Sharp Threshold of Global Existence and Instability of Standing Wave for a Davey-Stewartson System

This paper concerns the sharp threshold of blowup and global existence of the solution as well as the strong instability of standing wave \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi(t,x) = e^{i\omega t} u(x)$$\end{document} for the system:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\phi_{t}+\Delta \phi+a|\phi|^{p-1}\phi+b E_{1}(|\phi|^{2})\phi\,=\,0,\quad t\,\geq 0,\quad x\in {\bf R}^{N}, \quad \quad \quad ({\rm DS})$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a > 0, b > 0, 1 < p < \frac{N+2}{(N-2)^{+}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\,\in\,\{2,3\}$$\end{document}. Firstly, by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp threshold for global existence and blowup of the solution to the Cauchy problem for (DS) provided \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+\frac{4}{N}\,\leq p\, < \frac{N+2}{(N-2)^{+}}$$\end{document} . Secondly, by using the scaling argument, we show how small the initial data are for the global solutions to exist. Finally, we prove the strong instability of the standing waves with finite time blow up for any ω > 0 by combining the former results.