  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.
