Asymptotic behavior for a dissipative nonlinear Schrodinger equation

We consider the Schrodinger equation with nonlinear dissipation begin{equation*} i partial _t u +Delta u=lambda|u|^{alpha}u end{equation*} in ${mathbb R}^N $, $Ngeq1$, where $lambdain {mathbb C} $ with $Imlambda<0$. Assuming $frac {2} {N+2}<alpha<frac2N$, we give a precise description of the long-time behavior of the solutions (including decay rates in $L^2$ and $L^infty $, and asymptotic profile), for a class of arbitrarily large initial data.