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Time‐periodic solutions of wave equations on ℝ 1 and ℝ 3
Author(s) -
Smiley Michael W.
Publication year - 1988
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.1670100410
Subject(s) - mathematics , uniqueness , hilbert space , mathematical analysis , wave equation , spectrum (functional analysis) , operator (biology) , mathematical physics , space (punctuation) , linearization , exponential dichotomy , differential equation , quantum mechanics , physics , nonlinear system , biochemistry , chemistry , linguistics , philosophy , repressor , transcription factor , gene
The classical Paiey‐Wiener theorem and Hilbert space methods are used to show the existence of time‐periodic solutions of the wave equation w tt − w rr +λ w = h , 0 < r < + ∞, which are radially symmetric and have exponential decay as r → + ∞. This problem is obtained when considering a one‐dimensional or three‐dimensional problem, and should be thought of as a linearization of a semilinear problem in which the associated linear operator has point spectrum (− ∞, λ). When λ ⩽ 0 there is uniqueness, otherwise there is a non‐trivial finite‐dimensional null space. Estimates on w , w t , w r are obtained, which show that in either case there is a continuous correspondence h → w , where w is a uniquely characterized solution.