# American Institute of Mathematical Sciences

July  2013, 33(7): 3211-3223. doi: 10.3934/dcds.2013.33.3211

## Persistence properties and infinite propagation for the modified 2-component Camassa--Holm equation

 1 Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China 2 Institute of Applied Physics & Computational Math., Beijing 100088

Received  April 2012 Revised  July 2012 Published  January 2013

In this paper, we mainly study persistence properties and infinite propagation for the modified 2-component Camassa--Holm equation. We first prove that persistence properties of the solution to the equation provided the initial potential satisfies a certain sign condition. Finally, we get the infinite propagation if the initial datas satisfy certain compact conditions, while the solution to system (1.1) instantly loses compactly supported, the solution has exponential decay as $|x|$ goes to infinity.
Citation: Xinglong Wu, Boling Guo. Persistence properties and infinite propagation for the modified 2-component Camassa--Holm equation. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3211-3223. doi: 10.3934/dcds.2013.33.3211
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