On Nonlinear Schrödinger-Type Equations with Nonlinear Damping

We consider equations of nonlinear Schrödinger-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 (nontrivial) scattering.