Time asymptotics for the polyharmonic wave equation in waveguides

Time asymptotics for the polyharmonic wave equation in waveguides

Let Ω denote an unbounded domain in ℝn having the form Ω=ℝl×D with bounded cross‐section D⊂ℝn−l, and let m∈ℕ be fixed. This article considers solutions u to the scalar wave equation ∂ 2tu(t,x) +(−Δ)mu(t,x) = f(x)e−iωt satisfying the homogeneous Dirichlet boundary condition. The asymptotic behaviour...

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Veröffentlicht in:Mathematical methods in the applied sciences 2003-02, Vol.26 (3), p.193-212
1. Verfasser: Lesky, P. H.
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Exact sciences and technology
Mathematical analysis
Mathematics
Partial differential equations
resonances
Sciences and techniques of general use
wave equation
waveguide
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description Let Ω denote an unbounded domain in ℝn having the form Ω=ℝl×D with bounded cross‐section D⊂ℝn−l, and let m∈ℕ be fixed. This article considers solutions u to the scalar wave equation ∂ 2tu(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.
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subjects Exact sciences and technology
Mathematical analysis
Mathematics
Partial differential equations
resonances
Sciences and techniques of general use
wave equation
waveguide
title Time asymptotics for the polyharmonic wave equation in waveguides
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