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September/October 2011 Global existence for the cubic nonlinear Schrödinger equation in lower order Sobolev spaces
Nakao Hayashi, Pavel I. Naumkin
Differential Integral Equations 24(9/10): 801-828 (September/October 2011).

Abstract

We consider the Cauchy problem for the cubic nonlinear Schrödinger equation \begin{equation} \begin{cases} iu_{t}+\frac{1}{2}u_{xx}=u^{3},\text{ }x \in \mathbf{R},\text{ }t>0, \\ u(0,x)=u_{0}(x),\text{ }x\in \mathbf{R.} \end{cases} \label{A} \end{equation} The aim of the present paper is to consider problem (0.1) in low-order Sobolev spaces, when the initial data $u_{0}\in \mathbf{H}^{\alpha }\cap \mathbf{H}^{0,\alpha }$ with $\alpha >\frac{1}{2}.$ In our previous paper [7] we proved the global existence of solutions to (0.1) if the initial data $u_{0}\in \mathbf{H}^{2}\cap \mathbf{H}^{0,2}$. Also we find the large-time asymptotics of solutions.

Citation

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Nakao Hayashi. Pavel I. Naumkin. "Global existence for the cubic nonlinear Schrödinger equation in lower order Sobolev spaces." Differential Integral Equations 24 (9/10) 801 - 828, September/October 2011.

Information

Published: September/October 2011
First available in Project Euclid: 20 December 2012

zbMATH: 1249.35306
MathSciNet: MR2850366

Subjects:
Primary: 35B40, 35Q55

Rights: Copyright © 2011 Khayyam Publishing, Inc.

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Vol.24 • No. 9/10 • September/October 2011
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