  We study the Cauchy problem for Schrodinger equations with repulsive
quadratic potential and power-like nonlinearity. The local problem is
well-posed in the same space as that used when a confining harmonic potential
is involved. For a defocusing nonlinearity, it is globally well-posed, and a
scattering theory is available, with no long range effect for any superlinear
nonlinearity. When the nonlinearity is focusing, we prove that choosing the
harmonic potential sufficiently strong prevents blow-up in finite time. Thanks
to quadratic potentials, we provide a method to anticipate, delay, or prevent
wave collapse; this mechanism is explicit for critical nonlinearity.
