The influence of advection on the propagation phenomena of reaction-diffusion equations with KPP-bistable nonlinearity

TL;DR

This paper investigates how advection influences the propagation of reaction-diffusion waves in a heterogeneous environment with mixed KPP and bistable nonlinearities, revealing conditions for propagation speeds, delays, and extinction.

Contribution

It provides a comprehensive analysis of propagation phenomena in reaction-diffusion-advection equations with heterogeneous nonlinearities, highlighting the effects of advection rate and initial data.

Findings

Propagation occurs with specific speeds depending on advection rate.

Logarithmic delay in the level sets during leftward spreading.

Extinction or propagation determined by initial data and nonlinearity type.

Abstract

This paper is devoted to propagation phenomena for a reaction-diffusion-advection equation in a one-dimensional heterogeneous environment, where heterogeneity is reflected by the nonlinearity term -- being KPP type on $(-\infty, -L]$ and being bistable type on $[L,+\infty)$ for some $L>0$. A comprehensive analysis is presented on the influence of advection and heterogeneous reactions, based on various values of the advection rate $c$. Denote by $c_m$ and $c_b$ the spreading speeds of KPP and bistable reactions, respectively. When $c>-c_m$, it is shown that propagation can always occur with leftward spreading speed $c_m+c$ and rightward spreading speed $\min\big(\max(c_b-c,0),c_m-c\big)$. Moreover, a logarithmic delay of the level sets in the left direction is discovered. When $c \le -c_m$, propagation phenomena are determined by the initial data and by the sign of $c_b$. In particular, when $c_b>0$, the leftward propagation speed is $c_b+c$ if the initial population is "large enough"; whereas extinction occurs if the initial value is located in the bistable region and is "relatively small". In addition, the attractiveness of the bistable traveling wave is obtained when the leftward spreading speed is $c_b+c$ and/or when the rightward spreading speed is $c_b-c$.