Asymptotic stability for a non-local problem in electromagnetism

In this paper, constitutive equations of non-local type are coupled with Maxwell equations and the resulting differential problem is studied. A weak formulation is given for an initial-boundary-value problem for Maxwell equations in a medium obeying such constitutive equations with perfectly conducting boundary, and it is shown that such a problem admits at most one solution. The uniqueness theorem is then shown to imply the density of the range of a certain operator in the space of solutions and this result, together with an a priori energy inequality, is used to prove existence of solutions. Then the study of asymptotic stability of solutions is addressed. In particular, solutions are shown to be L2 in time over (0, ∞). Finally, a brief description is given of the alternative problem arising when more general constitutive equations are used.