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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

TL;DR

This work advances the understanding of metastability in the one-dimensional Allen–Cahn equation with nonlinear, phase-dependent diffusion by deriving a reduced ODE system that governs the slow motion of multiple transition layers. Central to the approach is a detailed stationary analysis, including the existence of monotone heteroclinic waves, degenerate-case compactons, and ℓ-periodic stationary states, together with precise asymptotics for key quantities like $\theta^j$ and the energy-like integral. The derived layer dynamics, $h_j' = (\varepsilon/S_G)(\theta^{j+1}-\theta^j)$, generalizes Carr–Pego to nonlinear diffusion and shows that diffusion through $D$ critically controls the speed and pattern of layer interaction via $A_\alpha,A_\beta$, with exponential smallness in $\varepsilon$. Numerical experiments corroborate the theory, demonstrating close agreement between the reduced ODE model and full PDE simulations for multiple layer configurations and holding the promise of efficient predictions for metastable evolution in nonlinear diffusion settings.

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 $D\equiv1$.

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

TL;DR

This work advances the understanding of metastability in the one-dimensional Allen–Cahn equation with nonlinear, phase-dependent diffusion by deriving a reduced ODE system that governs the slow motion of multiple transition layers. Central to the approach is a detailed stationary analysis, including the existence of monotone heteroclinic waves, degenerate-case compactons, and ℓ-periodic stationary states, together with precise asymptotics for key quantities like and the energy-like integral. The derived layer dynamics, , generalizes Carr–Pego to nonlinear diffusion and shows that diffusion through critically controls the speed and pattern of layer interaction via , with exponential smallness in . Numerical experiments corroborate the theory, demonstrating close agreement between the reduced ODE model and full PDE simulations for multiple layer configurations and holding the promise of efficient predictions for metastable evolution in nonlinear diffusion settings.

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 , where is a strictly positive function and 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 .

Paper Structure

This paper contains 8 sections, 13 theorems, 194 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The stationary problem
  3. Stationary waves
  4. Existence of compactons in the degenerate case
  5. Periodic solutions
  6. Estimates for the $\ell$-periodic solutions
  7. Layer dynamics
  8. Numerical evidence

Key Result

Theorem 1.3

Assume that $f,D\in C^2(I)$ satisfy eq:ass-D-eq:ass-f-eq:ass-int0-eq:ass-int1. Let $w,r$ be as in eq:vstruct-eq:r and let $A\in(0,r\sqrt{2\lambda})$, with $\lambda$ defined in eq:lambda. If $u^\varepsilon$ is the solution of eq:D-model-eq:Neu-eq:initial with initial datum $u_0^{\varepsilon}$ satisfy for any $m>0$.

Figures (2)

  • Figure 1: Initial datum $u_0$ with $N=6$ located at -3.4, -2, -0.5, 0.8, 2.2, 3.2.
  • Figure 2: Numerical solutions to \ref{['eq:D-model']}-\ref{['eq:Neu']}-\ref{['eq:initial']} with $D(u)=(1+u^2)^{-1}$, $f(u)=u^3-u$ and $\varepsilon=0.1$. The initial datum $u_0$ with 6 transitions is depicted in Figure \ref{['fig:pde:icond']}. This Figure is taken from FHLP.

Theorems & Definitions (28)

  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3: FHLP
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • ...and 18 more