The Cauchy problem for the Novikov equation

Wei Yan1, Yongsheng Li2, Yimin Zhang3
1College of Mathematics and Information Science, Henan Normal University, Xinxiang, People’s Republic of China
2Department of Mathematics, South China University of Technology, Guangzhou, People’s Republic of China
3Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, People’s Republic of China

Tóm tắt

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.

Tài liệu tham khảo

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