Singular limits of a conserved Penrose-Fife phase field model with special heat flux laws and memory effects

A phase-field model of Penrose-Fife type for diffusive phase transitions with conserved order parameter is introduced. A Cauchy-Neumann problem is considered for the related parabolic system which couples a nonlinear Volterra integro-differential equation for the temperature T with a fourth order relation describing the evolution of the phase variable h. The latter equation contains a relaxation parameter w related to the speed of the transition process, which happens to be very small in the applications. Existence and uniqueness for this model as w>0 have been recently proved by the first author. Here, the asymptotic behaviour of the model is studied as w is let tend to zero. By a priori estimates and compactness arguments, the convergence of the solutions is shown. The approximating initial data have to be properly chosen. The problem obtained at the limit turns out to couple the original energy balance equation with an elliptic fourth order inclusion