Bounded and Almost Periodic Solvability of Nonautonomous Quasilinear Hyperbolic Systems

TL;DR

This paper studies boundary value problems for nonautonomous quasilinear hyperbolic systems, establishing conditions for existence, uniqueness, and almost periodicity of solutions, and addressing technical challenges like loss of smoothness.

Contribution

It introduces new conditions for the existence of bounded and almost periodic solutions, and develops a perturbation theorem for linear hyperbolic systems addressing smoothness loss.

Findings

Existence of small global classical solutions under small data assumptions.

Bounded solutions are almost periodic if data are almost periodic.

Stable dissipativity conditions ensure unique bounded solutions.

Abstract

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.