Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation

Yuanyuan Ren; Yongsheng Li; Wei Yan

doi:10.3934/cpaa.2018027

Article Contents

2018, Volume 17, Issue 2: 487-504. Doi: 10.3934/cpaa.2018027

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Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation

1.
School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China
2.
College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, China

Received: December 31, 2016

Revised: August 31, 2017

Published: June 2018

This work is supported by NSFC under grant numbers 11571118, 11771127 and 11401180, and also by the Fundamental Research Funds for the Central Universities of China under the grant number 2017ZD094.

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Abstract

In this paper, we investigate the Cauchy problem for the fourth order nonlinear Schrödinger equation

$i \partial_{t}u+\partial_{x}^{4}u=u^{2},\ \ (t,x)∈[0,T]× \mathbb{R}.$

Zheng (Adv. Differential Equations, 16(2011), 467-486.) has proved that the problem is locally well-posed in $H^{s}(\mathbb{R})$ with $-\frac{7}{4} <s≤q 0.$ In this paper, we aim at extending Zheng's work to a lower regularity index. We prove that the equation is locally well-posed in $H^{s}(\mathbb{R})$ when $s≥q -2$ and ill-posed when $s < -2$ in the sense that the solution map is discontinuous for $s <-2$. The key ingredient used in this paper is Besov-type space introduced by Bejenaru and Tao (Journal of Functional Analysis, 233(2006), 228-259.).

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Mathematics Subject Classification: Primary:35Q53;Secondary:35G25, 46E35.

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References

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