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

This paper is devoted to the Cauchy problem for a generalized Camassa-Holm equation. First, we prove that the solution u(x, t) to the generalized Camassa-Holm equation with compactly supported initial data u0(x) instantly loses compact support. In this sense, the localized disturbance represented by u0 propagates with an infinite speed. We further prove that the solution u(x, t) to the generalized Camassa-Holm equation has an exponential decay as |x| goes to infinity. Moreover, the asymptotic behaviors of the solution at infinity are investigated as the initial data decays exponentially or algebraically