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

Abstract

We study traveling wave solutions of the following chemotaxis systems,$$\begin{cases}u_t=Δu-χ_1\nabla(u\nabla v_1)+χ_2\nabla(u\nabla v_2)+u(a-bu),\ x\in\mathbb{R}^N\\ 0=Δv_1-λ_1v_1+μ_1u,\ x\in\mathbb{R}^N,\\ 0=Δv_2-λ_2v_2+μ_2u,\ x\in\mathbb{R}^N,\end{cases}$$where $u(x,t), v_1(x,t)$ and $v_2(x,t)$ represent the population densities of a mobile species, a chemoattractant, and a chemo-repulsion, respectively. In an earlier work, we proved that there is a constant $K\geq0$ such that if $b+χ_2μ_2>χ_1μ_1+K$, then the steady solution $(\frac{a}{b},\frac{aμ_1}{bλ_1},\frac{aμ_2}{bλ_2})$ is asymptotically stable with respect to positive perturbations. In this paper, we prove that if $b+χ_2μ_2>χ_1μ_1+K$, then there exist a number $c^*(χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)\geq 2\sqrt a$ such that for every $c\in ( c^*(χ_1,μ_1,λ_1,χ_2,μ_2,λ_2) , \infty)$ and $ξ\in S^{N-1}$, the system has a traveling wave solution $(u,v_1,v_2)=(U(x\cdotξ-ct),V_1(x\cdotξ-ct),V_2(x\cdotξ-ct))$ 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$$\lim_{(χ_1,χ_2)\to(0^+,0^+)}c^{*}(χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)=\begin{cases}2\sqrt a\ \text{if}\ a\leq \min\{λ_1, λ_2\}\\ \frac{a+λ_1}{\sqrt{λ_1}}\ \text{if}\ λ_1\leq \min\{a, λ_2\}\\ \frac{a+λ_2}{\sqrt{λ_2}}\ \text{if}\ λ_2\leq \min\{a, λ_1\}\end{cases},\forall λ_1,λ_2,μ_1,μ_2>0,$$and$$\lim_{x\to\infty}\frac{U(x)}{e^{-\sqrt{a}μx}}=1,$$ where $μ$ solves $\sqrt a(μ+\frac{1}μ)=c$ in the interval $(0 , \min\{1,\sqrt{\frac{λ_1}{a}},\sqrt{\frac{λ_2}{a}}\})$.