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May  2008, 7(3): 699-713. doi: 10.3934/cpaa.2008.7.699

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, Brazil
2. 

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

Received  November 2006 Revised  November 2007 Published  February 2008

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.
Keywords: Schrödinger-Debye system, Bourgain's method., well-posedness.
Mathematics Subject Classification: Primary: 35Q9.
Citation: Alexander Arbieto, Carlos Matheus. On the periodic Schrödinger-Debye equation. Communications on Pure & Applied Analysis, 2008, 7 (3) : 699-713. doi: 10.3934/cpaa.2008.7.699
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