Infinite propagation speed and asymptotic behavior for a generalized Camassa-Holm equation with cubic nonlinearity

This paper is devoted to the study of the infinite propagation speed and the asymptotic behavior for a generalized Camassa-Holm equation with cubic nonlinearity. First, we get the infinite propagation speed in the sense that the corresponding solution with compactly supported initial data does not have compact support any longer in its lifespan. Then, the asymptotic behavior of the solution at infinity is investigated. Especially, we prove that the solution decays algebraically with the same exponent as that of the initial data.