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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: M. Grinfeld and A. Novick-Cohen
Journal: Trans. Amer. Math. Soc. 351 (1999), 2375-2406
MSC (1991): Primary 35K22, 35K30, 58F12, 58F39
Published electronically: February 15, 1999
MathSciNet review: 1650085
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Abstract: 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 flow on the global attractor is shown to be semi-conjugate to the flow 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.


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Additional Information

M. Grinfeld
Affiliation: Department of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, Scotland, United Kingdom
Email: michael@conley.maths.strath.ac.uk

A. Novick-Cohen
Affiliation: Faculty of Mathematics, Technion-IIT, Haifa 32000, Israel
Email: amync@techunix.technion.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02445-9
PII: S 0002-9947(99)02445-9
Received by editor(s): September 24, 1996
Published electronically: February 15, 1999
Article copyright: © Copyright 1999 American Mathematical Society