The Cauchy problem for a generalized Dullin–Gottwald–Holm equation in Besov spaces

Abstract

This paper deals with the Cauchy problem for a generalized Dullin–Gottwald–Holm (DGH) equation. The local well-posedness of the generalized DGH equation is obtained in Besov space with and by the transport equations theory and the classical Friedrichs regularization method. Moreover, the local well-posedness in critical case is also considered. Then a lower bound for the maximal existence time of the solution is derived. Finally, the analytic of the solution is given.

Keywords:

• local well-posedness
• blow-up
• Dullin–Gottwald–Holm equation

AMS Subject Classifications:

• 35G25
• 35L05
• 35Q50

Acknowledgements

The author is very grateful to the anonymous reviewers for their careful read and useful suggestions, which greatly improved the presentation of the paper. This work is supported in part by NSFC grant 11301573 and in part by the Program of Chongqing Innovation Team Project in University under Grant No. KJTD201308.