ROBUST EXPONENTIAL ATTRACTORS AND CONVERGENCE TO EQUILIBRIA FOR NON-ISOTHERMAL CAHN-HILLIARD EQUATIONS WITH DYNAMIC BOUNDARY CONDITIONS

Abstract. We consider a model of non-isothermal phase separation taking place in a confined container. The order parameter φ is governed by a viscous or non-viscous Cahn-Hilliard type equation which is coupled with a heat equation for the temperature θ. 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 α the viscosity coefficient, by ε a (small) relaxation parameter multiplying ∂tθ in the heat equation and by δ a small latent heat coefficient (satisfying δ ≤ λα, δ ≤ λε, λ, λ> 0) multiplying ∆θ in the Cahn-Hilliard equation and ∂tφ in the heat equation. Then, we construct a family of exponential attractorsMε,δ,α which is a robust perturbation of an exponential attractor M0,0,α of the (isothermal) viscous (α> 0) Cahn-Hilliard equation, namely, the symmetric Hausdorff distance between Mε,δ,α andM0,0,α goes to 0, for each fixed value of α> 0, as (ε, δ) goes to (0, 0), in an explicitly controlled way. Moreover, the robustness of thi