Global existence results for nonlinear Schrödinger equations with quadratic potentials

Rémi Carles

doi:10.3934/dcds.2005.13.385

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2005, Volume 13, Issue 2: 385-398. Doi: 10.3934/dcds.2005.13.385

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Global existence results for nonlinear Schrödinger equations with quadratic potentials

Rémi Carles^1,

1.
MAB, UMR CNRS 5466 and Université Bordeaux 1, 351 cours de la Libération, F-33 405 Talence cedex

Received: May 31, 2004

Revised: January 31, 2005

Published: January 2005

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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.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35A05, 35B30, 35B35.

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