The properties of solutions to the dissipative 2-component Camassa–Holm system

Abstract

The local well-posedness for the Cauchy problem of the dissipative 2-component Camassa–Holm system is established by using the Littlewood–Paley theory and a priori estimates of solutions to the transport equation. The blow-up results, the exact blow-up rate, and the global existence of solutions to the system are analyzed. Moreover, the infinite propagation speed of solutions is investigated. The novelty in this paper is that the effects of the diffusion coefficient and dissipative coefficient to the solutions are given in an apparent form.

Keywords:

• local well-posedness
• dissipative 2-component Camassa–Holm system
• blow-up
• global solution
• infinite propagation speed

AMS Subject Classifications:

• 35G25
• 35L15
• 35Q58

Acknowledgements

We are grateful to the anonymous referees for a number of valuable comments and suggestions.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This paper is supported by National Natural Science Foundation of P.R. China [grant number 11471263] and Fundamental Research Funds for the Central Universities [grant number SWJTU12CX061] and [grant number SWJTU09ZT36].