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.
Cite
@article{arxiv.2404.12878,
title = {John's blow up examples and scattering solutions for semi-linear wave equations},
author = {Louie Bernhardt and Volker Schlue and Dongxiao Yu},
journal= {arXiv preprint arXiv:2404.12878},
year = {2024}
}
Comments
Contributed article to MATRIX Annals, for the MATRIX workshop "Hyperbolic PDEs and Nonlinear Evolution Problems", held in Creswick, Australia, from September 17 -- 30, 2023