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-linear partial differential equation system and the long-time behaviour of its solutions. Namely, an equation ruling the evolution of the absolute temperature of the system (recently introduced in BONETTI, E., COLLI, P., FABRIZIO, M. & GILARDI, G. (2006) Modelling and long-time behaviour for phase transitions with entropy 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) Global solution to a singular integrodifferential system related to the entropy balance. Nonlinear Anal., 66, 1949–1979) is coupled with a generalization of the well-known Cahn–Hilliard equation for the order parameter . In particular, under suitable assumptions on the non-linearities involved, we prove that the elements of the -limit set (i.e. the cluster points) of the trajectories solve the steady-state system that is naturally associated to the evolution problem.