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Metastability and Layer Dynamics for the Hyperbolic Relaxation of the Cahn–Hilliard Equation

  • Raffaele FolinoEmail author
  • Corrado Lattanzio
  • Corrado Mascia
Original Article
  • 26 Downloads

Abstract

The goal of this paper is to accurately describe the metastable dynamics of the solutions to the hyperbolic relaxation of the Cahn–Hilliard equation in a bounded interval of the real line, subject to homogeneous Neumann boundary conditions. We prove the existence of an approximately invariant manifold\(\mathcal {M}_0\) for such boundary value problem, that is we construct a narrow channel containing \(\mathcal {M}_0\) and satisfying the following property: a solution starting from the channel evolves very slowly and leaves the channel only after an exponentially long time. Moreover, in the channel the solution has a transition layer structure and we derive a system of ODEs, which accurately describes the slow dynamics of the layers. A comparison with the layer dynamics of the classic Cahn–Hilliard equation is also performed.

Keywords

Cahn–Hilliard equation Metastability Layer dynamics Singular perturbations 

Notes

Acknowledgements

We thank the anonymous referee for the careful review and for the comments which helped us to improve the paper.

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Authors and Affiliations

  1. 1.Dipartimento di Ingegneria e Scienze dell’Informazione e MatematicaUniversità degli Studi dell’AquilaL’AquilaItaly
  2. 2.Dipartimento di MatematicaSapienza Università di RomaRomeItaly

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