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Polynomial stability of a coupled wave-heat network

Lassi Paunonen, David Seifert

TL;DR

The paper analyzes a coupled one-dimensional wave-heat network with multiple boundary couplings and proves energy decay to zero. By decomposing the network into a wave subnetwork and a heat subnetwork and applying abstract boundary-node theory (NiPaSe24) and interconnection results, it obtains resolvent bounds and positive transfer-function estimates. For classical solutions, the energy decays at the rate $E(t)=o(t^{-4})$, while all solutions satisfy $E(t)\to0$ as $t\to\infty$. This work provides a robust semigroup-theoretic framework for polynomial stability in complex PDE networks and demonstrates how boundary-node and interconnection techniques yield explicit decay rates.

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.

Polynomial stability of a coupled wave-heat network

TL;DR

The paper analyzes a coupled one-dimensional wave-heat network with multiple boundary couplings and proves energy decay to zero. By decomposing the network into a wave subnetwork and a heat subnetwork and applying abstract boundary-node theory (NiPaSe24) and interconnection results, it obtains resolvent bounds and positive transfer-function estimates. For classical solutions, the energy decays at the rate , while all solutions satisfy as . This work provides a robust semigroup-theoretic framework for polynomial stability in complex PDE networks and demonstrates how boundary-node and interconnection techniques yield explicit decay rates.

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 as for all classical solutions.
Paper Structure (6 sections, 4 theorems, 36 equations, 2 figures)

This paper contains 6 sections, 4 theorems, 36 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Boundary nodes and an abstract stability result
  3. Analysis of the wave-heat network
  4. Analysis of the wave network
  5. Analysis of the heat network
  6. Energy decay

Key Result

Theorem 1.1

The energy of every solution of the wave-heat system satisfies $E(t)\to0$ as $t\to\infty$, and for classical solutions we have $E(t)=o(t^{-4})$ as $t\to\infty$.

Figures (2)

  • Figure 1: A wave-heat network with heat parts (red) and wave parts (blue). The arrows indicate the directions in which the spatial coordinates increase.
  • Figure 2: Decomposition of the network into a wave network (the two blue edges) and a heat network (the three red edges) which interact via coupling at the three vertices.

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • proof : Proof of Theorem \ref{['thm:WaveMultiHeat']}
  • Remark 4.1