Junior Analysis Seminar: Istvan Kadar (Cambridge)

Systems of wave equations may fail to have global solutions, even for small initial data. Attempts to classify systems into stable, unstable categories work by identifying structural properties of the equations that can work as indicators of stability 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 only good for categorising behaviour close to null infinity and propose an alternative condition for semilinear equations that work for undifferentiated non-linearities as well. Furthermore, we give an example of a system satisfying the weak null condition which forms singularities in finite time.