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 $\chi ,\mu ,\lambda ,a,$ and $b$ are positive numbers, and $\tau \ge 0$. Among others, it is proved that if $b>2\chi \mu$ and $\tau \ge \frac{1}{2}(1-\frac{\lambda }{a}{)}_{+},$ then for every $c\ge 2\sqrt{a}$, (1) has a traveling wave solution $(u,v)(t,x)=(U^{\tau ,c}(x\cdot \xi -ct),V^{\tau ,c}(x\cdot \xi -ct))$ ($\forall \xi \in {\mathbb{R}}^N$) 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 considerably the results established in \cite{SaSh3}, and shows that (1) has a minimal wave speed $c_0^{\ast }=2\sqrt{a}$, which is independent of the chemotaxis.

arXiv admin note: text overlap with arXiv:1901.00045

References in corpus (2)

• Propagation in a Fisher-KPP equation with non-local advection *
• Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic Keller-Segel systems with logistic source?

Cited by in corpus (1)

• A reducing mechanism on wave speed for chemotaxis systems with degenerate diffusion