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

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  • 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.
    Mathematics Subject Classification: Primary: 35K55, 35B40, 35B45, 37L30; Secondary: 74N20.


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