Coupled reaction-diffusion equations with degenerate diffusivity: wavefront analysis

TL;DR

This paper analyzes traveling wave solutions in a coupled reaction-diffusion system with degenerate diffusivity, relevant for modeling bacterial growth patterns, establishing existence, threshold speeds, and regularity conditions.

Contribution

It proves the existence of monotone traveling wave solutions for all wave speeds in a half-line and discusses their regularity related to wave speed.

Findings

Existence of traveling wave solutions for all wave speeds in a closed half-line.

Threshold speed estimates for wave solutions.

Regularity conditions depending on wave speed.

Abstract

We investigate traveling wave solutions for a nonlinear system of two coupled reaction-diffusion equations characterized by double degenerate diffusivity: \[n_t= -f(n,b), \quad b_t=[g(n)h(b)b_x]_x+f(n,b).\] These systems mainly appear in modeling spatial-temporal patterns during bacterial growth. Central to our study is the diffusion term $g(n)h(b)$, which degenerates at $n=0$ and $b=0$; and the reaction term $f(n,b)$, which is positive, except for $n=0$ or $b=0$. Specifically, the existence of traveling wave solutions composed by a couple of strictly monotone functions for every wave speed in a closed half-line is proved, and some threshold speed estimates are given. Moreover, the regularity of the traveling wave solutions is discussed in connection with the wave speed.