Removal of uniform continuity of the damping function for exponential energy decay (nonradial case)

Determine whether exponential energy decay holds for solutions of the damped wave equation w_tt(x,t) + γ(x) w_t(x,t) + Δ w(x,t) = 0 on ℝ^d when the damping γ: ℝ^d → [0,∞) is bounded but not assumed uniformly continuous, under the hypothesis that the superlevel set {x ∈ ℝ^d : γ(x) > ε} satisfies the geometric control condition (GCC), without any radial symmetry assumption. This asks to remove the uniform continuity requirement in the Burq–Joly result and establish exponential decay in the general (nonradial) setting.

Background

Burq and Joly proved exponential energy decay for the classical damped wave equation on ℝ^d under the GCC when the damping function γ is uniformly continuous, and asked whether the uniform continuity assumption can be dropped. The present paper obtains a high-frequency uncertainty principle for the Fourier–Bessel transform and uses it to derive decay rates for the fractional damped wave equation, answering the removal-of-continuity question positively in the radial case.

Despite this progress, the authors state that the nonradial version of the removal-of-continuity question remains unresolved. Establishing exponential decay without uniform continuity for general (nonradial) damping functions under GCC would close this gap and connect uncertainty principles to control-theoretic decay in full generality.