Existence and uniqueness for the parabolic conserved phase field model with memory

A nonlinear system for the heat diffusion inside a material subject to a phase change is considered. The underlying model is a generalized version of the well-known Caginalp conserved phase-field system, where the Fourier law is replaced by the Coleman-Gurtin heat flux law and a linear growth is allowed for the latent heat density. The resulting problem couples a non-linear parabolic equation derived from the balance of energy with a fourth order parabolic inclusion which rules the evolution of the order parameter $\chi$. Homogeneous Neumann boundary conditions guarantee that the space-average of $\chi$ is conserved in time. Existence and uniqueness of the solution are proved.