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

Cecilia Cavaterra; Maurizio Grasselli; Hao Wu

doi:10.3934/cpaa.2014.13.1855

Article Contents

2014, Volume 13, Issue 5: 1855-1890. Doi: 10.3934/cpaa.2014.13.1855

This issue Previous Article Stabilization of the simplest normal parabolic equation Next Article Regular solutions and global attractors for reaction-diffusion systems without uniqueness

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: September 30, 2013

Revised: January 31, 2014

Published: August 2015

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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 Łojasiewicz-Simon inequality.

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Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 37L99, 80A22.

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