Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source

Rachidi B. Salako

doi:10.3934/dcds.2019260

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October 2019, 39(10): 5945-5973. doi: 10.3934/dcds.2019260

Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source

Rachidi B. Salako ^,

Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA

^* Corresponding author: Rachidi B. Salako (salako.7@osu.edu)

Received December 2018 Published July 2019

In this paper, we study traveling wave solutions of the chemotaxis system

$ \begin{matrix} \left\{ \begin{array}{*{35}{l}} {{u}_{t}} = \Delta u-{{\chi }_{1}}\nabla (u\nabla {{v}_{1}})+{{\chi }_{2}}\nabla (u\nabla {{v}_{2}})+u(a-bu),\qquad \ x\in \mathbb{R} \\ \tau {{\partial }_{t}}{{v}_{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{array} \right. & (0.1) \\\end{matrix} $

where

$ \tau>0,\chi_{i}> 0,\lambda_i> 0,\ \mu_i>0 $

(

$ i = 1,2 $

) and

$ \ a>0,\ b> 0 $

are constants. Under some appropriate conditions on the parameters, we show that there exist two positive constant

$ c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)<c^{**}(\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

$ \lim\limits_{z\to \infty}\frac{U(z)}{e^{-\mu z}} = 1, $

where

$ \mu\in (0,\sqrt a) $

is such that

$ c = c_\mu: = \mu+\frac{a}{\mu} $

. Moreover,

$ \lim\limits_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) = \infty $

and

$ \lim\limits_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) = c_{\tilde{\mu}^*}, $

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 (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} $

Keywords: Parabolic-parabolic-parabolic chemotaxis system, logistic source, classical solution, local existence, global existence, asymptotic stability, traveling wave solutions.

Mathematics Subject Classification: 35B35, 35B40, 35K57, 35Q92, 92C17.

Citation: Rachidi B. Salako. Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5945-5973. doi: 10.3934/dcds.2019260