Existence and stability of a solution blowing up on a sphere for an L 2-supercritical nonlinear Schrödinger equation

Duke Mathematical Journal

Volumn 134, Issue 2, 2006, Pages 199-258

  Raphaël, Pierre ^a  

^a UNIVERSITÉ PARIS SUD   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33748575882     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-06-13421-X     Document Type: Article

References (30)

1. H. BERESTYCKI and P.-L. LIONS, Nonlinear scalar field equations, I: Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313-345. MR 0695535
2. J. BERGH and J. LÖFSTRÖM, Interpolation Spaces: An Introduction, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976. MR 0482275
3. J. BOURGAIN, "Problems in Hamiltonian PDE's" in GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, special volume, part 1, Birkhäuser, Basel, 32-56. MR 1826248
4. J. BOURGAIN and W. WANG, Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), 197-215. MR 1655515
5. T. CAZENAVE, Semilinear Schrödinger Equations, Courant Lect. Notes Math. 10, Courant Inst. Math. Sci., New York Univ., New York, 2003. MR 2002047
6. T. CAZENAVE and P.-L. LIONS, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), 549-561. MR 0677997
7. T. CAZENAVE and F. B. WEISSLER, "Some remarks on the nonlinear Schrödinger equation in the critical case" in Nonlinear Semigroups, Partial Differential Equations and Attractors (Washington, D.C., 1987), Lecture Notes in Math. 1394, Springer, Berlin, 1989, 18-29. MR 1021011
8. M. CHRIST and A. KISELEV, Maximal functions associated to filiations, J. Funct. Anal. 179 (2001), 409-425. MR 1809116
9. 2 critical nonlinear Schrödinger equation, to appear in Phys. D., preprint, 2005.
10. Y. GIGA and R. V. KOHN, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989), 845-884. MR 1003437
11. J. GINIBRE and G. VELO, On a class of nonlinear Schrödinger equations, I: The Cauchy problem, general case, J. Funct. Anal. 32 (1979), 1-32. MR 0533218
12. _, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 309-327. MR 0801582
13. R. T. GLASSEY, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), 1794-1797. MR 0460850
14. T. KATO and G. PONCE, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891-907. MR 0951744
15. n, Arch. Rational Mech. Anal. 105 (1989), 243-266. MR 0969899
16. M. J. LANDMAN, G. C. PAPANICOLAOU, C. SULEM, and P.-L. SULEM, Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension, Phys. Rev. A (3) 38 (1988), 3837-3843. MR 0966356
17. P.-L. LIONS, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal. 49 (1982), 315-334. MR 0683027
18. F. MERLE, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J. 69 (1993), 427-454. MR 1203233
19. F. MERLE and P. RAPHAËL, Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation, Geom. Funct. Anal. 13 (2003), 591-642. MR 1995801
20. 2 critical nonlinear Schrödinger equation, Invent. Math. 156 (2004), 565-672. MR 2061329
21. _, Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. of Math. (2) 161 (2005), 157-222. MR 2150386
22. _, Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys. 253 (2005), 675-704. MR 2116733
23. 2 critical nonlinear Schrödinger equation, J. Amer. Math. Soc. 19 (2006), 37-90. MR 2169042
24. G. PERELMAN, On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré 2 (2001), 605-673. MR 1852922
25. P. RAPHAËL, Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation, Math. Ann. 331 (2005), 577-609. MR 2122541
26. R. S. STRICHARTZ, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705-714. MR 0512086
27. C. SULEM and P.-L. SULEM, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Appl. Math. Sci. 139, Springer, New York, 1999. MR 1696311
28. M. I. WEINSTEIN, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), 567-576. MR 0691044
29. _, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), 472-491. MR 0783974
30. V. E. ZAKHAROV and A. B. SHABAT, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34 (1972), 62-69. MR 0406174