We establish global well-posedness and scattering for solutions to the defocusing mass-critical (pseudoconformal) nonlinear Schrödinger equation iut + ∆u = |u| 4/n u for large spherically symmetric L 2x (R n ) initial data in dimensions n ≥ 3. After using the reductions in [32] to reduce to eliminating blowup solutions which are almost periodic modulo scaling, we obtain a frequency-localized Morawetz estimate and exclude a mass evacuation scenario (somewhat analogously to [9], [23], [36]) in order to conclude the argument.

We consider the Cauchy problem for a family of semilinear defocusing Schrödinger equations with monomial nonlinearities in one space dimension. We establish global well-posedness and scattering. Our analysis is based on a four-particle interaction Morawetz estimate giving a priori L 8t,x spacetime control on solutions.2000 Mathematics Subject Classification. 35Q55.

We investigate the global well-posedness, scattering and blow up phenomena when the 3-D quintic nonlinear Schrödinger equation, which is energy-critical, is perturbed by a subcritical nonlinearity λ 1 |u| p u. We find when the quintic term is defocussing, then the solution is always global no matter what the sign of λ 1 is. Scattering will occur either when the perturbation is defocussing and 4 3 < p < 4 or when the mass of the solution is small enough and 4 3 p < 4. When the quintic term is focusing, we show the blow up for certain solutions.