Global existence and long-time behaviour for a singular integro-differential phase-field system

This paper deals with a singular integro-differential PDE system describing phase transitions in terms of nonlinear evolution equations for micromotions and for the entropy. The model is derived from a non-convex free energy functional, possibly accounting for thermal memory effects. After recovering a global existence result for a related initial and boundary value problem, the long-time behaviour of the solutions is investigated. In particular, it is proved that the elements of the ω-limit set (i.e. the cluster points) of the solution trajectories solve the steady state system which is naturally associated to the evolution problem.