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
Full-text PDF Free Access
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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.
References
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- Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
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- Thierry Cazenave and Fred B. Weissler, Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Math. Z. 228 (1998), no. 1, 83–120. MR 1617975, DOI https://doi.org/10.1007/PL00004606
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- Adán J. Corcho and Carlos Matheus, Sharp bilinear estimates and well posedness for the 1-D Schrödinger-Debye system, Differential Integral Equations 22 (2009), no. 3-4, 357–391. MR 2492826
- Adán J. Corcho, Filipe Oliveira, and Jorge Drumond Silva, Local and global well-posedness for the critical Schrödinger-Debye system, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3485–3499. MR 3080171, DOI https://doi.org/10.1090/S0002-9939-2013-11612-6
- Lucas C. F. Ferreira, Existence and scattering theory for Boussinesq type equations with singular data, J. Differential Equations 250 (2011), no. 5, 2372–2388. MR 2756068, DOI https://doi.org/10.1016/j.jde.2010.11.013
- Tosio Kato and Hiroshi Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. Mat. Univ. Padova 32 (1962), 243–260. MR 142928
- Alan C. Newell and Jerome V. Moloney, Nonlinear optics, Advanced Topics in the Interdisciplinary Mathematical Sciences, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1992. MR 1163192
References
- Alexander Arbieto and Carlos Matheus, On the periodic Schrödinger–Debye equation, Commun. Pure Appl. Anal. 7 (2008), no. 3, 699–713. MR 2379450 (2009c:35425), DOI https://doi.org/10.3934/cpaa.2008.7.699
- Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press Inc., Boston, MA, 1988. MR 928802 (89e:46001)
- Pablo Braz e Silva, Lucas C. F. Ferreira, and Elder J. Villamizar-Roa, On the existence of infinite energy solutions for nonlinear Schrödinger equations, Proc. Amer. Math. Soc. 137 (2009), no. 6, 1977–1987. MR 2480279 (2010h:35365), DOI https://doi.org/10.1090/S0002-9939-09-09773-1
- Brigitte Bidégaray, On the Cauchy problem for some systems occurring in nonlinear optics, Adv. Differential Equations 3 (1998), no. 3, 473–496. MR 1751953 (2001c:78026)
- Brigitte Bidégaray, The Cauchy problem for Schrödinger–Debye equations, Math. Models Methods Appl. Sci. 10 (2000), no. 3, 307–315. MR 1753113 (2001a:78041), DOI https://doi.org/10.1142/S0218202500000185
- Thierry Cazenave and Fred B. Weissler, Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Math. Z. 228 (1998), no. 1, 83–120. MR 1617975 (99d:35149), DOI https://doi.org/10.1007/PL00004606
- A. J. Corcho and F. Linares, Well-posedness for the Schrödinger–Debye equation, Partial differential equations and inverse problems, Contemp. Math., vol. 362, Amer. Math. Soc., Providence, RI, 2004, pp. 113–131. MR 2091494 (2005k:35373), DOI https://doi.org/10.1090/conm/362/06608
- Adán J. Corcho and Carlos Matheus, Sharp bilinear estimates and well posedness for the 1-D Schrödinger–Debye system, Differential Integral Equations 22 (2009), no. 3-4, 357–391. MR 2492826 (2009m:35478)
- A. J. Corcho, F. Oliveira and J. D. Silva, Local and global well-posedness for the critical Schrödinger–Debye system, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3485–3499. MR 3080171
- Lucas C. F. Ferreira, Existence and scattering theory for Boussinesq type equations with singular data, J. Differential Equations 250 (2011), no. 5, 2372–2388. MR 2756068 (2012a:35205), DOI https://doi.org/10.1016/j.jde.2010.11.013
- Tosio Kato and Hiroshi Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. Mat. Univ. Padova 32 (1962), 243–260. MR 0142928 (26 \#495)
- Alan C. Newell and Jerome V. Moloney, Nonlinear optics, Advanced Topics in the Interdisciplinary Mathematical Sciences, Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1992. MR 1163192 (93i:78010)
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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
MR Author ID:
795159
Email:
lcff@ime.unicamp.br
Keywords:
Schrödinger-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
