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Exponentially slow motion of interface layers for the one-dimensional Allen–Cahn equation with nonlinear phase-dependent diffusivity

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Abstract

This paper considers a one-dimensional generalized Allen–Cahn equation of the form

$$\begin{aligned} u_t = \varepsilon ^2 (D(u)u_x)_x - f(u), \end{aligned}$$

where \(\varepsilon > 0\) is constant, \(D = D(u)\) is a positive, uniformly bounded below, diffusivity coefficient that depends on the phase field u, and f(u) is a reaction function that can be derived from a double-well potential with minima at two pure phases \(u = \alpha \) and \(u = \beta \). It is shown that interface layers (namely, solutions that are equal to \(\alpha \) or \(\beta \) except at a finite number of thin transitions of width \(\varepsilon \)) persist for an exponentially long time proportional to \(\exp (C/\varepsilon )\), where \(C > 0\) is a constant. In other words, the emergence and persistence of metastable patterns for this class of equations is established. For that purpose, we prove energy bounds for a renormalized effective energy potential of Ginzburg–Landau type. Numerical simulations, which confirm the analytical results, are also provided.

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Acknowledgements

We thank two anonymous referees whose comments greatly improved the quality of the paper. The work of R. G. Plaza was partially supported by DGAPA-UNAM, program PAPIIT, Grant IN-100318.

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Authors and Affiliations

  1. Departamento de Matemáticas y Mecánica, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Circuito Escolar s/n, Ciudad Universitaria, C.P. 04510, Cd. de México, Mexico

    Raffaele Folino, Luis Lopez Rios & Ramón G. Plaza

  2. Departamento de Matemática, Universidade Estadual de Maringá, Av. Colombo, 5790 Jd. Universitário, Maringá, PR, CEP 87020-900, Brazil

    César A. Hernández Melo

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  1. Raffaele Folino
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Correspondence to Raffaele Folino.

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Folino, R., Hernández Melo, C.A., Lopez Rios, L. et al. Exponentially slow motion of interface layers for the one-dimensional Allen–Cahn equation with nonlinear phase-dependent diffusivity. Z. Angew. Math. Phys. 71, 132 (2020). https://doi.org/10.1007/s00033-020-01362-0

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