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Invariant manifolds for metastable patterns in ut = ε2uxx—f(u)

Published online by Cambridge University Press:  14 November 2011

Jack Carr  and
Robert Pego
Jack Carr
Affiliation:
Department of Mathematics, Heriot Watt University, Riccarton, Edinburgh EH14 4AS, Scotland, U.K.
Robert Pego
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI, U.S.A.
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Synopsis

We consider the above equation on the interval 0 ≦ x ≦ 1 subject to Neumann boundary conditions with f(u) = F′(u) where F is a double well energy density function with equal minima. Our previous work [3] proved the existence and persistence of very slowly evolving patterns (metastable states) in solutions with two-phase initial data. Here we characterise these metastable states in terms of the global unstable manifolds of equilibria, as conjectured by Fusco and Hale [6].

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 116 , Issue 1-2 , 1990 , pp. 133 - 160
Copyright
Copyright © Royal Society of Edinburgh 1990

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