Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square

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November 2006, 15(4): 1137-1153. doi: 10.3934/dcds.2006.15.1137

Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square

S. Maier-Paape ^1, and Ulrich Miller ^2,

1.	Institut für Mathematik, RWTH Aachen, D-52062 Aachen
2.	Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany

Received December 2004 Revised January 2006 Published May 2006

Abstract
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We state an alternative for paths of equilibria of the Cahn-Hilliard equation on the square, bifurcating from the trivial solution at eigenfunctions of the form $w_{ij}=\cos(\pi ix)\cos(\pi j y)$, for $i,j \in \N$. We show that the paths either only connect the bifurcation point $m_{ij}$ with $-m_{ij}$ and are separated from all other paths with even more symmetry, or they contain a loop of nontrivial solutions connecting the bifurcation point $m_{ij}$ with itself. In any case the continua emerging at $m_{ij}$ and $-m_{ij}$ are equal. For fixed mass $m_0=0$ we furthermore prove that the continua bifurcating from the trivial solution at eigenfunctions of the form $w_{i0}+w_{0i}$ or $w_{ij}$, for $i,j \in \N$ are smooth curves parameterized over the interaction length related parameter $\lambda$.

Keywords: Cahn-Hilliard equation., Bifurcation theory.

Mathematics Subject Classification: Primary: 35B32,35Q55; Secondary: 35B4.

Citation: S. Maier-Paape, Ulrich Miller. Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1137-1153. doi: 10.3934/dcds.2006.15.1137

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