**
J. Rákosník (ed.), **

Function spaces, differential operators and nonlinear analysis.

Proceedings of the conference held in Paseky na Jizerou, September 3-9, 1995.

Mathematical Institute, Czech Academy of Sciences, and Prometheus Publishing House, Praha 1996

p. 125 - 140

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On systems of telegraph equations

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Juha Berkovits and Vesa Mustonen

Department of Mathematical Sciences, University of Oulu, Linnanmaa, P.O. Box 333, 90570 Oulu, Finland vesa.mustone@oulu.fi

**Abstract:** Let $\Omega=]0,2\pi[\times]0,\pi[$ and assume that $(t,x,s)\mapsto g(t,x,s)$ from $\Omega\times{\Bbb R}^n$ to ${\Bbb R}^n$ is a Carathéodory function which is $2\pi$-periodic in $t$. We shall discuss the nonresonance for the system of wave equations $$ \text{(WE)}\qquad \cases u_{tt}-u_{xx}-g(t,x,u)=h(t,x) \quad \text{in }\Omega

u(t,0)=u(t,\pi)=0,\quad t\in{\Bbb R}

u(t+2\pi,x)=u(t,x),\quad (t,x)\in{\Bbb R}_\times]0,\pi[, \endcases $$ i.e., the conditions for $g$ such that a solution $u\in L^2(\Omega,{\Bbb R}^n)$ exists for any given $h\in L^2(\Omega,{\Bbb R}^n)$.

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