Stability of the log-log bound for blow up solutions to the critical non linear Schr�dinger equation

Abstract

We consider finite time blow up solutions to the critical nonlinear Schrodinger equation Open image in new window with initial condition u0 ∈ H1. 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.