Large time behavior of solutions to the critical dissipative nonlinear Schrödinger equation with large data

We consider the Cauchy problem for the dissipative nonlinear Schrödinger equation with a critical cubic nonlinearity in one space dimension. We show the global uniform bound of dissipative solutions in the Gevrey class and its L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-decay order without any restriction of the size of smooth initial data.