Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 31, pp. 1–23. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu PYRAMIDAL TRAVELING FRONTS OF A TIME PERIODIC DIFFUSION EQUATION WITH DEGENERATE MONOSTABLE NONLINEARITY ZHEN-HUI BU, CHEN-LU WANG, XIN-TIAN ZHANG Abstract. This article focuses on the nonplanar traveling fronts of degenerate monostable time periodic reaction-diffusion equations in Rn with n ≥ 3. By constructing a couple of proper supersolution and subsolution, we prove the existence of periodic pyramidal traveling front in R3 and then in Rn with n > 3. 1. Introduction In this article, we investigate nonplanar traveling fronts of the equation ∂u(x, t) ∂t = ∆u(x, t) + f(u(x, t), t), x ∈ Rn, t ∈ (0,+∞), (1.1) where n ≥ 3 is an integer, ∆ is the Laplace operator and the nonlinear reaction term f is degenerate monostable satisfying the hypotheses: (H1) f(u, t) ∈ C1+ι,ι/2([0, 1]×R,R) is T -periodic in t, where ι ∈ (0, 1) and T > 0; (H2) f(0, t) = f(1, t) = 0 with t ∈ R and f(u, t) > 0 in (0, 1) × R; fu(0, t) = 0 and fu(1, t) < 0 for all t ∈ R, where fu(0, t) = lim u→0+ f(u, t) u , fu(1, t) = lim u→1− f(u, t) u− 1 . Many diffusion phenomena in nature can be portrayed by reaction-diffusion equa- tions such as the movement of populations, propagation of burning flame and the spread of diseases in the air [5, 11]. As the special solutions of reaction-diffusion equations on an unbounded region, the traveling fronts can describe the propaga- tion phenomenon of reaction-diffusion equations well. According to whether the level set of traveling front is a hyperplane, the traveling fronts are classified into planar traveling fronts and nonplanar traveling fronts. Planar traveling fronts have been well studied in arbitrary dimensional space be- cause their simple form and good geometric properties [10, 13, 16, 17, 30]. However, owing to the effects of curvature and spatial dimension, many reaction phenomena cannot be accurately described by planar traveling fronts in Rn with n > 1, such as the conical premixed Bunsen flames [4] and the fertilization Ca2+ waves in ma- ture Xenopus laevis eggs [26]. Therefore, the multidimensional nonplanar traveling 2020 Mathematics Subject Classification. 35C07, 35K57, 35B08. Key words and phrases. Reaction-diffusion equation; time periodic; pyramidal traveling front; degenerate monostable nonlinear term. ©2023. This work is licensed under a CC BY 4.0 license. Submitted August 8, 2022. Published April 3, 2023. 1 2 Z.-H. BU, C.-L. WANG, X.-T. ZHANG EJDE-2023/31 fronts of reaction-diffusion equations have attracted more and more scholars’ at- tention. For the combustion case, Bonnet and Hamel [4] established the existence of two-dimensional V-shaped traveling fronts. Then, Hamel and Monneau [12] inves- tigated conical traveling fronts in Rn with n ≥ 3. For the Fisher-KPP monostable case, Hamel and Nadirashvili [13] showed the existence of an infinite-dimensional manifold of solutions. For the bistable case, using the method of the compari- son principle coupled with the supersolution and subsolution technique, Ninomiya and Taniguchi [18, 23] obtained the existence of the two-dimensional V-shaped traveling fronts and the three-dimensional pyramidal traveling fronts. Kurokawa and Taniguchi [15] further considered the existence of n-dimensional pyramidal traveling fronts with n ≥ 4. For more information on the higher dimensions of this case, one can refer to the literature of Taniguchi [24, 25]. Recently, the first author of this paper and Wang [7, 28] investigated the existence and stability of the three-dimensional pyramidal traveling fronts to the reaction-diffusion equations with combustion and degenerate Fisher-KPP nonlinearities without periodicity. To simulate real natural phenomena (e.g. seasonal cycles), the influence of time periodicity has been considered by researchers recently. Wang and Wu [29] and Sheng et al. [21] studied the existence and stability of two-dimensional periodic V-shaped and three-dimensional periodic pyramidal traveling fronts for reaction- diffusion equations with bistable time-periodic nonlinearity, respectively. El Smaily et al. [22] explored the traveling fronts of Fisher-KPP monostable reaction-diffusion equations with periodic advection in R2. Subsequently, Bu and Wang [6] studied the traveling fronts of reaction-advection-diffusion equations in space-time periodic medium in Rn (n ≥ 3). Then Zhang et al. [31] concerned the existence, uniqueness and stability of V-shaped traveling fronts to reaction-diffusion equations with ig- nition time-periodic nonlinearity. For more results about time-periodic nonplanar traveling fronts, we refer to [2, 20] and the references therein. In this article, we study the nonplanar traveling fronts of degenerate monostable time periodic reaction-diffusion equation (1.1) in Rn with n ≥ 3. Inspired by Wang and Bu [28] and Zhang et al. [31], we will use the super-sub solution method com- bined with the comparison principle. The sign of derivative of nonlinear term f at the equilibrium points 0 and 1 plays a key role in constructing the supersolution. In contrast to bistable and combustion cases, the derivative of the degenerate monos- table case that satisfies the hypotheses (H1) and (H2) at the equilibrium point 0 is zero and f(t+ T, u) = f(t, u) > 0 in R× (0, 1). Thus there will be some difficulties in constructing the supersolution. To overcome these difficulties, we will adopt the method of adding small perturbation to the planar traveling front to construct supersolution. From [3], we know that under the assumptions on f , equation (1.1) has a periodic planar traveling front Ψ(ξ, t) : R× R→ R with the wave speed c∗ > 0 satisfying Ψt + c∗Ψξ −Ψξξ − f(Ψ, t) = 0, Ψξ(ξ, t) > 0, (ξ, t) ∈ R2, Ψ(−∞, t) = 0,Ψ(+∞, t) = 1 uniformly in t ∈ R, Ψ(ξ, t+ T ) = Ψ(ξ, t), (ξ, t) ∈ R2 (1.2) and lim ξ→−∞ Ψξ(ξ, t) Ψ(ξ, t) = Λ2 = c∗ > Λ1 = 0, lim ξ→−∞ Ψξξ(ξ, t) Ψ(ξ, t) = Λ2 2 = c2∗ (1.3) EJDE-2023/31 PERIODIC PYRAMIDAL TRAVELING FRONTS 3 uniformly in t ∈ R, where Λ1 and Λ2 are roots of the equation λ2 − c∗λ = 0. In fact, c∗ > 0 is the critical speed of the periodic planar traveling fronts to (1.1). For any β ∈ (0, 1), we can easily obtain Π(βΛ2) = (βΛ2)2 − c∗(βΛ2) < 0. In addition, there exist positive constants L1, L2, L3, β1 such that L1e Λ2ξ ≤ Ψ(ξ, t), Ψξ(ξ, t), |Ψξξ(ξ, t)| ≤ L2e Λ2ξ, ∀ξ < 0, t ∈ R, (1.4) |Ψ(ξ, t)− 1|, Ψξ(ξ, t), |Ψξξ(ξ, t)| ≤ L3e −β1ξ, ∀ξ > 0, t ∈ R. (1.5) By the super-sub solution method, this paper firstly studies the existence of the three-dimensional periodic pyramidal traveling fronts of (1.1). That is, we investigate ∂u(x, t) ∂t = ∆u(x, t) + f(u(x, t), t), x ∈ R3, t > 0. (1.6) Then we establish the existence of n-dimensional periodic pyramidal traveling fronts of (1.1) with n ≥ 4. Assume c > c∗ and m∗ = √ c2−c2∗ c∗ . Denote x = (x1, x2, x3) ∈ R3. We assume that the traveling fronts travel towards −x3 direction with the speed of c > c∗. Let u(x1, x2, x3, t) = v(x1, x2, x3 + ct, t) = v(x1, x2, w, t). We still express v(x1, x2, w, t) as v(x1, x2, x3, t) for convenience. By substituting v into (1.6), it follows that vt = ∆v − cvx3 + f(v, t), x ∈ R3, t > 0, v(x, 0) = v0(x), x ∈ R3. (1.7) One of the purposes of this paper is to find the solution V (x, t) satisfying Vt − Vx1x1 − Vx2x2 − Vx3x3 + cVx3 − f(V, t) = 0, x ∈ R3, t ∈ R, (1.8) V (·, ·, ·, ·) = V (·, ·, ·, ·+ T ), x ∈ R3, t ∈ R. (1.9) Let l ≥ 3, and {(Aj , Bj)}1≤j≤l be a set of unit vectors in R2 such that AjBj+1 −Aj+1Bj > 0, j = 1, 2, . . . , l − 1; AlB1 −A1Bl > 0. (1.10) For each (x1, x2) ∈ R2, let hj(x1, x2) = m∗(x1Aj + x2Bj), 1 ≤ j ≤ l, h(x1, x2) = max 1≤j≤l hj(x1, x2) = m∗ max 1≤j≤l (x1Aj + x2Bj), then {x ∈ R3| − x3 = h(x1, x2)} is a pyramid in R3. Clearly, for any (x1, x2) ∈ R2, we have h(x1, x2) ≥ 0 and lim R→∞ inf x2 1+x2 2≥R2 h(x1, x2) =∞. Set Ωj = {(x1, x2) ∈ R2 : h(x1, x2) = hj(x1, x2)}, j = 1, 2, . . . , l, then R2 = ∪lj=1Ωj . From (1.10), the planes Ω1,Ω2, . . . ,Ωl are arranged in a coun- terclockwise direction. Let ∂Ωj be the boundary of Ωj . Denote E = ∪lj=1∂Ωj . Each side of the pyramid can be represented as Gj = {x ∈ R3 : −x3 = hj(x1, x2), (x1, x2) ∈ Ωj}, j = 1, 2, . . . , l. We denote Γj = { Gj ∩Gj+1, 1 < j < l − 1, Gl ∩G1, j = l. 4 Z.-H. BU, C.-L. WANG, X.-T. ZHANG EJDE-2023/31 Then Γ = ∪lj=1Γj represents the set of all edges of a pyramid, and the lateral surfaces of the pyramid consist of ∪lj=1Gj ⊂ R3. For each γ̄ ≥ 0, we define D(γ̄) = {x ∈ R3 : dist(x,Γ) ≥ γ̄}. Note that the above setting on a pyramid comes from Taniguchi [23]. For any 1 ≤ j ≤ l, it is obvious that Ψ( c∗c (x3 + hj(x1, x2)), t) is the solution of (1.8). We define ψ(x, t) = Ψ( c∗ c (x3 + h(x1, x2)), t) = max 1≤j≤l Ψ( c∗ c (x3 + hj(x1, x2)), t). (1.11) Then ψ(x, t) is a subsolution to (1.8). Furthermore, we have ψ x3 (x, t) > 0. Now we state the main result of this article in R3. The generalized result in Rn with n ≥ 4 will be given in Section 4. Theorem 1.1. Assume that (H1) and (H2) hold. For each c > c∗, equation (1.6) has a periodic nonplanar traveling front V (x, t) satisfying (1.8)-(1.9). Moreover, lim γ̄→∞ sup x∈D(γ̄),t∈[0,T ] |V (x, t)− ψ(x, t)| (ψ(x, t))β = 0, ∀β ∈ (0, 1), and Vx3 (x, t) > 0, (x, t) ∈ Rn × R. The rest of this article is organized as follows. Some preliminaries are given in Section 2. In Section 3, we construct the supersolution, and then prove the existence of periodic pyramidal traveling fronts in R3. That is, we give the proof of Theorem 1.1. We establish the existence of n-dimensional periodic pyramidal traveling fronts with n ≥ 4 in Section 4. In Section 5, the article ends with a short conclusion. 2. Preliminaries In this section, we give some preliminaries which are useful in the proof of the existence of three-dimensional periodic pyramidal traveling fronts to (1.6). Firstly, we mollify the original pyramid {x ∈ R3|−x3 = h(x1, x2)}, see [23]. Let function ρ̃(r) ∈ C∞[0,∞) satisfy the following properties: (1) ρ̃(r) > 0, ρ̃r(r) ≤ 0, r ≥ 0; (2) If r > 0 is small enough, ρ̃(r) = 1; (3) If r > 0 is large enough, say r > R0, ρ̃(r) = e−r, where R0 > 0 is a constant; (4) ∫ R2 ρ̃( √ x2 1 + x2 2) dx1 dx2 = 1. It is easy to check that∫ R2 ρ̃ (√ x2 1 + x2 2 ) dx1dx2 = 2π ∫ ∞ 0 rρ̃(r)dr = 1. Letting ρ(x1, x2) = ρ̃( √ x2 1 + x2 2), one gets ρ ∈ C∞(R2) and ∫ R2 ρ(x1, x2)dx1dx2 = 1. Set R0 > 1. For all nonnegative integers i1 ≥ 0 and i2 ≥ 0 with 0 ≤ i1 + i2 ≤ 3, we have |Di1x1 Di2x2 ρ(x1, x2)| ≤M∗ρ(x1, x2), (x1, x2) ∈ R2, EJDE-2023/31 PERIODIC PYRAMIDAL TRAVELING FRONTS 5 where M∗ > 0, Di1x1 = ∂i1 ∂x i1 1 and Di2x2 = ∂i2 ∂x i2 2 . Define ϕ̄(x1, x2) = ρ ∗ h. That is, ϕ̄(x1, x2) = ∫ R2 ρ(x1 − x′1, x2 − x′2)h(x′1, x ′ 2)dx′1dx ′ 2 = ∫ R2 ρ(x′1, x ′ 2)h(x1 − x′1, x2 − x′2)dx′1dx ′ 2 (2.1) for each (x1, x2) ∈ R2. The set {x ∈ R3| − x3 = ϕ̄(x1, x2)} is called the mollified pyramid of {x ∈ R3| − x3 = h(x1, x2)}. Let G(x1, x2) = c√ 1 + |∇ϕ̄(x1, x2)|2 − c∗, (2.2) where |∇ϕ̄(x1, x2)| = √ ϕ̄2 x1 (x1, x2) + ϕ̄2 x2 (x1, x2). The next two lemmas come from [23], which show some properties on the functions ϕ̄(x1, x2) and G(x1, x2) on R2. Lemma 2.1. Assume that ϕ̄ and G are defined in (2.1) and (2.2) respectively. Then sup (x1,x2)∈R2 |Di1x1 Di2x2 ϕ̄(x1, x2)| <∞, h(x1, x2) < ϕ̄(x1, x2) ≤ h(x1, x2) + 2πm∗ ∫ ∞ 0 r2ρ̃(r)dr, |∇ϕ̄(x1, x2)| < m∗, 0 < G(x1, x2) ≤ c− c∗, (x1, x2) ∈ R2, lim λ→∞ sup{G(x1, x2)|(x1, x2) ∈ R2,dist((x1, x2), E) ≥ λ} = 0, lim λ→∞ sup{ϕ̄(x1, x2)− h(x1, x2)|(x1, x2) ∈ R2,dist((x1, x2), E) ≥ λ} = 0. Lemma 2.2. There exist two positive constants a1 and a2 such that a1 = inf (x1,x2)∈R2 ϕ̄(x1, x2)− h(x1, x2) G(x1, x2) ≤ sup (x1,x2)∈R2 ϕ̄(x1, x2)− h(x1, x2) G(x1, x2) = a2 <∞. In addition, for any integers i1 ≥ 0 or i2 ≥ 0 satisfying 2 ≤ i1 + i2 ≤ 3, there exists a constant K > 0 such that sup (x1,x2)∈R2 | Di1x1 Di2x2 ϕ̄(x1, x2) G(x1, x2) | < K, and |ϕ̄x1x1 (x1, x2)|, |ϕ̄x2x2 (x1, x2)| ≤ m∗M∗, (x1, x2) ∈ R2. (2.3) Secondly, we study the eigenfunction at equilibrium point 1. Assume that Λ0 is the eigenvalue of the linearized periodic system Ῡ′(t)− fu(1, t)Ῡ(t) = Λ0Ῡ(t), t ∈ R, Ῡ(t+ T ) = Ῡ(t), t ∈ R. (2.4) By a direct calculation, we have Ῡ(t) = eΛ0t+ ∫ t 0 fu(1,s)ds, Λ0 = − 1 T ∫ t 0 fu(1, s)ds > 0. (2.5) 6 Z.-H. BU, C.-L. WANG, X.-T. ZHANG EJDE-2023/31 We define ν(t) = kῩ(t), P1 = mint∈[0,T ] ν(t), and P2 = maxt∈[0,T ] ν(t), where k is a positive constant such that P1 > 1. Next, we give some properties about the reaction term f . By assumptions (H1) and (H2), we can choose ε1 ∈ (0, 1) small enough such that fu(u, t) < − 1 16 Π(βΛ2), −ε1 ≤ u ≤ ε1, t > 0, (2.6) |fu(u, t)− fu(1, t)| ≤ 1 2 Λ0, 1− ε1 ≤ u ≤ 1 + ε1, t > 0. (2.7) Finally, we construct an auxiliary function ω(x) ∈ C∞(R) such that ω(x) = 1, if x ≥ 1, 0 < ω(x) < 1, 0 < ω′(x) < 1, ω′′(x) < 0, if − 1 < x < 1, ω(x) = 0, if x ≤ −1, (2.8) which will be used in constructing the supersolution. 3. Existence of periodic pyramidal traveling fronts in R3 In this section, we first use the idea of perturbation to construct a suitable super- solution. And then we prove the existence of three-dimensional periodic pyramidal traveling front V (x, t) to (1.6). Obviously, 1 αh(αx1, αx2) = h(x1, x2) for any α ∈ (0, 1). Let z3 = αx3, z′ = (z1, z2) = (αx1, αx2) = αx′, z = αx and $(x) = c∗ c ( x3 + 1 α ϕ̄(αx1, αx2) ) = c∗ c z3 + ϕ̄(z′) α , (3.1) %(x) = x3 + 1 α ϕ̄(αx1, αx2)√ 1 + |∇ϕ̄(αx1, αx2)|2 = z3 + ϕ̄(z′) α √ 1 + |∇ϕ̄(z′)|2 . (3.2) Using Lemma 2.1, we can obtain{ c c∗ $(x) < %(x) < $(x), if %(x) < 0, $(x) < %(x) < c c∗ $(x), if %(x) > 0. (3.3) By a direct calculation, this indicates $x3 = c∗ c , $x3x3 = 0, $xi = c∗ c ϕ̄zi , $xixi = α c∗ c ϕ̄zizi , %x3 = 1√ 1 + |∇ϕ̄(z′)|2 , %x3x3 = 0 and %xi = ( √ 1 + |∇ϕ̄(z′)|2)−1ϕ̄zi − α%Ci(z′), %xixi = αDi(z ′)− α2%Ei(z ′), where Ci(z ′) = √ 1 + |∇ϕ̄(z′)|2 ∂ ∂zi ( √ 1 + |∇ϕ̄(z′)|2)−1, Di(z ′) = ∂ ∂zi ((√ 1 + |∇ϕ̄(z′)|2 )−1 ϕ̄zi ) − Ci(z ′)√ 1 + |∇ϕ̄(z′)|2 ϕ̄zi , Ei(z ′) = ∂Ci(z ′) ∂zi − C2 i (z′), EJDE-2023/31 PERIODIC PYRAMIDAL TRAVELING FRONTS 7 for i = 1, 2. Let σ(x1, x2) = G(αx1, αx2) = G(z′), where α > 0 is a constant, which will be determined later. Then σxi(x1, x2) = αGzi(z ′) and σxixi(x1, x2) = α2Gzizi(z ′), i = 1, 2. 3.1. Construction of the supersolution. Motivated by Wang and Bu [28] and Zhang et al. [31], we construct an appropriate supersolution in this subsection. Lemma 3.1. For each β ∈ (0, 1), there exist positive constants ε+ 0 (β) and α+ 0 (β, ε) such that, for any 0 < ε < ε+ 0 (β) and 0 < α < α+ 0 (β, ε), the function ψ(x, t;β, ε, α) = Ψ(%(x), t) + εσ(x′)(ω($(x))ν(t) + (1− ω($(x)))Ψβ($(x), t)) is a supersolution of (1.8)-(1.9) on R3 × (−∞,+∞). Moreover, lim γ→∞ sup x∈D(γ),t∈[0,T ] |ψ(x, t;β, ε, α)− ψ(x, t)| ψ(x, t)β ≤ 2ε, (3.4) ψ(x, t) < ψ(x, t;β, ε, α), (x, t) ∈ R3 × [0, T ], (3.5) ψx3 (x, t;β, ε, α) > 0, (x, t) ∈ R3 × [0, T ]. (3.6) Proof. Firstly, we prove that ψ(x, t;β, ε, α) is the supersolution of (1.8)-(1.9). We always assume 0 < α < ε < ε1, and denote ψ(x, t;β, ε, α), $(x), %(x) and Ψ(%(x), t) by ψ(x, t), $, % and Ψ(%, t), respectively. A direct calculation yields L(ψ) = ψt(x, t)− ψx1x1 (x, t)− ψx2x2 (x, t)− ψx3x3 (x, t) + cψx3 (x, t)− f(ψ(x, t), t) = Ψt(%, t) + εσ(x′)[ω($)ν′(t) + (1− ω($))βΨβ−1($, t)Ψt($, t)] − 2∑ i=1 Ψ%%(%, t)% 2 xi −Ψ%%(%, t)% 2 x3 − 2∑ i=1 Ψ%(%, t)%xixi − 2∑ i=1 εσxixi(x ′)[ω($)ν(t) + (1− ω($))Ψβ($, t)] − 2 2∑ i=1 εσxi(x ′) [ ω′($)$xiν(t)− ω′($)$xiΨ β($, t) + (1− ω($))βΨβ−1($, t)Ψ$($, t)$xi ] − εσ(x′) [ ω′′($) ( 2∑ i=1 $2 xi ) ν(t) +$2 x3 ω′′($)ν(t) − ω′′($) ( 2∑ i=1 $2 xi ) Ψβ($, t) − ω′′($)$2 x3 Ψβ($, t) + ω′($) ( 2∑ i=1 $xixi ) ν(t) − ω′($)Ψβ($, t) ( 2∑ i=1 $xixi ) − 2ω′($)βΨβ−1($, t)Ψ$($, t) 2∑ i=1 $2 xi − 2ω′($)βΨβ−1($, t)Ψ$($, t)$2 x3 8 Z.-H. BU, C.-L. WANG, X.-T. ZHANG EJDE-2023/31 + (1− ω($))β(β − 1)Ψβ−2($, t)Ψ2 $($, t) ( 2∑ i=1 $2 xi +$2 x3 ) + (1− ω($))βΨβ−1($, t)Ψ$$($, t) ( 2∑ i=1 $2 xi +$2 x3 ) + (1− ω($))βΨβ−1($, t)Ψ$($, t) 2∑ i=1 $xixi ] + c%x3Ψ$($, t) + c$x3εσ(x′)ω′($)(ν(t)−Ψβ($, t)) + c$x3εσ(x′)(1− ω($))βΨβ−1($, t)Ψ$($, t)− f(ψ, t) = Ψt(%, t) + εσ(x′)[ω($)ν′(t) + (1− ω($))βΨβ−1($, t)Ψt($, t)] + ( − 2∑ i=1 %2 xi − 1 1 + |∇ϕ̄(z′)|2 ) Ψ%%(%, t)−Ψ%(%, t) 2∑ i=1 %xixi − εσ(x′) {∑2 i=1 σxixi(x ′) σ(x′) ω($)ν(t) + (1− ω($)) [∑2 i=1 σxixi(x ′) σ(x′) Ψβ($, t) + 2 ∑2 i=1 σxi(x ′) σ(x′) βΨβ−1($, t)Ψ$($, t) c∗ c ϕ̄zi + β(β − 1)Ψβ−2($, t)Ψ2 $($, t) ( 2∑ i=1 $2 xi + c2∗ c2 Big) + βΨβ−1($, t)Ψ$$ ( 2∑ i=1 $2 xi + c2∗ c2 ) + βΨβ−1($, t)Ψ$($, t) 2∑ i=1 $xixi − c∗βΨβ−1($, t)Ψ$($, t) ]} − εσ(x′) [ ω′′($) ( ν(t)−Ψβ($, t) )( 2∑ i=1 $2 xi + c2∗ c2 ) − 2ω′($)βΨβ−1($, t)Ψ$($, t) ( 2∑ i=1 $2 xi + c2∗ c2 ) + ω′($) ( ν(t)−Ψβ($, t) )( 2 ∑2 i=1 σxi(x ′)$xi σ(x′) + 2∑ i=1 $xixi − c∗ )] − f(ψ, t) = ( − 2∑ i=1 %2 xi − 1 1 + |∇ϕ̄(z′)|2 + 1 ) Ψ%%(%, t)−Ψ%(%, t) 2∑ i=1 %xixi − εσ(x′) { α2 ∑2 i=1Gzizi(z ′) σ(x′) ω($)ν(t) EJDE-2023/31 PERIODIC PYRAMIDAL TRAVELING FRONTS 9 + (1− ω($)) [ α2 ∑2 i=1Gzizi(z ′) σ(x′) Ψβ($, t) + 2αβ c∗ c ∑2 i=1Gzi(z ′)ϕ̄zi(zi) σ(x′) Ψβ−1($, t)Ψ$($, t) + β(β − 1)Ψβ−2($, t)Ψ2 $($, t) c2∗ c2 ( |∇ϕ̄(z′)|2 + 1 ) + β c2∗ c2 Ψβ−1($, t)Ψ$$($, t) ( |∇ϕ̄(z′)|2 + 1 ) + αβ c∗ c Ψβ−1($, t)Ψ$($, t)∆ϕ̄(z′)− c∗βΨβ−1($, t)Ψ$($, t) ]} − εσ(x′) [ ω′′($) ( ν(t)−Ψβ($, t) )( |∇ϕ̄(z′)|2 + 1 )c2∗ c2 − 2ω′($)β c2∗ c2 Ψβ−1($, t)Ψ$($, t) ( |∇ϕ̄(z′)|2 + 1 ) + ω′($) ( ν(t)−Ψβ($, t) ) × ( 2α ∑2 i=1Gzi(z ′)ϕ̄zi(zi) σ(x′) c∗ c + α c∗ c ∆ϕ̄(z′)− c∗ )] + ( c√ 1 + |∇ϕ̄(z′)|2 − c∗)Ψ%(%, t) + εσ(x′)[ω($)ν′(t) + (1− ω($))βΨβ−1($, t)Ψt($, t)] + f(Ψ(%, t), t)− f(ψ, t). Let B1 = sup z′∈R2 ∑2 i=1 |Gzizi(z′)| G(z′) , B2 = sup z′∈R2 ∑2 i=1 |Gzi(z′)| G(z′) . (3.7) Lemmas 2.1 and 2.2 imply that there exist constants Bi > 0 (i = 3, 4, 5, 6) such that 2∑ i=1 %2 xi + 1 1 + |∇ϕ̄(z′)|2 − 1 ≤ αB3G(αx′)|%(x)|+ α2B4G(αx′)%2(x) = ασ(x′) ( B3|%(x)|+ αB4% 2(x) ) ≤ εσ(x′) ( B3|%(x)|+ αB4% 2(x) ) , (3.8) and | 2∑ i=1 %xixi | ≤ αB5G(αx′) + αB6G(αx′)%(x) = ασ(x′)(B5 +B6|%(x)|) ≤ εσ(x′) ( B5 +B6|%(x)| ) . (3.9) We divide the remaining part of the proof into three cases: 1. % < −X ′, 2. % > X ′′, 3. −X ′ ≤ % < X ′′, where X ′ > 0 and X ′′ > 0 are sufficiently large constants which will be determined later. Case 1: % < −X ′, where X ′ > 0 is a sufficiently large constant. We assume that $ < −1 without loss of generality, then ω = 0 by the definition of ω, and hence we 10 Z.-H. BU, C.-L. WANG, X.-T. ZHANG EJDE-2023/31 can obtain L(ψ) = ( 1− 2∑ i=1 %2 xi − 1 1 + |∇ϕ̄(z′)|2 ) Ψ%%(%, t)−Ψ%(%, t) 2∑ i=1 %xixi − εσ(x′)Ψβ($, t) [ α2 ∑2 i=1Gzizi(z ′) σ(x′) + 2αβ c∗ c ∑2 i=1Gzi(z ′)ϕ̄zi(z ′) σ(x′) Ψ$($, t) Ψ($, t) + β(β − 1) Ψ2 $($, t) Ψ2($, t) c2∗ c2 ( |∇ϕ̄(z′)|2 + 1 ) + c∗ c αβ Ψ$($, t) Ψ($, t) 4ϕ̄(z′) + β Ψ$$($, t) Ψ($, t) c2∗ c2 ( |∇ϕ̄(z′)|2 + 1 ) − c∗β Ψ$($, t) Ψ($, t) ] + ( c√ 1 + |∇ϕ̄(z′)|2 − c∗ ) Ψ%(%, t) + εσ(x′)βΨβ−1($, t)Ψt($, t) + f(Ψ(%, t), t)− f(ψ, t). When % < 0, inequality (3.3) yields % < $. From (3.8) and (3.9), it follows that L(ψ) ≥ −εσ(x′)Ψβ($, t)(B3|%|+B4% 2) |Ψ%%(%, t)| Ψβ(%, t) − εσ(x′)Ψβ($, t)(B5 +B6|%|) |Ψ%(%, t)| Ψβ(%, t) − εσ(x′)Ψβ($, t) [ α2B1 + 2αB2m∗ Ψ$($, t) Ψ($, t) + α Ψ$($, t) Ψ($, t) 2∑ i=1 |ϕ̄zizi(z′)| + β c2∗ c2 | − (Ψ$($, t) Ψ($, t) )2 + Ψ$$($, t) Ψ($, t) | ( |∇ϕ̄(z′)|2 + 1 ) + β2 c 2 ∗ c2 (Ψ$($, t) Ψ($, t) )2( |∇ϕ̄(z′)|2 + 1 ) − β2 (Ψ$($, t) Ψ($, t) )2 + β2 (Ψ$($, t) Ψ($, t) )2 − c∗β Ψ$($, t) Ψ($, t) − βΨt($, t) Ψ($, t) ] + ( c√ 1 + |∇ϕ̄(z′)|2 − c∗)Ψ%(%, t)− f(ψ, t) + f(Ψ(%, t), t). Since limϑ→−∞Ψ(ϑ, t) = 0 uniformly for t ∈ [0, T ], by (1.3) and Lemma 2.1, we have lim $→−∞ f(Ψ($, t), t) Ψ($, t) = fu(0, t) = 0, lim $→−∞ [( − c2∗ c2 Ψ2 $($, t) Ψ2($, t) + c2∗ c2 Ψ$$($, t) Ψ($, t) ) (|∇ϕ̄(z′)|+ 1) ] = 0, lim $→−∞ [ β2 (Ψ$($, t) Ψ($, t) )2 − c∗β Ψ$($, t) Ψ($, t) ] = Π(βΛ2), lim $→−∞ [ βc∗ Ψ$($, t) Ψ($, t) − βΨ$$($, t) Ψ($, t) ] = 0, EJDE-2023/31 PERIODIC PYRAMIDAL TRAVELING FRONTS 11 uniformly in t ∈ [0, T ]. Thus there exists a sufficiently large constant X1 > 0 such that for any t ∈ [0, T ] and $ < −X1, −β f(Ψ($, t), t) Ψ($, t) < − 1 16 Π(βΛ2), Ψ$($, t) Ψ($, t) < 3 2 Λ2,∣∣− ( Ψ$($, t) Ψ($, t) )2 + Ψ$$($, t) Ψ($, t) ∣∣ < − 1 16 Π(βΛ2), β2( Ψ$($, t) Ψ($, t) )2 − c∗β Ψ$($, t) Ψ($, t) < 1 2 Π(βΛ2), βc∗ Ψ$($, t) Ψ($, t) − βΨ$$($, t) Ψ($, t) < − 1 16 Π(βΛ2). Inequalities (1.4)-(1.5) imply that there exists a sufficiently large constant X2 > 0 such that (B3|%|+B4% 2) |Ψ%%(%, t)| Ψβ(%, t) < − 1 16 Π(βΛ2), (B5 +B6|%|) |Ψ%(%, t)| Ψβ(%, t) < − 1 16 Π(βΛ2) for any t ∈ [0, T ] and % < −X2. In addition, we can choose α1 ∈ (0, β) small enough such that α2B1 + 3αB2m∗Λ2 + 3αm∗M∗Λ2 < − 1 16 Π(βΛ2), ∀α ∈ (0, α1). It follows from (2.6) that there exists a sufficiently large constant X3 > 0 such that −ε1 < Ψ(%, t) + εσ(x′)Ψβ($, t) < ε1 for any 0 < ε < ε1 2(c−c∗) . Therefore, f(ψ, t)− f(Ψ(%, t), t) = fu(Ψ(%, t) + θεσ(x′)Ψβ($, t), t)εσ(x′)Ψβ($, t) < − 1 16 Π(βΛ2) for % < −X3 and t ∈ [0, T ], where θ ∈ (0, 1). Let X ′ = max{ cc∗ , c c∗ X1, X2, c c∗ X3}. Thus when % < −X ′, we have L(ψ) ≥ −εσ(x′)Ψβ($, t)(B3|%|+B4% 2) |Ψ%%(%, t)| Ψβ(%, t) − εσ(x′)Ψβ($, t)(B5 +B6|%|) |Ψ%(%, t)| Ψβ(%, t) − εσ(x′)Ψβ($, t) [ α2B1 + 3αB2m∗Λ2 + 3αm∗M∗Λ2 + β c2∗ c2 ∣∣∣− (Ψ$($, t) Ψ($, t) )2 + Ψ$$($, t) Ψ($, t) ∣∣∣(|∇ϕ̄(z′)|2 + 1 ) + β2 c 2 ∗ c2 (Ψ$($, t) Ψ($, t) )2( |∇ϕ̄(z′)|2 + 1 ) − β2 (Ψ$($, t) Ψ($, t) )2 + β2 (Ψ$($, t) Ψ($, t) )2 − c∗β Ψ$($, t) Ψ($, t) + c∗β Ψ$($, t) Ψ($, t) − βΨ$$($, t) Ψ($, t) − β f(Ψ($, t), t) Ψ($, t) ] − εσ(x′)fu(Ψ(%, t) + θεσ(x′)Ψβ($, t), t)Ψβ($, t) 12 Z.-H. BU, C.-L. WANG, X.-T. ZHANG EJDE-2023/31 ≥ −εσ(x′)Ψβ($, t) ( 1 16 Π(βΛ2) + 1 16 Π(βΛ2) + 1 16 Π(βΛ2) + 1 16 Π(βΛ2) − 1 2 Π(βΛ2) + 1 16 Π(βΛ2) + 1 16 Π(βΛ2) + 1 16 Π(βΛ2) ) > 0. Case 2: % > X ′′ > 0, where X ′′ is a sufficiently large constant. Let $ > 1, (1.5) implies that there exists a sufficiently large constant X ′1 > 0 such that for any % > X ′1 and t ∈ [0, T ] (B3|%|+B4% 2)|Ψ%%(%, t)| < 1 8 P1Λ0, (B5 +B6|%|)|Ψ%(%, t)| < 1 8 P1Λ0. Since lim%→+∞Ψ(%, t) = 1, there exists a sufficiently large constant X ′2 > 0, such that for any ε ∈ (0, ε1 P2(c−c∗) ), we have 1− ε1 < Ψ(%, t) + θεσ(x′)ν(t) < 1 + ε1, % > X ′2, t ∈ [0, T ]. Therefore, for each % > X ′1, inequality (2.7) yields (fu(1, t)− fu(Ψ(%, t) + θεσ(x′)ν(t), t))ν(t) > −1 2 P1Λ0, t ∈ [0, T ]. In addition, from (3.7), one has εσ(x′)Λ0ν ′(t)− εα2 2∑ i=1 Gzizi(z ′)ν(t) ≥ εσ(x′)(Λ0 − α2B1)P1, for x′ ∈ R2 and t ∈ [0, T ]. Choosing X ′′ = max { X ′1, X ′ 2, c c∗ } , then for any % > X ′′2 , one has L(ψ) = ( 1− 2∑ i=1 %2 xi − 1 1 + |∇ϕ̄(z′)|2 ) Ψ%%(%, t)−Ψ%(%, t) 2∑ i=1 %xixi − c∗Ψ%(%, t) + f(Ψ(%, t), t)− f(ψ, t) + εσ(x′)ν′(t)− εα2 2∑ i=1 Gzizi(z ′)ν(t) = ( 1− 2∑ i=1 %2 xi − 1 1 + |∇ϕ̄(z′)|2 ) Ψ%%(%, t)−Ψ%(%, t) 2∑ i=1 %xixi − c∗Ψ%(%, t) − fu(Ψ(%, t) + θεσ(x′)ν(t), t)εσ(x′)ν(t)− εα2 2∑ i=1 Gzizi(z ′)ν(t) + εσ(x′)ν(t) ( fu(1, t) + Λ0 ) ≥ −εσ(x′)(B3|%|+B4% 2)|Ψ%%(%, t)| − εσ(x′)(B5 +B6|%|)|Ψ%(%, t)| − εσ(x′) 1 2 P1Λ0 + εσ(x′) ( Λ0 − α2B1 ) P1 ≥ εσ(x′) [ − 1 8 P1Λ0 − 1 8 P1Λ0 − 1 2 P1Λ0 + (Λ0 − α2B1)P1 ] > 0, if 0 < α < √ Λ0 4B1 . EJDE-2023/31 PERIODIC PYRAMIDAL TRAVELING FRONTS 13 Case 3: −X ′ < % < X ′′, where X ′ and X ′′ are defined in Cases 1 and 2. Let ψ∗ = min −X′≤%≤X′′,t∈[0,T ] Ψ%(%, t), Q0 = sup u∈[−ε1,1+ε1],t∈[0,T ] |fu(u, t)|, Q1 = sup %∈R,t∈[0,T ] |Ψ%(%, t)|, Q2 = sup %∈R,t∈[0,T ] |%||Ψ%(%, t)|, Q3 = sup %∈R,t∈[0,T ] |%||Ψ%%(%, t)|, Q4 = sup %∈R,t∈[0,T ] %2|Ψ%%(%, t)|, Q5 = sup %∈R,t∈[0,T ] ∣∣Ψ%(%, t) Ψ(%, t) ∣∣, Q6 = sup %∈R,t∈[0,T ] ∣∣Ψ%%(%, t) Ψ(%, t) ∣∣. Since ν(t) = keΛ0t+ ∫ t 0 fu(1,s)ds and ν′(t) = keΛ0t+ ∫ t 0 fu(1,s)ds(Λ0+fu(1, t)), ν′(t) is bounded following from the boundedness of ν(t). By Ψt = Ψ%%− c∗Ψ%+f(Ψ(%), t), we have Ψt is also bounded and max $∈R,t∈R |ω($)ν′(t) + (1− ω($)βΨβ−1($, t)Ψt($, t)| ≤ C0 for some constant C0 > 0. Therefore we obtain L(ψ) = ( 1− 2∑ i=1 %2 xi − 1 1 + |∇ϕ̄(z′)|2 ) Ψ%%(%, t)−Ψ%(%, t) 2∑ i=1 %xixi − εσ(x′) { α2 ∑2 i=1Gzizi(z ′) σ(x′) ω($)ν(t) + (1− ω($)) [ α2 ∑2 i=1Gzizi(z ′) σ(x′) Ψβ($, t) + 2αβ c∗ c ∑2 i=1Gzi(z ′)ϕ̄zi(z ′) σ(x′) Ψβ−1($, t)Ψ$($, t) + αβ c∗ c 2∑ i=1 ϕ̄zizi(z ′)Ψβ−1($, t)Ψ$($, t) + βΨβ−1($, t)Ψ$$($, t) (c∗ c )2(|∇ϕ̄(z′)|2 + 1 )]} − εσ(x′) [(c∗ c )2(|∇ϕ̄(z′)|2 + 1 ) ω′′($) ( ν(t)−Ψβ($, t) ) + 2α c∗ c ∑2 i=1Gzi(z ′)ϕ̄zi(zi) σ(x′) ω′($) ( ν(t)−Ψβ($, t) ) + α c∗ c 2∑ i=1 ϕ̄zizi(z ′)ω′($) ( ν(t)−Ψβ($, t) )] + f(Ψ(%, t), t)− f(ψ, t) + ( c√ 1 + |∇ϕ̄(z′)|2 − c∗)Ψ%(%, t) + εσ(x′)[ω($)ν′(t) + (1− ω($))βΨβ−1($, t)Ψt($, t)] ≥ −εσ(x′)(B3|%|+B4% 2)|Ψ%%(%, t)| − εσ(x′)(B5 +B6|%|)|Ψ%(%, t)|+ σ(x′)Ψ%(%, t) − εσ(x′) { α2 ∑2 i=1 |Gzizi(z′)| σ(x′) [ω($)ν(t) + (1− ω($))Ψβ($, t)] 14 Z.-H. BU, C.-L. WANG, X.-T. ZHANG EJDE-2023/31 + 2α ∑2 i=1 |Gzi(z′)ϕ̄zi(z′)| σ(x′) Ψ$($, t) Ψ($, t) + α 2∑ i=1 |ϕ̄zizi(z′)| Ψ$($, t) Ψ($, t) + |Ψ$$($, t) Ψ($, t) | } − εσ(x′)|ω′′($)| ( ν(t)−Ψβ($, t) ) − εσ(x′)2α ∑2 i=1 |Gzi(z′)ϕ̄zi(zi)| σ(x′) ω′($) ( ν(t)−Ψβ($, t) ) − εσ(x′)α 2∑ i=1 |ϕ̄zizi(z′)|ω′($) ( ν(t)−Ψβ($, t) ) − εσ(x′)C0 − εσ(x′)fu [ Ψ(%, t) + θεσ(x′)(ω($)ν(t) + (1− ω($))Ψβ($, t)) · (ω($)ν(t) + (1− ω($))Ψβ($, t)) ] ≥ σ(x′)(−αB3Q3 − αB4Q4 − αB5Q1 − αB6Q2 − αB1P2 − 2αB1m∗Q5 − 2αm∗M∗Q5 − εN2 − εA+ u∗ − εC0 −Q0P2) > 0, where A = ( sup x∈R |ω′′(x)|+ 2B2m∗ + 2M∗m∗ ) P2, α < α2 = u∗ 2 (B3Q3 +B4Q4 +B5Q1 +B6Q2 +B1P2 + 2B1m∗Q5 + 2m∗M∗Q5) , ε < ε2 = u∗ 2 (N2 +A+ C0 +Q0P2) . To sum up, combining the above Cases 1– 3, ψ is the supersolution of (1.8)-(1.9) on R3 × (−∞,+∞). Secondly, we prove (3.5). Let ϑ(x) = c∗ c (x3 + h(x′)), η(x) = x3 + h(x′)√ 1 + |∇ϕ̄(αx′)|2 . Recall that $(x) = c∗ c (x3 + ϕ̄(z′)/α), %(x) = x3 + ϕ̄(z′)/α√ 1 + |∇ϕ̄(αx′)|2 , ψ(x, t)− ψ(x, t) = Ψ(%(x), t)−Ψ(ϑ(x), t) + εσ(x′)(ω($)ν(t) + (1− ω($))Ψβ($, t)), h(x′) ≤ ϕ̄(αx′)/α. We divide the proof into two cases. Case 1: %(x) ≥ ϑ(x) for any x ∈ R3. Since the function Ψ(ξ, t) is monotonically increasing in ξ, it is obvious that ψ(x, t) < ψ(x, t) for any (x, t) ∈ R3× (−∞,+∞). Case 2: %(x) < ϑ(x) for any x ∈ R3. %(x)− ϑ(x) = x3 + ϕ̄(z′)/α√ 1 + |∇ϕ̄(z′)|2 − c∗ c (x3 + h(x′)) EJDE-2023/31 PERIODIC PYRAMIDAL TRAVELING FRONTS 15 = ( 1√ 1 + |∇ϕ̄(z′)|2 − c∗ c ) (x3 + h(x′)) + ϕ̄(z′)− h(z′) α √ 1 + |∇ϕ̄(z′)|2 < 0. Since 1√ 1+|∇ϕ̄(αx′)|2 > c∗ c , we have x3 + h(x′) < − ϕ̄(αx′)− h(αx′) α √ 1 + |∇ϕ̄(z′)|2 /( 1√ 1 + |∇ϕ̄(z′)|2 − c∗ c ) ≤ −a1c∗ α < 0, where a1 is defined in Lemma 2.2. Thus %(x) < ϑ(x) ≤ −a1c 2 ∗ cα < 0, c c∗ $(x) < %(x) < $(x). Furthermore, ψ(x, t)− ψ(x, t) = Ψ(%(x), t)−Ψ(η(x), t) + Ψ(η(x), t)−Ψ(ϑ(x), t) + εσ(x′)(ω($)ν(t) + (1− ω($))Ψβ($, t)) ≥ Ψ(η(x), t)−Ψ(ϑ(x), t) + εσ(x′)(ω($)ν(t) + (1− ω($))Ψβ($, t)) ≥ ( 1√ 1 + |∇ϕ̄(αx′)|2 − c∗ c ) (x3 + h(x′))Ψ$(θη(x) + (1− θ)ϑ(x), t) + εσ(x′)Ψβ($(x), t) = ( 1√ 1 + |∇ϕ̄(αx′)|2 − c∗ c ) c c∗ ϑ(x)Ψ$(θη(x) + (1− θ)ϑ(x), t) + εσ(x′)Ψβ($(x), t), where θ ∈ (0, 1). Since x3 + h(x′) < 0, we have η(x) < %(x) < ϑ(x) < $(x) < 0, Ψ$(θη(x) + (1− θ)ϑ(x), t) ≤ L2e −Λ2|θη(x)+(1−θ)ϑ(x)| ≤ L2e Λ2ϑ(x) and Ψβ($(x), t) ≥ Lβ1 eβΛ2$(x) > L1e βΛ2$(x) > L1e βΛ2ϑ(x). Thus we have ψ(x, t)− ψ(x, t) = ( 1√ 1 + |∇ϕ(αx′)|2 − c∗ c ) c c∗ ϑ(x)Ψ$(θη(x) + (1− θ)ϑ(x), t) + εσ(x′)Ψβ($(x), t) ≥ σ(x′) (L2 c∗ ϑ(x)eΛ2ϑ(x) + εL1e Λ2βϑ(x) ) ≥ σ(x′)eΛ2βϑ(x) ( L2 c∗(1− β)2Λ2 2ϑ(x) sup ω>0 (ω2e−ω) + εL1 ) ≥ σ(x′)eΛ2βϑ(x) ( − 4L2αc c3∗e 2(1− β)2a1Λ2 2 + εL1 ) > 0, if α < α3 = εa1L1c 3 ∗e 2(1− β)2Λ2 2 4L2c . In conclusion, we can obtain ψ(x, t) > ψ(x, t) for any x ∈ R3 and t ∈ [0, T ]. 16 Z.-H. BU, C.-L. WANG, X.-T. ZHANG EJDE-2023/31 Finally, we prove (3.4). We just need to prove lim γ̄→∞ sup x∈D(γ̄),t∈[0,T ] |Ψ(%(x), t)−Ψ(ϑ(x), t)| Ψβ($(x), t) = 0. (3.10) We prove it by a contradiction argument. Assume that (3.10) is not true, then there exist a positive number ε∗, sequences {γ̄n}n∈N ∈ R and {xn}n∈N ∈ R3 such that lim n→∞ γ̄n =∞, xn ∈ D(γ̄n), (3.11) |Ψ(%(x), t)−Ψ(ϑ(x), t)| Ψβ($(x), t) ≥ ε∗. (3.12) We denote xn = (x′n, xn,3) ∈ R3, with x′n = (xn,1, xn,2) ∈ R2. Obviously, %(xn) = xn,3 + ϕ̄(αx′n) α√ |∇ϕ̄(αx′n)|2 + 1 = xn,3 + h(x′n) + ϕ̄(αx′n)−h(αx′n) α√ |∇ϕ̄(αx′n)|2 + 1 . Now we consider two cases and prove them separately. Case 1: limn→+∞ dist(x′n, E) =∞. In this case, we can obtain limn→+∞G(x′n) = 0 and limn→+∞ |ϕ̄(x′n) − h(x′n)| = 0. Hence we can obtain limn→+∞ |%(xn) − ϑ(xn)| = 0. If ϑ(xn) = c∗ c (xn,3+h(x′n))→ +∞ as n→ +∞, then %(xn)→ +∞ and therefore lim n→∞ |Ψ(%(xn), t)−Ψ(ϑ(xn), t)| Ψβ($(xn), t) = 0, which contradicts with (3.12). If ϑ(xn) = c∗ c (xn,3 + h(x′n))→ −∞ as n→ +∞, then %(xn)→ −∞. Since h(x′n) < 1 α ϕ̄(αx′n) ≤ h(x′n) + 2πm∗ α ∫ ∞ 0 r2ρ̃(r)dr, and 1 < √ |∇ϕ̄(αx′n)|2 + 1 < c c∗ , it can be obtained that, when n is sufficiently large, there are 0 > $(xn) > %(xn) > c c∗ $(xn) and ϑ(xn) < $(xn) < ϑ(xn) + 2πm∗c∗ αc ∫ ∞ 0 r2ρ̃(r)dr. Letting n→ +∞, we have |Ψ(%(xn), t)−Ψ(ϑ(xn), t)| Ψβ($(xn), t) = |(%(xn)− ϑ(xn)) ·Ψ′((1− θ)ϑ(xn) + θ%(xn), t)| Ψβ($(xn), t) ≤ L2e Λ2((1−θ)ϑ(xn)+θ%(xn)) Lβ1 e βΛ2$(xn) |%(xn)− ϑ(xn)| ≤ L2e Λ2$(xn) Lβ1 e βΛ2$(xn) [ ( c c∗ + 1)|$(xn)|+ 2πm∗c∗ αc ∫ ∞ 0 r2ρ̃(r) dr ] → 0, which contradicts with (3.12). EJDE-2023/31 PERIODIC PYRAMIDAL TRAVELING FRONTS 17 If ϑ(xn) = c∗ c (xn,3 + h(x′n)) is bounded for each n ∈ N, then we have $(xn) is also bounded for each n ∈ N. Since limn→+∞ |%(xn)− ϑ(xn)| = 0, it holds lim n→∞ |Ψ(%(xn), t)−Ψ(ϑ(xn), t)| Ψβ($(xn), t) = 0, which also contradicts with (3.12). Case 2: dist(x′n, E) is uniformly bounded in k. From (3.11), we can easily obtain (xn,3 + h(x′n))→ ±∞ as n→ +∞. If (xn,3 + h(x′n))→ +∞ as n→ +∞, then ϑ(xn) = c∗ c (xn,3 + h(x′n))→ +∞ as n→ +∞ and %(xn) = xn,3 + h(x′n) + ϕ̄(αx′n)−h(αx′n) α√ |∇ϕ̄(αx′n)|2 + 1 ≥ ϑ(xn)→ +∞ as n→ +∞. So we can obtain lim n→∞ |Ψ(%(xn), t)−Ψ(ϑ(xn), t)| Ψβ($(xn), t) = 0, which contradicts with (3.12). If (xn,3 + h(x′n))→ −∞ as n→ +∞, then ϑ(xn) = c∗ c (xn,3 + h(x′n))→ −∞ as n→ +∞ and %(xn) = xn,3 + h(x′n) + ϕ̄(αx′n)−h(αx′n) α√ |∇ϕ̄(αx′n)|2 + 1 ≤ ϑ(xn) + 1√ |∇ϕ̄(αx′n)|2 + 1 2πm∗ α ∫ ∞ 0 r2ρ̃(r)dr → −∞ as n→ +∞. Similar to the argument in Case 1, we have |Ψ(%(xn), t)−Ψ(ϑ(xn), t)| Ψβ($(xn), t) ≤ 0, which contradicts (3.12). Summing up, (3.4) is true. In conclusion, letting ε+ 0 (β) = { ε1 P2 , ε1 c− c∗ , ε2}, α+ 0 (ε, β) = { ε, α1, α2, α3, √ Λ0 4B7 } , we complete the proof. � 3.2. Existence. In this subsection, we give the proof of Theorem 1.1. That is, we prove the existence of three-dimensional periodic pyramidal traveling front. Theorem 3.2. Assume that (H1) and (H2) hold. For each c > c∗, equation (1.1) has a periodic nonplanar traveling front V (x, t) satisfying (1.8)-(1.9) and ψ(x, t) < V (x, t) < ψ(x, t;β, ε, α), x ∈ R3, t ∈ [0, T ]. Moveover, lim γ̄→∞ sup x∈D(γ̄),t∈[0,T ] |V (x, t)− ψ(x, t)| (ψ(x, t))β = 0 (3.13) and Vx3 (x, t) > 0 for all (x, t) ∈ R3 × [0, T ]. 18 Z.-H. BU, C.-L. WANG, X.-T. ZHANG EJDE-2023/31 Proof. According to the parabolic estimation, there exists a constant C > 0 such that the solution ψ(x, t;ψ0) of Eq. (1.7) with the initial value ψ0(x, t) ∈ [0, 1] satisfies ‖ψ(·, ·;ψ0)‖ C2+θ,1+ θ 2 (R3×[T,+∞)) < C, where 0 < θ < 1. Since ψ(x, t) is the subsolution of (1.7) for any x ∈ R3 and t ∈ [0, T ], and ψ(x, t+ T ) = ψ(x, t), we have 0 < ψ(x, t+ kT ;ψ) ≤ ψ(x, t+ (k + 1)T ;ψ) < 1, x ∈ R3, t ∈ [0, T ] from the maximum principle. Thus ψ(x, t + kT ;ψ) monotonically increasing con- verges to V (·, ·) under the norm ‖ · ‖C2,1 loc(R3×[0,T ]) as k →∞. That is lim k→∞ ∥∥ψ(x, t+ kT ;ψ)− V (x, t) ∥∥ C2,1 loc (R3×[0,T ]) = 0. Meanwhile, since ψ(x, t;β, ε, α) is a supersolution, it can be obtained that ψ(x, t) < V (x, t) < ψ(x, t;β, ε, α) by the comparison principle. Since ψ x3 (x, t) > 0, it follows that Vx3 (x, t) ≥ 0. By the strong maximum principle, we can obtain Vx3 (x, t) > 0. From (3.4), we have lim γ̄→∞ sup x∈D(γ̄),t∈[0,T ] |V (x, t)− ψ(x, t)| (Ψ(x, t))β ≤ 2ε. (3.14) Then we fix β ∈ (0, 1) and let β ∈ (β, 1). From (3.14), for any 0 < ε < min{ε+ 0 (β), ε+ 0 (β)} and 0 < α < min{α+ 0 (β, ε), α+ 0 (β, ε)}, we can easily get lim γ̄→∞ sup x∈D(γ̄),t∈[0,T ] |V (x, t)− ψ(x, t)| Ψβ( c∗c (x3 + ϕ(αx′) α ), t) ≤ 2ε. (3.15) Fix α ∈ (0,min{α+ 0 (β, ε), α+ 0 (β, ε)}). Then there exists γ′ > 0 such that for any γ̄ > γ′, sup x∈D(γ̄),t∈[0,T ] |V (x, t)− ψ(x, t)| Ψβ( c∗c (x3 + ϕ̄(αx′) α ), t) ≤ 4ε. (3.16) We divide the remaining part of the proof into two cases. Case 1: |x3 + h(x′)| > K1, where K1 > 0 is sufficiently large. Obviously if x ∈ R3 satisfies |x3+h(x′)| > K2, we can obtain dist(x,Γ) > γ′, whereK2 > 0 is sufficiently large. Fix K3 > 0 such that dist(x,Γ) > γ′ and Ψβ(ϑ(x)) > 4 5 if |x3 + h(x′)| > K3. Because |V (x, t)− ψ(x, t)| Ψβ( c∗c (x3 + ϕ̄(αx′) α ), t) ≥ |V (x, t)− ψ(x, t)| (ψ(x), t)β Ψβ (c∗ c (x3 + h(x′)), t ) for any t ∈ [0, T ] and x ∈ R3 with |x3 + h(x′)| > K3, we have |V (x, t)− ψ(x, t)| (ψ(x, t))β ≤ |V (x, t)− ψ(x, t)| Ψβ( c∗c (x3 + ϕ̄(αx′) α ), t) 1 Ψβ( c∗c (x3 + h(x′)), t) ≤ 5ε. Let K4 > 2πm∗ α ∫∞ 0 r2ρ̃(r)dr large enough satisfy L−β1 Lβ2 e Λ2(β−β)ϑe 2πm∗ α Λ2β ∫∞ 0 r2ρ̃(r)dr < 5 4 , ∀ϑ < −K4, EJDE-2023/31 PERIODIC PYRAMIDAL TRAVELING FRONTS 19 where (1.4) gives the definitions of L1 and L2. Since |V (x, t)− ψ(x, t)| Ψβ( c∗c (x3 + ϕ̄(αx′) α ), t) = (ψ(x), t)β (ψ(x), t)β |V (x, t)− ψ(x, t)| Ψβ( c∗c (x3 + ϕ̄(αx′) α ), t) = Ψβ( c∗c (x3 + h(x′)), t) Ψβ( c∗c (x3 + ϕ̄(αx′) α ), t) |V (x, t)− ψ(x, t)| (ψ(x), t)β ≥ Ψβ( c∗c (x3 + h(x′)), t) Ψβ( c∗c (x3 + h(x′) + 2πm∗ α ∫∞ 0 r2ρ̃(r)dr), t) |V (x, t)− ψ(x, t)| (ψ(x), t)β , then from (1.4), for any x ∈ R3 that satisfies x3 + h(x′) < −K4, we can obtain |V (x, t)− ψ(x, t)| (ψ(x, t))β ≤ 5ε, t ∈ [0, T ]. Case 2: |x3+h(x′)| ≤ K1 and K > 0 is sufficiently large such that dist(x′, E) > K, where K1 = max{K2,K3,K4}. When K > 0 is sufficiently large for all x ∈ R3 with dist(x′, E) > K and |x3 + h(x′)| ≤ K1, one has Ψβ( c∗c (x3 + h(x′)), t) Ψβ( c∗c (x3 + ϕ̄(αx′) α ), t) > 4 5 . Thus we obtain |V (x, t)− ψ(x, t)| (ψ(x, t))β ≤ |V (x, t)− ψ(x, t)| Ψβ( c∗c (x3 + h(x′)), t) Ψβ( c∗c (x3 + ϕ̄(αx′) α ), t) Ψβ( c∗c (x3 + ϕ̄(αx′) α ), t) ≤ 5ε. According to the definition of D(γ̄), there exists γ∗ > 0 such that D(γ∗) ⊂ {x ∈ R3 : |x3 + h(x′)| > K1 or |x3 + h(x′)| ≤ K1 and dist(x′, E) > K}. Thus Cases 1 and 2 imply |V (x, t)− ψ(x, t)| (ψ(x, t))β ≤ 5ε, x ∈ D(γ∗), t ∈ [0, T ]. Therefore, sup x∈D(γ̄),t∈[0,T ] |V (x, t)− ψ(x, t)| (ψ(x, t))β ≤ 5ε, ∀γ > γ∗. Hence (3.13) holds by the arbitrariness of ε. The proof is complete. � 4. Periodic pyramidal traveling fronts in Rn with n ≥ 4 In this section, we investigate the existence of periodic nonplanar traveling front to (1.1) in Rn (n ≥ 4). We use the same notation as above. We denote s = (s1, s2, ...sn) ∈ Rn and s′ = (s1, s2, . . . , sn−1) ∈ Rn−1. Assume that the traveling fronts travel towards −sn direction at the speed of c > c∗. Let u(s, t) = v(s′, sn + ct, t) = v(s′, w, t). 20 Z.-H. BU, C.-L. WANG, X.-T. ZHANG EJDE-2023/31 We still express v(s′, w, t) as v(s′, sn, t) for convenience. Substitute v into (1.1), then vt = ∆v − cvsn + f(v, t), s ∈ Rn, t > 0, v(s, 0) = v0(s), s ∈ Rn. The purpose of this section is to find a functon V (s, t) satisfying the equations Vt −∆V + cVsn − f(V, t) = 0, s ∈ Rn, t ∈ R, (4.1) V (s, t) = V (s, t+ T ), s ∈ Rn, t ∈ R. (4.2) Let l ≥ 3 be a given integer and {Aj}lj=1 ⊂ Rn be a set of unit vectors such that {Ai 6= Aj}, if i 6= j. Then Aj = (A1,j , A2,j , . . . , An−1,j) satisfies |Aj | = n−1∑ i=1 A2 i,j = 1, j = 1, 2, . . . , l. Therefore (m∗Aj , 1) ∈ Rn is a normal vector of {s ∈ Rn|−sn = m∗(Aj , s ′)}, where (Aj , s ′) = ∑n−1 i=1 Ai,jsi. Let hj(s ′) = m∗(Aj , s ′), 1 ≤ j ≤ l, h(s′) = max 1≤j≤l hj(s ′) = m∗ max 1≤j≤l (Aj , s ′), then {s ∈ Rn| − sn = h(s′)} is a pyramid in Rn. Similar to the previous works, we define Ωj , Gj ,Γj ,D(γ), E as in Section 1 by replacing (x1, x2) and (x1, x2, x3) with s′ and s, respectively. Let ∂Ωj be the boundary of Ωj . For any 1 ≤ j ≤ l, it’s obvious that Ψ( c∗c (sn + hj(s ′)), t) is the solution of (4.1). Define ψ(s, t) = Ψ (c∗ c (sn + h(s′)), t ) = max 1≤j≤l Ψ (c∗ c (sn + hj(s ′)), t ) , then ψ(s, t) is the subsolution of (4.1). Let function ρ̃(r) ∈ C∞[0,∞) satisfy the following properties: (1) ρ̃(r) > 0, ρ̃r(r) ≤ 0, r ≥ 0; (2) If r > 0 is small enough, ρ̃(r) = 1; (3) If r > 0 is large enough, say r > R0, ρ̃(r) = e−r, where R0 > 1 is a constant; (4) ∫ Rn−1 ρ̃(|s′|) ds′ = 1. It is obvious that∫ Rn−1 ρ̃(|s′|)ds′ = (n− 1)π n−1 2 Γ(n+1 2 ) ∫ ∞ 0 rn−2ρ̃(r)dr. Let ρ(s′) = ρ̃(|s′|), one has ∫ Rn−1 ρ(s′)ds′ = 1. For all nonnegative integers j1, . . . , jn−1 satisfying 0 ≤ ∑n−1 q=1 jq ≤ 3, we have |Dj11 . . .Djn−1 n−1 ρ(s′)| ≤M∗ρ(s′), s′ ∈ Rn−1, where M∗ is a positive constant. Define ϕ̄(s′) = ρ ∗ h, then for each s′ ∈ Rn−1, ϕ̄(s′) = ∫ R2 ρ(s′′)h(s′ − s′′)ds′′ = ∫ R2 ρ(s′ − s′′)h(s′′)ds′′. (4.3) EJDE-2023/31 PERIODIC PYRAMIDAL TRAVELING FRONTS 21 The set {s ∈ Rn| − sn = ϕ̄(s′)} is called the mollified pyramid of {s ∈ Rn| − sn = h(s′)}. Let G(s′) = c√ 1 + |∇ϕ̄(s′)|2 − c∗, (4.4) where |∇ϕ̄(s′)| = √∑n−1 i=1 ϕ̄ 2 si(s ′). The next two lemmas come from the [15, Lemma 2.2 and Prop. 2.3] and from [28, Remark 2.3]. Lemma 4.1. Let ϕ̄(s′) and G(s′) be as defined in (4.3) and (4.4) respectively. Then for any fixed (j1, . . . , jn−1) 6= (0, . . . , 0) with jq ≥ 0 (q = 1, . . . , n − 1), one has sup s′∈Rn−1 |Dj1s1D j2 s2 . . . ,D jn−1 sn−1 ϕ̄(s′)| <∞, h(s1, s2, . . . , sn−1) < ϕ̄(s′) ≤ h(s′) + (n− 1)π n−1 2 Γ(n+1 2 ) m∗ ∫ ∞ 0 rn−1ρ̃(r)dr, |∇ϕ̄(s′)| < m∗, 0 < G(s′) ≤ c− c∗, ∀s′ ∈ Rn−1. Lemma 4.2. There exist two constants b1 and b2 such that 0 < b1 = inf s′∈Rn−1 ϕ̄(s′)− h(s′) G(s′) ≤ sup s′∈Rn−1 ϕ̄(s′)− h(s′) G(s′) = b2 <∞. Moreover, for every integer jq ≥ 0 (q = 1, . . . , n−1) with 2 ≤ j1+j2+· · ·+jn−1 ≤ 3, there exists a constant K > 0 such that sup s′∈Rn−1 ∣∣Dj1s1Dj2s2 . . .Djn−1 sn−1 ϕ̄(s′) G(s′) ∣∣ < K, |ϕ̄sisi(s′)| ≤ m∗M∗, i = 1, 2, . . . , n− 1, s′ ∈ Rn−1. Proceeding as in the previous sections, we obtain the following lemma and the- orem. Lemma 4.3. For each β ∈ (0, 1), there exist positive constants ε+ 0 (β) and α+ 0 (β, ε) such that, for any 0 < ε < ε+ 0 (β) and 0 < α < α+ 0 (β, ε), the function ψ(s, t;β, ε, α) = Ψ(%(s), t) + εσ(s′)(ω($(s))ν(t) + (1− ω($(s)))Ψβ($(s), t)) is a supersolution of (4.1)-(4.2) on Rn × (−∞,+∞). In addition, lim γ→∞ sup s∈D(γ),t∈[0,T ] |ψ(s, t;β, ε, α)− ψ(s, t)| ψ(s, t)β ≤ 2ε, ψ(s, t) < ψ(s, t;β, ε, α), (s, t) ∈ Rn × [0, T ], ψsn(s, t;β, ε, α) > 0, (s, t) ∈ Rn × [0, T ]. Theorem 4.4. Assume that (H1) and (H2) hold. Then for each c > c∗, equa- tion (1.1) has a periodic nonplanar traveling front V (s, t) satisfying (4.1)-(4.2). Moreover, lim γ̄→∞ sup s∈D(γ̄),t∈[0,T ] |V (s, t)− ψ(s, t)| (ψ(s, t))β = 0, ∀β ∈ (0, 1), Vsn(s, t) > 0, (s, t) ∈ Rn × R. 22 Z.-H. BU, C.-L. WANG, X.-T. ZHANG EJDE-2023/31 5. Conclusion Since the environment changes over time, it is of great practical significance to study the effect of time period on the dynamical behavior of reaction-diffusion equations. In this paper, we mainly consider the time periodic reaction-diffusion equation with degenerate monostable nonlinearity. We prove the existence of pe- riodic pyramidal traveling fronts in Rn with n ≥ 3. Due to the degeneration at the equilibrium point 0 and f(u, t) = f(u, t+ T ) > 0 on (0, 1)× R, the dynamical properties of degenerate monostable periodic nonlinearity are essentially different from the bistable and combustion nonlinear terms. For the purpose of obtaining the existence of nonplanar traveling fronts, we use the super-sub solution method combined with comparison principle. It is worth noting that we adopt the method of adding small perturbation to the planar traveling front to overcome the difficul- ties in constructing the supersolution. 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Zhen-Hui Bu (corresponding author) College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China Email address: buzhenhui14@163.com Chen-Lu Wang College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China Email address: nwsuafwcl@163.com Xin-Tian Zhang College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China Email address: zhangxintian808@163.com 1. Introduction 2. Preliminaries 3. Existence of periodic pyramidal traveling fronts in R3 3.1. Construction of the supersolution 3.2. Existence 4. Periodic pyramidal traveling fronts in R n with n4 5. Conclusion Acknowledgments References