Global well-posedness and exponential stability for heterogeneous anisotropic Maxwell's equations under a nonlinear boundary feedback with delay

Abstract

We consider an initial–boundary value problem for the Maxwell's system in a bounded domain with a linear nonhomogeneous anisotropic instantaneous material law subject to a nonlinear Silver–Müller-type boundary feedback mechanism incorporating both an instantaneous damping and a time-localized delay effect. By proving the maximal monotonicity property of the underlying nonlinear generator, we establish the global well-posedness in an appropriate Hilbert space. Further, under suitable assumptions and geometric conditions, we show the system is exponentially stable.