Time-Periodic Solutions for Hyperbolic-Parabolic Systems

TL;DR

This paper establishes the existence and uniqueness of time-periodic solutions for a coupled hyperbolic-parabolic system with partial damping, broadening understanding of wave-heat interactions and stability in complex systems.

Contribution

It introduces novel a priori estimates and a constructive approach for periodic solutions in heat-wave systems without wave interior dissipation, addressing an open theoretical problem.

Findings

Constructed periodic solutions for coupled heat-wave systems.

Identified geometric constraints related to wave domain control.

Mitigated regularity loss by trading time and space derivatives.

Abstract

Time-periodic weak solutions for a coupled hyperbolic-parabolic system are obtained. A linear heat and wave equation are considered on two respective $d$-dimensional spatial domains that share a common $(d-1)$-dimensional interface $\Gamma$. The system is only partially damped, leading to an indeterminate case for existing theory (Galdi et al., 2014). We construct periodic solutions by obtaining novel a priori estimates for the coupled system, reconstructing the total energy via the interface $\Gamma$. As a byproduct, geometric constraints manifest on the wave domain which are reminiscent of classical boundary control conditions for wave stabilizability. We note a ``loss" of regularity between the forcing and solution which is greater than that associated with the heat-wave Cauchy problem. However, we consider a broader class of spatial domains and mitigate this regularity loss by trading time and space differentiations, a feature unique to the periodic setting. This seems to be the first constructive result addressing existence and uniqueness of periodic solutions in the heat-wave context, where no dissipation is present in the wave interior. Our results speak to the open problem of the (non-)emergence of resonance in complex systems, and are readily generalizable to related systems and certain nonlinear cases.