Blow-up criteria for linearly damped nonlinear Schrödinger equations

We consider the Cauchy problem for linearly damped nonlinear Schrödinger equations {document}i∂ₜ u + Δ u + i a u = ± |u|^α u, (t,x) ∈ [0,∞) × RN,{document} where {document}$ a>0 ${document} and {document}α>0{document} . We prove the global existence and scattering for a sufficiently large damping parameter in the energy-critical case. We also prove the existence of finite time blow-up {document}H¹{document} solutions to the focusing problem in the mass-critical and mass-supercritical cases.