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arXiv · 2503.14934

Layer dynamics for the Allen-Cahn equation with nonlinear phase-dependent diffusion

José Alejandro Butanda Mejía·Daniel Castañon Quiroz·Raffaele Folino·Luis Fernando Lopez Ríos
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Abstract

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 $D\equiv1$.

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José Alejandro Butanda Mejía, Daniel Castañon Quiroz, Raffaele Folino, Luis Fernando Lopez Ríos. 2025-03-19. Layer dynamics for the Allen-Cahn equation with nonlinear phase-dependent diffusion. https://doi.org/10.3934/dcds.2025132

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