Blow-up criteria for linearly damped nonlinear Schr\"odinger equations

We consider the Cauchy problem for linearly damped nonlinear Schr\"odinger equations $$i\partial_t u + \Delta u + i a u= \pm |u|^\alpha u, \quad (t,x) \in [0,\infty) \times \mathbb{R}^N,$$ where $a>0$ and $\alpha>0$. 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 $H^1$ solutions to the focusing problem in the mass-critical and mass-supercritical cases.