# Singular limits of reaction diffusion equations and geometric flows with discontinuous velocity

## TL;DR

This paper studies the behavior of reaction diffusion equations with space-dependent velocities approaching discontinuities, showing solutions converge to stable states separated by a front with discontinuous velocity using viscosity solutions.

## Contribution

It introduces a framework for analyzing the singular limit of reaction diffusion equations with discontinuous velocities via viscosity solutions and geometric flow theory.

## Key findings

- Solutions converge to stable equilibria away from a propagating front.
- The front propagates with a discontinuous velocity.
- The convergence is global in time.

## Abstract

We consider the singular limit of a bistable reaction diffusion equation in the case when the velocity of the traveling wave solution depends on the space variable and converges to a discontinuous function. We show that the family of solutions converges to the stable equilibria off a front propagating with a discontinuous velocity. The convergence is global in time by applying the weak geometric flow uniquely defined through the theory of viscosity solutions and the level-set equation.