Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on R^N

Abstract

In this paper, we study the spreading speeds and traveling wave solutions of the PDE $$\{\begin{array}{l}{u}_t=\Delta u-\chi \nabla \cdot (u\nabla v)+u(1-u),x\in {\mathbb{R}}^{N}0=\Delta v-v+u,x\in {\mathbb{R}}^N,\end{array}$$ where $u(x,t)$ and $v(x,t)$ represent the population and the chemoattractant densities, respectively, and $\chi$ is the chemotaxis sensitivity. It has been shown in an earlier work by the authors of the current paper that, when $0<\chi <1$, 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, if $0<\chi <\frac{1}{2}$, 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 $0<\chi <1$, then there are nonnegative constants $c_{-}^{\ast }(\chi )\le c_{+}^{\ast }(\chi )$ such that for every nonnegative initial function $u_0(\cdot )$ with nonempty compact support $$\underset{t\to \infty }{\lim }\underset{|x|\le ct}{\sup }[|u(x,t;u_0)-1|+|v(x,t;u_0)-1|]=0\forall 0<c<c_{-}^{\ast }(\chi )$$ and $$\underset{t\to \infty }{\lim }\underset{|x|\ge ct}{\sup }[u(x,t;u_0)+v(x,t;u_0)]=0\forall c>c_{+}^{\ast }(\chi ).$$ We also show that if $0<\chi <\frac{1}{2}$, there is a positive constant $c^{\ast }(\chi )$ such that for every $c\ge c^{\ast }(\chi )$, 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 $U(\cdot )$ and $V(\cdot )$ satisfying $(U(-\infty ),V(-\infty ))=(1,1)$ and $(U(\infty ),V(\infty ))=(0,0)$. Moreover, $$\underset{\chi \to 0}{\lim }{c}^{\ast }(\chi )=\underset{\chi \to 0}{\lim }{c}_{+}^{\ast }(\chi )=\underset{\chi \to 0}{\lim }{c}_{-}^{\ast }(\chi )=2.$$ We first give a detailed study in the case $N=1$ and next we extend these results to the case $N\ge 2$.

Cited by in corpus (2)

• Existence of Traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source
• Existence of Traveling wave solutions of parabolic-parabolic chemotaxis systems