  We study the time-asymptotic behavior of solutions of the Schr\"odinger
equation with nonlinear dissipation \begin{equation*}
  \partial _t u = i \Delta u + \lambda |u|^\alpha u \end{equation*} in
${\mathbb R}^N $, $N\geq1$, where $\lambda\in {\mathbb C}$, $\Re \lambda <0$
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 $\frac2N$.
