Well-posedness And Blow-up Phenomena For The Camassa-Holm Equation In H^s,p?R?

The manuscript is devoted to the study of the well-posedness and blow-up phenom-ena for the Camassa—Holm equation in HS,P(R),which is open and unsolved problem for p?2.By the Kato's theory of semigroup,we first prove that the Cauchy problem of the Camassa—Holm equation is local well-posed in space Hs,p(R),s>1+1/p,p?(1,?).Second,we give the precise blow-up criteria,and show the blow-up phenomena of the equation.Finally,we prove that the equation has global solution when the initial data satisfy some conditions.