Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 82, pp. 1–16. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu REGULAR TRAVELING WAVES FOR A REACTION-DIFFUSION EQUATION WITH TWO NONLOCAL DELAYS HAIQIN ZHAO, SHI-LIANG WU Abstract. This article concerns regular traveling waves of a reaction-diffusion equation with two nonlocal delays arising from the study of a single species with immature and mature stages and different ages at reproduction. Establishing a necessary condition on the regular traveling waves, we prove the uniqueness of noncritical regular traveling waves, regardless of being monotone or not. Under a quasi-monotone assumption and among other things, we further show that all noncritical monotone traveling waves are exponentially stable, by establishing two comparison theorems and constructing an auxiliary lower equation. 1. Introduction It is well known that many species, such as Ixodes ticks, may exhibit totally different ages at successful reproduction [10]. To understand the influence of dif- ferent ages at reproduction on the evolution of such populations, Lou and Zhang [10] considered a species (Ixodes scapularis ticks) with two classes of individuals, 1 and 2, with different ages at reproduction, τ1 and τ2, respectively. Based on the Mckendrick-von Foerster equation, they derived the following reaction-diffusion equation with two nonlocal delays: Mt = DMxx − (µ+ g(M)M + pe−d1τ1 ∫ R Γ1(D1τ1, x− y)f(M(y, t− τ1))dy + (1− p)e−d2τ2 ∫ R Γ2(D2τ2, x− y)f(M(y, t− τ2))dy, x ∈ R, t > 0, (1.1) where Γi(Diτi, x), i = 1, 2, are the kernel functions which satisfy (H0) Γi(Diτi, ·) ∈ C(R,R), Γi(Diτi, x) = Γi(Diτi,−x) > 0, ∫ R Γi(Diτi, y)dy = 1, and ∫ R Γi(Diτi, y)eλ|y|dy <∞ for any λ > 0. In [10], Γi(Diτi, x) = √ 4πDiτie − x2 4Diτi for i = 1, 2. In this model, M(x, t) represents the total adult tick density at time t and location x; D > 0 is the diffusion coefficient of mature individuals; p ∈ (0, 1) and 1− p are the proportions of eggs at birth rate for classes 1 and 2, respectively; di > 0 and Di > 0 are the death rate and diffusion coefficients of immature individuals for class i, respectively; µ and the function g(·) denote the density-independent and density-dependent death rate of adults, respectively. The functions f(·) and g(·) satisfy the following assumptions 2020 Mathematics Subject Classification. 35R10, 35B40, 92D30. Key words and phrases. Regular traveling fronts; reaction-diffusion equation; nonlocal delay; uniqueness; stability. ©2022. This work is licensed under a CC BY 4.0 license. Submitted August 26, 2022. Published December 12. 2022. 1 2 H. ZHAO, S.-L. WU EJDE-2022/82 (H1) f, g ∈ C1(R+,R+), f(0) = g(0) = 0, g′(u) > 0 for u > 0, and there exists u∗ > 0 such that f ′(u) > 0 for u ∈ (0, u∗) and f ′(u) < 0 for u ∈ (u∗,+∞). (H2) ( pe−d1τ1 + (1 − p)e−d2τ2 ) f ′(0) > µ and f(u)/u is non-increasing in u ∈ (0,+∞). It is clear that f is an unimodal function. In biological literature, two types of such functions which have been widely used are the Richer type function: f(N) = aNe−bN , a, b > 0, and the Beverton-Holt type function: f(N) = aN 1+bNm , m > 1, a, b > 0. From (H1) and (H2), equation (1.1) has exactly two equilibria 0 and M∗ > 0. As mentioned in [14], a traveling wave is called a regular traveling wave if it decays exponentially at minus infinity (see also Definition 2.1 below). Under the assumptions (H1) and (H2), the authors in [10] established the existence and non-existence of monostable regular traveling waves of (1.1) in the case where M∗ ≤ u∗ (i.e. quasi-monotone case) and M∗ > u∗ (i.e. non-quasi-monotone case). In particular, the regular traveling wave is increasing in the quasi-monotone case, while it may be non-monotone in the non-quasi-monotone case. They also give some results on the existence, uniqueness and stability of bistable traveling fronts of (1.1) with quasi-monotonicity. However, to the best of our knowledge, there has been no results on uniqueness and stability of the monostable regular traveling waves for such reaction-diffusion equations with two nonlocal delays. The aim of this paper is to solve these problems. We remark that, in recent years, there have been a few of interesting results de- voted to the uniqueness of monotone traveling waves for various diffusion equations [2, 3, 6, 16, 18, 20, 22]. For example, Wang et al. [16] proved the uniqueness of monotone traveling waves for a class of delayed reaction-diffusion equations. How- ever, such method, if not impossible, can not be applied to study the uniqueness of non-monotone traveling waves. In this paper, we adapt nontrivially the technique developed in Diekmann and Kapper [5] and Aguerrea [1] to study the uniqueness of the regular traveling waves of (1.1) which may not be monotone. More precisely, based on establishing a necessary condition on the regular traveling waves (Lemma 2.2), we prove the uniqueness of all noncritical regular traveling waves regardless of whether they are monotone or not (Theorem 2.4). In addition to the uniqueness, another interesting problem is the stability of traveling waves. In recent years, the squeezing technique and the weighted energy method have been widely used to study the stability of monostable traveling waves for reaction-diffusions with single delay, see [4, 8, 9, 11, 12, 16]. We will use a different approach [7, 13, 17] to study the stability of the traveling waves of the reaction-diffusion equation (1.1) with two nonlocal delays. More precisely, by es- tablishing two comparison theorems and constructing a lower auxiliary equation, we show that all noncritical monotone traveling waves (traveling fronts for short) of (1.1) are exponentially stable. In particular, the exponential convergence rate is also obtained (Theorem 3.3). The rest of this article is organized as follows. In Section 2, we first establish a necessary condition on the regular traveling waves. Then, we prove the uniqueness of all noncritical regular traveling waves of (1.1) with or without quasi-monotone as- sumptions. Section 3 is devoted to the exponential stability of noncritical traveling waves. EJDE-2022/82 TRAVELING WAVES FOR A REACTION-DIFFUSION EQUATION 3 2. Uniqueness In this section, we prove the uniqueness of all noncritical regular traveling waves. We first give the definition of (regular) traveling waves [14, 21]. Definition 2.1. A bounded solution u(x, t) of (1.1) is called a traveling wave (solution) with speed c, if u(x, t) = U(x+ ct) for some function U(·) which satisfies U(−∞) = 0 and lim inf ξ→+∞ U(ξ) ≥ ε for some constant ε > 0, (2.1) where c and U(·) are called the wave speed and the wave profile, respectively. Further, if there exists a constant β > 0 such that limξ→−∞ U(ξ)e−βξ = q for some q > 0, then U(x+ ct) is called a regular traveling wave solution. It is clear that the wave profile U(·) of the traveling wave of (1.1) satisfies DU ′′(ξ)− cU ′(ξ)− µU(ξ)− g(U(ξ))U(ξ) + ω1 ∫ R Γ1(D1τ1, y)f(U(ξ − y − cτ1))dy + ω1 ∫ R Γ2(D2τ2, y)f(U(ξ − y − cτ2))dy = 0, (2.2) where ω1 := pe−d1τ1 and ω2 := (1− p)e−d2τ2 . As in [10] one can easily verify that the characteristic equation λ = Dν2 − µ+ ω1f ′(0)e−τ1λγ1(ν) + ω2f ′(0)e−τ2λγ2(ν) admits a principal eigenvalue λ(ν) > 0 with limν→0+ λ(ν) ν = limν→+∞ λ(ν) ν = +∞, where γi(ν) := ∫ R Γi(Diτi, y)e−νydy, i = 1, 2. Moreover, there exist c∗, ν∗ > 0 such that c∗ = λ(ν∗) ν∗ = inf ν>0 λ(ν) ν , and for any c > c∗, there exists a unique ν1 := ν1(c) ∈ (0, ν∗) such that λ(ν1) = cν1, and λ(ν) < cν for any ν ∈ (ν1, ν∗). We define ∆(c, ν) := Dν2 − cν − ν + ω1f ′(0)e−cτ1νγ1(ν) + ω2f ′(0)e−cτ2νγ2(ν). It follows from [10, Lemma 4.3] that ∆(c∗, ν∗) = 0, ∂ ∂ν ∆(c∗, ν) ∣∣ ν=ν∗ = 0. Furthermore, if c > c∗, then the equation ∆(c, ν) = 0 has two positive real roots ν1 := ν1(c) < ν∗ < ν2 := ν2(c) such that ∆(c, ν) > 0 for ν ∈ R \ (ν1, ν2) and ∆(c, ν) < 0 for ν ∈ (ν1, ν2). Using the method in the proof of Lou and Zhang [10, Theorem 4.5], one has the following result. Lemma 2.2. Assume (H1) and (H2). Then for each c > c∗, equation (1.1) admits a traveling wave U(x+ ct) which satisfies M− ≤ lim inf ξ→+∞ U(ξ) ≤ lim sup ξ→+∞ U(ξ) ≤M+ 4 H. ZHAO, S.-L. WU EJDE-2022/82 for two constants M± > 0 and limξ→−∞ U(ξ)e−ν1ξ = 1. Moreover, if M∗ ≤ u∗, then U(ξ) is non-decreasing and U(+∞) = u∗. From Definition 2.1, we see that the traveling waves obtained in Lemma 2.2 are regular traveling waves. We now give a necessary condition on the regular traveling waves, which will play a critical role in proving uniqueness. Lemma 2.3. Assume (H1), (H2), and c > c∗. If W (x + ct) is a regular traveling wave of (1.1) with limξ→−∞W (ξ)e−βξ = q for some β, q > 0, then lim ξ→−∞ W (ξ)e−ν1ξ = q. Proof. It suffices to show β = ν1, which is done in two steps. Step 1. We show that ∆(c, β) = 0. Denote λ± = c± √ c2+4D(µ+L) 2 for some constant L > 0. It is clear that λ− < 0 < λ+ are two roots of the equation: Dλ2 − cλ − (µ + L) = 0. Take L > 0 large enough such that λ+ > α. Let µ1 := µ+ L and define H(W )(ξ) := [L− g(W (ξ))]W (ξ) + ω1 ∫ R Γ1(D1τ1, y)f(W (ξ − y − cτ1))dy + ω1 ∫ R Γ2(D2τ2, y)f(W (ξ − y − cτ2))dy. Then by (2.2), we obtain DW ′′(ξ)− cW ′(ξ)− µ1W (ξ) +H(W )(ξ) = 0. (2.3) It follows that W (ξ) = 1 D(λ+ − λ−) [ ∫ ξ −∞ eλ−(ξ−s)H(W )(s)ds+ ∫ ∞ ξ eλ+(ξ−s)H(W )(s)ds ] , which implies that W ′(ξ) = 1 D(λ+ − λ−) [ λ− ∫ ξ −∞ eλ−(ξ−s)H(W )(s)ds + λ+ ∫ ∞ ξ eλ+(ξ−s)H(W )(s)ds ] . (2.4) Since W (−∞) = 0, by (2.4), it is easy to verify that W ′(−∞) = 0. Integrating both sides of (2.3), we obtain DW ′(ξ) = cW (ξ)− ∫ ξ −∞ [−µ1W (s) +H(W )(s)]ds. (2.5) Let V (ξ) = W (ξ)e−βξ. Then V (−∞) = q > 0 and W ′(ξ) = V ′(ξ)eβξ+βV (ξ)eβξ. From (2.5), we infer that DV ′(ξ) = −DβV (ξ) + cV (ξ)− e−βξ ∫ ξ −∞ [−µ1W (s) +H(W )(s)]ds. (2.6) It is easy to verify that lim ξ→−∞ e−βξH(W )(ξ) = lim ξ→−∞ e−βξ { [L− g(W (ξ))]W (ξ) + ω1 ∫ R Γ1(D1τ1, y)f(W (ξ − y − cτ1))dy EJDE-2022/82 TRAVELING WAVES FOR A REACTION-DIFFUSION EQUATION 5 + ω2 ∫ R Γ2(D2τ2, y)f(W (ξ − y − cτ2))dy } = Lq + q$(β), where $(β) := ω1f ′(0)e−cτ1βγ1(β) + ω2f ′(0)e−cτ2βγ2(β), by which we obtain lim ξ→−∞ e−βξ ∫ ξ −∞ [−µ1W (s) +H(W )(s)]ds = 1 β lim ξ→−∞ e−βξ[−µ1W (ξ) +H(W )(ξ)] = q β [−µ+$(β)]. (2.7) From (2.6) it follows that lim ξ→−∞ V ′(ξ) = q Dβ [−Dβ2 + cβ + µ−$(β)] = − q Dβ ∆(c, β). (2.8) Since V (ξ) is bounded, limξ→−∞ V ′(ξ) = 0; therefore ∆(c, β) = 0. Step 2. We complete the proof by showing ∆(c, α) > 0 for any α ∈ (0, β). Denote Z(ξ) = W (ξ)e−αξ for α ∈ (0, β). Then Z(±∞) = 0. Thus, we may assume that Z(ξ) attains its maxmimum at ξ1 ∈ R. Hence, 0 = Z ′(ξ1) = W ′(ξ1)e−αξ1 − αW (ξ1)e−αξ1 and W (s− y − cτi) = Z(s− y − cτi)eα(s−y−cτi) ≤ Z(ξ1)eαse−α(y+cτi), for each x, y ∈ R, i = 1, 2. By assumption (H2), we see that f(u) ≤ f ′(0)u for u ≥ 0, and hence, for any s ∈ R, H(W )(s) ≤ LW (s) + ω1 ∫ R Γ1(D1τ1, y)f ′(0)W (s− y − cτ1)dy + ω2 ∫ R Γ2(D2τ2, y)f ′(0)W (s− y − cτ2)dy ≤ { L+ f ′(0) ∫ R Γ1(D1τ1, y)[ω1e −αcτ1 + ω2e −αcτ2 ]e−αydy } Z(ξ1)eαs = { L+ [ ω1e −αcτ1γ1(α) + ω2e −αcτ2γ2(α) ] f ′(0) } Z(ξ1)eαs =: L1Z(ξ1)eαs. Combining (2.4) and noting that λ+ > α, it holds αW (ξ1) = W ′(ξ1) ≤ L1Z(ξ1) D(λ+ − λ−) [ λ− ∫ ξ1 −∞ eλ−(ξ1−s)eαsds+ λ+ ∫ ∞ ξ1 eλ+(ξ1−s)eαsds ] = L1Z(ξ1)eαξ1 D(λ+ − λ−) [ λ− α− λ− + λ+ λ+ − α ] = L1W (ξ1)α D(α− λ−)(λ+ − α) . (2.9) Then L+ [ ω1e −αcτ1γ1(α) + ω2e −αcτ2γ2(α) ] f ′(0) 6 H. ZHAO, S.-L. WU EJDE-2022/82 ≥ D(α− λ−)(λ+ − α) = −Dα2 − cα+ µ+ L, and hence, ∆(c, α) ≥ 0 for any α ∈ (0, β). Since ∆(c, α) is strictly convex, we obtain ∆(c, α) > 0 for any α ∈ (0, β). Therefore, it must be β = ν1. This completes the proof. � We are now ready to give the uniqueness of the regular traveling waves of (1.1). Theorem 2.4. Assume (H1), (H2), and that f ′(u) ≤ f ′(0) for u ≥ 0. Let U1(x+ct) and U2(x+ct) be two regular traveling waves of (1.1) with speed c > c∗. Then there exists ξ0 such that U1(ξ + ξ0) = U2(ξ). Proof. By a translation in Lemma 2.2, we can assume that limξ→−∞ Ui(ξ)/e ν1ξ = 1 for i = 1, 2. We define Π(ξ) := [U1(ξ)− U2(ξ)]/eν1ξ, ξ ∈ R. Obviously, we have W (±∞) = 0. Next, we show that Π(·) ≡ 0. We first prove Π(ξ) ≤ 0, ∀ξ ∈ R. Suppose for the contrary that maxξ∈R Π(ξ) > 0. Since Π(±∞) = 0, there exists ξ1 such that Π(ξ1) = maxξ∈R Π(ξ) > 0 and Π(ξ) < Π(ξ1) for any ξ < ξ1. Then Π′(ξ1) = 0 and Π′′(ξ1) ≤ 0. Those imply that (U1 − U2)(ξ1) = Π(ξ1)eν1ξ1 > 0, (U1 − U2)′(ξ1) = [Π′(ξ1) + ν1Π(ξ1)]eν1ξ1 = ν1Π(ξ1)eν1ξ1 , (U1 − U2)′′(ξ1) = [Π′′(ξ1) + 2ν1Π′(ξ1) + ν2 1Π(ξ1)]eν1ξ1 ≤ ν2 1Π(ξ1)eν1ξ1 . Nothing that g(u)u is increasing, it holds g(U1(ξ1))U1(ξ1) − g(U2(ξ1))U2(ξ1) ≥ 0. Using f ′(u) ≤ f ′(0) for u ≥ 0, direct computations show that (cν1 −Dν2 1)Π(ξ1)eν1ξ1 ≤ c(U1 − U2)′(ξ1)−D(U1 − U2)′′(ξ1) = −µ(U1 − U2)(ξ1)− [g(U1(ξ1))U1(ξ1)− g(U2(ξ1))U2(ξ1)] + ω1 ∫ R Γ1(D1τ1, y)[f(U1(ξ1 − y − cτ1))− f(U2(ξ1 − y − cτ1))]dy + ω2 ∫ R Γ2(D2τ2, y)[f(U1(ξ1 − y − cτ2))− f(U2(ξ1 − y − cτ2))]dy ≤ −µΠ(ξ1)eν1ξ1 + ω1f ′(0) ∫ R Γ1(D1τ1, y) ×max{0, U1(ξ1 − y − cτ1)− U2(ξ1 − y − cτ1)}dy + ω2f ′(0) ∫ R Γ2(D2τ2, y) max{0, U1(ξ1 − y − cτ2)− U2(ξ1 − y − cτ2)}dy = −µΠ(ξ1)eν1ξ1 + ω1f ′(0) ∫ R Γ1(D1τ1, y) ×max{0,Π(ξ1 − y − cτ1)e−ν1y}dye−cτ1ν1eν1ξ1 + ω2f ′(0) ∫ R Γ2(D2τ2, y) max{0,Π(ξ1 − y − cτ2)e−ν1y}dye−cτ2ν1eν1ξ1 , which yields (cν1 −Dν2 1)Π(ξ1) EJDE-2022/82 TRAVELING WAVES FOR A REACTION-DIFFUSION EQUATION 7 ≤ −µΠ(ξ1) + ω1f ′(0) ∫ R Γ1(D1τ1, y) max{0,Π(ξ1 − y − cτ1))e−ν1y}dye−cτ1ν1 + ω2f ′(0) ∫ R Γ2(D2τ2, y) max{0,Π(ξ1 − y − cτ2))e−ν1y}dye−cτ2ν1 . In view of (cν1 −Dν2 1)Π(ξ1) = −µΠ(ξ1) + ω1f ′(0)e−cτ1ν1γ1(ν1)Π(ξ1) + ω2f ′(0)e−cτ2ν1γ2(ν1)Π(ξ1) = −µΠ(ξ1) + ω1f ′(0)e−cτ1ν1 ∫ R Γ1(D1τ1, y)e−ν1yΠ(ξ1)dy + ω2f ′(0)e−cτ2ν1 ∫ R Γ2(D2τ2, y)e−ν1yΠ(ξ1)dy, we conclude that ω1f ′(0)e−cτ1ν1 ∫ R Γ1(D1τ1, y)[e−ν1yΠ(ξ1)−max{0,Π(ξ1 − y − cτ1))e−ν1y}]dy + ω2f ′(0)e−cτ2ν1 ∫ R Γ2(D2τ2, y)[e−ν1yΠ(ξ1)−max{0,Π(ξ1 − y − cτ2))e−ν1y}]dy ≤ 0. In view of Π(y) ≤ Π(ξ1) for all y ∈ R, we obtain ω1f ′(0)e−cτ1ν1 ∫ R Γ1(D1τ1, y)[e−ν1yΠ(ξ1)−max{0,Π(ξ1 − y − cτ1))e−ν1y}]dy = 0. Consequently, there exists y1 > 0 such that e−ν1y1Π(ξ1) = max{0,Π(ξ1 − y1 − cτ1)e−ν1y1}. Since Π(ξ1) > 0, we obtain Π(ξ1) = Π(ξ1−y1−cτ1), which contradicts Π(ξ) < Π(ξ1) for ξ < ξ1. Hence, Π(ξ) ≤ 0, ∀ξ ∈ R. Similarly, one can easily show that Π(ξ) ≥ 0, ∀ξ ∈ R. Thus, Π(·) ≡ 0, i.e. U1(·) ≡ U2(·). Since we have made possible translations of U1(ξ) and U2(ξ), it holds U1(·+ ξ0) ≡ U2(·) for some ξ0 ∈ R. This completes the proof. � 3. Stability This section is devoted to the stability of monotone traveling waves under the quasi-monotone assumption, i.e. M∗ ≤ u∗. In this case, the traveling wave U(x+ct) with speed c > c∗ given in Lemma 2.2 is non-decreasing and U(+∞) = M∗. For studying the stability of the traveling front U(x+ ct) with speed c > c∗, in addition to (H1)-(H2), we need the following assumption: (H3) M∗ ≤ u∗ and µ > −g′(M∗)M∗ − g(M∗) + ( ω1 + ω2 ) f ′(M∗); (H4) Γ1(·) and Γ2(·) have compact supports. Recall that ω1 := pe−d1τ1 and ω2 := (1− p)e−d2τ2 . For simplicity, we may assume [−ri, ri] = supp Γi for some ri > 0, i = 1, 2. It is easy to see that the second part of (H3) means that the unique positive equilibrium is stable. Consider the linear equation M ′(t) = −(µ+ g′1(M∗))M(t) + f ′(M∗)[ω1M(t− τ1) + ω2M(t− τ2)]. (3.1) Clearly, its characteristic equation has the from λ = −(µ+ g′1(M∗)) + f ′(M∗)[ω1e −λτ1 + ω2e −λτ2 ]. (3.2) 8 H. ZHAO, S.-L. WU EJDE-2022/82 Lemma 3.1. The characteristic problem (3.2) has a principal eigenvalue λ̌ < 0. Proof. The existence of the principal eigenvalue λ̌ of (3.2) follows from [15, Theorem 5.1]. It remains to show λ̌ < 0. Suppose on the contrary that λ̌ ≥ 0. Then λ̌ = −(µ+ g′1(M∗)) + f ′(M∗)[ω1e −λ̌τ1 + ω2e −λ̌τ2 ] ≤ −(µ+ g′1(M∗)) + f ′(M∗)[ω1 + ω2] < 0. This contradiction implies that λ̌ < 0, and the lemma follows. � Let X be the set of all bounded and continuous functions from R to R equipped with compact open topology. Denote τ = max{τ1, τ2} and C = C([−τ, 0],X ). In the sequel, we assume that U(x + ct) is a traveling front of (1.1) connecting 0 and M∗ with speed c > c∗. Take ν ∈ (ν1(c), ν∗). From the discussions in Section 2, we see that cν − λ(ν) > 0 for any ν ∈ (ν1(c), ν∗). We now define the weight function W (x) = e−νx. (3.3) We make the following assumptions on the initial data u0(x, s): (H5) u0 ∈ C, 0 < u0(x, s) ≤ M∗ for any x ∈ R, s ∈ [−τ, 0], and for any given x0 ∈ R, inf { u0(x, s) ∣∣x ≥ x0 and s ∈ [−τ, 0] } > 0, and sup x∈R,s∈[−τ,0] ∣∣u0(x, s)− U(x+ cs) ∣∣W (x) <∞. (3.4) We first give the results on the existence, uniqueness and positivity of solutions for the Cauchy problem of (1.1). Lemma 3.2. Assume (H1)-(H3), (H5). Then the Cauchy problem of (1.1) with the initial value u0(x, s) admits a unique solution u(x, t) satisfying 0 < u(x, t) ≤ M∗ for all (x, t) ∈ R× [−τ,+∞). Proof. The existence and uniqueness of solutions can be found in [10, Section 4]. Moreover, CM∗ := {ψ ∈ C : 0 ≤ ψ(·) ≤ M∗} is positively invariant. It remains to show that u(x, t) > 0 for all (x, t) ∈ R× [−τ,+∞). Suppose for the contrary that there exists (x0, t0) ∈ R × [0, τ ] such that u(x0, t0) = 0. Clearly, t0 ∈ (0, τ ]. Then ut(x0, t0) ≤ 0 and uxx(x0, t0) ≥ 0. Consequently, it follows from (1.1) that 0 ≥ ω1 ∫ R Γ1(D1τ1, x0 − y)f(u(y, t0 − τ1))dy + ω2 ∫ R Γ2(D2τ2, x0 − y)f(u(y, t0 − τ2))dy, which yields 0 = ω1 ∫ R Γ1(D1τ1, x0 − y)f(u(y, t0 − τ1))dy + ω2 ∫ R Γ2(D2τ2, x0 − y)f(u(y, t0 − τ2))dy. (3.5) Without loss of generality, we assume τ2 = τ = max{τ1, τ2}. By (3.5), we have∫ R Γ2(D2τ2, y)f(u0(x0 − y, t0 − τ2))dy = 0. Thus, f(u0(x0, t0−τ2)) = 0, which implies that u0(x0, t0−τ2) = 0. This contradicts to the assumption of u0. Then, u(x, t) > 0 for all (x, t) ∈ R×[0, τ ]. By induction, we conclude that u(x, t) > 0 for all (x, t) ∈ R× [−τ,+∞). The proof is complete. � EJDE-2022/82 TRAVELING WAVES FOR A REACTION-DIFFUSION EQUATION 9 Now, we are ready to state the stability of the traveling front. Theorem 3.3. Assume (H0)–(H5). Then the unique solution u(x, t) of (1.1) with the initial data u0(x, s), s ∈ [−τ, 0] satisfies 0 < u(x, t) ≤ M∗ for (x, t) ∈ R × R+, and there exist positive constants ρ, C and T1 such that sup x∈R |u(x, t)− U(x+ ct)| ≤ Ce−ρt for all t ≥ T1. To prove this theorem, we need some lemmas. In particular, we need to establish two comparison theorems and construct an auxiliary sub-system. In the sequel, we always make all the assumptions in Theorem 3.3. Given c1 ≥ 0, ζ ∈ R ∪ {−∞}, and 0 ≤ t1 < T1 ≤ +∞. Take χ = max{r1, r2}+ cmax{τ1, τ2}. We further define the following regions: For ζ = −∞, Ω1 ζ = ∅, Ω2 ζ = R× [t1 − τ, t1], Ω3 ζ = R× (t1, T1]; and for ζ ∈ R, Ω1 ζ = { (x, t) ∈ R2 ∣∣ζ − χ ≤ x+ c1t ≤ ζ, t ∈ [t1 − τ, T1] } , Ω2 ζ = { (x, t) ∈ R2 ∣∣x+ c1t > ζ, t ∈ [t1 − τ, t1] } , Ω3 ζ = { (x, t) ∈ R2 ∣∣x+ c1t > ζ, t ∈ (t1, T1] } . We denote Ωζ = Ω1 ζ ∪Ω2 ζ ∪Ω3 ζ . Then, we have the following comparison theorems, which can be proved by using similar methods as in [19, Lemma 5.2]. Lemma 3.4. Assume (H0)–(H4), and that w±(x, t) : Ωζ → R+ are two continuous functions satisfying (i) 0 ≤ w+(x, t), w−(x, t) ≤M∗ for all (x, t) ∈ Ωζ ; (ii) w+(x, t) ≥ w−(x, t) for all (x, t) ∈ Ω1 ζ ∪ Ω2 ζ ; (iii) G(w+)(x, t) ≥ 0 ≥ G(w−)(x, t) for all (x, t) ∈ Ω3 ζ , where G(w±)(x, t) := ∂w± ∂t −Dw±xx − ω1 ∫ R Γ1(D1τ1, y)f(w±(x− y, t− τ1))dy + (µ+ g(w±))w± − ω2 ∫ R Γ2(D2τ2, y)f(w±(x− y, t− τ2))dy. Then w+(x, t) ≥ w−(x, t) for all (x, t) ∈ Ω3 ζ . Lemma 3.5. Assume (H0), (H4), and that for all Q1, Q2 > 0, the functions w±(x, t) : Ωζ → R+ are continuous and satisfying: (i) 0 ≤ w+(x, t) and w−(x, t) ≤M∗ for all (x, t) ∈ Ωζ ; (ii) w+(x, t) ≥ w−(x, t) for all (x, t) ∈ Ω1 ζ ∪ Ω2 ζ ; (iii) F(w+)(x, t) ≥ 0 ≥ F(w−)(x, t) for all (x, t) ∈ Ω3 ζ , where F(w±)(x, t) := ∂w± ∂t −Dw±xx −Q1 ∫ R Γ1(D1τ1, y)w±(x− y, t− τ1)dy + µw± −Q2 ∫ R Γ2(D2τ2, y)w±(x− y, t− τ2)dy. Then w+(x, t) ≥ w−(x, t) for all (x, t) ∈ Ω3 ζ . 10 H. ZHAO, S.-L. WU EJDE-2022/82 Let us define u+ 0 (x, s) = max{u0(x, s), U(x+ cs)}, u−0 (x, s) = min{u0(x, s), U(x+ cs)} for (x, s) ∈ R× [−τ, 0]. One can easily see that 0 ≤ u−0 (x, s) ≤ u0(x, s), U(x+ cs) ≤ u+ 0 (x, s) ≤ u∗(s) (3.6) for (x, s) ∈ R× [−τ, 0]. Let u−(x, t) and u+(x, t) be the solution of (1.1) with the initial data u−0 (x, s) and u+ 0 (x, s), respectively. Applying the comparison principle for quasi-monotone systems, we have 0 ≤ u−(x, t) ≤ u(x, t), U(x+ ct) ≤ u+(x, t) ≤M∗. (3.7) Hence, |u(x, t)− U(x+ ct)| ≤ max { |u+(x, t)− U(x+ ct)|, |u−(x, t)− U(x+ ct)| } (3.8) for any (x, t) ∈ R× R+. We denote W+(x, t) = u+(x, t)− U(x+ ct), W−(x, t) = U(x+ ct)− u−(x, t) (3.9) for any (x, t) ∈ R × [−τ,∞). Thus, we only need to show the exponential conver- gence of W±(x, t) to 0. Fix any ε0 ∈ (0,−λ̆) and choose ε > 0 such that ε0 ≥ ε+ ω1εe ε0τ1 + ω2εe ε0τ2 . (3.10) We define the function g1(u) = g(u)u for u ≥ 0. Then we can take δ0 ∈ (0,M∗) such that for any u2 ≥ u1 ≥M∗ − δ0, g1(u2)− g1(u1) ≥ [g′1(M∗)− ε](u2 − u1), f(u2)− f(u1) ≤ [f ′(M∗) + ε](u2 − u1). (3.11) Since M∗ ≥ u+(x, t) ≥ U(x + ct) and U(+∞) = M∗, it is clear that U(x + ct) and u+(x, t) are close to M∗ as x + ct � 1. To show that u−(x, t) is also close to M∗ when x + ct and t are sufficiently large, we need to construct an auxiliary equation. Take κ0 ∈ (0, 1] such that for any κ ∈ [κ0, 1], µ > g′(M∗)M∗ + g(M∗) + ( ω1 + ω2 ) κf ′(M∗). Consider the equation M ′(t) = −(µ+ g(M))M + ω1κf(M(t− τ1)) + ω2κf(M(t− τ2)), (3.12) where κ ∈ [κ0, 1]. It is easy to see that, for each κ ∈ [κ0, 1], (3.12) has a unique positive equilibrium Mκ ∗ ∈ (0,M∗] which is globally stable. We can further choose κ1 ∈ [κ0, 1) such that M∗ − δ0 2 < Mκ1 ∗ < M∗. (3.13) Proof of Theorem 3.3. In view of 0 ≤ V ±(x, s) ≤ |u0(x, s) − U(x + cs)|, ∀x ∈ R, s ∈ [−τ, 0], it follows from (A1) that V ±(x, s)W (x) is uniformly bounded on R. To prove the convergence of U±(x, t) to U(x+ ct), we use the following 2 claims: Claim 1. There exists L1 > 0 such that 0 ≤W±(x, t) ≤ L1e ν(x+ct)−ρ1t for all x ∈ R, t > 0, (3.14) where ρ1 = cν − λ(ν) > 0. EJDE-2022/82 TRAVELING WAVES FOR A REACTION-DIFFUSION EQUATION 11 We only prove the assertion for W+(x, t), since the assertion for W−(x, t) can be obtained similarly. By (3.4), there exists sufficiently large L1 > 0 such that 0 ≤W+(x, s)W (x) ≤ sup x∈R,s∈[−τ,0] ∣∣u0(x, s)− U(x+ cs) ∣∣W (x) ≤ L1e −λ(ν)τ , which implies that W+(x, s) ≤ L1e νxeλ(ν)s for all x ∈ R, s ∈ [−τ, 0]. We define W̄ (x, t) = L1e νx+λ(ν)t, ∀x ∈ R, t ≥ −τ. It is obvious that W̄ (x, s) ≥W+(x, s) for ∀x ∈ R, s ∈ [−τ, 0]. In view of λ(ν) = Dν2 − µ+ ω1f ′(0)e−τ1λ(ν)γ1(ν) + ω2f ′(0)e−τ2λ(ν)γ2(ν), one can easily verify that W̄t = DW̄xx − µW̄ + ω1f ′(0) ∫ R Γ1(D1τ1, y)W̄ (x− y, t− τ1)dy + ω2f ′(0) ∫ R Γ2(D2τ2, y)W̄ (x− y, t− τ2)dy. (3.15) On the other hand, from (1.1) and (3.9) it follows that W+ t = DW+ xx − µW+ − [g(u+(x, t))u+(x, t)− g(U(x+ ct))U(x+ ct)] + ω1 ∫ R Γ1(D1τ1, y)[f(u+(x− y, t− τ1))− f(U(x− y + ct− cτ1))]dy + ω2 ∫ R Γ2(D2τ2, y)[f(u+(x− y, t− τ2))− f(U(x− y + ct− cτ2))]dy. (3.16) Since g is increasing on (0,+∞) and f ′(u) ≤ f ′(0) for all u ≥ 0, we have W+ t ≤ DW+ xx − µW+ + ω1f ′(0) ∫ R Γ1(D1τ1, y)W+(x− y, t− τ1)dy + ω2f ′(0) ∫ R Γ2(D2τ2, y)W+(x− y, t− τ2)dy. (3.17) Consequently, using Lemma 3.5 with c1 = c > 0, ζ = −∞, T1 =∞, Q1 = ω1f ′(0), and Q2 = ω2f ′(0), we obtain W+(x, t) ≤ W̄ (x, t) for (x, t) ∈ R× (0,∞), that is, W+(x, t) ≤ L1e νx+λ(ν)t = L1e ν(x+ct)−ρ1t for (x, t) ∈ R× (0,∞). This proves claim 1. Claim 2. There exist γ∗ ∈ R and t∗ > 0 such that U(x + ct), u±(x, t) ≥ M∗ − δ0 for any x+ ct ≥ γ∗ and t ≥ t∗. In view of u+(x, t) ≥ U(x + ct) and U(+∞) = M∗, it is clear that there exists γ∗ ∈ R such that u+(x, t) ≥ U(x+ ct) ≥ M∗+M κ1 ∗ 2 ≥Mκ1 ∗ = M∗ − δ0 for x+ ct ≥ γ∗ and t > 0. To prove Claim 2, it remains to show that there exists t∗ > 0 such that u−(x, t) ≥ M∗ − δ0 for any x+ ct ≥ γ∗ and t ≥ t∗. We first consider the case γ∗ ≤ x+ ct ≤ γ∗ + χ. By Claim 1, |u−(x, t)− U(x+ ct)| ≤ L1e ν(γ∗+χ)−ρ1t ≤ L1e ν(γ∗+χ)e−ρ1(γ∗−x)/c, (3.18) from which, we can take Y0 < 0 so that L1e ν(γ∗+χ)e−ρ1(γ∗−x)/c ≤ (M∗ −Mκ1 ∗ )/2 for x ≤ Y0. (3.19) 12 H. ZHAO, S.-L. WU EJDE-2022/82 Thus, we conclude from (3.13), (3.18), and (3.19) that for x ≤ Y0 and t ≥ 0 with x+ ct ∈ [γ∗, γ∗ + χ], we have u−(x, t) ≥ U(x+ ct)− L1e ν(γ∗+χ)e−ρ1(γ∗−x)/c ≥ M∗ +Mκ1 ∗ 2 − M∗ −Mκ1 ∗ 2 = Mκ1 ∗ ≥M∗ − δ0, which implies that u−(x, t) ≥Mκ1 ∗ ≥M∗− δ0 for (x, t) ∈ Ω0, where t0 := (γ∗+χ− Y0)/c and Ω0 := {(x, t) ∈ R2|γ∗ ≤ x+ ct ≤ γ∗ + χ and t ≥ t0}. We now define the following regions: Ω1 = {(x, t) ∈ R2 ∣∣Y0 − χ ≤ x ≤ Y0, t ∈ [−τ, t0 + τ ]}, Ω2 = {(x, t) ∈ R2 ∣∣x ≥ Y0, t ∈ [−τ, 0]}, Ω3 = {(x, t) ∈ R2 ∣∣x > Y0, t ∈ (0, t0 + τ ]}. By (H5) and Lemma 3.1, there exists ζ ∈ (0,Mκ1 ∗ ) such that u−(x, t) ≥ ζ for (x, t) ∈ Ω1 ∪ Ω2, and (ω1 + ω2)f(ζ)− (µ+ g(ζ)ζ > 0. We define u(x, t) = ζ for all (x, t) ∈ Ω1 ∪ Ω2 ∪ Ω3. One can easily verify that u−(x, t) ≥ u(x, t) for (x, t) ∈ Ω1 ∪ Ω2 and ut −Duxx + (µ+ g(u(x, t))u(x, t)− ω1 ∫ R Γ1(D1τ1, x− y)f(u(y, t− τ1))dy − ω2 ∫ R Γ2(D2τ2, x− y)f(u(y, t− τ2))dy = (µ+ g(ζ)ζ − (ω1 + ω2)f(ζ) ≤ 0. According to Lemma 3.4 with c1 = 0, t1 = 0, T1 = t0 + τ and ζ = γ∗ + χ, we have u−(x, t) ≥ u(x, t) = ζ for (x, t) ∈ Ω3. Let us now define the following regions: Ω4 = {(x, t) ∈ R2 ∣∣γ∗ ≤ x+ ct ≤ γ∗ + χ, t ≥ t0}, Ω5 = {(x, t) ∈ R2 ∣∣x+ ct ≥ γ∗ + χ, t ∈ [t0, t0 + τ ]}, Ω6 = {(x, t) ∈ R2 ∣∣x+ ct > γ∗ + χ, t > t0 + τ}, and the function u1(x, t) = T (t) for (x, t) ∈ Ω4 ∪Ω5 ∪Ω6, where T (t) is the unique solution of the initial value problem T ′(t) = −(µ+ g(T (t))T (t) + κ1ω1f(T (t− τ1)) + κ1ω2f(T (t− τ2)), t > t0 + τ, T (θ) = ζ, θ ∈ [t0, t0 + τ ]. It is obvious that T (t) ∈ [0,Mκ1 ∗ ] for any t > 0 and limt→∞ T (t) = Mκ1 ∗ . Further, we have u−(x, t) ≥Mκ1 ∗ ≥ T (t) for (x, t) ∈ Ω4 = Ω0, u−(x, t) ≥ ζ = T (t) for (x, t) ∈ Ω5 ⊆ Ω1 ∪ Ω3, and (u1)t −D(u1)xx + (µ+ g(u1)u1 − ω1 ∫ R Γ1(D1τ1, x− y)f(u1(y, t− τ1))dy EJDE-2022/82 TRAVELING WAVES FOR A REACTION-DIFFUSION EQUATION 13 − ω2 ∫ R Γ2(D2τ2, x− y)f(u1(y, t− τ2))dy = T ′(t) + (µ+ g(T (t))T (t)− ω1f(T (t− τ1))− ω2f(u1(t− τ2)) = (κ1 − 1)ω1f(T (t− τ1)) + (κ1 − 1)ω2f(u1(t− τ2)) ≤ 0, ∀t > t0 + τ. Using Lemma 3.4 again with c1 = c, t1 = t0 + τ , T1 =∞, and ζ = γ∗ + χ, we have u−(x, t) ≥ u1(x, t) = T (t) for (x, t) ∈ Ω6. In view of limt→∞ T (t) = Mκ1 ∗ , there exists t∗ > t0 + τ such that u−(x, t) ≥Mκ1 ∗ − δ0 2 ≥M∗ − δ0 for x+ ct > γ∗ + χ, t ≥ t∗. Therefore, U−(x, t) ≥M∗ − δ0 for any x+ ct ≥ γ∗ and t ≥ t∗. Completion of the proof. We first consider the case x + ct ≤ γ∗ + χ. In this case, from claim 1, there exists C0 > 0 such that |u±(x, t)− U(x+ ct)| ≤ L1e ν(x+ct)−ρ1t ≤ C0e −ρ1t, t > 0. (3.20) Next, we consider the case x+ ct > γ∗ + χ. We denote Ω1 γ∗ = {(x, t) ∈ R2 ∣∣γ∗ ≤ x+ ct ≤ γ∗ + χ, t ≥ t∗t}, Ω2 γ∗ = {(x, t) ∈ R2 ∣∣x+ ct > γ∗ + χ, t ∈ [t∗, t∗ + τ ]}, Ω3 γ∗ = {(x, t) ∈ R2 ∣∣x+ ct > γ∗ + χ, t > t∗ + τ}. Noting that χ = max{r1, r2} + cmax{τ1, τ2} and [−ri, ri] = supp Γi, i = 1, 2, we have x− y + c(t− τi) = x+ ct− y − cτi ≥ γ∗ + χ− ri − cτi ≥ γ∗ for any y ∈ [−ri, ri] and (x, t) ∈ R × [0,∞) with x + ct > γ∗ + χ. It then follows from Claim 2 and (3.11) that ∂W− ∂t = Uξ(x+ ct)− u−t (x, t) = DW−xx − µW− − (g1(U(x+ ct))− g1(u−(x, t))) + ω1 ∫ R Γ1(D1τ1, y)[f(U(x− y + c(t− τ1)))− f(u−(x− y, t− τ1))]dy + ω2 ∫ R Γ2(D2τ2, y)[f(U(x− y + c(t− τ2)))− f(u−(x− y, t− τ2))]dy ≤ DW−xx − (µ+ g′1(M∗)− ε)W− + ω1[f ′(M∗) + ε] ∫ R Γ1(D1τ1, x− y)W−(y, t− τ1)dy + ω2[f ′(M∗) + ε] ∫ R Γ2(D2τ2, x− y)W−(y, t− τ2)dy. 14 H. ZHAO, S.-L. WU EJDE-2022/82 Similarly, we obtain ∂W+ ∂t ≤ DW+ xx − (µ+ g′1(M∗)− ε)W+ + ω1[f ′(M∗) + ε] ∫ R Γ1(D1τ1, x− y)W+(y, t− τ1)dy + ω2[f ′(M∗) + ε] ∫ R Γ2(D2τ2, x− y)W+(y, t− τ2)dy. (3.21) Let ε0 = min{ρ1,−λ̆−ε0}, and choose L2 > 0 such that L2 ≥ max{C0,M∗e ε0(t∗+τ)}. Now, we define Ṽ (x, t) = L2e −ε0t, ∀(x, t) ∈ Ω1 ξ∗ ∪ Ω2 ξ∗ ∪ Ω3 ξ∗ . By (3.20), we obtain W±(x, t) ≤ C0e −ρ1t for (x, t) ∈ Ω1 ξ∗ , W±(x, t) ≤M∗ for (x, t) ∈ Ω2 ξ∗ . (3.22) Thus W±(x, t) ≤ Ṽ (x, t) for all (x, t) ∈ Ω1 ξ∗ ∪ Ω2 ξ∗ . From (3.10), we obtain ε0Ṽ (x, t) = ε0L2e −ε0t ≥ εL2e −ε0t + ω1εe −ε0(t−τ1) + ω2εe −ε0(t−τ2) = εṼ (x, t) + ω1ε ∫ R Γ1(D1τ1, x− y)Ṽ (y, t− τ1)dy + ω2ε ∫ R Γ2(D2τ2, x− y)Ṽ (y, t− τ2)dy. (3.23) Noting that λ̌ = −(µ+ g′1(M∗)) + f ′(M∗)[ω1e −λ̌τ1 + ω2e −λ̌τ2 ], it follows from (3.23) and ε0 < −λ̆− ε0 < −λ̆ that Ṽt(x, t) = −ε0L2e −ε0t ≥ (ε0 + λ̆)L2e −ε0t = L2e −ε0t { − (µ+ g′1(M∗)) + f ′(M∗)[ω1e −λ̌τ1 + ω2e −λ̌τ2 ] + ε0 } = −(µ+ g′1(M∗))Ṽ (x, t) + f ′(M∗)ω1 ∫ R Γ1(D1τ1, x− y)Ṽ (y, t− τ1)dy + f ′(M∗)ω2 ∫ R Γ2(D2τ2, x− y)Ṽ (y, t− τ2)dy + ε0Ṽ (x, t) ≥ −(µ+ g′1(M∗)− ε)Ṽ (x, t) + [f ′(M∗) + ε]ω1 ∫ R Γ1(D1τ1, x− y)Ṽ (y, t− τ1)dy + [f ′(M∗) + ε]ω2 ∫ R Γ2(D2τ2, x− y)Ṽ (y, t− τ2)dy + ε0Ṽ (x, t), by which and (3.23), we obtain ∂Ṽ ∂t ≥ DṼxx − (µ+ g′1(M∗)− ε)Ṽ EJDE-2022/82 TRAVELING WAVES FOR A REACTION-DIFFUSION EQUATION 15 + ω1[f ′(M∗) + ε] ∫ R Γ1(D1τ1, x− y)Ṽ (y, t− τ1)dy + ω2[f ′(M∗) + ε] ∫ R Γ2(D2τ2, x− y)Ṽ (y, t− τ2)dy. Consequently, using Lemma 3.5 with c1 = c, t1 = t0 + τ , T1 = ∞ and ζ = γ∗ + χ, Q1 = ω1[f ′(M∗) + ε], and Q2 = ω2[f ′(M∗) + ε], we deduce that W±(x, t) ≤ Ṽ (x, t) = L2e −ε0t for all (x, t) ∈ Ω3 ξ∗ . Take M := max {C0, L2}. 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Xiao; Regular traveling waves for a nonlocal diffusion equation, J. Differential Equations, 258 (2015), 191–223. [22] Z. Yu; Uniqueness of critical traveling wave for delayed lattice equation, Proc. Amer. Math. Soc., 140 (2012), 3853–3859. Haiqin Zhao School of Mathematics and Statistics, Xidian University, Xian, Shaanxi 710071, China Email address: zhaohaiqin@xidian.edu.cn Shi-Liang Wu School of Mathematics and Statistics, Xidian University, Xian, Shaanxi 710071, China Email address: slwu@xidian.edu.cn 1. Introduction 2. Uniqueness 3. Stability Acknowledgments References