Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion

Abstract

We are interested in the existence and stability of traveling waves of arbitrary amplitudes to a chemotaxis model with porous medium diffusion. We first make a complete classification of traveling waves under specific relations among the biological parameters. Then we show all these traveling waves are asymptotically stable under appropriate perturbations. The proof is based on a Cole-Hopf transformation and the energy method.

Analysis of degenerate Burgers' equations involving small perturbation and large wave amplitude

“…The purpose of this paper is to provide a first attempt to break down these barriers and prove the existence and stability of shock waves to system (1.1). Moreover, we employ the same strategy to study the porous medium diffusion as in Ghani et al [21]. The porous medium problem was also studied in Choi and Kim [22] for the existence of chemotactic traveling waves with compact support…”

“…The purpose of this paper is to provide a first attempt to break down these barriers and prove the existence and stability of shock waves to system (). Moreover, we employ the same strategy to study the porous medium diffusion as in Ghani et al [21]. The porous medium problem was also studied in Choi and Kim [22] for the existence of chemotactic traveling waves with compact support $$$$ {\displaystyle \begin{array}{cc}\hfill & {r}_t=\left(\gamma \left(s,r\right)\left({r}_x-\frac{r}{s}{s}_x\right)\right),\hfill \\ {}\hfill & {s}_t=-\omega \left(s,r\right)r,\hfill \end{array}} $$$$ where $$$ r $$$ represents population density of bacteria, $$$ s $$$ represents resource (food) density that bacteria consume.…”