Mathematische Annalen

, Volume 331, Issue 3, pp 577–609

Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation

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Abstract.

We consider finite time blow up solutions to the critical nonlinear Schrödinger equation Open image in new window with initial condition u0H1. Existence of such solutions is known, but the complete blow up dynamic is not understood so far. For initial data with negative energy, finite time blow up with a universal sharp upper bound on the blow up rate corresponding to the so-called log-log law has been proved in [10], [11]. We focus in this paper onto the positive energy case where at least two blow up speeds are known to possibly occur. We establish the stability in energy space H1 of the log-log upper bound exhibited in the negative energy case, and a sharp lower bound on blow up rate in the other regime which corresponds to known explicit blow up solutions.

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References

  1. 1.
    Berestycki, H., Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82, 313–345 (1983)Google Scholar
  2. 2.
    Bourgain, J.: Global solutions of nonlinear Schrödinger equations. American Mathematical Society Colloquium Publications, 46. American Mathematical Society, Providence, RI, 1999Google Scholar
  3. 3.
    Bourgain, J., Wang, W.: Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(1–2), 197–215 (1997) (1998)Google Scholar
  4. 4.
    Cazenave, Th., Weissler, F.: Some remarks on the nonlinear Schrödinger equation in the critical case. Nonlinear semigroups, partial differential equations and attractors (Washington, DC, 1987), 18–29, Lecture Notes in Math., 1394, Springer, Berlin, 1989Google Scholar
  5. 5.
    Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. J. Funct. Anal. 32, 1–32 (1979)Google Scholar
  6. 6.
    Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Theor. 46, 113–129 (1987)Google Scholar
  7. 7.
    Kwong, M. K.: Uniqueness of positive solutions of Δu-u+up=0 in Rn. Arch. Rational Mech. Anal. 105, 243–266 (1989)CrossRefGoogle Scholar
  8. 8.
    Landman, M. J., Papanicolaou, G. C., Sulem, C., Sulem, P.-L.: Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension. Phys. Rev. A (3) 38, 3837–3843 (1988)Google Scholar
  9. 9.
    Merle, F.: Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J. 69, 427–454 (1993)CrossRefGoogle Scholar
  10. 10.
    Merle, F., Raphael, P.: Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation. To appear in Ann. of Math.Google Scholar
  11. 11.
    Merle, F., Raphael, P.: Sharp upper bound on the blow up rate for critical nonlinear Schrödinger equation. Geom. Funct. Anal. 13, 591–642 (2003)CrossRefGoogle Scholar
  12. 12.
    Merle, F., Raphael, P.: On universility of blow-up profile for L2 critical non linear Schrödinger equation. Invent. Math. 156, 565–572 (2004)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Perelman, G.: On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D. Ann. H. Poincaré 2, 605–673 (2001)Google Scholar
  14. 14.
    Weinstein, M.I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16, 472–491 (1985)Google Scholar
  15. 15.
    Weinstein, M.I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87 , 567–576 (1983)MATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de Cergy-PontoiseCergy-PontoiseFrance

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