Time‐periodic solutions of wave equations on ℝ 1 and ℝ 3

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.