On the existence of axisymmetric traveling fronts in LotkaVolterra competitiondiffusion systems in $\mathbb{R}^3$
ZhiCheng Wang  School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China (email) Abstract: This paper is concerned with the following twospecies LotkaVolterra competitiondiffusion system in the threedimensional spatial space \[ \left\{ \begin{array}{l} \frac \partial {\partial t}u_1 (\mathbf{x}, t)=\Delta u_1(\mathbf{x}, t) + u_1 (\mathbf{x}, t)\left[ 1\ u_{1 }(\mathbf{x}, t)k_1u_2(\mathbf{x}, t)\right] , \\ \frac \partial {\partial t}u_2(\mathbf{x}, t)=d\Delta u_2(\mathbf{x}, t)+ru_2(\mathbf{x}, t)\left[ 1u_2(\mathbf{x}, t)k_2u_1(\mathbf{x}, t)\right] , \end{array} \right. \] where $\mathbf{x}\in \mathbb{R}^3$ and $t>0$. For the bistable case, namely $k_1,k_2>1$, it is well known that the system admits a onedimensional monotone traveling front $\mathbf{\Phi}(x+ct)=\left(\Phi_1(x+ct),\Phi_2(x+ct)\right)$ connecting two stable equilibria $\mathbf{E}_u=(1,0)$ and $\mathbf{E}_v=(0,1)$, where $c\in\mathbb{R}$ is the unique wave speed. Recently, twodimensional Vshaped fronts and highdimensional pyramidal traveling fronts have been studied under the assumption that $c>0$. In this paper it is shown that for any $s>c>0$, the system admits axisymmetric traveling fronts \[ \mathbf{\Psi}(\mathbf{x}^\prime, x_3+st)=\left(\Phi_1(\mathbf{x}^\prime, x_3+st),\Phi_2(\mathbf{x}^\prime, x_3+st)\right) \] in $\mathbb{R}^3$ connecting $\mathbf{E}_u=(1,0)$ and $\mathbf{E}_v=(0,1)$, where $\mathbf{x}^\prime\in\mathbb{R}^2$. Here an axisymmetric traveling front means a traveling front which is axially symmetric with respect to the $x_3$axis. Moreover, some important qualitative properties of the axisymmetric traveling fronts are given. When $s$ tends to $c$, it is proven that the axisymmetric traveling fronts converge locally uniformly to planar traveling wave fronts in $\mathbb{R}^3$. The existence of axisymmetric traveling fronts is obtained by constructing a sequence of pyramidal traveling fronts and taking its limit. The qualitative properties are established by using the comparison principle and appealing to the asymptotic speed of propagation for the resulting system. Finally, the nonexistence of axisymmetric traveling fronts with concave/convex level sets is discussed.
Keywords: LotkaVolterra competitiondiffusion system, bistability, axisymmetric
traveling front, existence, nonexistence, qualitative properties.
Received: July 2015; Revised: April 2016; Available Online: December 2016. 
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