Asymptotic Justification of the Conserved Phase-Field Model with Memory

We consider a conserved phase-field model with memory in which the Fourier heat conduction law is replaced by a constitutive assumption of Gurtin-Pipkin type; the system is conserved in the sense that the initial mass of the order parameter is preserved during the evolution. We investigate a Cauchy-Neumann problem for this model which couples an integro-differential equation with a nonlinear fourth-order equation for the phase field. Here we assume that the heat flux memory kernel has a decreasing exponential as principal part and we study the behaviour of solutions when this kernel converges to a Dirac mass. We show that the solution to the conserved phase-field model with memory converges to a solution to the phase-field problem without memory under suitable assumptions on the data.