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Asymptotic characterization of standing waves and of the static limit for a class of wave equations of higher order with a variable coefficient and time‐independent incitation
Author(s) -
Winter Matthias
Publication year - 1995
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.1670180205
Subject(s) - mathematics , eigenvalues and eigenvectors , limit (mathematics) , mathematical analysis , infinity , operator (biology) , order (exchange) , characterization (materials science) , zero (linguistics) , mathematical physics , combinatorics , physics , quantum mechanics , biochemistry , chemistry , linguistics , philosophy , finance , repressor , transcription factor , optics , economics , gene
We consider the equation (−1) m ∇ m ( p ∇ m u ) + ∂ t 2u = ƒ in ℝ n × (0, ∞) for arbitrary positive integers m and n and under the assumptions p − 1, ƒ ϵ C 0 ∞ (ℝ n ) and p > 0. Even if the differential operator (−1) m ∇ m ( p ∇ m u ) has no eigenvalues, the solution u ( x,t ) may increase as t → ∞ for 2 m ≥ n . For this case, we derive necessary and sufficient conditions for the convergence of u ( x,t ) as t → ∞. Furthermore, we characterize the functions occurring in these conditions as solutions of the homogeneous static equation (−1) m ∇ m ( p ∇ m u ) = 0, which satisfy appropriate asymptotic conditions at infinity. We also give an asymptotic characterization of the static limit.