The paper concerns boundary value problems for general nonautonomous first-order quasilinear hyperbolic systems in a strip. We construct small global classical solutions, assuming that the right-hand sides are small. In the case that all data of the quasilinear problem are almost periodic, we prove that the bounded solution is also almost periodic. For the nonhomogeneous version of a linearized problem, we provide stable dissipativity conditions ensuring a unique bounded continuous solution for any smooth right-hand sides. In the autonomous case, this solution is two times continuously differentiable. In the nonautonomous case, the continuous solution is differentiable under additional dissipativity conditions, which are essential. A crucial ingredient of our approach is a perturbation theorem for general linear hyperbolic systems. One of the technical complications we overcome is the “loss of smoothness” property of hyperbolic PDEs.

本文涉及带中一般非自治一阶拟线性双曲系统的边值问题。我们假设右边很小，则构造小的全局经典解。在所有拟线性问题的数据几乎都是周期的情况下，我们证明了有界解也几乎是周期的。对于线性化问题的非齐次形式，我们提供稳定的耗散条件，从而确保对任何平滑右手边都有唯一的有界连续解。在自主情况下，该解决方案是两次连续可微的。在非自治情况下，连续解在必要的附加耗散条件下是可微的。我们方法的重要组成部分是一般线性双曲系统的扰动定理。