We prove that no finite time blow up can occur for nonlinear Schrödinger
equations with quadratic potentials, provided that the potential has a
sufficiently strong repulsive component. This is not obvious in
general, since the energy associated to the linear equation is not
positive. The proof relies essentially on two arguments: global
in time Strichartz estimates, and a refined analysis of the linear
equation, which makes it possible to
control the nonlinear effects.