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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:256

Article

Yan, Wei¹  Li, Yongsheng²  Zhang, Yimin³ 

[1] Henan Normal Univ, Coll Math & Informat Sci, Xinxiang 453007, Henan, Peoples R China

[2] S China Univ Technol, Dept Math, Guangzhou 510640, Guangdong, Peoples R China

[3] Chinese Acad Sci, Wuhan Inst Phys & Math, Wuhan 430071, Hubei, Peoples R China

关键词: Cauchy problem;    Generalized Camassa-Holm equation;    Besov spaces;    Osgood Lemma;    Blow-up criterion;   

DOI  :  10.1016/j.jde.2014.01.023

来源: Elsevier

【 摘 要 】

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.

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