This section is devoted to the uniqueness of the traveling wave solution. We shall follow the method developed in [5]. For a smooth function ϕ, we let
L[ϕ](x) := −cϕ′(x)− D2[ϕ](x)−
∑2 i=−2
J (i)[b(ϕ(x− i)) − dϕ(x)].
First, we define the notion of super-sub-solutions as follows.
Definition 2.5.1 A non-constant smooth function ϕ : [a− 2, b + 2] → (0, 1) is called a supersolution (subsolution, resp.) of (2.1.6) on [a, b] for a wave speed c, if L[ϕ](x) ≥ 0 (L[ϕ](x) ≤ 0, resp.) for x ∈ (a, b).
Definition 2.5.2 A non-constant smooth function ϕ : [a − 2, ∞) → (0, 1) is called a supersolution (subsolution, resp.) of (2.1.6) on [a,∞) for a wave speed c, if L[ϕ](x) ≥ 0 (L[ϕ](x) ≤ 0, resp.) for x ∈ (a, ∞).
Lemma 2.5.1 Assume (H1). Let (c, U ) be a solution of (P) and V (x) be a subsolution (supersolution, resp.) of (2.1.6) on [a, b] for the same speed c, where a < b. If V (x) < U (x) (V (x) > U (x), resp.) for x∈ [a − 2, a) ∪ (b, b + 2], then V (x) < U(x) (V (x) > U(x), resp.) for x∈ [a, b].
Proof . Since the case for supersolution is similar, we only consider the case when V (x) is a subsolution. We introduce
g(t) := max
x∈[a−2,b+2]{V (x) − U(x − t)}.
Since U (∞) = 0 and U(−∞) = 1, we can choose ζ ∈ R such that g(ζ) = 0. Let y ∈ [a− 2, b + 2] be the maximum value in [a − 2, b + 2] such that V (y) − U(y − ζ) = 0. We claim that y ∈ [a − 2, a) ∪ (b, b + 2].
Suppose on the contrary that y ∈ [a, b]. Then we have
V (y) = U (y− ζ), V′(y) = U′(y− ζ), V (y − 1) ≤ U(y − 1 − ζ),
V (y− 2) ≤ U(y − 2 − ζ), V (y + 1) < U(y + 1 − ζ), V (y + 2) < U(y + 2 − ζ).
Hence we have L[V ](y) > L[U ](y− ζ). By the strictly inequality, without loss of generality we may assume that y ∈ (a, b). This contradicts that U(x) is a solution of (2.1.6) and V (x) is a subsolution of (2.1.6) on [a, b]. Therefore, y ∈ [a − 2, a) ∪ (b, b + 2].
By hypothesis, we have U (y) > V (y) = U (y− ζ). It follows from the monotonicity of U that ζ < 0. Hence U (x− ζ) < U(x) for all x ∈ R. Since g(ζ) = 0, we deduce that V (x) < U (x) for all x∈ [a, b]. The proof is completed.
Lemma 2.5.2 Assume (H1). Let (c, U ) be a solution of (P) and ϕ(x) be a subsolution (or supersolution) of (2.1.6) with the same speed c on [a,∞) for some constant a. If
xlim→∞
ϕ′(x)
ϕ(x) = lim
x→∞
U′(x) U (x) = Λ, then there exists A∈ [−∞, ∞] such that
xlim→∞W (ξ, x) = A + Λξ, ∀ξ ∈ R, where W (ξ, x) := ln[U (x + ξ)]− ln[ϕ(x)].
Proof. Given a subsolution ϕ(x) of (2.1.6) on [a,∞) for some constant a. By the definition of W (ξ, x), we obtain
(2.5.1) lim
x→∞[W (ξ, x)−W (0, x)] = lim
x→∞{ln[U(x+ξ)]−ln[U(x)]} = lim
x→∞
∫ x+ξ x
U′(t)
U (t)dt = Λξ
for all ξ ∈ R. It follows from (2.5.1) that either the limit limx→∞W (ξ, x) exists for all ξ ∈ R or it does not exist for all ξ ∈ R.
Suppose that the limit limx→∞W (ξ, x) does not exist for all ξ ∈ R. By (2.5.1), we can choose an appropriate ξ such that
A := lim sup
x→∞ W (ξ, x) > 0 > B := lim inf
x→∞ W (ξ, x).
Indeed, ξ can be chosen as
ξ =−[lim sup
x→∞ W (0, x) + lim inf
x→∞ W (0, x)]/(2Λ).
Take α, β such that B < β < 0 < α < A. Then we may choose two sequences {xi} and {yi} such that
ilim→∞xi =∞, a − 2 ≤ xi < yi < xi+1, W (ξ, xi) = α, W (ξ, yi) = β, ∀i ∈ N.
Since
xlim→∞Wx(ξ, x) = lim
x→∞
{U′(x + ξ)
U (x + ξ) − ϕ′(x) ϕ(x)
}
= 0,
we can choose a fixed integer i large enough such that
W (ξ, x) > 0, if x∈ [xi− 2, xi]∪ [xi+1, xi+1+ 2],
i.e., U (x + ξ) > ϕ(x) for all x∈ [xi − 2, xi]∪ [xi+1, xi+1+ 2]. Since W (ξ, yi) = β < 0, we obtain that U (yi+ ξ) < ϕ(yi). But, by Lemma 2.5.1, it is impossible, since yi is between xi and xi+1. Therefore, the limit limx→∞W (ξ, x) exists for all ξ ∈ R. We conclude that limx→∞W (ξ, x) = A + Λξ for all ξ∈ R, where A := limx→∞W (0, x).
The case when ϕ(x) is a supersolution is similar, the lemma follows.
With this lemma, we are ready to prove Theorem 2.1.5.
Proof of Theorem 2.1.5. Suppose that the roots of Φ(· ; c) = 0 are given by Λ and λ with Λ > λ. We choose ω < 0 such that max{λ, (1 + α)Λ} < ω < Λ, where the constant α is defined in (H2). Since Φ(λ; c) is a convex function in λ, we have Φ(ω; c) < 0. Following [5], we define
ϕ±(x; ϵ, δ) := δ((1∓ ϵ)eΛx± ϵeωx), where δ > 0, ϵ∈ (0, eω], x≥ −2. We may easily check
0 < ϕ±(x; ϵ, δ) < 2δeΛx, if δ > 0, ϵ∈ (0, eω], x≥ 0.
(2.5.2)
By a simple computation, we have
On the other hand, by (2.5.2), we have
0 < ϕ+(x + i; ϵ, δ) < 2δeΛ(x+i) ≤ 2δe−2ΛeΛx,
Therefore, we can easily deduce the following facts.
(a1) For every ϵ ∈ (0, eω], there exists δϵ > 0 such that ϕ+(x; ϵ, δ) is a supersolution on [2,∞), if δ ∈ (0, δϵ].
(a2) For every ϵ∈ (0, eω] and δ = 1, there exists xϵ ≥ 0 such that ϕ+(x; ϵ, 1) is a superso-lution on [xϵ,∞).
Now, we consider ϕ(x) := ϕ+(x; ϵ, δ) with ϵ = eω and δ = 1. By Lemma 2.5.2, there exists A∈ [−∞, ∞] such that
xlim→∞{ln[U(x + ξ)] − ln[ϕ(x)]} = A + Λξ, ∀ξ ∈ R.
We claim that A >−∞. Suppose not. Then we have
xlim→∞{ln[U(x + ξ)] − ln[ϕ(x)]} = −∞, ∀ξ ∈ R.
(2.5.4)
Fix ϵ = eω > 0, let δϵ be the constant defined in (a1). Then it follows from Theorem 2.1.2 and U (∞) = 0 that there exists η > 0 such that U(η) < δϵ and
(2.5.5) U′(x)
U (x) > Λ + ϵ(ω− Λ), if x ≥ η − 2.
By the fact (a1), the function ˆϕ(x) := ϕ+(x; ϵ, U (η)) is a supersolution on [2,∞).
Note that ˆϕ(0) = ϕ+(0; ϵ, U (η)) = U (η). Also, from (2.5.3) and (2.5.5) it follows that ϕˆ′(x)
ϕ(x)ˆ ≤ Λ + ϵ(ω − Λ) < U′(x + η)
U (x + η), ∀x ∈ [−2, 0].
By an integration, we obtain ˆϕ(x) > U (x + η), if x ∈ [−2, 0). On the other hand, since [ ˆϕ(x)/ϕ(x)]≡ U(η), it follows from (2.5.4) that
xlim→∞{ln[U(x + η)] − ln[ˆϕ(x)]} = −∞.
So we can choose T > 0 such that ˆϕ(x) > U (x+η), ∀x ∈ [T, ∞). Therefore, by Lemma 2.5.1, we have ˆϕ(x) > U (x + η) for all x∈ [−2, ∞). This contradicts ˆϕ(0) = U(η). Therefore, we conclude that A >−∞. Thus
xlim→∞ln
[U (x) ϕ(x) ]
= lim
x→∞{ln[U(x)] − ln[ϕ(x)]} = A.
It follows from the definition of ϕ(x) that the limit L := limx→∞[U (x)e−Λx] exists and L > 0.
Moreover, using the function ϕ−, (2.5.3) and (H2), we can prove, by a similar reasoning as above, that L <∞. This proves the theorem.
Chapter 3
Two-front entire solutions
3.1 Introduction
In this chapter, we study the following discrete diffusive equation with convolution type nonlinearity.
ut(x, t) =D2[u](x, t)− du(x, t) +∑
i∈Z
J (i)b(u(x− i, t)), x∈ R, t ∈ R, (3.1.1)
where d > 0, J (i) = J (−i) ≥ 0, ∑
i∈ZJ (i) = 1, and
D2[u](x, t) := D[u(x + 1, t) + u(x− 1, t) − 2u(x, t)]
for some positive constant D. Throughout this chapter, we shall always assume that the function b(·) is an increasing smooth function on [0, 1] such that
(P1) b(0) = b(a)− ad = b(1) − d = 0, where 0 < a < 1, (P2) b(t) < dt for 0 < t < a, b(t) > dt for a < t < 1, (P3) max{b′(0), b′(1)} < d < b′(a) (bistable nonlinearity), (P4) ∫1
0[b(u)− du]du > 0 (unbalanced case).
When J (0) = 1 and J (i) = 0 for all i̸= 0, (3.1.1) is reduced to the classical equation ut(x, t) =D2[u](x, t) + f (u(x, t)), f (u) := b(u)− du,
which has been studied recently in [17, 18].
We also note that (3.1.1) is the continuum version of the following lattice dynamical system:
(3.1.2) u′n(t) = D[un+1(t) + un−1(t)− 2un(t)]− dun(t) +∑
i∈Z
J (i)b(un−i(t)), n ∈ Z, t ∈ R.
For (3.1.2), in ecology, un represents the population density at site n, D is the migration coefficient, d is the death rate and the nonlinear function b is the birth function of population density which is interacting with neighbors by the nonnegative weighted function J , if the habitat is divided into discrete regions and the population density is measured at the representative point in each region. In this model, we assume that the migration only happens to the nearest neighbors and the interaction happens with finite or infinite range.
We say that {un(t)} is a traveling wavefront solution of (3.1.2) connecting two different equilibria{u±} ⊂ {0, a, 1} with speed c, if un(t) = U (n + ct) for n∈ Z and t ∈ R for some
Similarly, we can define the notion of traveling wavefront solution of (3.1.1) by setting u(x, t) = U (x + ct), then U also satisfies the equation (3.1.3).
Recently, a more general version of (3.1.2) including time delay was studied in [25, 23].
In [25], they studied (3.1.2) with time delay for the bistable case. They proved that the problem admits a unique (up to a translation) strictly monotone increasing traveling wavefront solution connecting from 0 to 1 with a positive wave speed when D ≥ D0 for a certain positive constant D0, under the following extra assumption
(3.1.4) ∑
More precisely, from [25, Theorem 1.1], under the above assumptions, there exist a unique speed ˆc > 0 and a unique (up to translations) wave profile U (x) such that
{ cUˆ ′(x) =D2[U ](x)− dU(x) +∑
i∈ZJ (i)b(U (x− i)), x ∈ R, U (−∞) = 0, U(+∞) = 1, 0 < U < 1, U′ > 0 in R,
(3.1.5)
if D≥ D0. Note that a propagation failure occurs when D is small enough.
The monostable case for (3.1.2) with time delay was considered in [23]. In the present setting, it corresponding to the case for connecting two equilibria {a, 1} or {0, a}. They obtained the existence of the asymptotic speed of propagation, the existence and (partial) uniqueness of the traveling wavefront and the minimal speed of the traveling wavefront for
the delayed lattice dynamical system under the following extra condition at the unstable equilibrium a, namely,
(3.1.6) b′(a)(u− a) − M|u − a|1+α ≤ b(u) − da ≤ b′(a)(u− a) + M|u − a|1+α for u∈ [0, 1]
for some constants M > 0 and α ∈ (0, 1]. In fact, by [23, Theorem 1.2], there exist two constants c∗, c∗ with c∗ > 0 > c∗ such that for any c1, c2 (with c1 ≥ c∗, c2 ≤ c∗) there exist V1(x) and W2(x) satisfying the following equations:
{ c1V1′(x) =D2[V1](x)− dV1(x) +∑
i∈ZJ (i)b(V1(x− i)), x ∈ R, V1(−∞) = a, V1(+∞) = 1, a < V1 < 1, V1′ > 0 in R.
(3.1.7)
and {
c2W2′(x) = D2[W2](x)− dW2(x) +∑
i∈ZJ (i)b(W2(x− i)), x ∈ R, W2(−∞) = 0, W2(+∞) = a, 0 < W2 < a, W2′ > 0 in R,
(3.1.8)
where c∗ (c∗, resp.) is the minimal (maximal, resp.) speed of (3.1.7) ((3.1.8), respectively).
The study of traveling wavefront solutions are important in many applications. It can describe certain dynamical behavior of the studied problem such as (3.1.2). But, the dynamics of reaction-diffusion equations or its discrete analogue is so rich that there might be other interesting patterns. Recently it is found that two-front entire solutions exist in many problems. Here an entire solution is meant by a solution defined for all (x, t) ∈ R2. In particular, traveling wavefront solutions are also entire solutions. For the study of entire solutions, we refer the reader to, for instance, [14, 17, 18, 19, 20, 22, 26, 28] and reference therein.
In a very interesting work by Morita and Ninomiya [26], they gave three different types of entire solutions for a bistable reaction-diffusion equation (see also [18] for the discrete diffusive case). The purpose of this work is to construct these three types of entire solutions for (3.1.1). Although the main idea and the methods of proofs in this part are from [18, 26], there are certain difficulties in dealing with (3.1.1) (or (3.1.2)) due to the convolution type nonlinearity. For example, in the construction of super/sub solutions, we need to derive some estimations. In these estimations, the compactness (finite range interaction) assumption is needed in this study. So, from now on, besides the assumptions (3.1.4) and (3.1.6), we shall assume that
(3.1.9) J (i) = 0 for |i| > m for some m ∈ N.
In fact, to construct these two-front entire solutions it is crucial to have a precise information on the asymptotic behavior of wave tails. More precisely, we need the following estimates for solutions U, V1, W2 of (3.1.5), (3.1.7), (3.1.8) respectively.
First, there exists a positive constant η such that
Furthermore, there are positive constants K, k, γ, δ such that
keλy ≤ U(y) ≤ Keλy, ∀ y ≤ m; γe−µy ≤ 1 − U(y) ≤ δe−µy, ∀ y ≥ −m, (3.1.11)
where λ is the unique positive root of the characteristic equation ˆ
and µ is the unique positive root of the equation
−ˆcµ = D(eµ+ e−µ− 2) − d + b′(1)
Furthermore, there exist positive constants N , ρ such that ρ[V1(y)− a] ≤ V1′(y)≤ Neλ1y on (−∞, 0],
The above asymptotic behavior of wave tail at the unstable equilibrium can be found in [15]. But, due to the technical difficulty arising from the convolution type nonlinearity, we need to assume that m = 2. As for the wave tail at the stable equilibrium, the method developed in [9] is well applicable here for any finite m.
Based on these asymptotic behaviors, we prove the following theorems on two-front entire solutions.
Theorem 3.1.1 Let (3.1.9) be in force with m = 2 and let (ˆc, U (x)) be a solution of (3.1.5). Then for any real number θ there exists an entire solution u(x, t) of (3.1.1) such that a constant ω and an entire solution u(x, t) of (3.1.1) such that
t→−∞lim { sup
Moreover, there exists θ ∈ R such that
tlim→∞{sup
x∈R|u(x, t) − U(x + ˆct + θ)|} = 0.
(3.1.22)
Theorem 3.1.3 Let (3.1.9) be in force with m = 2. For any c2 < c∗ with −c2 < ˆc, let (ˆc, U (x)) and (c2, W2(x)) be solutions of (3.1.5) and (3.1.8) respectively. Then there exist a constant ω and an entire solution u(x, t) of (3.1.1) such that
t→−∞lim { sup
The above constructed entire solutions have some common characters. When −t ≫ 1, they behave as two traveling wavefronts on the opposite sides or on the same side of x-axis.
Note that, different from the previous works, we choose the distinguishing line of the initial conditions in the above theorems to be the mid-points of two front-positions of traveling wavefronts. For example, in Theorem 3.1.2, x =−c1t and x =−c2t are front-positions for two traveling wavefronts V1(x + c1t) and W2(x + c2t), respectively. It is nature to choose the distinguishing line to be x =−(c1+ c2)t/2 in (3.1.21).
We organize this chapter as follows. In section 2, we give some proofs of the asymptotic behaviors of the traveling wavefronts stated above and some useful functions. Next, in section 3, we offer the proofs of Theorem 3.1.1, Theorem 3.1.2 and Theorem 3.1.3 by constructing suitable super/sub solutions.
3.2 Preliminaries
In this section, we first study the asymptotic behaviors of a solution U (y) of (3.1.5) as y→ ±∞. Since the behavior near y = ∞ is similar to the one near y = −∞, we shall only give the details for y =−∞. For this, we use the following notation
N [uj](t) := u′j(t)− D[uj+1(t) + uj−1(t)− 2uj(t)]
+duj(t)−
∑m i=−m
J (i)b(uj−i(t)), j ∈ Z, t ∈ R.
First, we have the following strong comparison principle.
Lemma 3.2.1 Let c ∈ R, j0 ∈ Z and t0 ∈ R. Assume that uj(t) and vj(t) are bounded and continuous in the set {(j, t) ∈ Z × R| j ≤ j0− ct + m, t ∈ [t0,∞)} and satisfy
N [uj](t)≥ N [vj](t) when j < j0− ct, t > t0, uj(t0)≥ vj(t0) when j < j0− ct0,
uj(t)≥ vj(t) when j0− ct ≤ j ≤ j0− ct + m, t ≥ t0.
Then uj(t) ≥ vj(t) for all j < j0 − ct, t > t0. In addition, if uj1(t0) > vj1(t0) for some j1 < j0− ct0, then uj(t) > vj(t) for all j < j0− ct, t > t0.
Since the proof is exactly the same as the one for [9, Lemma 1], we omit it here.
Using this comparison principle (Lemma 3.2.1), we can follow the proof of [9, Theorem 2] to prove the following theorem on the asymptotic behavior.
Theorem 3.2.1 Assume that (c,{uj(t)}) is a traveling wave solution of (3.1.2) connecting from 0 to 1 with positive speed c. Then there exists two positive constants C1, C2 such that
(3.2.1) C1 ≤ uj(t)
eΛ(j+ct) ≤ C2, ∀j + ct ≤ −m, t ≥ 0 where Λ is the positive root of the following characteristic equation
P (c, λ) := cλ− D(eλ+ e−λ− 2) + d − b′(0)
∑m i=−m
J (i)e−iλ= 0.
By the definition of Λ, the function ψ(x) := eΛx is a solution of the following equation
cψ′(x)− D[ψ(x + 1) + ψ(x − 1) − 2ψ(x)] + dψ(x) − b′(0)
∑m i=−m
J (i)ψ(x− i) = 0.
In the construction of sub/supersolutions, ψ(x) play an important role. Indeed, we define u+j(t; ϵ1, θ, ϵ3) := ϵ1ψ(0) + θψ(Λ)eΛ(j+ct) − ϵ3ψ(2Λ)e2Λ(j+ct), j ∈ Z, t ∈ R.
where ϵ1 ≥ 0, ϵ3 ≥ 0, θ ∈ R. Hereafter the function b is suitably defined so that it is smooth with b, b′, b′′ bounded inR. Since P (c, 0) > P (c, Λ) = 0 > P (c, 2Λ) (due to the fact that b′(0) < d), we have
N [u+j](t)≥ 0 when j + ct ≤ −m, t ∈ R,
if
0≤ ϵ1 ≤ E1, ϵ3 = E3θ2, |θ| ≤ E2, where
E1 := P (c, 0)
2Lψ(0), E3 := 8Lψ(Λ)2e2Λm
−P (c, 2Λ)ψ(2Λ), E2 := ψ(Λ)
E3ψ(2Λ), L := max
u∈R |b′′(u)|.
Similarly, by defining
u−j (t; ϵ1, θ, ϵ3) :=−ϵ1ψ(0) + θψ(Λ)eΛ(j+ct)+ ϵ3ψ(2Λ)e2Λ(j+ct), we also get
N [u−j](t)≤ 0 when j + ct ≤ −m, t ∈ R.
Then Theorem 3.2.1 can be proved by using the comparison principle as given in the proof of [9, Theorem 2]. We omit the details here.
Now, for a solution U of (3.1.5), using un(t) = U (n + ˆct) we obtain from (3.2.1) that C1eλy ≤ U(y) ≤ C2eλy, ∀ y ≤ −m,
where λ is the unique positive root of the equation (3.1.12). Hence the first part of (3.1.11) follows. We remark that this process can be carried out as long as the equilibrium is stable.
Therefore, all of the exponential tail behaviors of U, V1, W2 near the stable equilibria {0, 1}
in (3.1.11), (3.1.14) and (3.1.15) can be derived similarly.
As for the exponential tail behavior near the unstable equilibrium a, we refer to [15, Theorem 5]. There it is assumed that m = 2. Therefore, we have the exponential tail behaviors of V1, W2 near the equilibrium a in (3.1.14) and (3.1.15) for c1 > c∗ and c2 < c∗ when m = 2.
For the estimates related the first derivatives of U, V1, W2, we recall from [15, Theorem
exist and are positive. Here we need to assume that m = 2. This result is based on [15, Theorem 1] and is applicable to the case of stable equilibrium. Therefore, we also have the limits
exist and are positive. Then the estimates (3.1.10) and (3.1.16)-(3.1.19) can be derived.
Next, we give some useful functions which were constructed in [17]. Given positive constants α, c, M and consider p(t) and q(t) solutions of
p′(t) = c + M eαp(t), q′(t) = c− Meαq(t), t ≤ 0, (3.2.2)
p(0)≤ 0, q(0) < min{0, ln(c/M)/α}, e−αq(0) − e−αp(0)= 2M/c.
(3.2.3)
Indeed, p(t) and q(t) can be solved explicitly by
p(t) = p(0) + ct− ln[1 + Meαp(0)(1− ecαt)/c]/α, q(t) = q(0) + ct− ln[1 − Meαq(0)(1− ecαt)/c]/α.
Furthermore, there exists a positive constant κ such that
−κecαt/2≤ q(t) − ct − ω < 0 < p(t) − ct − ω ≤ κecαt/2, if t≤ 0,
Finally, we give the following definitions about a supersolution and a subsolution.
Definition 3.2.1 A function u(x, t) is called a supersolution (subsolution, resp.) of (3.1.1) on (x, t) ∈ R × (−∞, −T ] for some T ∈ R, if L[u](x, t) ≥ 0 (L[u](x, t) ≤ 0, resp.) for all
The following useful lemma can be found in [8, 14, 17].
Lemma 3.2.2 Suppose that u(x, t) and u(x, t) are a subsolution and a supersolution of (3.1.1) on (x, t) ∈ R × (−∞, −T ] for some T ∈ R, respectively and satisfy that u(x, t) ≤ u(x, t) on (x, t) ∈ R × (−∞, −T ]. Then there exists an entire solution u(x, t) of (3.1.1) such that
u(x, t)≤ u(x, t) ≤ u(x, t) for all (x, t) ∈ R × (−∞, −T ].
With this lemma, the construction of entire solutions is reduced to finding a suitable pair of super/sub solutions.