# Propagating speeds of bistable transition fronts in spatially periodic media

## TL;DR

This paper investigates the speeds of transition fronts in spatially periodic bistable reaction-diffusion equations, establishing properties of front speeds and profiles, and relating general transition front speeds to pulsating front speeds.

## Contribution

It demonstrates continuity and differentiability of front speeds with respect to direction and bounds the speeds of general transition fronts between pulsating front speeds.

## Key findings

- Front speeds are continuous and differentiable in direction.
- Transition front speeds are bounded by pulsating front speeds.
- General transition front speeds exceed the infimum and are below the supremum of pulsating front speeds.

## Abstract

This paper is concerned with the propagating speeds of transition fronts in $R^N$ for spatially periodic bistable reaction-diffusion equations. The notion of transition fronts generalizes the standard notions of traveling fronts. Under the a priori assumption that there exist pulsating fronts for every direction $e$ with nonzero speeds, we show some continuity and differentiability properties of the front speeds and profiles with respect to the direction $e$. Finally, we prove that the propagating speed of any transition front is larger than the infimum of speeds of pulsating fronts and less than the supremum of speeds of pulsating fronts.