# American Institute of Mathematical Sciences

June  2018, 38(6): 3085-3097. doi: 10.3934/dcds.2018134

## 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  April 2018

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.

Citation: Qiaoyi Hu, Zhixin Wu, Yumei Sun. Liouville theorems for periodic two-component shallow water systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3085-3097. doi: 10.3934/dcds.2018134
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