"Bootstrap Domain of Dependence": Bounds and Time Decay of Solutions of the Wave Equation

TL;DR

This paper introduces a new 'bootstrap domain-of-dependence' concept that provides bounds and super-algebraic decay estimates for wave solutions scattered by obstacles, applicable even to trapping obstacles, using real-frequency analysis.

Contribution

It develops a novel approach to estimate wave decay and bounds based on boundary scattering history, without relying on complex resonance theory, applicable to trapping obstacles.

Findings

Establishes super-algebraic decay for wave solutions with arbitrary trapping obstacles.

Provides bounds on wave solutions using only short-time boundary data.

Demonstrates decay estimates without complex-variables scattering frameworks.

Abstract

This article introduces a novel "bootstrap domain-of-dependence" concept, according to which, for all time following a given illumination period of arbitrary duration, the wave field scattered by an obstacle is encoded in the history of boundary scattering events for a time-length equal to the diameter of the obstacle, measured in time units. Resulting solution bounds provide estimates on the solution values in terms of a short-time history record, and they establish super-algebraically fast decay (i.e., decay faster than any negative power of time) for a wide range of scattering obstacle--including certain types of "trapping" obstacles whose periodic trapped orbits span a set of positive volumetric measure, and for which no previous fast-decay theory was available. The results, which do not rely on consideration of the Lax-Phillips complex-variables scattering framework and associated resonance-free regions in the complex plane, utilize only real-valued frequencies, and follow from use of Green functions and boundary integral equation representations in the frequency and time domains, together with a certain $q$-growth condition on the frequency-domain operator resolvent.