Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 46, pp. 1–18. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GLOBAL STABILITY OF TRAVELING WAVES FOR DELAY REACTION-DIFFUSION SYSTEMS WITHOUT QUASI-MONOTONICITY SI SU, GUO-BAO ZHANG Abstract. This article concerns the global stability of traveling waves of a reaction-diffusion system with delay and without quasi-monotonicity. We prove that the traveling waves (monotone or non-monotone) are exponentially stable in L∞(R) with the exponential convergence rate t−1/2e−µt for some con- stant µ > 0. We use the Fourier transform and the weighted energy method with a suitably weight function. 1. Introduction This article is devoted to studying the delay reaction-diffusion system ∂ ∂t u1(x, t) = d1 ∂2 ∂x2 u1(x, t)− αu1(x, t) + h(u2(x, t− τ1)), ∂ ∂t u2(x, t) = d2 ∂2 ∂x2 u2(x, t)− βu2(x, t) + g(u1(x, t− τ2)). (1.1) Here u1(x, t) and u2(x, t) stand for the spatial density of the bacterial population and the infective human population at point x ∈ R and time t ≥ 0, respectively. Both bacteria and humans are assumed to diffuse, d1 and d2 are diffusion coeffi- cients; the term αu1 is the natural death rate of the bacterial population and the nonlinearity h(u2) is the contribution of the infective humans to the growth rate of the bacterial; βu2 is the natural diminishing rate of the infective population due to the finite mean duration of the infectious population and the nonlinearity g(u1) is the infection rate of the human population under the assumption that the total susceptible human population is constant during the evolution of the epidemic, and τ1, τ2 are time delays. Wu and Hsu [23] have already established the existence and qualitative features of solutions of (1.1). For the particular case τi = 0, i = 1, 2, system (1.1) becomes the non-delay reaction-diffusion system ∂ ∂t u1(x, t) = d1 ∂2 ∂x2 u1(x, t)− αu1(x, t) + h(u2(x, t)), ∂ ∂t u2(x, t) = d2 ∂2 ∂x2 u2(x, t)− βu2(x, t) + g(u1(x, t)). (1.2) 2010 Mathematics Subject Classification. 35C07, 35B35, 92D30. Key words and phrases. Delay reaction-diffusion system; traveling waves; global stability; Fourier transform; weighted energy method. c©2020 Texas State University. Submitted December 8, 2019. Published May 19, 2020. 1 2 S. SU, G.-B. ZHANG EJDE-2020/46 Hsu and Yang [6] studied the existence, uniqueness, monotonicity and asymptotic behavior of monostable traveling wave solutions of (1.2). For τ1 = 0 and h(u2) = γu2 in (1.1), Freedman and Zhao [3] presented a threshold result for the global dynamics of the epidemic system ∂ ∂t u1(x, t) = d1 ∂2 ∂x2 u1(x, t)− αu1(x, t) + γu2(x, t), ∂ ∂t u2(x, t) = d2 ∂2 ∂x2 u2(x, t)− βu2(x, t) + g(u1(x, t− τ)). (1.3) The epidemic model (1.3) with τ = 0 was first proposed and analyzed by Capasso and Maddalena [1]. When d2 = 0, system (1.3) becomes ∂ ∂t u1(x, t) = d1 ∂2 ∂x2 u1(x, t)− αu1(x, t) + γu2(x, t), ∂ ∂t u2(x, t) = −βu2(x, t) + g(u1(x, t− τ)). (1.4) Thieme and Zhao [20] investigated the existence of spreading speed and minimal wave speed of (1.4) in the quasi-monotone case. The results in [20] were then ex- tended by Wu and Liu [22] to the non-quasi-monotone case by constructing two auxiliary monotone integral equations. Yang, Li and Wu [25, 26] studied the sta- bility of traveling wave solutions of (1.4) in both the quasi-monotone case and the non-quasi-monotone case by using the weighted energy method. When τ = 0 in (1.4), Xu and Zhao [24] proved the existence, uniqueness (up to translation) and globally exponential stability of bistable traveling wave fronts of (1.4), and Zhao and Wang [31] proved the existence and non-existence of monostable traveling wave fronts of (1.4). More recently, Hsu, Yang and Yu [7] studied the existence and exponential sta- bility of traveling wave solutions for general delay reaction-diffusion systems ∂ ∂t u1(x, t) = d1 ∂2 ∂x2 u1(x, t) + h(u1(x, t), u1(x, t− τ̂1), u2(x, t− τ2)), ∂ ∂t u2(x, t) = d2 ∂2 ∂x2 u2(x, t) + g(u2(x, t), u1(x, t− τ1), u2(x, t− τ̂2)). (1.5) When system (1.5) is monotone, by applying the techniques of weighted energy method and the comparison principle, they showed that the traveling wave solutions of (1.5) are exponentially stable provided that the initial perturbations around the traveling wave fronts belong to a suitable weighted Sobolev space. To the best of our knowledge, global stability for traveling wave solutions of (1.1)-(1.5) without monotonicity have not been considered. The purpose of this article is to establish the global stability of traveling waves of (1.1) with τ1 = τ2, without quasi-monotonicity. The stability of traveling waves for various evolution equations has been exten- sively studied. We refer the readers to [4, 5, 9, 10, 11, 13, 14, 15, 18, 19, 21, 26] for reaction-diffusion equations and to [8, 12, 17, 27, 28, 29, 30] for nonlocal dispersal equations. Note that when the evolution equations are non-monotone, the compar- ison principle is not applicable. Thus, the frequently used methods for the stability of traveling waves, such as the squeezing technique, the method of combination of the comparison principle and the weighted energy method are not applicable. Recently, the weighted energy method without the comparison principle was used to prove the stability of traveling waves of nonmonotone equations, see Chern et al. EJDE-2020/46 GLOBAL STABILITY FOR NON-MONOTONE TRAVELING WAVES 3 [2], Huang et al. [8], Li et al. [9], Wu et al. [21], Yang et al. [26], Zhang and Ma [28] and Zhang et al. [30]. In particular, Yang et al. [26] studied the stability of traveling waves of (1.4) without quasi-monotonicity. Zhang, Li and Feng [30] further investi- gated the stability of traveling waves of (1.4) by replacing d ∂2 ∂x2u1 with d(J∗u1−u1). However, local stability of traveling waves has been obtained only for perturbations around the traveling wave with properly small weighted norm. Recently, Mei et al. [16] established the global stability for the oscillatory traveling waves of local Nicholson’s blowflies equations by using the anti-weighted energy method together with the Fourier transform. Zhang [29] applied this method to a nonlocal dispersal equation with time delay and obtained the global stability of traveling waves. Mo- tivated by [12, 16, 29], we shall extend this method to the study of global stability of traveling waves of reaction-diffusion system (1.1) without quasi-monotonicity. The rest of this article is organized as follows. In Section 2, we present some preliminaries and summarize our main results. Section 3 is dedicated to the global stability of traveling waves of (1.1) by the Fourier transform and the weighted energy method, when h(u) and g(u) are not monotone. 2. Preliminaries and statement main results Throughout this article, we assume that τ1 = τ2 = τ in (1.1), that the initial data satisfies ui(x, s) = ui0(x, s), x ∈ R, s ∈ [−τ, 0], i = 1, 2. (2.1) Now we state some basic assumptions on the nonlinearities g and h. (H1) g ∈ C2([0,K1],R), g(0) = h(0) = 0, K2 = g(K1)/β > 0, h ∈ C2([0,K2],R), h(g(K1)/β) = αK1, h(g(u)/β) > αu for u ∈ (0,K1), where K1 is a positive constant. (H2) |g′(u)| ≤ g′(0) and |h′(v)| ≤ h′(0) for u, v ∈ [0,+∞). From (H1), we see that the spatially homogeneous system of (1.1) admits two constant equilibria (u1−, u2−) = (0, 0) =: 0 and (u1+, u2+) = (K1,K2) =: K. A traveling wave solution (in short, traveling wave) of (1.1) is a special trans- lation invariant solution of the form (u1(x, t), u2(x, t)) = (φ1(x + ct), φ2(x + ct)), where c > 0 is the wave speed. If φ1 and φ2 are monotone, then (φ1, φ2) is called a traveling wavefront. Substituting (φ1(x + ct), φ2(x + ct)) into (1.1) and letting ξ = x+ ct, we obtain the following wave profile system with boundary conditions cφ′1(ξ) = d1φ ′′ 1(ξ)− αφ1(ξ) + h(φ2(ξ − cτ)), cφ′2(ξ) = d2φ ′′ 2(ξ)− βφ2(ξ) + g(φ1(ξ − cτ)), (φ1, φ2)(−∞) = (u1−, u2−), (φ1, φ2)(+∞) = (u1+, u2+). (2.2) It is clear that the characteristic function for (2.2) with respect to the trivial equilibrium 0 can be represented by ∆1(λ, c) := f1(c, λ)− f2(c, λ) for c ≥ 0 and λ ∈ C, where f1(c, λ) := (d1λ 2 − cλ− α)(d2λ 2 − cλ− β), f2(c, λ) := h′(0)g′(0)e−2cλτ . 4 S. SU, G.-B. ZHANG EJDE-2020/46 For convenience, we denote λ±1 = c± √ c2 + 4d1α 2d1 , λ±2 = c± √ c2 + 4d2β 2d2 , λcm = min{λ+ 1 , λ + 2 }. It is clear that f1(c, λ±1 ) = f1(c, λ±2 ). According to [23, Lemma 2.1], we have the following result. Lemma 2.1. There exist a positive number c∗ such that the following items hold. (i) If c ≥ c∗, then the equation ∆1(λ, c) = 0 has two positive real roots λ1 := λ1(c) and λ2 := λ2(c) with 0 < λ1(c) ≤ λ2(c) < λcm. (ii) If c = c∗, then λ∗ = λ1(c∗) = λ2(c∗) and if c > c∗, then λ1(c) < λ2(c) and ∆1(·, c) > 0 in (λ1(c), λ2(c)). When g′(u) ≥ 0 for u ∈ [0,K1] and h′(v) ≥ 0 for v ∈ [0,K2], system (1.1) is a quasi-monotone system. The existence of traveling wave fronts has been obtained by Wu and Hsu, see [23, Theorem 2.3]. When the condition g′(u) ≥ 0 for u ∈ [0,K1] or h′(v) ≥ 0 for v ∈ [0,K2] does not hold, system (1.1) is a non-quasi-monotone system. The existence of traveling waves can also be obtained by using auxiliary equations and Schauder’s fixed point theorem [22, 26], if we assume the following conditions: (H3) There exist K± = (K±1 ,K ± 2 )� 0 with K− < K < K+ and four continuous and twice piecewise continuous differentiable functions g± : [0,K+ 1 ] → R and h± : [0,K+ 2 ]→ R such that (i) K±2 = g±(K±1 )/β, h±( 1 β g ±(K±1 )) = αK±1 , and h±( 1 β g ±(u)) > αu for u ∈ (0,K±1 ); (ii) g±(u) and h±(v) are non-decreasing on [0,K+ 1 ] and [0,K+ 2 ], respec- tively; (iii) (g±)′(0) = g′(0), (h±)′(0) = h′(0) and 0 < g−(u) ≤ g(u) ≤ g+(u) ≤ g′(0)u for u ∈ [0,K+ 1 ], 0 < h−(v) ≤ h(v) ≤ h+(v) ≤ h′(0)v for v ∈ [0,K+ 2 ]. Proposition 2.2. Assume that (H1) and (H3) hold, τ ≥ 0, and let c∗ be de- fined as in Lemma 2.1. Then for every c > c∗, system (1.1) has a traveling wave (φ1(ξ), φ2(ξ)) satisfying (φ1(−∞), φ2(−∞)) = (0, 0) and K−1 ≤ lim inf ξ→+∞ φ1(ξ) ≤ lim sup ξ→+∞ φ1(ξ) ≤ K+ 1 , 0 ≤ lim inf ξ→+∞ φ2(ξ) ≤ lim sup ξ→+∞ φ2(ξ) ≤ K+ 2 . Notation. C > 0 denotes a generic constant, while Ci (i = 1, 2, . . . ) represents a specific constant. Let ‖ · ‖ and ‖ · ‖∞ denote 1-norm and ∞-norm of the matrix (or vector), respectively. Let I be an interval, typically I = R. Denote by L1(I) the space of integrable functions defined on I, and W k,1(I)(k ≥ 0) the Sobolev space of the L1-functions f(x) defined on the interval I whose derivatives dn dxn f(n = 1, . . . , k) also belong to L1(I). Let L1 w(I) be the weighted L1-space with a weight function w(x) > 0 and norm ‖f‖L1 w(I) = ∫ I w(x)|f(x)|dx . EJDE-2020/46 GLOBAL STABILITY FOR NON-MONOTONE TRAVELING WAVES 5 Let W k,1 w (I) be the weighted Sobolev space with norm ‖f‖Wk,1 w (I) = k∑ i=0 ∫ I w(x) ∣∣dif(x) dxi ∣∣dx. Let T > 0 be a number and B be a Banach space. We denote by C([0, T ];B) the space of the B-valued continuous functions on [0, T ], and by L1([0, T ];B) the space of the B-valued L1-functions on [0, T ]. The corresponding spaces of the B- valued functions on [0,∞) are defined similarly. For any function f(x), its Fourier transform is F [f ](η) = f̂(η) = ∫ R e−ixηf(x)dx and the inverse Fourier transform is F−1[f̂ ](x) = 1 2π ∫ R eixη f̂(η)dη, where i is the imaginary number, i2 = −1. To obtain stability of traveling waves of (1.1), we need the following assumptions. (H4) d1 > d2, α > β and max{h′(0), g′(0)} > β. (H5) The initial data (u10(x, s), u20(x, s)) ≥ (0, 0) satisfies lim x→±∞ (u10(x, s), u20(x, s)) = (u1±, u2±) uniformly in s ∈ [−τ, 0]. We consider the function ∆2(λ, c) = d2λ 2 − cλ− β + max{h′(0), g′(0)}e−λcτ . It is easy to see that there exist λ∗ > 0 and c∗ > 0, such that ∆2(λ∗, c∗) = 0 and ∂∆2(λ,c) ∂λ |(λ∗,c∗) = 0. When c > c∗, the equation ∆2(λ, c) = 0 has two positive real roots λ\1(c) and λ\2(c) with 0 < λ\1(c) < λ∗ < λ\2(c). When λ ∈ (λ\1(c), λ\2(c)), ∆2(λ, c) < 0. Moreover, (λ\1)′(c) < 0 and (λ\2)′(c) > 0. Since (λ\1)′(c) < 0, there exists a positive number c\ such that when c > c\ > c∗, λ\1(c) < √ α−β d1−d2 . Define the weight function w(ξ) > 0 as w(ξ) = e−2λξ, where λ > 0 satisfies λ\1(c) < λ < min {√ α−β d1−d2 , λ \ 2(c) } . Now we present the main result on global stability of traveling waves. Theorem 2.3. Assume that (H1)–(H5) hold. For any given traveling wave (φ1(x+ ct), φ2(x+ ct)) of (1.1) with speed c ≥ max{c∗, c\} connecting (0, 0) and (K1,K2), whether it is monotone or non-monotone, if the initial data satisfy ui0(x, s)− φi(x+ cs) ∈ Cunif [−τ, 0] ∩ C([−τ, 0];W 2,1 w (R)), i = 1, 2, ∂s(ui0 − φi) ∈ L1([−τ, 0];L1 w(R)), i = 1, 2, then there exists τ0 > 0 such that for any τ ≤ τ0, the solution (u1(x, t), u2(x, t)) of (1.1)-(2.1) converges to the traveling wave (φ1(x+ ct), φ2(x+ ct)) with sup x∈R |ui(x, t)− φi(x+ ct)| ≤ Ct−1/2e−µt, t > 0, 6 S. SU, G.-B. ZHANG EJDE-2020/46 where C and µ are two positive constants, and Cunif [r, T ] is the space of uniformly continuous functions, Cunif [r, T ] := { u ∈ C2([r, T ]× R) : lim x→+∞ u(x, t) exists uniformly in t ∈ [r, T ], lim x→+∞ ux(x, t) = lim x→+∞ uxx(x, t) = 0 uniformly for t ∈ [r, T ] } . 3. Global stability of traveling waves In this section we prove Theorem 2.3. Let (φ1(x+ct), φ2(x+ct)) = (φ1(ξ), φ2(ξ)) be a given traveling wave with speed c ≥ c∗ and define Ui(ξ, t) := ui(x, t)− φi(x+ ct) = ui(ξ − ct, t)− φi(ξ), i = 1, 2, Ui0(ξ, s) := ui0(x, s)− φi(x+ cs) = ui0(ξ − cs, s)− φ(ξ), i = 1, 2. Then from (1.1) and (2.2), Ui(ξ, t) satisfies U1t + cU1ξ − d1U1ξξ + αU1 = P1(U2(ξ − cτ, t− τ)), U2t + cU2ξ − d2U2ξξ + βU2 = P2(U1(ξ − cτ, t− τ)), Ui(ξ, s) = Ui0(ξ, s), (ξ, s) ∈ R× [−τ, 0], i = 1, 2. (3.1) The nonlinear terms are P1(U2) := h(φ2 + U2)− h(φ2) = h′(φ̃2)U2, P2(U1) := g(φ1 + U1)− g(φ1) = g′(φ̃1)U1, (3.2) for some φ̃i between φi and φi+Ui, with φi = φi(ξ−cτi) and Ui = Ui(ξ−cτi, t−τi). We first prove the existence and uniqueness of solution (U1(ξ, t), U2(ξ, t)) to the initial value problem (3.1) in the space Cunif [−τ,+∞)× Cunif [−τ,+∞). Proposition 3.1. Assume that (H1) and (H2) hold. If the initial perturbation satisfies (U10(ξ, s), U20(ξ, s)) ∈ Cunif [−τ, 0]× Cunif [−τ, 0] for c ≥ c∗, then a solution (U1, U2) of the perturbed equation (3.1) is unique, exists globally in time, and belongs to Cunif [−τ,+∞)× Cunif [−τ,+∞). Proof. When t ∈ [0, τ ], we have t−τ ∈ [−τ, 0] and Ui(ξ−cτ, t−τ) = Ui0(ξ−cτ, t−τ), i = 1, 2, which imply that (3.1) is linear. Thus, the solution of (3.1) can be explicitly and uniquely solved: U1(ξ, t) = e−αt ∫ ∞ −∞ G1(η, t)U10(ξ − η, 0)dη + ∫ t 0 e−α(t−s) ∫ ∞ −∞ G1(η, t− s)P1(U20(ξ − η − cτ, s− τ)) dη ds, U2(ξ, t) = e−βt ∫ ∞ −∞ G2(η, t)U20(ξ − η, 0)dη + ∫ t 0 e−β(t−s) ∫ ∞ −∞ G2(η, t− s)P2(U10(ξ − η − cτ, s− τ)) dη ds (3.3) for t ∈ [0, τ ], where Gi(η, t) is the heat kernel Gi(η, t) = 1√ 4πdit exp ( − (η + ct)2 4dit ) , i = 1, 2. EJDE-2020/46 GLOBAL STABILITY FOR NON-MONOTONE TRAVELING WAVES 7 Since Ui0(ξ, s) ∈ Cunif [−τ, 0], i = 1, 2, namely, limξ→+∞ Ui0(ξ, s) = Ui0(∞, s) and limξ→+∞ Ui0,ξ(ξ, s) = limξ→+∞ Ui0,ξξ(ξ, s) = 0 uniformly in s ∈ [−τ, 0], we immediately prove the following uniform convergence lim ξ→+∞ U1(ξ, t) = e−αt ∫ ∞ −∞ G1(η, t) lim ξ→+∞ U10(ξ − η, 0)dη + ∫ t 0 e−α(t−s) ∫ ∞ −∞ G1(η, t− s) lim ξ→+∞ P1(U20(ξ − η − cτ, s− τ)) dη ds = e−αtU10(∞, 0) ∫ ∞ −∞ G1(η, t)dη + ∫ t 0 e−α(t−s)P1(U20(∞, s− τ)) ∫ ∞ −∞ G1(η, t− s) dη ds = e−αtU10(∞, 0) + ∫ t 0 e−α(t−s)P1(U20(∞, s− τ))ds =: g1(t), uniformly for t ∈ [0, τ ], and lim ξ→+∞ U2(ξ, t) = e−βt ∫ ∞ −∞ G2(η, t) lim ξ→+∞ U20(ξ − η, 0)dη + ∫ t 0 e−β(t−s) ∫ ∞ −∞ G2(η, t− s) lim ξ→+∞ P2(U10(ξ − η − cτ, s− τ)) dη ds = e−βtU20(∞, 0) ∫ ∞ −∞ G2(η, t)dη + ∫ t 0 e−β(t−s)P2(U10(∞, s− τ)) ∫ ∞ −∞ G2(η, t− s) dη ds = e−βtU20(∞, 0) + ∫ t 0 e−β(t−s)P2(U10(∞, s− τ))ds =: g2(t), uniformly for t ∈ [0, τ ], where we used that ∫∞ −∞Gi(η, t− s)dη = 1 for i = 1, 2. Furthermore, we obtain lim ξ→+∞ ∂kξU1(ξ, t) = e−αt ∫ ∞ −∞ ∂kηG1(η, t) lim ξ→+∞ U10(ξ − η, 0)dη + ∫ t 0 e−α(t−s) ∫ ∞ −∞ ∂kηG1(η, t− s) lim ξ→+∞ P1(U20(ξ − η − cτ, s− τ)) dη ds = e−αtU10(∞, 0) ∫ ∞ −∞ ∂kηG1(η, t)dη + ∫ t 0 e−α(t−s)P1(U20(∞, s− τ)) ∫ ∞ −∞ ∂kηG1(η, t− s) dη ds = 0, uniformly for t ∈ [0, τ ], k = 1, 2, 8 S. SU, G.-B. ZHANG EJDE-2020/46 and lim ξ→+∞ ∂kξU2(ξ, t) = e−βt ∫ ∞ −∞ ∂kηG2(η, t) lim ξ→+∞ U20(ξ − η, 0)dη + ∫ t 0 e−β(t−s) ∫ ∞ −∞ ∂kηG2(η, t− s) lim ξ→+∞ P2(U10(ξ − η − cτ, s− τ)) dη ds = e−βtU20(∞, 0) ∫ ∞ −∞ ∂kηG2(η, t)dη + ∫ t 0 e−β(t−s)P2(U10(∞, s− τ)) ∫ ∞ −∞ ∂kηG2(η, t− s) dη ds = 0, uniformly for t ∈ [0, τ ], k = 1, 2. Here we used that Gi(±∞, t− s) = 0, ∂ηGi(η, t− s) ∣∣ η=±∞ = 0, i = 1, 2. Thus, we have proved that (U1, U2) ∈ Cunif [−τ, τ ]× Cunif [−τ, τ ]. Now we consider (3.1) for t ∈ [τ, 2τ ]. Since t− τ ∈ [0, τ ] and Ui(ξ, t− τ) is solved already in (3.3), P1(U2(ξ−cτ, t−τ)) and P2(U1(ξ−cτ, t−τ)) are known for (3.1) with t ∈ [0, 2τ ], namely, the equation (3.1) is linear for t ∈ [0, 2τ ]. As showed before, we can similarly prove the existence and uniqueness of the solution (U1(ξ, t), U2(ξ, t)) to (3.1) for t ∈ [0, 2τ ], and particularly (U1, U2) ∈ Cunif [−τ, 2τ ]× Cunif [−τ, 2τ ]. By repeating this process for t ∈ [nτ, (n+ 1)τ ] with n ∈ Z+, we prove that there exists a unique solution (U1, U2) ∈ Cunif [−τ, (n+1)τ ]×Cunif [−τ, (n+1)τ ] for (3.1), and step by step, we finally prove the uniqueness and existence global in time of the solution (U1, U2) ∈ Cunif [−τ,∞)× Cunif [−τ,∞) for (3.1). � Now we state a stability result for the perturbed equation (3.1), which automat- ically implies Theorem 2.3. Proposition 3.2 (Stability of traveling waves). Assume that (H1), (H2), (H4) and (H5) hold. If Ui0 ∈ Cunif [−τ, 0] ∩ C([−τ, 0];W 2,1 w (R)), i = 1, 2, and ∂sUi0 ∈ L1([−τ, 0];L1 w(R)) for i = 1, 2, then there exists τ0 > 0 such that for any τ ≤ τ0, when c ≥ min{c∗, c\}, it holds sup ξ∈R |Ui(ξ, t)| ≤ Ct−1/2e−µt, t > 0, i = 1, 2, (3.4) for some µ > 0 and C > 0. To prove Proposition 3.2, we first investigate the time-exponential decay estimate of Ui(ξ, t) at ξ = +∞, i = 1, 2. Lemma 3.3. There exist τ0 > 0 and a large number x0 � 1 such that when τ ≤ τ0, the solution Ui(ξ, t) of (3.1) satisfies sup ξ∈[x0,+∞) |Ui(ξ, t)| ≤ Ce−µ1t, t > 0, i = 1, 2, for some µ1 > 0 and C > 0. EJDE-2020/46 GLOBAL STABILITY FOR NON-MONOTONE TRAVELING WAVES 9 Proof. Denote z+ i (t) := Ui(∞, t), z+ i0(s) := Ui0(∞, s), s ∈ [−τ, 0], i = 1, 2. Since (U1, U2) ∈ Cunif [−τ,+∞)× Cunif [−τ,+∞), we have lim ξ→+∞ Ui(ξ, t) = z+ i (t) exists uniformly for t, and lim ξ→+∞ Uiξ(ξ, t) = lim ξ→+∞ Uiξξ(ξ, t) = 0 uniformly for t. Let us take the limits in (3.1) as ξ → +∞. Then we have dz+ 1 dt + αz+ 1 − h′(u2+)z+ 2 (t− τ) = Q1(z+ 2 (t− τ)), dz+ 2 dt + βz+ 2 − g′(u1+)z+ 1 (t− τ) = Q2(z+ 1 (t− τ)), z+ i (s) = z+ i0(s), s ∈ [−τ, 0], i = 1, 2, where Q1(z+ 2 ) = h(u2+ + z+ 2 )− h(u2+)− h′(u2+)z+ 2 , Q2(z+ 1 ) = g(u1+ + z+ 1 )− g(u1+)− g′(u1+)z+ 1 . Then by [9, Lemma 3.8], there exist positive constants τ0, µ1 and C such that when τ ≤ τ0, |Ui(∞, t)| = |z+ i (t)| ≤ Ce−µ1t, t > 0, i = 1, 2, provided that |z+ i0| � 1, i = 1, 2. Furthermore, by the continuity and the uniform convergence of Ui(ξ, t) as ξ → +∞, there exists a large x0 � 1 such that sup ξ∈[x0,+∞) |Ui(ξ, t)| ≤ Ce−µ1t, t > 0, i = 1, 2, provided that supξ∈[x0,+∞) |Ui0(ξ, s)| � 1 for s ∈ [−τ, 0]. Such a smallness for the initial perturbation (U10, U20) near ξ → +∞ can be easily verified, since lim x→+∞ (u10(x, s), u20(x, s)) = (K1,K2) uniformly in s ∈ [−τ, 0], which implies lim ξ→+∞ Ui0(ξ, s) = lim ξ→+∞ [ui0(ξ, s)− φi(ξ)] = Ki −Ki = 0 uniformly for s ∈ [−τ, 0], i = 1, 2. The proof is complete. � Next we establish the a priori decay estimate of supξ∈(−∞,x0] |Ui(ξ, t)|. We shall use the anti-weighted technique [2, 8] together with Fourier transform to treat this problem. First of all, we shift Ui(ξ, t) to Ui(ξ + x0, t) by the constant x0 given in Lemma 3.3, and then introduce the transformation Vi(ξ, t) = √ w(ξ)Ui(ξ + x0, t) = e−λξUi(ξ + x0, t), i = 1, 2. 10 S. SU, G.-B. ZHANG EJDE-2020/46 Substituting U = w−1/2V in (3.1) yields V1t + ρ1(c)V1ξ − d1V1ξξ + ρ2(c)V1 = P̃1(V2(ξ − cτ, t− τ)), V2t + ρ3(c)V2ξ − d2V2ξξ + ρ4(c)V2 = P̃2(V1(ξ − cτ, t− τ)), (ξ, t) ∈ R× [0,+∞), Vi(ξ, s) = √ w(ξ)Ui0(ξ + x0, s) =: Vi0(ξ, s), ξ ∈ R, s ∈ [−τ, 0], i = 1, 2, (3.5) where ρ1(c) := c− 2d1λ, ρ2(c) := cλ− d1λ 2 + α, ρ3(c) := c− 2d2λ, ρ4(c) := cλ− d2λ 2 + β, P̃1(V2) = e−λξP1(U2), P̃2(V1) = e−λξP2(U1). By (3.2), P̃1(V2) satisfies P̃1(V2(ξ − cτ, t− τ)) =e−λξP1(U2(ξ − cτ + x0, t− τ)) =e−λξh′(φ̃2)U2(ξ − cτ + x0, t− τ) =e−λcτh′(φ̃2)V2(ξ − cτ, t− τ) (3.6) and P̃2(V1) satisfies P̃2(V1(ξ − cτ, t− τ)) = e−λcτg′(φ̃1)V1(ξ − cτ, t− τ). (3.7) Furthermore, by (H2), we have |P̃1(V2(ξ − cτ, t− τ))| ≤ h′(0)e−λcτ |V2(ξ − cτ, t− τ)|, |P̃2(V1(ξ − cτ, t− τ))| ≤ g′(0)e−λcτ |V1(ξ − cτ, t− τ)|. Taking (3.6) and (3.7) into (3.5), we see that the coefficient h′(φ̃2) and g′(φ̃1) on the right side of (3.5) is variable and can be negative. Thus, the classical methods, such as the monotone technique and the Fourier transform cannot be applied directly to establish the decay estimate for (V1, V2). A new method should be introduced. The main ideas of this method can be described as follows. (i) We replace h′(φ̃2) in the first equation of (3.5) with a constant h′(0), and g′(φ̃1) in the second equation of (3.5) with a constant g′(0), and then consider the following linear delayed reaction-diffusion system V + 1t + ρ1(c)V + 1ξ − d1V + 1ξξ + ρ2(c)V + 1 = h′(0)e−λcτV + 2 (ξ − cτ, t− τ), V + 2t + ρ3(c)V + 2ξ − d2V + 2ξξ + ρ4(c)V + 2 = g′(0)e−λcτV + 1 (ξ − cτ, t− τ), V + i (ξ, s) = √ w(ξ)Ui0(ξ + x0, s) =: V + i0 (ξ, s), i = 1, 2, (3.8) where ξ ∈ R, t ∈ [0,+∞) and s ∈ [−τ, 0]. Then we investigate the decay estimate of (V + 1 , V + 2 ) by applying the Fourier transform to (3.8); (ii) We prove that the solution (V1, V2) of (3.5) can be bounded by the solution (V + 1 , V + 2 ) of (3.8). Lemma 3.4 (Positiveness). When (V + 10(ξ, s), V + 20(ξ, s)) ≥ (0, 0) for (ξ, s) ∈ R × [−τ, 0], then (V + 1 (ξ, t), V + 2 (ξ, t)) ≥ (0, 0) for (ξ, t) ∈ R× [0,+∞). Proof. When t ∈ [0, τ ], we have t− τ ∈ [−τ, 0] and h′(0)e−λcτV + 2 (ξ − cτ, t− τ) = h′(0)e−λcτV + 20(ξ − cτ, t− τ)dy ≥ 0. (3.9) EJDE-2020/46 GLOBAL STABILITY FOR NON-MONOTONE TRAVELING WAVES 11 Applying (3.9) to the first equation of (3.8), we obtain V + 1t + ρ1(c)V + 1ξ − d1V + 1ξξ + ρ2(c)V + 1 ≥ 0, (ξ, t) ∈ R× [0, τ ], V + 10(ξ, s) ≥ 0, ξ ∈ R, s ∈ [−τ, 0]. By the comparison principle, we have V + 1 (ξ, t) ≥ 0, (ξ, t) ∈ R× [0, τ ]. (3.10) Similarly, we obtain V + 2t + ρ3(c)V + 2ξ − d2V + 2ξξ + ρ4(c)V + 2 ≥ 0, (ξ, t) ∈ R× [0, τ ], V + 20(ξ, s) ≥ 0, ξ ∈ R, s ∈ [−τ, 0]. Using the comparison principle again, we obtain V + 2 (ξ, t) ≥ 0, (ξ, t) ∈ R× [0, τ ]. (3.11) When t ∈ [nτ, (n+ 1)τ ], n = 1, 2, . . . , repeating the above procedure step by step, we can similarly prove (V + 1 (ξ, t), V + 2 (ξ, t)) ≥ (0, 0), (ξ, t) ∈ R× [nτ, (n+ 1)τ ]. (3.12) Combining (3.10), (3.11) and (3.12), we obtain (V + 1 (ξ, t), V + 2 (ξ, t)) ≥ (0, 0) for (ξ, t) ∈ R× [0,+∞). The proof is complete. � Now we establish the following crucial boundedness estimate for (V1, V2). Lemma 3.5. Let (V1(ξ, t), V2(ξ, t)) and (V + 1 (ξ, t), V + 2 (ξ, t)) be the solutions of (3.5) and (3.8), respectively. If |Vi0(ξ, s)| ≤ V + i0 (ξ, s) for (ξ, s) ∈ R× [−τ, 0], i = 1, 2, (3.13) then |Vi(ξ, t)| ≤ V + i (ξ, t) for (ξ, t) ∈ R× [0,+∞), i = 1, 2. Proof. First of all, we prove |Vi(ξ, t)| ≤ V + i (ξ, t) for t ∈ [0, τ ], i = 1, 2. In fact, when t ∈ [0, τ ], namely, t− τ ∈ [−τ, 0], it follows from (3.13) that |Vi(ξ − cτ, t− τ)| = |Vi0(ξ − cτ, t− τ)| ≤ V + i0 (ξ − cτ, t− τ) = V + i (ξ − cτ, t− τ) for (ξ, t) ∈ R× [0, τ ]. (3.14) Then by |h′(φ̃2)| < h′(0) and |g′(φ̃1)| < g′(0) and (3.14), we obtain h′(0)e−λcτV + 2 (ξ − cτ, t− τ)± h′(φ̃2)e−λcτV2(ξ − cτ, t− τ) ≥ h′(0)e−λcτV + 2 (ξ − cτ, t− τ)− |h′(φ̃2)|e−λcτ |V2(ξ − cτ, t− τ)| ≥ 0 for (ξ, t) ∈ R× [0, τ ] (3.15) and g′(0)e−λcτV + 1 (ξ − cτ, t− τ)± g′(φ̃1)e−λcτV1(ξ − cτ, t− τ) ≥ 0 for (ξ, t) ∈ R× [0, τ ]. Let v−i (ξ, t) := V + i (ξ, t)− Vi(ξ, t), v+ i (ξ, t) := V + i (ξ, t) + Vi(ξ, t), i = 1, 2. 12 S. SU, G.-B. ZHANG EJDE-2020/46 We are going to estimate v±i (ξ, t). From (3.5), (3.6), (3.8) and (3.15), we see that v−1 (ξ, t) satisfies v−1t + ρ1(c)v−1ξ − d1v − 1ξξ + ρ2(c)v−1 ≥ 0, (ξ, t) ∈ R× [0, τ ], v−10(ξ, s) = V + 10(ξ, s)− V10(ξ, s) ≥ 0, ξ ∈ R, s ∈ [−τ, 0]. By the comparison principle, we obtain v−1 (ξ, t) ≥ 0, (ξ, t) ∈ R× [0, τ ], namely, V1(ξ, t) ≤ V + 1 (ξ, t), (ξ, t) ∈ R× [0, τ ]. (3.16) Similarly, one has v−2t + ρ3(c)v−2ξ − d2v − 2ξξ + ρ4(c)v−2 ≥ 0, (ξ, t) ∈ R× [0, τ ], v−20(ξ, s) = V + 20(ξ, s)− V20(ξ, s) ≥ 0, ξ ∈ R, s ∈ [−τ, 0]. Applying the comparison principle again, we have v−2 (ξ, t) ≥ 0, (ξ, t) ∈ R× [0, τ ], i.e., V2(ξ, t) ≤ V + 2 (ξ, t), (ξ, t) ∈ R× [0, τ ]. (3.17) On the other hand, v+ 1 (ξ, t) satisfies v+ 1t + ρ1(c)v+ 1ξ − d1v + 1ξξ + ρ2(c)v+ 1 ≥ 0, (ξ, t) ∈ R× [0, τ ], v+ 10(ξ, s) = V + 10(ξ, s) + V10(ξ, s) ≥ 0, ξ ∈ R, s ∈ [−τ, 0]. Then the comparison principle implies that v+ 1 (ξ, t) = V + 1 (ξ, t) + V1(ξ, t) ≥ 0, (ξ, t) ∈ R× [0, τ ]; that is, − V + 1 (ξ, t) ≤ V1(ξ, t), (ξ, t) ∈ R× [0, τ ]. (3.18) Similarly, v+ 2 (ξ, t) satisfies v+ 2t + ρ3(c)v+ 2ξ − d2v + 2ξξ + ρ4(c)v+ 2 ≥ 0, (ξ, t) ∈ R× [0, τ ], v+ 20(ξ, s) = V + 20(ξ, s) + V20(ξ, s) ≥ 0, ξ ∈ R, s ∈ [−τ, 0]. Therefore, we can prove that v+ 2 (ξ, t) = V + 2 (ξ, t) + V2(ξ, t) ≥ 0, (ξ, t) ∈ R× [0, τ ], namely − V + 2 (ξ, t) ≤ V2(ξ, t), (ξ, t) ∈ R× [0, τ ]. (3.19) Combining (3.16) and (3.18), we obtain |V1(ξ, t)| ≤ V + 1 (ξ, t) for (ξ, t) ∈ R× [0, τ ], (3.20) and combining (3.17) and (3.19), we prove |V2(ξ, t)| ≤ V + 2 (ξ, t) for (ξ, t) ∈ R× [0, τ ], (3.21) Next, when t ∈ [τ, 2τ ], namely, t − τ ∈ [0, τ ], based on (3.20) and (3.21) we can similarly prove |Vi(ξ, t)| ≤ V + i (ξ, t) for (ξ, t) ∈ R× [τ, 2τ ], i = 1, 2. Repeating this procedure, we then further prove |Vi(ξ, t)| ≤ V + i (ξ, t), (ξ, t) ∈ R× [nτ, (n+ 1)τ ], n = 1, 2, . . . , EJDE-2020/46 GLOBAL STABILITY FOR NON-MONOTONE TRAVELING WAVES 13 which implies |Vi(ξ, t)| ≤ V + i (ξ, t) for (ξ, t) ∈ R× [0,∞), i = 1, 2. The proof is complete. � In the following, we derive the stability of traveling waves for the linear system (3.8) by using the weighted method and by carrying out the crucial boundedness estimate on the fundamental solutions. Now let us recall the properties of the solutions to the delayed ODE system. Lemma 3.6 ([12, Lemma 3.1]). Let z(t) be the solution to the scalar differential equation with delay d dt z(t) = Az(t) +Bz(t− τ), t ≥ 0, τ > 0, z(s) = z0(s), s ∈ [−τ, 0]. (3.22) where A,B ∈ CN×N , N ≥ 2, and z0(s) ∈ C1([−τ, 0],CN ). Then z(t) = eA(t+τ)eB1t τ z0(−τ) + ∫ 0 −τ eA(t−s)eB1(t−τ−s) τ [z′0(s)−Az0(s)]ds, where B1 = Be−Aτ and eB1t τ is the so-called delayed exponential function in the form eB1t τ =  0, −∞ < t < −τ, I, −τ ≤ t < 0, I +B1 t 1! , 0 ≤ t < τ, I +B1 t 1! +B2 1 (t−τ)2 2! , τ ≤ t < 2τ, . . . . . . I +B1 t 1! +B2 1 (t−τ)2 2! + · · ·+Bm1 [t−(m−1)τ ]m m! , (m− 1)τ ≤ t < mτ, . . . . . . where 0, I ∈ CN×N , and 0 is zero matrix and I is the identity matrix. Lemma 3.7 ([12, Theome 3.1]). Suppose µ(A) := µ1(A)+µ∞(A) 2 < 0, where µ1(A) and µ∞(A) denote the matrix measure of A induced by the matrix 1-norm ‖ · ‖1 and ∞-norm ‖ · ‖∞, respectively. If ν(B) := ‖B‖+‖B‖∞ 2 ≤ −µ(A), then there exists a decreasing function ετ = ε(τ) ∈ (0, 1) for τ > 0 such that any solution of system (3.22) satisfies ‖z(t)‖ ≤ C0e −ετσt, t > 0, where C0 is a positive constant depending on initial data z0(s), s ∈ [−τ, 0] and σ = |µ(A)| − ν(B). In particular, ‖eAteB1t τ ‖ ≤ C0e −ετσt, t > 0, where eB1t τ is defined in Lemma 3.6. It can be seen from the proof of [12, Theome 3.1] that µ1(A) = lim θ→0+ ‖I + θA‖ − 1 θ = max 1≤j≤N [ Re(ajj) + N∑ j 6=i |aij | ] , 14 S. SU, G.-B. ZHANG EJDE-2020/46 µ∞(A) = lim θ→0+ ‖I + θA‖∞ − 1 θ = max 1≤i≤N Re(aii) + N∑ i 6=j |aij |  . Next, we shall estimate the decay rate for the solution V +(ξ, t). Lemma 3.8. Let the initial data V + i0 (ξ, s), i = 1, 2, be such that V + i0 ∈ C([−τ, 0];W 2,1(R)), ∂sV + i0 ∈ L 1([−τ, 0];L1(R)), i = 1, 2. Then ‖V + i (t)‖L∞(R) ≤ Ct−1/2e−µ2t for c ≥ max{c∗, c\}, i = 1, 2, where µ > 0 and C > 0. Proof. Taking Fourier transform in (3.8) and denoting the transform of V +(ξ, t) by V̂ +(η, t), we obtain V̂ + 1t (η, t) = −(d1|η|2 + ρ2(c) + iρ1(c)η)V̂ + 1 (η, t) + h′(0)e−cτ(λ+iη)V̂ + 2 (η, t− τ), V̂ + 2t (η, t) = −(d2|η|2 + ρ4(c) + iρ3(c)η)V̂ + 2 (η, t) + g′(0)e−cτ(λ+iη)V̂ + 1 (η, t− τ), V̂ + i (η, s) = V̂ + i0 (η, s), η ∈ R, s ∈ [−τ, 0], i = 1, 2. (3.23) Let A(η) = ( −(d1|η|2 + ρ2(c) + iρ1(c)η) 0 0 −(d2|η|2 + ρ4(c) + iρ3(c)η) ) , B(η) = ( 0 h′(0)e−cτ(λ+iη) g′(0)e−cτ(λ+iη) 0 ) . Then system (3.23) can be rewritten as V̂ + t (η, t) = A(η)V̂ +(η, t) +B(η)V̂ +(η, t− τ), (3.24) where V̂ +(η, t) = (V̂ + 1 (η, t), V̂ + 2 (η, t))T . By Lemma 3.6, the linear delayed system (3.24) has solution V̂ +(η, t) =eA(η)(t+τ)eB1(η)t τ V̂ + 0 (η,−τ) + ∫ 0 −τ eA(η)(t−s)eB1(η)(t−s−τ) τ [ ∂sV̂ + 0 (η, s)−A(η)V̂ + 0 (η, s) ] ds :=I1(η, t) + ∫ 0 −τ I2(η, t− s)ds, (3.25) where B1(η) = B(η)eA(η)τ . Let V +(ξ, t) := (V + 1 (ξ, t), V + 2 (ξ, t))T . Then by taking the inverse Fourier transform in (3.25), one has V +(ξ, t) = F−1[I1](ξ, t) + ∫ 0 −τ F−1[I2](ξ, t− s)ds = 1 2π ∫ ∞ −∞ eiξηeA(η)(t+τ)eB1(η)t τ V̂ + 0 (η,−τ)dη + 1 2π ∫ 0 −τ ∫ ∞ −∞ eiξηeA(η)(t−s)eB1(η)(t−s−τ) τ × [ ∂sV̂ + 0 (η, s)−A(η)V̂ + 0 (η, s) ] dη ds. (3.26) EJDE-2020/46 GLOBAL STABILITY FOR NON-MONOTONE TRAVELING WAVES 15 From the definition of µ(·) and ν(·), we have µ(A(η)) = µ1(A(η)) + µ∞(A(η)) 2 = max { −d1η 2 − ρ2(c),−d2η 2 − ρ4(c) } =− d2η 2 − cλ+ d2λ 2 − β, since d1 > d2, α > β and λ2 < α−β d1−d2 , and ν(B(η)) = max{h′(0), g′(0)}e−λcτ . By considering λ ∈ (λ\1(c), λ\2(c)), we obtain µ(A(η)) < 0 and µ(A(η)) + ν(B(η)) = −d2η 2 − cλ+ d2λ 2 − β + max{h′(0), g′(0)}e−λcτ < 0. Furthermore, we obtain |µ(A(η))| − ν(B(η)) =d2η 2 + cλ− d2λ 2 + β −max{h′(0), g′(0)}e−λcτ =−∆2(λ, c) + d2η 2, where ∆2(λ, c) = dλ2 − cλ − β + max{h′(0), g′(0)}e−λcτ < 0 for c ≥ max{c∗, c\}. It then follows from Lemma 3.7 that there exists a decreasing function ετ = ε(τ) ∈ (0, 1) such that ‖eA(η)(t+τ)eB1(η)t‖ ≤ C1e −ετ (|µ(A(η))|−ν(B(η)))t ≤ C1e −ετµ0te−ετdη 2t, where C1 is a positive constant and µ0 := −∆2(λ, c) > 0 with c > c\. By the definition of Fourier transform, we have sup η∈R ‖V̂ + 0 (η,−τ)‖ ≤ ∫ R ‖V + 0 (ξ,−τ)‖dξ = 2∑ i=1 ‖V + i0 (·,−τ)‖L1(R). Therefore, sup ξ∈R ‖F−1[I1](ξ, t)‖ = sup ξ∈R ∥∥ 1 2π ∫ ∞ −∞ eiξηeA(η)(t+τ)eB1(η)tV̂ + 0 (η,−τ)dη ∥∥ ≤C ∫ ∞ −∞ e−ετdη 2te−ετµ0t‖V̂ + 0 (η,−τ)‖dη ≤Ce−ετµ0t sup η∈R ‖V̂ + 0 (η,−τ)‖ ∫ ∞ −∞ e−ετdη 2tdη ≤Ce−µ2tt−1/2 2∑ i=1 ‖V + i0 (·,−τ)‖L1(R), (3.27) with µ2 := ετµ0. By using the property of Fourier transform, we obtain sup η∈R |diη2V̂ + i (η, t)| = sup η∈R ∣∣diF [V + iξξ](η, t) ∣∣ = di‖∂ξξV + i (·, t)‖L1(R) ≤ di‖V + i (·, t)‖W 2,1(R) and sup η∈R |(iη)V̂ + i (η, t)| = sup η∈R |F [∂ξV + i ](η, t)| 16 S. SU, G.-B. ZHANG EJDE-2020/46 ≤ ∫ R |∂ξV + i (ξ, t)|dξ = ‖∂ξV + i (·, t)‖L1(R), for i = 1, 2. Thus, sup η∈R ‖A(η)V̂ + 0 (η, s)‖ ≤ C 2∑ i=1 ‖V + i0 (·, s)‖W 2,1(R). Similarly, we can derive that sup ξ∈R ‖F−1[I2](ξ, t− s)‖ = sup ξ∈R ∥∥ 1 2π ∫ ∞ −∞ eiξηeA(η)(t−s)eB1(η)(t−s−τ)[∂sV̂ + 0 (η, s)−A(η)V̂ + 0 (η, s)]dη ∥∥ ≤ C ∫ ∞ −∞ e−ετdη 2(t−s)e−ετµ0(t−s)∥∥∂sV̂ + 0 (η, s)−A(η)V̂ + 0 (η, s) ∥∥dη ≤ Ce−ετµ0teετµ0s sup η∈R ‖∂sV̂ + 0 (η, s)−A(η)V̂ + 0 (η, s)‖ ∫ ∞ −∞ e−ετdη 2(t−s)dη ≤ Ce−ετµ0t(t− s)−1/2E(s), (3.28) where E(s) = ‖∂sV + 0 (·, s)‖L1(R) + ‖V + 0 (·, s)‖W 2,1(R). Furthermore, one has∫ 0 −τ (t− s)−1/2E(s)ds ≤ (1 + t)−1/2 ∫ 0 −τ (1 + t)1/2 (t− s)1/2 E(s)ds ≤ Ct−1/2 ( ‖∂sV + 0 (s)‖L1([−τ,0];L1(R)) + ‖V + 0 (s)‖L1([−τ,0];W 2,1(R)) ) . (3.29) Substituting (3.27), (3.28) and (3.29) in (3.26), we obtain the the decay rate 2∑ i=1 ‖V + i (t)‖L∞(R) ≤ Ct−1/2e−µ2t. This proof is complete. � Let us choose that V + i0 (ξ, s) such that V + i0 ∈ C([−τ, 0];W 2,1(R)), ∂sV + i0 ∈ L 1([−τ, 0];L1(R)), V + i0 (ξ, s) ≥ |Vi0(ξ, s)|, (ξ, s) ∈ R× [−τ, 0], i = 1, 2. Combining Lemmas 3.5 and 3.8, we obtain the convergence rates for V (ξ, t). Lemma 3.9. If Vi0 ∈ C([−τ, 0];W 2,1(R)) and ∂sVi0 ∈ L1([−τ, 0];L1(R)), then ‖Vi(t)‖L∞(R) ≤ Ct−1/2e−µ2t, for some µ2 > 0, i = 1, 2. EJDE-2020/46 GLOBAL STABILITY FOR NON-MONOTONE TRAVELING WAVES 17 Since Vi(ξ, t) = √ w(ξ)Ui(ξ + x0, t) = e−λξUi(ξ + x0, t) and √ w(ξ) = e−λξ ≥ 1 for ξ ∈ (−∞, 0], it follows that sup ξ∈(−∞,0] |Ui(ξ + x0, t)| ≤ ‖Vi(t)‖L∞(R) ≤ Ct−1/2e−µ2t. Thus, we obtain the following estimate for the unshifted U(ξ, t). Lemma 3.10. It holds that sup ξ∈(−∞,x0] |Ui(ξ, t)| ≤ Ct−1/2e−µ2t, i = 1, 2, for some µ2 > 0. Proof of Proposition 3.2. By Lemmas 3.3 and 3.10, we immediately obtain (3.4) for 0 < µ < min{µ1, µ2}. � Acknowledgements. G.-B. Zhang was supported by the NSF of Gansu Province (18JR3RA093), and by the NSF of China (11861056). The authors are grateful to the anonymous referees for their careful reading of this paper and for their valuable suggestions. References [1] V. Capasso, L. 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Ma; Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution type crossing-monostable nonlinearity, Z. Angew. Math. Phys., 65 (2014), 819-844. [29] G.-B. Zhang; Global stability of non-monotone traveling wave solutions for a nonlocal dis- persal equation with time delay, J. Math. Anal. Appl., 475 (2019), 605-627. [30] G.-B. Zhang, Y. Li, Z. Feng; Exponential stability of traveling waves in a nonlocal dispersal epidemic model with delay, J. Comput. Appl. Math., 344 (2018), 47-72. [31] X.-Q. Zhao, W. Wang; Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117-1128. Si Su College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China Email address: 18809422634@163.com Guo-Bao Zhang (corresponding author) College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China Email address: zhanggb2011@nwnu.edu.cn 1. Introduction 2. Preliminaries and statement main results Notation 3. Global stability of traveling waves Acknowledgements References