Exponential decay for Maxwell equations with a boundary memory condition

Abstract We study the asymptotic behavior of the solution of the Maxwell equations with the following boundary condition of memory type: (0.1) E τ ( t ) = η 0 H ( t ) × n + ∫ 0 ∞ η ( s ) H ( t − s ) × n d s . We consider a ‘Graffi’ type free energy and we prove that, if the kernel η satisfies the condition η ″ + κ η ′ > 0 and the domain Ω is strongly star shaped, then the energy of the solution exponentially decays. We also prove that the exponential decay of η is a necessary condition for the exponential decay of the solution.