Coupled reaction-diffusion equations with degenerate diffusivity: wavefront analysis

We investigate traveling wave solutions for a nonlinear system of two coupled reaction-diffusion equations characterized by double degenerate diffusivity: nt=-f(n,b),bt=[g(n)h(b)bx]x+f(n,b). These systems mainly appear in modeling spatio-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.