Liouville theorems for periodic two-component shallow water systems

Qiaoyi Hu; Zhixin Wu; Yumei Sun

doi:10.3934/dcds.2018134

Article Contents

2018, Volume 38, Issue 6: 3085-3097. Doi: 10.3934/dcds.2018134

This issue Previous Article Interface stabilization of a parabolic-hyperbolic pde system with delay in the interaction Next Article Exit time asymptotics for small noise stochastic delay differential equations

Liouville theorems for periodic two-component shallow water systems

1.
Department of Mathematics, South China Agricultural University, 510642 Guangzhou, China
2.
Department of Mathematics, DePauw University, 46135 Greencastle, IN, USA
3.
Department of Mathematics, Shandong University of Science and Technology, 266590 Qingdao, Shandong, China

Received: October 2017

Revised: January 2018

Published: June 2018

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Abstract

We establish Liouville-type theorems for periodic two-component shallow water systems, including a two-component Camassa-Holm equation (2CH) and a two-component Degasperis-Procesi (2DP) equation. More presicely, we prove that the only global, strong, spatially periodic solutions to the equations, vanishing at some point $(t_0, x_0)$, are the identically zero solutions. Also, we derive new local-in-space blow-up criteria for the dispersive 2CH and 2DP.

Keywords:

Periodic two-component shallow water systems,
Liouville-type theorem,
global solution,
blow-up.

Mathematics Subject Classification: 35G25, 35L05.

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References

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