Global solutions of nonlinear wave equations with large data

In this paper, we give a criterion on the Cauchy data for the semilinear wave equations satisfying the null condition in R+×R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb R}}^+\times {\mathbb {R}}^{3}$$\end{document} such that the data can be arbitrarily large, while the solution is still globally in time in the future.