High Energy Decay Estimates for Waves in a Locally Perturbed Medium

The acoustic wave equation (∂2t + K)ψ = 0, where K = −c2ρ∇ρ−1∇, is studied. The notion of a "Mourre estimate in a neighbourhood of infinity" is introduced and is used to obtain a criterion for the rapid decay of high-energy solutions. The function c(x) is required to satisfy a non-resonance condition: there is a vector field v(x) with v(x) → x as |x| → ∞ such that 2v • (∇c)I < c(∇Tv + ∇vT). An example of such a vector field is v(x) = x, where this condition reduces to x • ∇c < c. The rapid decay of energy implies that there are no "geometric optics" resonances or resonances at infinity in the system.