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

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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

## Abstract.

We consider finite time blow up solutions to the critical nonlinear Schrödinger equation with initial condition *u*_{0} ∈ *H*^{1}. 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 *H*^{1} 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.