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

Abstract

In this paper, we mainly study persistence properties and infinitepropagation for the modified 2-component Camassa--Holm equation. Wefirst prove that persistence properties of the solution to theequation provided the initial potential satisfies a certain signcondition. Finally, we get the infinite propagation if the initialdatas satisfy certain compact conditions, while the solution tosystem (1.1) instantly loses compactly supported, the solution hasexponential decay as $|x|$ goes to infinity.