Time asymptotics for the polyharmonic wave equation in waveguides

✍ Scribed by P. H. Lesky

• Publisher: John Wiley and Sons
• Year: 2003
• Tongue: English
• Weight: 178 KB
• Volume: 26
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.351

✦ Synopsis

Abstract

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 ∂u(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.