On the instability of resonances in parallel‐plane waveguides under local perturbations of the boundary

We study the large‐time asymptotics for solutions u ( x , t ) of the wave equation with Dirichlet boundary data, generated by a time‐harmonic force distribution of frequency ω, in a class of domains with non‐compact boundaries and show that the results obtained in [11] for a special class of local perturbations of Ω 0 ≔ ℝ 2 × (0,1) can be extended to arbitrary smooth local perturbations Ω of Ω 0 . In particular, we prove that u is bounded as t → ∞ if Ω does not allow admissible standing waves of frequency ω in the sense of [8]. This implies in connection with [8]. Theorem 3.1 that the logarithmic resonances of the unperturbed domain Ω 0 at the frequencies ω = π k ( k = 1, 2,…) observed in [14] can be simultaneously removed by small perturbations of the boundary. As a main step of our analysis, the determination of admissible solutions of the boundary value problem Δ U + κ 2 U = − f in Ω, U = 0 on ∂Ω is reduced to a compact operator equation.