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arXiv · 2010.09002

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

Thomas G. Anderson·Oscar P. Bruno
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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.

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Thomas G. Anderson, Oscar P. Bruno. 2020-10-18. "Bootstrap Domain of Dependence": Bounds and Time Decay of Solutions of the Wave Equation. https://arxiv.org/abs/2010.09002

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