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\partial_tu+\Delta u+\mu|x|^{-b}|u|^{\alpha}u+iau = 0 $, $ \mu \in{\mathbb C} $, $ b>0 $, $ {a}>0 $, $ \alpha>0 $. We establish lower and upper bound estimates of the life-span. 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 precised decay rates for large damping. Some of the results are new even for $ b = 0. $ [ABSTRACT FROM AUTHOR]