Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$

The current paper is devoted to the study of spreading speeds and traveling wave solutions of the following parabolic-elliptic chemotaxis system,

$\label{IntroEq0-2}\begin{cases}u_{t}=Δ{u}-χ\nabla·(u\nabla{v})+u(1-u),{x}∈\mathbb{R}^N,\\{0}=Δ{v}-v+u,{x}∈\mathbb{R}^N,\end{cases}$

where $u(x, t)$ represents the population density of a mobile species and $v(x, t)$ represents the population density of a chemoattractant, and $χ$ represents the chemotaxis sensitivity. We first give a detailed study in the case $N=1$. In this case, it has been shown in an earlier work by the authors of the current paper that, when $0 < χ < 1$, for every nonnegative uniformly continuous and bounded function $u_0(x)$, the system has a unique globally bounded classical solution $(u(x, t;u_0), v(x, t;u_0))$ with initial condition $u(x, 0;u_0)=u_0(x)$. Furthermore, it was shown that, if $0 < χ < \frac{1}{2}$, then the constant steady-state solution $(1, 1)$ is asymptotically stable with respect to strictly positive perturbations. In the current paper, we show that if $0 < χ < 1$, then there are nonnegative constants $c_ - ^*\left( \chi \right) \le c_ + ^*\left( \chi \right)$ such that for every nonnegative initial function $u_0(·)$ with non-empty and compact support ${\rm{supp}}(u_0)$,

$\mathop {\lim }\limits_{t \to \infty } \mathop {\sup }\limits_{|x| \le ct} [|u(x,t;{u_0}) - 1| + |v(x,t;{u_0}) - 1|] = 0\quad \forall {\mkern 1mu} {\mkern 1mu} 0 < c < c_ - ^*(\chi )$

and

$\mathop {\lim }\limits_{t \to \infty } \mathop {\sup }\limits_{|x| \le ct} [u(x,t;{u_0}) + v(x,t;{u_0})] = 0\quad \forall {\mkern 1mu} {\mkern 1mu} c > c_ + ^*(\chi ).$

We also show that if $0 < χ < \frac{1}{2}$, there is a positive constant $c^*(χ)$ such that for every $c \ge c^*(χ)$, the system has a traveling wave solution $(u(x, t), v(x, t))$ with speed $c$ and connecting $(1, 1)$ and $(0, 0)$, that is, $(u(x, t), v(x, t))=(U(x-ct), V(x-ct))$ for some functions $U(·)$ and $V(·)$ satisfying $(U(-∞), V(-∞))=(1, 1)$ and $(U(∞), V(∞))=(0, 0)$. Moreover, we show that

$\mathop {\lim }\limits_{\chi \to 0} {c^*}(\chi ) = \mathop {\lim }\limits_{\chi \to 0} c_ + ^*(\chi ) = \mathop {\lim }\limits_{\chi \to 0} c_ - ^*(\chi ) = 2.$

We then consider the extensions of the results in the case $N=1$ to the case $N \ge 2$.