## Differential and Integral Equations

- Differential Integral Equations
- Volume 18, Number 8 (2005), 947-960.

### The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension

#### Abstract

We show that the Cauchy problem for the quintic NLS on $\mathbf{R}$ is globally well posed in $H^s$ for $4/9<s\leq 1/2$. Since we work below the energy space we cannot immediately use the energy. Instead we use the "I-method" introduced by J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. This method allows us to define a modification of the energy functional that is "almost conserved" and thus can be used to iterate the local result.

#### Article information

**Source**

Differential Integral Equations, Volume 18, Number 8 (2005), 947-960.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060152

**Mathematical Reviews number (MathSciNet)**

MR2150447

**Zentralblatt MATH identifier**

1212.35459

**Subjects**

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx]

#### Citation

Tzirakis, Nikolaos. The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension. Differential Integral Equations 18 (2005), no. 8, 947--960. https://projecteuclid.org/euclid.die/1356060152