Time-Periodic Solutions for Hyperbolic-Parabolic Systems

Stanislav Mosny; Boris Muha; Sebastian Schwarzacher; Justin T. Webster

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arXiv · 2412.18801

Time-Periodic Solutions for Hyperbolic-Parabolic Systems

Stanislav Mosny·Boris Muha·Sebastian Schwarzacher·Justin T. Webster

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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.

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Stanislav Mosny, Boris Muha, Sebastian Schwarzacher, Justin T. Webster. 2024-12-25. Time-Periodic Solutions for Hyperbolic-Parabolic Systems. https://arxiv.org/abs/2412.18801

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