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


DOI: 10.1007/s00208-004-0596-0

Cite this article as:
Raphael, P. Math. Ann. (2005) 331: 577. doi:10.1007/s00208-004-0596-0


We consider finite time blow up solutions to the critical nonlinear Schrödinger equation 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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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

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