Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation

Geometric and Functional Analysis

Volumn 13, Issue 3, 2003, Pages 591-642

  Merle, Frank ^a   Raphael, Pierre ^b  

^a INSTITUT UNIVERSITAIRE DE FRANCE   (France)

^b UNIVERSITÉ DE CERGY PONTOISE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0041350485     PISSN: 1016443X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00039-003-0424-9     Document Type: Article

References (35)

1. [BL] H. BERESTYCKI, P.-L. LIONS, Nonlinear scalar field equations. I Existence of a ground state, Arch. Rational Mech. Anal. 82:4 (1983), 313-345.
2. [Bo1] J. BOURGAIN, Global Solutions of Nonlinear Schrödinger Equations, American Mathematical Society Colloquium Publications, 46. American Mathematical Society, Providence, RI, 1999.
3. [Bo2] J. BOURGAIN, Harmonic analysis and nonlinear partial differential equations, Proceedings of the International Congress of Mathematicians (Zurich, 1994), Birkhäuser, Basel 1,2 (1995), 31-44.
4. [BoW] J. BOURGAIN, W. WANG, Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa C1. Sci. (4) 25:1-2 (1997), 197-215.
5. [CW] T. CAZENAVE, F. WEISSLER, Some remarks on the nonlinear Schrödinger equation in the critical case. Nonlinear semigroups, partial differential equations and attractors (Washington, DC, 1987), Springer Lecture Notes in Math. 1394 (1989), 18-29.
6. [FP] G. FIBICH, G. PAPANICOLAOU, A modulation method for self-focusing in the perturbed critical nonlinear Schrödinger equation, Phys. Lett. A 239:3 (1998), 167-173.
7. [GNN] B. GIDAS, W.M. NI, L. NIRENBERG, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243.
8. [GS] B. GIDAS, J. SPRUCK, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6:8 (1981), 883-901.
9. [GiT] D. GILBARG, N.S. TRUDINGER, Elliptic Partial Differential Equations of Second Order (Reprint of the 1998 edition), Classics in Mathematics, Springer-Verlag, Berlin, 2001.
10. [GinV] J. GINIBRE, G. VELO, On a class of nonlinear Schrödinger equations, I The Cauchy problem, general case, J. Funct. Anal. 32:1 (1979), 1-32.
11. [GlM] L. GLANGETAS, F. MERLE, Existence of self-similar blow-up solutions for Zakharov equation in dimension two. I, Comm. Math. Phys. 160:1 (1994), 173-215.
12. [K] T. KATO, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Theor. 46:1 (1987), 113-129.
13. n, Arch. Rational Mech. Anal. 105:3 (1989), 243-266.
14. [LPSS] M.J. LANDMAN, G.G. PAPANICOLAOU, C. SULEM, P.-L. SULEM, Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension, Phys. Rev. A (3) 38:8 (1988), 3837-3843.
15. [Li1] P.-L. LIONS, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaŕ Anal. Non Linéaire 1:2 (1984), 109-145.
16. [Li2] P.-L. LIONS, The concentration-compactness principle in the calculus of variations. The locally compact case: II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1:4 (1984), 223-283.
17. [MM1] Y. MARTEL, F. MERLE, A liouville theorem for the critical generalized Korteweg-de Vries equation, Journal de Math. Pures et Appliquees 79 (2000), 339-425.
18. [MM2] Y. MARTEL, F. MERLE, Stability of blow up profile and lower bounds for blow up rate for the critical generalized KdV equation, Annals of Math., to appear.
19. 2-critical generalized KdV equation, J. Amer. Math. Soc., to appear.
20. [MM4] Y. MARTEL, F. MERLE, Instability of solitons for the critical generalized Korteweg-de Vries equation, Geom. funct. anal. 11:1 (2001), 74-123.
21. [Mel] F. MERLE, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc. 14:3 (2001), 555-578.
22. [Me2] F. MERLE, Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations, Proceeding of the International Congress of Mathematicians (Berlin, 1998), Doc. Math. extra vol. III (1998), 57-66.
23. [Me3] F. MERLE, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J. 69:2 (1993), 427-454.
24. [Me4] F. MERLE, Lower bounds for the blowup rate of solutions of the Zakharov equation in dimension two, Comm. Pure Appl. Math. 49:8 (1996), 765-794.
25. [MeR] F. MERLE, P. RAPHAEL, Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, preprint.
26. [N] H. NAWA, Asymptotic and limiting profiles of blowup solutions of the nonlinear Schrödinger equation with critical power, Comm. Pure Appl. Math. 52:2 (1999), 193-270.
27. 1 solution for the nonlinear Schrödinger equation, J. Differential Equations 92:2 (1991), 317-330.
28. [PSSW] G.C. PAPANICOLAOU, C. SULEM, P.-L. SULEM, X.P. WANG, Singular solutions of the Zakharov equations for Langmuir turbulence, Phys. Fluids B 3:4 (1991), 969-980.
29. [Pe] G. PERELMAN, On the blow up phenomenon for the critical nonlinear Schrödinger equation in ID, Annale Henri Poincaré, to appear.
30. [RX] J. RUSSELL, P. XINGBIN, On an elliptic equation related to the blow up phenomenon in the nonlinear Schrödinger equation, Proc. Roy. Soc. Ed. 123 A (1993), 763-782.
31. [SS] C. SULEM, P.-L. SULEM, The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse, Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999.
32. [W1] M.I. WEINSTEIN, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure. Appl. Math. 39 (1986), 51-68.
33. [W2] M.I. WEINSTEIN, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), 472-491.
34. [W3] M.I. WEINSTEIN, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567-576.
35. [ZS] V.E. ZAKHAROV, A.B. SHABAT, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media, Sov. Phys. JETP 34 (1972), 62-69.