Convergence of phase field to phase relaxation governed by the entropy balance with memory

The subject of the present paper consists in proving the convergence of a phase-field model, based on the entropy balance with memory, to phase relaxation. The well-posedness and the long-time behaviour of solutions for the nonlinear and singular phase-field system have been recently shown in [4, 5]. Here weprove the convergence of this solution to the solution of the corresponding phase relaxation problem, which turns also out to be unique and depend continuously on the data.