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Time Periodic Solutions Of Nonlinear Wave Equations

We study time periodic and time almost periodic solutions of the nonlinear wave equation ∂ 2 t ϕ − ∆ϕ + f (ϕ) = 0. We prove that if ϕ is a 2π/ω-periodic solution then ω 2 ≤ f (0). In addition, we show that there is a strictly positive number α such that e α|x| ϕ ∈ L 2 (R n × (0, 2π/ω)). In the second chapter we derive integral identities satisfied by almost periodic solutions and use these to prove several nonexistence results.