On the Camassa-Holm equation

This talk is focused on recent progress of studies for the Camassa-Holm equation. First, we will give a brief review on the derivations, well-posedness for the strong solution, blow-up phenomenon and existence of the weak solutions. Then, infinite propagation speed for the Camassa-Holm equation will be proved in the following sense: the corresponding solution u(x, t) with compactly supported initial datum u0(x) does not has compact x−support any longer in its lifespan. Moreover, we show that for any fixed time t > 0 in its lifespan, the corresponding solution u(x, t) behaves as: u(x, t) = L(t)e−x for x À 1, and u(x, t) = l(t)e for x ¿ −1, with a strictly increasing function L(t) > 0 and a strictly decreasing function l(t) < 0 respectively. Finally, some interesting open problems will be listed. References: 1. A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Comm. Math. Phys. 271, 511-522 (2007).