On traveling wave solutions in full parabolic Keller-Segel chemotaxis systems with logistic source

Abstract

This paper is concerned with traveling wave solutions of the following full parabolic Keller-Segel chemotaxis system with logistic source, \begin{equation} \begin{cases} u_t=Δu -χ\nabla\cdot(u\nabla v)+u(a-bu),\quad x\in\mathbb{R}^N \cr τv_t=Δv-λv +μu,\quad x\in \mathbb{R}^N, \end{cases}(1) \end{equation} where $χ, μ,λ,a,$ and $b$ are positive numbers, and $τ\ge 0$. Among others, it is proved that if $b>2χμ$ and $τ\geq \frac{1}{2}(1-\fracλ{a})_{+} ,$ then for every $c\ge 2\sqrt{a}$, (1) has a traveling wave solution $(u,v)(t,x)=(U^{τ,c}(x\cdotξ-ct),V^{τ,c}(x\cdotξ-ct))$ ($\forall\, ξ\in\mathbb{R}^N$) connecting the two constant steady states $(0,0)$ and $(\frac{a}{b},\fracμλ\frac{a}{b})$, and there is no such solutions with speed $c$ less than $2\sqrt{a}$, which improves considerably the results established in \cite{SaSh3}, and shows that (1) has a minimal wave speed $c_0^*=2\sqrt a$, which is independent of the chemotaxis.

arXiv admin note: text overlap with arXiv:1901.00045