Direct and inverse problems for a parabolic integro-differential system of Caginalp type

In this paper we consider some integro-differential systems of two parabolic PDE's coming from the Caginalp approach to phase transition models. The first (integro-differential) equation describes the evolution of the temperature and also accounts for memory effects through a memory kernel k. The latter equation, governing the evolution of the order parameter, is semilinear and of the fourth-order (in space). We prove some continuous dependence
and regularity results for the solution of the Cauchy problem associated to the PDE's. Taking advantage of these results, we prove a global in time conditional existence and uniqueness result for the identification problem consisting in recovering the memory kernel k.
appearing in the first equation