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

Abstract

We consider boundary value problems for quasilinear first-order one-dimensional hyperbolic systems in a strip. The boundary conditions are supposed to be of a smoothing type, in the sense that the $L^2$-generalized solutions to the initial-boundary value problems become eventually $C^2$-smooth for any initial $L^2$-data. We investigate small global classical solutions and obtain the existence and uniqueness result under the condition that the evolution family generated by the linearized problem has exponential dichotomy on R. We prove that the dichotomy survives under small perturbations in the leading coefficients of the hyperbolic system. Assuming that the coefficients of the hyperbolic system are almost periodic, we prove that the bounded solution is almost periodic also.

arXiv commentCorrected typos

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Cited by in corpus (3)

• Bounded and Almost Periodic Solvability of Nonautonomous Quasilinear Hyperbolic Systems
• On the bounded smooth solutions to exponentially stable linear nonautonomous hyperbolic systems
• Lyapunov function and smooth periodic solutions to quasilinear 1D hyperbolic systems