John's blow up examples and scattering solutions for semi-linear wave equations

TL;DR

This paper revisits Fritz John's classic semi-linear wave equation example, constructs global solutions from asymptotic data, and introduces a new blow-up result for finite energy solutions, clarifying the relation to John's theorem.

Contribution

It provides a detailed construction of global solutions from asymptotic data and introduces a novel blow-up result for finite energy solutions with sign conditions.

Findings

Constructed future global solutions from asymptotic data.

Established a new blow-up criterion for finite energy solutions.

Showed solutions blow up in the past when extended backwards in time.

Abstract

In light of recent work of the third author, we revisit a classic example given by Fritz John of a semi-linear wave equation which exhibits finite in time blow up for all compactly supported data. We present the construction of future global solutions from asymptotic data given in arXiv:2204.12870(2022) for this specific example, and clarify the relation of this result of Yu to John's theorem. Furthermore we present a novel blow up result for finite energy solutions satisfying a sign condition due to the first author, and invoke this result to show that the constructed backwards in time solutions blow up in the past.