Orthogonal decomposition and asymptotic behavior for nonlinear Maxwell's equations

We consider nonlinear Maxwell's equations in which the electric conductivity strongly depends on the electric field, acting like a switch-like function for the electromagnetic field. We combine multiplier techniques and Nakao's Method to show that the solution decays at polynomial rates if the conductivity satisfies suitable conditions. Under more restrictive hypotheses, exponential decay is obtained. As a byproduct, we also obtain the decay of the solution of quasi-stationary Maxwell's equations.