一类广义Camassa-Holm方程的无限传播速度与渐近行为（英文）

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

研究了一类广义的Camassa-Holm方程的Cauchy问题.首先,证明当初始值u_0(x)具有紧支集的情况下,方程的解u(x,t)不再具有紧支集.因此,由u_0(x)表示的具有紧支集的初始扰动的传播速度是无限的.其次,当x趋于无穷时,证明了方程的解u(x,t)具有指数衰减.最后,研究了当初始值为指数或代数衰减时,方程的解在无穷远处的渐近行为.

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 uo（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（xt） to the generalized Camassa-Holm equation has an exponential decay as Ixl goes to infinity. Moreover, the asymptotic behaviors of the solution at infinity axe investigated as the initial data decays exponentially or algebraically.