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On the existence of periodic solutions of a semilinear wave equation with a superlinear forcing term. (English) Zbl 0665.35050
This paper proves that there exists a positive integer T such that the semilinear wave equation \[ u_{tt}-u_{xx}+f(x,t,u)=0 \] has a \(2\pi\) /T-periodic solution if the forcing term satisfies the following conditions: (1) continuity; (2) periodicity: \(f(x,t+2\pi /T,u)=f(x,t,u)\), \(x\in [0,\pi]\), \(t,u\in R'\); (3) monotonicity, \(f(x,t,u_ 2)\geq f(x,t,u_ 1)\) if \(u_ 2\geq u_ 1\), for \(x\in [0,\pi]\), \(t\in R^ 1\); (4) growth condition.
Reviewer: J.H.Tian

35L70 Second-order nonlinear hyperbolic equations
35B10 Periodic solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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