Small data nonlinear wave equation numerology: The role of asymptotics

TL;DR

This paper introduces a new classification condition for small data nonlinear wave equations that better captures the effects of undifferentiated terms, leading to new well- and ill-posedness results and challenging existing conjectures.

Contribution

It proposes an alternative condition for semilinear wave equations that accounts for undifferentiated nonlinearities, improving classification accuracy.

Findings

Established global well-posedness for certain systems under the new condition.

Proved ill-posedness for some systems not critical by previous classifications.

Disproved the weak null conjecture with counterexamples satisfying the weak null condition.

Abstract

Systems of wave equations may fail to be globally well posed, even for small initial data. Attempts to classify systems into well and ill-posed categories work by identifying structural properties of the equations that can work as indicators of well-posedness. The most famous of these are the null and weak null conditions. As noted by Keir, related formulations may fail to properly capture the effect of undifferentiated terms in systems of wave equations. We show that this is because null conditions are good for categorising behaviour close to null infinity, but not at timelike infinity. In this paper, we propose an alternative condition for semilinear equations that work for undifferentiated non-linearities as well. We illustrate the strength of this new condition by proving global well and ill-posedness statements for some systems of equation that are not critical according to the our classification. Furthermore, we given two examples of systems satisfying the weak null condition with global ill-posedness due to undifferentiated terms, thereby disproving the weak null conjecture as stated in [DP18].