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

Non-isothermal viscous Cahn--Hilliard equation with inertial term and dynamic boundary conditions