Existence of Traveling wave solutions of parabolic-parabolic chemotaxis systems

The current paper is devoted to the study of traveling wave solutions of the following parabolic-parabolic chemotaxis systems, $$\{\begin{array}{l}{u}_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (u\mathrm{\nabla }v)+u(a-bu),x\in {\mathbb{R}}^N\tau v_t=\mathrm{\Delta }v-v+u,x\in {\mathbb{R}}^N,\end{array}$$ 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 $0<{\chi }_{\tau }^{\ast }<\frac{b}{2}$ such that for every $0<\chi <{\chi }_{\tau }^{\ast }$, there exist two positive numbers $2\sqrt{a}\le c^{\ast }(\chi ,\tau )<c^{\ast \ast }(\chi ,\tau )$ satisfying that for every $c\in [c^{\ast }(\chi ,\tau )c^{\ast \ast }(\chi ,\tau ))$ 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, $$\underset{\chi \to 0^{+}}{\lim }{c}^{\ast \ast }(\chi ,\tau )=\mathrm{\infty },$$ $$\underset{\chi \to 0^{+}}{\lim }{c}^{\ast }(\chi ,\tau )=\{\begin{array}{l}2\sqrt{a}\text{ if}0<a\le \frac{1+\tau a}{(1-\tau {)}_{+}}\\ \frac{1+\tau a}{(1-\tau {)}_{+}}+\frac{a(1-\tau {)}_{+}}{1+\tau a}\text{if}a\ge \frac{1+\tau a}{(1-\tau {)}_{+}},\end{array}$$ and $$\underset{x\to -\mathrm{\infty }}{\lim }\frac{U(x;\tau )}{e^{-\mu x}}=1,$$ where $\mu$ is the only solution of the equation $\mu +\frac{a}{\mu }=c$ in the interval $(0, \min{\sqrt a, \sqrt{\frac{1+\tau a}{(1-\tau)+}}})$\mathrm{.}Furthermore,ithodsthat$\lim{\tau\to 0^+}\chi_{\tau}^*=\frac{b}{2}$.