Well-posedness results for phase field systems with memory effects in the order parameter dynamics

We study two models arising in phase transition dynamics. Thestate of the system is described by the pair (\u3d1,\u3c7), where \u3d1 is the (relative) temperature and \u3c7 is the order parameter or phase field. The main difference between the two models relies on whether global constraints on \u3c7 are imposed or not: accordingly, the resulting models will be called conserved or nonconserved. Memory effects influencing both the heat flux and the dynamics of \u3c7 have been considered in a number of recent papers. Here we assume the Fourier law for the heat flux in the energy balance equation, while we consider memory effects in the order parameter dynamics. We analyze the well-posedness of corresponding Cauchy-Neumann problems for both conserved and nonconserved models. Various results are derived according to properties of the memory kernel involved