The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor

In this paper we establish a Morse decomposition of the stationary solutions of the one-dimensional viscous Cahn-Hilliard equation by explicit energy calculations. Strong non-degeneracy of the stationary solutions is proven away from turning points and points of bifurcation from the homogeneous state and the dimension of the unstable manifold is calculated for all stationary states. In the unstable case, the ow on the global attractor is shown to be semi-conjugate to the ow on the global attractor of the Chaffee-Infante equation, and in the metastable case close to the nonlocal reaction-diffusion limit, a partial description of the structure of the global attractor is obtained by connection matrix arguments, employing a partial energy ordering and the existence of a weak lap number principle.