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.

Ψ(x , x 3 + st) = Φ 1 (x , x 3 + st), Φ 2 (x , x 3 + st) in R 3 connecting Eu = (1, 0) and Ev = (0, 1), where x ∈ 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 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.

Introduction.
In this paper we study the existence of traveling wave solutions in the following two-specis Lotka-Volterra competition-diffusion system in the threedimensional spatial space: x ∈ R 3 , t > 0, (1) where k 1 , k 2 , r and d are positive constants, the variables u 1 (x, t) and u 2 (x, t) are the population densities of two competing species.In the field of population biology the Lotka-Volterra competition system is known as a physiological model describing competing interactions of multiple species.The related kinetic system of (1) is as follows: ( We note that systems (1) and ( 2) are normalized so that they have the equilibrium solutions E u = (1, 0) and E v = (0, 1).Obviously, E 0 = (0, 0) is also an equilibrium of systems ( 1) and (2).When k 1 , k 2 < 1 or k 1 , k 2 > 1, there exists the fourth equilibrium (co-existence state) E * = (E * ,1 , E * ,2 ), where In general, the species u 1 is called a strong (weak, resp.)competitor if k 2 > 1 (k 2 < 1, resp.).For the case 0 < k 1 < 1 < k 2 (or 0 < k 2 < 1 < k 1 ), one species is superior than the other.In this case, there is only one stable equilibrium and it is called the monostable case.For the case k 1 , k 2 > 1, both E u and E v are stable and it is called the bistable (strong competition) case.We call the case when k 1 , k 2 < 1 the weak competition (coexistence) case.See Guo and Wu [15] and Morita and Tachibana [37].
Traveling wave solutions of (1) in the one-dimensional spatial space have been extensively studied in the literature.We refer to a nice survey by Guo and Wu [15], see also Li et al. [32], Lin and Li [34], Morita and Tachibana [37], Zhao and Ruan [56] and the references therein.In this paper we are interested in the bistable case (or the strong competition case), namely, k 1 , k 2 > 1.As mentioned above, in this case both E u and E v are stable.Following from Conley and Gardner [7], Gardner [11], Kan-on [27], Kan-on and Fang [30], and Volpert et al. [50], we know that in the bistable case system (1) admits a one-dimensional traveling wave front Φ (x + ct) = (Φ 1 (x + ct) , Φ 2 (x + ct)) connecting E v and E u , where x ∈ R, t > 0, and the wave speed c ∈ R. The traveling wave front Φ (ξ) = (Φ 1 (ξ) , Φ 2 (ξ)) with ξ = x + ct is unique up to translation and satisfies Φ 1 (ξ) > 0 and Φ 2 (ξ) < 0 for any ξ ∈ R. In particular, the wave speed c is also unique.It should be pointed out that though the existence of traveling wave fronts is well known, there are few conclusions on the sign of the wave speed c of traveling wave solutions of (1) in the bistable case since it is difficult to determine the sign of c.Recently, Guo and Lin [14] gave some sufficient criteria about the sign of the wave speed under some parameter restrictions by using the result of Kan-on [27] (see also Alcahrani et al. [1] for the sign of wave speed for near-degenerate bistable competition models).For the positive stationary solutions in the bistable case, we refer to Kan-on [29] for the instability of stationary solutions and Kan-on [28] for the standing waves.Other than traveling wave solutions, there are solutions with two fronts approaching each other from both ends of the real line, which are called entire solutions and are constructed by Morita and Tachibana [37].Here an entire solution means a solution defined for all t ∈ R.Moreover, we refer to Guo and Wu [16] for a two-component lattice dynamical system arising in strong competition models and Lin and Li [34] for a Lotka-Volterra competition-diffusion system with nonlocal delays, respectively.
The above results on the existence of traveling wave solutions of ( 1) are only about one-dimensional traveling wave solutions (or planar traveling wave solutions in high-dimensional spaces).However, it is observed that in high-dimensional spaces propagating wave fronts may change shape and evolve to new nonplanar traveling waves.Therefore, it is interesting but challenging to study possible nonplanar traveling waves.In the past decade, multi-dimensional traveling fronts have attracted a lot of attention and new types of nonplanar traveling waves have been obtained for the scalar reaction-diffusion equation with various nonlinearities.For the combustion nonlinearity, see Bonnet and Hamel [3], Hamel and Monneau [17], and Hamel et al. [18].For the Fisher-KPP case, see Brazhnik and Tyson [4], Hamel and Roquejoffre [21] and Huang [26].For the bistable case (in particular the Allen-Cahn equation), see Fife [10], Hamel et al. [19,20], Ninomiya and Taniguchi [39,40] and Gui [13] for V-form front solutions with m = 2, Chen et al. [6], Hamel et al. [19,20] and Taniguchi [48] for cylindrically symmetric traveling fronts with m ≥ 3, and Taniguchi [46,47,48] and Kurokawa and Taniguchi [31] for traveling fronts with pyramidal shapes with m ≥ 3.For traveling fronts with V-shape, pyramidal shape and conical shape for a bistable reaction-diffusion equation with time-periodic nonlinearity, we refer to Wang and Wu [54], Sheng et al. [44] and Wang [52], respectively.For non-connected traveling fronts and non-convex traveling fronts, we refer to del Pino et al. [41,42].We also refer to a survey by Witelski et al. [55] on axisymmetric traveling waves of semi-linear elliptic equations.Other related works can be found in Chapuisat [5], El Smaily et al. [9], Fife [10], Hamel and Roquejoffre [22], and Morita and Ninomiya [36].
Recently, there have been important progresses on the study of nonplanar traveling wave solutions in systems of reaction-diffusion equations.By using bifurcation theory, Haragus and Scheel [23,24,25] studied almost planar waves (V-form waves) in reaction-diffusion systems in which the interface region is close to hyperplanes (the angle of the interface is close to π).By developing the arguments of Ninomiya and Taniguchi [39,40], Wang [51] established the existence and stability of twodimensional V-form curved fronts for bistable reaction-diffusion systems for any admissible wave speed.In particular, the result of Wang [51] are applicable to system (1) with k 1 , k 2 > 1 in R 2 .Furthermore, the existence, uniqueness and stability of pyramidal traveling fronts of bistable reaction-diffusion systems in R 3 were established in Wang et al. [53] by extending the arguments of Taniguchi [46,47].The result of [53] is also applicable to system (1) with k 1 , k 2 > 1 in R 3 .At the same time, Ni and Taniguchi [38] also established the existence of pyramidal traveling fronts of (1) with k 1 , k 2 > 1 in R m (m ≥ 3).We refer to [53,38] and the next section for details on the pyramidal traveling fronts of (1).
In this paper we are interested in the axisymmetric traveling wave solutions of (1) with k 1 , k 2 > 1 in x ∈ R 3 .Though axisymmetric traveling wave solutions in scalar bistable reaction-diffusion equations have been studied before (see Hamel et al. [19,20] and Taniguchi [48]), here we would like to emphasize that there is no result about the axisymmetric traveling wave solutions of bistable reaction-diffusion systems in R 3 up to now (according to our best knowledge).The purpose of the current paper is to establish the existence of axisymmetric traveling fronts with wave speed s > c > 0 for (1) in R 3 and to show some important qualitative properties of the axisymmetric traveling fronts.In addition, we will show the nonexistence of axisymmetric traveling fronts.Now we state the main results of this paper.
Theorem 1.1.Assume that k 1 , k 2 > 1 and c > 0, where c is the wave speed of the planar traveling wave front Φ (x 3 and |e| = 1.Then for any s > c, there exists a function In addition, one has of Theorem 1.1, we call the function Ψ(x) an axisymmetric traveling front of (1).As stated previously, it is difficult to determine the sign of the wave speed c of traveling wave solutions of (1) with k 1 , k 2 > 1.For the reader's convenience, here we recall some sufficient conditions from Guo and Lin [14] to ensure c > 0. That is, one has c > 0 if one of the following conditions holds: In the following we evaluate the limit of axisymmetric traveling fronts Ψ(x) with speed s > c as s → c and the nonexistence of axisymmetric traveling fronts.
Theorem 1.4.For s < c, there is no axisymmetric traveling front Ψ(x) of (1) satisfying (4), lim x3→+∞ Ψ(0, The result of Theorem 1.3 corresponds to Remark 1.7 of Hamel et al. [19] for the scalar bistable equation, see also Hamel and Monneau [17, Theorems 1.1 and 1.6] for the combustion equation.As reported by Hamel et al. [19], in the terminology of Haragus and Scheel [24], there is no exterior corner for system (1), while the solutions given in Theorem 1.1 are interior corners.Theorem 1.4 further implies that there must be s > c for an interior corner.See also Haragus and Scheel [24,Theorem 1.1].
In this paper we prove Theorems 1.1-1.4 by using the results of Wang et al. [53] and Ni and Taniguchi [38] on the pyramidal traveling fronts of (1).Set u * 2 = 1 − u 2 and transform system (1) into (for the sake of simplicity, we drop the symbol * ) x ∈ R 3 , t > 0. ( Correspondingly, the equilibria , respectively, where ) is a traveling wave front of ( 5) connecting E 1 = (1, 1) and E 0 = (0, 0), where ξ = x•e+ct, e ∈ R 3 with |e| = 1.In particular, U(−∞) = E 0 , U(+∞) = E 1 , U 1 (ξ) > 0 and U 2 (ξ) > 0 for any ξ ∈ R. To complete the proof of Theorem 1.1, we need only to prove that there exists an axisymmetric traveling front W(x) of ( 5) satisfying ( 14) and (ii)-(vii) of Theorem 3.1 in Section 3. In order to obtain such a function W(x), we use the results of Wang et al. [53] to construct a sequence of pyramidal traveling fronts of (5), and then take a limit for the sequence of pyramidal traveling fronts.Thus, the limit function is just the expected solution.This step is similar to that in Taniguchi [48].However, due to the effect of the coupled nonlinearity, we cannot use the arguments of Taniguchi [48] and Hamel et al. [19] to prove qualitative properties of the axisymmetric traveling front W(x) (namely (ii)-(vii) of Theorem 3.1).Therefore, in this paper we develop a new method to show (ii)-(vii) of Theorem 3.1 in Section 3, where a crucial procedure is to use the comparison principle and appeal to the spreading speed of solutions of the resulting equations (systems).The proof of Theorem 1.2 can be completed by using the result of Theorem 1.1.
In Section 4, we prove Theorems 1.3 and 1.4, which imply the nonexistence of axisymmetric traveling fronts.Before giving the proofs of Theorems 1.1-1.4,we first show the existence results and some qualitative properties of pyramidal traveling fronts of (5) in Section 2. In Section 5, we give the proofs of two important lemmas, which are listed in Section 4. Finally, in Section 6 we give a discussion on the obtained results of this paper.

2.
Preliminaries.In this section we state the existence results on the pyramidal traveling fronts of (5) in R 3 , which has been established by Ni and Taniguchi [38] and Wang et al. [53].Then we show some properties of the pyramidal traveling fronts which are very important to establish the axisymmetric traveling wave solutions in next section.Suppose be the planar traveling wave front of ( 5) connecting E 0 and E 1 .Assume c > 0.
For two vectors c = (c 1 , c 2 ) and where BC R 3 , R 2 denotes the set of bounded and continuous functions defined on R 3 .Fix s > c > 0. Assume that the solutions travel towards the −x 3 direction without loss of generality.Take Then we have the following initial value problem where x ∈ R 3 , t > 0.
Let n ≥ 3 be a given integer and Let {A j = (A j , B j )} n j=1 be a set of unit vectors in R 2 such that Then Γ := ∪ n j=1 Γ j represents the set of all edges of a pyramid.Define )) be the solution of ( 6) with v 0 = v − .Then there exists a function V(x) ∈ C 2 R 3 such that The following theorem is obtained by Wang et al. [53,Theorem 1.1], see also Ni and Taniguchi [38].
Theorem 2.1.For each s > c > 0, there exists a solution u(x, t) = V(x , x 3 + st) of (5 Moreover, for any the solution u(x, t; u 0 ) of (5) with initial value u 0 satisfies The next two lemmas show the monotonicity of the pyramidal traveling front V.
It follows from Lemma 2.3 that ∂ ∂x3 V(x) 0 for any x ∈ R 3 .In the following we further show that if the initial value v 0 is even in x 1 , then the solution v(x, t; v 0 ) is also even in x 1 .Furthermore, if v 0 is nondecreasing in x 1 ≥ 0, then the solution v(x, t; v 0 ) is also nondecreasing in x 1 ≥ 0.Here we use a method which is different from that in Taniguchi [48,Lemma 3.5].
Proof.Let ε > 0. We show that F (x + ε, t) − F (x, t) ≥ 0. We have By direct calculations, we have In view of x ≥ 0, we obtain This completes the proof.
Proof.The proof is divided into three steps. Step , where BU C R 3 , R 2 denotes the set of bounded and uniformly continuous functions defined on R 3 .Define

By results in Daners and
McLeod [8] we know that T (t) is a strongly continuous analytic semigroup of contractions on X which is generated by D∆ X , where D = 1 0 0 d and ∆ X is the X-realisation of ∆.In particular, for any u (x) ∈ X we have We identify an element For any w (t) ∈ S ∞ , consider the following initial-valued problem where The existence and uniqueness of solutions of ( 11) are well known.In particular, the solution u (x, t) of ( 11) satisfies the following integral equation where e −Mt =diag e −M1t , e −M2t , H (w) = H 1 (w) , H 2 (w) and Since H i (w) are nondecreasing on w j (j = 1, 2), it is easy to show that By virtue of (10), we further obtain that u (t) ∈ C ([0, ∞) , X).
In view of the evenness of v 0 (x) and w (x, t) in x 1 , we know that the solution u (x, t) of ( 11) is also even on x 1 ∈ R. In particular, we have Consequently, we have that the solution u (x, t) of ( 11) satisfies the following parabolic problem where Ω = x ∈ R 3 , x 1 > 0 , ∂ ∂n denotes the outside normal derivative.It is known that the solution u(x, t) = (u 1 (x, t), u 2 (x, t)) of ( 12) satisfies the following integral equation H i (w (x, y, z, s)) dxdydzds, where i = 1, 2. By the uniqueness of solutions of ( 12), we have u (x, t) ≡ u (x, t) on Ω × [0, ∞).Following Lemma 2.4, we know that the solution u (x, t) of ( 11) is nondecreasing on x 1 ∈ (0, ∞).Define an operator A by u (t) = Aw (t).Then we know that A maps S ∞ into S ∞ .
Step 2. Fix T 0 > 0. Denote we can show that A mapping S T0 into S T0 is a contraction map.Thus, the contraction mapping principle implies that there exists a unique u (t) ∈ S T0 such that satisfying (13).
Step 3. Let v (x, t) := u (x 1 , x 2 , x 3 − st, t) for any x ∈ R 3 and t > 0, where u (x, t) is the solution of (13).Then it is easy to see that v (x, t) is the solution of (6).Since u ∈ S ∞ , we have that v (x, t) is symmetric on x 1 and is nondecreasing in x 1 ∈ [0, +∞).This completes the proof.
Corollary 1. Suppose that v − (x) is even in x 1 ∈ R and x 2 ∈ R, respectively.Then the pyramidal traveling front V(x) defined by Theorem 2.1 satisfies Proof.Since v − (x) is even in x 1 and x 2 , respectively, it is easy to see that v − (x) is nondecreasing in x 1 > 0 and x 2 > 0, respectively.It follows from Lemma 2.5 that v(x, t; v − ) is even in x 1 and x 2 and is nondecreasing in x 1 > 0 and x 2 > 0, respectively.Thus, V(x) is even in x 1 and x 2 and is nondecreasing in x 1 > 0 and x 2 > 0, respectively.By (7), we obtain Therefore, we have By Theorem 2.1 we have Applying the maximum principle (Potter and Weinberger [43]) for the scalar equation yields This completes the proof.
3. Axisymmetric traveling fronts.In this section we establish the existence of axisymmetric traveling fronts of ( 5) in R 3 .The method is to take the limit of a sequence of pyramidal traveling fronts.Consequently, we show some important qualitative properties of the axisymmetric traveling fronts.Let It is not difficult to show that the plane for any k ∈ N and 1 ≤ i ≤ 2 k .Replacing h(x ) by h k (x ) in Theorem 2.1, we obtain a sequence of pyramidal traveling fronts of ( 5), namely, Denote the edge of the pyramid Since v k,− (x) is nondecreasing in x 1 ∈ (0, ∞) and x 2 ∈ (0, ∞) and is even in x 1 ∈ R and x 2 ∈ R, respectively, by Theorem 2.1, Lemma 2.3 and Corollary 1 we obtain , where . By Lemmas 2.2 and 2.3 and Corollary 1 we have that V k (x) satisfies: Then we have the following theorem for the function (14) In addition, one has (vii) ∂ ∂ν W (x) 0 for any x ∈ R 3 , where It can be shown that W(x) satisfies ( 14) and (i) and (ii) of Theorem 3.1.In view of we can prove (iii) of Theorem 3.1.In the following we prove (iv)-(vii) of Theorem 3.1 by a sequence of lemmas.Following the properties (a)-(f) of V(x) k , we have: Now we show that W 1 (x) ≡ θ 1 for x ∈ R 3 .On the contrary we assume that (a) If θ 1 = 0, it follows from the second equation of ( 14) that which implies that W (x 1 , x 2 , x 3 − st) is a solution of the following Fisher-KPP equation Then u(x, t) is also a supersolution of (15).In particular, u(x, t) ≤ u(x, 0) ≤ By the classical results of Aronson and Weinberger [2, Corollary 1] on the asymptotic speed of propagation for the Fisher-KPP equation, we know that the solution u(x, t; ϕ) of ( 15) with initial value ϕ satisfies lim satisfies that ϕ(•) ≡ 0 and supp ϕ is compact, where τ * = 2 √ dr.However, the comparison principle implies that if ϕ further satisfies ϕ(x) < u(x, 0) for any x ∈ R 3 .It is obvious that (17) contradicts the fact (16).Therefore, Then by the first equation of ( 14) we have Following the above arguments, we conclude that W 1 (x) ≡ θ 1 for x ∈ R 3 .This completes the proof.
In the following we prove (v) of Theorem 3.1.To reach the aim, we first present a lemma, which will be proved in Section 5. Consider the following reaction-diffusion system in R Lemma 3.3.There exists κ * > 0 such that for any κ ≥ κ * system (18) admits an increasing traveling wave front (ρ 1 (x + κt) , ρ 2 (x + κt)) connecting the equilibria E 0 and E * , and for any κ < κ * system (18) does not admit an increasing traveling wave front (ρ 1 (x + κt) , ρ 2 (x + κt)) connecting the equilibria E 0 and E * .Now we prove (v) of Theorem 3.1, namely, we have the following lemma.
In order to complete the proof of Theorem 3.1 (vi), we need further to show that To do this, it is sufficient to show that there must be (α 1 , α 2 ) = E 1 .Our method is to assume (α 1 , α 2 ) = E * and derive a contradiction, which is similar to the argument in Lemma 3.4.To obtain the contradiction, we first consider the spreading speed for a cooperation reaction-diffusion system, namely, the below (23).Since we are working in three-dimensional spatial space, the results of Liang and Zhao [33] on the spreading speed of the monotone semiflow are not applicable, so we use the theory of Thieme and Zhao [49].Before presenting the lemma, we introduce some notations, which come from Thieme and Zhao [49].Let σ > 0 and u(x, t) : R This is equivalent to the statement that for any ε > 0, there exists some t > 0 such that |u(x, s) − u * | < ε whenever s > t and |x| ≤ σs.Let is the unique positive root of the equation , where θ 2 satisfies the assumption (H).Define Consider the following system x ∈ R 3 , t > 0.
(23) Due to the strict convexity of the function v − k1 is the unique positive equilibrium of system (23).In particular, Here we emphasize that the reason that we can establish the spreading speed for solutions of system (23) is that the theory of Thieme and Zhao [49] only works for scalar nonlinear integral equations and system (23) exactly enables us to reduce it to a scalar integral equation for the second component u 2 .But it is difficult to reduce any component of solutions of the original Lotka-Volterra competitiondiffusion system to a scalar integral equation.
We postpone to prove the lemma in subsection 5.2.Now we are in the position to prove (21).Lemma 3.7.One has Proof.As mentioned above, we need only to show lim x3→+∞ W(0, 0, x 3 ) = E 1 .By Lemma 3.5, we have that Therefore, it is sufficient to show that the latter is impossible.
Up to now, we have completed the proof of Theorem 3.1 (vi).Theorem 3.1 (vii) can be easily proved by using the results of Theorem 3.1 (ii)-(v) and the maximum principle.Here we omit the details of the proof.Thus, we have completed the proof of Theorem 3.1.Theorem 3.8.Let s > c > 0 and denote the axisymmetric traveling front W(x) defined in Theorem 3.1 by where (U, c) is the planar traveling wave front of (5) connecting E 0 and E 1 .
Proof.Note that there exists as n → ∞.
By (iii) of Theorem 3.1, we have . Due to m n * → 0 as n → ∞, we have that W sn (x 1 , x 2 , x 3 ) converges to U(x 3 ) uniformly in any compact set Ω ⊂ R 3 as n → ∞.Consequently, we have that In view of U 2 (0) = θ 2 and U i (x) ≥ 0 for any x ∈ R, similar to the proof of Theorem 3.1 we can show that U(+∞) = E 1 , U(−∞) = E 0 and U i (x) > 0 for any x ∈ R , where i = 1, 2. It then follows from the uniqueness of planar traveling wave fronts of ( 5) connecting two equilibria E 0 and E 1 that U(x) ≡ U(x) in x ∈ R.This completes the proof.
4. Nonexistence of axisymmetric traveling fronts.In this section we prove Theorems 1.3 and 1.4, which imply the nonexistence of axisymmetric traveling fronts.Here we give only the proof of Theorem 1.4.Theorem 1.3 can be similarly proved.
Proof of Theorem 1.4.We prove it by a contradiction argument.On the contrary, we assume that for s < c, there exists an axisymmetric traveling front Ψ(x) satisfying (4), lim x3→+∞ Ψ(0, 0, Let U(x 3 ) = W(0, 0, x 3 ) for any x 3 ∈ R. Then we have On the other hand, following Wang [51, Lemma 4.2] (see also Lin and Li [34]), we obtain that the function is a subsolution of (33), where ρ, δ, β are appropriate positive constants, + is a monotone vector-valued function and satisfies Q(±∞) 0, ξ − ∈ R is an arbitrary number.In view of there exists a sufficiently large ξ − > 0 so that u − (x, 0) ≤ u + (x, 0) for all x ∈ R. Applying the comparison principle (see also Wang [51, Sections 2 and 5]), we have which is a contradiction, this completes the proof of Theorem 1.4.
In this case we have Combining the above cases (i)-(iii), we obtain κ * ≥ inf α>0 Φ (α) = inf α>0 λ(α) α > 0. This completes the proof of Lemma 3.3.Lemma 3.6.In this subsection we prove Lemma 3.6 by using the theory of Thieme and Zhao [49].For the reader's convenience, in subsection 5.2.1 we state some results of Thieme and Zhao [49] on the spreading speed for scalar nonlinear integral equations, which are needed in the next subsection.In subsection 5.2.2, we use these results to prove Lemma 3.6 in details.
We say that u 0 is admissible if for every σ, λ > 0 with K (σ, λ) < 1, there exists some γ > 0 such that Now we state two theorems of Thieme and Zhao [49].
Proof.By (39), we have and Consequently, we have where In the following we define appropriate k(x, t) so that we can rewrite u 2 (x, t) in the form of (38) with the special case F (u, x, t) = f (u)k(x, t).In particular, we show that assumption (B) holds in this case and u 0 2 (x, t) is admissible.Following (41), we have A direct calculation yields It is easy to show that k (z, t) satisfies the assumption (B).In particular, we have for σ > 0.
Then we have K (σ, λ) < +∞ for λ ∈ 0, λ (σ) and K (σ, λ) → +∞ as λ → λ (σ) − 0. In particular, we have K (σ, 0) = k * > 1 for any σ ≥ 0 and It is easy to see that for any x ∈ R 3 and t > 0, which implies that u 0 2 (x, t) converges to 0 uniformly in x ∈ R 3 as t → +∞.Give σ > 0 and λ ∈ 0, λ (σ) with K (σ, λ) < 1.In this case we have dλ 2 − λσ − r (1 − E * 2 + Λ 2 ) < 0 and λ 2 − λσ − E * 1 < 0. It follows from the assumption of the lemma that there exists γ > 0 such that φ 1 (x) + φ 2 (x) ≤ γe −λ|x| for any x ∈ R 3 .For any e ∈ R 3 with |e| = 1, we have φ 1 (x) + φ 2 (x) ≤ γe λe•x for any x ∈ R 2 ) E * 1 + λσ − λ 2 e λ(σt−|x|) , ∀ x ∈ R 3 , t ≥ 0, which implies that u 0 2 (x, t) is admissible.Let F (u, x, s) = f (u)k(x, s).It is easy to show that the assumption (A) holds.In addition, we have the integral equation It follows from Thieme and Zhao [49, Proposition 2.1] that for any bounded u 0 2 (x, t), the integral equation (42) has a unique solution which is bounded on (x, t) ∈ R 3 × [0 + ∞).It is obvious that the second component u 2 (x, t) of the solution of system (39) is a solution of (42).Thus, the result of the lemma for u 2 (x, t) follows from Theorems 5.1 and 5.2.Consequently, the result of the lemma for u 1 (x, t) follows from an argument on (40), see [49,Theorem 4.4].This completes the proof.Now we prove Lemma 3.6.Proof of Lemma 3.6: Assume that φ = (φ 1 , φ 2 ) ∈ C R 3 , E 0 , E * * is compactly supported with φ 1 (•) + φ 2 (•) ≡ 0. It is easy to show that φ satisfies the condition of Lemma 5.3.By the comparison principle, we have u(x, t; φ) ∈ E 0 , E * * for any x ∈ R 3 and t > 0. By the definition of f , we have that the solution u(x, t; φ) of ( 23) is also a solution of (39).Applying Lemma 5.3, we know that the conclusions of Lemma 3.6 hold.This completes the proof of Lemma 3.6.6. Discussion.Under the assumptions that k 1 , k 2 > 1 and c > 0, in this paper we have established the existence of axisymmetric traveling fronts of a two-species Lotka-Volterra competition-diffusion system in R 3 for any s > c and demonstrated some important qualitative properties, such as monotonicity, of the axisymmetric traveling fronts.When s tends to c, we showed that the axisymmetric traveling fronts converge locally uniformly to the planar traveling wave fronts in R 3 .Furthermore, we showed the nonexistence of axisymmetric traveling fronts with convex level set.Note that the nonexistence results of Theorems 1.3 and 1.4 remain valid for two-dimensional V-shaped traveling fronts and high-dimensional pyramidal traveling fronts.
Due to the effect of the coupled nonlinearity, in this paper we did not consider the behavior of level sets of the axisymmetric traveling fronts at infinity.We conjecture that the level set admits an asymptotic behavior similar to that for the scalar equations, see Hamel et al. [19,20] and Taniguchi [48].Another natural problem is the uniqueness and stability of the axisymmetric traveling fronts.We leave these for our future studies.In addition, in the current paper we only considered the case c > 0. For the case c = 0, it should be expected that there exist more complex dynamics, such as those obtained for the balanced Allen-Cahn equation.This case is very interesting and remains open.