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

摘要

AbstractWe study the large‐time asymptotics for solutionsu(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 thatuis bounded ast→ ∞ 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 Ω0at 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= −fin Ω,U= 0 on ∂Ω is reduc