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Maximum principles and the method of upper and lower solutions for time-periodic problems of the telegraph equations. (English) Zbl 1108.35021

Summary: This paper deals with the existence of bounded time-periodic solutions for the nonlinear telegraph equation \[ u_{tt}-u_{xx}+cu_t= F(t,x,u),\quad (t,x)\in \mathbb{R}^2, \] where \(c>0\) is a constant, \(F\in C (\mathbb{R}^3,\mathbb{R})\) is \(2\pi\)-periodic in \(t\). We build a maximum principle for the time-periodic solutions of the corresponding linear telegraph equation. Using this maximum principle, we develop a method of the upper and lower solutions for the time-periodic problem of the nonlinear telegraph equation and obtain some existence and uniqueness results.

MSC:

35B50 Maximum principles in context of PDEs
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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