THE CAHN-HILLIARD'S EQUATION WITH BOUNDARY NONLINEARITY AND HIGH VISCOSITY

Abstract

The paper studies in less general scales of Banach spaces the dynamics generated by a Cahn-Hilliard type equation in a smooth open bounded domain of any space dimensions. The equation on the boundary satisfy nonlinear conditions. It establishes local well posedness of the problem and a priori uniform on the domain boundedness and existence in the large of the solutions is studied. It also discusses the asymptotic behaviour of the solutions in the form of existence of a global attractor. An adequate notion of upper semicontinuity of the attractor in the limit of high viscosity is considered and the limit attractor is found to correspond to finite dimensional processes. These processes are depicted by limits of the spatial average solutions of the problem.