Exponentially slow motion of interface layers for the one-dimensional Allen-Cahn equation with nonlinear phase-dependent diffusivity
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| Main Authors: | , , , |
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| Format: | Preprint |
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2019
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| _version_ | 1866913354530947072 |
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| author | Folino, Raffaele Melo, César Hernández Ríos, Luis López Plaza, Ramón |
| author_facet | Folino, Raffaele Melo, César Hernández Ríos, Luis López Plaza, Ramón |
| contents | This paper considers a one-dimensional generalized Allen-Cahn equation of the form \[ u_t = \varepsilon^2 (D(u)u_x)_x - f(u), \] 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 $u = α$ and $u = β$. It is shown that interface layers (namely, solutions that are equal to $α$ or $β$ 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 \emph{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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1911_06926 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Exponentially slow motion of interface layers for the one-dimensional Allen-Cahn equation with nonlinear phase-dependent diffusivity Folino, Raffaele Melo, César Hernández Ríos, Luis López Plaza, Ramón Analysis of PDEs 35K20, 35K57, 35B36, 82B26 This paper considers a one-dimensional generalized Allen-Cahn equation of the form \[ u_t = \varepsilon^2 (D(u)u_x)_x - f(u), \] 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 $u = α$ and $u = β$. It is shown that interface layers (namely, solutions that are equal to $α$ or $β$ 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 \emph{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. |
| title | Exponentially slow motion of interface layers for the one-dimensional Allen-Cahn equation with nonlinear phase-dependent diffusivity |
| topic | Analysis of PDEs 35K20, 35K57, 35B36, 82B26 |
| url | https://arxiv.org/abs/1911.06926 |