×

Global self-similar solutions of a class of nonlinear Schrödinger equations. (English) Zbl 1180.35493

Summary: For a certain range of the value \(p\) in the nonlinear term \(|u|^{p}u\), we mainly study the global existence and uniqueness of global self-similar solutions to the Cauchy problem for some nonlinear Schrödinger equations by harmonic analysis.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35C05 Solutions to PDEs in closed form
35A25 Other special methods applied to PDEs
PDFBibTeX XMLCite
Full Text: DOI EuDML

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. · Zbl 0706.35127 · doi:10.1016/0362-546X(90)90023-A
[2] J. Ginibre and G. Velo, “On a class of nonlinear Schrödinger equations,” Journal of Functional Analysis, vol. 32, pp. 1-71, 1979. · Zbl 0396.35028 · doi:10.1016/0022-1236(79)90076-4
[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. · Zbl 0535.35069
[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. · Zbl 0586.35042
[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. · Zbl 0876.35080 · doi:10.1142/S0129055X97000154
[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. · Zbl 0916.35109 · doi:10.1007/PL00004606
[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. · Zbl 0928.35159 · doi:10.1016/S0021-7824(99)80004-X
[8] H. Pecher and W. von Wahl, “Time dependent nonlinear Schrödinger equations,” Manuscripta Mathematica, vol. 27, no. 2, pp. 125-157, 1979. · Zbl 0399.35030 · doi:10.1007/BF01299292
[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. · Zbl 0713.35016
[10] P. Sjölin, “Regularity of solutions to nonlinear equations of Schrödinger type,” Tohoku Mathematical Journal, vol. 45, no. 2, pp. 191-203, 1993. · Zbl 0777.35074 · doi:10.2748/tmj/1178225916
[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.
[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. · Zbl 1161.35502
[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. · Zbl 0863.35031
[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. · Zbl 0143.34701 · doi:10.1090/S0002-9904-1963-11025-3
[16] F. Ribaud and A. Youssfi, “Self-similar solutions of the nonlinear wave equation,” preprint, 2007. · Zbl 0933.35140
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.