We are interested in the behavior of solutions to the damped inhomogeneous nonlinear Schr\"odinger equation $ i\partial_tu+\Delta u+\mu|x|^{ -b}|u|^{ \alpha}u+iau=0$, $\mu \in\mathbb{ C} $, $b>0$, $a \in \mathbb{ C}$ such that $\Re \textit{ e}(a) \geq 0$, $\alpha>0$. We establish lower and upper bound estimates of the life-span. In particular for $a\geq 0$, we obtain explicit values $a_*,\; a^*$ such that if $a<a_*$ then blow up occurs, while for $a>a^*,$ global existence holds. Also, we prove scattering results with precise decay rates for large damping. Some of the results are new even for $b=0.$
Categories: math.AP