The first initial-boundary-value problem for a nonlinear model of the electrodynamics of vibrating elastic media

The first initial-boundary-value problem for nonlinear differential equations describing the interactions of a vibrating electroconductive body and the electromagnetic field is studied. We assume that the motion of the body occurs at velocities that are much smaller than the velocity of propagation of the electromagnetic waves through the elastic medium. The model under study consists of two coupled differential equations; one of them is the hyperbolic equation (an analogue of the Lamé system) and the other is the parabolic equation (an analogue of the diffusion Maxwell system). We prove an existence and uniqueness result. The proof is based on the classical Faedo-Galerkin method.