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Transactions of the American Mathematical Society

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Invariant manifolds of interior multi-spike states for the Cahn-Hilliard equation in higher space dimensions


Authors: Peter W. Bates, Giorgio Fusco and Jiayin Jin
Journal: Trans. Amer. Math. Soc. 369 (2017), 3937-3975
MSC (2010): Primary 35B25, 35K55, 37D10; Secondary 34G20, 37L25
DOI: https://doi.org/10.1090/tran/6817
Published electronically: November 16, 2016
MathSciNet review: 3624398
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Abstract: We construct invariant manifolds of interior multi-spike states for the nonlinear Cahn-Hilliard equation and then investigate the dynamics on it. An equation for the motion of the spikes is derived. It turns out that the dynamics of interior spikes has a global character and each spike interacts with all the others and with the boundary. Moreover, we show that the speed of the interior spikes is super slow, which indicates the long time existence of dynamical multi-spike solutions in both positive and negative time. This result is the application of an abstract result concerning the existence of truly invariant manifolds with boundary when one has only approximately invariant manifolds. The abstract result is an extension of one by P. Bates, K. Lu, and C. Zeng to the case of a manifold with boundary, consisting of almost stationary states.


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Additional Information

Peter W. Bates
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48823
Email: bates@math.msu.edu

Giorgio Fusco
Affiliation: Dipartimento di Matematica Pura ed Applicata, Università degli Studi dell’Aquila, Via Vetoio, 67010 Coppito, L’Aquila, Italy
Email: fusco@dm.univaq.it

Jiayin Jin
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48823
Email: jinjiayi@msu.edu

DOI: https://doi.org/10.1090/tran/6817
Keywords: Invariant manifold theory, infinite-dimensional dynamical systems, Cahn-Hilliard equation, multi-spike solutions
Received by editor(s): October 8, 2014
Received by editor(s) in revised form: May 25, 2015
Published electronically: November 16, 2016
Additional Notes: The first author was supported in part by NSF grants 0908348 and 1413060. All authors would like to acknowledge the support of the IMA, where this work was initiated.
Article copyright: © Copyright 2016 American Mathematical Society

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