Traveling waves of a full parabolic attraction-repulsion chemotaxis systems with logistic sources

In this paper, we study traveling wave solutions of the chemotaxis systems \begin{equation} \begin{cases} u_{t}=\Delta u -\chi_1\nabla( u\nabla v_1)+\chi_2 \nabla(u\nabla v_2 )+ u(a -b u), \qquad \ x\in$\mathbb{R}$ \ \tau\partial_tv_1=(\Delta- \lambda_1 I)v_1+ \mu_1 u, \qquad \ x\in$\mathbb{R}$, \ \tau\partial v_2=(\Delta- \lambda_2 I)v_2+ \mu_2 u, \qquad \
x\in$\mathbb{R}$, \end{cases} (0.1) \end{equation} where $\tau>0,\chi_{i}> 0,\lambda_i> 0,\ \mu_i>0$($i=1,2$)and$\ a>0,\ b> 0$areconstants,and$N$ is a positive integer. Under some appropriate conditions on the parameters, we show that there exist two positive constant $0<c^{\ast }(\tau ,{\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)<c^{\ast \ast }(\tau ,{\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)$ such that for every $c^{}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)\leq c<c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)$,$(0.1)$ has a traveling wave solution $(u,v_1,v_2)(x,t)=(U,V_1,V_2)(x-ct)$ connecting $(\frac{a}{b},\frac{a{\mu }_1}{b{\lambda }_1},\frac{a{\mu }_2}{b{\lambda }_2})$ and $(0,0,0)$ satisfying $$\underset{z\to \mathrm{\infty }}{\lim }\frac{U(z)}{e^{-\mu z}}=1,$$ where $\mu \in (0,\sqrt{a})$ is such that $c=c_{\mu }:=\mu +\frac{a}{\mu }$. Moreover, $$\underset{({\chi }_1,{\chi }_2)\to (0^{+},0^{+}))}{\lim }{c}^{\ast \ast }(\tau ,{\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)=\mathrm{\infty }$$ and $$\underset{({\chi }_1,{\chi }_2)\to (0^{+},0^{+}))}{\lim }{c}^{\ast }(\tau ,{\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)=c_{{\tilde{\mu }}^{\ast }},$$ where $\tilde{\mu}^={\min{\sqrt{a}, \sqrt{\frac{\lambda_1+\tau a}{(1-\tau){+}}},\sqrt{\frac{\lambda_2+\tau a}{(1-\tau){+}}}}}$. We also show that $(0.1)$ has no traveling wave solution connecting $(\frac{a}{b},\frac{a{\mu }_1}{b{\lambda }_1},\frac{a{\mu }_2}{b{\lambda }_2})$ and $(0,0,0)$ with speed $c<2\sqrt{a}$.