# American Institute of Mathematical Sciences

March  2009, 2(1): 113-147. doi: 10.3934/dcdss.2009.2.113

## Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions

 1 Department of Mathematics, University of Missouri, Columbia, MO, 65211, United States 2 Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 6086, SP2MI, 86962 Chasseneuil Futuroscope Cedex

Received  February 2008 Revised  August 2008 Published  January 2009

We consider a model of non-isothermal phase separation taking place in a confined container. The order parameter $\phi$ is governed by a viscous or non-viscous Cahn-Hilliard type equation which is coupled with a heat equation for the temperature $\theta$. The former is subject to a nonlinear dynamic boundary condition recently proposed by physicists to account for interactions with the walls, while the latter is endowed with a standard (Dirichlet, Neumann or Robin) boundary condition. We indicate by $\alpha$ the viscosity coefficient, by $\varepsilon$ a (small) relaxation parameter multiplying $\partial _{t}\theta$ in the heat equation and by $\delta$ a small latent heat coefficient (satisfying $\delta \leq \lambda \alpha$, $\delta \leq \overline{\lambda }\varepsilon$, $\lambda , \overline{\lambda }>0$) multiplying $\Delta \theta$ in the Cahn-Hilliard equation and $\partial _{t}\phi$ in the heat equation. Then, we construct a family of exponential attractors $\mathcal{M}_{\varepsilon ,\delta ,\alpha }$ which is a robust perturbation of an exponential attractor $\mathcal{M} _{0,0,\alpha }$ of the (isothermal) viscous ($\alpha >0$) Cahn-Hilliard equation, namely, the symmetric Hausdorff distance between $\mathcal{M} _{\varepsilon ,\delta ,\alpha }$ and $\mathcal{M}_{0,0,\alpha }$ goes to 0, for each fixed value of $\alpha >0,$ as $( \varepsilon ,\delta)$ goes to $(0,0),$ in an explicitly controlled way. Moreover, the robustness of this family of exponential attractors $\mathcal{M}_{\varepsilon ,\delta ,\alpha }$ with respect to $( \delta ,\alpha ) \rightarrow ( 0,0) ,$ for each fixed value of $\varepsilon >0,$ is also obtained. Finally, assuming that the nonlinearities are real analytic, with no growth restrictions, the convergence of solutions to single equilibria, as time goes to infinity, is also proved.
Citation: Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113
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