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

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.

Contributed article to MATRIX Annals, for the MATRIX workshop "Hyperbolic PDEs and Nonlinear Evolution Problems", held in Creswick, Australia, from September 17 -- 30, 2023