Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Global existence for Schrödinger-Debye system for initial data with infinite $ {L}^{2}$-norm


Authors: Adán J. Corcho and Lucas C. F. Ferreira
Journal: Quart. Appl. Math. 73 (2015), 253-264
MSC (2010): Primary 35Q55, 35Q60; Secondary 35A01, 35A02, 35B40, 35B65
DOI: https://doi.org/10.1090/S0033-569X-2015-01371-4
Published electronically: March 16, 2015
MathSciNet review: 3357494
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Abstract: In this paper we study global-in-time existence for the Cauchy problem associated to the Schrödinger-Debye system for a class of initial data with infinite $ L^{2}$-norm, namely weak-$ L^{p}$ spaces. This model appears in nonlinear optics as a perturbation of the classical nonlinear Schrödinger equation (NLS). Our results exhibit differences between both models in that setting, e.g. the Debye perturbation imposes restrictions in the spatial dimension. We also analyze the asymptotic stability of the solutions.


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Additional Information

Adán J. Corcho
Affiliation: Universidade Federal do Rio de Janeiro, Instituto de Matemática/Departamento de Matemática, Av. Athos da Silveira Ramos 149, Centro de Tecnologia-Bloco C. Ilha do Fundão, CEP 21941-909, Rio de Janeiro-RJ, Brazil
Email: adan@im.ufrj.br

Lucas C. F. Ferreira
Affiliation: Universidade Estadual de Campinas, IMECC - Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas-SP, Brazil
Email: lcff@ime.unicamp.br

DOI: https://doi.org/10.1090/S0033-569X-2015-01371-4
Keywords: Schr\"odinger-Debye system, global existence, asymptotic stability, singular data
Received by editor(s): February 18, 2013
Received by editor(s) in revised form: May 15, 2013
Published electronically: March 16, 2015
Additional Notes: The first author was partially supported by CNPq (Grant: Edital Universal-482129/2009-3), Brazil
The second author was supported by Fapesp-SP/Brazil and CNPq/Brazil.
Article copyright: © Copyright 2015 Brown University

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