In this paper we consider an integro-differential system consisting
of a parabolic and a hyperbolic equation related to phase
transition models. The first equation is integro-differential
and of hyperbolic type. It describes the evolution of the
temperature and also accounts for memory effects through a
memory kernel $k$ via the Gurtin-Pipkin heat flux law.
The latter equation, governing the evolution of the order
parameter, is semilinear, parabolic and of the fourth order
(in space). We prove a local in time existence result and
a global uniqueness result for the identification problem
consisting in recovering the memory kernel $k$ appearing in
the first equation.