Dynamics of threshold solutions for energy critical NLW with inverse square potential

被引:0
作者
Kai Yang
Xiaoyi Zhang
机构
[1] SouthEast University,School of Mathematics
[2] University of Iowa,Department of Mathematics
来源
Mathematische Zeitschrift | 2022年 / 302卷
关键词
Inverse square potential; Ground state solution; Energy critical; NLW;
D O I
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学科分类号
摘要
We consider the focusing energy critical NLW with inverse square potential in dimensions d=3,4,5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d= 3, 4, 5$$\end{document}. Solutions on the energy surface of the ground state are characterized. We prove that solutions with kinetic energy less than that of the ground state must scatter to zero or belong to the stable/unstable manifold of the ground state. In the latter case they converge to the ground state exponentially in the energy space as t→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow \infty $$\end{document} or t→-∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow -\infty $$\end{document}. When the kinetic energy is greater than that of the ground state, we show that all solutions with finite mass blow up in finite time in both time directions in d=3,4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=3,4$$\end{document}. In d=5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=5$$\end{document}, a finite mass solution can either have finite lifespan or lie on the stable/unstable manifolds of the ground state. The proof relies on the detailed spectral analysis of the linearized operator, local invariant manifold theory, and a global Virial analysis.
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页码:353 / 389
页数:36
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