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 epsilon and permeability mu, we show strong stability of the underlying semigroup by checking the spectral criteria of the Arendt-Batty-Lyubich-Vu Theorem. If epsilon = mu = 1, Omega is the cube (0, pi)3 and supp sigma contains a strip, the semigroup is polynomially stable of rate 2. To derive this result, we establish the resolvent estimate of the Borichev-Tomilov Theorem using an 1 orthonormal basis of eigenfunctions of the Maxwell operator for sigma = 0.