Wavefront solutions of degenerate quasilinear reaction–diffusion systems with mixed quasi-monotonicity

Wavefront solutions of degenerate quasilinear reaction–diffusion systems with mixed quasi-monotonicity

Nonlinear Analysis 182 (2019) 75–96 Contents lists available at ScienceDirect Nonlinear Analysis www.elsevier.com/locate/na Wavefront solutions of ...

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Nonlinear Analysis 182 (2019) 75–96

Contents lists available at ScienceDirect

Nonlinear Analysis www.elsevier.com/locate/na

Wavefront solutions of degenerate quasilinear reaction–diffusion systems with mixed quasi-monotonicity Weihua Ruan Department of Mathematics, Statistics and Computer Science, Purdue University Northwest, Hammond, IN 46323, United States

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Article history: Received 12 March 2018 Accepted 3 December 2018 Communicated by Enzo Mitidieri Keywords: Degenerate quasilinear parabolic equations Wavefront solutions Upper and lower solutions

abstract This paper is concerned with the existence of wavefront solutions to a system of degenerate quasilinear reaction–diffusion equations of mixed quasi-monotone properties in the form ∂ui /∂t = ∇ · (Di (ui ) ∇ui ) + fi (u)

− ∞ < x < ∞,

t>0

for i = 1, . . . , n. The important features of this system are that some of the diffusion coefficients Di (ui ) are density dependent and may vanish at certain value of ui , and that each function fi is quasi-monotone increasing for some components of u = (u1 , . . . , un ) and decreasing for other components of u. Such systems model reaction–diffusion processes with density driven diffusion mechanism. Under certain general conditions we prove the existence of a traveling wave solution that is between a pair of coupled upper and lower solutions. A predator–prey model with nonlinear diffusion is used as an illustration of application. The presence of wavefront solutions flowing toward the coexistence states is established by constructing appropriate upper and lower solutions. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, reaction–diffusion equations with nonlinear diffusion mechanism have attracted much attentions. This is because nonlinear diffusion processes, especially degenerate ones, are commonly found in the nature. Well-known examples include diffusion in porous media and density driven diffusion in population dynamics. It has long been observed that degenerate diffusion processes behave very differently from nondegenerate ones. One striking difference is that the rate of diffusion is infinite for the nondegenerate diffusion, but it is finite for degenerate ones. This paper deals with the wavefront solutions to the quasilinear reaction–diffusion system ∂ui /∂t = ∇ · (Di (ui ) ∇ui ) + fi (u) E-mail address: [email protected] https://doi.org/10.1016/j.na.2018.12.003 0362-546X/© 2018 Elsevier Ltd. All rights reserved.

for x ∈ R,

t > 0,

i = 1, . . . , n.

(1.1)

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Of particular concern is the case where the diffusion coefficients, Di (ui ), may depend on the density ui and may vanish at certain value of ui . This system arises in the modeling of population dynamics with density driven diffusion, and it has recently been investigated in a number of articles [20–24]. A wavefront is a solution in the form u (x, t) = w (x + ct), where w (s) ≡ (w1 (s) , . . . , wn (s)) satisfies ′

(Di (wi ) wi′ ) (s) − cwi′ (s) + fi (w (s)) = 0

for s ∈ R.

Without increasing complexity in the following analysis, we consider a more general system in which functions fi , i = 1, . . . , n, may also depend on s. Thus we consider the following problem ′

(Di (wi ) wi′ ) − cwi′ + fi (s, w) = 0 s ∈ R,

i = 1, . . . , n.

(1.2)

with the asymptotic boundary conditions lim wi (s) = u− i ,

s→−∞

lim wi (s) = u+ i

s→∞

i = 1, . . . , n

(1.3)

( ) + ( + ) − + n for some constant vectors u− ≡ u− 1 , . . . , un , u ≡ u1 , . . . , un ∈ R . We assume that the functions fi , i = 1, . . . , n, possess the following mixed quasi-monotone property. (Hm ) For each i, there are disjoint subsets of positive integers ki and kˆi such that ki ∪ kˆi = {1, . . . , n} \ {i}

(1.4)

and the function fi (s, w) is monotone nondecreasing in each wj for j ∈ ki and is monotone nonincreasing in each wj for j ∈ kˆi . In the sequel, we also use the notation ( ) w = wi , [wj ]j∈ki , [wj ]j∈kˆi for any vector w = (w1 , . . . , wn ), where [wj ]j∈ki represents the components wj with j ∈ ki , and [wj ]j∈kˆi represents the components wj with j ∈ kˆi . The goal of this paper is to prove the existence of wavefront solutions between coupled upper and lower solutions. Upper and lower solutions for (1.2) with the mixed monotonicity (Hm ) are defined by Definition 1.1. Let functions fi , i = 1, . . . , n, satisfy the mixed quasi-monotonicity property (Hm ). A pair ˜ ≡ (w ˆ ≡ (w of C 2 vector functions w ˜1 , . . . , w ˜n ) and w ˆ1 , . . . , w ˆn ) are called coupled upper and lower solutions − if w ˜i ≥ w ˆi ≥ ui and ( ) ′ ˜ ki , [w] ˆ kˆi ≤ 0, (Di (w ˜i ) w ˜i′ ) − cw ˜i′ + fi s, w ˜i , [w] (1.5) ( ) ′ ˆ ki , [w] ˜ kˆi ≥ 0, (Di (w ˆi ) w ˆi ) − cw ˆi′ + fi s, w ˆi , [w] for each i = 1, . . . , n. Traveling wave solutions to systems of parabolic equations modeling chemical processes and population dynamics have been actively studied by many researches (cf. e.g. [2,3,5–7,13,18,19,24,25,27–30]). Some of works use the method of upper and lower solutions and iteration process [6,18,19,24,25]. There are also a number of papers on traveling wave solution for reaction–diffusion equations with nonlinear diffusions [9– 12,26]. However, few papers in the literature deal with wavefront solutions to degenerate quasilinear parabolic systems. The main goal of this paper is to prove the existence of a solution w = (w1 , . . . , wn ) to the ˆ and w, ˜ under certain differential equation in (1.2) between a pair of coupled upper and lower solutions w

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additional conditions (Hypotheses (H1 ) below), and give conditions under which the solutions also satisfy the asymptotic boundary conditions in (1.3). This work is a continuation of the recent paper [24] written by the author and collaborators. The system in [24] are quasilinear but nondegenerate in the sense that the diffusion coefficients Di (ui ) is positively bounded below for ui between a pair of upper and lower solutions. That is, there are positive constants δˆ and δ˜ such that δˆ ≤ Di (w) ≤ δ˜ for w ˆi ≤ w ≤ w ˜i , i = 1, . . . , n. In this paper we remove this restriction and allow nonlinear diffusion to be degenerate. The main assumptions are the following Hypotheses (H1 ). (i) Di (w) is a nonnegative nondecreasing differentiable function that satisfies Di (w) > 0 for w > u− i . Also, 3 − ′ Di (w) / (Di (w)) is nonincreasing and is bounded if w ≥ ui + ε for any ε > 0. (ii) c > 0 and for any i = 1, . . . , n, w ˜i and w ˆi are both bounded and nondecreasing in R. (iii) Functions f1 , . . . , fn are and their partial derivatives (fi )s and (fi )wj for i, j = 1, . . . , n are bounded ∏n ¯i ], where wi = inf s∈R w ˆi (s) and w ¯i = sups∈R w ˜i (s). and continuous for s ∈ R and w ∈ i=1 [wi , w (iv) There are positive constants ρ and ε0 such that lim inf e−ρs Di (w ˆi (s)) ≥ ε0 , s→−∞

lim sup eρs Di′ (w ˆi (s)) ≤ ε0

for i = 1, . . . , n.

s→−∞

Note that (H1 )–(i) implies that Di (w ˆi (s)) > 0 for all s ∈ R and all i ∈ {1, . . . , n} if w ˆi (s) > u− i in R. Our main result is ˜ w ˆ ∈ C 2 (R) satisfy (1.5). Then there is a solution w Theorem 1.1. Let (Hm ) and (H1 ) hold. Suppose w, ˆ ≤w≤w ˜ in R. to (1.2) that satisfies w The proof of this theorem is given in Section 2. The main idea is to use a change of the independent variable to remove the degeneracy, and then use results in [24] to complete the proof. The rest of the paper is organized as follows: In Section 2 we prove Theorem 1.1 through a sequence of lemmas about a continuous mapping on a compact and convex invariant set in a Banach space. Since some part of the proofs are the same as those in [24], we only indicate the changes. In Section 3, we discuss the properties of the solution of (1.2) as s → ±∞. In Section 4 we use the results of Sections 2 and 3 to study a predator–prey model with degenerate diffusion. Numerical examples are also presented to show the profiles of the solution. 2. Existence of wavefront between upper and lower solutions This section is devoted to proving Theorem 1.1. We use Schauder’s Fixed Point Theorem as the main approach. To carry out, we first construct a nonlinear mapping whose fixed point is a solution of (1.2), and then show that the mapping is continuous in a certain Banach space and it is invariant in a compact convex set. To construct the mapping, we first make a change of variable in Eq. (1.2). Let vi = ϕi (wi ) where ϕi is a function that satisfies ϕ′i = Di , and denote Ji = ϕ−1 so that wi = Ji (vi ). Then (1.2) can be written in the i form vi′′ (s) − cJi′ (vi ) vi′ (s) + fi (s, J (v) (s)) = 0 s ∈ R,

i = 1, . . . , n

(2.1)

where J (v) = (J1 (v1 ) , . . . , Jn (vn )) = w.

(2.2)

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The main difficulty with Eq. (2.1) is that Ji′ (vi ) is not bounded if the function Di (wi ) vanishes at some values of wi . We make a change of variable s ↦→ t to resolve this difficulty. The new variable t is chosen to be an increasing C 2 function of s and satisfies the conditions t = 1 − e−4ρs

for s ≤ 0,

t′ ≥ ρ,

t′′ ≤ 0

and t′′ = 0

for s > 0

for s > 1

(2.3)

where ρ is the positive constant in (H1 )-(iv). In what follows, we use the upper dot to denote differentiation with respect to t, and use the apostrophe to denote differentiation with respect to s. (Except in Ji′ (vi ) which means differentiation with respect to vi .) By the Chain Rule we derive vi′ = v˙ i t′ ,

2

vi′′ = v¨i (t′ ) + v˙ i t′′ .

In terms of the new variable, t, Eq. (2.1) becomes ) ( t′′ Fi (t, v) cJi′ (vi ) − v˙ i + =0 v¨i − 2 2 t′ (t′ ) (t′ )

(2.4)

where Fi (t, v) = fi (s (t) , J (v)) . We let Bi (t, vi ) and Gi (t, v) denote the functions Bi (t, vi ) =

cJi′ (vi (t)) t′′ − 2, ′ t (t′ )

Gi (t, v) =

Fi (t, v) 2

(t′ )

.

(2.5)

Then Eq. (2.4) can be written as v¨i − Bi (t, vi ) v˙ i + Gi (t, v) = 0

for t ∈ R,

i = 1, . . . , n.

We first prove the following properties of Bi and Gi . Lemma 2.1. Suppose Hypotheses (Hm ) and (H1 ) hold. Then Bi (t, vi ) is nonnegative, Bi (t, vi ) and Gi (t, v) ˆ ≤ v ≤ ϕ (w), ˜ and there is a positive constant β such that are bounded for t ∈ R and ϕ (w) |Bi (t, v1,i (t)) − Bi (t, v2,i (t))| ≤ β |v1,i (t) − v2,i (t)| , |Gi (t, v1 (t)) − Gi (t, v2 (t))| ≤ β |v1 (t) − v2 (t)| ˆ and ϕ (w). ˜ for all t ∈ R and vi and v2 between ϕ (w) Here, ϕ (w) for any vector w = (w1 , . . . , wn ) denotes the vector function (ϕ1 (w1 ) , . . . , ϕn (wn )). Proof . We first show that Bi and Gi are bounded functions. To see that Gi is bounded, it suffices to note that t′ = 4ρe−4ρs ≥ 4ρ for s ≤ 0 and t′ ≥ ρ for s > 0. Hence 2

|Gi (t, v)| = |Fi (t, v)| / (t′ ) ≤ |Fi (t, v)| /ρ. ˆ and w. ˜ The boundedness of Fi (t, v) follows from the continuity of fi , ϕi and the boundedness of w To show that Bi is bounded, observe that for 2

−1 < t′′ / (t′ ) = −e4ρs < 0

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2

for s ≤ 0, t′′ = 0 for s > 1 and t is a C 2 function with t′ ≥ ρ in R. Hence t′′ / (t′ ) is bounded for all s ∈ R . Furthermore, by the monotone nondecreasing property of Di we see that 0≤

cJi′ (vi (t)) 1 1 = ≤ , t′ 4ρe−4ρs Di (wi (s)) 4ρe−4ρs Di (w ˆi (s))

for s ≤ 0, and 0≤

c c cJi′ (vi (t)) ≤ ≤ , t′ ρDi (w ˆi (s)) ρDi (w ˆi (0))

for s > 0. Hence, 0 ≤ Bi (t, vi ) ≤ for t ≤ 0 and

1 4ρe−ρs Di

(w ˆi (s))

+1

⏐ ⏐ ⏐ t′′ ⏐ c ⏐ ⏐ 0 ≤ Bi (t, vi ) ≤ + max ⏐ ⏐ ρDi (w ˆi (0)) 0≤s≤1 ⏐ (t′ )2 ⏐

for t > 0. It is easy to see that Bi is bounded for t > 0. Since by (H)–(iv), e−ρs Di (w ˆi (s)) is bounded away from zero for −∞ < s ≤ 0, it follows that B (t, vi ) is also bounded for any t ≤ 0 and all vi that satisfies ϕi (w ˆ i ) ≤ vi ≤ ϕ i ( w ˜i ). This proves the nonnegativity and boundedness of Bi (t, vi ). To show that Gi satisfies the Lipschitz condition, we let zj,i denote the components of zj , and let ′ wj,i = ϕ−1 i (zj,i ) for j = 1, 2. By (H1 )–(iii) and the inequality t ≥ ρ, |Gi (t, z1 ) − Gi (t, z2 )| =

1 2

(t′ )

|Fi (t, z1 ) − Fi (t, z2 )|

ˆ ≤ z1 , z2 ≤ ϕ (w). ˜ Here J (z) = (J1 (z1 ) , . . . , Jn (zn )). By the for any t ∈ R and z1 , z2 that satisfy ϕ (w) Mean Value Theorem, Fi (t, z1 ) − Fi (t, z2 ) =

n ∑

(fi )wj (θ) Jj′ (σj ) (z1,j − z2,j )

j=1

where θ is between ϕ (z1 ) and ϕ (z2 ) and σj is between z1,j and z2,j . By the boundedness of the derivatives (fi )wj and applying the Inverse Function Theorem, we have |Fi (t, z1 ) − Fi (t, z2 )| ≤ M

n ∑ |z1,j − z2,j | ( ) Dj ϕ−1 j (σj ) j=1

for some positive constant M . Since Dj is nondecreasing and z1,j and z2,j are between ϕj (w ˆj ) and ϕj (w ˜j ), it follows that ( ) Dj ϕ−1 ˆj ) . j (σj ) ≥ Dj (w Hence

n M ∑ |z1,j − z2,j | |Gi (t, z1 ) − Gi (t, z2 )| ≤ . 2 ˆj ) (t′ ) j=1 Dj (w 2

To show that Gi satisfies the Lipschitz condition, it suffices to show that (t′ ) Dj (w ˆj ) is bounded below by a positive constant for all s ∈ R. For s > 0, since t′ ≥ ρ and Dj and w ˆj are monotone nondecreasing, it follows that 2 (t′ ) Dj (w ˆj ) ≥ ρ2 Dj (w ˆj (0)) > 0. For s ≤ 0, by (H)–(iv) 2

lim inf (t′ ) Dj (w ˆj (s)) = 16ρ2 lim inf e−8ρs Dj (w ˆj (s)) ≥ 4ρlim inf e−ρs Dj (w ˆj (s)) ≥ 4ρε0 . s→−∞

s→−∞

s→−∞

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2

2

Hence (t′ ) Dj (w ˆj ) is positively bounded below for s ≤ 0. This proves that (t′ ) Dj (w ˆj ) is positively bounded below on R. Hence, Gi satisfies the Lipschitz condition. To show that Bi satisfies the Lipschitz condition, we use the definition of Bi in (2.5) to derive |Bi (t, v1,i ) − Bi (t, v2,i )| =

c ′ |J (v1,i ) − Ji′ (v2,i )| . t′ i

(2.6)

Since Ji = ϕ−1 and ϕ′i = Di , by the Inverse Function Theorem i Ji′ (v) = where w = ϕ−1 (v). Hence, Ji′′ (v) = −

1 Di (w)

Di′ (w) dw Di′ (w) = − 2 3 (Di (w)) dv (Di (w))

( ) which exists whenever v > ϕi u− i . Hence c |Bi (t, v1,i ) − Bi (t, v2,i )| ≤ ′ t

⏐ ⏐ ⏐ D′ (θ ) ⏐ ⏐ ⏐ i i ⏐ ⏐ |v − v2,i | ⏐ (Di (θi ))3 ⏐ 1,i

where θi is between w ˆi and w ˜i . It remains to show that the quantity ⏐ ⏐ c ⏐⏐ Di′ (θi ) ⏐⏐ M (t, θi ) ≡ ′ ⏐ ⏐ t ⏐ (Di (θi ))3 ⏐ is uniformly bounded for t ∈ R and w ˆ i ≤ θi ≤ w ˜i . For s > 0, t′ ≥ ρ and by (H1 )–(i) 0≤

Di′ (θi ) (Di (θi ))

3



Di′ (w ˆi ) (Di (w ˆi ))

Thus M (t, θi ) ≤

3



Di′ (w ˆi (0)) (Di (w ˆi (0)))

3.

c Di′ (w ˆi (0)) 3 ρ (Di (w ˆi (0)))

For s ≤ 0, by (H1 )–(i), (iv) and (2.3), Di′ (w ˆi (s)) c 3 −4ρs 4ρe s→−∞ (Di (w ˆi (s))) 1 c ρs ′ lim sup ˆi (s)) ≤ 3 · lim supe Di (w 4ρ s→−∞ (e−ρs Di (w s→−∞ ˆi (s))) c −2 ≤ ε . 4ρ 0

lim supM (t, θi ) ≤ lim sup s→−∞

Hence, M (t, θi ) is again bounded. This completes the proof that Bi satisfies the Lipschitz condition.



For any vector function z (s) = (z1 (s) , . . . , zn (s)) that is bounded in R we define a mapping K: z ↦→ v so that v (s) is a solution to the problem v¨i − Bi (t, zi (t)) v˙ i − βvi + G∗i (t, z (s)) = 0 t ∈ R,

i = 1, . . . , n,

(2.7)

where β is the positive constant in Lemma 2.1, and G∗i (t, z) = Gi (t, z) + βzi .

(2.8)

It is clear that a fixed point of K is a solution of Problem (2.1). We first give an integral representation of K.

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2.1. Integral representation of K To write a solution of Problem (2.7) in an integral form, we need the following lemmas regarding the linear ordinary differential equations y¨ − a (t) y˙ − by = 0, (2.9) and y¨ − a (t) y˙ − by + f (t) = 0

(2.10)

for −∞ < t < ∞, where b > 0 is a constant, a (t) > 0 is a function and f is a bounded continuous function. Lemma 2.2. Suppose b > 0 and there are nonnegative constants a ˆ and a ˜ such that a ˆ ≤ a (t) ≤ a ˜. Also ∗ suppose that there is a t ∈ R such that inf {a (t) : t ≥ t∗ } > 0.

(2.11)

Then there are positive solutions y − (t) and y + (t) to Eq. (2.9) such that y − (t) is monotone decreasing having the limits lim y − (t) = ∞, lim y − (t) = 0, t→−∞

t→∞

+

and y (t) is monotone increasing having the limits lim y + (t) = 0,

t→−∞

Furthermore inequalities

lim y + (t) = ∞.

t→∞

+ − ˜− < 0 < λ ˆ + ≤ y˙ (t) ≤ λ ˜+ ˆ − ≤ y˙ (t) ≤ λ λ y − (t) y + (t)

(2.12)

hold for t ∈ R, where

) ) ( ( √ √ ˜± = 1 a ˆ± = 1 a λ ˜2 + 4b , λ ˆ2 + 4b . ˜± a ˆ± a 2 2 If in addition, limt→∞ a (t) = a ¯ for some a ¯ ∈ R, then ) √ y˙ ± (t) 1( 2 + 4b . = a ¯ ± a ¯ t→∞ y ± (t) 2 lim

(2.13)

(2.14)

Similarly, if limt→−∞ a (t) = a for some a ∈ R then ) √ y˙ ± (t) 1( 2 + 4b . = a ± a t→−∞ y ± (t) 2 lim

(2.15)

The proof of this lemma is the same as the proof of Lemma 2.1 of [24] except in [24] it is assumed that a ˆ > 0. However, an examination of the proof shows that it is valid even if a ˆ = 0 and b > 0. The only difference is the relation ∫ t a (τ ) dτ ≥ a ˆt

A (t) = 0

on Page 30 of [24], which is used to show that A (t) → ∞ as t → ∞. This relation should be changed to ∫ t A (t) = a (τ ) dτ ≥ (t − t∗ ) inf {a (τ ) : τ ≥ t∗ } for t ≥ t∗ (2.16) 0

which still implies A (t) → ∞ as t → ∞ by (2.11). Using functions y − and y + we can give an integral representation of a special solution to Eq. (2.10).

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Lemma 2.3. Suppose b > 0 and there are nonnegative constants a ˆ and a ˜ such that a ˆ ≤ a (t) ≤ a ˜ and (2.11) − + holds. Furthermore, suppose that f is continuous and bounded in R. Let y (t) and y (t) be the functions in Lemma 2.2. Then the function y (t) defined by [ ] ∫ t ∫ ∞ 1 − + −A(τ ) + − −A(τ ) y (t) = + y (t) y (τ ) e f (τ ) dτ + y (t) y (τ ) e f (τ ) dτ , (2.17) y˙ (0) − y˙ − (0) −∞ t where A satisfies A˙ (t) = a (t) and A (0) = 0, is a solution of Eq. (2.10) that satisfies lim

t→−∞

y (t) y (t) = lim + = 0. − t→∞ y (t) y (t)

(2.18)

The proof of this lemma is given by the proof of Lemma 2.2 in [24] with the only change in the proof of lim e−A(t) = 0.

t→∞

This still holds true since A (t) → ∞ as t → ∞. The following properties of y − and y + are useful. Lemma 2.4. Let the hypothesis of Lemma 2.2 holds. Then the following identities hold true: [ − ] y (t) y˙ + (t) − y + (t) y˙ − (t) e−A(t) = y˙ + (0) − y˙ − (0) and ∫ ∞ ∫ t 1 1 y − (τ ) e−A(τ ) dτ = − y˙ − (t) e−A(t) , y + (τ ) e−A(τ ) dτ = y˙ + (t) e−A(t) . b b t −∞

(2.19) (2.20)

These identities are given by (A6) and (A7) in [24]. Let ˆb and ˜b be upper and lower bounds of Bi (t, vˆi (t)) and Bi (t, v˜i (t)), t ∈ R and ˆb ≤ min (Bi (t, vˆi (t)) , Bi (t, v˜i (t))) ,

˜b ≥ max (Bi (t, vˆi (t)) , Bi (t, v˜i (t)))

for all t ∈ R and i = 1, . . . , n. Then 0 ≤ ˆb ≤ ˜b < ∞. Let µ be a constant that satisfies { − + } 0 < µ < min −˜ ρ , ρˆ , −β/ˆ ρ− , β/˜ ρ+ where ˆb ±

√ ˆb2 + 4β

(2.21)

√ ˜b2 + 4β , ρ˜ = . (2.22) ρˆ = 2 2 Let X denote the Banach space of bounded continuous functions z ∈ C (R; Rn ) endowed with the norm { } |z|X = sup |z (t)| e−µ|t| . ±

±

˜b ±

t∈R

For each z ∈ X, we define Kz = v as the vector function with the components [ ∫ t 1 − vi (t) = + y (t; z ) yi+ (τ ; zi ) e−Ai (τ ) G∗i (τ, z (τ )) dτ i ξi (zi ) − ξi− (zi ) i∫ −∞ ] ∞ − + −Ai (τ ) ∗ +yi (t; zi ) yi (τ ; zi ) e Gi (τ, z (τ )) dτ

(2.23)

t

where yi± (·; zi ) for i = 1, . . . , n are and ξi± (zi ) = y˙ i± (0; zi ). By Lemma

the functions given in Lemma 2.2 with a (t) = Bi (t, zi (t)) and b = β, 2.3 , vi satisfies the differential equation in (2.7) and has the properties lim

t→−∞

vi (t) vi (t) = lim + = 0. t→∞ yi (t; zi ) (t; zi )

yi−

We next show that the set of functions in X which are between the upper and lower solutions is invariant for the mapping K. The main approach is the same as that in [24], with Lemmas 2.1 and 2.2 in [24] replaced by Lemmas 2.2 and 2.3, respectively. We only include the changes in the proofs here and indicate which parts of the proofs are the same as those in [24].

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2.2. An invariant set for K Let ˆ≤z≤v ˜} . S = {z ∈ X : v We show that S is invariant for K. ˙ L∞ (R) is uniformly bounded for Lemma 2.5. Let (Hm ) and (H1 ) hold. Then KS ⊂ S. Furthermore, |v| v ∈ KS. Proof . By Lemma 2.1, G∗i (t, z) ≡ Gi (t, z) + βzi is nondecreasing in zi . Also, by (Hm ), G∗i (t, z) is nondecreasing in zj for any j ∈ ki , and is nonincreasing in zj for any j ∈ kˆi . Let z ∈ S and let ¨i + Bi (t, vˆi ) vˆ˙ i + βˆ ˆ i (t) = −vˆ G vi , ˆ i (t; zi ) = (Bi (t, zi ) − Bi (t, vˆi )) vˆ˙ i , H

˜ i (t) = −v¨˜i + Bi (t, v˜i ) v˜˙ i + β˜ G vi , ˜ i (t; zi ) = (Bi (t, zi ) − Bi (t, v˜i )) v˜˙ i H

ˆ satisfy the equations for i = 1, . . . , n. Then by (1.5)˜ v and v ¨i − Bi (t, zi ) vˆ˙ i − βˆ ˆi + H ˆ i = 0, vˆ vi + G ˜i + H ˜ i = 0. ¨i − Bi (t, zi ) v˜˙ i − β˜ v˜ vi + G ˜ and v ˆ are bounded, it follows that In addition, since v lim

t→−∞

v˜i (t) v˜i (t) vˆi (t) vˆi (t) = lim = 0. = lim = lim yi− (t) t→−∞ yi− (t) t→∞ yi+ (t) t→∞ yi+ (t)

Thus by Lemma 2.3 [ ∫ t ( ) 1 − ˆ i (τ ) + H ˆ i (τ ; zi ) dτ y (t; z ) yi+ (τ ; zi ) e−Ai (τ ) G i i − + ξi (zi ) − ξi (zi ) ∫ −∞ ∞ ( ) ] − + −Ai (τ ) ˆ ˆ +yi (t; zi ) yi (τ ; zi ) e Gi (τ ) + Hi (τ ; zi ) dτ , [ t ∫ t ( ) 1 ˜ i (τ ) + H ˜ i (τ ; zi ) dτ yi− (t; zi ) yi+ (τ ; zi ) e−Ai (τ ) G v˜i (t) = + − ξi (zi ) − ξi (zi ) ∫ −∞ ] ∞ ( ) + + −Ai (τ ) ˜ ˜ +yi (t; zi ) yi (τ ; zi ) e Gi (τ ) + Hi (τ ; zi ) dτ .

vˆi (t) =

t

Since vˆi ≤ zi ≤ v˜i in R for all i = 1, . . . , n, by the monotonicity of Gi , it follows that ( ) ( ) ˆ i (t) ≤ G∗ t, vˆi , [ˆ ˜ i (t) . G v]ki , [˜ v]kˆi ≤ G∗i (t, z) ≤ G∗i t, v˜i , [˜ v]ki , [ˆ v]kˆi ≤ G i Also by the Inverse Function Theorem, Ji′ (vi ) =

1 1 ( ) = Di (wi ) Di ϕ−1 i (vi )

which is nonincreasing in vi , it follows that Bi (t, zi ) is nonincreasing in zi . Hence ˆ i (τ ; zi ) ≤ 0, H

˜ i (τ ; zi ) ≥ 0 H

and by (2.23), vˆi (t) ≤

[ ∫ t 1 − y (t; z ) yi+ (τ ; zi ) e−Ai (τ ) G∗i (τ, z (τ )) dτ i ξi+ (zi ) − ξi− (zi ) i∫ −∞ ] ∞ +yi+ (t; zi ) yi− (τ ; zi ) e−Ai (τ ) G∗i (τ, z (τ )) dτ ≤ v˜i (t) , t

for i = 1, . . . , n. This implies vˆi (t) ≤ vi (t) ≤ v˜i (t) for all t ∈ R and all i = 1, . . . , n. Thus v ∈ S.

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84

˙ being uniformly bounded in KS is the same as the corresponding part of the proof of The proof of |v| ˙ Lemma 2.4 in [24]. The only differences are the notations, with v′ , ξi,1 , ξi,2 , yi,1 , yi,2 , and Gi replaced by v, ξi− , ξi+ , yi− , yi+ , and G∗i , respectively. We omit the rest part of the proof here. ■ 2.3. Continuity of K in X Lemma 2.6. Let (Hm ) and (H1 ) hold. Then there is a constant M > 0 such that |Kz1 − Kz2 |X < M |z1 − z2 |X

for all z1 , z2 ∈ X.

Proof . The proof of Lemma 2.5 in [24] are valid with obvious notation changes as in the proof of Lemma 2.5 ˜1, λ ˜2, λ ˆ1, λ ˆ 2 replaced by Bi (t, z1,i ), Bi (t, z2,i ), λ ˜−, λ ˜+ , λ ˆ − , and λ ˆ + respectively. We omit and cJi′ (zj,i ), λ the proof here. ■ 2.4. Precompactness of KS in X Let Sˆ = {z ∈ S : |zi (t1 ) − zi (t2 )| ≤ L′ |t1 − t2 | for t1 , t2 ∈ R, i = 1, . . . , n}

(2.24)

ˆ The next lemma shows that Sˆ ˙ L∞ (Rn ) in KS. By Lemma 2.5 KS ⊂ S. where L′ is the uniform bound of |v| is precompact in X. Lemma 2.7. Sˆ defined by (2.24) is precompact in X. The proof of this lemma is the same as the proof of Lemma 2.6 in [24] with no change. We omit it here. Proof of Theorem 1.1. By Lemmas 2.2–2.6, the operator K is continuous in the Banach space X, invariant in a closed convex set S and the image KS is precompact in X. Thus, by Schauder’s Fixed Point Theorem, K has a fixed point in S. Let v = (v1 , . . . , vn ) be a fixed point of K. Then each vi (t) satisfies (2.4). Changing back to the variable s, it follows that vi (s) satisfies (2.1). Finally, by the equivalence between ˆ (s) ≤ w (s) ≤ w ˜ (s) hold for s ∈ R. Eqs. (1.2) and (2.1), wi = ϕ−1 i (vi ) satisfies (1.2) and the relations w This concludes the proof of the theorem. ■ 3. Properties of the solution as s → ±∞ Theorem 1.1 only concludes the existence of a solution w to the system of Eqs. (1.2) that is between the ˆ and w. ˜ In general, the solution is not unique and need not pair of coupled upper and lower solutions, w ˆ w⟩ ˜ is large enough, it may contain satisfy the boundary conditions in (1.3). For example, if the sector ⟨w, equilibria of the equations that do not satisfy one or more boundary conditions. In this section, we derive properties of the upper and lower limits lim sups→±∞ w (s) and lim inf s→±∞ w (s) which can be useful in the proof of existence of solutions that satisfy the boundary conditions in (1.3). We first consider the boundary properties of the solution of (2.10) given by (2.17). Lemma 3.1. Let the hypothesis of Lemma 2.3 hold, and let y (t) be the solution of Eq. (2.10) defined by (2.17). Suppose f (t) is bounded in R. Then lim sup y (t) ≤ t→∞

1 lim sup f (t) , b t→∞

lim inf y (t) ≥ t→∞

1 lim inf f (t) , b t→∞

(3.1)

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and

1 lim sup f (t) , b t→−∞

lim sup y (t) ≤ t→−∞

lim inf y (t) ≥ t→−∞

85

1 lim inf f (t) . b t→−∞

(3.2)

Proof . Let L = lim supt→∞ f (t). Then, for any ε > 0 there is T > 0 such that f (t) < L + ε for all t > T . Hence, by the positivity of y + , ∫ t ∫ T ∫ t + −A(τ ) + −A(τ ) y (τ ) e f (τ ) dτ < y (τ ) e f (τ ) dτ + (L + ε) y + (τ ) e−A(τ ) dτ. −∞

−∞

T

Since y + satisfies (2.9), it follows that d ( −A(t) + ) e y˙ (t) = by + (t) e−A(t) > 0. dt Hence y˙ + (t) e−A(t) is increasing. By (2.12) this implies y + (τ ) e−A(τ ) is positively bounded below for τ > 0. It follows that ∫ t lim y + (τ ) e−A(τ ) dτ = ∞. t→∞

−∞

Hence, by (2.20), ∫T lim

−∞

t→∞

y + (τ ) e−A(τ ) f (τ ) dτ y˙ + (t) e−A(t)

∫T = lim

t→∞

y + (τ ) e−A(τ ) f (τ ) dτ = 0. ∫t b −∞ y + (τ ) e−A(τ ) dτ

−∞

Also, by l’Hˆ opital’s Rule, ∫t lim ∫ tT

t→∞

y + (τ ) e−A(τ ) dτ

−∞

y + (τ ) e−A(τ ) dτ

This implies that there is a T1 > T such that ∫T + y (τ ) e−A(τ ) f (τ ) dτ < ε, 0 < −∞ y˙ + (t) e−A(t)

= 1.

∫t 1 − ε < ∫ tT

y + (τ ) e−A(τ ) dτ

−∞

y + (τ ) e−A(τ ) dτ

< 1.

Hence ∫t −∞

y + (τ ) e−A(τ ) f (τ ) dτ y˙ + (t) e−A(t)

∫T

y + (τ ) e−A(τ ) f (τ ) dτ

∫t

y + (τ ) e−A(τ ) f (τ ) dτ y˙ + (t) e−A(t) y˙ + (t) e−A(t) { } ε (L + ε) L + 2ε < max + (1 − ε) , b b b =

−∞

+

T

for t > T1 . Also, since f (τ ) < L + ε for τ > T , by (2.20), ∫ ∞ ∫ ∞ L+ε − y − (τ ) e−A(τ ) f (τ ) dτ < (L + ε) y − (τ ) e−A(τ ) dτ = − y˙ (t) e−A(t) b t t If t > T . Thus by (2.17) and (2.19), [ { } ] 1 L + 2ε L+ε + ε (L + ε) − −A(t) − −A(t) y (t) < + y (t) y˙ (t) e max + (1 − ε) , − y (t) y˙ (t) e y˙ (0) − y˙ − (0) b b b b { } [ ] ε L+ε L + 2ε 1 ≤ max + (1 − ε) , y − (t) y˙ (t) e−A(t) − y + (t) y˙ − (t) e−A(t) + − b b b y˙ (0) − y˙ (0) { } ε L+ε L + 2ε = max + (1 − ε) , . b b b

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for any t > T1 . This leads to { lim sup y (t) ≤ max t→∞

ε L+ε L + 2ε + (1 − ε) , b b b

} .

Since ε is arbitrary, it follows that lim sup y (t) ≤ t→∞

L 1 = lim sup f (t) . b b t→∞

This proves the first inequality in (3.1). The proof of the second inequality in (3.1) is similar. To prove the first inequality in (3.2), we let ε > 0 and let T < 0 be the constant such that t < T implies f (t) < L + ε, where L = lim supt→−∞ f (t) . Hence, by (2.17) ∫ ∞ ∫ T ∫ ∞ y − (τ ) e−A(τ ) f (τ ) dτ y − (τ ) e−A(τ ) dτ + y − (τ ) e−A(τ ) f (τ ) dτ < (L + ε) T

t

t

for any t < T . Observe that both y − (t) and e−A(t) are decreasing, and by Lemma 2.2, y − (t) → ∞ as t → −∞. Thus ∫ ∞ lim y − (τ ) e−A(τ ) dτ = ∞. t→−∞

Hence

t

∫∞ − y (τ ) e−A(τ ) f (τ ) dτ lim T ∫ ∞ − = 0, t→−∞ y (τ ) e−A(τ ) dτ t

and by l’Hˆ opital’s Rule

∫T − y (τ ) e−A(τ ) dτ lim ∫ t∞ − = 1. t→−∞ y (τ ) e−A(τ ) dτ t

So there is a T1 < T such that ∫∞ − y (τ ) e−A(τ ) f (τ ) dτ 0 < T∫∞ − < ε, y (τ ) e−A(τ ) dτ t

∫T − y (τ ) e−A(τ ) dτ 1 − ε < ∫ t∞ − < 1. y (τ ) e−A(τ ) dτ t

Using (2.20) we derive ∫∞ − ∫T − ∫∞ − y (τ ) e−A(τ ) f (τ ) dτ y (τ ) e−A(τ ) f (τ ) dτ y (τ ) e−A(τ ) f (τ ) dτ T ∫ t ∫ t = + ∞ ∞ −y˙ − (t) e−A(t) b t y − (τ ) e−A(τ ) dτ b t y − (τ ) e−A(τ ) dτ { } ε L+ε L + 2ε < max + (1 − ε) , . b b b Also, since f (τ ) < L + ε for τ < T , by (2.20) ∫ t ∫ y + (τ ) e−A(τ ) f (τ ) dτ < (L + ε) −∞

t

y + (τ ) e−A(τ ) dτ =

−∞

L+ε + y˙ (t) e−A(t) . b

Hence, by (2.17) [ { }] 1 L+ε − ε L+ε L + 2ε + −A(t) + − −A(t) y (t) < + y (t) y˙ (t) e − y (t) y˙ (t) e max + (1 − ε) , y˙ (0) − y˙ − (0) b b b b { } ε L+ε L + 2ε ≤ max + (1 − ε) , . b b b This leads to

{ lim sup y (t) ≤ max t→−∞

ε L+ε L + 2ε + (1 − ε) , b b b

Since ε is arbitrary,

L 1 = lim sup f (t) . b b t→−∞ t→−∞ The proof of the second inequality in (3.2) is similar. ■ lim sup y (t) ≤

} .

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Remark 3.1. In case limt→∞ f (t) or limt→−∞ f (t) exists, Lemma 3.1 ensures that limt→∞ y (t) or limt→−∞ y (t) also exists. Consider a solution w = (w1 , . . . , wn ) given by Theorem 1.1. For simplicity in notation, for any function u (t), we denote u+ = lim inf u (t) , t→∞

u ¯+ = lim sup u (t) , t→∞

u− = lim inf u (t) , t→−∞

u ¯− = lim sup u (t) . t→−∞

For any function f (t, z) defined for t ∈ R, we use the notations f + (z) = lim f (t, z) , t→∞

f − (z) = lim f (t, z) t→−∞

if the involved limits exist. 3.1. Properties of w as s → +∞ The next theorem gives the properties of w given by Theorem 1.1 as s → +∞. Theorem 3.1. Let (Hm ) and (H1 ) hold, and let w be the solution given by the proof of Theorem 1.1. Suppose limt→+∞ fi (t, w) exists for i = 1, . . . , n. Then the inequalities ( ( [ +] [ +] ) [ +] [ +] ) ¯ kˆ ≤ 0, ¯ k , w kˆ ≥ 0, fi+ w+ , w k , w fi+ w ¯i+ , w (3.3) i i

i

i

i

hold for i = 1, . . . , n. If the limit w+ = lims→+∞ w (s) exists, then w+ is a zero of f . I.e., fi+ (w+ ) = 0 for each i = 1, . . . , n. Proof . Let v = (v1 , . . . , vn ) be the fixed point of the operator K defined in the proof of Theorem 1.1. We apply Lemma 3.1 to each vi . Note that vi satisfies (2.7) with zi = vi . Since vi is bounded between ϕi (w ˆi ) + + and ϕi (w ˜i ), v¯i and v i exist. By Lemma 3.1, 1 1 v¯i+ ≤ lim sup G∗i (t, v (t)) , v + lim inf G∗ (t, v (t)) . i ≥ β t→+∞ β t→+∞ i Since G∗i (t, v) = βvi + G∗i (t, v) which is nondecreasing in vi and [vj ]j∈ki , and is nonincreasing in [vj ]j∈kˆi , it follows that ( ( [ +] [ +] ) [ +] [ +] ) 1 1 ¯ k , v kˆ , ¯ k , v kˆ = v¯i+ + G+ v¯i+ , v v¯i+ ≤ lim sup G∗i t, v¯i+ , v i i i i i β t→+∞ β ( ) ( ) [ [ ] [ ] ] [ ] 1 1 + + + + ∗ + + + + ¯ ¯ v+ lim inf G t, v , v = v G v , v ≥ , v + , v ˆi ˆi i i i i ki k ki k β t→+∞ i β i provided that the limit limt→+∞ Gi (t, v) exists for each i. By (2.3), t′ (s) = t′ (1) is a positive constant for s > 1. Hence, by (2.5) 1 1 + lim Gi (t, v) = lim fi (s, J (v)) = 2 fi (J (v)) ′ t→+∞ s→+∞ (t′ )2 (t (1)) + exists for each i. Hence, G+ i (v) exists and is proportional to fi (w). By Lemma 3.1, ( ) ( [ +] [ +] [ +] [ +] ) + + + ¯ ¯ kˆ ≤ 0. ≥ 0, G v , v k , v G+ v ¯ , v , v ˆ i i i i k k i

i

G+ i

i

i

fi+

Using the relations between (v) and (w), we see that these inequalities are the same as in (3.3). + + In the case where w exists, then w ¯i = w + i for each i. Hence, by (3.3) ) ( + + fi w1 , . . . , wn+ = 0 for i = 1, . . . , n. This completes the proof.



We pose the following hypothesis that improves the results of Theorem 3.1.

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Hypothesis (H2 ). ˜ and w ˆ have limits w ˜ + and w ˆ + as s → +∞. (i) The upper and lower solutions w (ii) For each i, fi = fi (w), and for each vector w−i = (w1 , . . . , wi−1 , wi+1 , . . . , wn ), the equation fi (wi , w−i ) = 0 has only one positive solution wi = hi (w−i ) such that hi (w−i ) is continuous, and fi (wi , w−i ) > 0 if 0 < wi < hi (w−i ) and fi (wi , w−i ) < 0 if wi > hi (w−i ). Theorem 3.2. Suppose Hypotheses (Hm ), (H1 ) and (H2 ) hold. Then there are two constant vectors, ( ) ¯ = W ¯ 1, . . . , W ¯ n that satisfy the inequalities W = (W 1 , . . . , W n ) and W ˆ + ≤ W ≤ w+ ≤ w ¯ + ≤ W ¯≤ w w ˜ +, and the equations

(3.4)

( ) ( [ ] ) [ ] ¯ ˆ =0 ¯ ¯ i, W , [W] , W fi W , [W] = f W ˆ i i k k k k i i

(3.5)

i

i

for i = 1, . . . , n. (Here, inequalities between vectors are componentwise.) Proof . By (Hm ), fi is nondecreasing in wj for each j ∈ ki , and is nondecreasing in wj for each j ∈ kˆi . We show that the function hi in (H2 )–(ii) also has the same monotone properties. Fix integers i, j ∈ {1, . . . , n} ′ such that i ̸= j, and let w−i be a vector. Suppose wj′ > wj . We let w−i to denote the vector that has the ( ′ ) ′ same components as w−i except its jth component is replaced by wj . If wi > hi w−i , then ( ) ′ 0 > fi wi , w−i ≥ fi (wi , w−i ) . ( ′ ) This implies that wi > hi (w−i ). Hence hi w−i ≥ hi (w−i ). That is, hi is monotone nondecreasing in wj . In a similar way we can show that hi is nonincreasing in wj for each j ∈ kˆi . { } { (m) } ¯ and W(m) for Let w be a solution given by Theorem 1.1. We construct two sequences W ¯ (m) and W(m) , ¯ (m) and W (m) for i = 1, . . . , n denote the components of W m = 0, 1, . . . as follows. Let W i i (0) (0) + + + + ¯ respectively. We first define W =w ˜ , and W =w ˆ for each i, where w ˜ and w ˆ are components of i

i

i

i

i

i

˜ + and w ˆ + , respectively. We then define the sequences for m ≥ 1 iteratively, by w ([ ([ ] [ ] ) ] [ ] ) (m+1) (m+1) (m) (m) (m) (m) ¯ ¯ ¯ , W , W Wi = hi W , Wi = hi W ˆi k

ki

ki

ˆi k

(3.6)

{ } { (m) } ¯ for i = 1, . . . , n. We show that W is nonincreasing and W(m) is nondecreasing in m. To show that ¯ (1) ≤ W ¯ (0) , we observe that since w ˜ and w ˆ have limits as s → +∞, w ˜ ′, w ˜ ′′ , w ˆ ′ and W(0) ≤ W(1) and W ˆ ′′ all converge to 0 as s → +∞. Thus from (1.5) we see w ( ( [ +] [ +] ) [ +] [ +] ) ˜ k , w ˆ kˆ ≤ 0, fi w ˆ k , w ˜ kˆ ≥ 0. fi w ˜i+ , w ˆi+ , w i

Hence, w ˜i+ ≥ hi

([

˜+ w

] ki

i

i

[ +] ) ¯ (1) , ˆ kˆ = W , w i i

w ˆi+ ≤ hi

i

([ ] [ ] ) (1) ˆ+ k , w ˜ + kˆ = W i . w i

i

(3.7)

(3.8)

˜+ ≥ w ˆ + , it follows that Furthermore, since w ([ ] [ ] ) ([ ] [ ] ) ˜+ k , w ˆ + kˆ ≥ hi w ˆ+ k , w ˜ + kˆ . hi w i

i

i

i

¯ (1) ≥ W (1) for each i. Suppose This means W i i ¯ (m) ≤ W ¯ (m−1) ≤ W ¯ (0) . W(0) ≤ W(m−1) ≤ W(m) ≤ W

(3.9)

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89

Then, by the monotonicity of hi , ([ ([ ] [ ] ) ] ) ] [ (m−1) (m−1) ¯ ¯ (m) , W(m) ¯ (m) , ¯ (m+1) = hi W , W ≤ h W =W W i i i ˆi ˆi ki k k ki ([ ([ ] [ ] ) ] [ ] ) (m+1) (m) ¯ (m) ¯ (m−1) Wi = hi W(m) , W ≥ hi W(m−1) , W = Wi , ˆi k

ki

and ¯ (m+1) = hi W i

([

¯ (m) W

] ki

ˆi k

ki

([ ] ) ] [ [ ] ) (m+1) (m) (m) (m) ¯ , W , W ≥ hi W . = Wi ˆi k

ˆi k

ki

Hence, by induction, the inequalities ¯ (m) ≤ · · · ≤ W ¯ (1) ≤ w ˆ + ≤ W(1) ≤ · · · ≤ W(m) ≤ · · · ≤ W ˜+ w ¯ be the limits of W(m) and W ¯ (m) , hold for all m. Therefore, both sequences converge. Let W and W respectively. Then by the continuity of hi , we see that ([ ] ) ( [ ] ) ¯ ˆ , W ¯ ¯ i = hi W for each i. W i = hi [W]ki , W , [W] ˆi k k k i

i

This is equivalent to (3.5). It remains to show (3.4). It suffices to show that ¯ (m) ¯+ ≤ W W(m) ≤ w+ ≤ w

for each m = 1, 2. . . . .

(3.10)

The inequalities in (3.10) clearly hold for m = 0. Suppose the inequalities hold for some m. Then for any ε > 0 there is T > 0 such that s > T implies (m)

Wi

¯ (m) + ε for i = 1, . . . , n. − ε < wi (s) < W i

Fix an i ∈ {1, . . . , n} and define a new variable η by ∫ s dτ η= ≡ φ (s) D (w i i (τ )) 0

(3.11)

for s > 0.

( +) It is clear that η > 0 for any s > 0 and since Di (wi (τ )) is bounded above by Di w ˜i > 0, η → ∞ if s → ∞. Let y (η) = wi (s). By differentiation, yη = Di (wi ) wi′ ,



yηη = Di (wi ) [Di (wi ) wi′ ] .

Thus y (η) is a solution to the differential equation ( ( )) yηη − cyη + Di (y) fi y, w−i φ−1 (η) = 0

for η > 0

and it satisfies y¯+ = lim sup y (η) = w ¯i+ , η→+∞

y + = lim inf y (η) = w+ i . η→+∞

Let u (x, t) = y (x + ct). Then u satisfies the initial–boundary value problem ( ( )) ut − uxx = Di (u) fi u, w−i φ−1 (x + ct) for (x, t) ∈ (¯ η , ∞) × R+ , + u (¯ η , t) = y (¯ η + ct) for t ∈ R , u (x, 0) = y (x) for x ∈ (¯ η , ∞) ,

(3.12)

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90

where η¯ = φ (T ). We consider the initial–boundary value problem ( ] ) [ (m) ] [ ¯ u ˜t − u ˜xx = Di (˜ u ) fi u ˜, W + ε k , W(m) − ε , ˆi i k ( [ ) ] [ ] ¯ (m) + ε ˆ , u ˆt − u ˆxx = Di (ˆ u ) fi u ˆ, W(m) − ε , W k

for (x, t) ∈ (¯ η , ∞) × R+ ,

i

ki

u ˜ (¯ η , t) = u ˆ (¯ η , t) = y (¯ η + ct)

for t ∈ R+ ,

u ˜ (x, 0) = u ˆ (x, 0) = y (x)

for x ∈ (¯ η , ∞)

where ε = (ε, ε, . . . , ε) is an n − 1-dimensional constant vector. Note that x > η¯ and t > 0 imply that φ−1 (x + ct) > φ−1 (¯ η ) = T. Hence by (3.11), (m)

Wj

( ) ¯ (m) + ε − ε < wj φ−1 (x + ct) < W j

for j = 1, . . . , n.

Using the monotone properties of fi we obtain ( [ ] ) ( [ ( ] [ )] [ ( )] ) (m) (m) ¯ ≥ fi u −ε ˜, w φ−1 (x + ct) k w φ−1 (x + ct) kˆ , fi u +ε , W ˜, W ˆi i i k ki ( [ ] [ ] ) ( [ ( )] [ ( )] ) ¯ (m) + ε ≤ fi u ˆ, w φ−1 (x + ct) k w φ−1 (x + ct) kˆ , fi u ˆ, W(m) − ε , W ˆi k

ki

i

i

So u ˜ and u ˆ are upper and lower solutions of (3.12). By the comparison principle [1, Proposition 2.1], u ˆ (x, t) ≤ u (x, t) ≤ u ˜ (x, t)

for (x, t) ∈ (¯ η , ∞) × R+ .

Furthermore, by [1, Theorem 5.1] ([ ] ) ] [ ¯ (m) + ε , W(m) − ε , W ˆi x→∞ t→∞ k ki ([ ) ] [ ] ¯ (m) + ε , W(m) − ε lim lim sup u ˜ (x, t) = hi W . lim lim inf u ˆ (x, t) = hi

x→∞

ˆi k

ki

t→∞

Hence ([ ] [ ] ) (m) (m) ¯ +ε lim lim inf u (x, t) ≥ hi W −ε , W , ˆi x→∞ t→∞ ki k ([ ) ] [ ] ¯ (m) + ε , W(m) − ε lim lim sup u (x, t) ≤ hi W .

x→∞

ˆi k

ki

t→∞

This implies that ([ ([ ] [ ] ) ] [ ] ) (m) (m) + + (m) (m) ¯ ¯ hi W −ε , W +ε ≤ wi ≤ w ¯i ≤ hi W +ε , W −ε . ˆi k

ki

ˆi k

ki

Since hi is continuous and ε is arbitrary, we see that ([ ([ ] [ ] ) ] [ ] ) (m) + (m) ¯ (m) ¯ hi W(m) , W ≤ w+ ≤ w ¯ ≤ h W W , i i i ki

ˆi k

ki

ˆi k

for each i. This proves ¯ (m+1) ¯+ ≤ W W(m+1) ≤ w+ ≤ w and completes the induction proof of (3.10). Finally, (3.4) follows from taking the limit m → ∞ in (3.10). This completes the proof of the theorem.



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3.2. Properties of w as s → −∞ Similar properties of w given in Theorems 3.1 and 3.2 need not hold for s → −∞ because t′ = 4ρe−4ρs → − ∞ as s → −∞. Hence G− i (v) = 0 for all i and v. Therefore, no information about fi (w) is given from the corresponding properties of G− i (v). However, if some of the equations are nondegenerate, then the same properties hold for the corresponding components. Theorem 3.3. Let (Hm ) and (H1 ) hold. Suppose there is a subset N of the integers {1, . . . , n} such that for each i ∈ N there is a δi > 0 such that Di (w) > δi for w ˆi ≤ w ≤ w ˜i and lims→−∞ fi (s, w) exist. Then ( ) ( [ −] [ −] [ −] [ −] ) ¯ kˆ ≤ 0 for i ∈ N. ¯ k , w kˆ ≥ 0, fi− w− , w (3.13) fi− w ¯i− , w i , w k i

i

i

i

If (H2 )-(i) also holds and (H2 )-(ii) holds for each i ∈ N , then there are two constant vectors, W = ( ) ¯ = W ¯ 1, . . . , W ¯ n that satisfy the relations (W 1 , . . . , W n ) and W ¯i ≤ w w ˆi− ≤ W i ≤ w− ¯i− ≤ W ˜i− i ≤w ¯i = w W ¯i− for i ̸∈ N. W i = w− i ,

for i ∈ N,

and Eqs. (3.5) are satisfied for each i ∈ N . Proof . The proof is similar to those of Theorems 3.1 and 3.2. We only highlight the main steps. Suppose the hypothesis of the theorem holds for i ∈ N . Then Ji (zi ) is bounded for zi in between ϕi (w ˆi ) and ϕi (w ˜i ). ∗ Construct an operator Ki corresponding to Eq. (2.1), and apply Lemma 3.1 to the fixed point, vi , of Ki∗ , we obtain ( ( )] )] [ ] [ −] [ ] [ −] 1[ − 1[ − βv i + Fi− v − β¯ vi + Fi− v¯i− , v¯j− j∈k , v − v− ¯j j∈kˆ . , v¯i− ≤ i ≥ i , v j j∈ki , v j ˆ j∈ki i i β β Hence,

) ( [ ] [ ] ≥ 0, Fi− v¯i− , v¯j− j∈k , v − j j∈k ˆ i

i

( ) [ −] [ −] Fi− v − , v , v ¯ ≤ 0. i j j∈k j j∈k ˆ i

i

These inequalities are equivalent to those in (3.13). If (H2 )-(i) holds and (H2 )-(ii) holds for an i ∈ N , then the variable η defined by ∫ 0 dτ ≡ φ (s) for s < 0 η= s Di (wi (τ )) tends to ∞ as s → −∞. Let y (η) = wi (s). By differentiation, y (η) is a solution to the differential equation ( ( )) yηη + cyη + Di (y) fi y, w−i φ−1 (η) = 0 for η > 0. Let u (x, t) = y (x − ct). Then u satisfies the initial–boundary value problem (3.12) with ct changed to −ct. All the remaining proof of Theorem 3.2 after (3.12) is valid upon changing ct to −ct, changing the superscript “+” to “−”, and changing “i ∈ {1, . . . , n}” to “i ∈ N ”. This proves the second part of the theorem. ■ 4. Application As an illustration of application of results in Sections 2 and 3, we consider a two-species predator– prey model. This model is also investigated in [24] with nondegenerate diffusion. We extend the result to a degenerate case. Traveling wave solutions for predator–prey models with linear or nonlinear diffusions have been under investigation by a number of researchers [4,8,14–17]. However, few work deals degenerate nonlinear diffusion.

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We consider a Lotka–Volterra predator–prey model with degenerate nonlinear diffusion for the predator species, ut − D1 uxx = u (a1 − b1 u − c1 v) , x ∈ R, t > 0 vt − D2 (v m vx )x = v (−a2 + b2 u − c2 v) , where ai , bi , and ci for i = 1, 2 are positive constants, D1 and D2 are positive constants, and m is a nonnegative constant. Using scalings of x, t, u and v, we can assume that a1 = b1 = c1 = D2 = 1 and write the equations as ut − uxx = u (1 − u − v) , (4.1) vt − D (v m vx )x = αv (−β + γu − v) , where D, α and β are positive constants. The system has a unique positive constant solution (u∗ , v ∗ ) for any γ > β, where 1+β γ−β u∗ = , v∗ = 1+γ 1+γ and the trivial and semitrivial equilibria (0, 0) and (1, 0). We seek a traveling wave solution connecting (1, 0) and (u∗ , v ∗ ). It is easy to verify that for the corresponding ODE system in which u and v are independent of x, (1, 0) is a saddle point and (u∗ , v ∗ ) is an attractor. Let u1 = (1 − u) /k,

u2 = v/k

where k = (γ − β) / (1 + γ). Then u1 and u2 satisfy the differential equations (u1 )t − (u1 )xx = (u1 − u2 ) (ku1 − 1) , (u2 )t − Dk m (um 2 (u2 )x )x = αku2 (1 + γ − γu1 − u2 ) and the equilibria (1, 0) and (u∗ , v ∗ ) for (u, v) correspond to (0, 0) and (1, 1) for (u1 , u2 ), respectively. Let (w1 (s) , w2 (s)) be the traveling wave solution such that u1 (x, t) = w1 (x + ct) ,

u2 (x, t) = w2 (x + ct) .

Then (w1 , w2 ) satisfies w1′′ − cw1′ + (w1 − w2 ) (kw1 − 1) = 0, ′ d (w2m w2′ ) − cw2′ + aw2 (1 + γ − γw1 − w2 ) = 0, lim (w1 , w2 ) = (0, 0) , lim (w1 , w2 ) = (1, 1) ,

ξ→−∞

s ∈ R,

(4.2)

ξ→∞

m

where d = Dk and a = αk. We need a result regarding the solution to the problem ′

d (um u′ ) − cu′ + au (1 − u) = 0 in R, lim u (s) = 0, lim u (s) = 1. s→−∞

(4.3)

s→+∞

By a result in [2], there is a positive constant c∗ such that the above problem has a solution for all c > c∗ , and the solution u (s) is strictly increasing in s. Furthermore, the solution u (s) satisfies the relation u′ (s) ∼ ρu (s)

as u → 0

(4.4)

for some positive constant ρ. (It can be shown that ) √ 1( c − c2 − 4ad ρ= 2 −1 for the nondegenerate case, m = 0, and √ it is proportional to c for the degenerate case, m > 0. Indeed, in ′ the latter case, by the scaling s = s a/d, one can write the differential equation in (4.3) as ( ) d du m du u − a′ ′ + u (1 − u) = 0 ds′ ds′ ds

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√ where a′ = c/ ad. Then relations (18) and (22) in [2] imply un du ∼ Aun+1 a′ ds′ where A = (a′ )

−2

. This leads to (4.4) with −1

ρ = a′ A = (a′ )

√ =

adc−1 .

Hence, ρ is proportional to c−1 .) As a result, u (s) ∼ εeρs as s → −∞ for some constant ε > 0. These results will be useful in the following analysis. Note that the equation is invariant under any translation s ↦→ s + η for any η ∈ R. So the solution is not unique. The next theorem shows that when the predation rate γ is larger than the predator death rate β, then there is a wavefront solution connecting the prey-only state (1, 0) and the coexistent state (u∗ , v ∗ ) for c sufficiently large. Theorem 4.1. Suppose that m ≥ 0 and β < γ < 1. Then there is a constant c∗ > 0 such that (4.2) has a solution for every c > c∗ , where c∗ is the critical value such that Problem (4.3) has a strictly increasing solution if and only if c > c∗ . Proof . Upper and lower solutions (w ˜1 , w ˜2 ) and (w ˆ1 , w ˆ2 ) of (4.2) satisfy the inequalities w ˜1′′ − cw ˜1′ + (w ˜1 − w ˆ2 ) (k w ˜1 − 1) ≤ 0, ′′ w ˆ 1 − cw ˆ1 − w ˜2 ) (k w ˆ1 − 1) ≥ 0 ˆ1′ + (w ′ d (w ˜2m w ˜2′ ) − cw ˜2′ + aw ˜2 (1 + γ − γ w ˆ1 − w ˜2 ) ≤ 0, m ′ ′ ′ d (w ˆ2 w ˆ2 ) − cw ˆ2 + aw ˆ2 (1 + γ − γ w ˜1 − w ˆ2 ) ≥ 0

in R.

(4.5)

We choose w ˆ1 = 0, w ˜2 = 1 + γ and let w ˆ2 be a solution of the problem (4.3). Finally, we let w ˜1 be a solution to the problem w′′ − cw′ + (w − w ˆ2 (s)) (kw − 1) = 0 in R, (4.6) lim w (s) = 0, lim w (s) = 1. s→−∞

s→+∞

To see that the above problem has a solution, we observe that 0 and 1 are ordered upper and lower solutions of this problem, respectively. Therefore by Theorem 1.1 a solution w exists that satisfies 0 ≤ w (s) ≤ 1 in R. Since w ˆ2− = 0, by Theorem 3.3 ( − ) ( ) w ¯− kw ¯ − 1 ≥ 0, w− kw− − 1 ≤ 0. Since 0 < k < 1 and 0 ≤ w ≤ 1, it follows that k w ¯ − , kw− < 1. Hence w ¯ − ≤ 0 and w− ≥ 0. This implies + − − − that w = w ¯ = w = 0. Similarly, since w ˆ2 = 1, by Theorem 3.1 ( + )( + ) ( + )( ) w ¯ − 1 kw ¯ − 1 ≥ 0, w − 1 kw+ − 1 ≤ 0. Since k w ¯ + , kw+ < 1, it follows that w ¯ + ≤ 1 and w+ ≥ 1. This implies that w+ = w ¯ + = w+ = 1. Hence Problem (4.6) has a solution, and we use it as w ˜1 . To see that w ˆ1 , w ˜1 , w ˆ2 and w ˜2 are coupled upper and lower solutions of (4.2), we observe that by definition, w ˆ1 = 0 ≤ w ˜1 and w ˆ2 ≤ 1 < 1 + γ = w ˜2 . Also, since w ˆ1 = 0, w ˆ2 ≤ 1 and k < 1, w ˜1′′ − cw ˜1′ + (w ˜1 − w ˆ2 ) (k w ˜1 − 1) = 0, w ˆ1′′ − cw ˆ1′ + (w ˆ1 − w ˜2 ) (k w ˆ1 − 1) = w ˜2 ≥ 0, ′

d (w ˜2m w ˜2′ ) − cw ˜2′ + aw ˜2 (1 + γ − γ w ˆ1 − w ˜2 ) = −aγ w ˜2 w ˆ1 ≤ 0, ′

d (w ˆ2m w ˆ2′ ) − cw ˆ2′ + aw ˆ2 (1 + γ − γ w ˜1 − w ˆ2 ) = aγ w ˆ2 (1 − w ˜1 ) ≥ 0.

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Thus all inequalities in (4.5) hold. Furthermore, as a strictly increasing solution of (4.3), there is a constant ρ > 0 such that w ˆ2 (s) ∼ εeρs as s → −∞. Hence lim inf e−ρms dw ˆ2m (s) ∼ dεm > 0, s→−∞

lim sup eρms mdw ˆ2m−1 (s) ∼ lim sup mdεm−1 eρs = 0. s→−∞

s→−∞

So (H1 )–(iv) also holds. By Theorem 1.1, Problem (4.2) has a solution (w1 , w2 ) that satisfies the inequalities 0 ≤ w1 (s) ≤ 1,

w ˆ2 (s) ≤ w2 (s) ≤ w ˜2 (s)

in R.

It remains to show that (w1 , w2 ) satisfies the boundary conditions lim (w1 (s) , w2 (s)) = (0, 0) ,

s→−∞

lim (w1 (s) , w2 (s)) = (1, 1) .

(4.7)

s→∞

Since lim w ˆ1 (s) = lim w ˜1 (s) = 0,

s→−∞

s→−∞

it follows that lims→−∞ w1 (s) = 0. Consider lims→−∞ w2 (s). Since the first equation in (4.2) is ¯ 1, W nondegenerate and it is easy to verify that (H2 ) holds, by Theorem 3.3, there exist constants W 1 such that ¯1 ≤ w w ˆ1− ≤ W 1 ≤ w− ¯1− ≤ W ˜1− 1 ≤w and ( ) f1 W 1 , w − 2 = 0,

) ( ¯ 1, w ¯2− = 0. f1 W

¯ 1 = 0 . Therefore By the definition of w ˆ1 and w ˜1 , it follows that w ˆ1− = w ˜1− = 0, Hence W 1 = W ( ) w− w ¯2− = f1 0, w ¯2− = 0. 2 = f1 (0, w 2 ) = 0, It follows that w2− = 0. This means lims→−∞ w2 (s) = 0. We next prove the second boundary condition in (4.7). By construction, w ˆ1+ = 0, w ˜1+ = w ˆ2+ = 1, and + w ˜2 = 1 + γ. Also, the equation f1 (w1 , w2 ) = 0 has a unique positive solution w1 = w2 on the interval [0, 1] for any w2 ≥ 0, and the equation f2 (w1 , w2 ) = 0 has a unique positive solution w2 = 1 + γ − γw1 for any w1 ∈ [0, 1]. It is easy to verify that (H2 ) is satisfied. By Theorem 3.2, there are vectors (W 1 , W 2 ) and ( ) ¯ 1, W ¯ 2 such that W ( ) ¯ 1, W ¯ 2 ≤ (1, 1 + γ) (0, 1) ≤ (W 1 , W 2 ) ≤ W and ( ) ( ) ( ) ( ) ¯ 1 , W = f1 W , W ¯ 2 = f2 W ¯ 1 , W = f2 W , W ¯ 2 = 0. f1 W 2 1 2 1 These equations lead to ¯1 = W , W 2

¯ 2, W1 = W

¯ 2 = 1 + γ − γW , W 1

¯ 1. W 2 = 1 + γ − γW

The linear system has a unique solution ¯1 = W = W ¯ 2 = W = 1. W 1 2 Hence, by Theorem 3.2, w+ ¯1+ = w+ ¯2+ = 1. This proves the second equation in (4.7). 1 =w 2 =w



Numerical solutions for unknowns u and v for (4.1) with D = 1, m = 2, α = 0.2, β = 0.2, γ = 0.8 are computed. The graph for t = 500 and 750 is shown in Fig. 1. It is apparent that the solution takes the form of a wavefront and propagates to left as time increases. (We also compared the solution curves with those that have m = 10, but the curves are not distinguishable.)

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Fig. 1. 2-species predator–prey model.

Acknowledgments This work was partially supported by the Simons Foundation grant #245488. The author would like to thank Mr. Miguel Aragon, a student at Purdue University Northwest, for writing the Matlab codes that generated the numerical solutions and their graphs for the models presented in Section 4. References [1] D.G. Aronson, H.F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. 446

96

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(1975) 5–49. [2] C. Atkinson, G.E.H. Reuter, C.J. Ridler-Rowe, Traveling wave solutions for some nonlinear diffusion equations, SIAM J. Math. Anal. 12 (6) (1981) 880–892. [3] A. Boumenir, V. Nguyen, Perron theorem in monotone iteration method for traveling waves in delayed reaction–diffusion equations, J. Differential Equations 244 (2008) 1551–1570. [4] S.R. Dunbar, Traveling wave solutions of diffusive Lotka–Volterra equations, J. Math. Biol. 17 (1983) 11–32. [5] N. Fei, J. Carr, Existence of travelling waves with their minimal speed for a diffusing Lotka–Volterra system, Nonlinear Anal.: Real World Appl. 4 (2003) 503–524. [6] W. Feng, X. Lu, Traveling waves and competitive exclusion in models of resource competition and mating interference, J. Math. Anal. Appl. 424 (2015) 542–562. [7] Wei Feng, Weihua Ruan, Xin Lu, On existence of wavefront solutions in mixed monotone reaction–diffusion systems, Discrete Contin. Dyn. Syst. Ser. B 21 (3) (2016) 815–836. [8] R.A. Gardner, Existence of traveling wave solutions of predator–prey systems via the connection index, SIAM J. Appl. Math. 44 (1984) 56–79. [9] B.H. Gilding, The correspondence between travelling-wave solutions of a nonlinear reaction-convection–diffusion equation and an integral equation, Differential Integral Equ. 9 (5) (1996) 919–947. [10] B.H. Gilding, R. Kersner, The characterization of reaction-convection–diffusion processes by travelling waves, J. Differential Equations 124 (1996) 27–79. [11] B.H. Gilding, R. Kersner, Travelling waves in nonlinear diffusion-convection reaction. 60, 2004. [12] B.H. Gilding, R. Kersner, A fisher/kpp-type equation with density-dependent diffusion and convection: travelling-wave solutions, J. Phys. A 38 (2005) 3367–3379. [13] K. Hong, P. Weng, Stability and traveling waves of diffusive predator–prey model with age-structure and nonlocal effect, J. Appl. Anal. Comput. 2 (2012) 173–192. [14] C.H. Hsu, C.R. Yang, T.H. Yang, T.S. Yang, Existence of traveling wave solutions for diffusive predator–prey type systems, J. Differential Equations 252 (2012) 3040–3075. [15] W. Huang, Traveling wave solutions for a class of predator–prey systems, J Dyn. Differential Equations 24 (2012) 633–644. [16] J.H. Huang, G. Lu, S.G. Ruan, Existence of traveling wave solutions in a diffusive predator–prey model, J. Math. Biol. 46 (2003) 132–152. [17] Huiru Li, Haibin Xiao, Traveling wave solutions for diffusive predator–prey type systems with nonlinear density dependence, Comput. Math. Appl. 74 (10) (2017) 2221–2230. [18] G. Lin, W. Li, M. Ma, Traveling wave solutions in delayed reaction diffusion system with applications to multi-species models, Discrete Cont. Dyn. Syst. -B 13 (2010) 393–414. [19] Guo Lin, Shigui Ruan, Traveling wave solutions for delayed reaction–diffusion systems and applications to diffusive Lotka– Volterra competition models with distributed delays, J. Dynam. Differential Equations 26 (3) (2014) 583–605. [20] C.V. Pao, Dynamics of food-chain models with density-dependent diffusion and ratio-dependent reaction function, J. Math. Anal. Appl. 433 (2016) 355–374. [21] C.V. Pao, W.H. Ruan, Quasilinear parabolic and elliptic systems with mixed quasimonotone functions, J. Differential Equations 255 (2013) 1515–1553. [22] C.V. Pao, W.H. Ruan, Existence and dynamics of quasilinear parabolic systems with time delays, J. Differential Equations 258 (2015) 3248–3285. [23] C.V. Pao, W.H. Ruan, Dynamics of degenerate quasilinear reaction diffusion systems with nonnegative initial functions, J. Differential Equations 263 (11) (2017) 7709–7752. [24] W.H. Ruan, W. Feng, Xin Lu, Wavefront solutions of quasilinear reaction–diffusion systems with mixed quasi-monotonicity, Appl. Anal., published online: 04 Dec. 2017. http://dx.doi.org/10.1080/00036811.2017.1408077. [25] W.H. Ruan, W. Feng, Xin. Lu, On traveling wave solutions in general reaction–diffusion systems with time delays, J. Math. Anal. Appl. 448 (2017) 376–400. [26] J.L. V´ azquez, Porous medium flow in a tube. traveling waves and kpp behavior, Commun. Contemp. Math. 9 (2007) 731–751. [27] A. Volpert, V. Volpert, V. Volpert, Traveling wave solutions of parabolic systems, Transl. Math. Monograhs 140 (1994). [28] Z-C. Wang, W-T. Li, S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations 238 (2007) 153–200. [29] J. Wu, X. Zou, Traveling wave fronts of reaction–diffusion systems with delay, J. Dynam. Differential Equations 3 (2001) 651–687. [30] J. Wu, X. Zou, Erratum to: Traveling wave fronts of reaction–diffusion systems with delay, J. Dynam. Differential Equations 20 (2) (2008) 531–533.