Convergence to stationary solutions for a parabolic-hyperbolic phase-field system

A parabolic-hyperbolic nonconserved phase-field model is here analyzed. This is an evolution system consisting of a parabolic equation for the relative temperature v which is nonlinearly coupled with a semilinear damped wave equation governing the order parameter chi. The latter equation is characterized by a nonlinearity phi(chi) with cubic growth. Assuming homogeneous Dirichlet and Neumann boundary conditions for v and chi, we prove that any weak solution has an omega-limit set consisting of one point only. This is achieved by means of adapting a method based on the Lojasiewicz-Simon inequality. We also obtain an estimate of the decay rate to equilibrium.