Blow-up criteria for linearly damped nonlinear Schrödinger equations

We consider the Cauchy problem for linearly damped nonlinear Schrödinger equations $i\partial_t u + Δu + i a u= \pm |u|^αu, \quad (t,x) \in [0,\infty) \times \mathbb{R}^N,$ where $a>0$ and $α>0$. 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.

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