Existence, uniqueness, stability, and monotonicity of traveling waves for repulsion/attraction chemotaxis models with logistic type source

Abstract

This paper is devoted to the study of existence, uniqueness, stability, and monotonicity of traveling wave solutions to the following parabolic-elliptic chemotaxis system with logistic type source \begin{equation}\label{E:main-abstract-eq}\tag{CM} \begin{cases} u_t=u_{xx}-\chi(u^m v_x)_x +u(1-u^\alpha),\quad &x\in\mathbb{R}\cr 0=v_{xx}-v+u^\gamma,\quad&x\in\mathbb{R}, \end{cases} \end{equation} where $m,\alpha,\gamma\ge 1$ and $\chi\in\mathbb{R}$. System (CM) can be used to describe the evolution of a biological species influenced by a chemical substance produced by the species itself. In this context, the function $u$ denotes the population density of the biological species, and $v$ denotes the concentration of the chemical agent. Traveling wave solutions of (CM) connecting the two constant solutions $(1,1)$ and $(0,0)$ are among important types of solutions, which characterize the front propagation phenomena in (CM). The existence of such traveling wave solutions to (CM) with $m=\alpha=\gamma=1$ has been studied in several papers. However, there is little study on the uniqueness, stability, and monotonicity of traveling wave solutions of (CM) in literature and there is also no study on the existence of traveling wave solutions of (CM) connecting $(1,1)$ and $(0,0)$ for general $m,\alpha,\gamma\ge 1$. In the current paper, we prove the existence of traveling wave solutions of (CM) connecting $(1,1)$ and $(0,0)$ for any $\chi\le 0$ with speed $c$ large than some number $c^*_{\chi,m,\gamma}$, or for $0<\chi<1/2$ with any speed $c>2$. We prove that the traveling wave solutions are monotone when $\chi\le 0$. We also prove the uniqueness and stability of traveling wave solutions of (CM) connecting $(1,1)$ and $(0,0)$ when the speed $c$ is larger than some number $c^{**}_{\chi,m,\alpha, \gamma}(\ge c^*_{\chi,m,\gamma})$.