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January/February 2009 Global well-posedness and scattering for a class of nonlinear Schröodinger equations below the energy space
Monica Visan, Xiaoyi Zhang
Differential Integral Equations 22(1/2): 99-124 (January/February 2009).

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 $0<s_0(n,p)<1$.

Citation

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Monica Visan. Xiaoyi Zhang. "Global well-posedness and scattering for a class of nonlinear Schröodinger equations below the energy space." Differential Integral Equations 22 (1/2) 99 - 124, January/February 2009.

Information

Published: January/February 2009
First available in Project Euclid: 20 December 2012

zbMATH: 1240.35525
MathSciNet: MR2483014

Subjects:
Primary: 35Q55

Rights: Copyright © 2009 Khayyam Publishing, Inc.

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Vol.22 • No. 1/2 • January/February 2009
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