Universal attractor for some singular phase transition systems

A singular parabolic system describing the thermal diffusion in a substance possibly subject to a phase transition is introduced. The physical process is described by the variables ϑ (absolute temperature) and χ (order parameter). The latter may have, or not, conserved total mass with respect to time. In both cases, after recalling and sometimes improving some known well-posedness results, the long-time behavior of the system is studied. It is shown that the process is dissipative and the compact universal attractor is constructed. It turns out to attract the trajectories of the system in a rather strong metric which is strictly linked to the constraints imposed to both variables. The techniques used in the proofs seem likely to be applied to other types of evolution systems containing maximal monotone nonlinearities.