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is globally wellposed in time. More precisely, we obtain a unique solution u = uφ ∈ CH1([0,∞[) such that for all time, u(t) depends continuously on the data φ (in fact, the dependence is even real analytic here). Moreover, there is scattering for t→∞. The same statement holds for radial data φ ∈ H, s ≥ 1 and proves in particular global existence of classical solutions in the radially symmetric case. Also this issue was open. Thus this is the analogue for NLS of the result for the wave equation with quintic nonlinearity obtained by M. Struwe [Str] in the radial case (and by M. Grillakis [Gr], [S-S], in general). In the case of the wave equation, the proof is based on the following two different facts:

Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case