Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 54, pp. 1–22. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu PROPAGATING INTERFACE IN REACTION-DIFFUSION EQUATIONS WITH DISTRIBUTED DELAY HAOYU WANG, GE TIAN Abstract. This article concerns the limiting behavior of the solution to a reaction-diffusion equation with distributed delay. We firstly consider the quasi-monotone situation and then investigate the non-monotone situation by constructing two auxiliary quasi-monotone equations. The limit behaviors of solutions of the equation can be obtained from the sandwich technique and the comparison principle of the Cauchy problem. It is proved that the propagation speed of the interface is equal to the minimum wave speed of the correspond- ing traveling waves. This makes possible to observe the minimum speed of traveling waves from a new perspective. 1. Introduction We consider the limiting behavior (as ε→ 0) of the solution uε(t, x) : [−ετ,∞)× RN → R for the reaction-diffusion equation with distributed delay: ∂tu ε(t, x) = ε∆uε(t, x) + 1 ε [ ∫ τ 0 k(s)g(uε(t− εs, x))ds− uε(t, x) ] , uε(s, x) = uε0( s ε , x), (1.1) where t > 0, x ∈ RN , s ∈ [−ετ, 0], and τ > 0 is a given delay parameter, the function k(·) satisfies k(·) > 0, ∫ τ 0 k(s)ds = 1, The kernel k(·) in model (1.1) has a biological explanation. Specifically, in pop- ulation dynamics, it is sometimes assumed that all juveniles mature sexually at the exact same age τ . However, this approximate age τ is not always realistic. Because of individual differences and the influence of the external environment, the time required for an individual from birth to maturity is not a fixed con- stant. Therefore, many scholars [13, 14, 15, 16, 28] put forward the concept of distributed delay, and described such a dynamic process through the distribution of maturity time weighted by the probability density function. The birth function g(·) : [0,∞)→ [0,∞) is of the class C2 and satisfies the following assumptions: (H1) g(0) = g(1) − 1 = 0, g′(1) < 1 < g′(0), and u < g(u) ≤ g′(0)u for any u ∈ (0, 1). 2010 Mathematics Subject Classification. 35B25, 35K57, 35C07, 35R35, 92D25. Key words and phrases. Reaction-diffusion equations; distributed delay; traveling wave; propagating interface. c©2021. This work is licensed under a CC BY 4.0 license. Submitted January 17, 2021. Published June 21, 2021. 1 2 H. WANG, G. TIAN EJDE-2021/54 Clearly, assumption (H1) shows that (1.1) is a monostable system. There are typical examples of function g(u) which satisfies the assumption above. One is g(u) = ρue−au with ρ > 0 and a > 0, and another is g(u) = ρu a+au% with ρ > 0, % > 1 and a > 0. As we know, when the diffusion coefficient is very small or the reaction term is very large, the solutions of some types of reaction diffusion equations usually generate internal transition layers (which is also called interface). This property is related to the traveling wave solutions of corresponding reaction diffusion equations. In particular, for the famous Fisher-KPP equation [11, 21] ut(t, x) = ∆u(t, x) + u(t, x)(1− u(t, x)), ∀t ≥ 0, x ∈ RN , it admits the traveling wave solution connecting two equilibria 0 and 1 with the wave speed c ≥ c∗ = 2. Taking the scale uε(t, x) ≡ u ( t ε , x ε ) , ε ∈ (0, 1), we obtain ∂tu ε = ε∆uε + 1 ε uε(1− uε), ∀(t, x) ∈ (0,∞)× RN , uε(0, x) = uε0(x), ∀x ∈ RN . (1.2) For the limiting behavior of uε, it seems reasonable to guess that the family {uε}ε>0 converges in some sense to 0 or 1 as ε → 0. By a formal analysis, we can see that the diffusion term ε∆uε can be negligible in the very early stage, since it is very small compared with the reaction term. As a result, (1.2) can be approximated by the ordinary differential equation duε dt = ε−1uε(1− uε). Obviously, this ODE has the equilibria 0 and 1, the value of uε quickly becomes close to either 1 or 0 in most part of RN , which creates a steep transition layer. As soon as the interface develops, the diffusion term starts to increase gradually to balance the reaction term, then the interface ceases development and starts to propagate in a much slower time scale. There are some studies about this phenomenon. For monostable case, Freidlin [12] investigated this asymptotic problem from the perspective of probability the- ory. Then Evans and Souganidis [10] studied it by a direct partial differential equation approach (i.e., geometric optics, Hamilton-Jacobi technics [5, 6]), similar methods were used in [9, 26, 27]. By the method of comparison principle, Alfaro and Ducrot [1, 2] proved the generation and motion of interface properties, and further estimated the thickness of the transition layers. Furthermore, Hilhorst et al. [18] applied the Hamilton Jacobi technics to consider the interface problem for the degenerate Fisher equation ut = ε∆um + 1 ε u(1− u), ∀(t, x) ∈ [0, T ]× Ω, ∂(um) ∂ν = 0, ∀(t, x) ∈ [0, T ]× ∂Ω, u(x, 0) = u0(x), ∀x ∈ Ω, EJDE-2021/54 PROPAGATING INTERFACE IN REACTION-DIFFUSION EQUATIONS 3 where Ω ∈ RN (N ≥ 1), m ≥ 2. For the bistable case, Chen et al. [7] gave a rigorous analysis of both the generation and the motion of interface, ∂tu ε = ∆uε + 1 ε2 g(uε), ∀(t, x) ∈ (0,∞)× RN , uε(0, x) = u0(x), ∀x ∈ RN . We refer the readers to [4, 8, 19, 23] for some other systems. Note that, there is a key factor that can not be ignored in the mathematical model: time delay, which usually represents resource regeneration time, maturity cycle, breastfeeding time, feedback time in biological model and the latency in epidemic model. Consider this effect in the model, Alfaro and Ducrot [3] gave some results about the propagating interface of the monostable equation ∂tu ε(t, x) = ε∆uε(t, x) + 1 ε [g(uε(t− ετ, x))− uε(t, x)], ∀t > 0, x ∈ RN , uε(s, x) = uε0 (s ε , x ) , for − ετ ≤ s ≤ 0, x ∈ RN , where g(u) is an increasing function on the interval (0, 1). However, there seems no results for the distributed delay case. Therefore, this paper is devoted to inves- tigating the propagating interface of (1.1) with distributed delay. Before demonstrating the main theorem, we give some notation. For c > 0, denote Hc := Ut≥0({t}×Hc,t) as the smooth solution of the free boundary problem V = c on Hc,t, Hc,t ∣∣ t=0 = H0 (1.3) with V the normal velocity of Hc,t in the exterior direction, the initial interface is defined as H0 = ∂H0. Assume the region enclosed by H0, namely Ω0, is convex, these solutions do exist for all t ≥ 0. By a slight abuse of notation, we consider Hc,t for all t ≥ −ετ , with ε > 0 small enough. For each t ≥ −ετ , we denote Ωc,t as the region enclosed by the hypersurface Hc,t. In addition, assume that (H2) g(·) is non-decreasing on [0, 1]. Next, we give the assumption on the initial condition. Assumption 1.1 (Initial condition). Assume that u0(s, x) : [−τ, 0]× RN → [0, 1] is a uniformly continuous function satisfying the following conditions: (i) there exists w0 ∈ BUC2(RN ,R) such that Ω0 := {x ∈ RN : w0(x) > 0} is a nonempty smooth bounded and convex domain, and w0(x) ≤ u0(s, x), ∀(s, x) ∈ [−τ, 0]× RN ; (1.4) (ii) there exists δ0 > 0 such that |∇w0(x)ν∂Ω0 (x)| ≥ δ0, ∀x ∈ H0 := ∂Ω0, (1.5) where ν∂Ω0 denotes the outward unit normal vector to Ω0 at x ∈ H0; (iii) there exists v0 ∈ BUC2(RN , [0, 1)) such that supp v0 = Ω0, (1.6) u0(s, x) ≤ v0(x), ∀(s, x) ∈ [−τ, 0]× RN . (1.7) 4 H. WANG, G. TIAN EJDE-2021/54 Theorem 1.2. Suppose g(u) satisfies (H1) and (H2). Let the initial data u0(s, x) satisfy Assumption 1.1. For every ε > 0, let uε(t, x) : [−ετ,∞) × RN → R be the solution of (1.1). Then (i) for each c ∈ (0, c∗) and each t0 > 0, we have lim ε→0+ sup t≥t0 sup x∈Ωc,t |uε(t, x;u0)− 1| = 0; (ii) for each c > c∗ and each t0 > 0, we have lim ε→0+ sup t≥t0 sup x∈RN\Ωc,t |uε(t, x;u0)| = 0, where c∗ is the minimal wave speed of the corresponding traveling waves of (1.1). The proof of the above theorem is inspired by the method in [3]. Here we would like to emphasize that the sublinear condition g(u) ≤ g′(0)u in assumption (H1) is not required explicitly in [3, Theorem 1.3], but it is an essential condition for the existence of monostable traveling wave solutions of (1.1). Furthermore, we extend the above result to the non-monotone case. Since the equation is lack of the monotonicity, the comparison is invalid, the method in [3] is not applicable. To overcome this issue, we first construct two auxiliary quasi- monotone equations. The propagating interface of the equation is estimated by using the sandwich technique and the comparison theorems of the Cauchy problem. We modify assumption (H2) into the following one: (H2’) There exists α ∈ (0, 1) such that g(·) is increasing on [0, α], non-increasing and positive on [α,+∞). Furthermore, g(u) satisfies (a) g(u) ≤ g′(0)u− ku2 on [0, α], where k > 0 is a fixed constant; (b) there exists a positive constant δ∗ such that g(u) ≥ g′(0)u − ρu2 on [0, δ∗], where ρ ∈ (0, g ′(0) 2δ∗ ] is a fixed constant. Then we construct a function g+(u) = g′(0) Mu M + u , ∀u ∈ [0,+∞), where the constant M > 0 will be given later. By calculating, for any u ∈ [0, α], it holds g+(u)− g(u) ≥g′(0) Mu M + u − g′(0)u+ ku2 = −g ′(0)u2 M + u + ku2 = u2 ( k − g′(0) M + u ) ≥ 0, if and only if M > 0 is chosen sufficiently large. This implies g+(α) ≥ g(α) := maxu≥0 g(u). Since g+(u) is increasing on [0,+∞), we immediately obtain that g+(u) ≥ g+(α) ≥ g(u) on [α,+∞). Obviously, g+(u) = u has a unique positive root u∗+ = M(g′(0)− 1). Furthermore, we construct another function g−(u) = g′(0) Nu N + u , ∀u ∈ [0,+∞), EJDE-2021/54 PROPAGATING INTERFACE IN REACTION-DIFFUSION EQUATIONS 5 where the constant N > 0 will be given later. For each 0 ≤ u ≤ δ∗, it holds g(u)− g−(u) ≥ g′(0)u− ρu2 − g′(0) Nu N + u = g′(0)u2 N + u − ρu2 = u2 ( g′(0) N + u − ρ ) ≥ u2 ( g′(0) N + u − g′(0) 2δ∗ ) = g′(0)u2 2δ∗ −N − u 2δ∗(N + u) ≥ 0, if and only if N > 0 is chosen sufficiently small. Since g−(u) is increasing in u ∈ [0,+∞), we can further take N > 0 small enough such that g−(u∗+) = g′(0) Nu∗+ N + u∗+ ≤ g′(0)N < g(u), ∀u ∈ [δ∗, u∗+]. Thus, we have that g−(u) ≤ g(u) ≤ g+(u) for all u ∈ [0, u∗+]. Clearly, g−(u) = u has a unique positive root u∗− satisfying u∗− < 1 < u∗+. Besides, it follows from the definition of g±(u) that g(u) and g±(u) have the same linearization at zero. From the definition of g±(u), we can obtain the following two auxiliary systems: ∂tu ε(t, x) = ε∆uε(t, x) + 1 ε [ ∫ τ 0 k(s)g+(uε(t− εs, x))ds− uε(t, x) ] , t > 0, x ∈ RN , uε(s, x) = ū0 (s ε , x ) , s ∈ [−ετ, 0], x ∈ RN (1.8) and ∂tu ε(t, x) = ε∆uε(t, x) + 1 ε [ ∫ τ 0 k(s)g−(uε(t− εs, x))ds− uε(t, x) ] , t > 0, x ∈ RN , uε(s, x) = u0 (s ε , x ) , s ∈ [−ετ, 0], x ∈ RN . (1.9) Theorem 1.3. Suppose g(u) satisfies (H1) and (H2’). Let the initial data u0(s, x) satisfy Assumption 1.1. For every ε > 0, let uε(t, x) : [−ετ,∞) × RN → R be the solution of (1.1). Then (i) for each c ∈ (0, c∗) and each t0 > 0, we have u∗− ≤ lim ε→0+ inf t≥t0 inf x∈Ωc,t uε(t, x;u0) ≤ lim ε→0+ sup t≥t0 sup x∈Ωc,t uε(t, x;u0) ≤ u∗+; (ii) for each c > c∗ and each t0 > 0, we have lim ε→0+ sup t≥t0 sup x∈RN\Ωc,t |uε(t, x;u0)| = 0, where c∗ is the minimal wave speed of the corresponding traveling waves of (1.1). Note that, the well-posedness of equations (1.1), (1.8) and (1.9) can be proved by a method similar to that of [33, Theorem 2.3], which mainly use the theory of abstract functional differential equations [25, 35], we omit the proof. This paper is 6 H. WANG, G. TIAN EJDE-2021/54 organized as follows: In Section 2, we prove the generation and the motion of inter- face of system (1.1) when the birth function g(·) is non-decreasing. In Section 3, on the basis of the above result, when g(·) is non-monotone, the propagating interface in (1.1) is estimated. Next we give some notation. Let X = BUC(RN ,R) be the Banach space of all bounded and uniformly continuous functions from RN to R with the supremum norm ‖ · ‖X . Let X+ := {v ∈ X : ϕ(x) ≥ 0, x ∈ RN}. Then X is a Banach lattice under the partial ordering induced by X+. Let C = C([−τ, 0], X) be the Banach space of continuous functions from [−τ, 0] into X with the supremum norm and let C+ = {ϕ ∈ C : ϕ(s) ∈ X+, s ∈ [−τ, 0]}. Then C+ is a positive cone of C. Let C0 = C([−τ, 0],R). Usually, we identify an element ϕ ∈ C as a function from [−τ, 0]× RN into R defined by ϕ(s, x) = ϕ(s)(x). We define [a, b]C := {ϕ ∈ C : a ≤ ϕ(s, x) ≤ b, ∀(s, x) ∈ [−τ, 0]× RN}. In addition, [a, b]C0 := C0 ∩ [a, b]C . For any continuous function w : [−τ, T ) → X,T > 0, we define wt ∈ C, t ∈ [0, T ), by wt(s) = w(t + s), s ∈ [−τ, 0]. Then t 7→ wt is a continuous function from [0, T ) to C. 2. Monotone case In this section we investigate the generation and propagation of interface in equation (1.1), when g(·) is non-decreasing, namely, (H1) and (H2) hold. Some important notation is given at first. We defined the function D̃(t, x) := { −dist(x,Hc,t) for x ∈ Ωc,t, dist(x,Hc,t) for x ∈ RN\Ωc,t, (2.1) where dist(x,Hc,t) is the distance from x to the hypersurface Hc,t. We remark that D̃ = 0 on Hc, |∂D̃(t,x) ∂x | = 1 in a neighborhood of Hc. Let T > 0 be given. Choose D0 > 0 small enough so that D̃ is smooth in the tubular neighborhood of Hc {(t, x) ∈ [−ετ, T ]× RN : |D̃(t, x)| < 3D0}. Then, for any s ∈ R, define a smooth increasing function Υ(s) as Υ(s) :=  s if |s| ≤ D0, −2D0 if s ≤ −2D0, 2D0 if s > 2D0. On the basis of the above definitions, the cut-off signed distance function D(t, x) is represented as follows: D(t, x) := Υ(D̃(t, x)). (2.2) From the definition of D(t, x), we know |D(t, x)| < D0 ⇒ |∇D(t, x)| = 1, (2.3) furthermore, the free boundary problem (1.3) implies |D(t, x)| < D0 ⇒ ∂tD(t, x) + c = 0. (2.4) By the mean value theorem, there exists a constant N > 0 such that |∂tD(t, x) + c| ≤ N |D(t, x)|, ∀(t, x) ∈ [−ετ, T ]× RN . (2.5) EJDE-2021/54 PROPAGATING INTERFACE IN REACTION-DIFFUSION EQUATIONS 7 In addition, there exists a constant C > 0 such that |∇D(t, x)|+ |∆D(t, x)| ≤ C, ∀(t, x) ∈ [−ετ, T ]× RN . (2.6) 2.1. Preliminaries. Proposition 2.1 (Comparison principle [33, Theorem 2.3.]). Let τ > 0, T > 0 and g : R → R a non-decreasing and continuous function be given. Let u, v ∈ C([−τ, T ]× RN ) be two bounded functions. Assume (∂t −∆ + 1)u(t, x)− ∫ τ 0 k(s)g(u(t− s, x))ds ≤ 0, (∂t −∆ + 1)v(t, x)− ∫ τ 0 k(s)g(v(t− s, x))ds ≥ 0, (2.7) for almost every (t, x) ∈ (0, T )× RN , and u(s, x) ≤ v(s, x), ∀(s, x) ∈ [−τ, 0]× RN . (2.8) Then u(t, x) ≤ v(t, x) for all (t, x) ∈ [−τ, T ]× RN . Bistable approximations of g. For η ∈ (0, 1], we introduce a non-decreasing and bounded map gη : R→ R of the class C2 such that gη(u) = g(u), ∀u ∈ [0, 1], gη(−η) = −η, g′η(−η) < 1, gη(u) < u, ∀u ∈ (−η, 0) ∪ (1,∞), gη(u) > u, ∀u ∈ (−∞,−η) ∪ (0, 1). (2.9) In addition, we assume that gη satisfies ∀(η, η′) ∈ (0, 1]2, η < η′ ⇒ gη′(u) ≤ gη(u), ∀u ∈ R. (2.10) Traveling waves. We consider the one dimensional bistable system (∂t −∆ + 1)u(t, x) = ∫ τ 0 k(s)gη(u(t− s, x))ds, t > 0, x ∈ R, u(s, x) = ϕ0 ∈ [−η, 1]C , ∀s ∈ [−τ, 0], (2.11) which admits the solution uη ≡ uη(t, x;ϕ0) : [−τ,∞)×R→ [−η, 1]. System (2.11) generates a strongly continuous and non-decreasing semiflow {Qη(t)}t≥0 as [Qη(t)ϕ](s, x) = (uη)t(s, x;ϕ0), ∀(s, x) ∈ [−τ, 0]× R. From the definition of gη, for t ≥ 0, there is Qη(t)[0, 1]C ⊂ [0, 1]C , where Q(t) := Qη(t)|[0,1]C does not depend upon η, Qη presents a semiflow generated by a bistable dynamic and Q is a semiflow generated by the corresponding monostable type. Lemma 2.2 (Bistable traveling waves). For each η ∈ (0, 1], the following results hold: (i) there exists a unique speed cη such that (2.11) has a traveling wave solution (Uη, cη) ∈ C2(R)× R whose profile Uη is non-increasing and satisfies U ′′η (z) + cηU ′ η(z) + ∫ τ 0 k(s)gη(Uη(z + cηs))ds− Uη(z) = 0, ∀z ∈ R, Uη(−∞) = 1, Uη(+∞) = −η; (2.12) 8 H. WANG, G. TIAN EJDE-2021/54 (ii) there exist two positive constants µ and M such that |1− Uη(z)|+ | − η − Uη(−z)| ≤Meµz, ∀z ≤ 0, |U ′η(z)|+ |U ′′η (z)| ≤Me−µ|z|, ∀z ∈ R; (2.13) (iii) there exists some constant γ > 0 such that, for any ϕ0 ∈ [−η, 1]C with lim inf x→−∞ min s∈[−τ,0] ϕ0(s, x) > 0, lim sup x→+∞ max s∈[−τ,0] ϕ0(s, x) < 0, (2.14) one can find C = C(ϕ0) > 0 and ξ = ξ(ϕ0) ∈ R such that |uη(t, x;ϕ0)− Uη(x− cηt+ ξ)| ≤ Ce−γt, ∀(t, x) ∈ [0,+∞)× R. Proof. Item (i) can be found in [33, Theorem 5.5] (also in [22, Theorem 5.1(iii)]). The proof of (ii), can be found in [34, Theorem 3.5]. The proof of (iii) can be found in [33, Theorem 4.5]. � We refer to [24] for the existence of the monotone traveling waves of the one dimensional monostable system (∂t −∆ + 1)u(t, x) = ∫ τ 0 k(s)g(u(t− s, x))ds, t > 0, x ∈ R. (2.15) Lemma 2.3 (Monostable traveling waves [24]). There exists c∗ > 0 such that equation (2.15) has a traveling wave solution (Uc, c) ∈ C2(R) × (0,∞) with 0 < Uc < 1, if and only if c ≥ c∗. In addition, the waves are non-increasing for c ≥ c∗. Let (U∗, c∗) be the traveling wave of (2.15) with minimal wave speed, i.e. (U∗)′′(z) + c∗(U∗)′(z) + ∫ τ 0 k(s)g(U∗(z + c∗s))ds− U∗(z) = 0, ∀z ∈ R, U∗(−∞) = 1, U∗(+∞) = 0. (2.16) Lemma 2.4 (Convergence of speeds). Let {g+ η }η∈(0,1] satisfy (2.9) and (2.10). Then the family {cη}η∈(0,1] is decreasing and cη ↗ c∗ as η ↘ 0. The proof of the above lemma is similar to that of [3, Lemma 2.4.], we omit it. 2.2. Generation of interface. We considering the two differential equations with delay d dt v(t) = ∫ τ 0 k(s)g(vt(−s))ds− v(t), t > 0, v0(·) = φ(·) ∈ [0, 1]C0 , (2.17) and d dt v(t) = ∫ τ 0 k(s)gη(vt(−s))ds− v(t), t > 0, v0(·) = φ(·) ∈ [−η, 1]C0 . (2.18) Lemma 2.5. For each φ ∈ C0, (2.18) has a unique global (mild) solution vη = vη(·;φ) : [−τ,∞) → R and the semiflow Vη(t)φ = Vη(t;φ) := (vη)t(·;φ) is strongly continuous and monotone increasing on C0. It further satisfies the following prop- erties: (i) for each t ≥ 0, Vη(t)[−η, 1]C0 ⊂ [−η, 1]C0 ; EJDE-2021/54 PROPAGATING INTERFACE IN REACTION-DIFFUSION EQUATIONS 9 (ii) for each t ≥ 0, Vη(t)[0, 1]C0 ⊂ [0, 1]C0 . The restriction V (t) = Vη(t) |[0,1]C0 does not depend upon η and, for φ ∈ [0, 1]C0 , the map t 7→ V (t)φ = V (t;φ) is the mild solution vt(·;φ) of (2.17). The above lemma follows straightforwardly from [29], we omit its proof. Lemma 2.6. The following holds: (i) for φ ∈ [0, 1]C0\{0}, we have limt→∞ V (t)φ = 1 in C0; (ii) there exists δ1 > 0, M > 0 and λ > 0 such that, for all φ ∈ C0, ‖1− φ‖L∞(−τ,0) ≤ δ1 ⇒ ‖1− V (t)φ‖L∞(−τ,0) ≤Me−λt, ∀t ≥ 0. Proof. Firstly, we give the proof of case (i). Consider a special situation, if there exists ζ ∈ (0, 1) such that φ(s) ≥ ζ, for all s ∈ [−τ, 0]. Since the semiflow corre- sponding to (2.17) is monotonically increasing and satisfies V (t)[0, 1]C0 ⊂ [0, 1]C0 , we can find a solution with the initial data ζ, i.e., V (t; ζ) = vt(·; ζ). Since g(ζ) > ζ and the mapping t 7→ v(t; ζ) is non-decreasing, then limt→+∞ v(t, x) = 1, which also indicates that ‖V (t)ζ−1‖∞ = sup−τ≤s≤0 |v(t+s, ζ)−1| → 0 as t→ +∞. Next, we consider the general situation. Since φ ∈ [0, 1]C0\{0}, there exists −τ ≤ A < B ≤ 0 and β > 0 such that φ(s) ≥ β1[A,B](s), ∀s ∈ [−τ, 0]. It follows from (2.17) that d dt (etv(t;φ)) = et ∫ τ 0 k(s)g(vt(−s))ds ≥ 0, which implies that etv(t;φ) is non-decreasing with respect to t > 0. In addition, for t ∈ [ 0, B−A2 ] , we have d dt (etv(t;φ)) = et ∫ τ 0 k(s)g(vt(−s))ds ≥et ∫ −A+B 2 −B k(s)g(β)ds >g(β) ∫ −A+B 2 −B k(s)ds. Integrating the above inequality on [0, B−A2 ], we have v (B −A 2 , φ ) > B −A 2 g(β)e A−B 2 ∫ −A+B 2 −B k(s)ds > 0. Consequently, etv(t;φ) ≥ e B−A 2 v (B −A 2 , φ ) > 0, ∀t > B −A 2 > 0, that is, v(t;φ) ≥ e−(t−B−A2 )v (B −A 2 , φ ) > 0, ∀t > B −A 2 > 0. Through the special case discussed above, we can verify that (i) is holds. Now we prove (ii). The characteristic equation corresponding to (2.17) around 1 is ∆(λ) := λ+ 1− g′(1) ∫ τ 0 k(s)e−λsds = 0. 10 H. WANG, G. TIAN EJDE-2021/54 Since g′(1) < 1, all roots of ∆(λ) = 0 have strictly negative real parts, then we obtain the conclusion (see [17, 31]). This completes the proof. � Proposition 2.7. Let φ ≥ 0 in C0\{0} be given. There exists λ > 0 such that, for all α > 0, there exists ε0 = ε0(α) > 0 such that, for all ε ∈ (0, ε0), 1− εαλ/2 ≤ V (α ln |ε|+ t; ε ln |ε|φ)(s) ≤ 1, ∀(s, t) ∈ [−τ, 0]× [0,∞). Proof. Let φ ≥ 0 belong to C0\{0}. Since g′(0) > 1, let δ ∈ (0, 1) and ρ > 1 satisfy g(u) ≥ ρu, ∀u ∈ [0, δ]. (2.19) Choose δ as an initial data, it follows from Lemma 2.4 that, there exist M > 0 and λ > 0 such that 0 ≤ 1− V (t; δ)(s) ≤Me−λt, ∀(s, t) ∈ [−τ, 0]× [0,+∞). (2.20) Let α > 0, choose a sufficiently small ε0 > 0 such that, for all ε ∈ (0, ε0), there is ε| ln ε|φ ∈ [0, δ]C0 and ε λα 2 M < 1. Note that φ ≥ 0 holds in C0\{0}, then there exists −τ < A < B < 0 and β > 0 such that ε| ln ε|φ ≥ ε| ln ε|β1[A,B](s), ∀s ∈ [−τ, 0]. Applying the argument in Lemma 2.4 and (2.19), there exists ζ > 0 such that, for sufficiently small ε > 0, it holds vε(t) := v(t; ε| ln ε|φ) ≥ ζε| ln ε|, ∀t ∈ [τ, 2τ ]. (2.21) Next, for all t ∈ (0, 2τ ], there is d dt (etvε(t)) = et ∫ τ 0 k(s)g(ε| ln ε|φ(t− s))ds ≤ e2τε| ln ε|‖φ‖∞‖g′‖∞ := Cε| ln ε|. Integrating the above inequality from 0 to t ∈ (0, 2τ ], for sufficiently small ε > 0, it yields vε(t) ≤ 2τ(φ(0) + C)ε| ln ε| < δ. Thus, we define tε := sup{t > 2τ : vε(t̃) ≤ δ, ∀t̃ ∈ [2τ, t]}. It follows from (2.17) and (2.19) that v′ε(t) ≥ ρ ∫ τ 0 k(s)vε(t− s)ds− vε(t), ∀t ∈ [2τ, tε]. (2.22) Since ρ > 1, there exists a > 0 such that a + 1 = ρ ∫ τ 0 k(s)e−asds. Then the mapping h : t 7→ Aε| ln ε|eat, A := ζ/e2aτ satisfies h′(t) = ρ ∫ τ 0 k(s)h(t− s)ds− h(t), ∀t ∈ [2τ, tε], (2.23) where h(t) ≤ ζε| ln ε|, t ∈ [τ, 2τ ]. From (2.21)–(2.23), we obtain vε(t) ≥ Aε| ln ε|eat, ∀t ∈ [2τ, tε]. It follows from vε(t ε) = δ that tε ≤ 1 a ln δ Aε| ln ε| . (2.24) Since the mapping t 7→ vε(t) is increasing, then, vε(t ε) = δ yields vε(t ε + t) ≥ δ, ∀t ∈ [0,+∞). EJDE-2021/54 PROPAGATING INTERFACE IN REACTION-DIFFUSION EQUATIONS 11 From (2.24), for sufficiently small ε > 0, there holds tε ≤ α| ln ε|, thus, vε(α| ln ε|+ t+ s) ≥ δ, ∀(s, t) ∈ [−τ, 0]× [0,+∞). Note that the semiflow corresponding to (2.17) is increasing in C0, then one has 0 ≤ 1− vε(α| ln ε|+ t+ s) ≤ 1− Vε(α| ln ε|+ t; δ)(s). Combining this with (2.20), we obtain 0 ≤ 1− vε(α| ln ε|+ t+ s) ≤Me−λ(α| ln ε|+t) ≤Mελα ≤ ελα2 . This completes the proof. � Lemma 2.8. For each t > 0, the map φ ∈ C0 7→ Vη(t;φ) ∈ C0 provided in Lemma 2.4 is of class C2. For each φ0 ∈ C0 and each φ ∈ C0, the map t ∈ [0,∞) 7→ ∂φVη(t;φ0) · φ ∈ C0 is the mild solution of the non-autonomous equation dv dt (t) = L(t, φ0)vt, t > 0, v(s) = φ(s), s ∈ [−τ, 0], (2.25) wherein, for each t > 0, L(t, φ0) : C0 → R is defined by L(t, φ0)φ := ∫ τ 0 k(s)g′η (Vη(t;φ0)(−s))φ(−s)ds− φ(0). (2.26) Moreover, for each φ0 ∈ C0 and each φ ∈ C0, the map t ∈ [0,∞) 7→ ∂2 φ,φVη(t;φ0) · (φ, φ) is the solution of dv dt (t) = L (t, φ0) vt + G (t;φ0;φ) , t > 0, v(s) = 0, s ∈ [−τ, 0], where the map t→ G(t;φ0;φ) is defined by G(t;φ0;φ) := ∫ τ 0 k(s)g′′η (Vη(t;φ0)(−s)) [∂φVη(t;φ0)(−s) · φ(−s)]2. (2.27) The proof of the above lemma is similar to that of [17, 31], we omit it. Proposition 2.9. Assume g(·) satisfies (H1) and (H2). (i) There exist Ĉ1 > 0 and γ1 > 0 such that, for all φ0 ∈ C0, e−τ−(t+s) ≤ ∂φVη(t;φ0)1(s) ≤ Ĉ1e γ1(t+s), for all (s, t) ∈ [−τ, 0]× [0,∞). (ii) There exist Ĉ2 > 0 and γ2 > 0 such that, for all φ0 ∈ C0, |∂φφVη(t;φ0) · (1, 1)(s)| ≤ Ĉ2e γ2(t+ s), for all (s, t) ∈ [−τ, 0]× [0,∞). (iii) There exist Ĉ3 > 0 and γ3 > 0 such that, for all φ0 ∈ C0, |∂φφVη(t;φ0) · (1, 1)(s)| ≤ Ĉ3e γ3t∂φVη(t;φ0) · 1(s), for all (s, t) ∈ [−τ, 0]× [0,∞). 12 H. WANG, G. TIAN EJDE-2021/54 Proof. (i) Let φ0 ∈ C0. Firstly, Vη(t) is monotonically increasing and satisfies ∂φVη(t;φ0) · 1(s) ≥ 0, ∀(s, t) ∈ [−τ, 0]× [0,+∞). (2.28) From (2.25) and (2.26), there exists w(t) := ∂φVη(t;φ0) · 1(0) satisfying w′(t) ≥ −w(t), ∀t ≥ 0, i.e., w(t) ≥ e−t for all t ≥ 0. Then, for any (s, t) ∈ [−τ, 0]× [0,+∞) which satisfies t+ s ≥ 0, we have ∂φVη(t;φ0)1(s) ≥ e−(t+s). For any (s, t) ∈ [−τ, 0]× [0,+∞) satisfying t+ s < 0, we have ∂φVη(t;φ0)1(s) ≥ e−(τ+t+s). Therefore, we prove that the inequality on the left is valid. Next, we choose a constant `� 1 such that 0 ≤ g′η(u) ≤ `, ∀u ∈ R. (2.29) It follows from (2.25) and (2.26) that w′(t) ≤ ` ∫ τ 0 k(s)w(t− s)ds− w(t), t > 0, w(s) = 1, s ∈ [−τ, 0]. (2.30) Let h(t) = e(`−1)t for all t > 0. Obviously, h(t) is an increasing function which satisfies 0 < h(t− s) ≤ h(t) for all s > 0. Note that ∫ τ 0 k(s) [h(t− s)− h(t)] ds ≤ 0, then we obtain h′(t)− ` ∫ τ 0 k(s)h(t− s)ds+ h(t) ≥ h′(t)− ` ∫ τ 0 k(s)h(t)ds+ h(t) ≥ h′(t)− `h(t) + h(t) = 0, (2.31) where t > 0, h(s) ≥ 1, s ∈ [−τ, 0]. Thus, (2.30) and (2.31) indicates w(t) ≤ e(`−1)t for all t ≥ 0. As discussed above, the inequality on the right holds. (ii) It follows from (2.27) and (i) that, there exists a constant A > 0 such that for any φ0 ∈ C0, |G(t;φ; 1)| ≤ Ae2γ1(t−τ), ∀t ≥ 0. Thus, function w(t) := ∂φφVη(t;φ0) · (1, 1)(0) satisfies w′(t) ≤ ` ∫ τ 0 k(s)w(t− s)ds− w(t) +Ae2γ1(t−τ), (2.32) where, t > 0, w(s) = 1, s ∈ [−τ, 0]. If K̃eµ̃t is a supper solution of (2.32), where the constant K̃ > 0, µ̃ > 0 will be given later, then we have µ̃ ≥ ` ∫ τ 0 k(s)e−µ̃sds− 1 + A K̃ e(2γ1−µ̃)t−2γ1τ , ∀t > 0, (2.33) if and only if µ̃ > 2γ1 and K̃ > 0 is chosen sufficiently large. By an argument as in (i), there exist constants K > 0 and µ > 0 such that, for any φ0 ∈ C0, s ∈ [−τ, 0] and t ≥ 0, it holds ∂φφVη(t;φ0) · (1, 1)(s) ≤ Keµ(t+s). Furthermore, since g′η(u) ≥ 0 for all u ∈ R, We can get w′(t) ≥ ∫ τ 0 k(s)g′η (Vη(t;φ0)(−s))φ(−s)ds− w(t)−Ae2γ1(t−τ) ≥− w(t)−Ae2γ1(t−τ). EJDE-2021/54 PROPAGATING INTERFACE IN REACTION-DIFFUSION EQUATIONS 13 Constructing a sub-solution −K̃eµ̃t, it is a lower bound of w(t). In conclusion, we obtain the boundedness of the second derivative. (iii) This result can be obtained directly from (i) and (ii). � Proposition 2.10 (Sub-solution). Assume g satisfies (H1) and (H2). Let u0(s, x) and w0(x) satisfy Assumption 1.1. Then there exist K > 0, α > 0 and ε0 > 0 such that, for all ε ∈ (0, ε0), max { 0; vη ( t ε ;w0(x)− εKτ −Kt )} ≤ uε(t, x) for all (t, x) ∈ [−ετ, αε ln |ε|] × RN . Here, uε(t, x) denotes the solution of (1.1), vη = vη(·;φ) : [−τ,∞)→ R denotes the solution of (2.18). Proof. Define Lεη[u](t, x) := ∂tu(t, x)− ε∆u(t, x)− 1 ε [ ∫ τ 0 k(s)gη(u(t− εs, x))ds− u(t, x) ] . Obviously, Lεη[uε](t, x) ≡ 0. Let u(t, x) := vη ( t ε ;w0(x)− εKτ −Kt ) , then, for any t > 0, x ∈ RN , we have Lεη[u](t, x) = −V ε(t, x) [ K + ε∆w0(x) + ε W ε(t, x) V ε(t, x) |∇w0(x)|2 ] + 1 ε [(dvη dt + vη )( t ε ;w0(x)− εKτ −Kt ) − ∫ τ 0 k(s)gη ( vη ( t ε − s;w0(x)− εKτ −K(t− εs) )) ds ] , where V ε(t, x) = ∂φVη ( t ε ;w0(x)− εKτ −Kt ) · 1(0), W ε(t, x) = ∂φφVη ( t ε ;w0(x)− εKτ −Kt ) · (1, 1)(0). Then we have(dvη dt + vη )( t ε ;w0(x)− εKτ −Kt ) − ∫ τ 0 k(s)gη ( vη ( t ε − s;w0(x)− εKτ −K(t− εs) )) ds ≤ (dvη dt + vη )( t ε ;w0(x)− εKτ −Kt ) − ∫ τ 0 k(s)gη ( vη ( t ε − s;w0(x)− εKτ −Kt )) ds = ∫ τ 0 k(s)gη ( vη ( t ε − s;w0(x)− εKτ −Kt )) ds − ∫ τ 0 k(s)gη ( vη ( t ε − s;w0(x)− εKτ −Kt )) ds = 0, 14 H. WANG, G. TIAN EJDE-2021/54 where we used the monotonicity of vη and gη(·). It follows from Proposition 2.9 that, for all ε ∈ (0, 1), t > 0, x ∈ RN, Lεη[u](t, x) ≤− V ε(t, x) [ K + ε∆w0(x) + ε W ε(t, x) V ε(t, x) |∇w0(x)|2 ] ≤− V ε(t, x) [ K − ε‖∆w0‖L∞ − ε‖∇w0‖2L∞K̂eγ t ε ] . For any ε ∈ (0, 1), t ∈ (0, γ−1ε‖ ln ε‖) and x ∈ RN , we have Lεη[u](t, x) ≤ −V ε(t, x) [ K − ε‖∆w0‖L∞ − ε‖∇w0‖2L∞Ĉ ] ≤ 0, if K > 0 is sufficiently large. Next, for all s ∈ [−ετ, 0], from (1.4), it holds that u(s, x) = w0(x)− εKτ −Ks ≤ w0(x) ≤ u0 (s ε , x ) = uε(s, x). Finally, the comparison principle in Proposition 2.1 indicates that u(t, x) ≤ uε(t, x), ∀(t, x) ∈ [−ετ, γ−1ε| ln ε| × RN ]. Using that uε(t, x) ≥ 0, we then complete the proof. � Proposition 2.11 (Generation of interface from below). Let the initial data u0(x) satisfy Assumption 1.1. Denote by D(0, x) the smooth cut-off signed distance func- tion to H0 (where d(0, x) < 0 if and only if x ∈ Ω0). Then there exists δ0 > 0, α0 > 0, ρ0 > 0 and ε0 > 0 such that, for all ε ∈ (0, ε0) and all (s, x) ∈ [−τ, 0]×RN , we have 1− ερ0 ≤ uε(α0ε| ln ε|+ ετ + εs, x) ≤ 1, provided that D(0, x) ≤ −δ0ε| ln ε|. Proof. Choose K > 0 and α > 0 as in Proposition 2.10. Let α0 = α/2, ρ0 = α0λ. For φ = α0 ∈ C0\{0}, choose λ as in Proposition 2.7. From Assumption (ii), the mean value theorem provides the existence of a constant δ0 > 0, such that for sufficiently small ε > 0, there is D(0, x) ≤ −δ0ε| ln ε| ⇒ w0(x) ≥ 4α0ε| ln ε| For any −τ <≤ s ≤ 0, define T̂ = α0ε| ln ε| + ετ + εs ∈ [α0ε| ln ε|, α0ε| ln ε| + ετ ]. Choose x satisfying D(0, x) ≤ −δ0ε| ln ε|. When ε is sufficiently small, it holds that 0 ≤ T̂ ≤ 2α0ε| ln ε| = αε| ln ε| and w0(x)− εKτ −KT̂ ≥ α0ε| ln ε|. Then vη(T̂ /ε, w0(x)− εKτ −KT̂ ) ≥ vη(T̂ /ε, α0ε| ln ε|) ≥ 1− ερ0 , since vη(t;φ) is an increasing semiflow. By Proposition 2.10, we obtain that uε(T̂ , x) ≥ vη(T̂ /ε, α0ε| ln ε|) ≥ 1− ερ0 . This completes the proof. � EJDE-2021/54 PROPAGATING INTERFACE IN REACTION-DIFFUSION EQUATIONS 15 2.3. Propagation of interface. Lower barriers via bistable approximation: For η ∈ (0, 1], we denote (Uη, cη) as the traveling wave solution of system (2.11), which satisfy U ′′η (z) + cηU ′ η(z) + ∫ τ 0 k(s)gη(Uη(z + cηs))ds− Uη(z) = 0, ∀z ∈ R, Uη(−∞) = 1, Uη(0) = 0, Uη(+∞) = −η. (2.34) Define the sub-solution of (2.11) as u−η (t, x) := Uη (Dη(t, x) + ε| ln ε|p(t) ε ) − q(t), (2.35) where p(t) = −e−βt/ε + eQt + P, (2.36) q(t) = σ ( βe−βt/ε + εQeQt ) , (2.37) the positive constants β, σ, P,Q are determined in the proof. Proposition 2.12 (Sub-solution). There exist positive constants β, σ,Q, for all P > 1 and sufficiently small ε > 0, it holds εLεη[u−η ](t, x) = ε∂tu − η (t, x)−ε2∆u−η (t, x)− ∫ τ 0 k(s)gη(u−η (t−εs, x))ds+u−η (t, x) ≤ 0, for all t > 0, x ∈ RN . Proof. For simplicity we denote z := Dη(t, x) + ε| ln ε|p(t) ε . From the definition of D(t, x) in (2.4), we have D(t− εs, x) = D(t, x) + cεs+ εΘε(t, x), where Θε(t, x) vanishes close to the interface and is O(1): |D(t, x)| ≤ D0, ‖Θε‖L∞ ≤ B ⇒ Θε(t, x) = 0 (2.38) for some constant B > 0. Next, since p(t) is increasing and Uη(z) is decreasing, it holds u−(t− εs, x) = Uη (Dη(t− εs, x) + ε| ln ε|p(t− εs) ε ) − q(t− εs) ≥ Uη (Dη(t, x) + ε| ln ε|p(t) ε + cs+ Θε(t, x) ) − q(t− εs). Since gη(u) is non-decreasing, we have gη(u−(t− εs, x)) ≥ gη ( Uη (Dη(t, x) + ε| ln ε|p(t) ε + cs+ Θε(t, x) ) − q(t− εs) ) = gη ( Uη (Dη(t, x) + ε| ln ε|p(t) ε + cs+ Θε(t, x) )) − q(t− εs)(gη)′(θ), for some constant θ satisfying Uη (z′ + cs+ Θε)− q(t− εs) ≤ θ ≤ Uη (z′ + cs) , 16 H. WANG, G. TIAN EJDE-2021/54 where z′ := Dη(t,x)+ε| ln ε|p(t) ε . Hence, it yields gη(u−(t− εs, x)) ≥ gη (Uη (z′ + cs+ Θε))− q(t− εs)(gη)′(θ) ≥ gη(U(z′ + cs))− q(t− εs)(gη)′(θ) + Θε(t, x)(gη ◦ U)′(z′ + cs+ ωΘε(t, x)) for some ω ∈ [0, 1]. By calculations, we have εLεη[u−η ](t, x) = (ε| ln ε|p′(t) + ∂tD) · U ′ − εq′(t)− U ′′ · (∇D)2 − εU ′ ·∆D − (∫ τ 0 k(s)gη(u−η (t− εs, x))ds− u−η (t, x) ) ≤ (ε| ln ε|p(t) + ∂tD) · U ′ − εq′(t)− U ′′ · (∇D)2 − εU ′ ·∆D − ∫ τ 0 k(s) [ gη(U(z′ + cs))− q(t− εs)(gη)′(θ) + Θε(t, x)(gη ◦ U)′(z′ + cs+ ωΘε(t, x)) ] ds+ Uη(z)− q(t) = E1 + E2 + E3, where E1 = ε| ln ε|p′(t) · U ′(z) + ∫ τ 0 k(s)q(t− εs)g′η(θ)ds− q(t)− εq′(t), E2 = (∂tD(t, x) + c− ε∆D(t, x))U ′(z) + (1− |∇D(t, x)|2)U ′′(z), E3 = − ∫ τ 0 k(s)Θε(t, x)(gη ◦ U)′(z′ + cs+ ωΘε(t, x))ds. Using (2.36) and (2.37), we have E1 = βe− β ε t [ | ln ε|U ′ + σ (∫ τ 0 k(s)g′η(θ)εeβsds− 1 + β )] + εQeQt [ | ln ε|U ′ + σ (∫ τ 0 k(s)g′η(θ)εe−Qsds− 1− εQ )] = βe−βt/εe1 + εQeQte2. From the definition of gη, we have g′η(−η) < 1 and g′η(1) < 1. Consequently, we can fix small % > 0 and β > 0 such that∫ τ 0 k(s)g′η(u)εeβsds− 1 + β < 0, ∀u ∈ [−η − %,−η + %] ∪ [1− %, 1 + %]. On the one hand, since U(−∞) = 1, U(+∞) = −η and Uη(z′+cs+Θε)−q(t−εs) ≤ θ ≤ Uη(z′ + cs), there exists a sufficient large z∗ such that θ ∈ [−η − %,−η + %] ∪ [1− %, 1 + %] once |z| ≥ z∗ (In order to control −q(t− ετ), we choose a sufficiently small σ). Since U ′(z) ≤ 0, we obtain e1 ≤ −σβ in the region {|z| > z∗}. On the other hand, in the region {|z| ≤ z∗}, we have U ′(z) ≤ −ς for some ς > 0, then e1 ≤ −ς| ln ε| + C. As a result, it yields e1 ≤ −σβ. We could get e2 ≤ −σβ by a similar argument. Therefore, it holds E1 ≤ ( βe− β ε t + εQeQt ) (−σβ) ≤ −σβεQ. To show εLεη[u−η ](t, x) ≤ 0, we divide the discussion into two situations. On the one hand, when |D(t, x)| < D0, it follows from (2.3) and (2.4) that E2 = EJDE-2021/54 PROPAGATING INTERFACE IN REACTION-DIFFUSION EQUATIONS 17 −ε∆D(t, x)U ′(z). In addition, (2.38) yields E3 = 0. Hence, εLεη[u−η ](t, x) ≤ −σβεQ+ ε‖U ′‖L∞(R) · ‖∆D(t, x)‖L∞(R) ≤ 0, provided that Q > 0 is large enough. On the other hand, when |D(t, x)| ≥ D0, we can use the exponential decay of the derivatives of U to control E2 and E3. Indeed in this region, |z| ≥ D0/(2ε). Hence, combining the exponential decay of U ′ and U ′′, (2.5) and (2.6), we have a bound |E2| ≤ C2e −C2D0/(2ε) for some C2 > 0. Also (2.38) indicates that |z′ + cs+ ωΘε(t, s)| ≥ D0/(2ε)− cτ − ωB ≥ D0 4ε , which yields |E3| ≤ C3e −C3 D0 4ε for some C3 > 0. Hence, εLεη[u−η ](t, x) ≤ −σβεQ+ Ce−C D0 4ε ≤ 0, if 0 < ε� 1. This completes the proof. � Lemma 2.13. There exists P > 1 such that, for sufficiently small ε > 0, it holds u−η (t, x) ≤ uε(t+ α0ε| ln ε|+ ετ, x), ∀ − ετ ≤ t ≤ 0, x ∈ RN , where α0ε| ln ε| denotes the “generation of interface from below time” appearing in Proposition 2.11. Proof. We consider two cases. On the one hand, if D(t, x) ≥ −ε| ln ε|p(t), from the definition of Uη, we have u−η (t, x) ≤ 0. On the other hand, for any (t, x) ∈ [−ετ, 0]×RN , if D(t, x) < −ε| ln ε|p(t), it follows from Proposition 2.11 (Generation of interface) that D(0, x) ≤ −δ0ε| ln ε| ⇒ 1− ερ0 ≤ uε(α0ε| ln ε|+ ετ + εt, x) ≤ 1, t ∈ [−ετ, 0]. (2.39) Then it holds D(0, x) = D(t, x) +O(t) ≤ −ε| ln ε|p(t) + Cετ ≤ −ε(−eβτ + e−Aτ + P ) + Cετ ≤ −δ0ε| ln ε|, where ε > 0 is sufficiently small and P is sufficiently large. From (2.39), we just need to prove that u−η (t, x) ≤ 1− ερ0 . From the definition of q(t), we obtain that u−η (t, x) ≤ 1− ερ0 . This completes the proof. � Proof of Theorem 1.2(i). From Proposition 2.12 and Lemma 2.13, by the compar- ison principle, we obtain u−η (t− α0ε| ln ε| − ετ, x) ≤ uε(t, x), ∀t ≥ α0ε| ln ε|+ ετ, x ∈ RN . (2.40) Note that u−η (t, x) is defined in (2.35) and Uη(−∞) = 1, then the conclusion in Theorem 1.2(i) can be immediately obtained by Lemma 2.6 and (2.40). � 18 H. WANG, G. TIAN EJDE-2021/54 Upper barriers: Let (U∗, c∗) denote the traveling wave of (2.15) with minimal wave speed, which is given in Lemma 2.3. It satisfies (U∗)′′(z) + c∗(U∗)′(z) + ∫ τ 0 k(s)g(U∗(z + c∗s))ds− U∗(z) = 0, ∀z ∈ R, (U∗)′(z) ≤ 0, ∀z ∈ R, U∗(−∞) = 1, U∗(+∞) = 0. (2.41) Next, we study the upper estimate on uε(t, x) of system (1.1). Proposition 2.14 (Super-solution). There exists κ ∈ R such that, for all ε > 0 small enough, uε(t, x) ≤ U∗ (D(0, x)− c∗t ε − κ ) , ∀(t, x) ∈ [−ετ,∞)× RN . Proof. From Assumption 1.1(iii), we know that ‖v0‖∞ < 1, so there exists κ ∈ R such that ‖v0‖∞ ≤ U∗(c∗τ − κ). Without loss of generality, here we choose κ = 0, then ‖v0‖∞ ≤ U∗(c∗τ). (2.42) Let x0 ∈ ∂Ω0 = H0 and n0 be the outward unit normal vector to H0 at x0, then define u+(t, x) := U∗ ( (x− x0) · n0 − c∗t ε ) and z = (x−x0)·n0−c∗t ε . By calculating, it yields εLεη[u+](t, x) = ∂tu +(t, x)− ε∆u+(t, x)− 1 ε (∫ τ 0 k(s)g(u+(t− εs, y))ds− u+(t, x) ) = −c ∗ ε (U∗)′(z)− 1 ε (U∗)′′(z)− 1 ε (∫ τ 0 k(s)g(U∗(z + c∗s))ds− U∗(z) ) = 0, where (t, x) ∈ (0,+∞)× RN . Next, we prove that uε(s, x) = u0 (s ε , x ) ≤ U∗ ( (x− x0) · n0 − c∗s ε ) = u(s, x), for all (s, x) ∈ [−ετ, 0]× RN . It follows from Assumption 1.1(iii) that ū0 (s ε , x ) ≤ v0(x). With the decrease of U∗, we have U∗ ( (x− x0) · n0 ε + c∗τ ) ≤ U∗ ( (x− x0) · n0 − c∗s ε ) . Thus, we need only to prove that v0(x) ≤ U∗ ( (x− x0) · n0 ε + c∗τ ) , ∀x ∈ RN . (2.43) When (x− x0) · n0 ≤ 0, (2.42) implies ‖v0‖∞ ≤ U∗(c∗τ) ≤ U∗ ( (x− x0) · n0 ε + c∗τ ) . When (x − x0) · n0 > 0, unequality (1.6) and the convexity of Ω0 imply that v0(x) = 0, thus (2.43) is obvious. EJDE-2021/54 PROPAGATING INTERFACE IN REACTION-DIFFUSION EQUATIONS 19 Finally, from the comparison principle, we obtain that uε(t, x) ≤ U∗ ( (x− x0) · n0 − c∗t ε ) , ∀(t, x) ∈ [−ετ,∞]× RN for every x0 ∈ ∂Ω0. This completes the proof. � Proof of Theorem 1.2(ii). We obtain the conclusion from Proposition 2.14. � 3. Non-monotone case Since the auxiliary systems (1.8) and (1.9) are monotonically increasing, the conclusion in Section 2 is applicable. Hence, we can get the following lemmas. Lemma 3.1 (Upper system). Suppose g(u) satisfies (H1) and (H2’), and the initial data ū0(s, x) : [−τ, 0] × RN → [0, u∗+] is continuous and satisfies Assumption 1.1. For each ε > 0, let uε+(t, x) : [−ετ,∞) × RN → R be the solution of (1.8). Then the following convergence results hold: (i) for each c ∈ (0, c∗) and each t0 > 0, we have lim ε→0+ sup t≥t0 sup x∈Ωc,t |u∗+ − uε+(t, x; ū0)| = 0, c ∈ (0, c∗), (ii) for each c > c∗ and each t0 > 0, we have lim ε→0+ sup t≥t0 sup x∈RN\Ωc,t |uε+(t, x; ū0)| = 0, c > c∗, where c∗ is the minimal wave speed of the corresponding traveling waves of (1.8). Lemma 3.2 (Lower system). Suppose g satisfies (H1) and (H2) ′ , and the initial data u0(s, x) : [−τ, 0] × RN → [0, u∗−] is continuous and satisfies Assumption 1.1. For each ε > 0, let uε−(t, x) : [−ετ,∞] × RN → R be the solution of (1.9). Then the following convergence results hold: (i) for each c ∈ (0, c∗) and each t0 > 0, we have lim ε→0+ sup t≥t0 sup x∈Ωc,t |u∗− − uε−(t, x;u0)| = 0, c ∈ (0, c∗), (ii) for each c > c∗ and each t0 > 0, we have lim ε→0+ sup t≥t0 sup x∈RN\Ωc,t |uε−(t, x;u0)| = 0, c > c∗, where c∗ is the minimal wave speed of the corresponding traveling waves of (1.9). Next, we give a comparison principle whose proof can be found in [25, 35, 32]. Lemma 3.3. Suppose g satisfies (H1) and (H2’), and for any u0 ∈ C[0,u∗+], (1.1), (1.8) and (1.9) have unique solutions uε(t, x;u0), uε+(t, x;u0) and uε−(t, x;u0) with uε, uε+, u ε − ∈ C([−ετ,∞] × RN ), respectively. In addition, for any u0, u0, ū0 ∈ C[0,u∗+], if u0 ≤ u0 ≤ ū0, then 0 ≤ uε−(t, x;u0) ≤ uε(t, x;u0) ≤ uε+(t, x; ū0) for all t ≥ 0, x ∈ RN . 20 H. WANG, G. TIAN EJDE-2021/54 Proof of Theorem 1.3. Let u0(s, x) = min{u0(s, x), uε−} for all (s, x) ∈ [−τ, 0]×RN . Then by Lemma 3.3, we have uε−(t, x;u0) ≤ uε−(t, x;u0) ≤ uε(t, x;u0) ≤ uε+(t, x;u0) for any (t, x) ∈ [−ετ,∞)× R. Now we consider two cases to complete the proof. Case (i): For each c ∈ (0, c∗) and each t0 > 0, it follows from Lemma 3.1(i) and Lemma 3.2(i) that u∗− ≤ lim ε→0+ inf t≥t0 inf x∈Ωc,t uε−(t, x;u0), lim ε→0+ sup t≥t0 sup x∈Ωc,t uε+(t, x;u0) ≤ u∗+, which further implies that u∗− ≤ lim ε→0+ inf t≥t0 inf x∈Ωc,t uε−(t, x;u0) ≤ lim ε→0+ sup t≥t0 sup x∈Ωc,t uε+(t, x;u0) ≤ u∗+. Case (ii): For each c > c∗ and each t0 > 0, it follows from Lemma 3.1(ii) that lim ε→0+ sup t≥t0 sup x∈RN\Ωc,t |uε+(t, x;u0)| = 0. Then it is clear that lim ε→0+ sup t≥t0 sup x∈RN\Ωc,t |uε(t, x;u0)| = 0. This completes the proof. � Acknowledgments. 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Applied Math- ematical Science, Vol. 119, Spring-Verlag, New York, 1996. Haoyu Wang School of Information Science and Engineering, Lanzhou University, Lanzhou, Gansu,730000, China. School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China Email address: wanghaoyu17@lzu.edu.cn 22 H. WANG, G. TIAN EJDE-2021/54 Ge Tian (corresponding author) School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China Email address: tiang17@lzu.edu.cn 1. Introduction 2. Monotone case 2.1. Preliminaries 2.2. Generation of interface 2.3. Propagation of interface 3. Non-monotone case Acknowledgments References