Well-posedness and long-time behaviour for a singular phase field system of conserved type

In this paper, we study the well-posedness of a singular non-linearpartial differential equation system and the long-time behaviourof its solutions. Namely, an equation ruling the evolution ofthe absolute temperature of the system (recently introducedin BONETTI, E., COLLI, P., FABRIZIO, M. & GILARDI, G. (2006)Modelling and long-time behaviour for phase transitions withentropy balance and thermal memory conductivity. Discrete Contin.Dyn. Syst. Ser. B, 6, 1001–1026 (electronic) and BONETTI,E., COLLI, P., FABRIZIO, M. & GILARDI, G. (2007) Globalsolution to a singular integrodifferential system related tothe entropy balance. Nonlinear Anal., 66, 1949–1979) iscoupled with a generalization of the well-known Cahn–Hilliardequation for the order parameter . In particular, under suitableassumptions on the non-linearities involved, we prove that theelements of the -limit set (i.e. the cluster points) of thetrajectories solve the steady-state system that is naturallyassociated to the evolution problem.