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Time-periodic solutions to a nonlinear wave equation with periodic or anti-periodic boundary conditions. (English) Zbl 1186.35112

Summary: This paper is concerned with the existence of time-periodic solutions to the nonlinear wave equation with \(x\)-dependent coefficients \(u(x)y_{tt} - (u(x)y_x)_x+au(x)y+|y|^{p-2}\), \(y=f(x, t)\) on \((0, \pi )\times \mathbb{R}\) under the periodic or anti-periodic boundary conditions \(y(0,t)=\pm y(\pi,t), y_x(0,t)=\pm y_x(\pi,t)\) and the time-periodic conditions \(y(x,t+T)=y(x, t), y_t(x, t+T)=y_t(x,t)\). Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. A main concept is the notion ‘weak solution’ to be given in §2. For \(T=2\pi/k\) \((k\in\mathbb{R}\)), we establish the existence of time-periodic solutions in the weak sense by investigating some important properties of the wave operator with \(x\)-dependent coefficients.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35Q35 PDEs in connection with fluid mechanics
35B10 Periodic solutions to PDEs
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