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# Spreading speeds of a parabolic-parabolic chemotaxis model with logistic source on $\mathbb{R}^{N}$

Dedicated to Professor Georg Hetzer on the occasion of his 75th Birthday

• The current paper is concerned with the spreading speeds of the following parabolic-parabolic chemotaxis model with logistic source on ${{\mathbb R}}^{N}$,

$\begin{equation} \begin{cases} u_{t} = \Delta u - \chi\nabla\cdot(u\nabla v)+ u(a-bu),\quad x\in{{\mathbb R}}^N, \\ {v_t} = \Delta v-\lambda v+\mu u,\quad x\in{{\mathbb R}}^N, \end{cases}\;\;\;\;\;\;\;\;\;\;\;\;\;\left(1\right) \end{equation}$

where $\chi, \ a,\ b,\ \lambda,\ \mu$ are positive constants. Assume $b>\frac{N\mu\chi}{4}$. Among others, it is proved that $2\sqrt{a}$ is the spreading speed of the global classical solutions of (1) with nonempty compactly supported initial functions, that is,

$\lim\limits_{t\to\infty}\sup\limits_{|x|\geq ct}u(x,t;u_0,v_0) = 0\quad \forall\,\, c>2\sqrt{a}$

and

$\liminf\limits_{t\to\infty}\inf\limits_{|x|\leq ct}u(x,t;u_0,v_0)>0 \quad \forall\,\, 0<c<2\sqrt{a}.$

where $(u(x,t;u_0,v_0), v(x,t;u_0,v_0))$ is the unique global classical solution of (1) with $u(x,0;u_0,v_0) = u_0$, $v(x,0;u_0,v_0) = v_0$, and ${\rm supp}(u_0)$, ${\rm supp}(v_0)$ are nonempty and compact. It is well known that $2\sqrt{a}$ is the spreading speed of the following Fisher-KPP equation,

$u_t = \Delta u+u(a-bu),\quad \forall\,\ x\in{{\mathbb R}}^N.$

Hence, if $b>\frac{N\mu\chi}{4}$, the chemotaxis neither speeds up nor slows down the spatial spreading in the Fisher-KPP equation.

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

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