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 Lojasiewicz-Simon inequality.
Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions / C. Cavaterra, M. Grasselli, H. Wu. - In: COMMUNICATIONS ON PURE AND APPLIED ANALYSIS. - ISSN 1534-0392. - 13:5(2014), pp. 1855-1890.
Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions
C. CavaterraPrimo
;
2014
Abstract
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 Lojasiewicz-Simon inequality.File | Dimensione | Formato | |
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