Singular limits of a conserved Penrose-Fife model with special heat flux law 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 \teta with a fourth order relation describing the evolution of the phase variable \chi. The lattere equation contains two relaxation parameters \mu and \epsilon, respectively related to the speed of transition process and to the interfacial energy, which happen to be very small in the applications. Existence and uniqueness for this model as \epsilon,\mu>0 have been recently proved by the first author. Here, the asymptotic behaviour of the model is studied as either \mu or \epsilon or both are let tend to zero. By a-priori estimates and compactness arguments, the convergence of the solutions is shown. In the cases when \muvaneshes, anyway, the approximating (i.e.,as \epsilon tends to zero) initial data have to be properly chosen. The problems obtained at the limit turn out to couple the original energy balance equation with a Hele-Shaw like system for \chi (as \epsilon tends to zero). or an elliptic fourth (as \mu tends to zero0 or second order inclusion (as both \mu,\epsilon tend to zero).