On the periodic Schrödinger-Debye equation

Alexander Arbieto; Carlos Matheus

doi:10.3934/cpaa.2008.7.699

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2008, Volume 7, Issue 3: 699-713. Doi: 10.3934/cpaa.2008.7.699

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On the periodic Schrödinger-Debye equation

Alexander Arbieto^1, and
Carlos Matheus^2,

1.
IMPA - Instituto de Matemática Pura e Aplicada, Est. D. Castorina, Jardim Botânico, Rio de Janeiro, RJ 22460-320
2.
IMPA - Instituto de Matemática Pura e Aplicada, Est. D. Castorina, Jardim Botânico, Rio de Janeiro, RJ 22460-3

Received: November 2006

Revised: November 2007

Published: May 2008

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Abstract

We study local and global well-posedness of the initial value problem for the Schrödinger-Debye equation in the periodic case. More precisely, we prove local well-posedness for the periodic Schrödinger-Debye equation with subcritical nonlinearity in arbitrary dimensions. Moreover, we derive a new a priori estimate for the $H^1$ norm of solutions of the periodic Schrödinger-Debye equation. A novel phenomenon obtained as a by-product of this a priori estimate is the global well-posedness of the periodic Schrödinger-Debye equation in dimensions $1$ and $2$ without any smallness hypothesis of the $H^1$ norm of the initial data in the "focusing" case.

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Mathematics Subject Classification: Primary: 35Q99.

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