Convergence towards equilibria for a hyperbolic system arising in ferroelectricity

In this paper we analyze the asymptotic behavior of a hyperbolic system describing the evolution of the electromagnetic field inside a ferroelectric material according with a model proposed by Greenberg, MacCamy and Coffman. The system consists of a linear damped wave equation for a field u coupled with a semilinear damped wave equation for the polarization field p. Assuming homogeneous Dirichlet and Neumann boundary conditions for u and p and that the nonlinearity phi is analytic, we prove that any weak solution converges to a stationary state. This is done by adapting an argument already used by Haraux-Jendoubi and based on the Simon- Lojasiewicz inequality. Furthermore, another goal of the paper is to provide the estimate of the decay rate to equilibrium. We give two results in this direction, according to different assumptions on phi.