Coupled reaction-diffusion equations with degenerate diffusivity: wavefront analysis
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2023
- DOI:
- 10.48550/arXiv.2311.05385
- arXiv:
- arXiv:2311.05385
- Bibcode:
- 2023arXiv231105385M
- Keywords:
-
- Mathematics - Analysis of PDEs;
- Mathematics - Classical Analysis and ODEs;
- 35C07;
- 35K55;
- 35K57