The Cauchy problem of a weakly dissipative shallow water equation

ABSTRACT

The local well-posedness for the Cauchy problem of a weakly dissipative shallow water equation is established using the Littlewood–Paley theory and a priori estimates of solutions to the transport equation. The $H^1(\mathbb{R})$ norm estimates, the blow-up mechanisms, and the global existence of solutions to the problem are investigated. The novelty is that the effects of the weakly dissipative coefficients and nonlinear index to guarantee the global existence of solutions are presented.

KEYWORDS:

• Local well-posedness
• weakly dissipative shallow water equation
• blow-up
• global solution

AMS SUBJECT CLASSIFICATIONS:

• 35G25
• 35L15

Acknowledgements

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

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by National Natural Science Foundation of P.R. China [grant number 11471263] and Scientific Research Foundation of North University of China [grant number 2017030] and [grant number 13011920].