## Differential and Integral Equations

- Differential Integral Equations
- Volume 22, Number 1/2 (2009), 99-124.

### Global well-posedness and scattering for a class of nonlinear Schröodinger equations below the energy space

#### Abstract

We prove global well posedness and scattering for the nonlinear Schröodinger
equation with power-type nonlinearity $$ \begin{cases} i u_t +\Delta u = |u|^p
u, \quad \frac{4}{n} < p < \frac{4}{n-2},\\ u(0,x) = u_0(x)\in
H^s({\mathbb R}^n), \quad n\geq 3, \end{cases} $$ below the energy space, i.e.,
for $s<1$. In [15], J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T.
Tao established polynomial growth of the $H^s_x$-norm of the solution, and hence
global well posedness for initial data in $H^s_x$, provided $s$ is sufficiently
close to $1$. However, their bounds are insufficient to yield scattering. In
this paper, we use the *a priori* interaction Morawetz inequality to show
that scattering holds in $H^s({\mathbb R}^n)$ whenever $s$ is larger than some
value $0<s_0(n,p)<1$.

#### Article information

**Source**

Differential Integral Equations Volume 22, Number 1/2 (2009), 99-124.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356038556

**Mathematical Reviews number (MathSciNet)**

MR2483014

**Subjects**

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

#### Citation

Visan, Monica; Zhang, Xiaoyi. Global well-posedness and scattering for a class of nonlinear Schröodinger equations below the energy space. Differential Integral Equations 22 (2009), no. 1/2, 99--124. https://projecteuclid.org/euclid.die/1356038556.