Global existence of solution to the Camassa-Holm equation

In this paper, we prove the result of global existence of solution to Camassa-Holm equation under the assumption that v0≔u0-u0″ has unique zero point and that v0 is negative at -∞. In this situation v0 clearly changes sign.Thus our result completes the picture described in [Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)XXVI (1998) 303], where the global existence of solutions is shown under the assumption that v0 does not change sign. We also give the sufficient and necessary condition of global existence of the solution when v0 is odd, with unique zero point.