Well-posedness and long time behavior of the non-isothermal viscous Cahn-Hilliard equation with dynamic boundary conditions

Abstract

We consider a model of non-isothermal phase transition taking place in a confined container. The order parameter φ is governed by a Cahn-Hilliard type equation which is coupled with a nonlinear heat equation for the temperature θ. The former is subject to a nonlinear dynamic boundary condition recently proposed by some physicists to account for interactions of the material with the walls. The latter is endowed with a boundary condition which can be a standard one (Dirichlet, Neumann or Robin). We thus formulate a class of initial and boundary value problems whose local existence and uniqueness is proven by means of a Faedo-Galerkin approximation scheme. The local solution becomes global owing to suitable a priori estimates. Then we analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems. In particular, we demonstrate the existence of a finite dimensional global attractor as well as of an exponential attractor. © 2008 International Press.