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Metastable patterns for a reaction-diffusion model with mean curvature-type diffusion

Raffaele Folino, Ramón G. Plaza, Marta Strani

Abstract

Reaction-diffusion equations are widely used to describe a variety of phenomena such as pattern formation and front propagation in biological, chemical and physical systems. In the one-dimensional model with a balanced bistable reaction function, it is well-known that there is persistence of metastable patterns for an exponentially long time, i.e. a time proportional to $\exp(C/\e)$ where $C,\e$ are strictly positive constants and $\e^2$ is the diffusion coefficient. In this paper, we extend such results to the case when the linear diffusion flux is substituted by the mean curvature operator both in Euclidean and Lorentz--Minkowski spaces. More precisely, for both models, we prove existence of metastable states which maintain a transition layer structure for an exponentially long time and we show that the speed of the layers is exponentially small. Numerical simulations, which confirm the analytical results, are also provided.

Metastable patterns for a reaction-diffusion model with mean curvature-type diffusion

Abstract

Reaction-diffusion equations are widely used to describe a variety of phenomena such as pattern formation and front propagation in biological, chemical and physical systems. In the one-dimensional model with a balanced bistable reaction function, it is well-known that there is persistence of metastable patterns for an exponentially long time, i.e. a time proportional to where are strictly positive constants and is the diffusion coefficient. In this paper, we extend such results to the case when the linear diffusion flux is substituted by the mean curvature operator both in Euclidean and Lorentz--Minkowski spaces. More precisely, for both models, we prove existence of metastable states which maintain a transition layer structure for an exponentially long time and we show that the speed of the layers is exponentially small. Numerical simulations, which confirm the analytical results, are also provided.

Paper Structure

This paper contains 8 sections, 12 theorems, 145 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Slow motion in the case of a saturating diffusion
  3. Slow motion in the case of an unbounded diffusion
  4. Layer Dynamics
  5. Numerical solutions
  6. Numerical experiment no. 1
  7. Numerical experiment no. 2: initial datum without layer structure
  8. Numerical experiment no. 3: degenerate potential

Key Result

Lemma 2.1

Let $u\in C([0,T],H^2(a,b))$ be solution of eq:Q-model-eq:Neu, with $Q$ as in eq:curv+. If $E_\varepsilon$ is the functional defined in eq:energy+, then for any $t\in(0,T)$.

Figures (5)

  • Figure 1: Numerical solutions to \ref{['eq:Q-model']}-\ref{['eq:Neu']}-\ref{['eq:initial']} with $F(u)=\frac{1}{4}(u^2-1)^2$, $\varepsilon=0.1$ and an initial profile $u_0$ with a 6-transition layer structure. The transition points are located at $(-3.4,-2,-0.5,0.8,2.2,3.2)$. In Figure \ref{['1left']}$Q$ is given by \ref{['eq:curv+']}, while in Figure \ref{['1right']}$Q$ is determined by \ref{['eq:curv-']}.
  • Figure 2: Numerical solutions to \ref{['eq:Q-model']}-\ref{['eq:Neu']}-\ref{['eq:initial']} with $Q$ given by \ref{['eq:curv+']}, $F(u)=\frac{1}{4}(u^2-1)^2$, $\varepsilon=0.01$ and discontinuous initial datum $u_0$, which has two jumps in $-0.05$ and $0.05$.
  • Figure 3: Numerical solutions to \ref{['eq:Q-model']}-\ref{['eq:Neu']}-\ref{['eq:initial']} with $Q$ given by \ref{['eq:curv-']}, $F(u)=\frac{1}{4}(u^2-1)^2$, $\varepsilon=0.1$ and initial datum $u_0(x)=[\cos(\frac{\pi x}{2})+\sin(\frac{\pi x}{2})]/100$.
  • Figure 4: Numerical solutions to \ref{['eq:Q-model']}-\ref{['eq:Neu']}-\ref{['eq:initial']} with $Q$ given by \ref{['eq:curv+']}, $F(u)=\frac{1}{8}(u^2-1)^4$, $\varepsilon=0.1$ and initial datum $u_0$ with a 6-transition layer structure and transition points located at $(-3.4,-2,-0.5,0.8,2.2,3.2)$, just as in the first numerical experiment (see Figure \ref{['fig:6trans']}).
  • Figure 5: Numerical solutions to \ref{['eq:Q-model']}-\ref{['eq:Neu']}-\ref{['eq:initial']} with $Q$ given by \ref{['eq:curv-']}, $F(u)=\frac{1}{12}(u^2-1)^6$, $\varepsilon=0.1$ and initial datum $u_0$ with a 6-transition layer structure and transition points located at $(-3.4,-2,-0.5,0.8,2.2,3.2)$.

Theorems & Definitions (24)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4: metastable dynamics with strong saturating diffusion
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 14 more