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Local and global well-posedness for the critical Schrödinger-Debye system


Authors: Adán J. Corcho, Filipe Oliveira and Jorge Drumond Silva
Journal: Proc. Amer. Math. Soc. 141 (2013), 3485-3499
MSC (2010): Primary 35Q55, 35Q60; Secondary 35B65
DOI: https://doi.org/10.1090/S0002-9939-2013-11612-6
Published electronically: June 18, 2013
MathSciNet review: 3080171
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Abstract: We establish local well-posedness results for the Initial Value Problem associated to the Schrödinger-Debye system in dimensions $ N=2, 3$ for data in $ H^s\times H^{\ell }$, with $ s$ and $ \ell $ satisfying $ \max \{0, s-1\} \le \ell \le \min \{2s, s+1\}$. In particular, these include the energy space $ H^1\times L^2$. Our results improve the previous ones obtained by B. Bidégaray, and by A. J. Corcho and F. Linares. Moreover, in the critical case ($ N=2$) and for initial data in $ H^1\times L^2$, we prove that solutions exist for all times, thus providing a negative answer to the open problem mentioned by G. Fibich and G. C. Papanicolau concerning the formation of singularities for these solutions.


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Additional Information

Adán J. Corcho
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro-UFRJ, Ilha do Fundão, 21945-970, Rio de Janeiro-RJ, Brazil
Email: adan@im.ufrj.br

Filipe Oliveira
Affiliation: Centro de Matemática e Aplicações, FCT-UNL, Monte da Caparica, Portugal
Email: fso@fct.unl.pt

Jorge Drumond Silva
Affiliation: Center for Mathematical Analysis, Geometry and Dynamical Systems, Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal
Email: jsilva@math.ist.utl.pt

DOI: https://doi.org/10.1090/S0002-9939-2013-11612-6
Keywords: Perturbed nonlinear Schr\"odinger equation, Cauchy problem, global well-posedness
Received by editor(s): February 14, 2011
Received by editor(s) in revised form: December 15, 2011
Published electronically: June 18, 2013
Additional Notes: The first author was supported by CAPES and CNPq (Edital Universal-482129/2009-3), Brazil
The second author was partially supported by FCT/Portugal through Financiamento Base 2008-ISFL-1-297
The third author was partially supported by the Center for Mathematical Analysis, Geometry and Dynamical Systems through the Fundação para a Ciência e Tecnologia (FCT/Portugal) program POCTI/FEDER
Communicated by: James E. Colliander
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.