Maxwell equations with localized internal damping: strong and polynomial stability

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 supp $\sigma$ contains a strip, the semigroup is polynomially stable of rate $\frac{1}{2}$. 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 $\sigma = 0$.