On the bounded smooth solutions to exponentially stable linear nonautonomous hyperbolic systems

TL;DR

This paper studies the existence and uniqueness of bounded, smooth solutions, including periodic and almost periodic solutions, for nonautonomous hyperbolic systems under exponential stability and dissipativity conditions.

Contribution

It establishes conditions for the existence of bounded smooth solutions to nonautonomous hyperbolic systems, highlighting the role of exponential stability and dissipativity in regularity.

Findings

Existence of bounded $C^1$ and $C^2$ solutions under exponential stability.

Periodic solutions exist under $L^2$-exponential stability.

Number of dissipativity conditions relates to solution regularity.

Abstract

We investigate global bounded solutions of higher regularity to boundary value problems for a general linear nonautonomous first order 1D hyperbolic system in a strip. We establish the existence of such solutions under the assumption of exponential stability and certain dissipativity (or non-resonant) conditions. Our results demonstrate the existence and uniqueness of $C^1$- and $C^2$-bounded solutions, in particular, periodic and almost periodic solutions. The number of imposed dissipativity conditions is related to the regularity of solutions. This connection arises due to the nonautonomous nature of the hyperbolic system under consideration, in contrast to autonomous settings. In the general case of global bounded smooth solutions we assume exponential stability in $H^1$. In the case of periodic solutions, the weaker assumption of $L^2$-exponential stability proves to be sufficient.