Analytical properties for the fifth order Camassa-Holm (FOCH) model

This paper devotes to present analysis work on the fifth order Camassa-Holm (FOCH) model recently proposed in [28]. Firstly, we establish the local and global existence of the solution to the FOCH model. Secondly, the infinite propagation speed is obtained, in other words, when the initial data u0(x) has a compact support, the corresponding solution u(x, t) does not have a compact x-support any longer in its lifespan. Finally, we discuss the long time behavior of the support of momentum density with a compactly supported initial data.