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

2025/03/19 by Mejía, José Alejandro Butanda, Quiroz, Daniel Castañon, Folino, Raffaele +1
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.14934

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=ε²(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≡1.

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