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},\cr { v_t}=\Delta v -\lambda v+\mu u,\quad x\in \mathbb{ R}^{ N}. \end{ cases}(1) \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, $$ \lim_{ t\to\infty}\sup_{ |x|\geq ct}u(x,t;u_0,v_0)=0\quad \forall\,\, c>2\sqrt{ a} $$ and $$ \liminf_{ t\to\infty}\inf_{ |x|\leq ct}u(x,t;u_0,v_0)>0 \quad \forall\,\, 0<c<2\sqrt{ a}. $$ 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.
Categories: math.AP