Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2008, Article ID 836124, 9 pages
http://dx.doi.org/10.1155/2008/836124
Research Article

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

Department of Mathematics and Information Science, Zhejiang University of Science and Technology, Hangzhou 310023, China

Received 11 November 2007; Accepted 4 March 2008

Academic Editor: Thomas Bartsch

Copyright © 2008 Yaojun YE. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. T. Cazenave and F. B. Weissler, “The Cauchy problem for the critical nonlinear Schrödinger equation in Hs,” Nonlinear Analysis. Theory, Methods & Applications, vol. 14, no. 10, pp. 807–836, 1990. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. J. Ginibre and G. Velo, “On a class of nonlinear Schrödinger equations,” Journal of Functional Analysis, vol. 32, pp. 1–71, 1979. View at Google Scholar
  3. 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. H. Pecher and W. von Wahl, “Time dependent nonlinear Schrödinger equations,” Manuscripta Mathematica, vol. 27, no. 2, pp. 125–157, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. P. Sjölin, “Regularity of solutions to nonlinear equations of Schrödinger type,” Tohoku Mathematical Journal, vol. 45, no. 2, pp. 191–203, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  11. 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. View at Google Scholar · View at MathSciNet
  12. B. Guo and B. Wang, “The global Cauchy problem and scattering of solutions for nonlinear Schrödinger equations in Hs,” Differential and Integral Equations, vol. 15, no. 9, pp. 1073–1083, 2002. View at Google Scholar · View at MathSciNet
  13. 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. C. X. Miao, Harmonic Analysis and Applications to Partial Differential Equations, Science Press, Beijing, China, 1999.
  15. 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. F. Ribaud and A. Youssfi, “Self-similar solutions of the nonlinear wave equation,” preprint, 2007.