## Differential and Integral Equations

- Differential Integral Equations
- Volume 24, Number 9/10 (2011), 801-828.

### Global existence for the cubic nonlinear Schrödinger equation in lower order Sobolev spaces

Nakao Hayashi and Pavel I. Naumkin

#### 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.

#### Article information

**Source**

Differential Integral Equations, Volume 24, Number 9/10 (2011), 801-828.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356012886

**Mathematical Reviews number (MathSciNet)**

MR2850366

**Zentralblatt MATH identifier**

1249.35306

**Subjects**

Primary: 35B40: Asymptotic behavior of solutions 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

#### Citation

Hayashi, Nakao; Naumkin, Pavel I. Global existence for the cubic nonlinear Schrödinger equation in lower order Sobolev spaces. Differential Integral Equations 24 (2011), no. 9/10, 801--828. https://projecteuclid.org/euclid.die/1356012886