Asymptotic behavior for a dissipative nonlinear Schr\"odinger equation

We consider the Schr\"odinger equation with nonlinear dissipation \begin{equation*} i \partial _t u +\Delta u=\lambda|u|^{\alpha}u \end{equation*} in ${\mathbb R}^N$, $N\geq1$, where $\lambda\in {\mathbb C}$ with $\Im\lambda<0$. 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.