Time asymptotics for the polyharmonic wave equation in waveguides

Let Ω denote an unbounded domain in ?ⁿ having the form Ω=?^l×D with bounded cross‐section D??^n?l, and let m∈? be fixed. This article considers solutions u to the scalar wave equation ?u(t,x) +(?Δ)^mu(t,x) = f(x)e^?iωt 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.