Zero resonances in local perturbations of parallel‐plane waveguides

In a joint paper with K. Morgenröther [9] we have studied the propagation of scalar waves in domains of the type Ω = Ω o −B̄ with Ω o :=ℝ 2 × (0, 1), where B is a smooth bounded domain with B̄ ⊏ Ω o . In particular, we have shown that the solution of Neumann's initial and boundary value problem for the wave equation with time‐independent right‐hand side ƒ increases with a logarithmic rate as t → ∞ if the integral over ƒ does not vanish. The main purpose of the present paper is to extend this result to arbitrary smooth local perturbations of Ω o .