Space-time Gevrey smoothing effect for the dissipative nonlinear Schrödinger equations

We study the global Cauchy problem for the dissipative nonlinear Schrödinger equations in the setting of the fractional Sobolev space Hs,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^s,$$\end{document}0<s<min(n/2,1).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<s<\min (n/2,1).$$\end{document} In particular, we show the space-time Gevrey smoothing effect for global solutions to the dissipative nonlinear Scrödinger equations with data which belong to the exponential weighted Sobolev space with large norm. The proof of main theorem of this study is based on the a priori estimate for Hs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^s$$\end{document} solutions and a continuation method for analytic solutions has been introduced in Hoshino (J Dyn Differ Equ 4:2339–2351, 2019).