"Domain-of-dependence" Bounds and Time Decay of Solutions of the Wave Equation

Thomas G. Anderson, Oscar P. Bruno

Published 2020-10-18Version 1

This article introduces certain "domain-of-dependence" bounds on the solutions of the Dirichlet initial and boundary-value problem for the wave equation in the exterior of a bounded obstacle, and it uses them to study the associated solution decay problem. This approach yields estimates on the Neumann traces of the ensuing wave equation solutions (scattered fields) in a number of contexts, including in geometrical settings where the obstacle is "trapping." These results lead to concrete superalgebraically-fast solution decay estimates for a wide range of obstacles satisfying a certain resolvent-norm $q$-growth condition in the limit of large real frequencies. The results follow from use of Green functions and boundary integral equation representations in the frequency and time domains. In particular this theory establishes the first rapid decay estimates (a) For solutions of scattering in connected-trapping contexts, as well as (b) For scattering in contexts where periodic trapped orbits span the full volume of a physical cube; and, finally (c) That do not utilize the Lax-Phillips complex variables approach to decay via resolvent bounds on resonance-free regions in the complex plane.