030311004].