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

## TL;DR

This paper studies boundary value problems for quasilinear hyperbolic systems, proving existence, uniqueness, and almost periodicity of solutions under certain conditions, with solutions becoming smooth over time.

## Contribution

It establishes the existence and uniqueness of small global classical solutions and shows that almost periodic coefficients lead to almost periodic solutions.

## Key findings

- Solutions become $C^2$-smooth over time
- Existence and uniqueness of solutions under exponential dichotomy
- Almost periodic coefficients imply almost periodic solutions

## 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.