Maxwell equations with localized internal damping: strong and polynomial stability

Serge Nicaise; Roland Schnaubelt; Serge Nicaise; Roland Schnaubelt

doi:10.3934/cam.2025034

Communications in Analysis and Mechanics

2025, Volume 17, Issue 4: 849-877. doi: 10.3934/cam.2025034

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Maxwell equations with localized internal damping: strong and polynomial stability

Serge Nicaise ^{1
,
,},
Roland Schnaubelt ²

1.
CÉRAMATHS/DMATHS and FR CNRS 2037, Université Polytechnique Hauts-de-France, 59313 Valenciennes Cedex 9, France
2.
Department of Mathematics, Karlsruhe Institute of Technology, P.O. Box 6980, 76049 Karlsruhe, Germany

Received: 28 March 2025 Revised: 21 August 2025 Accepted: 29 August 2025 Published: 10 October 2025
35L60, 35Q

We study the Maxwell system with localized conductivity $ \sigma $ and the boundary conditions of a perfect conductor on a simply connected domain $ \Omega $, assuming that there are no electric charges off the support of $ \sigma $. For matrix-valued permittivity $ \varepsilon $ and permeability $ \mu $, we show strong stability of the underlying semigroup by checking the spectral criteria of the Arendt–Batty–Lyubich–Vũ Theorem. If $ \varepsilon = \mu = 1 $, $ \Omega $ is the cube $ (0, \pi)^3 $ and $ \text{supp}\, \sigma $ contains a strip, the semigroup is polynomially stable of rate $ \frac12 $. To derive this result, we establish the resolvent estimate of the Borichev–Tomilov Theorem using an orthonormal basis of eigenfunctions of the Maxwell operator for $ \sigma = 0 $.
- stability,
- Maxwell system,
- localized conductivity,
- ABLV theorem,
- Borichev–Tomilov theorem
Citation: Serge Nicaise, Roland Schnaubelt. Maxwell equations with localized internal damping: strong and polynomial stability[J]. Communications in Analysis and Mechanics, 2025, 17(4): 849-877. doi: 10.3934/cam.2025034

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Abstract

We study the Maxwell system with localized conductivity $ \sigma $ and the boundary conditions of a perfect conductor on a simply connected domain $ \Omega $, assuming that there are no electric charges off the support of $ \sigma $. For matrix-valued permittivity $ \varepsilon $ and permeability $ \mu $, we show strong stability of the underlying semigroup by checking the spectral criteria of the Arendt–Batty–Lyubich–Vũ Theorem. If $ \varepsilon = \mu = 1 $, $ \Omega $ is the cube $ (0, \pi)^3 $ and $ \text{supp}\, \sigma $ contains a strip, the semigroup is polynomially stable of rate $ \frac12 $. To derive this result, we establish the resolvent estimate of the Borichev–Tomilov Theorem using an orthonormal basis of eigenfunctions of the Maxwell operator for $ \sigma = 0 $.

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