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 \({\mathbb {R}}^{1+3}\) ${\mathbb{R}}^{1+3}$ . 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.