Asymptotic behavior for a Schrodinger equation with nonlinear subcritical dissipation

We study the time-asymptotic behavior of solutions of the Schrodinger equation with nonlinear dissipation begin{equation*} partial _t u = i Delta u + lambda |u|^alpha u end{equation*} in ${mathbb R}^N $, $Ngeq1$, where $lambdain {mathbb C}$, $Re lambda <0$ and $0<alpha<frac2N$. We give a precise description of the behavior of the solutions (including decay rates in $L^2$ and $L^infty $, and asymptotic profile), for a class of arbitrarily large initial data, under the additional assumption that $alpha $ is sufficiently close to $frac2N$.