Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case

Inventiones Mathematicae

Volumn 166, Issue 3, 2006, Pages 645-675

  Kenig, Carlos E ^a   Merle, Frank ^b  

^a UNIVERSITY OF CHICAGO   (United States)

^b UNIVERSITÉ DE CERGY PONTOISE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33750526878     PISSN: 00209910     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00222-006-0011-4     Document Type: Article

References (30)

1. Aubin. T.: Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl., IX. Ser. 55, 269-296 (1976)
2. Bahouri, H., Gérard, P.: High frequency approximation of solutions to critical nonlinear wave equations. Am. J. Math. 121, 131-175 (1999)
3. Berestycki, H., Cazenave, T.: Instabilité des états stationnaircs dans les équations de Schrödinger et de Klein-Gordon non linéaires. C. R. Acad. Sci., Paris. Sér. I, Math. 293, 489-492 (1981)
4. Bergh, J., Lofstrom, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Berlin, New York: Springer 1976
5. Bourgain, J.: Global well-posedness of defocusing critical nonlinear Schrödinger equation in the radial case. J, Am. Math. Soc. 12, 145-171 (1999)
6. Bourgain, J.: New global well-posedness results for nonlinear Schrödinger equations. AMS Colloquium Publications, 46, 1999
7. Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10. New York: New York University Courant Institute of Mathematical Sciences 2003
8. s. Nonlinear Anal., Theory Methods Appl. 14, 807-836 (1990)
9. 3. To appear in Ann. Math.
10. Gérard, P.: Description du défaut de compacité de l'injection de Sobolev. ESAIM Control Optim. Cale. Var. 3, 213-233 (1998)
11. Gerard, P., Meyer, Y., Oru, R: Inégalités de Sobolev précisées, Séminaire sur les Équations aux Dérivées Partielles. 1996-1997, Exp. No. IV, 11. École Polytech. 1997
12. Glassey, R.T.: On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys. 18, 1794-1797 (1977)
13. Grillakis, M.G.: On nonlinear Schrödinger equations. Commun. Partial Differ. Equations 23, 1827-1844 (2000)
14. Keel, M., Tao, T.: Endpoint Strichartz estimates. Am. J. Math. 120, 955-980 (1998)
15. Keraani, S.: On the defect of compactness for the Strichartz estimates of the Schrödinger equations. J. Differ. Equations 175, 353-392 (2001)
16. Keraani, S.: On the blow up phenomenon of the critical Schrödinger equation. J. Funct. Anal. 235, 171-192 (2006)
17. Merle, R: Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J. 69, 427-454 (1993)
18. Merle, P.: On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass. Comm. Pure Appl. Math. 45, 203-254 (1992)
19. Merle, R: Existence of blow-up solutions in the energy space for the critical generalized KdV equation. J. Am. Math. Soc. 14, 555-578 (2001)
20. 2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity. J. Differ. Equations 84, 205-214 (1990)
21. 2 solutions of the critical nonlinear Schrödinger equation in 2D. Int. Math. Res. Not. 8, 399-425 (1998)
22. 1 solution for the nonlinear Schrödinger equation. J. Differ. Equations 92, 317-330 (1991)
23. 2 supercritical nonlinear Schrödinger equation. Duke Math. J. 134(2), 199-258 (2006)
24. 1+4. To appear in Amer. J. Math. Preprint. http://arkiv.org/ abs/math.AP/0501462
25. Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353-372 (1976)
26. Tao, T.: Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data. New York J. Math. 11, 57-80 (2005) (electronic)
27. Tao, T., Visan, M.: Stability of energy-critical nonlinear Schrödinger equations in high dimensions. Electron. J. Differ. Equ. 118, 28 (2005) (electronic)
28. Visan, M.: The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions. Preprint, http://arkiv.org/abs/math.AP/0508298
29. Weinstein, M.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87, 567-576 (1982/83)
30. Zhang, J.: Sharp conditions of global existence for nonlinear Schrödinger and KleinGordon equations. Nonlinear Anal., Theory Methods Appl. 48, 191-207 (2002)