Finite traveling wave solutions in a degenerate cross-diffusion model for bacterial colony with volume filling

This work deals with the properties of the traveling wave solutions of a double degenerate cross-diffusion model \begin{document}$\begin{eqnarray*} \frac{\partial b}{\partial t} & = & D_b\nabla·\{n^pb(1-b)\nabla b\}+ n^qb^l, \\ \frac{\partial n}{\partial t} & = & D_n\nabla^2n-n^qb^l, \end{eqnarray*}$ \end{document} where \begin{document} $p≥q 0, q>1, l>1$ \end{document} . This system accounts for degenerate diffusion at the population density \begin{document} $n=b=0$ \end{document} and \begin{document} $b=1$ \end{document} modeling the growth of certain bacteria colony with volume filling. The existence of the finite traveling wave solutions is proven which provides partial answers to the spatial patterns of the colony. In order to overcome the difficulty of traditional phase plane analysis on higher dimension, we use Schauder fixed point theorem and shooting arguments in our paper.