The Cauchy problem for the generalized Camassa-Holm equation in Besov space

In this paper we consider the Cauchy problem for the generalized Camassa-Holm equation ut+uQux+∂x(1−∂x2)−1[2ku+Q2+3Q2(Q+1)uQ+1+Q2uQ−1ux2]=0 in Besov space. First, we prove that the solutions to the Cauchy problem for the generalized Camassa-Holm equation do not depend uniformly continuously on the initial data in Hs(R) with s<3/2 when k=0. Second, combining the real interpolations among inhomogeneous Besov spaces with Lemma 5.2.1 of [6] which is called Osgood Lemma (a substitute for Gronwall inequality), we show that the Cauchy problem for the generalized Camassa-Holm equation is locally well-posed in B2,13/2. Finally, we give a blow-up criterion.