Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony (Q2480628)

The authors investigate the existence of monotone finite traveling wave solutions for a model which describes the growth pattern of a bacteria on the surface of thin agar plates, namely the system \[ \begin{cases}\frac{\partial b}{\partial t}=D \frac{\partial}{\partial x}\left(n^pb \frac{\partial b}{\partial x}\right)+n^qb^l,\\ \frac{\partial n}{\partial t}= \frac{\partial^2n}{\partial x^2}-n^qb^b,\end{cases}\tag{S} \] where \((x,t)\in\mathbb{R}\times \mathbb{R}^+\), \(D\) is a positive constant, \(q>1\), \(p\geq 0\) and \(l>1\). For this aim, they use the Schauder fixed point theorem and some shooting and comparison arguments that reduce the above problem to the solvability of a first-order singular boundary value problem. Numerical simulations for \((S)\) with \(D=p=q=l=1\), subject to Neumann boundary condition and initial data are also presented.