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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

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