摺積型非線性項之網格動態系統研究

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(1)國立臺灣師範大學數學系博士班博士論文. 指導教授：郭忠勝博士. On a lattice dynamical system with convolution type nonlinearity 摺積型非線性項之網格動態系統研究. 研究生：林穎志. 中華民國ㄧ ○ ㄧ年五月.

(2) 中文摘要在本篇論文中，我們研究非線性項為摺積型的網格動態系統。本論文分成兩個部份。在第一部分，我們考慮在非線性項為單穩定摺積型的網格動態系統中的行進波，我們討論行進波的漸近行為、單調性及唯一性。在第二部份中，我們考慮在非線性項為雙穩定摺積型的網格動態系統下之雙波峰全域解。我們建構三種不同類型之雙波峰全域解。在當時間趨近負無窮大時，每一種雙波峰全域解的行為近似於連接三個平衡態中的其中兩個平衡態的兩個行進波。. 關鍵字：網格動態系統、摺積型、行進波、全域解.

(3) ABSTRACT In this thesis, we study a lattice dynamical system with convolution type nonlinearity. This thesis is divided into two parts. In the first part, we consider traveling front solutions of a lattice dynamical system with monostable convolution type nonlinearity. We discuss the asymptotic behaviors, monotonicity and uniqueness of traveling waves. Then, in the second part, we consider two-front entire solutions of a lattice dynamical system with bistable convolution type nonlinearity. We construct three different types of two-front entire solutions. Each of these entire solutions behaves as two traveling fronts connecting two of those three equilibria as time approaches minus infinity.. Keywords：lattice dynamical system、convolution type、traveling wave、 entire solution.

(4) On a lattice dynamical system with convolution type nonlinearity Ying-Chih Lin Advisor: Dr. Jong-Shenq Guo Department of Mathematics, National Taiwan Normal University.

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(6) Contents 1 Introduction 1.1 1.2. 1. Traveling wave solution in the monostable case . . . . . . . . . . . . . . . . Two-front entire solutions . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 2. 2 Traveling wave solution in the monostable case 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5 5 9. 2.3 2.4 2.5. Asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monotonicity and the proof of Theorem 2.1.4 . . . . . . . . . . . . . . . . . Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23 24 27. 3 Two-front entire solutions 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33 33. 3.2 3.3. Preliminaries . . . . . . Entire solutions . . . . . 3.3.1 Proof of Theorem 3.3.2 Proof of Theorem 3.3.3. . . . . . . . . 3.1.1 . 3.1.2.. . . . .. . . . .. . . . .. 38 41 41 45. Proof of Theorem 3.1.3. . . . . . . . . . . . . . . . . . . . . . . . .. 53. 4 References. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 55. 3.

(7) Chapter 1 Introduction In this thesis, we study a lattice dynamical system with convolution type nonlinearity. This thesis is divided into two parts. In the first part, we consider traveling front solutions of a lattice dynamical system with monostable convolution type nonlinearity. We discuss the asymptotic behaviors, monotonicity and uniqueness of traveling waves. Then, in the second part, we consider two-front entire solutions of a lattice dynamical system with bistable convolution type nonlinearity. We construct three different types of two-front entire solutions. Each of these entire solutions behaves as two traveling fronts connecting two of those three equilibria as time approaches minus infinity.. 1.1. Traveling wave solution in the monostable case. We study the following lattice dynamical system (LDS) of convolution type: (1.1.1). u′j :=. ∑ duj = D(uj+1 + uj−1 − 2uj ) − duj + J(i)b(uj−i ), dt i∈Z. j ∈ Z,. where uj = uj (t), D, d > 0, ∑. J(i) = J(−i) ≥ 0,. J(i) = 1.. i∈Z. Here we assume that b is an increasing Lipschitz continuous function on [0, 1] such that b(0) = b(1) − d = 0.. b(u) > du if 0 < u < 1, So that we have the monostable nonlinearity.. In ecology, u represents the population density, d is the death rate and the nonlinear function b is the birth function of population density which is interacting with neighbors by the nonnegative weighted function J. 1.

(8) We are interested in the traveling wave solutions of (1.1.1). We say that {uj } is a traveling wave solution of (1.1.1) with speed c if uj (t) = U (j − ct) for j ∈ Z and t ∈ R for some function U (called wave profile). Then (c, U ) satisfies the following equation cU ′ (x) + D2 [U ](x) − dU (x) +. ∑. J(i)b(U (x − i)) = 0,. x ∈ R,. i∈Z. where D2 [U ](x) := D[U (x + 1) + U (x − 1) − 2U (x)]. We shall only study (1.1.1) with short range interaction so that J(i) = 0 for all |i| ≥ p with p = 3. Therefore, we shall study the following problem (P): { ∑ cU ′ (x) + D2 [U ](x) − dU (x) + 2i=−2 J(i)b(U (x − i)) = 0, x ∈ R, U (−∞) = 1, U (+∞) = 0, 0 ≤ U (·) ≤ 1 on R. For the existence of traveling wave of (1.1.1), it is already well-studied in [23, 29] for more general settings, including with time delay. Indeed, the method of [23] is by investigating the asymptotic speed of propagation, and it works also for the case of infinite range interaction. Note that we can also follow a (direct) method developed in [7] to derive the existence of solutions of (P) for any finite range interaction. The main purpose of this part is to investigate the asymptotic behavior of wave tails, the monotonicity of wave profiles and the uniqueness of traveling wave solution. This part has been published in the Discrete and Continuous Dynamical Systems Series A ([15]).. 1.2. Two-front entire solutions. In the second part, we study the following discrete diffusive equation with convolution type nonlinearity. (1.2.1) ut (x, t) = D2 [u](x, t) − du(x, t) +. ∑. J(i)b(u(x − i, t)),. x ∈ R, t ∈ R,. i∈Z. where d > 0, J(i) = J(−i) ≥ 0,. ∑ i∈Z. J(i) = 1, and. D2 [u](x, t) := D[u(x + 1, t) + u(x − 1, t) − 2u(x, t)] for some positive constant D. Here we assume that the function b(·) is an increasing smooth function on [0, 1] such that the nonlinearity is of unbalanced bistable type. 2.

(9) We say that u(x, t) is a traveling wavefront solution of (1.2.1) connecting two different equilibria {u± } ⊂ {0, a, 1} with speed c, if u(x, t) = U (x + ct) for x ∈ R and t ∈ R for some function U (called wave profile) such that U (±∞) = u± . Then (c, U ) satisfies the following equation ∑ cU ′ (y) = D2 [U ](y) − dU (y) + J(i)b(U (y − i)), y ∈ R, i∈Z. where D2 [U ](x) := D[U (x + 1) + U (x − 1) − 2U (x)]. In [25], they studied (1.2.1) for the bistable case. They proved that the problem admits a unique (up to a translation) strictly monotone increasing traveling wavefront solution connecting from 0 to 1 with a positive wave speed when D ≥ D0 for a certain positive constant D0 , under the following extra assumption } { ∫1 ∫1 ∑ 2 0 [b(u) − du]du 2 0 [b(u) − du]du , ∫1 . J(i) < max ∫1 b(u)du b(u)du − d i∈Z 0 0 More precisely, from [25, Theorem 1.1], under the above assumptions, there exist a unique speed cˆ > 0 and a unique (up to translations) wave profile U (x) such that { ∑ cÛ ′ (x) = D2 [U ](x) − dU (x) + i∈Z J(i)b(U (x − i)), x ∈ R, U (−∞) = 0, U (+∞) = 1, 0 0 in R, if D ≥ D0 . The monostable case for (1.2.1) was considered in [23]. In the present setting, it corresponding to the case for connecting two equilibria {a, 1} or {0, a}. In fact, by [23, Theorem 1.2], there exist two constants c∗ , c∗ with c∗ > 0 > c∗ such that for any c1 , c2 (with c1 ≥ c∗ , c2 ≤ c∗ ) there exist V1 (x) and W2 (x) satisfying the following equations: { ∑ c1 V1′ (x) = D2 [V1 ](x) − dV1 (x) + i∈Z J(i)b(V1 (x − i)), x ∈ R, (1.2.2) V1 (−∞) = a, V1 (+∞) = 1, a < V1 < 1, V1′ > 0 in R. and (1.2.3). {. ∑ c2 W2′ (x) = D2 [W2 ](x) − dW2 (x) + i∈Z J(i)b(W2 (x − i)), x ∈ R, W2 (−∞) = 0, W2 (+∞) = a, 0 < W2 < a, W2′ > 0 in R,. where c∗ (c∗ , resp.) is the minimal (maximal, resp.) speed of (1.2.2) ((1.2.3), respectively). Recently, there are many studies on the so-called two-front entire solutions. Here an entire solution is meant by a solution defined for all (x, t) ∈ R2 . In particular, traveling 3.

(10) wavefront solutions are also entire solutions. For two-front entire solutions, there is a very interesting work by Morita and Ninomiya [26], in which they gave three different types of entire solutions for a bistable reaction-diffusion equation (see also [18] for the discrete diffusive case). The above constructed entire solutions have the following common characters. When −t ≫ 1, they behave as two traveling wavefronts on the opposite sides or on the same side of x-axis. The purpose of this part is to construct these three types of entire solutions for our problem (1.2.1). This part has been accepted in the Osaka Journal of Mathematics ([16]).. 4.

(11) Chapter 2 Traveling wave solution in the monostable case 2.1. Introduction. In this chapter, we study the following lattice dynamical system (LDS) of convolution type: ∑ (2.1.1) u′j = D(uj+1 + uj−1 − 2uj ) − duj + J(i)b(uj−i ), j ∈ Z, i∈Z. where uj = uj (t), D, d > 0, ∑. J(i) = J(−i) ≥ 0,. J(i) = 1.. i∈Z. Hereafter the prime denotes the derivative with respect to the independent variable t. Throughout this chapter, we shall always assume that b is an increasing Lipschitz continuous function on [0, 1] such that (2.1.2). b(u) > du if 0 < u < 1,. b(0) = b(1) − d = 0.. The system (2.1.1) can be thought as the spatial discrete version of the following nonlocal partial differential equation: ∫ ∞ (2.1.3) ut = uxx − du + J ⋆ b(u), [J ⋆ b(u)](x, t) := J(y)b(u(x − y, t))dy. −∞. In ecology, u represents the population density, d is the death rate and the nonlinear function b is the birth function of population density which is interacting with neighbors by the nonnegative weighted function J. When J is the Dirac function, the equation (2.1.3) is reduced to the standard PDE: (2.1.4). ut = uxx + f (u),. f (u) := −du + b(u). 5.

(12) On the other hand, when the habitat is divided into discrete regions and the population density is measured at one point (e.g., center) in each region, then (2.1.3) is reduced to the system (2.1.1) in which the index j stands for the jth site in spatial domain. We are interested in the traveling wave solutions of (2.1.1). We say that {uj } is a traveling wave solution of (2.1.1) with speed c if uj (t) = U (j − ct) for j ∈ Z and t ∈ R for some function U (called wave profile). Then (c, U ) satisfies the following equation ∑ J(i)b(U (x − i)) = 0, x ∈ R, cU ′ (x) + D2 [U ](x) − dU (x) + i∈Z. where D2 [U ](x) := D[U (x + 1) + U (x − 1) − 2U (x)]. The spatial discrete version of the equation (2.1.4) has been studied very extensively for more general function f (cf. e.g., [1, 5, 6, 7, 13, 21, 30, 31, 32]). A related equation of convolution type is the following equation: vt = J ⋆ v − v + f (v). (2.1.5). with J compactly supported and satisfying ∫ J(−x) = J(x) ≥ 0,. J(y)dy = 1, R. and f monostable. Schumacher [27] has derived the existence of traveling wave solution for the equation (2.1.5). Here a solution v is a traveling wave solution with speed c if v(x, t) = V (x − ct) for some (wave profile) V . Carr and Chmaj [3] have obtained the uniqueness of traveling wave solution for (2.1.5). Moreover, the asymptotic behavior of traveling fronts and entire solutions of (2.1.5) are studied by Lv [22]. See also the works by Coville and Dupaigne [10, 11, 12]. For the same equation with bistable nonlinearity f , we refer the reader to [2]. We also refer to [1, 24, 25] for the corresponding discrete lattice case. In this chapter, we shall only study (2.1.1) with short range interaction so that J(i) = 0 for all |i| ≥ p with p = 3. Therefore, we shall study the following problem (P): (2.1.6). ′. cU (x) + D2 [U ](x) − dU (x) +. 2 ∑. J(i)b(U (x − i)) = 0,. x ∈ R,. i=−2. (2.1.7). U (−∞) = 1,. U (+∞) = 0,. 0 ≤ U (·) ≤ 1 on R.. For the existence of traveling wave of (2.1.1), it is already well-studied in [23, 29] for more general settings, including with time delay. In particular, under the assumptions 6.

(13) (H1) b is differentiable at 0 and 1 such that b′ (1) < d < b′ (0). (H2) b is differentiable at 0 and there exist constants M > 0, α ∈ (0, 1] such that b′ (0)u − M u1+α ≤ b(u) ≤ b′ (0)u + M u1+α , if u ∈ [0, 1], by the result of [23] (see also [29]), we have (i) There exists a positive constant cmin such that (P) admits a strictly decreasing solution if and only if c ≥ cmin . Moreover, if we assume the extra condition that b(u) ≤ b′ (0)u. for all u ∈ [0, 1],. then we have cmin = c∗ , where [ ] 2 ∑ 1 c∗ := inf D(er + e−r − 2) − d + b′ (0) J(i)eir . r>0 r i=−2 (ii) Every solution U (x) of (P) satisfies 0 cmin , the traveling wave is unique (up to a translation) under the additional condition (2.1.8). lim sup[U (x)e−Λ(c)x ] < ∞, x→∞. where Λ(c) be the larger (negative) root of the following characteristic equation (2.1.9). λ. Φ(λ; c) := cλ + D(e + e. −λ. ′. − 2) + b (0). 2 ∑. J(i)eiλ − d = 0.. i=−2. Indeed, the method of [23] is by investigating the asymptotic speed of propagation, and it works also for the case of infinite range interaction. Note that we can follow a (direct) method developed in [7] to derive the existence of solutions of (P) for any finite range interaction. The main purpose of this part is to investigate the asymptotic behavior of wave tails, the monotonicity of wave profiles and the uniqueness without the assumption (2.1.8). Now, we list the main theorems of this part as follows. First, in order to study the asymptotic behavior of wave tails, we need to study the following equation (2.1.10). ar(x) +. 2 ∑. [∫. x+i. a|i| exp. ] r(y)dy = 0,. x. i=−2. where a ̸= 0, a2 > 0, a1 > 0, a0 ∈ R. By using a method of [7], we have 7.

(14) Theorem 2.1.1 Let a ̸= 0, a2 > 0, a1 > 0, a0 ∈ R, and P (a, a2 , a1 , a0 , λ) := λa + a2 e2λ + a1 eλ + a0 + a1 e−λ + a2 e−2λ . Then (i) If P (a, a2 , a1 , a0 , λ) = 0 has no roots, then (2.1.10) has no solutions. (ii) If P (a, a2 , a1 , a0 , λ) = 0 has only one root Λ∗ , then (2.1.10) has only the trivial solution r(·) ≡ Λ∗ . (iii) If P (a, a2 , a1 , a0 , λ) = 0 has two real roots {Λ1 , Λ2 } with Λ1 < Λ2 , then all solutions of (2.1.10) are formed as r(x) =. θΛ1 eΛ1 x + (1 − θ)Λ2 eΛ2 x , θeΛ1 x + (1 − θ)eΛ2 x. θ ∈ [0, 1].. Next, the asymptotic behaviors of wave profiles near both tails are given as follows. Theorem 2.1.2 Assume (H1) and let (c, U ) be an arbitrary solution of (P). Then there exist constants Λ = Λ(c), σ = σ(c) with Λ(c) < 0 < σ(c) such that lim [U ′ (x)/U (x)] = Λ(c),. x→∞. lim [U ′ (x)/(U (x) − 1)] = σ(c),. x→−∞. where Λ(c) is a root of the characteristic equation Φ(λ; c) = 0 defined in (2.1.9) and σ(c) be the unique positive root of the following characteristic equation (2.1.11). Ψ(σ; c) := cσ + D(eσ + e−σ − 2) + b′ (1). 2 ∑. J(i)eiσ − d = 0.. i=−2. Note that Theorem 2.1.2 also implies that c ≥ c∗ for any solution (c, U ) of (P). In particular, we always have cmin ≥ c∗ . With this asymptotic behavior, we can derive the monotonicity of wave profile as follows. Theorem 2.1.3 Assume (H1) and let (c, U ) be an arbitrary solution of (P). Then U ′ (x) < 0 for x ∈ R. Moreover, combining this monotonicity property with an idea from [7], we can determine the tail behavior at x = ∞ more precisely as follows. Theorem 2.1.4 The limit Λ(c) in Theorem 2.1.2 is the larger (negative) root of the characteristic equation Φ(λ; c) = 0 for c > cmin . 8.

(15) Note that when c = c∗ there is the unique double root of Φ(λ; c) = 0. Finally, we prove the following theorem by using an idea from [5]. Theorem 2.1.5 Assume (H1) and (H2). Let (c, U ) be an arbitrary solution of (P) with c > cmin . Then the limit limx→∞ [U (x)e−Λ(c)x ] exists and is (finite) positive, where Λ(c) is the bigger root of the characteristic equation Φ(λ; c) = 0. Combining this theorem with the (partial) uniqueness result of [23], we conclude that the wave profile is unique (up to a translation) for each given admissible wave speed. Notice that (2.1.8) is not needed in our uniqueness result. It is automatic satisfied for each wave profile. We organize this chapter as follows. First, some preliminaries and the proof of Theorem 2.1.1 are given in section 2. Then we study the asymptotic behavior of the tails of wave profile in section 3. In section 4, we prove Theorem 2.1.3 (the monotonicity of wave profiles) and Theorem 2.1.4. Finally, we derive the uniqueness of wave profiles in section 5. The main idea and method of proofs are from [5, 7]. For the reader’s convenience, we provide some details of proofs for completeness. But, due to the convolution term some difficulties are presented. In particular, the proof of Proposition 2.2.1 is highly nontrivial comparing with the case treated in [7]. We remark that our results can be extended to any finite range interaction if the key proposition (Proposition 2.2.1 below) can be extended to general positive integer p.. 2.2. Preliminaries. In this section, we shall give some preliminaries for the asymptotic behavior of wave profiles near both wave tails. Also, we shall give the proof of Theorem 2.1.1. Lemma 2.2.1 Let (c, U ) be a solution of (P). Then there exists a positive constant K such that U (x + s) | U ′ (x) | + sup ≤ K. U (x) x∈R U (x) x∈R,|s|≤1 sup. Proof. Given a solution (c, U ) of (P). Since cU ′ (x) − (2D + d)U (x) ≤ 0, we obtain that U ′ (x) ≤ µU (x) for µ ≥ (2D + d)/c. This implies that e−µx U (x) is a nonincreasing function. Therefore, we have (2.2.1). U (x + s) ≤ U (x)eµs ≤ U (x)eµ , if x ∈ R, 0 ≤ s ≤ 1. 9.

(16) So we obtain that U (x + s) ≤ eµ . U (x) x∈R,0≤s≤1. (2.2.2). sup. Next, we focus on the case that x ∈ R and −1 ≤ s ≤ 0. By integrating (2.1.6) over [x, n], we get ∫. n. [U (y + 1) + U (y − 1) − 2U (y)]dy. c[U (n) − U (x)] + D + d. ∫. 2 ∑. x n. [U (y − i) − U (y)]dy +. J(i) x. i=−2. 2 ∑. ∫. n. [b(U (y − i)) − dU (y − i)]dy = 0.. J(i) x. i=−2. Since J(i) = J(−i) and b(s) ≥ ds if s ∈ [0, 1], by taking n → ∞, we have ∫. x+1. −cU (x) − D. [U (y) − U (y − 1)]dy − d x. 2 ∑. ∫. i=1. x+i. [U (y) − U (y − i)]dy ≤ 0.. J(i) x. Since U ≥ 0, we have ∫. ∫. x+1/2. cU (x) ≥ [D + dJ(1) + dJ(2)]. U (y − 1)dy − [D + dJ(1) + dJ(2)] x. x+2. U (y)dy. x. By (2.2.1), we have ∫. ∫. x+1/2. x−1/2. 1 U (y − 1)dy ≥ U (x − 1/2)e−µ/2 dy = e−µ/2 U (x − 1/2), 2 x x−1 ∫ x+2 ∫ x+2 U (y)dy ≤ U (x)e2µ dy = 2e2µ U (x). x. x. Hence 1 cU (x) ≥ e−µ/2 [D + dJ(1) + dJ(2)]U (x − 1/2) − 2e2µ [D + dJ(1) + dJ(2)]U (x). 2 This implies that [U (x − 1/2)/U (x)] is bounded uniformly for x ∈ R. Combining with (2.2.2), we conclude that U (x + s) ≤ M1 U (x) x∈R,|s|≤1 sup. for some positive constant M1 ∈ R. Moreover, dividing (2.1.6) by U (x) and using the Lipschitz continuity of b, we obtain that supx∈R |U ′ (x)/U (x)| ≤ M2 for some constant M2 . Therefore, the lemma is proved.. 10.

(17) Lemma 2.2.2 Let (c, U ) be a solution of (P). Then there exists a positive constant K such that | U ′ (x) | 1 − U (x + s) sup + sup ≤ K. 1 − U (x) x∈R 1 − U (x) x∈R,|s|≤1 Proof. First, we define V (x) = 1 − U (x). Then (2.1.6) can be re-written as ′. cV (x) + D2 [V ](x) − dV (x) +. (2.2.3). 2 ∑. J(i)[b(1) − b(1 − V (x − i))] = 0.. i=−2. Following the same method as that of Lemma 2.2.1, we can get V (x + s) ≤ eµ V (x) x∈R,0≤s≤1 sup. for some positive constant µ, Secondly, since V (−∞) = 1 − U (−∞) = 0 and 0 < V (·) < 1 on R, the quantity K(x) := max s≥0. V (x − s) V (x). is well-defined for each x ∈ R. We claim that K(x) < e2µ for all x ∈ R. Suppose not, then we can find x1 ∈ R and s0 > 0 such that V (x1 − s0 ) ≥ V (x1 )e2µ . Let y ∈ (−∞, x1 ) be the smallest value attained max(−∞,x1 ) V (·). So we have V ′ (y) = 0, max{V (y − 1), V (y − 2)} < V (y). On the other hand, since max V (·) ≤ V (x1 )e2µ ≤ V (x1 − s0 ) ≤ V (y),. [x1 ,x1 +2]. we have V (y + i) ≤ V (y) for i = 1, 2. Hence, by (2.1.2), ′. cV (y) + D[V (y + 1) + V (y − 1) − 2V (y)] − dV (y) +. 2 ∑. J(i)[b(1) − b(1 − V (y − i))]. i=−2. ≤ D[V (y + 1) + V (y − 1)] − (2D + d)V (y) + d − d(1 − V (y)) ≤ D[V (y − 1) − V (y)] < 0. This contradicts (2.2.3). So we have proved that K(x) < e2µ for all x ∈ R. This implies that V (x + s) ≤ e2µ . V (x) x∈R,|s|≤1 sup. 11.

(18) Finally, dividing (2.2.3) by V (x), it is easy to see that sup x∈R. |V ′ (x)| ≤K V (x). for some positive constant K. Hence the lemma follows. We shall follow the method of [7] to prove Theorem 2.1.1. In the course of proof, we need to analyze the following recurrence equation (2.2.4). an+3 an+2 + l1 an+2 +. l1 an+1. +. 1 an+1 an. = l,. n ∈ Z,. where l1 , l are positive constants. Recall that, for the case treated in [7], the recurrence equation is given by an+1 +. 1 = l, an. n ∈ Z,. for some positive constant l. It is easy to deduce the monotonicity of an , and we can easily obtain the convergence of an . But, the convergence of the sequence {an } defined by (2.2.4) is not trivial. By taking two consecutive equations from (2.2.4), we have (2.2.5). an+2 (an+3 − an+1 ) + l1 (an+2 − an+1 ) 1 l1 (an − an+1 ) + (an−1 − an+1 ) = 0, + an+1 an an+1 an an−1. n ∈ Z.. Moreover, we have ∞ (i) If {an }∞ n=−∞ is a positive sequence satisfying (2.2.4), then {an }n=−∞ is a bounded sequence and (2.2.5) holds.. (ii) If {an }∞ n=−∞ is a positive sequence satisfying (2.2.4) and any consecutive four terms are equal, then {an }∞ n=−∞ is a constant sequence. In the sequel, we shall prove the following very technical proposition. Proposition 2.2.1 If {an }∞ n=−∞ is a positive sequence satisfying (2.2.4), then both limits limn→∞ an and limn→−∞ an exist. To prove this proposition, we define Mn := max{an−2 , an−1 , an , an+1 , an+2 }, mn := min{an−2 , an−1 , an , an+1 , an+2 }. 12.

(19) Lemma 2.2.3 Let {an }∞ n=−∞ be a non-constant positive sequence satisfying (2.2.4). Then mn < a n < M n ,. ∀ n ∈ Z.. Proof. By the definitions of Mn and mn , we have mn ≤ an ≤ Mn for all n ∈ Z. Suppose that there exists k ∈ Z such that ak = Mk . This implies that ak ≥ ak+2 , ak ≥ ak+1 , ak ≥ ak−1 , ak ≥ ak−2 . By (2.2.5) with n = k − 1, we have ak−2 = ak−1 = ak = ak+1 = ak+2 . Therefore, {an }∞ n=−∞ is a constant sequence. We have reached a contradiction. The case that there exists k ∈ Z such that ak = mk can be treated similarly. Therefore, we have proved the lemma. By Lemma 2.2.3, we have Mn = max{an−2 , an−1 , an+1 , an+2 },. mn = min{an−2 , an−1 , an+1 , an+2 }, ∀n ∈ Z.. Indeed, in the next lemma, we have more precise information. Lemma 2.2.4 Let {an }∞ n=−∞ be a non-constant positive sequence satisfying (2.2.4). Then, for each integer k ∈ Z, we have 1. min{ak−2 , ak−1 } = mk < min{ak+1 , ak+2 }, if Mk = max{ak+1 , ak+2 }; 2. min{ak+2 , ak+1 } = mk < min{ak−1 , ak−2 }, if Mk = max{ak−1 , ak−2 }. Moreover, either Mk = max{ak+1 , ak+2 } or Mk = max{ak−1 , ak−2 }, and they are exclusive. Proof. Given any integer k. Suppose that Mk = max{ak+1 , ak+2 }. We claim that mk < min{ak+1 , ak+2 }. Suppose not. Then we have mk = min{ak+1 , ak+2 }. Since an is a nonconstant sequence, we have Mk > mk . This implies ak+1 ̸= ak+2 . Without loss of generality, we may assume ak+1 > ak+2 . So Mk = ak+1 and mk = ak+2 . By Lemma 2.2.3, we get ak+3 = Mk+1 > ak+1 , ak+4 = mk+2 < ak+2 . Continuing in this way, we obtain that · · · < ak+6 < ak+4 < ak+2 < ak+1 < ak+3 < ak+5 < · · · 13.

(20) Using (2.2.5) with n = k + 2t for any positive integer t, we have an+2 (an+3 − an+1 ) > l1 (an+1 − an+2 ) ⇒ an+2 (an+3 − an+2 ) + an+2 (an+2 − an+1 ) > l1 (an+1 − an+2 ) ⇒ an+2 (an+3 − an+2 ) > (an+1 − an+2 )(l1 + an+2 ) l1 l1 ⇒ an+3 − an+2 > (1 + )(an+1 − an+2 ) > (1 + )(an+1 − an ) an+2 ak+2 Repeating the above process, we get that ak+2t+1 − ak+2t ≥ (1 +. l1 ak+2. )t−1 (ak+3 − ak+2 ). for any positive integer t. This contradicts the boundedness of {an }. Therefore, mk < min{ak+1 , ak+2 }. By Lemma 2.2.3, we must have mk = min{ak−2 , ak−1 }. The case when Mk = max{ak−1 , ak−2 } is similar. Finally, it is clear that only one of the options can hold for each k. The proof is completed.. Lemma 2.2.5 Let {an }∞ n=−∞ be a non-constant positive sequence satisfying (2.2.4). Then either Mn = max{an+1 , an+2 }, mn = min{an−2 , an−1 } for all n ∈ Z and both {Mn } and {mn } are non-decreasing; or, Mn = max{an−1 , an−2 }, mn = min{an+2 , an+1 } for all n ∈ Z and both {Mn } and {mn } are non-increasing. Proof. Suppose that there is an integer k ∈ Z such that Mk = max{ak+1 , ak+2 }. By max{ak+1 , ak+2 } = Mk ≥ max{ak−1 , ak , ak+1 , ak+2 } and the definition of Mk+1 , we have Mk+1 ≥ max{ak+1 , ak+2 , ak+3 } ≥ max{ak−1 , ak , ak+1 , ak+2 , ak+3 } = Mk+1 . This and Lemma 2.2.3 imply that Mk+1 = max{ak+1 , ak+2 , ak+3 } = max{ak+2 , ak+3 }. 14.

(21) Next, we claim that Mk−1 = max{ak , ak+1 }. Suppose not. Then from Lemma 2.2.4 we have Mk−1 = max{ak−3 , ak−2 }. Moreover, we have mk−1 = min{ak , ak+1 }. Since, by assumption, mk = min{ak−2 , ak−1 }, we have mk = min{ak−2 , ak−1 } ≥ mk−1 = min{ak , ak+1 } ≥ mk . From this we deduce that mk = min{ak , ak+1 } = ak+1 , by Lemma 2.2.3. This contradicts Lemma 2.2.4. Therefore, we conclude that Mk−1 = max{ak , ak+1 }. By induction and Lemma 2.2.4, we get Mn = max{an+1 , an+2 }, mn = min{an−2 , an−1 }, ∀n ∈ Z. Since Mn+1 ≥ max{an+1 , an+2 } = Mn ,. mn+1 = min{an−1 , an } ≥ mn ,. both {Mn } and {mn } are non-decreasing sequences. The other case is similar. Therefore, the lemma is proved by combining Lemma 2.2.4. Proof of Proposition 2.2.1. By Lemma 2.2.5, without loss of generality, we may assume that Mn = max{an+1 , an+2 } and mn = min{an−2 , an−1 }, ∀n ∈ Z. So {Mn } and {mn } are bounded and non-decreasing sequences. Therefore, limn→±∞ Mn and limn→±∞ mn must exist. First, we consider limn→∞ an . We define M := limn→∞ Mn and m := limn→∞ mn . If M = m, then limn→∞ an exists (since mn < an < Mn ). Suppose that M > m. By hypothesis, Mn = max{an+1 , an+2 }, mn+3 = min{an+1 , an+2 }. This implies Mn + mn+3 = an+1 + an+2 . Since {Mn } and {mn } are non-decreasing sequences, an+1 + an+2 ≤ an+2 + an+3 ,. ∀n ∈ Z.. Hence {a2n } and {a2n+1 } are non-decreasing sequences. So the limits p := lim a2n , n→∞. q := lim a2n+1 n→∞. are well-defined. If p = q, then limn→∞ an exists. 15.

(22) Suppose that p ̸= q. Without loss of generality we may assume that p > q. Now, we claim that p = M, q = m. Since {a2n } and {a2n+1 } are non-decreasing sequences, we have a2n+1 ≤ q, a2n ≤ p for all n ∈ Z. This implies that Mn ≤ max{p, q} = p. Therefore, M ≤ p. On the other hand, a2n ≤ M2n implies p ≤ M . Therefore, M = p. Moreover, using the fact that M2n−1 + m2n+2 = a2n + a2n+1 , we get m = q. So limn→∞ a2n = M and limn→∞ a2n+1 = m. By (2.2.5) with n even and taking n → ∞, we obtain l1 (M − m) +. l1 (M − m) = 0. Mm. This implies that M = m, a contradiction. Therefore, limn→∞ an exists. Similarly, we can prove that limn→−∞ an exists. This completes the proof of the proposition. Now, we turn to the study of (2.1.10). First, we have Lemma 2.2.6 If r(·) ∈ L1loc (R) solves (2.1.10) in R, then r(·) ∈ L∞ (R) ∩ C ∞ (R). Proof. First, we may assume a > 0. (If a < 0, then we may define re(x) = −r(−x)). We define [ ] ∫ x v(x) := exp µx + r(y)dy 0. with µ := a0 /a. Since −ar(x) =. (2.2.6). 2 ∑. a|i| e−iµ. i=−2. v(x + i) , v(x). we get (2.2.7). ′. −av (x) = v(x)[−ar(x) − a0 ] = v(x). [∫. ∑. a|i| exp. i∈{±2,±1}. ]. x+i. r(y)dy > 0. x. This implies that v ′ (x) < 0 for all x ∈ R and so v(∞) exists and v(∞) ≥ 0. By integrating (2.2.7) over [x, M ], we get [∫ t+i ] ∫ M ∫ M ∑ ∑ −iµ v(t+i)dt. av(x)−av(M ) = a|i| v(t) exp r(y)dy dt = a|i| e i∈{±1,±2}. x. Sending M → ∞, we get. t. ∫. x+1/2. eµ v(t − 1)dt ≥. av(x) − av(∞) > a1 x. 16. i∈{±1,±2}. a1 µ e v(x − 1/2). 2. x.

(23) Therefore, we obtain that v(x − 1/2) ≤. 2a −µ e v(x), ∀x ∈ R. a1. It follows from (2.2.6) and the fact that v is non-increasing, we conclude that r(·) ∈ L∞ (R). Furthermore, r(·) ∈ C ∞ (R) by using (2.1.10). This proves the lemma. Lemma 2.2.7 A locally integrable solution of (2.1.10) that attains its global maximum or minimum must be a constant function. Proof. Let r be a locally integrable solution of (2.1.10). By differentiating (2.1.10), we get ′. ar (x) +. 2 ∑. [∫. ]. x+i. r(y)dy · [r(x + i) − r(x)] = 0.. a|i| exp x. i=−2. If we define. [∫. ]. x+1. r(y)dy ,. A(x) := exp x. then the above equality can be re-written as (2.2.8) ar′ (x) + a2 A(x + 1)A(x)[r(x + 2) − r(x)] + a1 A(x)[r(x + 1) − r(x)] + a1 [A(x − 1)]−1 [r(x − 1) − r(x)] + a2 [A(x − 1)A(x − 2)]−1 [r(x − 2) − r(x)] = 0. Suppose that r attains its global maximum. Without loss of generality and by a translation, we may assume that r(·) attains its global maximum r∗ at x = 0. Hence r′ (0) = 0 and r(±2) = r(±1) = r∗ . By induction, we easily deduce that r(j) = r∗ and r′ (j) = 0 for all j ∈ Z. By (2.1.10), we get a2 A(j + 1)A(j) + a1 A(j) + a1 /A(j − 1) + a2 /[A(j − 1)A(j − 2)] = −ar∗ − a0 for all j ∈ Z. Proposition 2.2.1 implies that A± := limj→±∞ A(j) exists such that (2.2.9). a2 (A± )2 + a1 (A± ) + a1 /(A± ) + a2 /[(A± )2 ] = −ar∗ − a0 .. Now, let {xi } be a sequence with lim r(xi ) = r∗ := inf {r(x)}.. i→∞. x∈R. We claim that r∗ = r∗ . By Lemma 2.2.6, {r(xi + ·)}∞ i=1 is a uniformly bounded and equicontinuous sequence. According to Ascoli-Arzela Theorem, we can extract a subsequence 17.

(24) (still called {r(xi +·)}∞ ˆ(·) uniformly in any compact subset i=1 ) such that limi→∞ r(xi +·) = r of R for some rˆ(·) ∈ C(R). It is easy to check that rˆ(·) is also a solution of (2.1.10) and rˆ(0) = r∗ is a global minimum of rˆ(·). A similar argument as above, we can get ˆ + 1)A(j) ˆ + a1 A(j) ˆ + −ar∗ − a0 = a2 A(j. a1 a2 + , ˆ ˆ ˆ − 2) A(j − 1) A(j − 1)A(j. ∀j ∈ Z.. Moreover, we also have a1 a2 −ar∗ − a0 = a2 (Aˆ± )2 + a1 (Aˆ± ) + , + ± ˆ ˆ (A± )2 A. (2.2.10). ∫ x+1 ˆ ˆ where A(x) := exp[ x rˆ(y)dy] and Aˆ± := limj→±∞ A(j). Finally, without loss of generality (by taking a subsequence if necessary), we may assume that xi ≥ −M for some positive constant M . Since ∫ j+1 r(y)dy = ln[A(j)] → ln[A+ ] as j → ∞, j. we have 1 n Therefore. ∫. x+n. r(y)dy → ln[A+ ] as n → ∞ uniformly in x ∈ [−M, ∞). x. ∫ ∫ 1 xi +n 1 xi +n ln[A ] = lim lim r(y)dy = lim lim r(y)dy n→∞ i→∞ n x i→∞ n→∞ n x i i ∫ n ∫ 1 n 1 lim r(xi + y)dy = lim rˆ(y)dy = ln[Aˆ+ ]. = lim n→∞ n 0 n→∞ n i→∞ 0 +. This implies that A+ = Aˆ+ . So by (2.2.9) and (2.2.10), we have r∗ = r∗ . Therefore r(·) is a constant function. The case when r(·) attains its global minimum can be also treated similarly. The lemma is proved. Lemma 2.2.8 If r(·) ∈ L1loc (R) solves (2.1.10), then r(±∞) := limx→±∞ r(x) exists and P (a, a2 , a1 , a0 , r(±∞)) = 0. Proof. First, we consider {xi } such that lim xi = ∞,. i→∞. lim r(xi ) = r∗ := lim sup r(x).. i→∞. x→∞. We claim that (2.2.11). lim. max {| r(·) − r∗ |} = 0.. i→∞ [xi −2,xi +2]. 18.

(25) Since the family {r(xi + ·)}∞ i=1 is uniformly bounded and equi-continuous, we can choose a subsequence (still denoted by {r(xi + ·)}) such that limi→∞ r(xi + ·) = rˆ(·) uniformly in any compact subset of R for some rˆ(·) ∈ C(R). This implies that rˆ(·) is also a solution of (2.1.10). By rˆ(0) = limi→∞ r(xi ) = r∗ and rˆ(y) = limi→∞ r(xi + y) ≤ r∗ for all y ∈ R, we have rˆ(0) = maxx∈R {ˆ r(x)}. By Lemma 2.2.7, we get rˆ(·) ≡ r∗ . Then (2.2.11) follows form the uniform convergence of {r(xi + ·)}. Next, we claim that r(∞) exists. Otherwise, we have r∗ := lim inf x→∞ r(x) < r∗ . Since lim. max {| r(·) − r∗ |} = 0, lim inf r(x) = r∗ < r∗ ,. i→∞ [xi −2,xi +2]. x→∞. we can find j ∈ N such that r ∗ + r∗ r(·) > in [xj , xj + 2] ∪ [xj+1 − 2, xj+1 ], 2. r∗ + r∗ min {r(·)} < . [xj ,xj+1 ] 2. Since r(·) is continuous, we can choose xˆ be the left-most point in [xj , xj+1 ] such that r(ˆ x) = min[xj ,xj+1 ] {r(·)}. From the above fact, we have xˆ ∈ [xj + 2, xj+1 − 2] such that r′ (ˆ x) = 0, r(ˆ x) ≤ min{r(ˆ x + 2), r(ˆ x + 1)}, r(ˆ x) < min{r(ˆ x − 1), r(ˆ x − 2)}. This leads a contradiction with (2.2.8). Therefore, r(∞) exists. Similarly, r(−∞) := limx→−∞ r(x) exists. Finally, by sending x → ±∞ in (2.1.10), we get P (a, a2 , a1 , a0 , r(±∞)) = 0. Then the lemma follows. Lemma 2.2.9 If r(·) ∈ L1loc (R) is a non-constant solution of (2.1.10), then r(−∞) < r(x) < r(∞) for all x ∈ R and ∫ ∞ ∫ [r(∞) − r(y)]dy + 0. 0 −∞. [r(y) − r(−∞)]dy < ∞.. Proof. According to Lemma 2.2.7, r(x) cannot attain its global maximum or minimum. This implies that r(∞) ̸= r(−∞) and Λ1 := min{r(−∞), r(∞)} < r(x) < Λ2 := max{r(−∞), r(∞)}, ∀x ∈ R. By Lemma 2.2.8, we have P (a, a2 , a1 , a0 , Λ1 ) = P (a, a2 , a1 , a0 , Λ2 ) = 0. 19.

(26) By the graph of P (a, a2 , a1 , a0 , λ), we have ∂ ∂ P (a, a2 , a1 , a0 , Λ1 ) < 0 0 such that (2.2.12). a + 2a2 e2(Λ1 +ϵ) + a1 e(Λ1 +ϵ) − a1 e−(Λ1 +ϵ) − 2a2 e−2(Λ1 +ϵ) < 0,. (2.2.13). a + 2a2 e2(Λ2 −ϵ) + a1 e(Λ2 −ϵ) − a1 e−(Λ2 −ϵ) − 2a2 e−2(Λ2 −ϵ) > 0.. Suppose that r(∞) = Λ1 and r(−∞) = Λ2 . By translation, we may assume that r(x) < Λ1 + ϵ, if x ≥ −4. If we define l := minx∈[−4,4] r(x), then we have l ∈ (Λ1 , Λ1 + ϵ). Compare the line h = l − kx with the curve h = r(x) for x > 0. If k ≥ ϵ and 0 < x ≤ 4, then r(x) ≥ l > l − kx. If k ≥ ϵ and x > 4, then r(x) > Λ1 > l − ϵ > l − kx. Combining these two cases, we get that r(x) > l − kx, ∀ x > 0, k ≥ ϵ. Therefore, the quantity δ := inf{k > 0 | r(x) ≥ l − kx, ∀x > 0} is well-defined and we can easily see that δ ∈ (0, ϵ). Moreover, there is a number x0 ∈ (4, ∞) such that r(x0 ) − (l − δx0 ) = 0 ≤ r(x) − (l − δx), ∀x > 0. This implies that r′ (x0 ) = −δ, r(x0 ± i) − r(x0 ) ≥ ∓iδ for i = 1, 2. Recall A(x) = exp[. ∫ x+1 x. r(y)dy] and using Λ1 < r(·) < Λ1 + ϵ in [0, ∞), we have. aδ = a2 A(x0 + 1)A(x0 )[r(x0 + 2) − r(x0 )] + a1 A(x0 )[r(x0 + 1) − r(x0 )] + a1 A(x0 − 1)−1 [r(x0 − 1) − r(x0 )] + a2 A(x0 − 1)−1 A(x0 − 2)−1 [r(x0 − 2) − r(x0 )] ≥ δ[−2a2 e2(Λ1 +ϵ) − a1 e(Λ1 +ϵ) + a1 e−(Λ1 +ϵ) + 2a2 e−2(Λ1 +ϵ) ]. This leads to a contradiction with (2.2.12). Therefore, r(∞) = Λ2 > r(−∞) = Λ1 . 20.

(27) Finally, we claim that ∫. 0. −∞. [r(y) − Λ1 ]dy < ∞.. Let R(x) := r(x) − Λ1 . For the ϵ above, there exists xϵ ∈ R such that r(·) ≤ Λ1 + ϵ on (−∞, xϵ + 4]. By a direct computation and the Mean Value Theorem, we have ∫ x+i ∑ a|i| exp[sgn{i}θi (x)] R(y)dy = 0, x ∈ R, (2.2.14) aR(x) + x. i∈{±1,±2}. where θ±2 (x) ∈ [2Λ1 , 2 max {r(·)}],. θ±1 (x) ∈ [Λ1 , max {r(·)}].. [x−2,x+2]. Integrating (2.2.14) over [−M, xϵ ], we get ∫ ∫ xϵ ∑ a|i| 0 = a R(x)dx + −M. ∫. i∈{±1,±2}. ∑. xϵ. = a. R(y)dy + −M. ∫. xϵ. = −M. {a +. ∑. ∫ a|i|. xϵ. −M xϵ. ∫. x+i. exp[sgn{i}θi (x)]R(y)dydx ∫. x y. exp[sgn{i}θi (x)]R(y)dxdy + O(1). −M. i∈{±1,±2} ∫ y. a|i|. [x−1,x+1]. y−i. exp[sgn{i}θi (x)]dx}R(y)dy + O(1), y−i. i∈{±1,±2}. where O(1) is uniformly bounded (independent of M ). According to the range of θi (x) and r(·) ≤ Λ1 + ϵ on (−∞, xϵ + 4], we have ∫ y ∑ a+ a|i| exp[sgn{i}θi (x)]dx y−i. i∈{±1,±2}. < a + 2a2 e2(Λ1 +ϵ) + a1 e(Λ1 +ϵ) − a1 e−(Λ1 +ϵ) − 2a2 e−2(Λ1 +ϵ) < 0. By taking M → ∞, we have −2(Λ1 +ϵ). ≤ Therefore,. [2a2 e  ∫ xϵ . ∫ xϵ −∞. −∞. . −a −. + a1 e. −(Λ1 +ϵ). ∑ i∈{±1,±2}. ∫ − a1 e. − 2a2 e. 2(Λ1 +ϵ). − a]. xϵ. R(y)dy −∞.   exp[sgn{i}θi (x)]dx R(y)dy = O(1).  y−i. ∫ a|i|. (Λ1 +ϵ). y. R(y)dy < ∞. We conclude that ∫ 0 [r(y) − Λ1 ]dy < ∞. −∞. 21.

(28) Similarly, using (2.2.13) we can deduce that ∫ ∞ [Λ2 − r(y)]dy < ∞. 0. The proof is now completed. Proof of Theorem 2.1.1. First, parts (i) and (ii) follows from Lemma 2.2.8 and Lemma 2.2.9. Next, we focus on the case that P (a, a2 , a1 , a0 , ·) = 0 has two real roots {Λ1 , Λ2 } with Λ1 < Λ2 . Suppose that r(x) is an arbitrary non-constant solution of (2.1.10). By Lemma 2.2.8 and Lemma 2.2.9, we have r(∞) = Λ2 > Λ1 = r(−∞). We define [∫ x ] u(x) = exp r(y)dy , 0 [∫ −∞ ] Λ1 x u1 (x) = θe , θ = exp (r(y) − Λ1 )dy , 0. u2 (x) = u(x) − u1 (x). By a direct computation, we have u′ (x) = u(x)r(x), u2 (0) = 1 − θ, 0 < θ < 1. Since −Λ1 x. u2 (x)e. [∫. ]. x. [∫. (r(y) − Λ1 )dy − exp. = exp 0. −∞. ] (r(y) − Λ1 )dy ,. 0. we obtain that u2 (x)e−Λ1 x > 0 for all x ∈ R and u2 (x)e−Λ1 x → 0 as x → −∞. Now we define rˆ(x) :=. u′2 (x) . u2 (x). It is easy to see that rˆ(x) is a solution of (2.1.10). We claim that rˆ(x) is a constant function. Suppose not. Then it follows from Lemma 2.2.9 that ∫ 0 [ˆ r(y) − Λ1 ]dy < ∞. −∞. But, ln[u2 (x)e. −Λ1 x. ∫ ] − ln[u2 (0)] = 0. x. (u2 (y)e−Λ1 y )′ dy = (u2 (y)e−Λ1 y ). By taking x → −∞, we have lim ln[u2 (x)e−Λ1 x ] < ∞,. x→−∞. 22. ∫. x. [ˆ r(y) − Λ1 ]dy. 0.

(29) a contradiction. Therefore, rˆ(x) must be a constant solution of (2.1.10). This also implies that either rˆ(x) ≡ Λ1 or rˆ(x) ≡ Λ2 . If rˆ(x) ≡ Λ1 , then u(x) = ceΛ1 x . This implies that r(·) is a constant function, a contradiction. Hence we have rˆ(x) ≡ Λ2 and so u2 (x) = u2 (0)eΛ2 x = (1−θ)eΛ2 x . Therefore, by adding two constant solutions r(x) ≡ Λ1 and r(x) ≡ Λ2 together, all solutions of (2.1.10) are given by r(x) =. θΛ1 eΛ1 x + (1 − θ)Λ2 eΛ2 x θeΛ1 x + (1 − θ)eΛ2 x. for θ ∈ [0, 1]. This proves the theorem.. 2.3. Asymptotic behavior. Let (c, U ) be a solution of (P). This section is devoted to the proof of Theorem 2.1.2, the asymptotic behavior of U near x = ±∞. Proof of Theorem 2.1.2. To study the behavior at x = ∞, we define ρ(x) := U ′ (x)/U (x). First, we claim that limx→∞ ρ(x) exists. Suppose not. Then we have that λ := lim sup ρ(x) > λ := lim inf ρ(x). x→∞. x→∞. By the definition of ρ(x), (2.1.6) can be re-written as ] [∫ x+2 ] [ b(U (x + 2)) (2.3.1) exp ρ(s)ds cρ(x) + J(2) U (x + 2) x [ ] [∫ x+1 ] b(U (x + 1)) + D + J(1) exp ρ(s)ds U (x + 1) x [ ] [∫ x−1 ] b(U (x − 1)) + D + J(1) ρ(s)ds exp U (x − 1) x [ ] [∫ x−2 ] b(U (x − 2)) + J(2) exp ρ(s)ds U (x − 2) x [ ] b(U (x)) + −2D − d + J(0) = 0. U (x) Choose λ1 , λ2 such that λ < λ1 < λ2 < λ and choose λ ∈ (λ1 , λ2 ) such that P (a, a2 , a1 , a0 , λ) ̸= 0. Let {ηi1 } → ∞ and {ηi2 } → ∞ as i → ∞ such that ρ(ηi1 ) = λ1 , ρ(ηi2 ) = λ2 and ηi2 < ηi1 for all i ∈ N. Since ρ(·) is a uniformly continuous function, we can choose that ξi to be the right-most point in (ηi2 , ηi1 ) such that ρ(ξi ) = λ. Since ρ(ξi ) − ρ(ηi1 ) = λ − λ1 > 0, we get ρ(·) ≤ λ on [ξi , ηi1 ] and δ := inf {ηi1 − ξi } > 0. i∈N. 23.

(30) By Lemma 2.2.1, {ρ(ξi +·)}∞ i=1 is uniformly bounded and equi-continuous. By Ascoli-Arzela Theorem, we can extract a subsequence (still denote ρ(ξi + ·)) such that ρ(ξi + ·) → r(·) uniformly in any compact subset of R for some function r ∈ C(R). Therefore, r(0) = lim ρ(ξi ) = λ, r(·) ≤ λ on [0, δ] i→∞. and r(x) is a solution of the equation (2.1.10) with the coefficients a = c, a2 = a−2 = J(2)b′ (0), a1 = a−1 = D + J(1)b′ (0), a0 = −2D − d + J(0)b′ (0). But, this contradicts Theorem 2.1.1, since all non-constant solutions of (2.1.10) are strictly increasing. Therefore, the limit Λ := limx→∞ ρ(x) exists. By taking x → ∞ in (2.3.1), we see that Λ is a root of (2.1.9). Since b′ (0) > d, we have Λ < 0. On the other hand, by the same argument as above and using Lemma 2.2.2, we can also prove that the limit U ′ (x) x→−∞ U (x) − 1. σ := lim. exists and satisfies (2.1.11). Note that σ ̸= 0, since b′ (1) < d. Indeed, (2.1.11) has a unique positive root and a unique negative root. Finally, it remains to determine the sign of σ. If σ < 0, then there exists M > 0 such that U ′ (x) > 0 if x ≤ −M . This contradicts U (−∞) = 1. Thus σ > 0 which is the unique positive root of the characteristic equation (2.1.11). This completes the proof of the theorem.. 2.4. Monotonicity and the proof of Theorem 2.1.4. In this section, we shall prove Theorems 2.1.3 and 2.1.4. Let (c, U ) be a solution of (P). For the notational convenience, we define W [U ](y) := cµU (y) + D2 [U ](y) − dU (y) + eµx T [U ](x) := c. ∫. 2 ∑. J(i)b(U (y − i)),. i=−2 ∞. e−µy W [U ](y)dy.. x. where µ is a constant satisfying µ ≥ (d + 2D)/c. Note that T [U ](x) = U (x) for all x ∈ R. We now prove the following strong comparison principle. Lemma 2.4.1 If (c, U1 ), (c, U2 ) are solutions of (P) and satisfy U1 ≤ U2 on R, then either U1 ≡ U2 or U1 < U2 on R. 24.

(31) Proof. Suppose that U1 < U2 on R does not hold. Then we can find x0 ∈ R such that U1 (x0 ) = U2 (x0 ). Since U1 (·), U2 (·) are solution of (P), we have T [U1 ](x0 ) = T [U2 ](x0 ). By the definition of T [U ](x), we can get ∫ ∞ e−µy {W [U1 ] − W [U2 ]}(y)dy = 0. x0. Since of W [U1 ] ≤ W [U2 ] on R, we have W [U1 ] ≡ W [U2 ] on [x0 , ∞). By the definition of W [U ] and the monotonicity of b, we get that U1 (y −1) = U2 (y −1) and U1 (y −2) = U2 (y −2) for y ∈ [x0 , ∞). Hence U1 ≡ U2 on [x0 − 2, ∞). Continuing in this way, we conclude that U1 ≡ U2 on R. The lemma is proved. Next, we use the sliding method to prove the following lemma. Lemma 2.4.2 If (c, U ) is a solution of (P) such that U ′ (x) ≤ 0 for all |x| ≫ 1, then U ′ (x) < 0 on R. Proof . By hypothesis, we can find M > 0 such that U ′ (x) ≤ 0 on R \ (−M, M ). Since U (∞) = 0, U (−∞) = 1 and U (·) is continuous, the set A := {ξ > 0 | U (x + η) ≤ U (x), ∀x ∈ R, ∀η ≥ ξ} is nonempty. Hence ξ ∗ := inf A is well-defined. Note that U (x + ξ ∗ ) ≤ U (x) for all x ∈ R. We claim that ξ ∗ = 0. Suppose ξ ∗ > 0. Since U (∞) ̸= U (−∞), by Lemma 2.4.1, we have U (x + ξ ∗ ) < U (x), ∀x ∈ R. Since U (x) is a continuous function, we can choose ϵ ∈ (0, ξ ∗ ) such that U (x + η) < U (x), if x ∈ [−M − 2ξ ∗ , M + 2ξ ∗ ] and η ∈ [ξ ∗ − ϵ, ξ ∗ ]. For x ∈ R \ (−M − ξ ∗ , M + ξ ∗ ) and η ∈ (0, ξ ∗ ], by the Mean Value Theorem, we obtain U (x + η) − U (x) = ηU ′ (f (x, η)), where x < f (x, η) < x + η. Hence U (x + η) ≤ U (x) for all x ∈ R \ (−M − ξ ∗ , M + ξ ∗ ). We conclude that U (x + η) ≤ U (x), ∀x ∈ R, ∀η ∈ [ξ ∗ − ϵ, ξ ∗ ]. But this contradicts the definition of ξ ∗ . Therefore, ξ ∗ = 0 and we have U ′ (x) ≤ 0 for all x ∈ R. Finally, differentiating U = T [U ], we get that U ′ (·) < 0 on R. The proof is completed. 25.

(32) Proof of Theorem 2.1.3. Let (c, U ) be a solution of (P). By Theorem 2.1.2, we have U ′ (x) U ′ (x) = Λ(< 0) and lim = σ(> 0). x→∞ U (x) x→−∞ U (x) − 1 lim. This implies that U ′ (x) < 0, if |x| ≫ 1. By Lemma 2.4.2, we have U ′ (x) < 0 for any x ∈ R. The theorem is proved. Now, we turn to the proof of Theorem 2.1.4. Proof of Theorem 2.1.4. Let c > cmin . Then the characteristic equation (2.1.9) always has two negative roots, denoted by λ(c) < Λ(c) < 0. We claim that U ′ (x) = Λ(c). x→∞ U (x) lim. Suppose on the contrary that. U ′ (x) = λ(c). x→∞ U (x) lim. Choose cˆ ∈ (cmin , c) and (ˆ c, Uˆ (x)) a solution of (P). Then, by Theorem 2.1.2, we have Uˆ ′ (x) ≥ λ(ˆ c). ˆ (x) x→∞ U lim. So we obtain lim [ln(Uˆ (x)/U (x))]′ = lim {[Uˆ ′ (x)/Uˆ (x)] − [U ′ (x)/U (x)]} ≥ λ(ˆ c) − λ(c) > 0.. x→∞. x→∞. This implies that limx→∞ ln[Uˆ (x)/U (x)] = +∞. Therefore, there exists a positive number M such that Uˆ (x) > U (x) ∀ x ≥ M.. (2.4.1) On the other hand, from. lim [U ′ (x)/(U (x) − 1)] = σ(c) > 0,. x→−∞. it follows that. [∫. ˆ (x) U. lim. x→−∞. [. U (x). 1 ds b(s) − ds. ]′. Uˆ ′ (x). Uˆ (x) − 1 U ′ (x) U (x) − 1 = lim · − · ˆ (x) − 1 b(Uˆ (x)) − dUˆ (x) U (x) − 1 b(U (x)) − dU (x) x→−∞ U σ(ˆ c) − σ(c) < 0. = b′ (1) − d 26. ].

(33) This implies that there exists M1 > 0 such that Uˆ (x) > U (x) ∀ x ≤ −M1 .. (2.4.2). By (2.4.1), (2.4.2) and Theorem 2.1.3, we obtain that Uˆ (x − M1 ) > U (x + M ), ∀x ∈ R.. (2.4.3). Note that both u1 (x, t) := Uˆ (x − M1 − cˆt) and u2 (x, t) := U (x + M − ct) are solutions of the following spatially continuous version of (2.1.1): ut (x, t) = D[u(x + 1, t) + u(x − 1, t) − 2u(x, t)] − du(x, t) +. ∑. J(i)b(u(x − i, t)).. i∈Z. By (2.4.3), we have u1 (·, 0) ≥ u2 (·, 0). The comparison principle implies that u1 (·, t) ≥ u2 (·, t) for all t ≥ 0. Writing u1 (x, t) ≥ u2 (x, t) by (2.4.4). Uˆ (x − (c + cˆ)t/2 − M1 + (c − cˆ)t/2) ≥ U (x − (c + cˆ)t/2 + M − (c − cˆ)t/2). fixing ξ := x − (c + cˆ)t/2 and letting t → ∞ in (2.4.4), it follows from c > cˆ that 0 = Uˆ (∞) ≥ U (−∞) = 1, a contradiction to (2.1.7). Therefore, the proof is completed.. 2.5. Uniqueness. This section is devoted to the uniqueness of the traveling wave solution. We shall follow the method developed in [5]. For a smooth function ϕ, we let L[ϕ](x) := −cϕ′ (x) − D2 [ϕ](x) −. 2 ∑. J(i)[b(ϕ(x − i)) − dϕ(x)].. i=−2. First, we define the notion of super-sub-solutions as follows. Definition 2.5.1 A non-constant smooth function ϕ : [a − 2, b + 2] → (0, 1) is called a supersolution (subsolution, resp.) of (2.1.6) on [a, b] for a wave speed c, if L[ϕ](x) ≥ 0 (L[ϕ](x) ≤ 0, resp.) for x ∈ (a, b). Definition 2.5.2 A non-constant smooth function ϕ : [a − 2, ∞) → (0, 1) is called a supersolution (subsolution, resp.) of (2.1.6) on [a, ∞) for a wave speed c, if L[ϕ](x) ≥ 0 (L[ϕ](x) ≤ 0, resp.) for x ∈ (a, ∞). 27.

(34) Lemma 2.5.1 Assume (H1). Let (c, U ) be a solution of (P) and V (x) be a subsolution (supersolution, resp.) of (2.1.6) on [a, b] for the same speed c, where a < b. If V (x) U (x), resp.) for x ∈ [a − 2, a) ∪ (b, b + 2], then V (x) U (x), resp.) for x ∈ [a, b]. Proof . Since the case for supersolution is similar, we only consider the case when V (x) is a subsolution. We introduce g(t) :=. max x∈[a−2,b+2]. {V (x) − U (x − t)}.. Since U (∞) = 0 and U (−∞) = 1, we can choose ζ ∈ R such that g(ζ) = 0. Let y ∈ [a − 2, b + 2] be the maximum value in [a − 2, b + 2] such that V (y) − U (y − ζ) = 0. We claim that y ∈ [a − 2, a) ∪ (b, b + 2]. Suppose on the contrary that y ∈ [a, b]. Then we have V (y) = U (y − ζ), V ′ (y) = U ′ (y − ζ), V (y − 1) ≤ U (y − 1 − ζ), V (y − 2) ≤ U (y − 2 − ζ), V (y + 1) L[U ](y − ζ). By the strictly inequality, without loss of generality we may assume that y ∈ (a, b). This contradicts that U (x) is a solution of (2.1.6) and V (x) is a subsolution of (2.1.6) on [a, b]. Therefore, y ∈ [a − 2, a) ∪ (b, b + 2]. By hypothesis, we have U (y) > V (y) = U (y − ζ). It follows from the monotonicity of U that ζ < 0. Hence U (x − ζ) < U (x) for all x ∈ R. Since g(ζ) = 0, we deduce that V (x) < U (x) for all x ∈ [a, b]. The proof is completed. Lemma 2.5.2 Assume (H1). Let (c, U ) be a solution of (P) and ϕ(x) be a subsolution (or supersolution) of (2.1.6) with the same speed c on [a, ∞) for some constant a. If ϕ′ (x) U ′ (x) = lim = Λ, x→∞ ϕ(x) x→∞ U (x) lim. then there exists A ∈ [−∞, ∞] such that lim W (ξ, x) = A + Λξ, ∀ξ ∈ R,. x→∞. where W (ξ, x) := ln[U (x + ξ)] − ln[ϕ(x)]. Proof. Given a subsolution ϕ(x) of (2.1.6) on [a, ∞) for some constant a. By the definition of W (ξ, x), we obtain ∫ x+ξ ′ U (t) (2.5.1) lim [W (ξ, x)−W (0, x)] = lim {ln[U (x+ξ)]−ln[U (x)]} = lim dt = Λξ x→∞ x→∞ x→∞ x U (t) 28.

(35) for all ξ ∈ R. It follows from (2.5.1) that either the limit limx→∞ W (ξ, x) exists for all ξ ∈ R or it does not exist for all ξ ∈ R. Suppose that the limit limx→∞ W (ξ, x) does not exist for all ξ ∈ R. By (2.5.1), we can choose an appropriate ξ such that A := lim sup W (ξ, x) > 0 > B := lim inf W (ξ, x). x→∞. x→∞. Indeed, ξ can be chosen as ξ = −[lim sup W (0, x) + lim inf W (0, x)]/(2Λ). x→∞. x→∞. Take α, β such that B < β < 0 < α < A. Then we may choose two sequences {xi } and {yi } such that lim xi = ∞, a − 2 ≤ xi < yi < xi+1 , W (ξ, xi ) = α, W (ξ, yi ) = β, ∀i ∈ N.. i→∞. Since. { lim Wx (ξ, x) = lim. x→∞. x→∞. U ′ (x + ξ) ϕ′ (x) − U (x + ξ) ϕ(x). } = 0,. we can choose a fixed integer i large enough such that W (ξ, x) > 0, if x ∈ [xi − 2, xi ] ∪ [xi+1 , xi+1 + 2], i.e., U (x + ξ) > ϕ(x) for all x ∈ [xi − 2, xi ] ∪ [xi+1 , xi+1 + 2]. Since W (ξ, yi ) = β < 0, we obtain that U (yi + ξ) < ϕ(yi ). But, by Lemma 2.5.1, it is impossible, since yi is between xi and xi+1 . Therefore, the limit limx→∞ W (ξ, x) exists for all ξ ∈ R. We conclude that limx→∞ W (ξ, x) = A + Λξ for all ξ ∈ R, where A := limx→∞ W (0, x). The case when ϕ(x) is a supersolution is similar, the lemma follows. With this lemma, we are ready to prove Theorem 2.1.5. Proof of Theorem 2.1.5. Suppose that the roots of Φ(· ; c) = 0 are given by Λ and λ with Λ > λ. We choose ω < 0 such that max{λ, (1 + α)Λ} < ω < Λ, where the constant α is defined in (H2). Since Φ(λ; c) is a convex function in λ, we have Φ(ω; c) < 0. Following [5], we define ϕ± (x; ϵ, δ) := δ((1 ∓ ϵ)eΛx ± ϵeωx ), where δ > 0, ϵ ∈ (0, eω ], x ≥ −2. We may easily check (2.5.2). 0 < ϕ± (x; ϵ, δ) < 2δeΛx , if δ > 0, ϵ ∈ (0, eω ], x ≥ 0. 29.

(36) By a simple computation, we have ϕ′± Λ(1 ∓ ϵ)eΛx ± ωϵeωx ±(ω − Λ)ϵ = = Λ + . ϕ± (1 ∓ ϵ)eΛx ± ϵeωx (1 ∓ ϵ)e(Λ−ω)x ± ϵ It follows that (2.5.3). ϕ′+ = Λ + ϵ(ω − Λ), −2≤x≤0 ϕ+ max. ϕ′− ϕ′ = Λ − ϵ(ω − Λ), lim ± = Λ. x→∞ ϕ± −2≤x≤0 ϕ− min. Next, we compute L[ϕ+ ](x) = −{δ(1 − ϵ)eΛx [cΛ + D(eΛ + e−Λ − 2) − d] + δϵeωx [cω + D(eω + e−ω − 2) − d] 2 ∑ + J(i)b(ϕ+ (x + i))} i=−2. = −{δ(1 − ϵ)e Φ(Λ; c) + δϵe Φ(ω; c) + Λx. ωx. 2 ∑. J(i)[b(ϕ+ (x + i)) − b′ (0)ϕ+ (x + i)]}. i=−2. = −δϵeωx Φ(ω; c) −. 2 ∑. J(i)[b(ϕ+ (x + i)) − b′ (0)ϕ+ (x + i)].. i=−2. On the other hand, by (2.5.2), we have 0 < ϕ+ (x + i; ϵ, δ) < 2δeΛ(x+i) ≤ 2δe−2Λ eΛx , if x ≥ 2, δ > 0, ϵ ∈ (0, eω ], i ∈ {±1, ±2, 0}. Hence, if 2δe−2Λ eΛx ≤ 1, then, by (H2), we have L[ϕ+ ](x) ≥ −δϵe Φ(ω; c) − M ωx. 2 ∑. J(i)[ϕ+ (x + i)]1+α. i=−2 −2Λ Λx 1+α. ≥ −δϵe Φ(ω; c) − M (2δe ωx. e ). ≥ δeωx {−ϵΦ(ω; c) − 21+α δ α M e−2Λ(1+α) e[(1+α)Λ−ω]x }. Therefore, we can easily deduce the following facts. (a1) For every ϵ ∈ (0, eω ], there exists δϵ > 0 such that ϕ+ (x; ϵ, δ) is a supersolution on [2, ∞), if δ ∈ (0, δϵ ]. (a2) For every ϵ ∈ (0, eω ] and δ = 1, there exists xϵ ≥ 0 such that ϕ+ (x; ϵ, 1) is a supersolution on [xϵ , ∞). 30.

(37) Now, we consider ϕ(x) := ϕ+ (x; ϵ, δ) with ϵ = eω and δ = 1. By Lemma 2.5.2, there exists A ∈ [−∞, ∞] such that lim {ln[U (x + ξ)] − ln[ϕ(x)]} = A + Λξ, ∀ξ ∈ R.. x→∞. We claim that A > −∞. Suppose not. Then we have (2.5.4). lim {ln[U (x + ξ)] − ln[ϕ(x)]} = −∞, ∀ξ ∈ R.. x→∞. Fix ϵ = eω > 0, let δϵ be the constant defined in (a1). Then it follows from Theorem 2.1.2 and U (∞) = 0 that there exists η > 0 such that U (η) < δϵ and (2.5.5). U ′ (x) > Λ + ϵ(ω − Λ), if x ≥ η − 2. U (x). ˆ By the fact (a1), the function ϕ(x) := ϕ+ (x; ϵ, U (η)) is a supersolution on [2, ∞). ˆ Note that ϕ(0) = ϕ+ (0; ϵ, U (η)) = U (η). Also, from (2.5.3) and (2.5.5) it follows that ϕˆ′ (x) U ′ (x + η) ≤ Λ + ϵ(ω − Λ) < , ∀x ∈ [−2, 0]. ˆ U (x + η) ϕ(x) ˆ By an integration, we obtain ϕ(x) > U (x + η), if x ∈ [−2, 0). On the other hand, since ˆ [ϕ(x)/ϕ(x)] ≡ U (η), it follows from (2.5.4) that ˆ lim {ln[U (x + η)] − ln[ϕ(x)]} = −∞.. x→∞. ˆ So we can choose T > 0 such that ϕ(x) > U (x+η), ∀x ∈ [T, ∞). Therefore, by Lemma 2.5.1, ˆ ˆ = U (η). Therefore, we we have ϕ(x) > U (x + η) for all x ∈ [−2, ∞). This contradicts ϕ(0) conclude that A > −∞. Thus [ ] U (x) lim ln = lim {ln[U (x)] − ln[ϕ(x)]} = A. x→∞ x→∞ ϕ(x) It follows from the definition of ϕ(x) that the limit L := limx→∞ [U (x)e−Λx ] exists and L > 0. Moreover, using the function ϕ− , (2.5.3) and (H2), we can prove, by a similar reasoning as above, that L < ∞. This proves the theorem.. 31.

(38) 32.

(39) Chapter 3 Two-front entire solutions 3.1. Introduction. In this chapter, we study the following discrete diffusive equation with convolution type nonlinearity. ∑ (3.1.1) ut (x, t) = D2 [u](x, t) − du(x, t) + J(i)b(u(x − i, t)), x ∈ R, t ∈ R, where d > 0, J(i) = J(−i) ≥ 0,. i∈Z. ∑ i∈Z. J(i) = 1, and. D2 [u](x, t) := D[u(x + 1, t) + u(x − 1, t) − 2u(x, t)] for some positive constant D. Throughout this chapter, we shall always assume that the function b(·) is an increasing smooth function on [0, 1] such that (P1) b(0) = b(a) − ad = b(1) − d = 0, where 0 < a < 1, (P2) b(t) < dt for 0 < t < a, b(t) > dt for a < t < 1, (P3) max{b′ (0), b′ (1)} < d < b′ (a) (bistable nonlinearity), ∫1 (P4) 0 [b(u) − du]du > 0 (unbalanced case). When J(0) = 1 and J(i) = 0 for all i ̸= 0, (3.1.1) is reduced to the classical equation ut (x, t) = D2 [u](x, t) + f (u(x, t)),. f (u) := b(u) − du,. which has been studied recently in [17, 18]. We also note that (3.1.1) is the continuum version of the following lattice dynamical system: ∑ J(i)b(un−i (t)), n ∈ Z, t ∈ R. (3.1.2) u′n (t) = D[un+1 (t) + un−1 (t) − 2un (t)] − dun (t) + i∈Z. 33.

(40) For (3.1.2), in ecology, un represents the population density at site n, D is the migration coefficient, d is the death rate and the nonlinear function b is the birth function of population density which is interacting with neighbors by the nonnegative weighted function J, if the habitat is divided into discrete regions and the population density is measured at the representative point in each region. In this model, we assume that the migration only happens to the nearest neighbors and the interaction happens with finite or infinite range. We say that {un (t)} is a traveling wavefront solution of (3.1.2) connecting two different equilibria {u± } ⊂ {0, a, 1} with speed c, if un (t) = U (n + ct) for n ∈ Z and t ∈ R for some function U (called wave profile) such that U (±∞) = u± . Then (c, U ) satisfies the following equation (3.1.3). cU ′ (y) = D2 [U ](y) − dU (y) +. ∑. J(i)b(U (y − i)),. y ∈ R,. i∈Z. where (as before) D2 [U ](x) := D[U (x + 1) + U (x − 1) − 2U (x)]. Similarly, we can define the notion of traveling wavefront solution of (3.1.1) by setting u(x, t) = U (x + ct), then U also satisfies the equation (3.1.3). Recently, a more general version of (3.1.2) including time delay was studied in [25, 23]. In [25], they studied (3.1.2) with time delay for the bistable case. They proved that the problem admits a unique (up to a translation) strictly monotone increasing traveling wavefront solution connecting from 0 to 1 with a positive wave speed when D ≥ D0 for a certain positive constant D0 , under the following extra assumption { ∫1 } ∫1 ∑ 2 0 [b(u) − du]du 2 0 [b(u) − du]du (3.1.4) J(i) < max , ∫1 . ∫1 b(u)du b(u)du − d i∈Z 0 0 More precisely, from [25, Theorem 1.1], under the above assumptions, there exist a unique speed cˆ > 0 and a unique (up to translations) wave profile U (x) such that { ∑ cÛ ′ (x) = D2 [U ](x) − dU (x) + i∈Z J(i)b(U (x − i)), x ∈ R, (3.1.5) U (−∞) = 0, U (+∞) = 1, 0 0 in R, if D ≥ D0 . Note that a propagation failure occurs when D is small enough. The monostable case for (3.1.2) with time delay was considered in [23]. In the present setting, it corresponding to the case for connecting two equilibria {a, 1} or {0, a}. They obtained the existence of the asymptotic speed of propagation, the existence and (partial) uniqueness of the traveling wavefront and the minimal speed of the traveling wavefront for 34.

(41) the delayed lattice dynamical system under the following extra condition at the unstable equilibrium a, namely, (3.1.6) b′ (a)(u − a) − M |u − a|1+α ≤ b(u) − da ≤ b′ (a)(u − a) + M |u − a|1+α for u ∈ [0, 1] for some constants M > 0 and α ∈ (0, 1]. In fact, by [23, Theorem 1.2], there exist two constants c∗ , c∗ with c∗ > 0 > c∗ such that for any c1 , c2 (with c1 ≥ c∗ , c2 ≤ c∗ ) there exist V1 (x) and W2 (x) satisfying the following equations: { ∑ c1 V1′ (x) = D2 [V1 ](x) − dV1 (x) + i∈Z J(i)b(V1 (x − i)), x ∈ R, (3.1.7) V1 (−∞) = a, V1 (+∞) = 1, a < V1 < 1, V1′ > 0 in R. and (3.1.8). {. ∑ c2 W2′ (x) = D2 [W2 ](x) − dW2 (x) + i∈Z J(i)b(W2 (x − i)), x ∈ R, W2 (−∞) = 0, W2 (+∞) = a, 0 < W2 < a, W2′ > 0 in R,. where c∗ (c∗ , resp.) is the minimal (maximal, resp.) speed of (3.1.7) ((3.1.8), respectively). The study of traveling wavefront solutions are important in many applications. It can describe certain dynamical behavior of the studied problem such as (3.1.2). But, the dynamics of reaction-diffusion equations or its discrete analogue is so rich that there might be other interesting patterns. Recently it is found that two-front entire solutions exist in many problems. Here an entire solution is meant by a solution defined for all (x, t) ∈ R2 . In particular, traveling wavefront solutions are also entire solutions. For the study of entire solutions, we refer the reader to, for instance, [14, 17, 18, 19, 20, 22, 26, 28] and reference therein. In a very interesting work by Morita and Ninomiya [26], they gave three different types of entire solutions for a bistable reaction-diffusion equation (see also [18] for the discrete diffusive case). The purpose of this work is to construct these three types of entire solutions for (3.1.1). Although the main idea and the methods of proofs in this part are from [18, 26], there are certain difficulties in dealing with (3.1.1) (or (3.1.2)) due to the convolution type nonlinearity. For example, in the construction of super/sub solutions, we need to derive some estimations. In these estimations, the compactness (finite range interaction) assumption is needed in this study. So, from now on, besides the assumptions (3.1.4) and (3.1.6), we shall assume that (3.1.9). J(i) = 0 for |i| > m for some m ∈ N.. In fact, to construct these two-front entire solutions it is crucial to have a precise information on the asymptotic behavior of wave tails. More precisely, we need the following estimates for solutions U, V1 , W2 of (3.1.5), (3.1.7), (3.1.8) respectively. 35.

(42) First, there exists a positive constant η such that (3.1.10). U ′ (y) ≥ η. y≥0 1 − U (y). U ′ (y) ≥ η, y≤0 U (y) inf. inf. Furthermore, there are positive constants K, k, γ, δ such that γe−µy ≤ 1 − U (y) ≤ δe−µy , ∀ y ≥ −m,. (3.1.11)keλy ≤ U (y) ≤ Keλy , ∀ y ≤ m;. where λ is the unique positive root of the characteristic equation (3.1.12). λ. −λ. cˆλ = D(e + e. m ∑. ′. − 2) − d + b (0). J(i)eiλ ,. i=−m. and µ is the unique positive root of the equation (3.1.13). −ˆ cµ = D(e + e µ. −µ. ′. − 2) − d + b (1). m ∑. J(i)eiµ .. i=−m. Next, for any c1 > c∗ and c2 < c∗ , let (c1 , V1 (x)) and (c2 , W2 (x)) be solutions of (3.1.7) and (3.1.8), respectively. Then there exist positive constants λi , µi , κi , γi , i = 1, 2, such that the following inequalities hold: (3.1.14). V1 (y) − a ≥ κ1 eλ1 y on (−∞, 0];. (3.1.15). W2 (y) ≥ κ2 eλ2 y on (−∞, 0];. 1 − V1 (y) ≥ γ1 e−µ1 y on [0, ∞). a − W2 (y) ≥ γ2 e−µ2 y on [0, ∞).. Furthermore, there exist positive constants N , ρ such that (3.1.16). ρ[V1 (y) − a] ≤ V1′ (y) ≤ N eλ1 y on (−∞, 0],. (3.1.17). ρ[1 − V1 (y)] ≤ V1′ (y) ≤ N e−µ1 y on [0, ∞),. (3.1.18). ρW2 (y) ≤ W2′ (y) ≤ N eλ2 y on (−∞, 0],. (3.1.19). ρ[a − W2 (y)] ≤ W2′ (y) ≤ N e−µ2 y on [0, ∞).. The above asymptotic behavior of wave tail at the unstable equilibrium can be found in [15]. But, due to the technical difficulty arising from the convolution type nonlinearity, we need to assume that m = 2. As for the wave tail at the stable equilibrium, the method developed in [9] is well applicable here for any finite m. Based on these asymptotic behaviors, we prove the following theorems on two-front entire solutions. 36.

(43) Theorem 3.1.1 Let (3.1.9) be in force with m = 2 and let (ˆ c, U (x)) be a solution of (3.1.5). Then for any real number θ there exists an entire solution u(x, t) of (3.1.1) such that (3.1.20) lim {sup |u(x, t) − U (x + cˆt + θ)| + sup |u(x, t) − U (−x + cˆt + θ)|} = 0. t→−∞ x≥0. x≤0. Moreover, u(x, t) → 1 as t → ∞ for any x. Theorem 3.1.2 Let (3.1.9) be in force with m = 2. For any c1 > c∗ and c2 < c∗ , let (c1 , V1 (x)) and (c2 , W2 (x)) be solutions of (3.1.7) and (3.1.8) respectively. Then there exist a constant ω and an entire solution u(x, t) of (3.1.1) such that (3.1.21). lim. t→−∞. {. |u(x, t) − V1 (x + c1 t + ω)|. sup x≥−(c1 +c2 )t/2. +. |u(x, t) − W2 (x + c2 t − ω)|} = 0.. sup x≤−(c1 +c2 )t/2. Moreover, there exists θ ∈ R such that lim {sup |u(x, t) − U (x + cˆt + θ)|} = 0.. (3.1.22). t→∞ x∈R. Theorem 3.1.3 Let (3.1.9) be in force with m = 2. For any c2 < c∗ with −c2 < cˆ, let (ˆ c, U (x)) and (c2 , W2 (x)) be solutions of (3.1.5) and (3.1.8) respectively. Then there exist a constant ω and an entire solution u(x, t) of (3.1.1) such that lim. t→−∞. { +. sup x≥(c2 −ˆ c)t/2. sup x≤(c2 −ˆ c)t/2. |u(x, t) − U (x + cˆt + ω)| |u(x, t) − W2 (−x + c2 t + ω)|} = 0.. Moreover, we have lim {inf u(x, t)} = a,. t→∞ x∈R. lim { sup |u(x, t) − 1|} = 0, ∀C > 0.. t→∞ x≥−C. The above constructed entire solutions have some common characters. When −t ≫ 1, they behave as two traveling wavefronts on the opposite sides or on the same side of x-axis. Note that, different from the previous works, we choose the distinguishing line of the initial conditions in the above theorems to be the mid-points of two front-positions of traveling wavefronts. For example, in Theorem 3.1.2, x = −c1 t and x = −c2 t are front-positions for two traveling wavefronts V1 (x + c1 t) and W2 (x + c2 t), respectively. It is nature to choose the distinguishing line to be x = −(c1 + c2 )t/2 in (3.1.21). We organize this chapter as follows. In section 2, we give some proofs of the asymptotic behaviors of the traveling wavefronts stated above and some useful functions. Next, in section 3, we offer the proofs of Theorem 3.1.1, Theorem 3.1.2 and Theorem 3.1.3 by constructing suitable super/sub solutions.. 37.