On a Conducting Electromagnetic Medium

In this article we consider a linear electromagnetic material characterized by a rate-type equation for the electric conduction in order to consider memory effects for this field. After deriving the restrictions imposed by the thermodynamic principles on the constitutive equations, we introduce the free energy useful to show the existence of a domain of dependence. Then, theorems of existence and uniqueness of weak and strong solutions are established for the initial-boundary value problem for the system of Maxwell's equations. Finally, we give an energy estimate.