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The Cauchy problem for the Novikov equation |
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| Authors: | Wei Yan Yongsheng Li Yimin Zhang |
| Affiliation: | 1. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, Henan, People’s Republic of China 2. Department of Mathematics, South China University of Technology, Guangzhou, 510640, Guangdong, People’s Republic of China 3. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, 430071, Hubei, People’s Republic of China |
| 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. |
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