We consider equations of nonlinear Schrodinger type augmented by nonlinear damping terms. We show that nonlinear damping prevents finite time blow-up in several situations, which we describe. We also prove that the presence of a quadratic confinement in all spatial directions drives the solution of our model to zero for large time. In the case without external potential we prove that the solution may not go to zero for large time due to (non-trivial) scattering.

17 pages. The case of partial confinement was removed, due to a flaw in the proof