Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source

The current paper is devoted to the study of traveling wave solutions of the following parabolic-elliptic-elliptic chemotaxis systems,

$\begin{equation}\label{main-eq-abstract}\begin{cases}u_{t} = Δ u- \nabla · ({ χ_1 u} \nabla v_1)+ \nabla · ({ χ_2 u} \nabla v_2) + u(a-bu), \;\;\;\;x∈\mathbb{R}^N, \\0 = Δ v_1-λ_1v_1+μ_1u, \;\;\;\;x∈\mathbb{R}^N, \\0 = Δ v_2-λ_2v_2+μ_2u, \;\;\;\; x∈\mathbb{R}^N, \end{cases}\;\;\;\;\;\;\;\;(0.1)\end{equation}$

where $a>0, \ b>0, $ $u(x, t)$ represents the population density of a mobile species, $v_1(x, t), $ represents the population density of a chemoattractant, $v_2(x, t)$ represents the population density of a chemorepulsion, the constants $χ_1≥ 0$ and $χ_2≥ 0$ represent the chemotaxis sensitivities, and the positive constants $λ_1, λ_2, μ_1$, and $μ_2$ are related to growth rate of the chemical substances. It was proved in an earlier work by the authors of the current paper that there is a nonnegative constant $K$ depending on the parameters $χ_1, μ_1, λ_1, χ_2, μ_2$, and $λ_2$ such that if $b+χ_2μ_2>χ_1μ_1+K$, then the positive constant steady solution $(\frac{a}{b}, \frac{aμ_1}{bλ_1}, \frac{aμ_2}{bλ_2})$ of (0.1) is asymptotically stable with respect to positive perturbations. In the current paper, we prove that if $b+χ_2μ_2>χ_1μ_1+K$, then there exists a positive number $c^{*}(χ_1, μ_1, λ_1, χ_2, μ_2, λ_2)≥ 2\sqrt{a}$ such that for every $ c∈ ( c^{*}(χ_1, μ_1, λ_1, χ_2, μ_2, λ_2)\ , \ ∞)$ and $ξ∈ S^{N-1}$, the system has a traveling wave solution $(u(x, t), v_1(x, t), v_2(x, t)) = (U(x·ξ-ct), V_1(x·ξ-ct), V_2(x·ξ-ct))$ with speed $c$ connecting the constant solutions $(\frac{a}{b}, \frac{aμ_1}{bλ_1}, \frac{aμ_2}{bλ_2})$ and $(0, 0, 0)$, and it does not have such traveling wave solutions of speed less than 2\sqrt a $. Moreover we prove that

$\begin{equation*}\lim\limits_{(χ_{1}, χ_2)?(0^+, 0^+)}c^{*}(χ_1, μ_1, λ_1, χ_2, μ_2, λ_2) = \begin{cases}\ 2\sqrt{a} \;\;\text{if}\;\; a≤ \min\{λ_1, λ_2\}\\\frac{a+λ_1}{\sqrt{λ_1}} \;\;\text{if}\;\; λ_1≤ \min\{a, λ_2\}\\\frac{a+λ_2}{\sqrt{λ_2}} \;\;\text{if}\;\; λ_2≤ \min\{a, λ_1\}\end{cases}\end{equation*}$

for every $ λ_1, λ_2, μ_1, μ_2>0$, and

$\begin{equation*}\lim\limits_{x?∞}\frac{U(x)}{e^{-\sqrt a μ x}} = 1, \end{equation*}$

where $μ$ is the only solution of the equation $μ+\frac{1}{μ} = \frac{c}{\sqrt{a}}$ in the interval $(0\ , \ \min\{1, \sqrt{\frac{λ_1}{a}}, \sqrt{\frac{λ_2}{a}}\})$.