Coupled reaction-diffusion equations with degenerate diffusivity: wavefront analysis

2023/11/09 by Muñoz-Hernández, Eduardo, Sovrano, Elisa, Taddei, Valentina
#35C07 #35K55 #35K57 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2311.05385

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), 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.

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