Price's law on Minkowski space in the presence of an inverse square potential

Abstract

We consider the pointwise decay of solutions to wave-type equations in two model singular settings. Our main result is a form of Price's law for solutions of the massless Dirac-Coulomb system in (3+1)-dimensions. Using identical techniques, we prove a similar theorem for the wave equation on Minkowski space with an inverse square potential. One novel feature of these singular models is that solutions exhibit two different leading decay rates at timelike infinity in two regimes, distinguished by whether the spatial momentum along a curve which approaches timelike infinity is zero or non-zero. An important feature of our analysis is that it yields a precise description of solutions at the interface of these two regions which comprise the whole of timelike infinity.