# American Institute of Mathematical Sciences

September  2014, 13(5): 1855-1890. doi: 10.3934/cpaa.2014.13.1855

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

 1 Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via C. Saldini, 50, I-20133 Milano 2 Dipartimento di Matematica, Politecnico di Milano, 20133 Milano 3 School of Mathematical Sciences, Fudan University, Han Dan Road 220, 200433 Shanghai

Received  October 2013 Revised  February 2014 Published  June 2014

We consider a non-isothermal modified viscous Cahn-Hilliard equation which was previously analyzed by M. Grasselli et al. Such an equation is characterized by an inertial term and it is coupled with a hyperbolic heat equation from the Maxwell-Cattaneo's law. 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 finite 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 Łojasiewicz-Simon inequality.
Citation: Cecilia Cavaterra, Maurizio Grasselli, Hao Wu. Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1855-1890. doi: 10.3934/cpaa.2014.13.1855
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