Electron. J. Differential Equations, Vol. 2020 (2020), No. 53, pp. 1-18.

Traveling wave solutions for fully parabolic Keller-Segel chemotaxis systems with a logistic source

Rachidi B. Salako, Wenxian Shen

This article concerns traveling wave solutions of the fully parabolic Keller-Segel chemotaxis system with logistic source,

where $\chi, \mu,\lambda,a,b$ are positive numbers, and $\tau\ge 0$. Among others, it is proved that if $b>2\chi\mu$ and $\tau \geq \frac{1}{2}(1-\frac{\lambda}{a})_{+} $, then for every $c\ge 2\sqrt{a}$, this system has a traveling wave solution $(u,v)(t,x)=(U^{\tau,c}(x\cdot\xi-ct),V^{\tau,c}(x\cdot\xi-ct))$ (for all $\xi\in\mathbb{R}^N $) connecting the two constant steady states $(0,0)$ and $(\frac{a}{b},\frac{\mu}{\lambda}\frac{a}{b})$, and there is no such solutions with speed $c$ less than $2\sqrt{a}$, which improves the results established in [30] and shows that this system has a minimal wave speed $c_0^*=2\sqrt a$, which is independent of the chemotaxis.

Submitted August 11, 2019. Published May 27, 2020.
Math Subject Classifications: 35B35, 35B40, 35K57, 35Q92, 92C17.
Key Words: Parabolic chemotaxis system; logistic source; traveling wave solution; minimal wave speed.

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Rachidi B. Salako
Department of Mathematics
The Ohio State University
Columbus, OH 43210-1174, USA
email: salako.7@osu.edu
Wenxian Shen
Department of Mathematics and Statistics
Auburn University
Auburn, AL 36849, USA
email: wenxish@auburn.edu

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