Notes on global existence for the nonlinear Schrödinger equation involves derivative

In this paper, we consider the scattering for the nonlinear Schrödinger equation with small, smooth, and localized data. In particular, we prove that the solution of the quadratic nonlinear Schrödinger equation with nonlinear term |u|2 involving some derivatives in two dimension exists globally and scatters. It is worth to note that there exist blow-up solutions of these equations without derivatives. Moreover, for radial data, we prove that for the equation with p-order nonlinearity with derivatives, the similar results hold for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p \geqslant \tfrac{{2d + 3}} {{2d - 1}} $\end{document} and d ≥ 2, which is lower than the Strauss exponents.