Global Well-posedness for the Fourth-order Defocusing Cubic Equation with Initial Data Lying in a Critical Sobolev Space

This paper is devoted to study the Cauchy problem of the fourth-order defocusing, cubic equation iut + Delta 2u = - divided by u divided by 2u in critical Sobolev space. We first prove that the problem is local well-posed in the critical Sobolev space H(center dot)sc(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{H}<^>{s_{c}}(\mathbb{R}<^>{N})$$\end{document}, 3 <= N <= 7. Using the argument of Dodson in [Rev. Mat. Iberoam., 2022, 38(4): 1087-1100], we further prove that the problem is globally well-posed in some critical Sobolev space Hps(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{p}<^>{s}(\mathbb{R}<^>{N})$$\end{document} for N = 6 and N = 7 with radial initial data.