Refinement of the $L^{2}$-decay estimate of solutions to nonlinear Schr\"odinger equations with attractive-dissipative nonlinearity

This paper is concerned with the $L^{2}$-decay estimate of solutions to nonlinear dissipative Schr\"odinger equations with power-type nonlinearity of the order $p$. It is known that the sign of the real part of the dissipation coefficient affects the long-time behavior of solutions, when neither size restriction on the initial data nor strong dissipative condition is imposed. In that case, if the sign is negative, then Gerelmaa, the first and third author [7] obtained the $L^{2}$-decay estimate under the restriction $p \le 1+1/d$. In this paper, we relax the restriction to $p \le 1+4/(3d)$ by refining an energy-type estimate. Furthermore, when $p < 1+ 4/(3d)$, using an iteration argument, the best available decay rate is established, as given by Hayashi, Li and Naumkin [11].