J. Differential Equations 255 (2013) 3505–3535

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Journal of Differential Equations

www.elsevier.com/locate/jde

Entire solutions in periodic lattice dynamical systems

Shi-Liang Wu a,∗,1, Zhen-Xia Shi b,2, Fei-Ying Yang c,3

a Department of Mathematics, Xidian University, Xi’an, Shaanxi 710071, People’s Republic of Chinab School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, People’s Republic of Chinac School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 July 2012Revised 16 May 2013Available online 13 August 2013

MSC:34K0534A3434E05

Keywords:Entire solutionSpatially periodic solutionPulsating traveling frontPeriodic lattice dynamical system

This paper deals with entire solutions of periodic lattice dynamicalsystems. Unlike homogeneous problems, the periodic equationstudied here lacks symmetry between increasing and decreasingpulsating traveling fronts, which affects the construction of entiresolutions. In the bistable case, the existence, uniqueness andLiapunov stability of entire solutions are proved by constructingdifferent sub- and supersolutions. In the monostable case, theexistence and asymptotic behavior of spatially periodic solutionsconnecting two steady states are first established. Some new typesof entire solutions are then constructed by combining leftward andrightward pulsating traveling fronts with different speeds and aspatially periodic solution. Various qualitative features of the entiresolutions are also investigated.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

In this paper, we consider entire solutions of the periodic lattice dynamical system:

u′j(t) = d j+1u j+1(t) + d ju j−1(t) − (d j+1 + d j)u j(t) + f j

(u j(t)

), (1.1)

* Corresponding author.E-mail address: slwu@xidian.edu.cn (S.-L. Wu).

1 Supported by the Fundamental Research Funds for the Central Universities (No. K5051370002) and the NSF of ShaanxiProvince of China (No. 2013JQ1012).

2 Supported by the Science Foundation for Youths of Lanzhou Jiaotong University (No. 2012018).3 Supported by the NSF of China (Nos. 11031003, 11271172) and the FRFCU (No. lzujbky-2011-k27).

0022-0396/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jde.2013.07.049

3506 S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535

where ( j, t) ∈ Z × R, d j > 0, d j = d j−N for all j ∈ Z, N is a positive integer, and the nonlinearityfunction f = { f j} j∈Z satisfies the following assumption:

(A) f j(·) ∈ C2([0,1],R), f j(0) = f j(1) = 0, f ′j(1) < 0 and f j(u) = f j−N (u) for all j ∈ Z and u ∈ [0,1].

Here and in what follows, f ′j(u) := d

du f j(u) for all j ∈ Z and u ∈ [0,1].System (1.1) can be viewed as a spatial-discrete version of the following one space-dimensional

periodic reaction–diffusion equation

ut = (a(x)ux

)x + f

(x, u(x, t)

), (x, t) ∈R

2, (1.2)

where the diffusion coefficient a(x) is an L-periodic function for some L > 0 and f (x, ·) = f (x + L, ·).On the other hand, it also arises in many different applied fields, see e.g., [11,24,44,45].

In the past decades, there have been extensive investigations on wave propagation in various evo-lution equations. Although most of the studies have been devoted to the evolution equations withspatially homogeneous environment, there are also quite a few important and interesting workson wave propagation in heterogeneous media, especially in periodic media, see e.g., Gärtner andFreidlin [14], Freidlin [12], Hudson and Zinner [23], Shigesada, Kawasaki and Teramoto [43], Wein-berger [51] and the survey paper by Xin [59]. For more recent related works, we refer to Berestyckiet al. [1–4], Hamel [19], Hamel and Roques [22], Liang et al. [28,29], Shen [38–42], Nadin [37],Xin [55–58], Weng and Zhao [52] and the references cited therein.

Recently, Guo and Hamel [15] studied the pulsating (or periodic) traveling fronts (see Definition 1.1below) of (1.1) with monostable nonlinearity. More precisely, they proved that a rightward pulsatingtraveling front exists if and only if the wave speed is above a positive minimal speed. They also con-sidered the monotonicity of the pulsating traveling fronts and the convergence of discretized minimalwave speeds to the continuous minimal wave speed. Guo and Wu [18] further proved the uniquenessand stability of certain monostable pulsating traveling fronts with non-minimal speed. In [9], Chenet al. studied the existence, uniqueness, and global stability of pulsating traveling fronts of a generalbistable periodic lattice dynamical system which includes (1.1) as a particular case. For related worksto (1.1) on homogeneous media with monostable or bistable nonlinearities, we refer to [5,6,10,34,61]and the references cited therein.

Although the pulsating traveling front is a key object characterizing the dynamics of lattice differ-ential equations, such as (1.1), it is not enough to understand the whole dynamics. In fact, travelingfront solutions are only special examples of the so-called entire solutions (see Definition 1.1 below).From the viewpoint of biology, the entire solutions provide some new spread and invasion waysof the epidemic and species, respectively [17,36]. Moreover, the entire solution can help us for themathematical understanding of transient dynamics and the structures of the global attractor [35]. Inthe recent years, some new types of entire solutions other than traveling front solutions are estab-lished for various evolution equations with spatially homogeneous environment, see e.g., [7,8,13,16,20,21,27,35,47,60] for reaction–diffusion equations with and without delays, [48,49] for delayed lat-tice differential equations with global interaction, [26,46] for nonlocal diffusion equations, [25,30] forreaction–advection–diffusion equations in cylinders, and [17,36,50,53,54] for reaction–diffusion modelsystems.

More recently, Liu and Li [31] studied the entire solutions of a reaction–advection–diffusion equa-tion with bistable nonlinearity in heterogeneous media. In [32], the authors further considered theentire solutions of the reaction–advection–diffusion equation with monostable nonlinearity in peri-odic excitable media. However, to the best of our knowledge, there has been no results for the entiresolutions in discrete periodic media. Resolving this issue represents a main contribution of our currentstudy.

For discrete periodic equations, the presence of the periodicity introduces an asymmetry betweenj and − j which impacts on the construction of entire solutions. To overcome this difficulty, weconstruct some new sub- and supersolutions to study the entire solutions of such equations. Be-sides, unlike homogeneous problems, the discrete periodic equation has N (> 1) wave profiles which

S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535 3507

introduce some additional difficulties for the construction of entire solutions. We introduce a transfor-mation ((1.5) below), which is very useful in the discrete periodic framework. A similar transformationhas been used by [18] in the study of uniqueness and stability of monostable pulsating traveling frontsof (1.1).

The purpose of this paper is to consider the entire solutions of (1.1) with monostable or bistablenonlinearity. Before to state our main results, we first give the following definition.

Definition 1.1.

(1) A pulsating (or periodic) traveling front of (1.1) refer to a solution u(t) = {u j(t)} j∈Z , t ∈ R, whichsatisfies {

u j(t + N/c1) = u j−N(t), j ∈ Z, t ∈R,

u j(t) → 1 as j → −∞, u j(t) → 0 as j → +∞, locally in t ∈ R,(1.3)

or {u j(t − N/c2) = u j−N(t), j ∈ Z, t ∈R,

u j(t) → 0 as j → −∞, u j(t) → 1 as j → +∞, locally in t ∈ R,(1.4)

where c1 and c2 are unknown constants (the wave speeds).(2) A function u(t) = {u j(t)} j∈Z , t ∈ R, is called an entire solution of (1.1) if for any j ∈ Z, u j(t) are

differential for all t ∈R and u(t) satisfies (1.1) for j ∈ Z and t ∈ R.(3) Let m ∈ N and p, p0 ∈ R

m . We say that the functions W p(t) = {W j;p(t)} j∈Z converge to a functionW p0 (t) = {W j;p0 (t)} j∈Z as p → p0 ∈ R

m in the sense of topology T if, for any compact set S ⊂Z×R, the functions W j;p(t) and d

dt W j;p(t) converge uniformly in S to W j;p0 (t) and ddt W j;p0 (t)

as p → p0.

According to Definition 1.1, we see that pulsating traveling fronts are special examples of entiresolutions. Throughout this paper, we always assume that

uc1(t) = {uc1, j(t)

}j∈Z and uc2(t) = {

uc2, j(t)}

j∈Z

are solutions of (1.1) with (1.3) and (1.4), respectively. If c1 > 0 (or < 0), then we say uc1 (t) is a right-ward (or leftward) pulsating traveling front of (1.1). Similarly, if c2 > 0 (or < 0), then we say uc2 (t) isleftward (or rightward) pulsating traveling front of (1.1). As mentioned above, with the appearing ofperiodicity (N � 2), the discrete equation (1.1) lack symmetry between the pulsating traveling frontsuc1 (t) and uc2 (t), which affects the choice of sub- and supersolutions, especially for the bistable case.

We now introduce the following transformation

φci , j(ξi) := uci , j(t), ξi = (−1)i j + cit for j ∈ Z, t ∈ R, i = 1,2. (1.5)

Then, by (1.1), (1.3) and (1.4), we have⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ciφ

′ci , j(ξi) = d j+1φci , j+1

(ξi + (−1)i) + d jφci , j−1

(ξi − (−1)i)

− (d j+1 + d j)φci , j(ξi) + f j(φci , j(ξi)

), j ∈ Z, ξi ∈ R,

φci , j(ξi) = φci , j−N(ξi), j ∈ Z, ξi ∈R,

φci , j(−∞) = 0, φci , j(+∞) = 1, j ∈ Z,

(1.6)

for i = 1,2.

3508 S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535

For any λ ∈R, define two matrixes Al(λ) = (alλ;k, j)N×N (l = 0,1), where

alλ; j, j = −(d j+1 + d j) + f ′

j(l), j = 1, . . . , N,

alλ; j, j+1 = d j+1e−λ, j = 1, . . . , N − 1,

alλ; j+1, j = d j+1eλ, j = 1, . . . , N − 1,

alλ;1,N = d1eλ, al

λ;N,1 = d1e−λ,

alλ;k, j = 0 if |k − j|� 2 and (k, j) /∈ {

(1, N), (N,1)}. (1.7)

Following [9, Theorem 1] and [15, Lemma 2.1], we see that Al(λ) has a principal eigenvalue Ml(λ)

which is associated to a strongly positive eigenvector vl1(λ) = (vl

1,1(λ), . . . , vl1,N(λ))T , Ml(λ) is convex

in R, Ml(±∞) = +∞, and

minj∈Z

f ′j(l) � Ml(0) � max

j∈Zf ′

j(l).

Let [Al(λ)]T be the transpose matrix of Al(λ). Then Ml(λ) is also the principal eigenvalue of [Al(λ)]T

and there is a strongly positive eigenvector, say vl2(λ) = (vl

2,1(λ), . . . , vl2,N (λ))T . In addition, since

[Al(λ)]T = Al(−λ) for λ ∈ R, we see that Ml(λ) = Ml(−λ) for λ ∈ R.We point out that some sufficient conditions on the existence of bistable pulsating traveling fronts

of (1.1) have been established in [9]. For example, if there exist some function f (·) and a constanta ∈ (0,1) such that f j(·) = f (·) for all j ∈ Z, f ′(0) < 0, f ′(1) < 0, f (a) = 0, f ′(a) > 0, f > 0 in (a,1)

and f < 0 in (0,a), it then follows from the argument of [9, Theorem 7] that (1.1) has two solutionsuc1 (t) = {uc1, j(t)} j∈Z and uc2 (t) = {uc2, j(t)} j∈Z which satisfy (1.3) and (1.4), respectively.

The main results of entire solutions for the bistable equation are as follows.

Theorem 1.2. Assume (A) and the following condition

(B) f ′j(0) < 0 for all j ∈ Z and either f ′

j(1) < f ′j(0) for all j ∈ Z or f ′

j(1) � f ′j(0) for all j ∈ Z.

If c1, c2 > 0, then for any θ1, θ2 ∈ R, there exists a unique entire solution Φθ1,θ2 (t) = {Φ j;θ1,θ2 (t)} j∈Z of (1.1)such that

limt→−∞

{supj�0

∣∣Φ j;θ1,θ2(t) − φc1, j(− j + c1t + θ1)∣∣ + sup

j�0

∣∣Φ j;θ1,θ2(t) − φc2, j( j + c2t + θ2)∣∣} = 0.

Furthermore, the following statements hold:

(i) Φ ′j;θ1,θ2

(t) > 0 and 0 < Φ j;θ1,θ2 (t) < 1 for all j ∈ Z and t ∈R.(ii) limt→+∞ sup j∈Z |Φ j;θ1,θ2 (t) − 1| = 0 and limt→−∞ sup| j|�N0

Φ j;θ1,θ2 (t) = 0 for any N0 ∈ N.(iii) lim| j|→+∞ supt�t0

|Φ j;θ1,θ2 (t) − 1| = 0 for any t0 ∈R.(iv) Φθ1,θ2 (t) converges to

{{φc2, j( j + c2t + θ2)

}j∈Z as θ1 → −∞ in the sense of topology T ;{

φc1, j(− j + c1t + θ1)}

j∈Z as θ2 → −∞ in the sense of topology T .

(v) For any j ∈ Z and t ∈R, Φ j;θ1,θ2 (t) is increasing with respect to (θ1, θ2) ∈R2 .

(vi) Φθ1,θ2 (t) depends continuously on (θ1, θ2) ∈ R2 in the sense of topology T .

S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535 3509

(vii) The entire solution Φθ1,θ2 (t) is Liapunov stable in the following sense: For any given ε > 0, there existsδ > 0 such that for any ϕ = {ϕ j} j∈Z with ϕ j ∈ [0,1] and sup j∈Z |ϕ j − Φ j+ j0;θ1,θ2 (t0)| < δ, the solutionu(t;ϕ) = {u j(t;ϕ)} j∈Z of (1.1) with initial value ϕ satisfies∣∣u j(t;ϕ) − Φ j+ j0;θ1,θ2(t + t0)

∣∣ < ε

for any j ∈ Z and t � 0, where j0 ∈ Z and t0 ∈ R are two constants.

Theorem 1.3. Assume (A) and the following condition

(B)′ f ′j(0) < 0 for all j ∈ Z and either f ′

j(0) < f ′j(1) for all j ∈ Z or f ′

j(0) � f ′j(1) for all j ∈ Z.

If c1, c2 < 0, then for any θ1, θ2 ∈ R, there exists a unique entire solution Wθ1,θ2 (t) = {W j;θ1,θ2 (t)} j∈Z of (1.1)such that

limt→−∞

{supj�0

∣∣W j;θ1,θ2(t) − φc2, j( j + c2t + θ2)∣∣ + sup

j�0

∣∣W j;θ1,θ2(t) − φc1, j(− j + c1t + θ1)∣∣} = 0.

Moreover, the statements (v)–(vii) in Theorem 1.2 hold and there further hold:

(i)′ W ′j;θ1,θ2

(t) < 0 and 0 < W j;θ1,θ2 (t) < 1 for all j ∈ Z and t ∈R.

(ii)′ limt→+∞ sup j∈Z W j;θ1,θ2 (t) = 0 and limt→−∞ sup| j|�N1|W j;θ1,θ2 (t) − 1| = 0 for any N1 ∈N.

(iii)′ lim| j|→+∞ supt�t1W j;θ1,θ2 (t) = 0 for any t1 ∈ R.

(iv)′ Wθ1,θ2 (t) converges to

{{φc2, j( j + c2t + θ2)

}j∈Z as θ1 → +∞ in the sense of topology T ;{

φc1, j(− j + c1t + θ1)}

j∈Z as θ2 → +∞ in the sense of topology T .

Remark 1.4. We note that Theorem 1.3 is a consequence of Theorem 1.2. In fact, when c1, c2 < 0, setci = −ci > 0, i = 1,2, φc1, j( j + c1t) = 1 − φc1, j(− j + c1t) = 1 − φc1, j(−( j + c1t)) and φc2, j(− j + c2t) =1 − φc2, j( j + c2t) = 1 − φc2, j(−(− j + c2t)). Then, by (1.6), we have⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

ciφ′ci , j(ξi) = d j+1φci , j+1

(ξi − (−1)i) + d jφci , j−1

(ξi + (−1)i) − (d j+1 + d j)φci , j(ξi)

+ g j(φci , j(ξi)

), j ∈ Z, ξi = (−1)i−1 j + cit ∈R,

φci , j(ξi) = φci , j−N(ξi), j ∈ Z, ξi ∈R,

φci , j(−∞) = 0, φci , j(+∞) = 1, j ∈ Z,

for i = 1,2, where g j(u) = − f j(1 − u). Hence, for any θ1, θ2 ∈ R, φc1, j( j + c1t − θ1) and φc2, j(− j +c2t − θ2) are pulsating traveling fronts of the following equation:

v ′j(t) = d j+1 v j+1(t) + d j v j−1(t) − (d j+1 + d j)v j(t) + g j

(v j(t)

). (1.8)

Note that g′j(u) = f ′

j(1 − u), j ∈ Z, u ∈ [0,1]. It is easy to see that if (A) and (B)′ hold, then thefunction g = {g j} j∈Z satisfies (A) and (B). Applying Theorem 1.2 to (1.8), we can get an entire solutionΦθ1,θ2 (t) = {Φ j;θ1,θ2 (t)} j∈Z of (1.8) which satisfies the statements (i)–(vii) in Theorem 1.2 and

limt→−∞

{supj�0

∣∣Φ j;θ1,θ2(t) − φc2, j(− j + c2t − θ2)∣∣ + sup

j�0

∣∣Φ j;θ1,θ2(t) − φc1, j( j + c1t − θ1)∣∣} = 0.

3510 S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535

Let Wθ1,θ2 (t) = 1 − Φθ1,θ2 (t). Then Wθ1,θ2 (t) is an entire solution of (1.1) satisfying all the statementsin Theorem 1.3.

When c1c2 > 0 and among other things, Theorems 1.2 and 1.3 guarantee that (1.1) has a familyof unique and Liapunov stable entire solutions which behave like two pulsating traveling fronts com-ing from both directions. But for c1c2 < 0, the pulsating traveling fronts uc1 (t) = {uc1, j(t)} j∈Z anduc2 (t) = {uc2, j(t)} j∈Z propagating along one direction of the j-axis. Hence, the supersolution definedfor c1c2 > 0 (see Lemma 3.3) is no longer valid and a new supersolution will be constructed (seeLemma 3.7).

Theorem 1.5. Assume c1c2 < 0 and c2 �= −c1 . The following statements hold:

(i) If (A) and (B) hold and c2 > −c1 , then (1.1) admits a unique and Liapunov stable entire solution Φ(t) ={Φ j(t)} j∈Z which satisfies

limt→−∞

{sup

j� c1−c22 t

∣∣Φ j(t) − φc1, j(− j + c1t + ω)∣∣ + sup

j� c1−c22 t

∣∣Φ j(t) − φc2, j( j + c2t + ω)∣∣} = 0,

(1.9)

and

0 < Φ j(t) < 1, limt→+∞ inf

| j|� J0Φ j(t) = 1 for any J0 ∈N, (1.10)

where ω is given by (3.32) with c0 = c2+c12 .

(ii) If (A) and (B)′ hold and c2 < −c1 , then (1.1) admits a unique and Liapunov stable entire solution Φ(t) ={Φ j(t)} j∈Z which satisfies

limt→−∞

{sup

j� c1−c22 t

∣∣Φ j(t) − φc2, j( j + c2t − ω)∣∣ + sup

j� c1−c22 t

∣∣Φ j(t) − φc1, j(− j + c1t − ω)∣∣} = 0,

(1.11)

and

0 < Φ j(t) < 1, limt→+∞ sup

| j|� J0

Φ j(t) = 0 for any J0 ∈ N, (1.12)

where ω is given by (3.32) with c0 = − c2+c12 .

Remark 1.6. (i) When there exist two fronts moving in the same direction with different speeds andglobally decreasing and increasing profiles, Theorem 1.5 implies that (1.1) has entire solutions whichbehave as t → −∞ like these two fronts moving in the same direction, the faster one then invadingthe slower one as t → +∞. This is an interesting phenomenon for the periodic equation.

(ii) In view of Remark 1.4, in Section 3, we only give the proof of Theorem 1.5 for the casec2 > −c1.

(iii) It is clear that if f j(·) = f (·) for all j ∈ Z and some function f (·) which satisfies f ′(1) < 0 andf ′(0) < 0, then the conditions (B) and (B)′ hold.

To consider the monostable case, we assume that

(C) 0 < f j(u) � f ′j(0)u for all ( j, u) ∈ Z× (0,1).

S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535 3511

Clearly, the assumption (C) implies f ′j(0) > 0 for all j ∈ Z. Now, we recall some results of monostable

pulsating traveling fronts in [15]. Under the assumptions (A) and (C), there exist c∗ > 0 and λ∗ > 0such that

c∗ = M0(λ∗)λ∗

= infλ>0

M0(λ)

λ

and for any c > c∗ , there exists a unique λ1 := λ1(c) ∈ (0, λ∗) such that M0(λ1) = cλ1 and M0(λ) <

cλ for any λ ∈ (λ1, λ∗]. From the argument of [15], one can further obtain the following result ofmonostable pulsating traveling fronts of (1.1), see also Liang and Zhao [29, Theorem 7.4].

Proposition 1.7. Assume (A) and (C). Then the following results hold:

(i) For any c1 > c∗ , (1.1) has a rightward pulsating traveling front uc1 (t) = {uc1, j(t)} j∈Z which satisfiesu′

c1, j(t) > 0, 0 < uc1, j(t) < 1, uc1, j(t) � eλ1(c1)(− j+c1t)v01, j(λ1(c1)) and

lim− j+c1t→−∞

uc1, j(t)e−λ1(c1)(− j+c1t) = v01, j

(λ1(c1)

)for all j ∈ Z and t ∈ R.

(ii) For any c2 > c∗ , (1.1) has a leftward pulsating traveling front uc2 (t) = {uc2, j(t)} j∈Z which satisfiesu′

c2, j(t) > 0, 0 < uc2, j(t) < 1, uc2, j(t) � eλ1(c2)( j+c2t)v02, j(λ1(c2)) and

limj+c2t→−∞

uc2, j(t)e−λ1(c2)( j+c2t) = v02, j

(λ1(c2)

)for all j ∈ Z and t ∈ R. Here and in the sequel, v0

i, j (i = 1,2) are extended by periodicity for j ∈ Z, i.e.

v0i, j+N = v0

i, j for i = 1,2 and j ∈ Z.

Recall that φci , j(ξi) = uci , j(t), ξi = (−1)i j + cit for j ∈ Z, t ∈ R, i = 1,2. It is easy to see thatφci , j(−∞) = 0, φci , j(+∞) = 1, φ′

ci , j(·) > 0, 0 < φci , j(·) < 1, and

limz→−∞φci , j(z)e−λ1(ci)z = v0

i, j

(λ1(ci)

), φci , j(z) � eλ1(ci)z v0

i, j

(λ1(ci)

)(1.13)

for all j ∈ Z, z ∈R and i = 1,2.For the monostable equations on homogeneous media, there are spatially independent solutions

connecting two steady states. Combining a spatially independent solution and traveling fronts withdifferent speeds, some new entire solutions can be constructed [16,20,25–27,48]. However, for theperiodic monostable equation (1.1), there are no such spatially independent solutions. To overcome thedeficiency, we establish the existence and asymptotic behavior of spatially periodic solutions connectingtwo steady states and then construct some new type of entire solutions by combining the spatiallyperiodic solution and rightward and leftward pulsating traveling fronts with different speeds. In fact,we have the following results.

Theorem 1.8. Assume (A) and (C). Then, system (1.1) has a spatially periodic solution Γ (t) = {Γ j(t)} j∈Z whichsatisfies Γ j(t) = Γ j−N (t), Γ j(−∞) = 0 and Γ j(+∞) = 1 for all j ∈ Z and t ∈ R. Furthermore,

limt→−∞Γ j(t)e−λ∗t = v∗

j , Γ ′j (t) > 0, and Γ j(t) � eλ∗t v∗

j , ∀ j ∈ Z, t ∈R, (1.14)

where λ∗ = M0(0) and v∗ = v01(0) = v0

2(0).

3512 S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535

For any N0 ∈ Z and a ∈ R, let’s denote the regions T iN0,a , i = 1, . . . ,6, by

T 1N0,a := (−∞, N0] × [a,∞), T 2

N0,a := [N0,∞) × [a,∞), T 3N0,a := Z× [a,∞),

T 4N0,a := [N0,∞) × (−∞,a], T 5

N0,a := (−∞, N0] × (−∞,a], T 6N0,a := Z× (−∞,a].

Theorem 1.9. Assume that (A) and (C) hold and f ′j(u) � f ′

j(0) for all ( j, u) ∈ Z × [0,1]. Then, for anyh1,h2,h3 ∈ R, c1, c2 > c∗ , and χ1,χ2,χ3 ∈ {0,1} with χ1 + χ2 + χ3 � 2, there exists an entire solutionU p(t) = {U j;p(t)} j∈Z of (1.1) such that

max{χ1φc1, j(− j + c1t + h1),χ2φc2, j( j + c2t + h2),χ3Γ j(t + h3)

}� U j;p(t)� min

{1,Π1( j, t),Π2( j, t),Π3( j, t)

}(1.15)

for ( j, t) ∈ Z×R, where p := pχ1,χ2,χ3 = (χ1c1,χ2c2,χ1h1,χ2h2,χ3h3) and

Π1( j, t) = χ1φc1, j(− j + c1t + h1) + χ2eλ1(c2)( j+c2t+h2)v02, j

(λ1(c2)

) + χ3eλ∗(t+h3)v∗j ,

Π2( j, t) = χ1eλ1(c1)(− j+c1t+h1)v01, j

(λ1(c1)

) + χ2φc2, j( j + c2t + h2) + χ3eλ∗(t+h3)v∗j ,

Π3( j, t) = χ1eλ1(c1)(− j+c1t+h1)v01, j

(λ1(c1)

) + χ2eλ1(c2)( j+c2t+h2)v02, j

(λ1(c2)

) + χ3Γ j(t + h3).

Furthermore, there holds:

(i) U ′j;p(t) > 0 and 0 < U j;p(t) < 1 for all j ∈ Z and t ∈ R.

(ii) limt→+∞ sup j∈Z |U j;p(t) − 1| = 0 and limt→−∞ sup| j|�N0U j;p(t) = 0 for any N0 ∈ N.

(iii) If χ1 = 1, lim j→−∞ supt�t0|U j;p(t)− 1| = 0 and if χ2 = 1, lim j→+∞ supt�t0

|U j;p(t)− 1| = 0 for anyt0 ∈R.

(iv) For every j ∈ Z, there exist D j > C j > 0 such that

C jeϑ(c1,c2)t � U j;p(t) � D je

ϑ(c1,c2)t

for all t −1, where

ϑ(c1, c2) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩min{c1λ1(c1), c2λ1(c2), λ

∗}, if (χ1,χ2,χ3) = (1,1,1);min{c2λ1(c2), λ

∗}, if (χ1,χ2,χ3) = (0,1,1);min{c1λ1(c1), λ

∗}, if (χ1,χ2,χ3) = (1,0,1);min{c1λ1(c1), c2λ1(c2)}, if (χ1,χ2,χ3) = (1,1,0).

(v) For any j ∈ Z, U j;p(t) is increasing with respect to hi , i = 1,2,3.(vi) U p(t) converges to 1 as hi → +∞ in T and uniformly on ( j, t) ∈ T i

N0,a for any N0 ∈ Z and a ∈ R,i = 1,2,3.

Remark 1.10. It should be mentioned that, if in addition to f ′j(u) � f ′

j(0) one assumes that each f j(·)is concave, then for any c1, c2 � c∗ , the function

u+j (t) := χ1φc1, j(− j + c1t + h1) + χ2φc2, j( j + c2t + h2) + χ3Γ j(t + h3)

could serve as a supersolution of (1.1) and hence the existence of entire solutions for the case c1 = c∗and/or c2 = c∗ holds also true.

S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535 3513

Moreover, according to the assumption χ1,χ2,χ3 ∈ {0,1} with χ1 + χ2 + χ3 � 2 in Theorem 1.9,we further denote the entire solution U p(t) = {U j;p(t)} j∈Z of (1.1) by

U p(t) :=

⎧⎪⎪⎪⎨⎪⎪⎪⎩U p0(t) = {U j;p0(t)} j∈Z, if (χ1,χ2,χ3) = (1,1,1);U p1(t) = {U j;p1(t)} j∈Z, if (χ1,χ2,χ3) = (0,1,1);U p2(t) = {U j;p2(t)} j∈Z, if (χ1,χ2,χ3) = (1,0,1);U p3(t) = {U j;p3(t)} j∈Z, if (χ1,χ2,χ3) = (1,1,0),

(1.16)

where p0 = (c1, c2,h1,h2,h3), p1 = (0, c2,0,h2,h3), p2 = (c1,0,h1,0,h3) and p3 = (c1, c2,h1,h2,0).Then we have the following convergence results.

Theorem 1.11. Let all the assumptions of Theorem 1.9 be satisfied. From (1.16), the following properties hold.

(i) For any N0 ∈ Z and a ∈ R, U p0(t) converges to

U pi (t) as hi → −∞ in T , and uniformly on ( j, t) ∈ T 3+iN0,a, i = 1,2,3.

(ii) For any N0 ∈ Z and a ∈ R, U p1 (t) converges to

{Γ (t + h3) as h2 → −∞ in T , and uniformly on ( j, t) ∈ T 5

N0,a;{φc2, j( j + c2t + h2)

}j∈Z as h3 → −∞ in T , and uniformly on ( j, t) ∈ T 6

N0,a.

(iii) For any N0 ∈ Z and a ∈ R, U p2 (t) converges to

{Γ (t + h3) as h1 → −∞ in T , and uniformly on ( j, t) ∈ T 4

N0,a;{φc1, j(− j + c1t + h1)

}j∈Z as h3 → −∞ in T , and uniformly on ( j, t) ∈ T 6

N0,a.

(iv) For any N0 ∈ Z and a ∈ R, U p3 (t) converges to⎧⎨⎩{φc2, j( j + c2t + h2)

}j∈Z as h1 → −∞ in T , and uniformly on ( j, t) ∈ T 4

N0,a;{φc1, j(− j + c1t + h1)

}j∈Z as h2 → −∞ in T , and uniformly on ( j, t) ∈ T 5

N0,a.

Remark 1.12. (i) In the monostable case, the existence and qualitative properties of entire solu-tions of (1.1) connecting the traveling wave fronts and spatially periodic solutions are established.Uniqueness and stability of such entire solutions remain open. One of the main difficulty is that anappropriate super–subsolution pair cannot be constructed to trap the entire solution since 0 is an“unstable steady state”. We will consider these in another paper.

(ii) We note that our results are easily extended to the following more general discrete periodicequation [9]:

u′j(t) =

∑k

a j,ku j+k(t) + f j(u j(t)

), j ∈ Z, t ∈R. (1.17)

The rest of the paper is organized as follows. In Section 2, we give some existence and compari-son theorems for solutions, supersolutions and subsolutions of (1.1). Section 3 is devoted the entiresolutions for the bistable dynamics. In Section 4, we study the spatially periodic solutions and entiresolutions for the monostable dynamics.

3514 S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535

2. Preliminaries

In this section, we give some existence and comparison theorems for solutions, supersolutions andsubsolutions of (1.1) which will be used in the sequel.

We make the following extension for the function f = { f j} j∈Z . For every j ∈ Z, define

f j : [−1,2] →R by

f j(u) =

⎧⎪⎨⎪⎩f ′

j(0)u, u ∈ [−1,0],f j(u), u ∈ [0,1],f ′

j(1)(u − 1), u ∈ [1,2].

Obviously, f j(u) ∈ C1([−1,2]) and | f ′j(u1) − f ′

j(u2)| � L|u1 − u2| for all u1, u2 ∈ [−1,2], where L =max( j,u)∈Z×[0,1] | f ′′

j (u)|. For convenience, we denote f by f in the remainder of this paper. Clearly,this extension does not affect the main results of this paper (Theorems 1.2, 1.3, 1.8, 1.9 and 1.11).

Let t0 ∈ R be any given constant. The definitions of supersolution and subsolution of (1.1) are givenas follows.

Definition 2.1. A sequence of continuous functions {u j(t)} j∈Z , t ∈ [t0, T ), T > t0, is called a supersolu-tion (or subsolution) of (1.1) on [t0, T ) if

u′j(t)� (or �)d j+1u j+1(t) + d ju j−1(t) − (d j+1 + d j)u j(t) + f j

(u j(t)

), (2.1)

for t ∈ [t0, T ).

Definition 2.2. A sequence of continuous functions {u j(t)} j∈Z , t ∈ (−∞, T ), is called a supersolution(or subsolution) of (1.1) on (−∞, T ) if for any t0 < T , {u j(t)} j∈Z is a supersolution (or subsolution)of (1.1) on [t0, T ).

By Definition 2.1, we have the following result. Its proof is standard and thus omitted, see e.g., [33,Lemma 4.1] and [34, Lemma 2.2].

Lemma 2.3.

(i) For any ϕ = {ϕ j} j∈Z with ϕ j ∈ [0,1], Eq. (1.1) admits a unique solution u(t;ϕ) = {u j(t)} j∈Z on [t0,+∞)

which satisfies u j(t0) = ϕ j and u j ∈ C1([t0,∞), [0,1]) for all j ∈ Z.(ii) Suppose {u+

j (t)} j∈Z and {u−j (t)} j∈Z are supersolution and subsolution of (1.1) on [t0,+∞), respectively

such that 0 � u+j (t), u−

j (t) � 2 and u+j (t0) � u−

j (t0) for j ∈ Z and t � t0 , then u+j (t) � u−

j (t) for allj ∈ Z and t � t0 , and

u+j (t) − u−

j (t) > e−σ (t−s)∑k∈Z

[u+

k (s) − u−k (s)

] [d(t − s)]| j−k|

| j − k|!

for all j ∈ Z and t > s � t0 , where σ = 2 max j∈Z d j + max( j,u)∈Z×[0,1] | f ′j(u)| and d = min j∈Z d j .

The following result gives the prior estimate of solutions of (1.1).

Lemma 2.4. Let u(t;ϕ) = {u j(t;ϕ)} j∈Z be a solution of (1.1) with initial value ϕ = {ϕ j} j∈Z with ϕ j ∈ [0,1].Then there exists a positive constant M, independent of t0 and ϕ , such that for any j ∈ Z and t > t0 + 1,∣∣u′

j(t;ϕ)∣∣ � M and

∣∣u′′j (t;ϕ)

∣∣ � M. (2.2)

S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535 3515

Proof. The proof is easy, so we omit it. �Lemma 2.5. Let w+

j ∈ C1([t0,+∞), [0,+∞)) and w−j ∈ C1([t0,+∞), (−∞,1]) be such that w+

j (t0) �w−

j (t0) for all j ∈ Z, and

d

dtw+

j (t)� d j+1 w+j+1(t) + d j w+

j−1(t) − (d j+1 + d j)w+j (t) + f ′

j(0)w+j (t),

d

dtw−

j (t)� d j+1 w−j+1(t) + d j w−

j−1(t) − (d j+1 + d j)w−j (t) + f ′

j(0)w−j (t),

for j ∈ Z and t > t0 . Then w+j (t) � w−

j (t) for all j ∈ Z and t � t0 .

Proof. The proof is similar to that of [33, Lemma 4.1] and thus omitted. �3. Entire solutions to bistable equations

In this section, we consider the entire solutions of (1.1) with bistable nonlinearity. Throughout thissection, we always assume that (1.1) has pulsating traveling fronts uc1 (t) = {uc1, j(t)} j∈Z and uc2 (t) ={uc2, j(t)} j∈Z with c1c2 �= 0.

From [15, Lemmas 2.2] and [9, Lemma 6], one can see that 0 < uci , j(·) < 1 and u′ci , j(·) > 0 (resp.

u′ci , j(·) < 0) if ci > 0 (resp. ci < 0), ∀ j ∈ Z, i = 1,2. Since φci , j(ξi) = uci , j(t), ξi = (−1)i j + cit for j ∈ Z,

t ∈ R, i = 1,2, we have

0 < φci , j(·) < 1 and φ′ci , j(·) > 0, ∀ j ∈ Z, i = 1,2.

Recall that Ml(λ) is the principal eigenvalue of the matrixes Al(λ) and [Al(λ)]T which is associatedto strongly positive eigenvectors vl

1(λ) = (vl1,1(λ), . . . , vl

1,N(λ))T and vl2(λ) = (vl

2,1(λ), . . . , vl2,N (λ))T ,

respectively, l = 0,1. It follows from [9, Theorem 2] that the characteristic equations associated with0 and 1

M0(λ) − ciλ = 0, λ > 0, i = 1,2, (3.1)

M1(λ) − ciλ = 0, λ < 0, i = 1,2, (3.2)

have exactly one root, respectively, say λ0i (> 0) and λ1

i (< 0), and there exist positive constants h+i

and h−i such that

limξ→−∞

φci , j(ξ)e−λ0i ξ

v0i, j(λ

0i )

= h−i and lim

ξ→+∞(1 − φci , j(ξ))e−λ1

i ξ

v1i, j(λ

1i )

= h+i , j ∈ Z, i = 1,2.

From (1.6), one can further show that

limξ→−∞

φ′ci , j(ξ)e−λ0

i ξ

v0i, j(λ

0i )

= λ0i h−

i and limξ→+∞

φ′ci , j(ξ)e−λ1

i ξ

v1i, j(λ

1i )

= −λ1i h+

i , j ∈ Z, i = 1,2.

Thus, we can choose positive constants k, K ,α,β such that

3516 S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535

keλ0i z � φci , j(z) � K eλ0

i z, ∀z � 0, j ∈ Z, i = 1,2, (3.3)

keλ1i z � 1 − φci , j(z)� K eλ1

i z, ∀z � 0, j ∈ Z, i = 1,2, (3.4)

infz�0

φ′ci , j(z)

φci , j(z)= α and inf

z�0

φ′ci , j(z)

1 − φci , j(z)= β, ∀ j ∈ Z, i = 1,2. (3.5)

3.1. The case: c1c2 > 0

We only consider the case c1, c2 > 0 (see Remark 1.4). Without loss of generality, we further as-sume that c2 � c1. It is easy to see that λ0

2 � λ01 > 0 and λ1

1 � λ12 < 0. In this subsection, we always

assume that (A) and (B) hold.To construct an appropriate supersolution, we need to define two functions p1(t) and p2(t). In the

remainder of this section, we set

N � max

{LK

α,

LK

βk

}and � = − 1

λ01

ln

(1 + N

c1

)< 0.

Consider the function

ω = ρ1 − 1

λ01

ln

(1 + N

c1eλ0

1ρ1

), ρ1 ∈ (−∞,0]. (3.6)

Since ω(ρ1) is increasing in ρ1 ∈ (−∞,0], its inverse function exists, say ρ1 = ρ1(ω) : (−∞,� ] →(−∞,0].

For any (ω, ω) ∈ (−∞,� ]2, let

ρ2 := ρ2(ω, ω) = ω + 1

λ01

ln

(1 + N

c1eλ0

1ρ1(ω)

).

Obviously, ω − ω = ρ2 − ρ1. Now, consider the system⎧⎪⎪⎨⎪⎪⎩p′

1(t) = c1 + Neλ01 p1(t), t < 0,

p′2(t) = c2 + Neλ0

1 p1(t), t < 0,(p1(0), p2(0)

) = (ρ1(ω),ρ2(ω, ω)

).

(3.7)

Solving the equation explicitly, we get

p1(t;ω) = ρ1(ω) + c1t − 1

λ01

ln

{1 + N

c1eλ0

1ρ1(ω)(1 − ec1λ0

1t)},

p2(t;ω, ω) = ρ2(ω, ω) + c2t − 1

λ01

ln

{1 + N

c1eλ0

1ρ1(ω)(1 − ec1λ0

1t)}.

Note that p2(t;ω, ω) − p1(t;ω) = (c2 − c1)t + ρ2(ω, ω) − ρ1(ω) = (c2 − c1)t + ω − ω and

p1(t;ω) − c1t − ω = p2(t;ω, ω) − c2t − ω = − 1

λ01

ln(1 − rec1λ0

1t/(1 + r))

(3.8)

for t � 0, where r = Nc eλ0

1ρ1(ω1) . We consider two cases:

1

S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535 3517

Case (i): c2 > c1. Given any M0 > 0. For any (ω1,ω2) ∈ (−∞,� ]2 with ω2 − ω1 � M0, letp1(t;ω1,ω2) = p1(t;ω) and p2(t;ω1,ω2) = p2(t;ω, ω) with ω = ω1 and ω = ω2. It is easy to seethat p2(t;ω1,ω2)� p1(t;ω1,ω2) for all t �−M0/(c2 − c1).

Case (ii): c2 = c1. Given any (ω1,ω2) ∈ (−∞,� ]2. If ω2 � ω1, let p1(t;ω1,ω2) = p1(t;ω) andp2(t;ω1,ω2) = p2(t;ω, ω) with ω = ω1 and ω = ω2. Then p2(t;ω1,ω2) � p1(t;ω1,ω2) � 0 forall t � 0. If ω1 � ω2, let p1(t;ω1,ω2) = p2(t;ω, ω) and p2(t;ω1,ω2) = p1(t;ω) with ω = ω2 andω = ω1. Then, p1(t;ω1,ω2) � p2(t;ω1,ω2) � 0 for all t � 0.

Furthermore, by (3.8), there exists a positive constant R0, independent of ω1 and ω2, such that

0 < p1(t;ω1,ω2) − c1t − ω1 = p2(t;ω1,ω2) − c2t − ω2 � R0ec1λ01t for t � 0.

In what follows, we denote pi(t) := pi(t;ω1,ω2) (i = 1,2) and

ΓM0 := {(x, y) ∈ (−∞,� ]2

∣∣ y − x � M0}.

Theorem 3.1. Given any M0 > 0. For any (ω1,ω2) ∈ ΓM0 , there exists an entire solution Φ(t) = {Φ j(t)} j∈Z :=Φω1,ω2 (t) = {Φ j;ω1,ω2 (t)} j∈Z of (1.1) such that

limt→−∞

{supj�0

∣∣Φ j(t) − φc1, j(− j + c1t + ω1)∣∣ + sup

j�0

∣∣Φ j(t) − φc2, j( j + c2t + ω2)∣∣} = 0. (3.9)

Furthermore, the following statements hold:

(i) Φ ′j(t) > 0 and 0 < Φ j(t) < 1 for all j ∈ Z and t ∈ R.

(ii) limt→+∞ sup j∈Z |Φ j(t) − 1| = 0 and limt→−∞ sup| j|�N0Φ j(t) = 0 for any N0 ∈N.

(iii) lim| j|→+∞ supt�t0|Φ j(t) − 1| = 0 for any t0 ∈ R.

(iv) For any j ∈ Z and t ∈ R, Φ j;ω1,ω2 (t) is increasing with respect to (ω1,ω2) ∈ ΓM0 .

Theorem 3.2. Given any M0 > 0. For any (ω1,ω2) ∈ ΓM0 , let Φ(t) = Φω1,ω2 (t) be the entire solution of (1.1)decided in Theorem 3.1. The following statements hold:

(i) If Φ(t) = {Φ j(t)} j∈Z is an entire solution of (1.1) satisfying (3.9), then Φ(t) = Φ(t).(ii) Φω1,ω2 (t) converges to

{{φc2, j( j + c2t + ω2)

}j∈Z as ω1 → −∞ in the sense of topology T ;{

φc1, j(− j + c1t + ω1)}

j∈Z as ω2 → −∞ in the sense of topology T .

(iii) Φω1,ω2 (t) depends continuously on (ω1,ω2) ∈ ΓM0 in the sense of the topology T .(iv) Φ(t) = Φω1,ω2 (t) is Liapunov stable in the sense of Theorem 1.2(vii).

The proofs of Theorems 3.1 and 3.2 will be given in the last part of this section. Based on Theo-rems 3.1 and 3.2, we can prove Theorem 1.2.

Proof of Theorem 1.2. For any θ1, θ2 ∈ R, there exists T1 < 0 such that c1T1 + θ1 < � and c2T1 +θ2 < � . Take

ω1 := c1T1 + θ1, ω2 := c2T1 + θ2 and M0 := |c1T1 − c2T1 + θ1 − θ2|.

3518 S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535

Clearly, (ω1,ω2) ∈ (−∞,� ]2 and ω2 − ω1 � M0. It follows from Theorems 3.1 and 3.2 that thereexists an entire solution Φω1,ω2(t) = {Φ j;ω1,ω2 (t)} j∈Z , which satisfies the conclusions of Theorems 3.1and 3.2. Let

Φθ1,θ2(t) = {Φ j;θ1,θ2(t)

}j∈Z := Φω1,ω2(t − T1) = {

Φ j;ω1,ω2(t − T1)}

j∈Z.

It is easy to verify that Φθ1,θ2 (t) is an entire solution of (1.1) which satisfies the assertions of Theo-rem 1.2. �

To prove Theorems 3.1 and 3.2, we need several lemmas.

Lemma 3.3. Given any M0 > 0. For any (ω1,ω2) ∈ ΓM0 , u(t) = {u j(t)} j∈Z defined by

u j(t) = φc1, j(− j + p1(t)

) + φc2, j(

j + p2(t))

is a supersolution of (1.1) on (−∞, T M0 ), where T M0 = −M0/(c2 − c1) if c2 > c1 and T M0 = 0 if c2 = c1 .

Proof. We only consider the case for c2 > c1 since the other case can be discussed similarly. Forconvenience, set

F j(u)(t) = u′j(t) − d j+1u j+1(t) − d ju j−1(t) + (d j+1 + d j)u j(t) − f j

(u j(t)

).

For any ( j, t) ∈ Z× (−∞,0), we have

F j(u)(t) = [p′

1(t) − c1]φ′

c1, j

(− j + p1(t)) + [

p′2(t) − c2

]φ′

c2, j

(j + p2(t)

) − G j(t)

= [φ′

c1, j

(− j + p1(t)) + φ′

c2, j

(j + p2(t)

)]Neλ0

1 p1(t) − G j(t),

where

G j(t) = f j(φc1, j

(− j + p1(t)) + φc2, j

(j + p2(t)

))− f j

(φc1, j

(− j + p1(t))) − f j

(φc2, j

(j + p2(t)

)).

Assume that t < −M0/(c2 − c1). Note that when c2 > c1, p2(t) � p1(t) � 0 for all t � −M0/

(c2 − c1). Now we estimate

H j(t) := G j(t)

φ′c1, j(− j + p1(t)) + φ′

c2, j( j + p2(t)), j ∈ Z.

We consider two cases.Case I: f ′

j(0) � f ′j(1) for all j ∈ Z. We claim that λ0

1 �−λ12 and λ0

2 � −λ11. In fact, by the definition

of the matrixes A0(λ) and A1(λ) (see (1.7)), we have A1(λ) � A0(λ) and hence M1(λ) � M0(λ). Definetwo functions

�i(λ) = Mi−1(λ) − ciλ, i = 1,2.

Clearly, �i(λ) is convex in R. Note that M1(λ) = M1(−λ) for λ ∈ R and λ01 and −λ1

2 are the uniquepositive roots of the equations �1(λ) = 0 and �2(−λ) = 0, respectively. Since

S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535 3519

�2(−λ) = M1(−λ) − c2(−λ) = M1(λ) + c2λ > �1(λ) = M0(λ) − c1λ

for λ > 0, we have λ01 � −λ1

2. Similarly, we can prove that λ02 � −λ1

1.In view of f j(u) ∈ C1([0,2],R) and | f ′

j(u1) − f ′j(u2)| � L|u1 − u2| for all u1, u2 ∈ [0,2], where

L = max( j,u)∈Z×[0,1] | f ′′j (u)|, it is easy to show that

G j(t)� Lφc1, j(− j + p1(t)

)φc2, j

(j + p2(t)

), j ∈ Z.

We divide Z into 3 parts.Case (i): j ∈ [p1(t),−p2(t)] ∩ Z. Note that λ0

2 � λ01 > 0. By (3.3) and (3.5), for j ∈ [p1(t),0] ∩ Z,

there holds

H j(t) �Lφc1, j(− j + p1(t))φc2, j( j + p2(t))

φ′c1, j(− j + p1(t)) + φ′

c2, j( j + p2(t))

� Lφc2, j( j + p2(t))

φ′c1, j(− j + p1(t))/φc1, j(− j + p1(t))

� LK eλ02( j+p2(t))

α� LK

αeλ0

1 p1(t), (3.10)

and for j ∈ [0,−p2(t)] ∩Z, there holds

H j(t) �Lφc1, j(− j + p1(t))

φ′c2, j( j + p2(t))/φc2, j( j + p2(t))

� LK eλ01(− j+p1(t))

α� LK

αeλ0

1 p1(t). (3.11)

Case (ii): j ∈ (−∞, p1(t)] ∩Z. In view of λ02 � −λ1

1, it follows from (3.3)–(3.5) that

H j(t)�Lφc2, j( j + p2(t))

φ′c1, j(− j + p1(t))

� LK eλ02 p2(t)

βkeλ11 p1(t)e−(λ1

1+λ02) j

� LK

βkeλ0

1 p1(t). (3.12)

Case (iii): j ∈ [−p2(t),+∞) ∩Z. Noting that λ01 � −λ1

2, we have

H j(t)�Lφc1, j(− j + p1(t))

φ′c2, j( j + p2(t))

� LK eλ01 p1(t)

βkeλ12 p1(t)e(λ1

2+λ01) j

� LK

βkeλ0

1 p1(t). (3.13)

Case II: f ′j(1) < f ′

j(0) for all j ∈ Z. In view of f j(u) = f j−N (u) for all ( j, u) ∈ Z×[0,1], there existsρ > 0 such that f ′

j(u) < f ′j(0) for all u ∈ (1 − ρ,1) and j ∈ Z. We may assume that φci , j(x) � 1 − ρ

for x � 0 and j ∈ Z by translations if necessary.As in the proof of Case I, we divide Z into three parts [p1(t),−p2(t)] ∩ Z, (−∞, p1(t)] ∩ Z and

[−p2(t),+∞) ∩ Z. In the part [p1(t),−p2(t)] ∩ Z, we obtain the same estimate as (3.10) and (3.11)for H j(t). For j ∈ (−∞, p1(t)] ∩Z, it is easy to verify that G j(t) � Lφ2

c2, j( j + p2(t)), and hence,

H j(t) �Lφ2

c2, j( j + p2(t))

φ′c2, j( j + p2(t))

� LK eλ02( j+p2(t))

α� LK

αeλ0

1 p1(t). (3.14)

Similarly, for j ∈ [−p2(t),+∞) ∩Z, we have

3520 S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535

H j(t)�Lφ2

c1, j(− j + p1(t))

φ′c1, j(− j + p1(t))

� LK

αeλ0

1 p1(t). (3.15)

Combining (3.10)–(3.15), there holds F j(u) � 0 for j ∈ Z and t ∈ (−∞,−M0/(c2 −c1)). Therefore, u(t)is a supersolution of (1.1) on (−∞, T M0 ). This completes the proof. �

Now, we give the proof of Theorem 3.1.

Proof of Theorem 3.1. For any (ω1,ω2) ∈ ΓM0 and n ∈ (−T M0 ,+∞) ∩N, let u(t) = {u j(t)} j∈Z definedby

u j(t) = max{φc1, j(− j + c1t + ω1),φc2, j( j + c2t + ω2)

}, t ∈R,

and Φn(t) = {Φnj (t)} j∈Z be the unique solution of the following initial value problem:

⎧⎨⎩d

dtΦn

j (t) = d j+1Φnj+1(t) + d jΦ

nj−1(t) − (d j+1 + d j)Φ

nj (t) + f j

(Φn

j (t)),

Φnj (−n) = u j(−n),

where j ∈ Z and t > −n. By Lemmas 2.3 and 3.3, we have

u j(t)� Φnj (t)� Φn+1

j (t) � u j(t) for j ∈ Z, −n � t < T M0 , (3.16)

and

u j(t) � Φnj (t) � 1 for j ∈ Z, t > −n. (3.17)

From the priori estimate (Lemma 2.4) and by a diagonal extraction process, there exists a subsequence{Φnk (t)}k∈N of Φn(t) such that Φnk (t) converges to a function Φ(t) = {Φ j(t)} j∈Z in T . In view ofΦn

j (t) � Φn+1j (t) for any t > −n, we have limn→+∞ Φn

j (t) = Φ j(t) for any ( j, t) ∈ Z × R. The limitfunction is unique, whence all of the functions Φn(t) converge to the function Φ(t) in T as n → +∞.Clearly, Φ(t) is an entire solution of (1.1). From (3.16) and (3.17), we have

u j(t) �Φ j(t)� u j(t) for all j ∈ Z, t < T M0 , (3.18)

and

u j(t)� Φ j(t)� 1 for all j ∈ Z, t ∈R. (3.19)

Since φ′ci , j(z) > 0 and 0 < φci , j(z) < 1 for all j ∈ Z, z ∈ R, i = 1,2, one can easily show that the

assertion (iv) holds. Moreover, using (3.18) and (3.19), the proofs of (ii)–(iii) and (3.9) in Theorem 3.1are straightforward and thus omitted.

We now prove (i). From Lemma 2.3, one can see that 0 < Φ j(t) < 1 for all j ∈ Z and t ∈ R. Letu(t;ϕ) = {u j(t;ϕ)} j∈Z be the unique solution of (1.1) with initial data u(0;ϕ) = ϕ . Then Φn(t) =u(t + n; u(−n)). Since for any ε > 0, u(· + ε) � u(·) on R, it follows that u(ε; u(−n)) � u(ε − n) �u(−n). By comparison and the uniqueness of solutions, we have Φn(t + ε) = u(t + n; u(ε; u(−n))) �u(t +n; u(−n)) = Φn(t) for any ( j, t) ∈ Z×[−n,+∞). Thus, it follows from the arbitrariness of ε thatΦn(t) is increasing on t . Therefore, Φ ′

j(t) � 0 for all ( j, t) ∈ Z ×R. Next, we show that Φ ′j(t) > 0 for

all j ∈ Z and t ∈R. In view of

S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535 3521

Φ ′′j (t) = d j+1Φ

′j+1(t) + d jΦ

′j−1(t) − (d j+1 + d j)Φ

′j(t) + f ′

j

(Φ j(t)

)Φ ′

j(t)

for j ∈ Z and t ∈ R, we have

Φ ′j(t) = Φ ′

j(r)e−μ j(t−r) +t∫

r

h j(s)e−μ j(t−s) ds, ∀r < t, (3.20)

where μ j = d j+1 + d j + max( j,u)∈Z×[0,1] | f ′j(u)| and

h j(t) = d j+1Φ′j+1(t) + d jΦ

′j−1(t) +

[max

( j,u)∈Z×[0,1]∣∣ f ′

j(u)∣∣ + f ′

j

(Φ j(t)

)]Φ ′

j(t).

Clearly, h j(t) � 0 for all j ∈ Z and t ∈ R. Suppose on the contrary that there exists a ( j0, t0) ∈ Z ×R

such that Φ ′j0(t0) = 0, then it follows from (3.20) that Φ ′

j0(r) = 0 for all r � t0. Hence Φ j0 (t) = Φ j0 (t0)

for all t � t0, which implies that limt→−∞ Φ j0 (t) = Φ j0 (t0). But following from (3.18) and (3.19),limt→−∞ Φ j0 (t) = 0 and Φ j0 (t0) > 0. This contradiction yields that Φ ′

j(t) > 0 for all j ∈ Z and t ∈ R.This completes the proof of Theorem 3.1. �In order to prove Theorem 3.2, we construct a sub–supersolution pair of (1.1) to trap the entire

solution Φ(t) = {Φ j(t)} j∈Z obtained in Theorem 3.1. For this, we first establish the following results.

Lemma 3.4. For any a ∈ (0,1/2], there exists η := η(a) > 0 such that

Φ ′j(t) � η for all a � Φ j(t)� 1 − a. (3.21)

Proof. Suppose that the assertion is not true. Then there exist a1 ∈ (0,1/2] and a sequence{( jk, tk)}k∈Z with a1 � Φ jk (tk) � 1 − a1 such that Φ ′

jk(tk) → 0 as k → +∞. In view of φci , j(+∞) = 1

for j ∈ Z and

max{φc1, j(− j + c1t + ω1),φc2, j( j + c2t + ω2)

}� Φ j(t) � φc1, j

(− j + p1(t)) + φc2, j

(j + p2(t)

), (3.22)

for all ( j, t) ∈ Z × (−∞, T M0 ), where T M0 � 0 is defined as in Lemma 3.3, there exists M1 ∈ R suchthat − jk + c1tk � M1 and jk + c2tk � M1 for all k ∈ Z, which implies that {tk}k∈Z is bounded fromabove, say T1.

Set Φk(t) := {Φkj (t)} j∈Z = {Φ j+ jk (t + tk)} j∈Z . From Lemma 2.4, there exists Φ∗(t) = {Φ∗

j (t)} j∈Zsuch that Φk(t) → Φ∗(t) in the sense of T as k → +∞ (up to extraction of some subsequence).It is easy to see that Φ∗(t) is an entire solution of (1.1), d

dt Φ∗0 (0) = 0, a1 � Φ∗

0 (0) � 1 − a1, andddt Φ

∗j (t) � 0 for all j ∈ Z and t ∈ R. Using a similar argument as in the proof of Theorem 3.1(i), we

can show that ddt Φ

∗0 (t) = 0 for all t � 0 and hence Φ∗

0 (t) = Φ∗0 (0) for all t � 0 which implies that

limt→−∞ Φ∗0 (t) = Φ∗

0 (0). Note that for all k ∈ Z and t ∈ (−∞, T M0 − T1),

− jk + p1(t + tk) = (− jk + c1tk + ω1) + [p1(t + tk) − c1(t + tk) − ω1

] + c1t

� M1 + ω1 + R0ec1λ01tec1λ0

1tk + c1t

� M1 + ω∗ + R0ec1λ01tec1λ0

1 T1 + c1t,

where ω∗ = max{ω1,ω2}. Similarly, there holds

3522 S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535

jk + p2(t + tk) � M1 + ω∗ + R0ec1λ01tec1λ0

1 T1 + c2t.

It follows from (3.22) that

Φ jk (t + tk) � φc1, jk

(− jk + p1(t + tk)) + φc2, jk

(jk + p2(t + tk)

)� φc1, jk

(M1 + ω∗ + R0ec1λ0

1tec1λ01 T1 + c1t

)+ φc2, jk

(M1 + ω∗ + R0ec1λ0

1tec1λ01 T1 + c2t

)for all k ∈ Z and t ∈ (−∞, T M0 − T1). In view of φci , j(·) = φci , j−N (·) and φci , j(−∞) = 0 for all j ∈ Z,i = 1,2, one can easily show that limt→−∞ Φ∗

0 (t) = 0 which contradicts to limt→−∞ Φ∗0 (t) = Φ∗

0 (0) �a1 > 0. This completes the proof. �Lemma 3.5. There exist δ0 > 0, ρ0 > 0 and σ0 > 0 such that for any γ ∈ R, δ ∈ (0, δ0] and σ � σ0 , thefunction U±(t) = {U±

j (t)} j∈Z defined by

U±j (t) = Φ j

(t + γ ± σδ

(1 − e−ρ0t)) ± δe−ρ0t

is a pair of supersolution and subsolution of (1.1) on [0,+∞).

Proof. We only prove U+j (t) is a supersolution, since the another case can be shown similarly. From

the conditions f ′j(0) < 0, f ′

j(1) < 0 for all j ∈ Z and f j(u) = f j−N (u) for all j ∈ Z and u ∈ [0,1], thereexist two constants ρ0 > 0 and a0 ∈ (0,1/2] such that

f ′j(u) � −ρ0 for all j ∈ Z and u ∈ [0,2a0] ∪ [1 − a0,1 + a0]. (3.23)

Take δ0 = a0 and

σ0 = ρ0 + max( j,u)∈Z×[0,1] | f ′j(u)|

ρ0η, (3.24)

where η = η(a0) is defined as in Lemma 3.4.Let γ ∈ R, δ ∈ (0, δ0], σ � σ0 and ξ(t) = t + γ + σδ(1 − e−ρ0t). Direct computations show that for

( j, t) ∈ Z× [0,+∞),

F j(U+)

(t) := d

dtU+

j (t) − d j+1U+j+1(t) − d j U

+j−1(t) + (d j+1 + d j)U+

j (t) − f j(U+

j (t))

= σδρ0Φ′j

(ξ(t)

)e−ρ0t − ρ0δe−ρ0t + f j

(Φ j

(ξ(t)

)) − f j(Φ j

(ξ(t)

) + δe−ρ0t)= δe−ρ0t[σρ0Φ

′j

(ξ(t)

) − ρ0 − f ′j

(Φ j

(ξ(t)

) + θδe−ρ0t)], (3.25)

where θ ∈ (0,1).Now, we consider two cases:Case (i): Φ j(ξ(t)) ∈ [0,a0] ∪ [1 − a0,1]. It is easy to see that

Φ j(ξ(t)

) + θδe−ρ0t ∈ [0,2a0] ∪ [1 − a0,1 + a0].It then follows from (3.23) and (3.25) that

F j(U+)

(t) � δe−ρ0t[−ρ0 − f ′j

(Φ j

(ξ(t)

) + θδe−ρ0t)] � 0.

S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535 3523

Case (ii): Φ j(ξ(t)) ∈ [a0,1−a0]. By Lemma 3.4, Φ ′j(ξ(t)) � η. Then, from (3.24) and (3.25), we have

F j(U+)(t) � 0.Combining the above two cases, we obtain F j(U+)(t) � 0 for all ( j, t) ∈ Z× [0,+∞), i.e., U+

j (t) isa supersolution of (1.1) on [0,+∞). This completes the proof. �Lemma 3.6. There exist δ∗ > 0, ρ∗ > 0 and σ∗ > 0 such that for any γ ∈ R, δ ∈ (0, δ∗] and σ � σ∗ , thefunction V ±(t) = {V ±

j (t)} j∈Z defined by

V ±j (t) = φc1, j

(− j + c1t + γ ± σδ(1 − e−ρ∗t)) ± δe−ρ∗t

is a pair of supersolution and subsolution of (1.1) on [0,+∞).

The proof is similar to that of Lemma 3.5. We omit it.We now give the proof of Theorem 3.2. In the sequel, let σ0, ρ0, δ0 and σ∗ , ρ∗ , δ∗ be positive

constants given in Lemmas 3.5 and 3.6, respectively.

Proof of Theorem 3.2. (i) Suppose that Φ(t) = {Φ j(t)} j∈Z is an entire solution of (1.1) satisfying (3.9).From (3.9), we have limt→−∞ sup j∈Z |Φ j(t) − Φ j(t)| = 0. Given any t1 < 0. Define

η = supj∈Z

∣∣Φ j(t1) − Φ j(t1)∣∣.

To prove Φ(·) = Φ(·), it suffices to show that η = 0. For any δ ∈ (0, δ0], there exists t2 < t1 such thatsup j∈Z |Φ j(t2) − Φ j(t2)| < δ, and hence,

Φ j(t2) − δ � Φ j(t2)� Φ j(t2) + δ for all j ∈ Z.

By Lemmas 2.3 and 3.5, we have

Φ j(t + t2 − σ0δ

(1 − e−ρ0t)) − δe−ρ0t � Φ j(t + t2)

� Φ j(t + t2 + σ0δ

(1 − e−ρ0t)) + δe−ρ0t

for all j ∈ Z and t � 0. Letting t = t1 − t2 > 0 in the above inequality, we get

Φ j(t1 − σ0δ

(1 − eρ0(t1−t2)

)) − δ � Φ j(t1) � Φ j(t1 + σ0δ

(1 − eρ0(t1−t2)

)) + δ.

Hence, for all j ∈ Z,

Φ j(t1 − σ0δ) − δ � Φ j(t1) � Φ j(t1 + σ0δ) + δ. (3.26)

Note that 0 < Φ ′j(t) � M3 := 2 max j∈Z d j + max( j,u)∈Z×[0,1] f j(u) for any j ∈ Z and t ∈ R. It follows

from (3.26) that

supj∈Z

∣∣Φ j(t1) − Φ j(t1)∣∣ � (1 + M3σ0)δ,

which implies that η � (1 + M3σ0)δ. Following the arbitrariness of δ, η = 0. Therefore, Φ(·) = Φ(·).(ii) We first prove that Φω1,ω2 (t) converges to {φc1, j(− j + c1t + ω1)} j∈Z as ω2 → −∞ in T . Take

a sequence {(ω1,ωk2)}k∈N satisfying (ω1,ω

k2) ∈ ΓM0 , ωk+1

2 < ωk2 < ω1 and ωk

2 → −∞ as k → −∞.

3524 S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535

From Theorem 3.1, for each k ∈ N, there exists an entire solution Φk(t) = {Φkj (t)} j∈Z := Φω1,ωk

2(t) =

{Φ j;ω1,ωk2(t)} j∈Z of (1.1) satisfying

φc1, j(− j + c1t + ω1) � max{φc1, j(− j + c1t + ω1),φc2, j

(j + c2t + ωk

2

)}� Φk+1

j (t)� Φkj (t)

� φc1, j(− j + p1

(t;ω1,ω

k2

)) + φc2, j(

j + p2(t;ω1,ω

k2

))= φc1, j

(− j + p1(t;ω1)) + φc2, j

(j + p2

(t;ω1,ω

k2

))(3.27)

for any j ∈ Z and t < T M0 . By Lemma 2.4 and the monotonicity of Φk(t) on k, there exists Ψ (t) ={Ψ j(t)} j∈Z such that Φk(t) → Ψ (t) in the sense of T as k → +∞. It then follows from (3.27) that forany j ∈ Z and t < T M0 ,

φc1, j(− j + c1t + ω1) � Ψ j(t)� φc1, j(− j + p1(t;ω1)

). (3.28)

Given any t1 < T M0 . Define

η = supj∈Z

∣∣Ψ j(t1) − φc1, j(− j + c1t1 + ω1)∣∣.

For any δ ∈ (0, δ∗], since p1(t;ω1) − c1t − ω1 → 0 as t → −∞, it follows from (3.28) that there existst2 < t1 such that for any j ∈ Z,

φc1, j(− j + c1t2 + ω1) � Ψ j(t2) � φc1, j(− j + c1t2 + ω1) + δ.

By comparison and Lemma 3.6, for any j ∈ Z and t > 0,

φc1, j(− j + c1(t + t2) + ω1

)� Ψ j(t + t2)

� φc1, j(− j + c1(t + t2) + ω1 + σδ

(1 − e−ρ0t)) + δe−ρ0t .

Letting t = t1 − t2 > 0 in the above inequality, we get

φc1, j(− j + c1t1 + ω1)� Ψ j(t1)

� φc1, j(− j + c1t1 + ω1 + σδ

(1 − e−ρ0(t1−t2)

)) + δe−ρ0(t1−t2)

� φc1, j(− j + c1t1 + ω1 + σδ) + δ, j ∈ Z,

which implies that

supj∈Z

∣∣Ψ j(t1) − φc1, j(− j + c1t1 + ω1)∣∣ � δ + σδ max

j∈Z,z∈Rφ′

c1, j(z).

According to the arbitrariness of δ, we obtain η = 0. Hence, Ψ j(t) = φc1, j(− j + c1t + ω1) for anyj ∈ Z and t ∈ R. Since Φω1,ω2 (t) is increasing with respect to ω2, Φω1,ω2 (t) converges to {φc1, j(− j +c1t + ω1)} j∈Z as ω2 → −∞ in T . Similarly, we can show that Φω1,ω2 (t) converges to {φc2, j( j + c2t +ω2)} j∈Z as ω1 → −∞ in T .

(iii) Given (ω01,ω0

2) ∈ ΓM0 . Take two sequences {(ωk+,1,ωk+,2)}k∈N and {(ωk−,1,ω

k−,2)}k∈N satisfying

(ωk±,1,ωk±,2) ∈ ΓM0 , limk→+∞(ωk±,1,ω

k±,2) → (ω01,ω0

2), and

S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535 3525

(ωk−,1,ω

k−,2

)�

(ωk+1

−,1 ,ωk+1−,2

)<

(ω0

1,ω02

)<

(ωk+1

+,1 ,ωk+1+,2

)�

(ωk+,1,ω

k+,2

), ∀k ∈N.

From Theorem 3.1, there exist entire solutions Φ0(t) := Φ0ω0

1,ω02(t) = {Φ0

j;ω01,ω0

2(t)} j∈Z and

Φk±(t) := Φk±;ωk±,1,ωk±,2

(t) = {Φk

±, j;ωk±,1,ωk±,2(t)

}j∈Z

of (1.1). By Lemma 2.4, there exists Φ±(t) = {Φ±, j(t)} j∈Z such that Φk±(t) → Φ±(t) in the sense ofT as k → +∞ (up to extraction of some subsequence). Clearly, Φ+(t) and Φ−(t) are entire solutions

of (1.1). In view of 0 < φ′ci , j(t) � M3/ci for all j ∈ Z, t ∈R, 0 < pi(t;ωk+,1,ω

k+,2)− cit −ωk+,i � R0ec1λ0

1t

for t � 0 and

max{φc1, j

(− j + c1t + ω01

), φc2, j

(j + c2t + ω0

2

)}� max

{φc1, j

(− j + c1t + ωk+,1

), φc2, j

(j + c2t + ωk+,2

)}� Φk

+, j(t) � φc1, j(− j + p1

(t;ωk

+,1,ωk+,2

)) + φc2, j(

j + p2(t;ωk

+,1,ωk+,2

))for any t < T M0 , j ∈ Z and k ∈N, it is easy to show that

limt→−∞

{supj�0

∣∣Φ+, j(t) − φc1, j(− j + c1t + ω0

1

)∣∣ + supj�0

∣∣Φ+, j(t) − φc2, j(

j + c2t + ω02

)∣∣} = 0.

By the uniqueness of entire solutions, we have Φ+(t) = Φ0(t). Similarly, one can prove thatΦ−(t) = Φ0(t). Moreover, for any (ω1,ω2) ∈ ΓM0 with (ω1,ω2) → (ω0

1,ω02), we can easily show

that Φω1,ω2 (t) → Φ0ω0

1,ω02(t) in the sense of T which implies that Φω1,ω2 (t) depends continuously

on (ω1,ω2) ∈ ΓM0 in the sense of the topology T .(iv) Given any ε > 0. Choose δ1 := δ1(ε) = ε

2M3> 0. Then, for all |z| � δ1,

supj∈Z,t∈R

∣∣Φ j(t) − Φ j(t + z)∣∣ � sup

j∈Z,t∈R∣∣Φ ′

j(t)∣∣|z|� M3δ1 �

ε

2. (3.29)

Let δ = min{ε/2, δ1/σ0, δ0}. For any ϕ = {ϕ j} j∈Z with ϕ j ∈ [0,1] and sup j∈Z |ϕ j −Φ j+ j0 (t0)| < δ, thereholds

Φ j+ j0(t0) − δ � ϕ j � Φ j+ j0(t0) + δ for all j ∈ Z.

By comparison and Lemma 3.5, we have

Φ j+ j0

(t + t0 − σ0δ

(1 − e−ρ0t)) − δe−ρ0t

� u j(t;ϕ)� Φ j+ j0

(t + t0 + σ0δ

(1 − e−ρ0t)) + δe−ρ0t (3.30)

for all j ∈ Z and t � 0. It then follows from (3.29) and (3.30) that

∣∣u j(t;ϕ) − Φ j+ j0(t + t0)∣∣ � M3σ0δ + δ � ε for all j ∈ Z and t � 0.

Now, we complete the proof of Theorem 3.2. �

3526 S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535

3.2. The case: c1c2 < 0 and c2 �= −c1

In this case where c1c2 < 0, the pulsating traveling fronts uc1 (t) = {uc1, j(t)} j∈Z and uc2 (t) ={uc2, j(t)} j∈Z propagate along one direction of the j-axis. Hence, the supersolution defined in theabove subsection is no longer valid. We shall construct a new supersolution for this case.

Assume c2 > −c1 and let c0 = c2+c12 . Consider the following initial value problem:

{p′(t) = c0 + N1eλ0

1 p(t), t < 0,

p(0) � 0,(3.31)

where N1 � LK max{1/α,1/(βk)}. We note that the argument about p(t) was first given by Guo andMorita [16], see also [13,30,47]. By solving (3.31), we have

p(t) − c0t − ω = − 1

λ01

ln

(1 − r1

1 + r1ec0λ0

1t)

, r1 = N1

c0eλ0

1 p(0),

where

ω := p(0) − 1

λ01

ln

(1 + N1

c0eλ0

1 p(0)

). (3.32)

It is easy to see that p(t) is increasing in t ∈ (−∞, 0] and

0 < p(t) − c0t − ω � k1ec0λ01t, t � 0 (3.33)

for some positive constant k1.

Lemma 3.7. Let (A) and (B) hold. Assume c1c2 < 0 and c2 > −c1 . Let c = c2−c12 . The functions u(t) =

{u j(t)} j∈Z and u(t) = {u j(t)} j∈Z defined by

u j(t) = φc1, j(− j + q1(t)

) + φc2, j(

j + q2(t)),

u j(t) = max{φc1, j(− j + c1t + ω),φc2, j( j + c2t + ω)

}are a pair of super- and subsolutions of (1.1) for t � 0, where q1(t) = −ct + p(t) and q2(t) = ct + p(t).

Proof. Obviously, u j(t) is a subsolution of (1.1) for t � 0. We now prove that u(t) is a supersolution.The proof is similar to that of Lemma 3.3. For the sake of completeness and reader’s convenience, wealso provide it here. Recall that

F j(u)(t) = u′j(t) − d j+1u j+1(t) − d ju j−1(t) + (d j+1 + d j)u j(t) − f j

(u j(t)

).

For any ( j, t) ∈ Z× (−∞,0), direct computations show that

F j(u)(t) = [φ′

c1, j

(− j + q1(t)) + φ′

c2, j

(j + q2(t)

)](Neλ0

1 p(t) − H j(t)),

where

S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535 3527

G j(t) = f j(φc1, j

(− j + q1(t)) + φc2, j

(j + q2(t)

))− f j

(φc1, j

(− j + q1(t))) − f j

(φc2, j

(j + q2(t)

)),

and

H j(t) := G j(t)

φ′c1, j(− j + q1(t)) + φ′

c2, j( j + q2(t)), j ∈ Z.

We consider two cases.Case I: f ′

j(0) � f ′j(1) for all j ∈ Z. Since c2 > −c1 and M1(·) � M0(·), we see that for λ� 0,

M1(−λ) − c2(−λ) = M1(λ) + c2λ� M0(λ) − c1λ,

and

M1(−λ) − c1(−λ) = M1(λ) + c1λ � M0(λ) − c2λ.

Hence, λ01 �−λ1

2 > 0 and λ02 � −λ1

1 > 0. It is easy to show that

G j(t) � Lφc1, j(− j + q1(t)

)φc2, j

(j + q2(t)

), j ∈ Z,

where L = max( j,u)∈Z×[0,1] | f ′′j (u)|.

We divide Z into 3 parts.Case (i): j ∈ [q1(t),−q2(t)] ∩Z. Note that λ0

2 � λ01 > 0. By (3.3) and (3.5), for j + ct � 0, there holds

H j(t) �Lφc2, j( j + q2(t))

φ′c1, j(− j + q1(t))/φc1, j(− j + q1(t))

� LK eλ02( j+ct+p(t))

α� LK

αeλ0

1 p(t), (3.34)

and for j + ct � 0, there holds

H j(t)�Lφc1, j(− j + q1(t))

φ′c2, j( j + q2(t))/φc2, j( j + q2(t))

� LK eλ01(− j−ct+p(t))

α� LK

αeλ0

1 p(t). (3.35)

Case (ii): j ∈ (−∞,q1(t)] ∩ Z. Then j + ct � p(t) � 0 for t � 0. In view of λ02 � −λ1

1 > 0, it followsfrom (3.3)–(3.5) that

H j(t) �Lφc2, j( j + q2(t))

φ′c1, j(− j + q1(t))

� LK eλ02 p(t)e(λ1

1+λ02)( j+ct)

βkeλ11 p(t)

� LK

βkeλ0

1 p(t). (3.36)

Case (iii): j ∈ [−q2(t),+∞) ∩Z. Then j + ct �−p(t) � 0 for t � 0. Since λ01 � −λ1

2 > 0, we have

H j(t)�Lφc1, j(− j + q1(t))

φ′ ( j + q2(t))� LK eλ0

1 p(t)e−(λ01+λ1

2)( j+ct)

βkeλ12 p(t)

� LK

βkeλ0

1 p(t). (3.37)

c2, j

3528 S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535

Case II: f ′j(1) < f ′

j(0) for all j ∈ Z. Since f j(u) = f j−N (u) for all ( j, u) ∈ Z × [0,1], there existsρ > 0 such that f ′

j(u) < f ′j(0) for all u ∈ (1 − ρ,1) and j ∈ Z. We may assume that φci , j(x) � 1 − ρ

for x � 0 and j ∈ Z by translations if necessary.As in the proof of Case I, we divide Z into three parts [q1(t),−q2(t)] ∩ Z, (−∞,q1(t)] ∩ Z and

[−q2(t),+∞) ∩ Z. In the part [q1(t),−q2(t)] ∩ Z, we obtain the same estimate as (3.34) and (3.35)for H j(t). For j ∈ (−∞,q1(t)] ∩Z, it is easy to verify that G j(t) � Lφ2

c2, j( j + q2(t)), and hence,

H j(t)�Lφ2

c2, j( j + q2(t))

φ′c2, j( j + q2(t))

� LK eλ02( j+ct+p(t))

α� LK

αeλ0

1 p(t). (3.38)

Similarly, for j ∈ [−q2(t),+∞) ∩Z, we have

H j(t) �Lφ2

c1, j(− j + q1(t))

φ′c1, j(− j + q1(t))

� LK

αeλ0

1 p(t). (3.39)

Therefore, combining (3.34)–(3.39), there holds F j(u) � 0 for all j ∈ Z and t � 0. Therefore, u(t) is asupersolution of (1.1) on (−∞,0). This completes the proof. �

Now, we give the proof of Theorem 1.5.

Proof of Theorem 1.5. We only consider the case where c1c2 < 0 and c2 > −c1 (see Remark 1.6).Using Lemma 3.7 and the comparison principle, one can prove the existence of entire solution Φ(t) ={Φ j(t)} j∈Z . The proof of unique and Liapunov stability of Φ(t) is similar to that of Theorem 1.2 so weomit it. Now we complete the proof of Theorem 1.5. �4. Entire solutions to monostable equations

In this section, we consider the entire solutions of (1.1) with monostable nonlinearity. We firstprove Theorem 1.8, i.e., the existence and asymptotic behavior of spatially periodic solutions of (1.1)connecting 0 and 1. Inspired by the works of Hamel and Nadirashvili [20] and Wang et al. [48], The-orems 1.9 and 1.11 are then proved by using the comparison principle and constructing appropriatesubsolutions and upper estimates.

4.1. Existence of spatially periodic solutions

This subsection is devoted to the spatially periodic solutions of system (1.1) connecting 0 and 1,that is, solutions of the problem:⎧⎪⎪⎪⎨⎪⎪⎪⎩

u′j(t) = d j+1u j+1(t) + d ju j−1(t) − (d j+1 + d j)u j(t)

+ f j(u j(t)

), j ∈ Z, t ∈R,

u j(t) = u j−N(t), j ∈ Z, t ∈R,

u j(−∞) = 0, u j(+∞) = 1, j ∈ Z.

(4.1)

In the subsection, we always assume that (A) and (C) hold. Define a matrix A = (ak, j)N×N , where

a j, j = −(d j+1 + d j) + f ′j(0), j = 1, . . . , N,

a j, j+1 = d j+1, j = 1, . . . , N − 1,

a j+1, j = d j+1, j = 1, . . . , N − 1,

S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535 3529

a1,N = d1, aN,1 = d1,

ak, j = 0 if |k − j|� 2 and (k, j) /∈ {(1, N), (N,1)

}.

Clearly, A = A0(0) (see (1.7)), and A has a principal eigenvalue λ∗ = M0(0), which is associated to astrongly positive eigenvector v∗ = (v∗

1, . . . , v∗N)T = v0

1(0).Denote

S = {φ(t) = {

φ j(t)}

j∈Z∣∣ φ j(·) ∈ C

(R, [0,1]) and φ j(·) = φ j−N(·) for all j ∈ Z

}.

For any φ(t) ∈ S , define an operator T = {T j} j∈Z by

T j(φ)(t) =t∫

−∞e−μ(t−s)H j(φ)(s)ds, j ∈ Z,

where μ = 2 max j∈Z d j + max( j,u)∈Z×[0,1] | f ′j(u)| and

H j(φ)(t) = μφ j(t) + d j+1φ j+1(t) + d jφ j−1(t) − (d j+1 + d j)φ j(t) + f j(φ j(t)

).

According to the periodicity of d j and f j(u), the following observation is straightforward.

Lemma 4.1.

(i) T : S → S.(ii) T (φ)(t) � T (ψ)(t) for φ,ψ ∈ S with φ(t) �ψ(t).

(iii) T (φ)(t) is increasing in R for φ ∈ S with φ(t) being increasing in R.

For any fixed ν ∈ (1,2] and sufficiently large q > 1, define two functions φ(t) = {φ j(t)} j∈Z andφ(t) = {φ

j(t)} j∈Z as follows:

φ j(t) = min{

1, v∗j eλ∗t} and φ

j(t) = max

{0, v∗

j eλ∗t − qv∗

j eνλ∗t}, j ∈ Z, t ∈R.

By direct computations, one can easily obtain the following result.

Lemma 4.2.

(i) 0 � φj(t) � φ j(t) � 1 for all j ∈ Z and t ∈ R.

(ii) T j(φ)(t) � φ j(t) and T j(φ)(t) � φj(t) for all j ∈ Z and t ∈R.

Proof of Theorem 1.8. Using the monotone iteration technique, we can show that (4.1) admits a solu-tion Γ (t) = {Γ j(t)} j∈Z which satisfies Γ (t) ∈ S , Γ ′

j (t) � 0 and

φj(t)� Γ j(t)� φ j(t) for all j ∈ Z, t ∈R.

Thus, Γ j(t) = Γ j−N (t), Γ j(t) � eλ∗t v∗j , and limt→−∞ Γ j(t)e−λ∗t = v∗

j for all j ∈ Z and t ∈ R. Further-more, it is easy to verify that Γ j(+∞) = 1 for all j ∈ Z. Using a similar argument as in Theorem 3.1(i),one can show that Γ ′

j (t) > 0 for all j ∈ Z and t ∈ R. This completes the proof of Theorem 1.8. �

3530 S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535

4.2. Proofs of Theorems 1.9 and 1.11

In the sequel, we always assume that the conditions of Theorem 1.9 hold. For any n ∈ N,h1,h2,h3 ∈R, c1, c2 > c∗ and χ1,χ2,χ3 ∈ {0,1} with χ1 + χ2 + χ3 � 2, we denote

ϕnj := max

{χ1φc1, j(− j − c1n + h1),χ2φc2, j( j − c2n + h2),χ3Γ j(−n + h3)

},

U j(t) := max{χ1φc1, j(− j + c1t + h1),χ2φc2, j( j + c2t + h2),χ3Γ j(t + h3)

},

where j ∈ Z and t ∈ R. Let Un(t) = {Unj (t)} j∈Z be the unique solution of the following initial value

problem of (1.1):⎧⎨⎩d

dtUn

j (t) = d j+1Unj+1(t) + d j U

nj−1(t) − (d j+1 + d j)Un

j (t) + f j(Un

j (t)),

Unj (−n) = ϕn

j ,

(4.2)

where j ∈ Z, t > −n. Then, by Lemma 2.3,

U j(t) � Unj (t)� 1 for all j ∈ Z and t �−n.

The following result gives an appropriate upper estimate of Un(t) = {Unj (t)} j∈Z .

Lemma 4.3. The function Un(t) = {Unj (t)} j∈Z satisfies

Unj (t) � min

{1,Π1( j, t),Π2( j, t),Π3( j, t)

}for all j ∈ Z and t �−n. Note that Πi( j, t), i = 1,2,3 are defined in Theorem 1.9.

Proof. We only prove Unj (t) � Π1( j, t) for all j ∈ Z and t � −n since the proof of the other inequali-

ties is similar. Assume χ1 = 1. Set

Znj (t) := Un

j (t) − φc1, j(− j + c1t + h1), j ∈ Z, t ∈R.

Clearly, 0 � Znj (t) � 1 for j ∈ Z, t ∈ R. By the condition f ′

j(u) � f ′j(0) for ( j, u) ∈ Z× [0,1] and direct

computations, we obtain⎧⎨⎩d

dtZn

j (t) � d j+1 Znj+1(t) + d j Zn

j−1(t) − (d j+1 + d j)Znj (t) + f ′

j(0)Znj (t),

Znj (−n) = ϕn

j − φc1, j(− j − c1n + h1),

where j ∈ Z and t > −n. Take

V j(t) := χ2eλ1(c2)( j+c2t+h2)v02, j

(λ1(c2)

) + χ3eλ∗(t+h3)v∗j , j ∈ Z, t � −n.

In view of Av∗ = λ∗v∗ and

[A0(λ1(c2)

)]Tv0

2

(λ1(c2)

) = M0(λ1(c2))

v02

(λ1(c2)

) = c2λ1(c2)v02

(λ1(c2)

),

we have

S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535 3531

d

dtV j(t) = d j+1 V j+1(t) + d j V j−1(t) − (d j+1 + d j)V j(t) + f ′

j(0)V j(t), (4.3)

where j ∈ Z and t � −n. According to Proposition 1.7 and Theorem 1.8, we get

φc2, j(z) � eλ1(c2)z v02, j

(λ1(c2)

)and Γ j(z)� eλ∗z v∗

j

for all j ∈ Z and z ∈R, which imply that

Znj (−n) = ϕn

j − φc1, j(− j − c1n + h1)

� χ2φc2, j( j − c2n + h2) + χ3Γ j(−n + h3)

� χ2eλ1(c2)( j−c2n+h2)v02, j

(λ1(c2)

) + χ3eλ∗(−n+h3)v∗j

= V j(−n).

It then follows from Lemma 2.5 that Znj (t) � V j(t) for all j ∈ Z and t �−n, that is,

Unj (t) � φc1, j(− j + c1t + h1) + χ2eλ1(c2)( j+c2t+h2)v0

2, j

(λ1(c2)

) + χ3eλ∗(t+h3)v∗j

= Π1( j, t).

When χ1 = 0, the assertion Unj (t) � Π1( j, t) reduces to

Unj (t) � χ2eλ1(c2)( j+c2t+h2)v0

2, j

(λ1(c2)

) + χ3eλ∗(t+h3)v∗j ,

which holds obviously. The proof is complete. �Now we give the proofs of Theorems 1.9 and 1.11.

Proof of Theorem 1.9. By Lemmas 2.3 and 4.3, we have

U j(t) � Unj (t)� Un+1

j (t) � min{

1,Π1( j, t),Π2( j, t),Π3( j, t)}

for any j ∈ Z and t � −n. Similar to the proof of Theorem 3.1, one can see that there exists a functionU p(t) = {U j;p(t)} j∈Z such that Un(t) converges to U p(t) in T as n → +∞ and U p(t) is an entiresolution of (1.1) satisfying (1.15).

The assertions (i)–(iii), (v) and (vi) in Theorem 1.9 are straightforward consequences of (1.15).Therefore, we only prove the assertion (iv).

(iv) We only prove the case for (χ1,χ2,χ3) = (1,1,1), since the proofs for other cases are similar.When (χ1,χ2,χ3) = (1,1,1), by (1.15), we have

max{φc1, j(− j + c1t + h1),φc2, j( j + c2t + h2),Γ j(t + h3)

}� U j;p(t) � eλ1(c1)(− j+c1t+h1)v0

1, j

(λ1(c1)

)+ eλ1(c2)( j+c2t+h2)v0

2, j

(λ1(c2)

) + eλ∗(t+h3)v∗j , (4.4)

for ( j, t) ∈ Z×R. Note that for any j ∈ Z,

3532 S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535

limt→−∞Γ j(t + h3)e−λ∗(t+h3) = v∗

j ,

limt→−∞φc1, j(− j + c1t + h1)e−λ1(c1)(− j+c1t+h1) = v0

1, j

(λ1(c1)

),

and

limt→−∞φc2, j( j + c2t + h2)e−λ1(c2)( j+c2t+h2) = v0

2, j

(λ1(c2)

).

Hence, the assertion follows from (4.4). The proof is complete. �Proof of Theorem 1.11. (i) We only prove the case that U p0 (t) converges to U p1 (t) in the sense oftopology T as h1 → −∞, and uniformly on (n, t) ∈ T 4

N0,a . The proofs for the other cases are similar.Note that Un(t) = {Un

j (t)} j∈Z is the unique solution of the value problem (4.2). For (χ1,χ2,χ3) =(1,1,1), we denote ϕn ={ϕn

j } j∈Z by ϕnp0

={ϕnj;p0

} j∈Z and Un(t)={Unj (t)} j∈Z by Un

p0(t)={Un

j;p0(t)} j∈Z ,

respectively. Similarly, when (χ1,χ2,χ3) = (0,1,1), we denote ϕn by ϕnp1

and Un(t) by Unp1

(t), re-spectively. Set

W n(t) = {W n

j (t)}

j∈Z := Unp0

(t) − Unp1

(t).

Then 0 � W nj (t) � 1 for all ( j, t) ∈ Z × [−n,+∞). Furthermore, by the assumption f ′

j(u) � f ′j(0) for

all ( j, u) ∈ Z× [0,1], we have

d

dtW n

j (t)� d j+1W nj+1(t) + d j W

nj−1(t) − (d j+1 + d j)W n

j (t) + f ′j(0)W n

j (t) (4.5)

for j ∈ Z and t > −n. Define the function W (t) = {W j(t)} j∈Z by

W j(t) := eλ1(c1)(− j+c1t+h1)v01, j

(λ1(c1)

), t ∈R.

According to

A0(λ1(c1))

v01

(λ1(c1)

) = M0(λ1(c1))

v01

(λ1(c1)

) = c1λ1(c1)v01

(λ1(c1)

),

we obtain

d

dtW j(t) = d j+1W j+1(t) + d j W j−1(t) − (d j+1 + d j)W j(t) + f ′

j(0)W j(t). (4.6)

By Proposition 1.7, there holds

0 � W nj (−n) = ϕn

j;p0− ϕn

j;p1� φc1, j(− j − c1n + h1)

� eλ1(c1)(− j−c1n+h1)v01, j

(λ1(c1)

) = W j(−n).

It then follows from Lemma 2.5 that

0 � W nj (t) = Un

j;p0(t) − Un

j;p1(t) � W j(t) � eλ1(c1)(− j+c1t+h1) max

j∈Zv0

1, j

(λ1(c1)

)for all ( j, t) ∈ Z× [−n,+∞). Since

S.-L. Wu et al. / J. Differential Equations 255 (2013) 3505–3535 3533

limn→+∞ Un

j;p0(t) = U j;p0(t) and lim

n→+∞ Unj;p1

(t) = U j;p1(t),

we get

0 � U j;p0(t) − U j;p1(t)� eλ1(c1)(− j+c1t+h1) maxj∈Z

v01, j

(λ1(c1)

)for all ( j, t) ∈ Z×R,

which implies that U p0 (t) converges to U p1 (t) as h1 → −∞ uniformly on ( j, t) ∈ T 4N0,a for any

N0 ∈ Z and a ∈ R. For any sequence h�1 with h�

1 → −∞ as � → +∞, the functions U p� (t) (here

p� := (c1, c2,h�1,h2,h3)) converge to a solution of (1.1) (up to extraction of some subsequence) in T ,

which turns out to be U p1 (t). The limit does not depend on the sequence h�1, whence all of the

functions U p0 (t) converge to U p1 (t) in T as h1 → −∞.The proofs of (ii)–(iv) are similar to that of (i), and omitted. We now complete the proof of Theo-

rem 1.11. �Acknowledgments

The authors thank the anonymous referee for their valuable comments and suggestions that helpthe improvement of the manuscript.

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