Global Self-similar Solutions of a Class of Nonlinear Schrödinger Equations

Global Self-similar Solutions of a Class of Nonlinear Schrödinger Equations

Yaojun YE

Source: Abstr. Appl. Anal. Volume 2008 (2008), 9 pages.

Abstract

For a certain range of the value $p$ in the nonlinear term ${|u|}^{p}u$, in this paper we mainly study the global existence and uniqueness of global self-similar solutions to the Cauchy problem for some nonlinear Schrödinger equations using the method of harmonic analysis.

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aaa/1220969158
Digital Object Identifier: doi:10.1155/2008/836124

back to Table of Contents

References

T. Cazenave and F. B. Weissler, ``The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$,'' Nonlinear Analysis. Theory, Methods & Applications, vol. 14, no. 10, pp. 807--836, 1990.

Mathematical Reviews (MathSciNet): MR1055532

Zentralblatt MATH: 0706.35127

J. Ginibre and G. Velo, ``On a class of nonlinear Schrödinger equations,'' Journal of Functional Analysis, vol. 32, pp. 1--71, 1979.

Mathematical Reviews (MathSciNet): MR533219

Digital Object Identifier: doi:10.1016/0022-1236(79)90077-6

J. Ginibre and G. Velo, ``Scattering theory in the energy space for a class of nonlinear Schrödinger equations,'' Journal de Mathématiques Pures et Appliquées, vol. 64, no. 4, pp. 363--401, 1985.

Mathematical Reviews (MathSciNet): MR839728

Zentralblatt MATH: 0535.35069

J. Ginibre and G. Velo, ``The global Cauchy problem for the nonlinear Schrödinger equation revisited,'' Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 2, no. 4, pp. 309--327, 1985.

Mathematical Reviews (MathSciNet): MR801582

Zentralblatt MATH: 0586.35042

M. Nakamura and T. Ozawa, ``Low energy scattering for nonlinear Schrödinger equations in fractional order Sobolev spaces,'' Reviews in Mathematical Physics, vol. 9, no. 3, pp. 397--410, 1997.

Mathematical Reviews (MathSciNet): MR1446653

Zentralblatt MATH: 0876.35080

Digital Object Identifier: doi:10.1142/S0129055X97000154

T. Cazenave and F. B. Weissler, ``Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations,'' Mathematische Zeitschrift, vol. 228, no. 1, pp. 83--120, 1998.

Mathematical Reviews (MathSciNet): MR1617975

Zentralblatt MATH: 0916.35109

Digital Object Identifier: doi:10.1007/PL00004606

F. Ribaud and A. Youssfi, ``Regular and self-similar solutions of nonlinear Schrödinger equations,'' Journal de Mathématiques Pures et Appliquées, vol. 77, no. 10, pp. 1065--1079, 1998.

Mathematical Reviews (MathSciNet): MR1661017

Zentralblatt MATH: 0928.35159

Digital Object Identifier: doi:10.1016/S0021-7824(99)80004-X

H. Pecher and W. von Wahl, ``Time dependent nonlinear Schrödinger equations,'' Manuscripta Mathematica, vol. 27, no. 2, pp. 125--157, 1979.

Mathematical Reviews (MathSciNet): MR526824

Zentralblatt MATH: 0399.35030

Digital Object Identifier: doi:10.1007/BF01299292

P. Sjögren and P. Sjölin, ``Local regularity of solutions to time-dependent Schrödinger equations with smooth potentials,'' Annales Academiae Scientiarum Fennicae. Series A I. Mathematica, vol. 16, no. 1, pp. 3--12, 1991.

Mathematical Reviews (MathSciNet): MR1127694

Zentralblatt MATH: 0713.35016

P. Sjölin, ``Regularity of solutions to nonlinear equations of Schrödinger type,'' Tohoku Mathematical Journal, vol. 45, no. 2, pp. 191--203, 1993.

Mathematical Reviews (MathSciNet): MR1215924

Digital Object Identifier: doi:10.2748/tmj/1178225916

Project Euclid: euclid.tmj/1178225916

Y. J. Ye, ``The global small solutions for a class of nonlinear Schrödinger equations,'' Acta Mathematicae Applicatae Sinica, vol. 29, no. 1, pp. 91--96, 2006.

Mathematical Reviews (MathSciNet): MR2227643

B. Guo and B. Wang, ``The global Cauchy problem and scattering of solutions for nonlinear Schrödinger equations in $H^s$,'' Differential and Integral Equations, vol. 15, no. 9, pp. 1073--1083, 2002.

Mathematical Reviews (MathSciNet): MR1919763

C. X. Miao, ``The global strong solution for Schrödinger equation of higher order,'' Acta Mathematicae Applicatae Sinica, vol. 19, no. 2, pp. 213--221, 1996, Chinese.

Mathematical Reviews (MathSciNet): MR1407564

Zentralblatt MATH: 0863.35031

C. X. Miao, Harmonic Analysis and Applications to Partial Differential Equations, Science Press, Beijing, China, 1999.

W. Littman, ``Fourier transforms of surface-carried measures and differentiability of surface averages,'' Bulletin of the American Mathematical Society, vol. 69, pp. 766--770, 1963.

Mathematical Reviews (MathSciNet): MR0155146

Zentralblatt MATH: 0143.34701

Digital Object Identifier: doi:10.1090/S0002-9904-1963-11025-3

Project Euclid: euclid.bams/1183525687

F. Ribaud and A. Youssfi, ``Self-similar solutions of the čommentPlease update the information of this reference, if possible. nonlinear wave equation,'' preprint, 2007.

previous :: next