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 u(t) + u(Q)u(x) + partial derivative(x)(1 - partial derivative(2)(x))(-1) [2ku + Q(2)+3Q/2(Q+1)u(Q+1) + Q/2u(Q-1)u(x)(2)] = 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 H-s(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 B-2,1(3/2). Finally, we give a blow-up criterion. (C) 2014 Elsevier Inc. All rights reserved.