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Existence of a countable set of periodic solutions of the problem of forced oscillations for a weakly nonlinear wave equation. (English. Russian original) Zbl 0683.35054

Math. USSR, Sb. 64, No. 2, 543-556 (1989); translation from Mat. Sb., Nov. Ser. 136(178), No. 4(8), 546-560 (1988).
Consider the boundary value problems of the semilinear wave equation \[ u_{tt}-u_{xx}=| u|^{p-2}+h(t,x),\quad 0<x<\pi \]
\[ u(t,0)=u(t,\pi)=0,\quad u(t+2\pi,x)=u(t,x) \] and \[ u_{tt}-u_{xx}=- | u|^{p-2}-h(t,x) \] where h(t,x) is a \(2\pi\)-periodic forced oscillation term. The solvability of the problem in the space of \(2\pi\)- periodic functions is proved.
Reviewer: J.Tian

MSC:

35L70 Second-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B10 Periodic solutions to PDEs
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