Exponential decay of quasilinear Maxwell equations with interior conductivity

We consider a quasilinear nonhomogeneous, anisotropic Maxwell system in a bounded smooth domain of R3 with a strictly positive conductivity subject to the boundary conditions of a perfect conductor. Under appropriate regularity conditions, adopting a classical L2-Sobolev solution framework, a nonlinear energy barrier estimate is established for local-in-time H3-solutions to the Maxwell system by a proper combination of higher-order energy and observability-type estimates under a smallness assumption on the initial data. Technical complications due to quasilinearity, anisotropy, the lack of solenoidality, and the fact that only partial dissipation is imposed on the system. Finally, provided the initial data are small, the barrier method is applied to prove that local solutions exist globally and exhibit an exponential decay rate