We analyze a conserved phase-field system, characterized by heat memory terms: memory kernels
$a$ and $b$ account for relaxation effects in the energy constitutive equation and in the
Gurtin-Pipkin heat conduction law, respectively. This model consists of a hyperbolic
integrodifferential equation for the temperature $\theta$ coupled with a nonlinear fourth order
evolution equation for the phase variable $\chi$. With appropriate initial and boundary data, we prove
that the system can be interpreted as a process. We show that it possesses an absorbing set, under
three -thermodynamically consistent- conditions on the memory kernel $a$.