Time-Periodic Solutions for Hyperbolic-Parabolic Systems

TL;DR

The paper tackles the existence and uniqueness of time-periodic weak solutions for a coupled heat-wave system with partial damping across an interface between two domains. It develops a constructive approach based on novel a priori estimates and energy reconstruction through the interface, leveraging geometric optics–type conditions (Generalized Optics and Graph Optics) and the E-property to control the hyperbolic component via the parabolic dissipation. Two main theorems establish periodic solvability under these geometric hypotheses, with sharper regularity results under stronger forcing assumptions, as well as very weak solutions in broader settings. The work clarifies how dissipation can propagate from the heat domain to the undamped wave domain through the interface, connects periodic solvability to boundary control ideas, and provides a framework that generalizes to related systems and potential nonlinear extensions, including explicit examples illustrating regularity and geometric constraints.

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 $Γ$. 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 $Γ$. 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.

Figures (11)

• Figure 1: Example configurations.
• Figure 2: Star-complemented domain $\Omega$, as taken from russell. Here, $\Gamma_0$ is the inactive (zero Dirichlet) portion of $\partial \Omega$ and $\Gamma_1$ is the active (or interface) portion.
• Figure 3: Graph setting.
• Figure 4: Triangular Domain, satisfying the Geometric Optics Condition. Described by $\mathbf{b} = (x, y)$.
• Figure 5: Horn Domain, satisfying the Generalized Optics Condition but not the Geometric Optics Condition. Described by $\mathbf{b} = (\beta x, y)$, with $\beta > 0$.

Theorems & Definitions (45)

• Remark 1.1
• Definition 1
• Remark 1.2
• Theorem 1.1
• proof
• Remark 1.3
• Remark 1.4
• Theorem 2.1
• Theorem 2.2
• Remark 2.1