一类广义Camassa-Holm方程的无限传播速度与渐近行为

研究了一类广义的Camassa-Holm方程的Cauchy问题.首先，证明当初始值u₀（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u₀（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.

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.