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Time asymptotics for the polyharmonic wave equation in waveguides
Author(s) -
Lesky P. H.
Publication year - 2003
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.351
Subject(s) - mathematics , resolvent , limiting , bounded function , mathematical analysis , wave equation , dirichlet boundary condition , domain (mathematical analysis) , scalar (mathematics) , amplitude , mathematical physics , boundary value problem , geometry , quantum mechanics , physics , mechanical engineering , engineering
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.