Critical local mass for rough damped waves: endpoint gaps, resolvent gauges, and pseudospectral transients

Abstract

We study damped waves on products $X=(\mathbb{R}/2π\mathbb{Z})\times Y$, with $Y$ compact and positive-dimensional, and with bounded measurable nonnegative damping. For transverse dampings $a=a(x)$, we prove that the lower interval mass $Θ_a(r)=\inf_x(2r)^{-1}\int_{x-r}^{x+r}a(y)dy$ is the complete high-frequency invariant at the exponential endpoint and at every critical slowly varying resolvent scale. At the endpoint, $\liminf_{r\downarrow0}Θ_a(r)>0$ is equivalent to exponential stability, and the imaginary-axis gap is comparable to this asymptotic mass, capped at the wave scale; the cap is certified by genuine overdamped real spectrum. At every critical scale-stable gauge $L$, the bound $Θ_a(r)\gtrsim L(1/r)^{-1}$ is equivalent to $\|(is-\mathcal{A})^{-1}\|\lesssim L(|s|)$, with necessity already visible from one transverse eigenfrequency per dyadic block. We show that this dictionary is genuinely critical: it fails at every sublinear power gauge, while saturated power dampings lie in a sharp two-exponent window. The critical profiles are realized by characteristic dampings of open dense sets of arbitrarily small measure. Under two-sided saturation, we determine the block pseudospectral portrait, including a sharp Lorentzian law on the elliptic side and confinement of all shallow spectrum, and prove the sharp transient plateau and frequency-localized decay laws. The sharp global decay envelope is reduced to a single resonance-inclusion problem and is proved whenever the confining boxes contain spectrum with dyadic density. The theory is stable under Dirichlet or Neumann cross-section boundaries and rough circle metrics, and it upgrades to a domination principle for arbitrary $L^\infty$ dampings dominating a transversally thick profile.