Infinite propagation speed and asymptotic behavior for a generalized fifth-order Camassa–Holm equation

Abstract

In this paper we consider the Cauchy problem for a generalized fifth-order Camassa–Holm equation. The infinite propagation speed is proved in the following sense: the corresponding solution u(x, t) with compactly supported initial data u0(x) does not have compact support in its lifespan. Moreover, the asymptotic behaviors of the solution at infinity are considered when the initial data decays exponentially and algebraically, respectively.