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$%S. Maier-Paape, U. Miller, K. Mischaikow, and T. Wanner, Rigorous numerics for the Cahn-Hilliard equation on the unit square, Rev. Mat. Complut. 21 (2008), no. 2, 351–426.%$

While the structure of the set of stationary solutions of the Cahn-Hilliard equation on one-dimensional domains is completely understood, only partial results are available for two-dimensional base domains. In this paper, we demonstrate how rigorous computational techniques can be employed to establish computer-assisted existence proofs for equilibria of the Cahn-Hilliard equation on the unit square. Our method is based on results by Mischaikow and Zgliczyński (Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, 2001), and combines rigorous computations with Conley index techniques. We are able to establish branches of equilibria and, under more restrictive conditions, even the local uniqueness of specific equilibrium solutions. Sample computations for several branches are presented, which illustrate the resulting patterns.

Key words: Cahn-Hilliard equation, stationary solutions, bifurcation diagram, continuation, computer-assisted proof.
2000 Mathematics Subject Classification: 37L10, 35B05, 35K35, 35K55.