Convergence of phase field to phase relaxation models governed by an entropy equation with memory

The subject of the present paper consists in proving the convergence of a phase-field model, based on the entropy equation with memory, to phase relaxation. The well-posedness and the long-time behaviour of solutions for the non-linear and singular phase-field system have been recently shown by Bonetti et al. (Preprint IMATI-CNR, 2005; Discrete Contin. Dyn. Syst. Ser. B, in press). Here, we study the asymptotic behaviour of such solutions as the interfacial energy coefficient tends to zero. The limit problem is a phase relaxation problem with memory, which is new. We prove well-posedness results through convergence under rather general assumptions. However, the case of a quadratic non-linearity for the latent heat is excluded. Such a situation is dealt for the problem without memory in a generalized setting by introducing an ad hoc logarithm.