Time asymptotics for the polyharmonic wave equation in waveguides

Let Ω denote an unbounded domain in ℝ n having the form Ω=ℝ l × D with bounded cross‐section D ⊂ℝ n − l , and let m ∈ℕ be fixed. This article considers solutions u to the scalar wave equation ∂   2 tu ( t , x ) +(−Δ) m u ( t , x ) = f ( x )e −i ωt satisfying the homogeneous Dirichlet boundary condition. The asymptotic behaviour of u as t →∞ is investigated. Depending on the choice of f , ω and Ω, two cases occur: Either u shows resonance, which means that ∣ u ( t , x )∣→∞ as t →∞ for almost every x ∈ Ω, or u satisfies the principle of limiting amplitude. Furthermore, the resolvent of the spatial operators and the validity of the principle of limiting absorption are studied. Copyright © 2003 John Wiley & Sons, Ltd.