Asymptotic behavior and life-span estimates for the damped inhomogeneous nonlinear Schrödinger equation

We are interested in the behavior of solutions to the damped inhomogeneous nonlinear Schrödinger equation $i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u+\mu \mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u+iau=0$, $\mu \in \mathbb{C}$, $b>0$, $a\in \mathbb{C}$ such that $\mathrm{\Re }\mathit{\text{e}}(a)\ge 0$, $\alpha >0$. We establish lower and upper bound estimates of the life-span. In particular for $a\ge 0$, we obtain explicit values $a_{\ast },a^{\ast }$ such that if $a<a_{\ast }$ then blow up occurs, while for $a>a^{\ast },$ global existence holds. Also, we prove scattering results with precise decay rates for large damping. Some of the results are new even for $b=0.$