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

Abstract

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=Δu-χ\nabla\cdot ( u\nabla v) + u(a-bu),\quad x\in\mathbb{R}^{N},\cr {v_t}=Δv -λv+μu,\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, $$\underset{t\to \infty }{\lim }\underset{|x|\ge ct}{\sup }u(x,t;u_0,v_0)=0\forall c>2\sqrt{a}$$ and $$\underset{t\to \infty }{\mathrm{liminf}}\underset{|x|\le ct}{\inf }u(x,t;u_0,v_0)>0\forall 0<c<2\sqrt{a}.$$ 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 $\mathrm{supp}(u_0)$, $\mathrm{supp}(v_0)$ are nonempty and compact. It is well known that $2\sqrt{a}$ is the spreading speed of the following Fisher-KPP equation, $$u_t=\Delta u+u(a-bu),\forall x\in {\mathbb{R}}^N.$$ Hence, if $b>\frac{N\mu \chi }{4}$, the chemotaxis neither speeds up nor slows down the spatial spreading in the Fisher-KPP equation.