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.