The global weak solution for the shallow water wave model of moderate amplitude

Abstract

In this paper, we firstly establish the existence theorem of the global weak solutions of the Cauchy problem for the shallow water wave model of moderate amplitude, then following the idea in Xin and Zhang’s work (see Xin and Zhang, 2002), we prove the uniqueness of global weak solutions using the localization analysis in the transport equation theory. Finally, several travelling wave solutions are derived.

Keywords:

• Existence
• uniqueness
• weak solutions
• the model equation for shallow water of moderate amplitude
• travelling wave solutions

AMS Subject Classifications:

• 35G25
• 35C07
• 35D30
• 35D40

Acknowledgements

The authors thank the reviewers for their comments and suggestions.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11471263]; Sichuan ProvinceUniversity Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing [grant number 2013QZJ02], [grant number 2014QYJ03]; 15 Scientific Research Foundation of the Education Department of Sichuan province Project [grant number 16ZA0265]; SUSER [grant number 2014RC03].