Some properties of solutions to the integrable Camassa–Holm type equation

In this paper, we study an integrable Camassa–Holm type equation. We proved that if the initial datum $u_0⁄\equiv 0$ is compactly supported in $[a,c]$; then the corresponding solution to the Camassa–Holm type equation has the following property: $u(x,t)=\left(\begin{array}{cc}0,& x>q(c,t);\\ l\left(t\right)e^x,& x<q(a,t).\end{array}\right)$Furthermore, $l\left(t\right)<0$ is a continuous non-vanishing function and strictly decreasing. Long time behavior for the support of momentum density is also studied.