Convergence to equilibrium for a parabolic-hyperbolic phase-field system with Neumann boundary conditions

This paper is concerned with the asymptotic behavior of global solutions to a parabolic–hyperbolic coupled system which describes the evolution of the relative temperature and the order parameter in a material subject to phase transitions. For the system with homogeneous Neumann boundary conditions for both the variables, under the assumption that the nonlinearities are real analytic functions, we prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable Lojasiewicz–Simon type inequality.