  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.
