Direct problem for a nonlinear evolutionary system

In this article, we consider the first initial boundary-value problem for an evolutionary system describing nonlinear interactions of electromagnetic and elastic waves. The system under study consists of three coupled differential equations, one of them is a hyperbolic equation (an analogue of the Lamé equations) and the other two equations form a parabolic system (an analogue of the diffusion Maxwell system). Existence and uniqueness results are established. We also prove the stability estimate of a weak solution.