February  2020, 13(2): 293-319. doi: 10.3934/dcdss.2020017

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

Department of Mathematics and Statistics, Auburn University, Auburn University, AL 36849, USA

* Corresponding author: Rachidi B. Salako

Received  February 2017 Revised  April 2018 Published  January 2019

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
$(0\ , \ \min\{1, \sqrt{\frac{λ_1}{a}}, \sqrt{\frac{λ_2}{a}}\})$
.
Citation: Rachidi B. Salako, Wenxian Shen. Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 293-319. doi: 10.3934/dcdss.2020017
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T. CieślakP. Laurencot and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system, Parabolic and Navier-Stokes Equations Banach Center Publications, Institute of Mathematics Polish Academy of Sciences Warszawa, 81 (2008), 105-117.  doi: 10.4064/bc81-0-7.  Google Scholar

[10]

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J. I. DiazT. Nagai and J.-M. Rakotoson, Symmetrization techniques on unbounded domains: Application to a chemotaxis system on ${{\mathbb{R}}^{N}}$, J. Differential Equations, 145 (1998), 156-183.  doi: 10.1006/jdeq.1997.3389.  Google Scholar

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[18]

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[22]

D. Horstmann and A. Stevens, A constructive approach to traveling waves in chemotaxis, J. Nonlin. Sci., 14 (2004), 1-25.  doi: 10.1007/s00332-003-0548-y.  Google Scholar

[23]

H. Y. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.  doi: 10.1016/j.jmaa.2014.09.049.  Google Scholar

[24]

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