Layer dynamics for the Allen-Cahn equation with nonlinear phase-dependent diffusion
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914164623015936 |
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| author | Mejía, José Alejandro Butanda Quiroz, Daniel Castañon Folino, Raffaele Ríos, Luis Fernando Lopez |
| author_facet | Mejía, José Alejandro Butanda Quiroz, Daniel Castañon Folino, Raffaele Ríos, Luis Fernando Lopez |
| contents | The goal of this paper is to describe the metastable dynamics of the solutions to the reaction-diffusion equation with nonlinear phase-dependent diffusion $u_t=\varepsilon^2(D(u)u_x)_x-f(u)$, where $D$ is a strictly positive function and $f$ is a bistable reaction term. We derive a system of ordinary differential equations describing the slow evolution of the metastable states, whose existence has been proved by Folino et al. (Z. Angew. Math. Phys., 2020). Such a system generalizes the one derived in the pioneering work of Carr and Pego (Comm. Pure Appl. Math., 1989) to describe the metastable dynamics for the classical Allen-Cahn equation, which corresponds to the particular case $D\equiv1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_14934 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Layer dynamics for the Allen-Cahn equation with nonlinear phase-dependent diffusion Mejía, José Alejandro Butanda Quiroz, Daniel Castañon Folino, Raffaele Ríos, Luis Fernando Lopez Analysis of PDEs The goal of this paper is to describe the metastable dynamics of the solutions to the reaction-diffusion equation with nonlinear phase-dependent diffusion $u_t=\varepsilon^2(D(u)u_x)_x-f(u)$, where $D$ is a strictly positive function and $f$ is a bistable reaction term. We derive a system of ordinary differential equations describing the slow evolution of the metastable states, whose existence has been proved by Folino et al. (Z. Angew. Math. Phys., 2020). Such a system generalizes the one derived in the pioneering work of Carr and Pego (Comm. Pure Appl. Math., 1989) to describe the metastable dynamics for the classical Allen-Cahn equation, which corresponds to the particular case $D\equiv1$. |
| title | Layer dynamics for the Allen-Cahn equation with nonlinear phase-dependent diffusion |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.14934 |