  We consider a non-isothermal modified Cahn--Hilliard equation which was
previously analyzed by M. Grasselli et al. Such an equation is characterized by
an inertial term and a viscous term and it is coupled with a hyperbolic heat
equation. The resulting system was studied in the case of no-flux boundary
conditions. Here we analyze the case in which the order parameter is subject to
a dynamic boundary condition. This assumption requires a more refined strategy
to extend the previous results to the present case. More precisely, we first
prove the well-posedness for solutions with bounded energy as well as for weak
solutions. Then we establish the existence of a global attractor. Finally, we
prove the convergence of any given weak solution to a single equilibrium by
using a suitable Lojasiewicz--Simon inequality.
