Asymptotic behavior for a dissipative nonlinear Schrödinger equation

We consider the Schrödinger equation with nonlinear dissipation \begin{equation*} i \partial _t u +Δu=λ|u|^αu \end{equation*} in ${\mathbb R}^N$, $N\geq1$, where $λ\in {\mathbb C}$ with $\Imλ<0$. Assuming $\frac {2} {N+2}<α<\frac2N$, 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.

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