Global Solutions with Asymptotic Self-Similar Behaviour for the Cubic Wave Equation

We construct a two-parameter family of explicit solutions to the cubic wave equation on R1+3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}<^>{1+3}$$\end{document}. Depending on the value of the parameters, these solutions either scatter to linear, blow-up in finite time, or exhibit a new type of threshold behaviour which we characterize precisely.