Continuity of pulsating wave speeds for bistable reaction-diffusion equations in spatially periodic media

TL;DR

This paper investigates the continuity of pulsating wave speeds in multi-dimensional reaction-diffusion equations within spatially periodic media, removing previous restrictions and establishing conditions for wave existence in complex media.

Contribution

It proves the continuity of wave speeds without the nonzero speed assumption and provides new conditions for pulsating wave existence in rapidly oscillating media.

Findings

Wave speeds are continuous with respect to propagation direction without extra conditions.

Sufficient conditions for pulsating wave existence in rapidly oscillating media.

Multiple stable steady states can coexist in these media.

Abstract

This paper is concerned with pulsating waves for multi-dimensional reaction-diffusion equations in spatially periodic media. First, assuming the existence of pulsating waves connecting two linearly stable steady states, we study the continuity of wave speeds with respect to the direction of propagation. The continuity was proved in [15] under the extra condition that the speeds are nonzero in all directions. Here, we revisit this continuity result without the extra condition. Secondly, we provide some sufficient conditions ensuring the existence of pulsating waves in rapidly oscillating media, which allow the equations to have multiple stable steady states.