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

Abstract

AbstractWe 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 Ω0 ≔ ℝ2 × (0,1) can be extended to arbitrary smooth local perturbations Ω of Ω0. 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 Ω0 at the frequencies ω = πk (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 ΔU + κ2U = − f in Ω, U = 0 on ∂Ω is reduced to a compact operator equation.