Existence of Traveling wave solutions of parabolic-parabolic chemotaxis systems

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 $\lim_{\tau\to 0^+}\chi_{\tau}^*=\frac{b}{2}$.