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

## TL;DR

This paper extends the understanding of metastable pattern persistence in reaction-diffusion models by incorporating mean curvature-type diffusion in Euclidean and Lorentz--Minkowski spaces, demonstrating exponentially slow layer movement.

## Contribution

It introduces and analyzes metastable states in reaction-diffusion equations with mean curvature diffusion, a novel extension of classical models.

## Key findings

- Existence of metastable states with transition layers lasting exponentially long.
- Layer speeds are exponentially small in the mean curvature diffusion models.
- Numerical simulations confirm analytical predictions.

## 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.