Uniform global attractors for non-isothermal viscous and non-viscous Cahn-Hilliard equations with dynamic boundary conditions

We consider a model of non-isothermal phase separation taking place in a con fined 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 non-linear 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 a the viscosity coefficient, by epsilon a (small) relaxation parameter multiplying partial derivative(t)theta in the heat equation and by delta a small latent heat coefficient (satisfying delta <= lambda alpha, lambda > 0) multiplying Delta theta in the Cahn-Hilliard equation and partial derivative(t)phi in the heat equation. We analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems. We first prove that the model generates a strongly continuous semigroup on a suitable phase space Y-K(alpha) (depending on the choice of the boundary conditions) which possesses the global attractor A(epsilon,delta,alpha) Our main results allow us to show that a proper lifting A(0.0.alpha,) alpha > 0, of the global attractor of the well-known viscous Cahn-Hilliard equation (that is, the system corresponding to (epsilon,delta) = (0, 0)) is upper semicontinuous at (0, 0) with respect to the family {A(epsilon,delta,alpha)}(epsilon,delta,alpha>0). We also establish that the global attractor A(0,0) of the non-viscous Cahn-Hilliard equation (corresponding to (epsilon, alpha) = (0, 0)) is upper semicontinuous at (0, 0) with respect to the same family of global attractors. Finally, the existence of exponential attractors M-epsilon,M-delta,M-alpha is also obtained in the cases epsilon not equal 0, delta not equal 0, alpha not equal 0, (0, delta, alpha), delta not equal 0, alpha not equal 0 and (epsilon,delta,alpha) = (0, 0, alpha), alpha > 0, respectively. This allows us to infer that, for each (epsilon,delta,alpha) epsilon [0, epsilon(0)] x [0, delta(0)] x [0, alpha(0)], A(epsilon,delta,alpha) has finite fractal dimension and this dimension is bounded, uniformly with respect to epsilon, delta and alpha. (C) 2008 Elsevier Ltd. All rights reserved.