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We consider boundary value problems for quasilinear first-orderone-dimensional hyperbolic systems in a strip. The boundary conditions aresupposed to be of a smoothing type, in the sense that the $L^2$-generalizedsolutions to the initial-boundary value problems become eventually $C^2$-smoothfor any initial $L^2$-data. We investigate small global classical solutions andobtain the existence and uniqueness result under the condition that theevolution family generated by the linearized problem has exponential dichotomyon R. We prove that the dichotomy survives under small perturbations in theleading coefficients of the hyperbolic system. Assuming that the coefficientsof the hyperbolic system are almost periodic, we prove that the boundedsolution is almost periodic also.

Classical Bounded and Almost Periodic Solutions to Quasilinear First-Order Hyperbolic Systems in a Strip