Electron. J. Diff. Equ., Vol. 2015 (2015), No. 07, pp. 1-17.

Local well-posedness and blow-up of solutions for wave equations on shallow water with periodic depth

Lili Fan, Hongjun Gao

Abstract:
In this article, we consider a nonlinear evolution equation for surface waves in shallow water over periodic uneven bottom. The local well-posedness in Sobolev space $H^s(\mathbb{S})$ with $s>3/2$ is established by applying Kato's theory. Then a blow up criterion is determined in $H^s(\mathbb{S})$, $s>3/2$. Finally, some blow-up results are given for a simplified model.

Submitted September 24, 2014. Published January 5, 2015.
Math Subject Classifications: 35Q53, 35B30, 35G25, 35B44.
Key Words: Shallow water equation; variable depth; well-posedness; blow-up.

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Lili Fan
School of Mathematical Science and Jiangsu Key Laboratory for NSLSCS
Nanjing Normal University, Nanjing 210023, China
email: fanlily89@126.com
Hongjun Gao
School of Mathematical Science and Jiangsu Key Laboratory for NSLSCS
Nanjing Normal University, Nanjing 210023, China.
email: gaohj@njnu.edu.cn

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