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This article concerns traveling wave solutions of the fully parabolic Keller-Segel chemotaxis system with logistic source, $$\displaylines{ u_t=\Delta u -\chi\nabla\cdot(u\nabla v)+u(a-bu),\quad x\in\mathbb{R}^N,\cr \tau v_t=\Delta v-\lambda v +\mu u,\quad x\in\mathbb{R}^N, }$$ 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.

This article concerns traveling wave solutions of the fully parabolic Keller-Segel chemotaxis system with logistic source, ... where χ, µ, λ, a, b are positive numbers, and τ ≥ 0. Among others, it is proved that if b > 2χµ and τ ≥ 1/2 (1 - λ/α)+, then for every c ≥ 2√α, this system has a traveling wave solution (u; v)(t; x) = (Uτ,c(x ⋅ξ - ct), Vτ,c (x ⋅ ξ - ct)) (for all ξ ∈ ℝN) connecting the two constant steady states (0; 0) and (a/b, µ/λ, a/b), and there is no such solutions with speed c less than 2√a, which improves the results established in [30], and shows that this system has a minimal wave speed c0* = 2√a, which is independent of the chemotaxis. [ABSTRACT FROM AUTHOR]