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The Cauchy problem for the Novikov equation

  • Wei Yan
  • Yongsheng Li
  • Yimin ZhangEmail author
Article

Abstract

In this paper we consider the Cauchy problem for the Novikov equation. We prove that the Cauchy problem for the Novikov equation is not locally well-posed in the Sobolev spaces \({H^s(\mathfrak{R})}\) with \({s < \frac{3}{2}}\) in the sense that its solutions do not depend uniformly continuously on the initial data. Since the Cauchy problem for the Novikov equation is locally well-posed in \({H^{s}(\mathfrak{R})}\) with s > 3/2 in the sense of Hadamard, our result implies that s =  3/2 is the critical Sobolev index for well-posedness. We also present two blow-up results of strong solution to the Cauchy problem for the Novikov equation in \({H^{s}(\mathfrak{R})}\) with s > 3/2.

Mathematics Subject Classification (2010)

Primary 35G25 35L05 Secondary 35R25 

Keywords

Cauchy problem Novikov equation Blow-up 

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Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceHenan Normal UniversityXinxiangPeople’s Republic of China
  2. 2.Department of MathematicsSouth China University of TechnologyGuangzhouPeople’s Republic of China
  3. 3.Wuhan Institute of Physics and MathematicsChinese Academy of SciencesWuhanPeople’s Republic of China

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