Existence of Traveling wave solutions of parabolic-parabolic chemotaxis systems

TL;DR

This paper establishes the existence of traveling wave solutions in parabolic-parabolic chemotaxis systems, identifying critical speeds and parameter conditions under which these solutions connect specific steady states.

Contribution

It provides new existence results for traveling wave solutions in chemotaxis models, including precise speed bounds and parameter dependencies, extending previous understanding of such systems.

Findings

Existence of traveling wave solutions for speeds between specific bounds.

Critical chemotaxis sensitivity thresholds depending on parameters.

Asymptotic behavior of wave speeds as chemotaxis sensitivity approaches zero.

Abstract

The current paper is devoted to the study of traveling wave solutions of the following parabolic-parabolic chemotaxis systems, $$ \begin{cases} u_{t}= \Delta u-\chi \nabla \cdot (u \nabla v) + u(a-bu),\quad x\in\mathbb{R}^N \tau v_t=\Delta v-v+u, \quad x\in\mathbb{R}^N, \end{cases} $$ where $u(x,t)$ represents the population density of a mobile species and $v(x,t)$ represents the population density of a chemoattractant, and $\chi$ represents the chemotaxis sensitivity. We prove that for every $\tau >0$, there is $0<\chi_{\tau}^*<\frac{b}{2}$ such that for every $0<\chi<\chi_{\tau}^*$, there exist two positive numbers $2\sqrt a \le c^{*}(\chi,\tau)<c^{**}(\chi,\tau)$ satisfying that for every $ c\in [ c^{*}(\chi,\tau)\,\ c^{**}(\chi,\tau))$ and $\xi\in S^{N-1}$, the system has a traveling wave solution $(u(x,t),v(x,t))=(U(x\cdot\xi-ct;\tau),V(x\cdot\xi-ct;\tau))$ with speed $c$ connecting the constant solutions $(\frac{a}{b},\frac{a}{b})$ and $(0,0)$, and it does not have such traveling wave solutions of speed less than $2\sqrt a$. Moreover, $$ \lim_{\chi\to 0^+}c^{**}(\chi,\tau)=\infty,$$ $$ \lim_{\chi\to 0^+}c^{*}(\chi,\tau)=\begin{cases} 2\sqrt{a}\qquad \qquad \qquad\ \text{if} \quad 0<a\leq \frac{1+\tau a}{(1-\tau)_+}\cr \frac{1+\tau a}{(1-\tau)_{+}}+\frac{a(1-\tau)_{+}}{1+\tau a}\quad \text{if} \quad a\geq \frac{1+\tau a}{(1-\tau)_+}, \end{cases} $$ and $$ \lim_{x\to -\infty}\frac{U(x;\tau)}{e^{-\mu x}}=1, $$ where $\mu$ is the only solution of the equation $\mu+\frac{a}{\mu}=c$ in the interval $(0, \min\{\sqrt a, \sqrt{\frac{1+\tau a}{(1-\tau)_+}}\})$. Furthermore, it hods that $\lim_{\tau\to 0^+}\chi_{\tau}^*=\frac{b}{2}$.