Polynomial stability of a coupled wave-heat network

TL;DR

This paper investigates the long-term energy decay in a complex wave-heat network, demonstrating a polynomial decay rate of t^{-4} for classical solutions by analyzing simpler subsystems and applying abstract coupled system results.

Contribution

It introduces a novel analysis of coupled wave-heat networks, establishing a specific polynomial decay rate for energy over time.

Findings

Energy decays at rate t^{-4} as t approaches infinity.

Analysis of simpler wave and heat networks informs the coupled system behavior.

Application of recent abstract coupled system results to complex networks.

Abstract

We study the long-time asymptotic behaviour of a topologically non-trivial network of wave and heat equations. By analysing the simpler wave and the heat networks separately, and then applying recent results for abstract coupled systems, we establish energy decay at the rate $t^{-4}$ as $t\to\infty$ for all classical solutions.