Bounded, almost-periodic, and periodic solutions to fully nonlinear telegraph equations. (English) Zbl 0762.35066

The author considers the nonlinear partial differential equations of hyperbolic type: \[ {\mathcal L}u+F(u_{xx},u_ x,u,u_{tt},u_{xt},u_ t)=f(x,t),\quad x\in(0,L),\quad t\in\mathbb{R}^ 1, \] where \({\mathcal L}u=u_{tt}+du_ t-au_{xx}\), \(a,d>0\), together with the boundary conditions \(u(0,t)=u(L,t)=0\).
The problem of existence of global in time solutions is studied. The existence theorem for the solutions belonging to the space of continuous functions is proved under some conditions on \(F\) when \(f(x,t)\) is sufficiently small. The existence theorems for periodic and almost- periodic solutions are proved when \(f(x,t)\) is a periodic or almost- periodic function of \(t\), respectively.


35L70 Second-order nonlinear hyperbolic equations
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