The Cauchy problem for the generalized Camassa-Holm equation

Abstract

In this paper, we consider the Cauchy problem for the generalized Camassa-Holm equation. By using the littlewood-Paley decomposition and nonhomogeneous Besov spaces, we prove that the Cauchy problem for the generalized Camassa-Holm equation is locally well posed in the Besov space with and which generalizes the local well-posedness result in [1]. We also establish the ill-posedness result of the generalized Camassa-Holm equation and give a blow-up criterion.

Keywords:

• Cauchy problem
• generalized Camassa-Holm equation
• Besov spaces
• ill-posedness

AMS Subject Classifications:

• 35G25
• 35L05
• 35R25

Acknowledgments

This work is supported by NSFC under grant numbers 11171116, 11226185, 11101418 and by NNSFC-NSAF under grant number 10976026. The second author is supported in part by the Fundamental Research Funds for the Central Universities under the grant number 2012ZZ0072.