UNIQUENESS AND STABILITY OF TIME-PERIODIC PYRAMIDAL FRONTS FOR A PERIODIC COMPETITION-DIFFUSION SYSTEM

. The existence, non-existence and qualitative properties of time periodic pyramidal traveling front solutions for the time periodic Lotka-Volterra competition-diﬀusion system have already been studied in R N with N ≥ 3. In this paper, we continue to study the uniqueness and asymptotic stability of such time-periodic pyramidal traveling front in the three-dimensional whole space. For any given admissible pyramid, we show that the time periodic pyramidal traveling front is uniquely determined and it is asymptotically stable under the condition that given perturbations decay at inﬁnity. Moreover, the time periodic pyramidal traveling front is uniquely determined as a combination of two-dimensional periodic V-form waves on the edges of the pyramid.


(Communicated by Junping Shi)
Abstract. The existence, non-existence and qualitative properties of time periodic pyramidal traveling front solutions for the time periodic Lotka-Volterra competition-diffusion system have already been studied in R N with N ≥ 3. In this paper, we continue to study the uniqueness and asymptotic stability of such time-periodic pyramidal traveling front in the three-dimensional whole space. For any given admissible pyramid, we show that the time periodic pyramidal traveling front is uniquely determined and it is asymptotically stable under the condition that given perturbations decay at infinity. Moreover, the time periodic pyramidal traveling front is uniquely determined as a combination of two-dimensional periodic V-form waves on the edges of the pyramid.
1. Introduction. Recently, there have been great progress on the study of multidimensional traveling wave solution of competition or cooperative-diffusive system in high-dimensional space and many new types of nonplanar traveling waves have been observed for autonomous reaction-diffusion systems. For example, see Wang [41] and Wang et al. [44] for the existence, uniqueness and stability of twodimensional traveling front solutions and three-dimensional pyramidal traveling front of bistable reaction-diffusion system for any admissible wave speed, also see Ni and Taniguchi [29] for the existence of pyramidal traveling fronts of competitiondiffusion system with constant coefficient in N -dimensional space (N ≥ 3). By utilizing the results in pyramidal traveling fronts, Wang et al. [45] has established the existence and qualitative properties of axisymmetric traveling fronts for competitiondiffusion system. More recently, for cooperative system in R 3 (N ≥ 3), Taniguchi [39] has showed that there exist traveling wave solutions associated with a given (N − 2)-dimensional smooth surface with the boundaries being compact set in R N −1 . Compared with autonomous systems, little attention has been paid to nonautonomous reaction-diffusion system. For time-periodic case, Bao et al. [3] recently proved the existence, uniqueness and stability of two-dimensional periodic V-form traveling fronts of time periodic Lotka-Volterra system for N = 2. More recently, the existence, non-existence and qualitative properties of time-periodic pyramidal traveling fronts for time periodic Lotka-Volterra competition-diffusion system in N -dimensional space (N ≥ 3) have been established, see [1]. The purpose of the current paper is to show the uniqueness and asymptotic stability of time-periodic pyramidal traveling fronts for the time periodic competition diffusion system in three-dimensional whole space.
We further assume that (A4) c > 0, where c is the wave speed in (1.3). Under the assumptions (A1)-(A4), the authors have established the existence of time periodic pyramidal traveling fronts of (1.1) in [1], see Theorem 1.1 later. Here we further study the uniqueness and asymptotic stability of time periodic pyramidal traveling wave for (1.1).
Without loss of generality, we assume that the solutions travel towards −x 3 direction. Let Then we have the following initial value problem We write the solution of (1.4) with initial value u 0 (x) ∈ C(R 3 , R 2 ) as (u 1 (x, t; u 0 ), u 2 (x, t; u 0 )). Since the curvature often accelerates the speed, we fix s > c > 0. For s < c, we have showed that there maybe not exist time periodic pyramidal traveling fronts of (1.1) in some special case, see [1,Theorem 1.3].
To describe the problems studied and the results obtained in the current paper, let n ≥ 3 be a given integer and (1.5) For any j = 1, ..., n, let x ∈ Ω j for each j = 1, · · · , n, and call ∪ n j=1 S j ⊂ R 3 the lateral surfaces of a pyramid. Denote Γ j := S j ∩ S j+1 , Γ n := S n ∩ S 1 , j = 1, · · · , n − 1. Then Γ := ∪ n j=1 Γ j represents the set of all edges of a pyramid. For each γ > 0, we define In particular, The existence of time periodic pyramidal traveling fronts is proved in [1].
Following from (1.8), we know that the nonplanar traveling wave (U 1 , U 2 ) has pyramidal structures and is characterized as a combination of time periodic planar traveling fronts on the lateral surface. The following theorem further shows that such time periodic pyramidal traveling front (U 1 , U 2 ) is unique and asymptotically stable.
We end the introduction with the following remarks. Firstly, Theorems 1.1-1.2 show that the time periodic pyramidal traveling front (U 1 , U 2 ) exists and is asymptotically stable. Since it is uniquely determined, now we call that (U 1 (x , x 3 + st, t), U 2 (x , x 3 + st, t)) is the pyramidal traveling wave associated with a pyramid −x 3 = h(x ). In the end of Section 4, we further characterize the time periodic pyramidal traveling fronts as a combination of two-dimensional periodic V-form traveling fronts on the edges of the pyramid, see Corollary 4.1 later.
Thirdly, in this paper we establish the uniqueness and stability of time periodic pyramidal fronts for competition-diffusion system (1.1) in N = 3. However, as the spatial dimension N becomes higher, the uniqueness and stability of N-dimensional periodic pyramidal traveling fronts of (1.1) with N ≥ 4 become more interesting and difficult and are left to be as interesting open problems.
The paper is organized as follows. In Section 2, we summarize some preliminaries. In Section 3, we prove some basic properties on the time-periodic pyramidal traveling fronts, which will be used to prove our main results. We then show that the time-periodic pyramidal traveling front is asymptotically stable and unique in Section 4.
is the solution of (2.11). Then there exist positive constants A, B and m > 0 such that and for any γ > 0, for any x 0 ∈ R 3 and R > 0.
In the current paper, we will extend the arguments of Wang et al [44], Taniguch [36] and Bao et al. [3] to study the uniqueness and stability of time periodic pyramidal fronts for (1.1) in R 3 under the assumptions (A1)-(A4). The main method is also to use the super-and subsolution technique coupled with comparison principle. Note that the supersolutions and subsolutions constructed later cannot be bounded from above by 1 and from below by 0 and the comparison principal on [0, 1] is invalid for such super-and subsolutions. To overcome this, we introduce an auxiliary system for (2.5) as that in [3]. Let C + 1 and C − 2 be positive constants defined in [1, Lemmas 2.1-2.2], we then have Consider the following initial value problem: Following from [1, Theorem 2.5] and [1, Corollary 2.6], the comparison principle holds for (2.14) for (u 1 , u 2 ) ∈ [−1, 2]. Thus we can construct some useful supersolutions and subsolutions of (2.14), which will be used to establish the stability of time periodic pyramidal traveling front (V 1 , V 2 ) in Section 4.
The following lemmas provide much more subsolutions and supersolutions of (2.14), which will be used to prove uniqueness and stability of time periodic pyramidal fronts in Section 4. Lemma 2.3. There exist positive constants ρ sufficiently large, β small enough and δ 1 such that for any 0 < δ < δ 1 , w + = (w + 1 , w + 2 ) defined by , t + δp i ς + , t e −βt , i = 1, 2 is a supersolution of (2.14), and , t − δq i ς − , t e −βt , i = 1, 2 is a subsolution of (2.14), where Proof. By Lemma 2 in [35], for any x ∈ R 2 , there is a positive constant m 0 such that It is easy to verify that there exist constants N ± i > 0 such that for any α ∈ (0, 1] and (x, t) ∈ R 3 × R + . We omit the rest of the proof, which is similar to that of [3,Lemma 4.5]. This completes the proof. Lemma 2.4. There exist positive constants ρ sufficiently large, β small enough and δ 2 such that, for any 0 < δ < δ 2 , W + defined by , t; ε, α + δp i (τ, t)e −βt is a supersolution of (2.14) for any δ > 0, where i = 1, 2 and The proof of the lemma is similar to that of Lemma 2.3. Following form [3,Lemma 4.7], we obtain the following lemma.
Lemma 2.5. There exist positive constants ρ sufficiently large, β small enough and δ 3 such that, for any 0 < δ < δ 3 , w j = ( w j 1 , w j 2 ) and w = ( w 1 , w 2 ) defined by are also subsolutions of (2.14), where j = 1, ..., n and 3. Basic properties. In this section, we will prove some basic properties and show that the time-periodic pyramidal traveling front (V 1 , V 2 ) convergences to twodimensional V-form traveling fronts on edges of the pyramid. To do this, we first study the exact formulation of the two-dimensional V-form front on each edge. For each j (1 ≤ j ≤ n) we consider a plane perpendicular to an edge Γ j = S j ∩ S j+1 . Then the cross section of −x 3 = max{h j (x ), h j+1 (x )} in this plane has a time periodic V-form front. Let E j = (E j 1 , E j 2 ) be the two-dimensional periodic V-form front as in Theorem 2.1 corresponding to the cross section −x 3 = max{h j (x ), h j+1 (x )}.
Next, by the same way in [44,36], we will give the precise definition of E j . Let A n+1 := A 1 and B n+1 := B 1 . Define and the traveling direction of a two-dimensional V-form wave E j is given by Let s j be the speed of E j and 2θ j (0 < θ j < π 2 ) be the angle between S j and S j+1 . Then s j sin θ j = c, sin θ j = m 2 * p 2 j + q 2 j q j 1 + m 2 * and s j = sq j m 2 * p 2 j + q 2 j .
We call E j a planar time periodic V-form traveling front corresponding to an edge Γ j .
Since E j (x, t) is strictly monotone increasing in x 3 for each j, we have E(x, t) is strictly monotone increasing in x 3 . In addition, E(x, t) also has the following properties.
) be the solution of (2.5) with the initial value v 0 . For any given t ∈ [0, T ] and k ∈ N, by Lemma 2.1, there exist constants A > 0 and B > 0 such that for any γ > 0 and t > 0. Then we obtain for any fixed k ∈ N, which implies and lim for any fixed k ∈ N.
. For any given ε 1 > 0, there exists k 1 ∈ N large enough such that, for any fixed k ≥ k 1 , Lemma 3.2 will play a key role in the following estimates and can be proved by the strategy in [36,Proposition 1] and [44,Proposition 4.5]. We provide the proof of Lemma 3.2 in Appendix for interested readers.
The equality (3.10) shows that time periodic pyramidal traveling front (V 1 , V 2 ) converges to two-dimensional time periodic V-form fronts ( E 1 , E 2 ) near the edges.

It then follows from [3, Lemma 4.3] that
Next, we prove (3.11). Since ∂ ∂x3 V i > 0 in R 3 , ∂ ∂x3 V i has a positive minimum on any compact subset of R 3 . Thus we need only to study ∂ ∂x3 V i as |x| → ∞. Fix i = 1, 2. Let Consequently, we have δ Applying the interior L p estimate for second order parabolic equations in [24,Theorem 7.22] to Thus, by virtue of the estimate on E j there exist R j > R j such that Apply the above argument to all j = 1, ..., n and i = 1, 2, we can obtain (3.11).
The assumption This completes the proof.
where 0 is a positive constant depending on δ and is independent of x.
Proof. Fix i = 1, 2. By the continuity of E i (x, t), there is 0 > 0 such that for any By the arbitrariness of and x 0 , we can obtain the results. This completes the proof.
Next, we consider the case that the initial value v 0 (x) = (v 10 (x), v 20 (x)) sat- 1 (x, t), v 2 (x, t)) as follows: t) , where i = 1, 2 and M is a positive constant that will be defined later. Recall that w is defined in Lemma 2.5 and w − is defined in Lemma 2.3, we also define w(x, t; δ) = max {w(x, t; δ), w(x, t; δ)} .
The proof is similar to that of [44,Lemma 4.14] and is omitted.
Then by comparison principal we have where v(x, t; v ± 0 ) denote the solution of (2.5) with initial values v ± 0 , respectively. Then applying Theorem 4.1 and 4.2 to v(x, t; v − 0 ) and v(x, t; v + 0 ), respectively, and combining (4.6) yield Hence, by (2.1) we can prove that Theorem 1.2 holds.
The following corollary shows that a three dimensional periodic pyramidal traveling front is uniquely determine as a combination of two-dimensional time periodic V-form fronts.

5.
Appendix. This appendix is devoted to the proof of Lemma 3.2.

Proof of Lemma 3.2. Let
Then I j is the projection of Γ j onto the x 1 − x 2 plane and ∪ n j=1 I j is the projection of Γ onto the x 1 − x 2 plane.

UNIQUENESS AND STABILITY OF TIME PERIODIC PYRAMIDAL FRONTS 273
For this change of variables, we have we have W i (x, 0) = W i0 (x), x ∈ R 3 , i = 1, 2.