Asymptotic behaviour of the energy for electromagnetic systems with memory. (English) Zbl 1054.35103
The authors consider initial boundary value problems for Maxwell’s equations with the perfect conductor boundary condition on the smooth boundary \(\partial\Omega\) of a bounded domain \(\Omega\) driven by a self-induced current density \(J\) of the form
\[
J\left(t,x\right)=-\int_{0}^{t}\sigma_{1}\left(t-s\right)\, E\left(t,x\right)\, ds
\]
and subject to material relations \(B\left(t,x\right)=\mu_{0}H\left(t,x\right)\),
\[
D\left(t,x\right)=\varepsilon_{0}E\left(t,x\right)+\int_{0}^{t}\varepsilon_{1}\left(t-s\right)\, E\left(t,x\right)\, ds ,
\]
\(t\in\mathbb{R}_{\geq0}\), \(x\in\Omega\), \(\varepsilon_{0},\mu_{0}\in\mathbb{R}_{>0}\). The integral kernel functions are assumed to be scalar-valued and required to satisfy \(\omega:\operatorname{Im} \widehat{\varepsilon}_{1}(\omega)- \operatorname{Re}\widehat{\sigma}_{1}(\omega)>0\) for \(\omega\in\mathbb{R}_{>0}\) to obtain long-time decay estimates for the solution. Polynomial energy decay is obtained for polynomially decaying kernel functions. The results are shown by utilizing suitable Lyapunov functionals.
Reviewer: Rainer Picard (Dresden)
MSC:
35Q60 | PDEs in connection with optics and electromagnetic theory |
78A25 | Electromagnetic theory (general) |
35B40 | Asymptotic behavior of solutions to PDEs |
45K05 | Integro-partial differential equations |