Front propagation in bistable reaction-diffusion-advection equations

Abstract

The paper deals with the existence and properties of front propagation between the stationary states $0$ and $1$ of the reaction-diffusion-advection equation $$ v_\tau + h(v)v_x = \big( D(v) v_x \big)_x + G(v), $$ where $G$ is a bistable reaction term and $D$ is a strictly positive diffusive process. We show that the additional transport term $h$ can cause the disappearance of such wavefronts and prove that their existence depends both on the local behavior of $G$ and $h$ near the unstable equilibrium and on a suitable sign condition on $h$ in $[0,1]$. We also provide an estimate of the wave speed, which can be negative even if $\int_0^1 G(u) D(u) du>0$, unlike what happens to the mere reaction-diffusion dynamic occurring when $h\equiv 0$.