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2013 Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space
Yi Wu
Anal. PDE 6(8): 1989-2002 (2013). DOI: 10.2140/apde.2013.6.1989

Abstract

In this paper, we prove that there exists some small ε>0 such that the derivative nonlinear Schrödinger equation (DNLS) is globally well-posed in the energy space, provided that the initial data u0H1() satisfies u0L2<2π+ε. This result shows us that there are no blow-up solutions whose masses slightly exceed 2π, even if their energies are negative. This phenomenon is much different from the behavior of the nonlinear Schrödinger equation with critical nonlinearity. The technique used is a variational argument together with the momentum conservation law. Further, for the DNLS on the half-line +, we show the blow-up for the solution with negative energy.

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Yi Wu. "Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space." Anal. PDE 6 (8) 1989 - 2002, 2013. https://doi.org/10.2140/apde.2013.6.1989

Information

Received: 9 March 2013; Revised: 11 September 2013; Accepted: 4 October 2013; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1298.35207
MathSciNet: MR3198590
Digital Object Identifier: 10.2140/apde.2013.6.1989

Subjects:
Primary: 35Q55
Secondary: 35A01 , 35B44

Keywords: Blow-up , global well-posedness , half-line , nonlinear Schrödinger equation with derivative

Rights: Copyright © 2013 Mathematical Sciences Publishers

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Vol.6 • No. 8 • 2013
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