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Periodic solutions of nonlinear telegraph equations. (English) Zbl 0547.35077

Dynamical systems, Proc. int. Symp., Gainesville/Fla. 1976, 193-210 (1977).
[For the entire collection see Zbl 0532.00013.]
The author considers the existence of generalized periodic solutions of the telegraph equations \(\beta u_ t+u_{tt}-u_{xx}=g(u)+h(t,x),\) where \(\beta \neq 0\), g is a continuous function and h is Lebesgue integrable on \(I^ 2=[0,2\pi]\times [0,2\pi].\) After the estimation of the periodic solutions of the linear equations, he studies nonlinear equations. According to the behavior of \(u^{-1}g(u)\) as \(u\to\infty \), the problem is divided into nonresonance and resonance cases. It should be noted that the results are not restricted to small nonlinearities although g must satisfy a linear growth condition.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B10 Periodic solutions to PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)

Citations:

Zbl 0532.00013