Long-time behaviour of the solution of Maxwell’s equations in dissipative generalized Lorentz materials (I): a frequency-dependent Lyapunov function approach

It is well known that electromagnetic dispersive structures such as metamaterials can be modelled by generalized Drude–Lorentz models. The present paper is the first of two articles dedicated to dissipative generalized Drude–Lorentz open structures. We wish to quantify the loss in such media in terms of the long-time decay rate of the electromagnetic energy for the corresponding Cauchy problem. By using an approach based on frequency-dependent Lyapunov estimates, we show that this decay is polynomial in time. These results extend to an unbounded structure the ones obtained for bounded media in Nicaise and Pignotti (Z Angew Math Phys 71(3):76, 2020) via a quite different method based on the notion of cumulated past history and semi-group theory. A great advantage of the approach developed here is to be less abstract and directly connected to the physics of the system via energy balances.