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

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.

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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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Correspondence to Raffaele Folino.

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Folino, R., Lattanzio, C. & Mascia, C. Metastability and Layer Dynamics for the Hyperbolic Relaxation of the Cahn–Hilliard Equation. J Dyn Diff Equat 33, 75–110 (2021). https://doi.org/10.1007/s10884-019-09806-6

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Keywords

  • Cahn–Hilliard equation
  • Metastability
  • Layer dynamics
  • Singular perturbations