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ISSN 1088-6826(e) ISSN 0002-9939(p)

On the existence of infinite energy solutions for nonlinear Schrödinger equations

Author(s): Pablo Braz e Silva; Lucas C. F. Ferreira; Elder J. Villamizar-Roa
Journal: Proc. Amer. Math. Soc. 137 (2009), 1977-1987.
MSC (2000): Primary 35Q55, 35D05, 35B40
Posted: January 21, 2009
MathSciNet review: 2480279
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Abstract | References | Similar articles | Additional information

Abstract: We derive new results about existence and uniqueness of local and global solutions for the nonlinear Schrödinger equation, including self-similar solutions. Our analysis is performed in the framework of weak- $L^{p}$ spaces.

References:

1.: Bergh, J., Löfström, J., Interpolation Spaces. An Introduction, Springer, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275 (58:2349)
2.: Cazenave, T., Vega, L., Vilela, M. C., A note on the nonlinear Schrödinger equation in weak $L^{p}$ spaces, Commun. Contemp. Math. 3 (2001), no. 1, 153-162. MR 1820017 (2002g:35192)
3.: Cazenave, T., Weissler, F. B., Some remarks on the nonlinear Schrödinger equation in the critical case, Nonlinear Semigroups, Partial Differential Equations, and Attractors (Washington, DC, 1987), 18-29, Lecture Notes in Math., 1394, Springer, Berlin, 1989. MR 1021011 (91a:35149)
4.: Cazenave, T., Weissler, F. B., The Cauchy problem for the nonlinear Schrödinger equation in $H^{1}$ , Manuscripta Math. 61 (1988), no. 4, 477-494. MR 952091 (89j:35114)
5.: Cazenave, T., Weissler, F. B., The Cauchy problem for the critical nonlinear Schrödinger equation in $H^{s}$ , Nonlinear Anal. 14 (1990), no. 10, 807-836. MR 1055532 (91j:35252)
6.: Cazenave, T., Weissler, F. B., Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Math. Z. 228 (1998), no. 1, 83-120. MR 1617975 (99d:35149)
7.: Christ, M., Colliander, J., Tao, T., A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order, J. Funct. Anal. 254 (2008), no. 2, 368-395. MR 2376575
8.: Ferreira, L. C. F., Villamizar-Roa, E. J., Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations, Differential Integral Equations 19 (2006), no. 12, 1349-1370. MR 2279332 (2008c:35122)
9.: Kato, T., On nonlinear Schrödinger equations. II. $H\sp s$ -solutions and unconditional well-posedness, J. Anal. Math. 67 (1995), 281-306. MR 1383498 (98a:35124a)
10.: Kato, T., Nonlinear Schrödinger equations, Schrödinger Operators (Sønderborg, 1988) , 218-263, Lecture Notes in Phys., 345, Springer, Berlin, 1989. MR 1037322 (91d:35202)
11.: Ginibre, J., Velo, G., On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal. 32 (1979), no. 1, 1-32. MR 533218 (82c:35057)
12.: Ginibre, J., Velo, G., On the global Cauchy problem for some nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 309-323. MR 778977 (87a:35164)
13.: Ginibre, J., Velo, G., The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 4, 309-327 MR 801582 (87b:35150)
14.: Grünrock, A., Herr, S., Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal. 39 (2008), no. 6, 1890-1920. MR 2390318
15.: Planchon, F., On the Cauchy problem in Besov spaces for a non-linear Schrödinger equation, Commun. Contemp. Math. 2 (2000), no. 2, 243-254. MR 1759790 (2001e:35157)
16.: Stein, E. M., Weiss, G., Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, no. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972 (46:4102)
17.: Vargas, A., Vega, L., Global wellposedness for D non-linear Schrödinger equation for data with an infinite $L^{2}$ norm, J. Math. Pures Appl. (9) 80 (2001), no. 10, 1029-1044. MR 1876762 (2002j:35290)

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Additional Information:

Pablo Braz e Silva
Affiliation: Departamento de Matemática, Universidade Federal de Pernambuco, 50740-540, Recife, PE, Brazil
Email: pablo@dmat.ufpe.br

Lucas C. F. Ferreira
Affiliation: Departamento de Matemática, Universidade Federal de Pernambuco, 50740-540, Recife, PE, Brazil
Email: lcff@dmat.ufpe.br

Elder J. Villamizar-Roa
Affiliation: Escuela de Matemáticas, Universidad Industrial de Santander, A.A. 678, Bucaramanga, Colombia
Email: jvillami@uis.edu.co

DOI: 10.1090/S0002-9939-09-09773-1
PII: S 0002-9939(09)09773-1
Received by editor(s): December 12, 2007
Posted: January 21, 2009
Additional Notes: The first author was partly supported during this work by CAPES/MECD-DGU Brazil/Spain, grant No. 117/06.
Communicated by: Hart F. Smith
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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