A Temperature-Dependent Phase-Field Model for Phase Separation and Damage

In this paper we study a model for phase separation and damage in thermoviscoelastic materials. The main novelty of the paper consists in the fact that, in contrast with previous works in the literature concerning phase separation and damage processes in elastic media, in our model we encompass thermal processes, nonlinearly coupled with the damage, concentration and displacement evolutions. More particularly, we prove the existence of “entropic weak solutions”, resorting to a solvability concept first introduced in Feireisl (Comput Math Appl 53:461–490, 2007) in the framework of Fourier–Navier–Stokes systems and then recently employed in Feireisl et al. (Math Methods Appl Sci 32:1345–1369, 2009) and Rocca and Rossi (Math Models Methods Appl Sci 24:1265–1341, 2014) for the study of PDE systems for phase transition and damage. Our global-in-time existence result is obtained by passing to the limit in a carefully devised time-discretization scheme.