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 manifoldM0 for such boundary value problem, that is we construct a narrow channel containing M0 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.

Metastability and Layer Dynamics for the Hyperbolic Relaxation of the Cahn–Hilliard Equation / Folino, Raffaele; Lattanzio, Corrado; Mascia, Corrado. - In: JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS. - ISSN 1040-7294. - 33:(2021), pp. 75-110. [10.1007/s10884-019-09806-6]

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

Folino Raffaele
;
Mascia Corrado
2021

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 manifoldM0 for such boundary value problem, that is we construct a narrow channel containing M0 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.
2021
Cahn–Hilliard equation; metastability; layer dynamics; singular perturbations
01 Pubblicazione su rivista::01a Articolo in rivista
Metastability and Layer Dynamics for the Hyperbolic Relaxation of the Cahn–Hilliard Equation / Folino, Raffaele; Lattanzio, Corrado; Mascia, Corrado. - In: JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS. - ISSN 1040-7294. - 33:(2021), pp. 75-110. [10.1007/s10884-019-09806-6]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1337877
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