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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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
Keywords:
Perturbed nonlinear Schrödinger 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.