Invariant manifolds of interior multi-spike states for the Cahn-Hilliard equation in higher space dimensions
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- by Peter W. Bates, Giorgio Fusco and Jiayin Jin PDF
- Trans. Amer. Math. Soc. 369 (2017), 3937-3975 Request permission
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.References
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Additional Information
- Peter W. Bates
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48823
- MR Author ID: 32495
- 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
- MR Author ID: 70195
- Email: fusco@dm.univaq.it
- Jiayin Jin
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48823
- Email: jinjiayi@msu.edu
- 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.
- © Copyright 2016 American Mathematical Society
- 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
- MathSciNet review: 3624398