The Cauchy problem for a family of generalized Camassa–Holm equations

Abstract

In this paper, we first use the energy method to establish the local well-posedness for a family of generalized Camassa–Holm equations in Sobolev spaces. Next, we give a precise blow-up criterion for the Cauchy problem on the family of equations. Then, we show the global existence of strong solutions to the family of equations under some certain sign conditions. Finally, we prove that the family of equations possesses some wave-breaking phenomena.

Keywords:

• a family of generalized Camassa–Holm equations
• local well-posedness
• global existence
• blow-up

AMS Subject Classifications:

• 35Q53
• 35A01
• 35B44
• 35B65

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by NNSFC [No. 11271382], RFDP [No. 20120171110014], the Macao Science and Technology Development Fund [No. 098/2013/A3], and the key project of Sun Yat-sen University.