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

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.