Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Global existence results for nonlinear Schrödinger equations with quadratic potentials

Pages: 385 - 398, Volume 13, Issue 2, July 2005      doi:10.3934/dcds.2005.13.385

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Rémi Carles - MAB, UMR CNRS 5466 and Université Bordeaux 1, 351 cours de la Libération, F-33 405 Talence cedex, France (email)

Abstract: We prove that no finite time blow up can occur for nonlinear Schrödinger equations with quadratic potentials, provided that the potential has a sufficiently strong repulsive component. This is not obvious in general, since the energy associated to the linear equation is not positive. The proof relies essentially on two arguments: global in time Strichartz estimates, and a refined analysis of the linear equation, which makes it possible to control the nonlinear effects.

Keywords:  Nonlinear Schrödinger equation, Mehler's formula, Strichartz estimates, global well-posedness, scattering.
Mathematics Subject Classification:  Primary: 35Q55; Secondary: 35A05, 35B30, 35B35.

Received: June 2004;      Revised: February 2005;      Available Online: April 2005.