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.
Categories: math.AP