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, doi: 10.3934/dcdss.2020017
References:
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show all references

References:
[1]

S. AiW. Huang and Z.-A. Wang, Reaction, diffusion and chemotaxis in wave propagation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1-21. doi: 10.3934/dcdsb.2015.20.1. Google Scholar

[2]

S. Ai and Z.-A. Wang, Traveling bands for the Keller-Segel model with population growth, Math. Biosci. Eng., 12 (2015), 717-737. doi: 10.3934/mbe.2015.12.717. Google Scholar

[3]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of KellerSegel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. Google Scholar

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[6]

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H. Berestycki and G. Nadin, Asymptotic spreading for general heterogeneous Fisher-KPP type, 2015. <hal-01171334v2>.Google Scholar

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M. Bramson, Convergence of solutions of the Kolmogorov equation to traveling waves, Mem. Amer. Math. Soc., 44 (1983), iv+190 pp. doi: 10.1090/memo/0285. Google Scholar

[9]

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]

J. I. Diaz and T. Nagai, Symmetrization in a parabolic-elliptic system related to chemotaxis, Advances in Mathematical Sciences and Applications, 5 (1995), 659-680. Google Scholar

[11]

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

[12]

E. Espejoand and T. Suzuki, Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34. doi: 10.1016/j.aml.2014.04.007. Google Scholar

[13]

R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar

[14]

M. Freidlin, On wave front propagation in periodic media. In: Stochastic Analysis and Applications, ed. M. Pinsky, Advances in Probablity and Related Topics, 7 (1984), 147–166. Google Scholar

[15]

M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and ramdom media, Soviet Math. Dokl., 20 (1979), 1282-1286. Google Scholar

[16]

A. Friedman, Partial Differential Equation of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. Google Scholar

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M. FunakiM. Mimura and T. Tsujikawa, Travelling front solutions arising in the chemotaxisgrowth model, Interfaces Free Bound, 8 (2006), 223-245. doi: 10.4171/IFB/141. Google Scholar

[18]

E. GalakhovO. Salieva and J. I. Tello, On a Parabolic-Elliptic system with Chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647. doi: 10.1016/j.jde.2016.07.008. Google Scholar

[19]

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[20]

M. A. Herrero and J. J. L. Velasquez, A blow-up mechanism for a chemotaxis model, Annali Della Scuola Normale Superiore di Pisa, Classe di Scienze, 24 (1997), 633-683. Google Scholar

[21]

D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multispecies chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlin. Sci., 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x. Google Scholar

[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]

K. Kanga and A. Steven, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72. doi: 10.1016/j.na.2016.01.017. Google Scholar

[25]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[26]

E. F. Keller and L. A. Segel, A Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[27]

A. KolmogorovI. Petrowsky and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskogo Gos. Univ., 1 (1937), 1-26. Google Scholar

[28]

J. LiT. Li and Z.-A. Wang, Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849. doi: 10.1142/S0218202514500389. Google Scholar

[29]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154. Google Scholar

[30]

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