On the Existence of Axisymmetric Traveling Fronts in Lotka-volterra Competition-diffusion Systems in R

This paper is concerned with the following two-species LotkaVolterra competition-diffusion system in the three-dimensional spatial space { ∂ ∂t u1(x, t) = ∆u1(x, t) + u1(x, t) [1− u1(x, t)− k1u2(x, t)] , ∂ ∂t u2(x, t) = d∆u2(x, t) + ru2(x, t) [1− u2(x, t)− k2u1(x, t)] , where x ∈ R3 and t > 0. For the bistable case, namely k1, k2 > 1, it is well known that the system admits a one-dimensional monotone traveling front Φ(x + ct) = (Φ1(x + ct),Φ2(x + ct)) connecting two stable equilibria Eu = (1, 0) and Ev = (0, 1), where c ∈ R is the unique wave speed. Recently, twodimensional V-shaped fronts and high-dimensional 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 Ψ(x′, x3 + st) = ( Φ1(x ′, x3 + st),Φ2(x ′, x3 + st) ) in R3 connecting Eu = (1, 0) and Ev = (0, 1), where x′ ∈ R2. Here an axisymmetric traveling front means a traveling front which is axially symmetric with respect to the x3-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 R3. 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. 2010 Mathematics Subject Classification. 35K57, 35B35, 35B40.