Entire solutions for a multi-type SIS nonlocal epidemic model in R or Z

J. Math. Anal. Appl. 394 (2012) 603–615

Contents lists available at SciVerse ScienceDirect

Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa

Entire solutions for a multi-type SIS nonlocal epidemic model in R or Z Shi-Liang Wu a,∗ , Peixuan Weng b a

Department of Mathematics, Xidian University, Xi’an, Shaanxi 710071, People’s Republic of China

b

School of Mathematics, South China Normal University, Guangzhou, Guangdong 510631, People’s Republic of China

article

abstract

info

Article history: Received 18 December 2011 Available online 15 May 2012 Submitted by Yuan Lou

In this paper, we study entire solutions for a multi-type SIS nonlocal epidemic model in R or Z. The existence and asymptotic behavior of spatially independent solutions are first established. Some new entire solutions are then constructed by combining traveling wave fronts with different speeds and a spatially independent solution. From the viewpoint of epidemiology, the results provide some new spread ways of the epidemic. © 2012 Elsevier Inc. All rights reserved.

Keywords: Entire solution Traveling wave front Spatially independent solution SIS nonlocal epidemic model

1. Introduction Due to the significant applications in several subjects, the problem on traveling wave solutions is an important issue in the study of various evolution equations. Another important topic in those equations is the interaction of traveling wave solutions, which is crucially related to the pattern formation problem, we refer to [1–3] for more details. Mathematically, this phenomenon can be described by the so-called entire solutions that are defined for all time t ∈ R and for all space points. Recently, there have been many works devoted to the interaction of traveling wave fronts and the entire solutions; see e.g., [4–12] for reaction–diffusion equations without delay, [13–15] for reaction–diffusion equations with nonlocal delay, [16,17] for delayed lattice differential equations with global interaction, [18,19] for nonlocal dispersal equations, [20–22] for a two-component Lotka–Volterra competition–diffusion system and [23] for a reaction–diffusion system modeling man–environment–man epidemics. In order to consider the spatial spread of a deterministic epidemic in multi-types of population, Rass and Radcliffe [24] presented the nonlocal epidemic model: m  ∂ ui (x, t )  = 1 − ui (x, t ) σk βi,k ∂t k =1



uk (x − y, t )pi,k (y)dy − µi ui (x, t ),

1 ≤ i ≤ m,

(1.1)

R

and its spatially discrete version: dui,j (t ) dt

m +∞    = 1 − ui,j (t ) σk βi,k uk,j−r (t )pi,k (r ) − µi ui,j (t ), k=1

1 ≤ i ≤ m,

(1.2)

r =−∞

where x ∈ R, j ∈ Z, t ∈ R, ui (x, t ) (or ui,j (t )) is the proportion of individuals for the ith population at position x (or j) who are infectious at time t , µi ≥ 0 is the combined death/emigration/recovery rate for infectious individuals, σk ≥ 0 is the

∗

Corresponding author. E-mail addresses: [email protected], [email protected] (S.-L. Wu).

0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.05.009

604

S.-L. Wu, P. Weng / J. Math. Anal. Appl. 394 (2012) 603–615

population size of the kth population, βi,k ≥ 0 is the infection rate of a type i susceptible by a type k infectious individual, and pi,k is the corresponding contact distribution. For more details about biological meaning, we refer to [24–26]. In [24, Chapter 8], Rass and Radcliffe analyzed completely the global dynamics of the spatially homogeneous m-dimensional system associated with (1.1) and (1.2). Weng and Zhao [25] and Zhang and Zhao [26] further considered the spreading speeds and traveling wave fronts of (1.1) and (1.2), respectively. However, the issue of the interaction of traveling wave fronts for such nonlocal systems is still open. In this paper, we consider the interaction of traveling wave fronts for systems (1.1) and (1.2) and establish some new entire solutions to describe the phenomenon. From the viewpoint of epidemiology, this provides a new spread way of the epidemic. For system (1.1), similar to [25], we make the following assumptions:

  (C1 ) pi,k (y) ≥ 0, pi,k (y) = pi,k (−y), ∀y ∈ R, R pi,k (y)dy = 1, and for any λ > 0, R eλy pi,k (y)dy < ∞, 1 ≤ i, k ≤ m. (C2 ) The matrix Λ := (σk βi,k )m×m is irreducible in the sense that for every i ̸= k, there exists a distinct sequence i1 , i2 , . . . , ir with i1 = i, ir = k such that σis+1 βis ,is+1 ̸= 0, 1 ≤ s ≤ r − 1. (C3 ) Either µi = 0 for some i ∈ {1, 2, . . . , m}, or µ = (µ1 , µ2 , . . . , µm ) ≫ 0 and ρ(Γ ) > 1, where Γ = (diag(µ))−1 Λ and ρ(Γ ) = max{|λ| : det(λI − Γ ) = 0} for µ ≫ 0. Throughout this paper, we always use the usual notations for the standard ordering in Rm and denote by ∥·∥ the Euclidean norm in Rm . Now, we recall the results on the traveling wave fronts obtained in [25]. From [25, Lemma 2.1 and Remark 2.1], system (1.1) has exactly two spatially homogeneous equilibria 0 := (0, . . . , 0) ∈ Rm and K := (K1 , . . . , Km ) ∈ Rm with 0 ≪ K ≤ 1 := (1, . . . , 1) ∈ Rm . For any λ ∈ R, define a matrix A(λ) = (Ai,k (λ))m×m , where

   λy  σ β  k i,k e pi,k (y)dy − µi , R  Ai,k (λ) =   σk βi,k eλy pi,k (y)dy,

i = k, (1.3) i ̸= k.

R

Since A(λ) is cooperative and irreducible, M (λ) := max{ℜα : det(α I − A(λ)) = 0} is a simple eigenvalue of A(λ) with a strongly positive eigenvector v(λ) = (v1 (λ), . . . , vm (λ)) (see, e.g., [27, Corollary 4.3.2]). Here and in what follows, ℜα denotes the real part of α . By [25, Lemma 3.3 and Theorem 3.2], there exists c∗ > 0 and λ∗ > 0 such that M (λ∗ ) M (λ) c∗ = = inf ,

λ∗

λ>0

λ

and for any c > c∗ , there exists a unique λ1 := λ1 (c ) ∈ (0, λ∗ ) such that M (λ1 ) = c λ1 , and M (λ) < c λ for any λ ∈ (λ1 , λ∗ ). Furthermore, one can obtain from the argument of [25] that (i) for every c ≥ c∗ , (1.1) admits a non-decreasing traveling wave front Φc (x + ct ) = (φc ,1 (x + ct ), . . . , φc ,m (x + ct )) satisfying Φc (−∞) = 0 and Φc (+∞) = K; and (ii) for any c > c∗ , limξ →−∞ Φc (ξ )e−λ1 (c )ξ = v(λ1 (c )). As mentioned above, there were many results on entire solutions for scalar evolution equations. But little has been done for systems of equations except the works of [20–23], where the existence of entire solutions for some specific twocomponent reaction–diffusion model systems has been established by using comparison principle and constructing a pair of super- and subsolutions. However, it seems difficult, if not impossible, to construct appropriate supersolutions for the m-component nonlocal systems (1.1) and (1.2). To overcome the difficulty, we shall extend the method used in [9] for a scalar KPP equation to the m-component systems (1.1) and (1.2). More precisely, the idea is to study the solutions un (x, t ) and unj (t ) of Cauchy problems for (1.1) and (1.2) starting at times −n(n ∈ N) with appropriate initial conditions, respectively. By constructing subsolutions and appropriate upper estimates, some new entire solutions satisfying some properties are obtained by passing the limit n → ∞. Although our method is similar to the work [9] for the scalar KPP equation, the technique details are different from those in [9]. For example, for the nonlocal system (1.1), since a lack of regularizing effect occurs, the sequence functions {un (x, t )} are not smooth enough with respect to x, and hence its convergence is not ensured. To obtain a convergent subsequence, we have to make {un (x, t )} possess a property which is similar to a global Lipschitz condition with respect to x (Lemma 3.4). For this, we impose the following assumption:

(C4 ) µ ≫ 0 and there exists L > 0 such that for any η > 0,  |pi,k (y + η) − pi,k (y)|dy ≤ Lη, 1 ≤ i, k ≤ m.

(1.4)

R

Clearly, if pi,k (·) ∈ C 1 (R) and pi,k (·) is compactly supported for all 1 ≤ i, k ≤ m, then the  inequality (1.4) holds. In what follows, we say that the functions Up (x, t ) = Up,1 (x, t ), . . . , Up,m (x, t ) converge to a function Up0 (x, t ) =

Up0 ,1 (x, t ), . . . , Up0 ,m (x, t ) as p → p0 ∈ Rn in the sense of the topology T if, for any compact set S ⊂ R2 , the functions





Up (x, t ) and ∂∂t Up (x, t ) converge uniformly in S to Up0 (x, t ) and ∂∂t Up0 (x, t ) as p → p0 . For any N ∈ Z and a ∈ R, let us denote the regions TNi ,a , i = 1, . . . , 6 by TN1,a := [N , +∞) × [a, +∞),

TN2,a := (−∞, N ] × [a, +∞),

TN3,a := R × [a, +∞),

TN4,a := (−∞, N ] × (−∞, a],

TN5,a := [N , +∞) × (−∞, a],

TN6,a := R × (−∞, a].

S.-L. Wu, P. Weng / J. Math. Anal. Appl. 394 (2012) 603–615

For convenience, we also define max{w1 , w2 , w3 }

=

605

max{u1 , u2 , u3 }, max{v1 , v2 , v3 }





and min{w1 , w2 , w3 }

=

min{u1 , u2 , u3 }, min{v1 , v2 , v3 } for wi = (ui , vi ), i = 1, 2, 3. Our main result on the entire solutions of (1.1) are as follows.





Theorem 1.1. Assume that (C1 )–(C4 ) hold.Then, for any h1 , h2 , h3 ∈ R, c1 , c2 > c∗ , and χ1 , χ2 , χ ∈ {0, 1} with χ1 +χ2 +χ ≥ 2, there exists an entire solution U (x, t ) := U1 (x, t ), . . . , Um (x, t ) of (1.1) such that max χ1 Φc1 x + c1 t + h1 , χ2 Φc2 −x + c2 t + h2 , χ Γ (t + h3 )













  ≤ U (x, t ) ≤ min K, Πχ1 (x, t ), Πχ2 (x, t ), Πχ (x, t ) ,

(x, t ) ∈ R2 ,

(1.5)

where Γ (t ) is the spatially independent solution of (1.1) given in Theorem 2.4 and

  ∗ Πχ1 (x, t ) = χ1 Φc1 x + c1 t + h1 + χ2 Qc2 eλ1 (c2 )(−x+c2 t +h2 ) v(λ1 (c2 )) + χ eλ (t +h3 ) v ∗ ,   ∗ Πχ2 (x, t ) = χ1 Qc1 eλ1 (c1 )(x+c1 t +h1 ) v(λ1 (c1 )) + χ2 Φc2 −x + c2 t + h2 + χ eλ (t +h3 ) v ∗ , Πχ (x, t ) = χ1 Qc1 eλ1 (c1 )(x+c1 t +h1 ) v(λ1 (c1 )) + χ2 Qc2 eλ1 (c2 )(−x+c2 t +h2 ) v(λ1 (c2 )) + χ Γ (t + h3 ).   Here, Qc = inf Q > 0 : Φc (ξ ) ≤ Qeλ1 (c )ξ v(λ1 (c )) for ξ ∈ R , and λ∗ = M (0), v ∗ = v(0). Furthermore, the following statements hold. (i) There exists a constant L′ > 0 such that for any η > 0, sup ∥U (x + η, t ) − U (x, t )∥ ≤ L′ η

and

(x,t )∈R2

sup ∥Ut (x + η, t ) − Ut (x, t )∥ ≤ L′ η.

(x,t )∈R2

(ii) ∂∂t Ui (x, t ) > 0 for all (x, t ) ∈ R2 , 1 ≤ i ≤ m. (iii) If χ = 0, then limt →+∞ sup|x|≤N ∥U (x, t ) − K∥ = 0 for any N ∈ N; if χ = 1, then limt →+∞ supx∈R ∥U (x, t ) − K∥ = 0, limt →−∞ sup|x|≤N ∥U (x, t )∥ = 0 for any N ∈ N. (iv) If χ = 1, then for every x ∈ R, U (x, t ) ∼ Γ (t + h3 ) ∼ v ∗ eλ (t +h3 ) as t → −∞. (v) If χ = 0, then for every x ∈ R, there exist B(x), D(x) ∈ Rm with D(x) ≫ B(x) ≫ 0 such that for all t ≪ −1, ∗

B(x)eϑ(c1 ,c2 )t ≤ U (x, t ) ≤ D(x)eϑ(c1 ,c2 )t , where ϑ(c1 , c2 ) = min c1 λ1 (c1 ), c2 λ1 (c2 ) . (vi) For any x, t ∈ R, U (x, t ) is increasing with respect to hi , i = 1, 2, 3. (vii) U (x, t ) converges to K as hi → +∞ in T and uniformly on (x, t ) ∈ TNi ,a for any N , a ∈ R, i = 1, 2, 3. (viii) If we denote U (x, t ) by Uc1 ,c2 ,h1 ,h2 ,h3 (x, t ) when χ1 = χ2 = χ = 1 (Similarly, we denote Uc1 ,c2 ,h1 ,h2 (x, t ), Uc1 ,h1 ,h3 (x, t ) and Uc2 ,h2 ,h3 (x, t )), then Uc1 ,c2 ,h1 ,h2 ,h3 (x, t ) converges to





(a) Uc2 ,h2 ,h3 (x, t ) as h1 → −∞ in T and uniformly on (x, t ) ∈ TN4,a for any N , a ∈ R; (b) Uc1 ,h1 ,h3 (x, t ) as h2 → −∞ in T and uniformly on (x, t ) ∈ TN5,a for any N , a ∈ R; and (c) Uc1 ,c2 ,h1 ,h2 (x, t ) as h3 → −∞ in T and uniformly on (x, t ) ∈ TN6,a for any N , a ∈ R. (ix) Uc1 ,c2 ,h1 ,h2 (x, t ) converges to Φc1 x + c1 t + h1 as h2 → −∞ in T and uniformly on (x, t ) ∈ TN5,a for any N , a ∈ R and   converges to Φc2 −x + c2 t + h2 as h1 → −∞ in T and uniformly on (x, t ) ∈ TN4,a for any N , a ∈ R; Uc2 ,h2 ,h3 (x, t )





converges to Φc2 −x + c2 t + h2 as h3 → −∞ in T and uniformly on (x, t ) ∈ TN6,a for any N , a ∈ R and converges to Γ (t + h3 ) as h2 → −∞ in T and uniformly on (x, t ) ∈ TN5,a for any N , a ∈ R. Similar results hold for Uc1 ,h1 ,h3 (x, t ).





(x) For any (c1∗ , c2∗ ) ̸= (c1 , c2 ) with ci∗ > c∗ , i = 1, 2, there is no (x0 , t0 ) ∈ R2 such that Uc ∗ ,c ∗ ,h1 ,h2 (·, ·) = Uc1 ,c2 ,h1 ,h2 (· + 1 2 x0 , · + t0 ) on R2 . (xi) For any h∗1 , h∗2 ∈ R, there exists (x0 , t0 ) ∈ R2 depending on c1 , c2 , h1 , h2 , h∗1 , h∗2 such that Uc1 ,c2 ,h∗ ,h∗ (·, ·) = Uc1 ,c2 ,h1 ,h2 (·+ 1

x0 , · + t0 ) on R2 .

2

For χ = 1, we denote the entire solution obtained in Theorem 1.1 by UΓ (x, t ). Clearly, the entire solutions UΓ (x, t ) and Uc1 ,c2 ,h1 ,h2 (x, t ) given in Theorem 1.1 are completely different, because they have different decay rates when t tends to minus infinity according to the assertions (iv) and (v) and Lemma 2.1. To obtain the existence and qualitative properties of entire solutions for (1.2). We replace the condition (C1 ) with its discrete version (C′1 ):

(C′1 ) pi,k (r ) ≥ 0, pi,k (r ) = pi,k (−r ), ∀r ∈ Z,

+∞

r =−∞

pi,k (r ) = 1, and for any λ > 0,

+∞

r =−∞

eλr pi,k (r ) < ∞, 1 ≤ i, k ≤ m.

606

S.-L. Wu, P. Weng / J. Math. Anal. Appl. 394 (2012) 603–615

Under the assumptions (C′1 ), (C2 ) and (C3 ), it is easy to see that system (1.2) also has exactly two spatially homogeneous equilibria 0 and K. For any λ ∈ R, define a matrix A¯ (λ) = (A¯ i,k (λ))m×m , where

A¯ i,k (λ) =

 +∞     eλr pi,k (r ) − µi , σk βi,k 

i = k,

+∞      σ β eλr pi,k (r ), k i , k 

i ̸= k.

r =−∞

(1.6)

r =−∞

¯ (λ) := max{ℜα : det(α I − A¯ (λ)) = 0} is a simple eigenvalue of A¯ (λ) with Since A¯ (λ) is cooperative and irreducible, M a strongly positive eigenvector v(λ) = (v 1 (λ), . . . , v m (λ)). By appealing to the theory of spreading speeds and traveling ¯ ∗ > 0 such that waves for monotone semiflows [28], it is shown in [26] that there exists c¯∗ > 0 and λ ¯ (λ) M = inf ¯λ∗ λ>0 λ ¯ (λ3 ) = c λ3 and M ¯ (λ) < c λ for any λ ∈ (λ3 , λ¯ ∗ ]. Moreover, ¯ ∗ ) such that M and for any c > c¯∗ , there exists λ3 := λ3 (c ) ∈ (0, λ (1.2) admits a non-decreasing traveling wave front Ψc (j + ct ) = (ψc ,1 (j + ct ), . . . , ψc ,m (j + ct )) satisfying Ψc (−∞) = 0 and Ψc (+∞) = K for every c ≥ c¯∗ . Using the monotone iteration technique coupled with the method of upper and lower solutions as in [25], we can further show that limξ →−∞ Ψc (ξ )e−λ3 (c )ξ = v¯ (λ3 (c )) for c > c¯∗ . We also say that the functions Up (t ) = {Uj,p (t )}j∈Z = {(U1,j,p (t ), . . . , Um,j,p (t ))}j∈Z converge to a function Up0 (t ) = {Uj,p0 (t )}j∈Z = {(U1,j,p0 (t ), . . . , Um,j,p0 (t ))}j∈Z as p → p0 ∈ Rn in the sense of the topology T if, for any compact set S ⊂ Z × R, the functions Uj,p (t ) and dtd Uj,p (t ) converge uniformly in S to Uj,p0 (t ) and dtd Uj,p0 (t ) as p → p0 . c¯∗ =

¯ (λ¯ ∗ ) M

The following theorem is our result on entire solutions for (1.2).

Theorem 1.2. Assume that (C′1 ), (C2 ) and (C3 ) hold. Then, for any h1 , h2 , h3 ∈ R, c1 , c2 > c¯∗ , and χ1 , χ2 , χ ∈ {0, 1} with χ1 + χ2 + χ ≥ 2, there exists an entire solution U (t ) = {Uj (t )}j∈Z = {(U1,j (t ), . . . , Um,j (t ))}j∈Z of (1.2) such that max χ1 Ψc1 j + c1 t + h1 , χ2 Ψc2 −j + c2 t + h2 , χ Γ (t + h3 )











  ¯ χ1 (j, t ), Π ¯ χ2 (j, t ), Π ¯ χ (j, t ) , ≤ Uj (t ) ≤ min K, Π



(j, t ) ∈ Z × R,

(1.7)

where Γ (t ) is the spatially independent solution of (1.2) given in Theorem 2.4 and

  ¯ χ1 (j, t ) = χ1 Ψc1 j + c1 t + h1 + χ2 Rc2 eλ3 (c2 )(−j+c2 t +h2 ) v¯ (λ3 (c2 )) + χ eλ∗ (t +h3 ) v ∗ , Π   ¯ χ2 (j, t ) = χ1 Rc1 eλ3 (c1 )(j+c1 t +h1 ) v¯ (λ3 (c1 )) + χ2 Ψc2 −j + c2 t + h2 + χ eλ∗ (t +h3 ) v ∗ , Π ¯ χ (j, t ) = χ1 Rc1 eλ3 (c1 )(j+c1 t +h1 ) v¯ (λ3 (c1 )) + χ2 Rc2 eλ3 (c2 )(−j+c2 t +h2 ) v¯ (λ3 (c2 )) + χ Γ (t + h3 ). Π   ¯ (0), v ∗ = v¯ (0). Here, Rc = inf R > 0 : Ψc (ξ ) ≤ Reλ3 (c )ξ v¯ (λ3 (c )) for ξ ∈ R , and λ∗ = M In particular, similar results as the assertions (ii)–(x) in Theorem 1.1 for U (·, ·) hold true for U (·) and (xi)′ if we denote {Uj (t )}j∈Z by {Uj;c1 ,c2 ,h1 ,h2 (t )}j∈Z when χ1 = χ2 = 1 and χ = 0, then for any h1 , h2 , h∗1 , h∗2 ∈ R, there exists (j0 , t0 ) ∈ Z × R, depending on c1 , c2 , h1 , h2 , h∗1 , h∗2 , such that Uj;c1 ,c2 ,h1 ,h2 (t ) = Uj+j0 ;c1 ,c2 ,h∗ ,h∗ (t + t0 ) for all (j, t ) ∈ Z × R 1

if and only if

c2 (h1 −h∗ )−c1 (h2 −h∗2 ) 1 c1 + c2

2

∈ Z.

The rest of the paper is organized as follows. In Section 2, we prove the existence and asymptotic behavior of spatially independent solutions connecting 0 and K. We mention that the existence of such solutions can be obtained by using the monotone dynamical systems theory (see e.g., [25]). However, these results do not give the exponential decay rate of the solution at minus infinity. To overcome the shortcoming, we shall use the standard monotone iteration technique to prove the existence and asymptotic behavior of the spatially independent solutions. Section 3 is devoted to the proof of Theorem 1.1. We first state some existences and comparison theorems, and give some prior estimates on solutions. We also establish a property for the solutions of Cauchy problem of (1.1) which is similar to a global Lipschitz condition with respect to x (Lemma 3.4). Then Theorem 1.1 is proved by using the comparison principle with appropriate subsolutions and upper estimates. In Section 4, we prove Theorem 1.2. We only sketch the outline and prove those different from Theorem 1.1. 2. Properties of spatially independent solutions In this section, we consider the spatially independent solutions connecting 0 and K of (1.1) and (1.2), that is, solutions of the spatially homogeneous system dui (t ) dt

m   σk βi,k uk (t ) − µi ui (t ), = 1 − ui (t ) k=1

1 ≤ i ≤ m.

(2.1)

S.-L. Wu, P. Weng / J. Math. Anal. Appl. 394 (2012) 603–615

607

We shall use the standard monotone iteration technique coupled with the method of upper and lower solutions to prove the existence and asymptotic behavior of solutions of (2.1). Define a matrix A = (Ai,k )m×m , where

 Ai,k =

σk βi,k − µi , σk βi,k ,

i = k, i ̸= k.

(2.2)

Since A is cooperative and irreducible, λ∗ := max{ℜα : det(α I − A) = 0} is a simple eigenvalue of A with a strongly positive ∗ ¯ (0), v ∗ = v(0) = v¯ (0). eigenvector v ∗ = (v1∗ , . . . , vm ). Note, A = A(0) = A¯ (0) and λ∗ = M (0) = M Lemma 2.1. For any c > c∗ , c λ1 (c ) > λ∗ and for any c > c¯∗ , c λ3 (c ) > λ∗ , where λ1 (c ) and λ3 (c ) are defined as in Section 1. Proof. We only prove that c λ1 (c ) > λ∗ for any c > c∗ , because the other assertion is similar. For any i ̸= k ∈ {1, . . . , m} and λ > 0, it follows from (C1 ) that Ai,k (λ) = σk βi,k



λy

e pi,k (y)dy = σk βi,k

∞



(eλy + e−λy )pi,k (y)dy > σk βi,k = Ai,k .

0

R

Similarly, we can show that Ai,k (λ) > Ai,k for any i = k ∈ {1, . . . , m} and λ > 0. Thus, A(λ) > A for any λ > 0. Recall that A(λ) and A are cooperative and irreducible,

λ∗ = max{ℜα : det(α I − A) = 0} and M (λ) = max{ℜα : det(α I − A(λ)) = 0}. It follows from [27, Corollary 4.3.2] that M (λ) > λ∗ for λ > 0. Therefore, for any c > c∗ , we have c λ1 (c ) = M (λ1 (c )) > λ∗ . The proof is complete.  In what follows, we denote W = [0, 1]m . Let C (R, Rm ) be the spaces of continuous real functions on R. Define the operator T = (T1 , . . . , Tm ) : C (R, W ) → C (R, Rm ) by t



Ti (u)(t ) =

e−(µi +δ)(t −s) Hi (u)(s)ds,

1 ≤ i ≤ m,

−∞

where

δ = max

1≤i≤m

m 

σk βi,k and Hi (u)(t ) = (1 − ui (t ))

k=1

m 

σk βi,k uk (t ) + δ ui (t ).

k=1

It is easy to verify that each Hi (·) is a nondecreasing map from C (R, W ) to C (R, R) with respect to the point-wise ordering. The following observation is straightforward. Lemma 2.2. (i) T : C (R, W ) → C (R, W ). (ii) If φ, ψ ∈ C (R, W ) with φ(·) ≥ ψ(·), then T (φ)(·) ≥ T (ψ)(·). (iii) If φ ∈ C (R, W ) is non-decreasing, then so is T (φ). For any fixed γ ∈ (1, 2], take



m 

1



max σk βi,k vk . (γ − 1)λ∗ 1≤i≤m k=1     Define two functions φ(t ) = φ 1 (t ), . . . , φ m (t ) and φ(t ) = φ (t ), . . . , φ (t ) as follows: 1 m     ∗ λ∗ t ∗ γ λ∗ t ∗ λ∗ t φ i (t ) = min Ki , vi e and φ (t ) = max 0, vi e − qvi e , t ∈ R. i q > max 1,

∗

Lemma 2.3. 0 ≤ φ(t ) ≤ φ(t ) ≤ K, T (φ)(t ) ≤ φ(t ) and T (φ)(t ) ≥ φ(t ) for all t ∈ R. Proof. Clearly, 0 ≤ φ(t ) ≤ φ(t ) ≤ K for all t ∈ R. We first show that T (φ)(t ) ≥ φ(t ) for all t ∈ R. Let t0 = − (γ −11)λ∗ ln q < 0. Obviously,

φ i (t ) = 0 for t > t0 and φ i (t ) = vi∗ eλ t − qvi∗ eγ λ ∗

∗t

for t ≤ t0 , 1 ≤ i ≤ m.

It is easy to see that T (φ)i (t ) ≥ 0 for all t ∈ R. Next, we show that Ti (φ)(t ) ≥ vi∗ eλ

∗t

Av ∗ = λ∗ v ∗ yields that

− qvi∗ eγ λ t for all t ≤ t0 . Note that ∗

σk βi,k vk∗ = (λ∗ + µi )vi∗ for all i = 1, . . . , m. For t ≤ t0 , direct calculation shows that  ∗  ∗ ∗ Hi (φ)(t ) ≥ (δ + λ∗ + µi )vi∗ eλ t − qeγ λ t − L2 e2λ t , m

k=1

608

S.-L. Wu, P. Weng / J. Math. Anal. Appl. 394 (2012) 603–615

where L2 = vi

m ∗

k=1

Ti (φ)(t ) ≥

σk βi,k vk∗ . Hence, for t ≤ t0 ,

t







e−(µi +δ)(t −s) (δ + λ∗ + µi )vi∗ eλ

∗s

− qeγ λ

∗s



− L2 e2λ

∗s



ds

−∞

= vi∗ eλ t − ∗

δ + λ ∗ + µi L2 ∗ ∗ qvi∗ eγ λ t − e2λ t ∗ ∗ δ + γ λ + µi δ + 2λ + µi

≥ vi∗ eλ t − qvi∗ eγ λ t . ∗

∗

Therefore, T (φ)(t ) ≥ φ(t ) for t ∈ R. Similarly, we can show that T (φ)(t ) ≤ φ(t ) for all t ∈ R. The proof is complete.



Theorem 2.4. Let (C2 ) and (C3 ) hold. Then, (2.1) has a non-decreasing solution Γ (t ) = Γ1 (t ), . . . , Γm (t ) which satisfies



∗t

lim Γ (t )e−λ

t →−∞

= v∗ ,



Γ ′ (t ) ≫ 0 and Γ (t ) ≤ eλ t v ∗ for all t ∈ R. ∗

Γ (+∞) = K,

Proof. By  using the monotone  iteration technique similar to that of [25, Theorem 4.1], we can get a non-decreasing solution Γ (t ) = Γ1 (t ), . . . , Γm (t ) of (2.1) which satisfies 0 ≤ φ(t ) ≤ Γ (t ) ≤ φ(t ) ≤ K ≤ 1 for all t ∈ R. ∗t

One can easily see that Γ (+∞) = K, limt →−∞ Γ (t )e−λ that Γ ′ (t ) ≫ 0 for all t ∈ R. In view of

Γi′′ (t ) = −Γi′ (t )

m 

σk βi,k Γk (t ) + (1 − Γi (t ))

k=1

 ≥ − µi +

m 

m 

= v ∗ , Γ ′ (t ) ≥ 0, and Γ (t ) ≤ eλ t v ∗ for all t ∈ R. Next, we show ∗

σk βi,k Γk′ (t ) − µi Γi′ (t )

k=1

 σk βi,k Γi′ (t ) := −σ Γi′ (t ),

k=1

we have

Γi′ (t ) ≥ Γi′ (τ )e−σ (t −τ ) ≥ 0 for any τ < t . Suppose for the contrary that there exist i0 ∈ {1, . . . , m} and t0 ∈ R such that Γi0′ (t0 ) = 0, then Γi0′ (τ ) = 0 for all τ < t0 , ∗t

which contradicts the fact limt →−∞ Γi0 (t )e−λ complete. 

= vi∗0 > 0. Therefore, Γi′ (t ) > 0 for all t ∈ R, 1 ≤ i ≤ m. The proof is

¯ (0) = Remark 2.5. We mention here that the spatially homogeneous system of (1.2) is also (2.1), and thus λ∗ = M M (0), v ∗ = v¯ (0) = v(0). 3. Proof of Theorem 1.1 This section is devoted to the proof of Theorem 1.1. We first state some existences and comparison theorems, and give some prior estimates on solutions. We also establish a property for the solutions of Cauchy problem of (1.1) which is similar to a global Lipschitz condition with respect to x. The technique is inspired by the works of Bates et al. [29] and Li et al. [18]. Finally, Theorem 1.1 is proved by using comparison principle with appropriate subsolutions and upper estimates. Throughout this section, we always assume (C1 )–(C4 ). 3.1. Preliminaries Consider the initial value problem of (1.1)

  m    ∂ ui (x, t ) = 1 − u (x, t ) σ β uk (x − y, t )pi,k (y)dy − µi ui (x, t ), i k i ,k ∂t R k=1   u(x, τ ) = ϕ(x) = (ϕ1 (x), . . . , ϕm (x)), 1 ≤ i ≤ m,

(3.1)

where x ∈ R, t > τ , τ ∈ R is a given constant. In the following, we denote the solution of (3.1) by u(x, t ; ϕ) and ∂ u (x,t ) ∂ u (x,t ) ut (x, t ) = ( 1∂ t , . . . , m∂ t ). Recall that W = [0, 1]m .

S.-L. Wu, P. Weng / J. Math. Anal. Appl. 394 (2012) 603–615

609

Definition 3.1. A function u ∈ C (R ×[τ , +∞), W ) is called an upper solution of (3.1) if it satisfies ut ∈ C (R ×[τ , +∞), Rm ) and for any x ∈ R and t > τ , m  ∂ ui (x, t )  ≥ 1 − ui (x, t ) σk βi,k ∂t k=1



uk (x − y, t )pi,k (y)dy − µi ui (x, t ),

1 ≤ i ≤ m.

R

A lower solution of (3.1) is defined by reversing the inequality. The following result on the existence and uniqueness of solutions of (3.1) and the comparison principle is a direct correspondence of [25, Theorems 2.1 and 2.2]. Proposition 3.2. (i) For any ϕ ∈ C R, W , (3.1) admits a unique solution u(x, t ; ϕ) satisfying u(·, τ ; ϕ) = ϕ(·), u(·, ·; ϕ) ∈





C R × [τ , +∞), W and ut (·, ·; ϕ) ∈ C (R × [τ , +∞), Rm ). (ii) Let w + (x, t ) and w − (x, t ) be an upper solution and a lower solution of (3.1), respectively. If w + (·, τ ) ≥ w − (·, τ ), then w+ (·, t ) ≥ w − (·, t ) for all t ≥ τ .





Lemma 3.3. Assume that u(x, t ; ϕ) is a solution of (3.1) with the initial value ϕ ∈ C R, W , then there exists a positive constant M1 , independent of τ and ϕ , such that for any x ∈ R and t > τ + 1,



∥ut (x, t ; ϕ)∥ ≤ M1 and



∥utt (x, t ; ϕ)∥ ≤ M1 .

The proof of Lemma 3.3 is direct. So we omit it. As mentioned in the induction, the solution u(x, t ; ϕ) of the Cauchy problem (3.1) is not smooth enough with respect to x. We show that u(x, t ; ϕ) possess such a property which is similar to a global Lipschitz condition with respect to x under the conditions (C1 )–(C4 ). Lemma 3.4. Let u(x, t ; ϕ) be the solution of (3.1) with initial value ϕ ∈ C R, W . If there exists a constant M > 0 such that for any η > 0, supx∈R ∥ϕ(x + η) − ϕ(x)∥ ≤ M η, then for any η > 0,





sup max {∥u(x + η, t ; ϕ) − u(x, t ; ϕ)∥, ∥ut (x + η, t ; ϕ) − ut (x, t ; ϕ)∥} ≤ M ′ η,

x∈R,t ≥τ

where M ′ > 0 is a constant which is independent of τ , ϕ and η. Proof. For any given η > 0, let w(x, t ) = u(x + η, t ; ϕ) − u(x, t ; ϕ). For simplicity, we denote u(x + η, t ; ϕ) and u(x, t ; ϕ) by u(x + η, t ) and u(x, t ), respectively. Clearly, |wi (x, τ )| = |ϕi (x + η) − ϕi (x)| ≤ M η for all x ∈ R, 1 ≤ i ≤ m. From Proposition 3.2, 0 ≤ ui (x + η, t ), ui (x, t ) ≤ 1 for all x ∈ R and t ≥ τ . It follows from (C4 ) that

  m m     ∂wi (x, t ) = σk βi,k uk (y, t ) pi,k (x + η − y) − pi,k (x − y) dy − wi (x, t ) σk βi,k uk (y, t )pi,k ∂t R R k=1 k=1  m    × (x + η − y)dy − ui (x, t ) σk βi,k uk (y, t ) pi,k (x + η − y) − pi,k (x − y) dy − µi wi (x, t ) R

k=1 m

≤2



σk βi,k



uk (y, t )|pi,k (x + η − y) − pi,k (x − y)|dy − hi (x, t )wi (x, t ), R

k=1

≤ M1′ η − hi (x, t )wi (x, t ), m for x ∈ R and t ≥ τ , where M1′ = 2L k=1 σk βi,k (L is defined in (C4 )) and  m  hi (x, t ) = µi + σk βi,k uk (y, t )pi,k (x + η − y)dy. R

k=1

Simple calculations show that, for x ∈ R and t ≥ τ ,

wi (x, t ) ≤ wi (x, τ )e−

t

τ hi (x,s)ds

≤ M ηe−µi (t −τ ) +

t

 + τ

t

 τ

M1′ ηe−

t

s hi (x,r )dr

ds

M1′ ηe−µi (t −s) ds

  = M ηe−µi (t −τ ) + M1′ η 1 − e−µi (t −τ ) /µi := wi (t ). Similarly, we can show that −wi (x, t ) ≤ w i (t ) for x ∈ R and t ≥ τ . Therefore, we get

|wi (x, t )| ≤ wi (t ) ≤ M2′ η := (M µi + M1′ )η/µi for all x ∈ R, t ≥ τ .

610

S.-L. Wu, P. Weng / J. Math. Anal. Appl. 394 (2012) 603–615

Moreover, for any x ∈ R and t ≥ τ , there holds

   ∂ ui  ∂ ui    ∂ t (x + η, t ; ϕ) − ∂ t (x, t ; ϕ)    m m   σk βi,k + µi |wi (x, t )| ≤2 σk βi,k |pi,k (x + η − y) − pi,k (x − y)|dy + R

k=1

 ≤ M3 η := 2L ′

m 

k=1

 σk βi,k +

k=1

Take M ′ =

m 

 σk βi,k + µi

M µi + M1′



µi

k=1

η.

√

m max{M2′ , M3′ } and the assertion holds. The proof is complete.



Lemma 3.5. Take u± (x, t )

   ± = (u± . If u+ ∈ C R × [τ , +∞), [0, +∞)m and u− ∈ C R × 1(x, t ), . . . , um (x, t ))  m [τ , +∞), (−∞, 1] satisfy u± , u+ (·, τ ) ≥ u− (·, τ ) and t ∈ C R × [τ , +∞), R  m  ∂ u+ + i (x, t ) 1 ≤ i ≤ m, ≥ σk βi,k u+ k (x − y, t )pi,k (y)dy − µi ui (x, t ), ∂t R k=1  m  ∂ u− − i (x, t ) 1 ≤ i ≤ m, σk βi,k u− ≤ k (x − y, t )pi,k (y)dy − µi ui (x, t ), ∂t R k=1  m

for x ∈ R and t > τ , then u+ (x, t ) ≥ u− (x, t ) for all x ∈ R and t ≥ τ . Proof. The proof is similar to that of [25, Theorem 2.2]. We omit it here.



3.2. Proof of Theorem 1.1 n For any n ∈ N, let U n (x, t ) = U1n (x, t ), . . . , Um (x, t ) be the unique solution of the following initial value problem





 n  m   ∂ Ui (x, t )  = 1 − Uin (x, t ) σk βi,k Ukn (x − y, t )pi,k (y)dy − µi Uin (x, t ), ∂t R k=1  n Ui (x, −n) = ϕin (x), 1 ≤ i ≤ m,

(3.2)

where x ∈ R and t > −n, and

      ϕ n (x) = max χ1 Φc1 x − c1 n + h1 , χ2 Φc2 −x − c2 n + h2 , χ Γ (−n + h3 ) .   Here, Φc (ξ ) = Φc x + ct is the traveling wavefront of (1.1) for c > c∗ decided in [25], and Γ (t ) is the solution of (2.1) decided in Theorem 2.4. Furthermore, we denote u(x, t ) := max χ1 Φc1 x + c1 t + h1 , χ2 Φc2 −x + c2 t + h2 , χ Γ (t + h3 ) .













From Proposition 3.2, u(x, t ) ≤ U n (x, t ) ≤ K for all x ∈ R and t ≥ −n. The following result gives an appropriate upper estimate of U n (x, t ). Lemma 3.6. The function U n (x, t ) satisfies U n (x, t ) ≤ min K, Πχ1 (x, t ), Πχ2 (x, t ), Πχ (x, t )





for x ∈ R and t ≥ −n, where Πχ1 (x, t ), Πχ2 (x, t ), Πχ (x, t ) are defined as in Theorem 1.1. Proof. We only prove U n (x, t ) ≤ Πχ1 (x, t ) for all x ∈ R and t ≥ −n. Assume χ1 = 1. Set n Z n (x, t ) = Z1n (x, t ), . . . , Zm (x, t ) := U n (x, t ) − Φc1 x + c1 t + h1 .









Simple calculations show that

  m n   ∂ Zi (x, t ) ≤  σ β Zkn (x − y, t )pi,k (y)dy − µi Zin (x, t ) k i ,k ∂t R k = 1  Z n (x, −n) = ϕ n (x) − Φ x − c n + h , c1 1 1 where x ∈ R and t > −n. Take

  ∗ V (x, t ) = V1 (x, t ), . . . , Vm (x, t ) := χ2 Qc2 eλ1 (c2 )(−x+c2 t +h2 ) v(λ1 (c2 )) + χ eλ (t +h3 ) v ∗ ,

S.-L. Wu, P. Weng / J. Math. Anal. Appl. 394 (2012) 603–615

611

where Qc is defined in Theorem 1.1. In view of Av ∗ = λ∗ v ∗ and A(λ1 (c2 ))v(λ1 (c2 )) = M (λ1 (c2 ))v(λ1 (c2 )) = c2 λ1 (c2 )v(λ1 (c2 )), it is easy to verify that m ∂ Vi (x, t )  = σk βi,k ∂t k=1



Vk (x − y, t )pi,k (y)dy − µi Vi (x, t ),

1 ≤ i ≤ m,

(3.3)

R

∗ where x ∈ R and t > −n. According to Φc2 (z ) ≤ Qc2 eλ1 (c2 )z v(λ1 (c2 )) and Γ (z ) ≤ eλ z v ∗ for all z ∈ R, we have ∗ V (x, −n) = χ2 Qc2 eλ1 (c2 )(−x−c2 n+h2 ) v(λ1 (c2 )) + χ v ∗ eλ (−n+h3 )

  ≥ χ2 Φc2 −x − c2 n + h2 + χ Γ (−n + h3 )   ≥ ϕ n (x) − Φc1 x − c1 n + h1 = Z n (x, −n). It then follows from Lemma 3.5 that Z n (x, t ) ≤ V (x, t ) for all x ∈ R and t ≥ −n, that is,

  ∗ U n (x, t ) ≤ Φc1 x + c1 t + h1 + χ2 Qc2 eλ1 (c2 )(−x+c2 t +h2 ) v(λ1 (c2 )) + χ eλ (t +h3 ) v ∗ = Πχ1 (x, t ). When χ1 = 0, the assertion U n (x, t ) ≤ Πχ1 (x, t ) reduces to ∗ U n (x, t ) ≤ χ2 Qc2 eλ1 (c2 )(−x+c2 t +h2 ) v(λ1 (c2 )) + χ eλ (t +h3 ) v ∗ ,

which holds obviously. The proof is complete.



Proof of Theorem 1.1. By Proposition 3.2 and Lemma 3.6, we have u(x, t ) ≤ U n (x, t ) ≤ U n+1 (x, t ) ≤ min K, Πχ1 (x, t ), Πχ2 (x, t ), Πχ (x, t )





for all x ∈ R and t ≥ −n. Note that the traveling wave Φc (ξ ) = (φc ,1 (ξ ), . . . , φc ,m (ξ )) of (1.1) satisfies the wave profile system c

dφc ,i (ξ ) dξ

= (1 − φc ,i (ξ ))

m 

σk βi,k

k=1



φc ,k (ξ − u)pik (u)du − µi φc ,i (ξ ),

1 ≤ i ≤ m.

(3.4)

R

Following from (3.4) and (2.1), we have that

µi +

m  k=1

|φc′ ,i (t )| ≤

c

σk βi,k and

|Γi′ (t )| ≤ µi +

m 

σk βi,k ,

1 ≤ i ≤ m,

k=1

for all t ∈ R. Consequently, there exists a constant M > 0 such that for any η > 0, supx∈R ∥ϕ n (x + η) − ϕ n (x)∥ ≤ M η. It follows from Lemma 3.4 that there exists a constant M ′ > 0, independent of n, such that for any η > 0, sup

x∈R,t ≥−n

max ∥U n (x + η, t ) − U n (x, t )∥, ∥Utn (x + η, t ) − Utn (x, t )∥ ≤ M ′ η.





(3.5)

Combining Lemma 3.3, by Arzela–Ascoli Theorem, there exists a subsequence {U nk (x, t )}k∈N of {U n (x, t )}n∈N such that  nk U (x, t ) converges to a function U (x, t ) = U1 (x, t ), . . . , Um (x, t ) in T . In view of U n (x, t ) ≤ U n+1 (x, t ) for any t ≥ −n, we have limn→+∞ U n (x, t ) = U (x, t ) for any (x, t ) ∈ R2 . The limit function is unique, whence all of the functions U n (x, t ) converge to the function U (x, t ) in T as n → +∞. Clearly, U (x, t ) is an entire solution of (1.1) satisfying (1.5). Now we prove the assertion (i). Since limn→+∞ U n (x, t ) = U (x, t ) and limn→+∞ U n (x + η, t ) = U (x + η, t ), it follows from (3.5) that sup ∥U (x + η, t ) − U (x, t )∥ ≤ M ′ η

and

(x,t )∈R2

sup ∥Utn (x + η, t ) − Utn (x, t )∥ ≤ M ′ η.

(x,t )∈R2

Take L′ = M ′ and the assertion follows. Next, we prove the assertion (ii). Since U n (x, t ) ≥ u(x, t ) ≥ u(x, −n) = ϕ n (x) = U n (x, −n) for all (x, t ) ∈ R ×[−n, +∞), we have ∂∂t U n (x, t ) ≥ 0 for (x, t ) ∈ R × (−n, +∞) by the order-preserving of the solution semiflow (see Proposition 3.2). This yields ∂∂t U (x, t ) ≥ 0 for all (x, t ) ∈ R2 . Simple calculation shows that for i = 1, . . . , m,

  m  ∂(Ui )t (x, t ) ≥ − µi + σk βi,k (Ui )t (x, t ), ∂t k=1

∀(x, t ) ∈ R2

612

S.-L. Wu, P. Weng / J. Math. Anal. Appl. 394 (2012) 603–615

which yields that

(Ui )t (x, t ) ≥ (Ui )t (x, τ )e−(µi +

m

k=1

σk βi,k )(t −τ )

≥ 0 for any τ ≤ t .

(3.6)

Suppose for the contrary that there exist i0 ∈ {1, . . . , m} and (x0 , t0 ) ∈ R2 such that (Ui0 )t (x0 , t0 ) = 0, it then follows from (3.6) that (Ui0 )t (x0 , τ ) = 0 for all τ ≤ t0 . Hence Ui0 (x0 , t ) = Ui0 (x0 , t0 ) for all t ≤ t0 , which implies that limt →−∞ Ui0 (x0 , t ) = Ui0 (x0 , t0 ). But following from (1.5), limt →−∞ Ui0 (x0 , t ) = 0 and Ui0 (x0 , t0 ) > 0. This contradiction yields that ∂∂t Ui (x, t ) > 0 for all (x, t ) ∈ R2 , 1 ≤ i ≤ m. The assertion (iv) follows from Lemma 2.1 and (1.5). Moreover, using (1.5), the proofs for (iii), (v)–(vii), and (xi) in Theorem 1.1 are straightforward and thus omitted. We now prove (viii). We first show that the statement (c) holds. For simplicity, we denote Uc1 ,c2 ,h1 ,h2 ,h3 (x, t ) by U (x, t ; h3 ) and denote Uc1 ,c2 ,h1 ,h2 (x, t ) by U (x, t ; −∞), respectively. For χ ∈ {0, 1}, we denote ϕ n (x) by ϕ n (x)χ and U n (x, t ) by U n (x, t )χ , respectively. Set

wn (x, t ) = (w1n (x, t ), . . . , wmn (x, t )) := U n (x, t )1 − U n (x, t )0 . Then 0 ≤ w n (x, t ) ≤ K for all (x, t ) ∈ R × [−n, +∞) and satisfies m ∂win (x, t )  ≤ σk βi,k ∂t k=1



wkn (x − y, t )pi,k (y)dy − µi win (x, t ),

1 ≤ i ≤ m,

(3.7)

R

where x ∈ R and t > −n. Note that ∗ 0 ≤ w n (x, −n) = ϕ n (x)1 − ϕ n (x)0 ≤ Γ (−n + h3 ) ≤ eλ (−n+h3 ) v ∗ ∗ and the function  un (x, t ) =  un1 (x, t ), . . . , unm (x, t ) := eλ (t +h3 ) v ∗ satisfies



m  ∂ uni (x, t ) = σk βi,k ∂t k=1





 unk (x − y, t )pi,k (y)dy − µi uni (x, t ),

1 ≤ i ≤ m.

R

It then follows from Proposition 3.2 that 0 ≤ w n (x, t ) ≤  un (x, t ) = eλ (t +h3 ) v ∗ for all (x, t ) ∈ R × [−n, +∞). Since n n limn→+∞ U (x, t )1 = U (x, t ; h3 ) and limn→+∞ U (x, t )0 = U (x, t ; −∞), we get ∗

∗ 0 ≤ U (x, t ; h3 ) − U (x, t ; −∞) ≤ eλ (t +h3 ) v ∗

for all (x, t ) ∈ R2 ,

which implies that U (x, t ; h3 ) converges to U (x, t ; −∞) as h3 → −∞ uniformly on (x, t ) ∈ R × (−∞, a] for any a ∈ R. For any sequence hn3 with hn3 → −∞ as n → +∞, the functions U (x, t ; hn3 ) converge to a solution of (1.1) (up to extraction of some subsequence) in T , which turns out to be U (x, t ; −∞). The limit does not depend on the sequence hn3 , whence all of the functions U (x, t ; h3 ) converge to U (x, t ; −∞) in T as h3 → −∞. Similarly, we can show that (a) and (b) hold. The proof of the assertion (ix) is similar to that of (viii). So we omit it here. Finally, we prove (x). We first claim that c λ1 (c ) is decreasing in c ∈ [c∗ , +∞). Let g (λ) = M (λ)/λ for λ > 0. From [28, Lemma 3.7], M (λ) is convex. Using the convexity of the function M, we can show that if g (λ1 ) = g (λ2 ) = γ > c∗ for some 0 < λ1 < λ2 < λ∗ , then M (λ) ≥ γ λ for λ ≥ λ2 , whence g (λ∗ ) ≥ γ > c∗ which is a contradiction. From this, it is easy to show that the function g is decreasing in (0, λ∗ ] which implies that λ1 (c ) is decreasing for c ∈ [c∗ , +∞). Note that A(λ) is increasing in λ ∈ R. If c1 > c2 ≥ c∗ , then λ1 (c1 ) < λ1 (c2 ) which yields that A(λ(c1 )) < A(λ(c2 )). In view of the fact that A(λ) is cooperative and irreducible, it then follows from [27, Corollary 4.3.2] that M (λ(  c1 )) < M (λ(c2 )) and hence c1 λ(c1 ) < c2 λ(c2 ). Therefore, c λ1 (c ) is decreasing in c ∈ [c∗ , +∞). Hence, ϑ(c1 , c2 ) = min c1 λ1 (c1 ), c2 λ1 (c2 ) = cmax λ1 (cmax ), where cmax = max{c1 , c2 }. Using (v) and (1.5), the rest proof of (x) is similar to that of [13, Theorem 1.1(vii)]. We omit the details. The proof of Theorem 1.1 is complete now.  4. Proof of Theorem 1.2 The proof of Theorem 1.2 is similar to that of Theorem 1.1. We only sketch the outline and prove those different from Theorem 1.1. Consider the initial value problem

  dui,j (t ) 

m +∞    = 1 − ui,j (t ) σk βi,k uk,j−r (t )pi,k (r ) − µi ui,j (t ),

dt ui,j (τ ) = ϕi,j (τ ),

k=1

1 ≤ i ≤ m,

r =−∞

where j ∈ Z, t > τ , τ ∈ R is a given constant.

(4.1)

S.-L. Wu, P. Weng / J. Math. Anal. Appl. 394 (2012) 603–615

613

Definition 4.1. A function u(t ) = {uj (t )}j∈Z with uj = (u1,j , . . . , um,j ) ∈ C 1 ([τ , ∞), W ) is called an upper solution of (4.1) if it satisfies for j ∈ Z and t > τ , dui,j (t ) dt

≥ (1 − ui,j (t ))

m 

σk βi,k

+∞ 

uk,j−r (t )pi,k (r ) − µi ui,j (t ),

1 ≤ i ≤ m.

r =−∞

k=1

A lower solution of (4.1) is defined by reversing the inequality. The following result follows from [26, Theorems 3.1–3.2]. Proposition 4.2. (i) For any ϕ = {ϕj }j∈Z with ϕj ∈ W , (4.1) admits a unique solution u(t ; ϕ) = {uj (t )}j∈Z on [τ , +∞) satisfies uj (τ ) = ϕj and uj ∈ C 1 ([τ , ∞), W ) for all j ∈ Z. (ii) Let wj+ (t ) j∈Z and wj− (t ) j∈Z be an upper solution and a lower solution of (4.1), respectively. If wj+ (τ ) ≥ wj− (τ ) for j ∈ Z, then wj+ (t ) ≥ wj− (t ) for all j ∈ Z and t ≥ τ .









Similar to Lemma 3.3, we have the following result which gives a prior estimate of solutions of (1.2). Lemma 4.3. Assume that u(t ; ϕ) = uj (t ; ϕ) j∈Z is a solution of (1.2) with initial value ϕ = {ϕj }j∈Z with ϕj ∈ W , then there exists a constant M > 0, independent of τ and ϕ , such that for any j ∈ Z and t > τ + 1,





∥u′j (t ; ϕ)∥ ≤ M and ∥u′′j (t ; ϕ)∥ ≤ M . ± + 1 m Lemma 4.4. Take wj± (t ) = (w1±,j (t ), . . . , wm and wj− ∈ C 1 [τ , +∞), (−∞, 1]m ,j (t )). If wj ∈ C [τ , +∞), [0, +∞) + − satisfy wj (τ ) ≥ wj (τ ) for all j ∈ Z, and



dwi+,j (t ) dt dwi−,j (t ) dt

≥

m 

σk βi,k

m  k=1

wk+,j−r (t )pi,k (r ) − µi wi+,j (t ),

1 ≤ i ≤ m,

wk−,j−r (t )pi,k (r ) − µi wi−,j (t ),

1 ≤ i ≤ m,





r =−∞

k=1

≤

+∞ 



σk βi,k

+∞  r =−∞

for j ∈ Z, t > τ , then there holds wj+ (t ) ≥ wj− (t ) for all j ∈ Z and t ≥ τ . For any n ∈ N, let U n (t ) = Ujn (t ) j∈Z = problem





n U1n,j (t ), . . . , Um ,j ( t )



 j∈Z

be the unique solution of the following initial value

 n m +∞     ∂ Ui,j (t ) = (1 − Uin,j (t )) σk βi,k Ukn,j−r (t )pi,k (r ) − µi Uin,j (t ), ∂t r =−∞ k = 1  U n (−n) = ϕ n , 1 ≤ i ≤ m, i ,j i,j

(4.2)

where j ∈ Z and t > −n, and

      ϕjn = max χ1 Ψc1 j − c1 n + h1 , χ2 Ψc2 −j − c2 n + h2 , χ Γ (−n + h3 ) .   Here, Ψc (ξ ) = Ψc j + ct is the traveling wavefront of (1.2) for c > c∗ decided in [26], and Γ (t ) is the solution of (2.1) decided in Theorem 2.4. Denote u(t ) = {uj (t )}j∈Z by       uj (t ) := max χ1 Ψc1 j + c1 t + h1 , χ2 Ψc2 −j + c2 t + h2 , χ Γ (t + h3 ) . From Proposition 4.2, uj (t ) ≤ Ujn (t ) ≤ 1 for all j ∈ Z and t ≥ −n. The following result gives an appropriate upper estimate of U n (t ) = Ujn (t ) j∈Z . Its proof is similar to that of Lemma 3.6.





Lemma 4.5. The function U n (t ) = Ujn (t ) j∈Z satisfies





¯ χ1 (j, t ), Π ¯ χ2 (j, t ), Π ¯ χ (j, t ) Ujn (t ) ≤ min K, Π 



¯ χ1 (j, t ), Π ¯ χ2 (j, t ), Π ¯ χ (j, t ) are defined as in Theorem 1.2. for all j ∈ Z and t ≥ −n, where Π

614

S.-L. Wu, P. Weng / J. Math. Anal. Appl. 394 (2012) 603–615

Proof of Theorem 1.2. We only prove (xi)′ , since the proofs of the other assertions are similar to those of Theorem 1.1 by virtue of Proposition 4.2 and Lemmas 4.3–4.5. When χ1 = χ2 = 1 and χ = 0, by (1.7), we have 0 ≤ Uj (t ) − Ψc1 j + c1 t + h1 ≤ Rc2 eλ3 (c2 )(c2 t +h2 ) v¯ (λ3 (c2 ))





for any j ≥ 0, which implies that lim sup ∥Uj;c1 ,c2 ,h1 ,h2 (t ) − Ψc1 (j + c1 t + h1 )∥ = 0.

(4.3)

t →−∞ j≥0

Similarly, there holds lim sup ∥Uj;c1 ,c2 ,h1 ,h2 (t ) − Ψc2 (−j + c2 t + h2 )∥ = 0.

(4.4)

t →−∞ j≤0

For any h1 , h2 , h∗1 , h∗2 ∈ R, suppose that there exists (j0 , t0 ) ∈ Z × R such that Uj;c1 ,c2 ,h1 ,h2 (t ) = Uj+j0 ;c1 ,c2 ,h∗ ,h∗ (t + t0 ) 1

2

for all (j, t ) ∈ Z × R.

Then, from (4.3), we get lim sup ∥Uj+j0 ;c1 ,c2 ,h∗ ,h∗ (t + t0 ) − Ψc1 (j + c1 t + h1 )∥ = 0 1

t →−∞ j≥0

2

and lim sup ∥Uj+j0 ;c1 ,c2 ,h∗ ,h∗ (t + t0 ) − Ψc1 ((j + j0 ) + c1 (t + t0 ) + h∗1 )∥ = 0.

t →−∞ j≥−j

1

2

0

Hence, lim

sup

t →−∞ j≥max{0,−j } 0

∥Ψc1 ((j + j0 ) + c1 (t + t0 ) + h∗1 ) − Ψc1 (j + c1 t + h1 )∥ = 0.

(4.5)

Let {tj }j∈N such that j + c1 tj = 0 for all j ∈ N, then (4.5) implies j0 + c1 t0 + h∗1 = h1 as n → +∞. Similarly, by (4.4), we obtain −j0 + c2 t0 + h∗2 = h2 . Then, there holds j0 =

c2 (h1 − h∗1 ) − c1 (h2 − h∗2 ) c1 + c2 c2 (h1 −h∗ )−c1 (h2 −h∗ )

and t0 =

(h1 − h∗1 ) + (h2 − h∗2 ) c1 + c2

(4.6) c2 (h1 −h∗ )−c1 (h2 −h∗ )

1 2 1 2 which yields ∈ Z. Conversely, we can show that, for any h1 , h2 , h∗1 , h∗2 ∈ R, if ∈ Z, then c1 + c2 c1 + c2 Uj;c1 ,c2 ,h1 ,h2 (t ) = Uj+j0 ;c1 ,c2 ,h∗ ,h∗ (t + t0 ) for all (j, t ) ∈ Z × R, where (j0 , t0 ) is given by (4.6). The proof of Theorem 1.2 is 1 2 complete. 

Acknowledgments The authors are grateful to the referee for a careful reading and helpful suggestions which led to an improvement of their original manuscript. The first author was supported by the NSF of China (11026127) and the Scientific Research Program Funded by Shaanxi Provincial Education Department (12JK0860). The second author was supported by the NSF of China (11171120), the Doctoral Program of Higher Education of China (20094407110001) and the NSF of Guangdong Province (10151063101000003). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

S.I. Ei, The motion of weakly interacting pulses in reaction–diffusion systems, J. Dynam. Differential Equations 14 (2002) 85–136. S.I. Ei, M. Mimura, M. Nagayama, Pulse–pulse interaction in reaction–diffusion systems, Physica D 165 (2002) 176–198. T. Kawahara, M. Tanaka, Interactions of traveling fronts: an exact solution of a nonlinear diffusion equation, Phys. Lett. A 97 (8) (1983) 311–314. X.F. Chen, J.S. Guo, Existence and uniqueness of entire solutions for a reaction–diffusion equation, J. Differential Equations 212 (2005) 62–84. X.F. Chen, J.S. Guo, H. Ninomiya, Entire solutions of reaction–diffusion equations with balanced bistable nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 1207–1237. Y. Fukao, Y. Morita, H. Ninomiya, Some entire solutions of Allen–Cahn equation, Taiwanese J. Math. 8 (2004) 15–32. Y. Guo, Entire solutions for a discrete diffusive equation, J. Math. Anal. Appl. 347 (2008) 450–458. J.S. Guo, Y. Morita, Entire solutions of reaction–diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst. 12 (2005) 193–212. F. Hamel, N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math. 52 (1999) 1255–1276. F. Hamel, N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in RN , Arch. Ration. Mech. Anal. 157 (2001) 91–163. H. Yagisita, Back and global solutions characterizing annihilation dynamics of traveling fronts, Publ. Res. Inst. Math. Sci. 39 (2003) 117–164. Y. Morita, H. Ninomiya, Entire solutions with merging fronts to reaction–diffusion equations, J. Dynam. Differential Equations 18 (2006) 841–861. W.T. Li, Z.C. Wang, J. Wu, Entire solutions in monostable reaction–diffusion equations with delayed nonlinearity, J. Differential Equations 245 (2008) 102–129.

S.-L. Wu, P. Weng / J. Math. Anal. Appl. 394 (2012) 603–615

615

[14] Z.C. Wang, W.T. Li, S. Ruan, Entire solutions in bistable reaction–diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc. 361 (2009) 2047–2084. [15] Z.C. Wang, W.T. Li, Dynamics of a nonlocal delayed reaction–diffusion equation without quasi-monotonicity, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010) 1081–1109. [16] Z.C. Wang, W.T. Li, S. Ruan, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal. 40 (2009) 2392–2420. [17] Z.C. Wang, W.T. Li, S. Ruan, Entire solutions in lattice delayed differential equations with nonlocal interaction: bistable case, Preprint. [18] W.T. Li, Y.J. Sun, Z.C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. 11 (2010) 2302–2313. [19] Y.J. Sun, W.T. Li, Z.C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations 251 (2011) 551–581. [20] J.S. Guo, C.H. Wu, Entire solutions for a two-component competition system in a lattice, Tohoku Math. J. 62 (2010) 17–28. [21] Y. Morita, K. Tachibana, An entire solution to the Lotka–Volterra competition-diffusion equations, SIAM J. Math. Anal. 40 (2009) 2217–2240. [22] M.X. Wang, G.Y. Lv, Entire solutions of a diffusive and competitive Lotka–Volterra type system with nonlocal delay, Nonlinearity 23 (2010) 1609–1630. [23] S.L. Wu, Entire solutions in a bistable reaction–diffusion system modeling man-environment-man epidemics, Nonlinear Anal. RWA 13 (2012) 1991–2005. [24] L. Rass, J. Radcliffe, Spatial Deterministic Epidemics, in: Math. Surveys Monogr., vol. 102, Amer. Math. Soc., Providence, RI, 2003. [25] P.X. Weng, X.Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations 229 (2006) 270–296. [26] F. Zhang, X.Q. Zhao, Spreading speed and traveling waves for a spatially discrete epidemic model, Nonlinearity 21 (2008) 97–112. [27] H.L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, in: Math. Surveys Monogr., vol. 41, Amer. Math. Soc., Providence, RI, 1995. [28] X. Liang, X.Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math. 60 (2007) 1–40. [29] P.W. Bates, et al., Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal. 138 (1997) 105–136.