Blow-up criteria for linearly damped nonlinear Schrödinger equations

Abstract

We consider the Cauchy problem for linearly damped nonlinear Schrodinger equations \begin{document}$ i\partial_t u + \Delta u + i a u = \pm |u|^\alpha u, \quad (t,x) \in [0,\infty) \times \mathbb R^N, $\end{document} where \begin{document}$ a>0 $\end{document} and \begin{document}$ \alpha>0 $\end{document} . We prove the global existence and scattering for a sufficiently large damping parameter in the energy-critical case. We also prove the existence of finite time blow-up \begin{document}$ H^1 $\end{document} solutions to the focusing problem in the mass-critical and mass-supercritical cases.