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

Abstract

This paper is devoted to studying the Cauchy problem of the fourth-order defocusing, cubic equation iu_t + Δ²u = − |u|²u in critical Sobolev space. We first prove that the problem is locally well-posed in the critical Sobolev space

{\dot{H}}^{s_c}({\mathbb{R}}^N)

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, 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

H_p^s({\mathbb{R}}^N)

\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.

Keywords

Global well-posedness / fourth-order cubic equation / critical Sobolev space

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Peking University