Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 22, pp. 1–24.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

PERIODIC TRAVELING WAVES AND ASYMPTOTIC

SPREADING OF A MONOSTABLE REACTION-DIFFUSION

EQUATIONS WITH NONLOCAL EFFECTS

BANG-SHENG HAN, DE-YU KONG, QIHONG SHI, FAN WANG

Abstract. This article concerns the dynamical behavior for a reaction-diffusion

equation with integral term. First, by using bifurcation analysis and center

manifold theorem, the existence of periodic steady-state solution are estab-
lished for a special kernel function and a general kernel function respectively.

Then, we prove the model admits periodic traveling wave solutions connect-

ing this periodic steady state to the uniform steady state u = 1 by applying
center manifold reduction and the analysis to phase diagram. By numerical

simulations, we also show the change of the wave profile as the coefficient of

aggregate term increases. Also, by introducing a truncation function, a shift
function and some auxiliary functions, the asymptotic behavior for the Cauchy

problem with initial function having compact support is investigated.

1. Introduction

In this article, we study the integro-differential equation

∂u

∂t
=
∂2u

∂x2
+ u{1 + αu− βu2 − (1 + α− β)(φ ∗ u)}, (t, x) ∈ (0,∞)× R, (1.1)

where α > 0, 0 < β < 1 + α and

(φ ∗ u)(t, x) :=

∫
R
φ(x− y)u(t, y)dy, t ∈ (0,∞), x ∈ R.

The kernel function φ(x) ∈ L1(R) satisfies the conditions:

(A1) φ(0) > 0, φ(x) = φ(−x), φ(x) ≥ 0,
∫
R φ(x)dx = 1 and

∫
R x

2φ(x)dx < +∞.

The equation with nonlocal term was first introduced by Britton [7, 8], when
considering model simpler than (1.1),

ut = uxx + u(1− φ ∗ u) in R× (0,∞). (1.2)

By using linear stability analysis and bifurcation analysis, he obtained that the
uniform steady state u ≡ 1 can be bifurcate to standing waves, periodic steady
states or periodic traveling waves. Later, Gourley [17] gave the existence of trav-
eling wave of equation (1.2) when the nonlocality is sufficiently weak. Recently,
there have been some great progress on traveling wave solutions of equation (1.2).
Particularly, Berestycki et al. [5] pointed out that, for all c ≥ c∗ = 2, equation

2010 Mathematics Subject Classification. 35C07, 35B10, 35B40, 35R09, 92D25.
Key words and phrases. Reaction-diffusion; nonlocal delay; periodic traveling wave;
asymptotic behavior; numerical simulation, critical exponent.
c©2021 Texas State University.

Submitted August 27, 2020. Published March 30, 2021.
1



2 B.-S. HAN, D.-Y. KONG, Q. SHI, F. WANG EJDE-2021/22

(1.2) admits traveling fronts connecting 0 to some unknown positive state, while
no such traveling wave solutions exists for c < c∗. Thereafter, through numerical
simulation, this unknown steady state is showed just the equilibrium u ≡ 1 for
some traveling front solutions in [31]. Alfaro and coville [2] gave a rigorously anal-
ysis proof (that is, they prove that equation (1.2) admits the rapid traveling front
connecting 0 to 1). More recently, Hamel and Ryzhik [20] proved that (1.2) exists
a periodic steady state due to the instability of the equilibrium. By using a center
manifold reduction, Faye and Holzer [13] proved that (1.2) admits modulated trav-
eling wave solutions. For more results about (1.2) (or similar nonlocal model), we
can refer to [1, 14, 15, 16, 23, 24, 29, 33, 34] and the references therein.

However, the advantage of individuals from local aggregation and competition
of individuals for space or resources still does not embody in model (1.2). Gour-
ley [19] considered model (1.1) which would be more realistic (in fact, the model
(1.1) was first mention in Britton [7, 8], however the research about (1.1) was first
introduced in [19]). In equation (1.1), the term αu denotes the advantage to indi-
viduals from local aggregation, the term −βu2 indicates the competition for space
and the nonlocal term −(1 + α − β)φ ∗ u represents the competition for food re-
sources. For more biological interpretation can be seen in [10, 11, 32, 30]. The
earlier researches about traveling wave solutions of equation (1.1) only consider
the kernel function with special form, for example the kernel with special form
φ(x) = λ

2 e
−λ|x|. Specially, Gourley et al. [19] studied the weakly nonlocal case

(i.e. λ is sufficiently large). By using an asymptotic explanation, they showed that
equation (1.1) exists traveling wave solutions connecting the two uniform steady
states and found this wave have a ‘hump’, which is differences with the wave of the
classical reaction-diffusion equation. Further, through the stability analysis and
numerical simulation, they verified that equation (1.1) indeed exists this kind of
wave. The case of the strongly nonlocal was researched by Billingham [6] (i.e. λ
is sufficiently small). By using numerical and asymptotic method, they showed
that in different, well-defined regions of parameter space, periodic traveling waves,
unsteady traveling waves and steady traveling waves develop from localized initial
conditions. And they also presented that equation (1.1) locally exists the traveling
wave solutions with speed c < 2 when λ is sufficiently large.

Recently, we considered traveling wave solutions of (1.1) whose kernel is without
limit (strong or weak), and the special kernel in [21]. We proved that (1.1) has
traveling wave solutions connecting 0 to an unknown positive steady state. We also
showed that this unknown steady state may be 1 or a periodic steady state. In
this article, we continue to study the properties of the solutions for (1.1) and try
to find some other types of traveling wave solutions. We will continue to study
the existence of periodic steady state for equation (1.1). Then, we will prove that
(1.1) has periodic traveling wave solution connecting this periodic steady state to
the uniform steady u = 1 (which is different from the one in [21]). We will also
present the change of the wave profile as α increased by numerical simulations.
Note that the initial condition here is different from the condition in [21]. In fact,
the solutions of (1.1) have a great relationship with the initial conditions. Also we
will study the asymptotic behavior of the Cauchy problem corresponding to the
(1.1), with a non-negative initial condition u0 ∈ L∞(R). It should be pointed out
that the periodic steady state we are considering here is quite different from that
in [21] (the causes are different and the amplitudes are different). In [21], it is



EJDE-2021/22 TRAVELING WAVES AND ASYMPTOTIC SPREADING 3

considered that the periodic steady state is caused by the natural growth rate µ.
And we mainly consider it caused by α and β. In addition, to clearly describe the
effect of the initial conditions on solution of (1.1), and the asymptotic spreading
speed, we present numerical simulations. Now we state our main results.

Theorem 1.1. (i) For a special kernel function φ(x) = 3a
2 e
−a|x| − e−|x|, if

α = αT , then the Turing bifurcation will occur in system (2.2) around the
unique positive equilibrium at the critical wave number σT , where σT and
αT are defined in (2.6).

(ii) For a general kernel function. Assume φ(x) satisfies (A1) and there exist
αc > 0, βc > 0 and σc > 0 such that αc, βc, σc satisfies
(a) f(0, σc, αc, βc) = 0.
(b) ∂σf(0, σc, αc, βc) = 0.
(c) ∂σσf(0, σc, αc, βc) < 0.
(d) −σ2

c + (αc − 2βc) < 0.
Then there exists ε0 > 0 such that for all ε ∈ (0, ε0] and

δ2 < − φ̂(σc)

2 + (1 + αc − βc)φ̂′′(σc)
ε2,

equation (1.1) has a periodic stationary solution of the form

uε,δ(x) = 1 +

√
φ̂(σc)

2 ε2 + (1 + 1+αc−βc
2 φ̂′′(σc))δ2

ς
cos((σc + δ)x) +O(|ε2 − δ2|),

where

f(λ, σ, α, β) = −σ2 − (1 + α− β)φ̂(σ) + (α− 2β)− λ, (1.3)

and λ, σ are given in (2.4), ς < 0 is defined in (2.10) below.

Obviously, there exist kernel functions, for example, φ(x) = 3
2ae
−a|x| − e−|x|,

where a ∈ (2/3,
√

2/3), and φ(x) = e−λ|x|/(2λ) with λ > 0.

Theorem 1.2. Assume the kernel φ(x) = 3ae−a|x|/2− e−|x| with a ∈ (2/3,
√

2/3)
and the conditions in Theorem 1.1(ii) are satisfied. Then for all s > s∗, there exists
a ε0 > 0 such that for all ε ∈ (0, ε0) and

δ2 < − φ̂(σc)

2 + (1 + αc − βc)φ̂′′(σc)
ε2,

equation (1.1) admits traveling wave solutions of the form

u(t, x) = U(x− εst, x) =
∑
n∈Z

Un(x− εst, x)e−in(σc+δ)x,

and satisfies the the boundary conditions

lim
ξ→−∞

U(ξ, x) = uε,δ(x), lim
ξ→+∞

U(ξ, x) = 1,

where

s∗ =

√
−2φ̂(σc)(1 +

1 + αc − βc
2

φ̂′′(σc))

and Un is defined by (3.3).



4 B.-S. HAN, D.-Y. KONG, Q. SHI, F. WANG EJDE-2021/22

Theorem 1.3. Let u be the solution of the Cauchy problem

ut = uxx + u{1 + αu− βu2 − (1 + α− β)(φ ∗ u)}, t > 0, x ∈ R,
u(0, x) = u0(x), on R,

(1.4)

where u0 ∈ L∞(R) and u0 6≡ 0. Then u has the following properties:

(i) lim inft→+∞

(
min|x|≤ct u(t, x)

)
> 0 for all 0 ≤ c ≤ 2.

(ii) If u0 is compactly supported, then

lim
t→+∞

(
max
|x|≥ct

u(t, x)

)
= 0 for all c > c∗,

where c∗ is the minimal speed of equation (4.12).

This article is organized as follows. In Section 2, we show the existence of
stationary periodic solution, and prove complete the proof of Theorem 1.1. In
Section 3, we prove the existence of periodic traveling wave solutions connecting
the stationary periodic state to the steady state u = 1 for (1.1); that is prove
Theorem 1.2. In Section 4 we study the asymptotic spreading speed of the Cauchy
problem (1.4); that is we prove Theorem 1.3.

2. Existence of a periodic steady state

In this section, we show the existence of stationary periodic solutions around
the steady state u = 1. Specially, in subsection 2.1, for a special kernel, we study
the bifurcation of the equation (1.1) could occur Turing bifurcation under some
conditions (that is to say equation (1.1) exist a periodic steady state, thus we
proved (i) of Theorem 1.1). In subsection 2.2, we prove that equation (1.1) has
stationary periodic solutions for a general kernel; thus we complete the proof of
Theorem 1.1 part (ii).

2.1. Special kernel. To discuss the bifurcation of the equation (1.1), we take the
kernel with a special form,

φ(x) =
3a

2
e−a|x| − e−|x|, (2.1)

where a ∈ ( 2
3 ,
√

2/3). We define

v(t, x) :=
(3a

2
e−a|x| ∗ u

)
(t, x) and w(t, x) :=

(
− e−|x| ∗ u

)
(t, x) .

Then equation (1.1) can be replaced by

ut = uxx + u(1 + αu− βu2 − (1 + α− β)(v + w)),

0 = vxx − a2v + 3a2u,

0 = wxx − w − 2u,

(2.2)

which has three equilibria (0, 0, 0), (−1/β,−3/β, 2/β) and (1, 3,−2). We are mainly
interested in the third equilibrium point from the biological point of view. Next,
we analyze system (2.2) to obtain the behavior of equation (1.1).



EJDE-2021/22 TRAVELING WAVES AND ASYMPTOTIC SPREADING 5

Now, linearizing system (2.2) near the point (1, 3,−2), then we obtain the linear
system

ũt = ũxx + (α− 2β)ũ− (1 + α− β)(ṽ + w̃),

0 = ṽxx − a2ṽ + 3a2ũ,

0 = w̃xx − w̃ − 2ũ.

(2.3)

Taking the test function of the formũṽ
w̃

 =

C1
σ

C2
σ

C3
σ

 eλt+iσx (2.4)

and substituting (2.4) into system (2.3), we obtain the characteristic equation for
λ, ∣∣∣∣∣∣

α− 2β − σ2 − λ −(1 + α− β) −(1 + α− β)
3a2 −a2 − σ2 0
−2 0 −1− σ2

∣∣∣∣∣∣ = 0,

which is equivalent to

λ =
(1 + α− β)

(
−a2 + (−3a2 + 2)σ2

)
(a2 + σ2)(1 + σ2)

− α+ 2β + σ2. (2.5)

Following [9, 19, 22, 25, 32, 35], we search for the Hopf bifurcation and the Turing
bifurcation of system (2.2). For the spatially homogeneous Hopf bifurcation, we
know that it occurs when Im(λ) 6= 0, Re(λ) = 0 at σ = 0 of equation (2.5), while
λ ∈ R for any α, β, σ, so the Hopf bifurcation cannot occur in system (2.2).

Next, we consider the spatially homogeneous Turing bifurcation, that is to prove
(i) of Theorem 1.1.

Proof Theorem 1.1(1). It is known that when Im(λ) = 0 and Re(λ) = 0 at σ = σT ,
system (2.3) will occur Turing bifurcation. For this purpose, let λ = 0, which
implies

α =
(2β + σ2)(a2 + σ2)(1 + σ2)− (1− β)(−a2 + (−3a2 + 2)σ2)

(a2 + σ2)(1 + σ2) + [−a2 + (−3a2 + 2)σ2]
.

Since

lim
σ→0+

α(σ) = lim
σ→+∞

α(σ) = +∞,

then there exists a σT ∈ (0,+∞) such hat

αT := α(σT ) = min
σ∈(0,+∞)

α(σ). (2.6)

Thus, when σ = σT and α = αT , the uniform steady state (1, 3,−2) lose stability
and has a new, non-uniformly steady state with the exp(iσcx)-like spatial structure,
i.e. system (2.2) has a Turing bifurcation at the critical wave number σT . The
bifurcation here is caused by α; of course we can also consider other factors (such
as β, σ or α and β working together, see Figure 1. This completes the proof. �



6 B.-S. HAN, D.-Y. KONG, Q. SHI, F. WANG EJDE-2021/22

α

λ

λ−α

0 1 2 3 4
−3

−2

−1

0

1

2

3

β

λ

λ−β

0 1 2 3 4
−2

0

2

4

6

8

σ

λ

λ−σ

0 1 2 3 4
−5

0

5

10

15

0

2

4

0

2

4

−5

0

5

10

α

λ−α−β

β

λ

Figure 1. Bifurcation diagram for (2.5). The top left panel shows
the relationship between λ and α for coefficients β = 0.5, a = 0.7,
and σ = 1. The top right panel shows the relationship between λ
and β for coefficients α = 0.2, a = 0.7, σ = 1. The bottom left
panel shows the relationship between λ and σ for coefficients α = 2,
β = 0.5, a = 0.7; The bottom right panel shows the relationship
between λ and α, β, for coefficients σ = 1, a = 0.7.

2.2. General kernel. In subsection 2.1, for a special kernel function, it is shown
that (1.1) can have Turing bifurcation under certain conditions. In this subsection,
we prove the existence of periodic stationary solutions around the steady state
u = 1 for the general kernel function. First set u(t, x) = 1 + v(t, σx), then (1.1) can
be written as

vt = σ2vxx+(α−2β)v−(1+α−β)(φσ ∗v)+(α−3β)v2−(1+α−β)v(φσ ∗v)−βv3,

where φσ(x) = 1
σφ( xσ ) and v is 2π-periodic in x. We define σ := σc+δ, α := αc+ε2,

β := βc + ε2

2 with 0 < ε� 1, 0 < δ � 1 and

B(σ, α, β)v := σ2vxx + (α− 2β)v − (1 + α− β)(φσ ∗ v),

Q(v, σ, α, β) := (α− 3β)v2 − (1 + α− β)v(φσ ∗ v)− βv3.

Then we obtain

vt = Bcv + C(ε, δ)v +Q
(
v, σc + δ, αc + ε2, βc +

ε2

2

)
, (2.7)

where

Bc = B(σc, αc, βc) and C(ε, δ) = B(σ, α, β)− Bc.
We define

Y := L2
per[0, 2π] = {u ∈ L2

loc(R)|u(x+ 2π) = u(x), x ∈ R},



EJDE-2021/22 TRAVELING WAVES AND ASYMPTOTIC SPREADING 7

W := D(B) = H2
per[0, 2π] = {u ∈ H2

loc(R)|u(x+ 2π) = u(x), x ∈ R}.

Since the linear operator Bc : W → Y is continuous and W is dense in Y and
compactly embedded into Y, the resolvent of Bc is compact, which implies the
spectrum σ(Bc) only includes eigenvalues, λ. It follows from (1.3) that

σ(Bc) =
{
λl ∈ C|λl = −σ2

c l
2 − (1 + αc − βc)φ̂(lσc) + (αc − 2βc), l ∈ Z

}
, (2.8)

so σ(Bc) ∩ iR = {0}, and the geometric multiplicity of λ = 0 is two, whose cor-
responding eigenvectors are e(x) = eix and ē(x) = e−ix. In addition, through
calculation, we can find that the algebraic multiplicity of λ = 0 is also two. We
define Yc as the subspace expanded by e and ē (i.e. Yc = {e, ē}) and the spectral
projection Gc : Yc → Yc as

Gc = 〈u, e〉e + 〈u, ē〉ē,

where 〈u, v〉 = 1
2π

∫ 2π

0
u(x)v(x)dx. From (2.8), we deduce that

‖(iν − Bc)−1‖(id−Gc)Y ≤
E

1 + |ν|
, ν ∈ R,

with some positive constant E > 0. So, by using the center manifold theorem (see
[26, 13]), we have the following conclusion.

Proposition 2.1. There exists U ⊂ Yc, V ⊂ (id − Gc)W, S ∈ R2, such that for
each n <∞, the Cn-map Φ : U × S → V satisfy the following properties:

(i) For each (ε, δ) ∈ S, each bounded solution v of (2.7) satisfies

v(t) = B(t)e + B(t)ē + Φ(B(t),B(t), ε, δ), ∀t ∈ R,

where B(t) and B(t) is a vector and only depends on t.
(ii) ‖Φ(B(t),B(t), ε, δ)‖W = O(|ε|2|B|+ |δ||B|+ |B|2).

(iii) dB
dt = g(B(t), B(t), ε, δ) = Bh(|B|2, ε, δ), where h is real-valued and a

Cn−1 −map in (B(t),B(t), ε, δ).

Lemma 2.2. The map h has the form

h(|B|2, ε, δ) =− φ̂(σc)

2
ε2 −

(
1 +

1 + αc − βc
2

φ̂′′(σc)
)
δ2

+ ς|B|2 +O(|δ|3 + |ε|2|δ|+ |B|4),

(2.9)

where

ς :=− 2(1 + αc − βc)φ̂(σc)− 2(αc − 3βc)

1 + βc

(
αc − 1− 5βc + φ̂(σc)

)
+

αc − 3βc − (1 + αc − βc)φ̂(σc)

4σ2
c − (αc − 2βc) + (1 + αc − βc)φ̂(2σc)

×
(

2(αc − 3βc)− (1 + αc − βc)φ̂(2σc) + φ̂(σc)
)
− 3βc < 0,

(2.10)

and φ̂ is the Fourier transform of φ.

The proof of the above lemma is similar to that of [13, Lemma 2.1], see Appendix
A for details.



8 B.-S. HAN, D.-Y. KONG, Q. SHI, F. WANG EJDE-2021/22

Proof of Theorem 1.1(ii). We define

Λ :=
φ̂(σc)

2 ε2 +
(
1 + 1+αc−βc

2 φ̂′′(σc)
)
δ2

ς
> 0.

Our goals is to find the nontrivial stationary solution B0 ∈ C that satisfies

h(|B|2, ε, δ) = 0. (2.11)

Up to a rescaling B0 =
√

ΛB̃0, (2.11) can be written as

Λ ·
(
−ς + ς|B̃0|2 +O(

√
Λ)
)
, as Λ→ 0.

It follows from the implicit function theorem that the solutions have the form

|B̃0| = 1 +O(
√

Λ), as Λ→ 0.

Thus

vε,δ(x) =
√

Λ cos((σc + δ)x) +O
(∣∣∣ φ̂(σc)

2
ε2 +

(
1 +

1 + αc − βc
2

φ̂′′(σc)
)
δ2
∣∣∣),

for some ε ∈ (0, ε0] and δ satisfies(
1 +

1 + αc − βc
2

φ̂′′(σc)
)
δ2 < − φ̂(σc)

2
ε2.

Further, we know that equation (1.1) has the periodic solution of the form

uε,δ(x) = 1 +
√

Λ cos((σc + δ)x) +O
(∣∣∣ φ̂(σc)

2
ε2 +

(
1 +

1 + αc − βc
2

φ̂′′(σc)
)
δ2
∣∣∣)

for some ε ∈ (0, ε0] and
(
1 + 1+αc−βc

2 φ̂′′(σc)
)
δ2 < − φ̂(σc)2 ε2. This completes the

proof. �

3. Traveling wave solutions connecting 1 to a periodic steady state

In the previous section, we show that equation (1.1) has periodic steady around
u = 1, in this section, we prove that (1.1) has traveling wave solutions connecting
this periodic steady to the uniform steady u = 1. Specifically, in subsection 3.1,
by using center manifold reduction (similar to [13]), we prove the existence of this
traveling wave solution. Then, we study the dynamic behavior of the solution for
equation (1.1) through numerical Simulation in subsection 3.2.

3.1. Existence of traveling wave solutions. In this subsection, we consider the
kernel function with the special form (2.1) and study the corresponding system
(2.2). Substituting u = 1 + ũ, v = 3 + ṽ, w = −2 + w̃ into (2.2) (in the case of no
confusion, we still write as u, v, w), we obtain

ut = uxx + (α− 2β)u− (1 + α− β)(v + w) + (α− 3β)u2

− (1 + α− β)u(v + w)− βu3,
0 = vxx − a2v + 3a2u,

0 = wxx − w − 2u.

(3.1)

Let ε2 = α− αc = 2(β − βc) and U = (u, v, w)T . Then (2.7) can be written as

DUt = Uxx +KcU + ε2KrU −
(

1 + αc − βc +
ε2

2

)
L(U)−

(
βc +

ε2

2

)
J(U), (3.2)



EJDE-2021/22 TRAVELING WAVES AND ASYMPTOTIC SPREADING 9

where D(1, 1) = 1 and the other items are zero, and

Kc =

αc − 2βc −(1 + αc − βc) −(1 + αc − βc)
3a2 −a2 0
−2 0 −1

 ,

Kr =

0 − 1
2 − 1

2
0 0 0
0 0 0

 , L(U) =

u(v + w)
0
0

 , J(U) =

u30
0

 .

Next, we find a solution of (3.2) of the form

U(t, x) = V (x− εst, x) = Σn∈ZVn(x− εst)e−inσcx. (3.3)

Let ζ = x− εst, plugging this into (3.2), we obtain1 0 0
0 0 0
0 0 0

−εs∂ζV un−εs∂ζV vn
−εs∂ζV wn

 =

 ∂ζζV
u
n − 2inσc∂ζV

u
n − n2σ2

cV
u
n

∂ζζV
v
n − 2inσc∂ζV

v
n − n2σ2

cV
v
n

∂ζζV
w
n − 2inσc∂ζV

w
n − n2σ2

cV
w
n


+

αc − 2βc −ι −ι
3a2 −a2 0
−2 0 −1

V unV vn
V wn

+ ε2

0 − 1
2 − 1

2
0 0 0
0 0 0

V unV vn
V wn


−
(
ι+

ε2

2

)Σp+q=nV
u
p V

v
q + V up V

w
q

0
0

− (βc +
ε2

2

)Σp+q+r=nV
u
p V

u
q V

u
r

0
0

 ,

(3.4)
where ι = 1 + αc − βc. Further, let X := (Xn)n∈Z and

Xn := (V un , ∂ζV
u
n , V

v
n , ∂ζV

v
n , V

w
n , ∂ζV

w
n )T ,

then (2.9) can be simplified to

∂ζXn = Nε
nXn −

(
1 + αc − βc +

ε2

2

)
Ln(X)−

(
βc +

ε2

2

)
Jn(X), n ∈ Z, (3.5)

where

Nε
n =



0 1 0 0 0 0

n2σ2
c − (αc − 2βc) 2inσc − εs ι+ ε2

2 0 ι+ ε2

2 0
0 0 0 1 0 0
−3a2 0 a2 + n2σ2

c 2inσc 0 0
0 0 0 0 0 1
2 0 0 0 1 + n2σ2

c 2inσc


and

Ln(X) =


0

Σp+q=nV
u
p V

v
q + V up V

w
q

0
0
0
0

 , Jn(X) =


0

Σp+q+r=nV
u
p V

v
q V

w
r

0
0
0
0

 .

By using a center manifold reduction (see Appendix C), we obtain

∂ζX = N εX −
(

1 + αc − βc +
ε2

2

)
L(X)−

(
βc +

ε2

2

)
J (X), X ∈ E , (3.6)



10 B.-S. HAN, D.-Y. KONG, Q. SHI, F. WANG EJDE-2021/22

where (N εX)n = (Nε
n)n, (L(X))n = Ln(X) and (J (X))n = Jn(X) for n ∈ N. It

is easy to see that there are only two center directions corresponding to the eigen-
values λε± (λε± := λε,±−,1) of Nε

1 (see Appendix B). Define ϕε± as the corresponding
eigenvectors of λε±, i.e.

Nε
1ϕ

ε
± = λε±ϕ

ε
±.

Similarly, ψε± ∈ C6 are denoted as the corresponding eigenvectors of the eigenvalue

λ
ε

± (which are the eigenvectors of the adjoint matrix (Nε
1 )∗, i.e.

(Nε
1 )∗ψε± = λ

ε

±ψ
ε
±.

By calculations, it is easy to check that

〈ψε±, ϕε±〉 = ±
√
ϑ21 − 4ϑ0ϑ2

ϑ2
ε and 〈ϕε±, ψε±〉 = 0.

We define Φε±,Ψ
ε
± ∈ E as

(Φε±)1 = φε±, (Φε±)n = 0C6 , n 6= 1;

(Ψε
±)1 = ψε±, (Ψε

±)n = 0C6 , n 6= 1.

Now, we define the spectral projection

Gεc = cε+〈Ψε
+, X〉lΦε+ + cε−〈Ψε

−, X〉lΦε−,

where

cε± =
±ϑ2

2lε
√
ϑ21 − 4ϑ0ϑ2

−O(1), as ε→ 0.

Obviously, there exists a constant d1 > 0, such that

σ(Nε|(id−Gεc )E) ⊂
{
λ ∈ C||R(λ)| ≥ d1

√
ε
}
.

Then the center manifold theorem [27, 12] can be used to obtain the following
result.

Proposition 3.1. For each 0 < r < 1/3 and sufficiently small ε > 0, there exist
Uε ⊂ Ec, Vε ⊂ (id − Gεc )E and a Cm-map Υε : Uε → Vε (for any m < ∞), such
that the following properties hold:

(i) All bounded solutions of (3.6) satisfy X = Xc + Υε(Xc);
(ii) ‖Υε(Xc)‖l = Oε(‖Xc‖2l );

(iii) The neighborhood Uε is of size O
(
ε

2
3+r
)
.

Next, by projecting equation (3.6) with Gεc , we obtain

dXc

dζ
=N εXc −

(
1 + αc − βc +

ε2

2

)
Gεc (L(Xc + Υε(Xc)))

−
(
βc +

ε2

2

)
Gεc (J (Xc + Υε(Xc))).

Let Xc = x+Φε+ + x−Φε− on the center manifold, then

X = x+Φε+ + x−Φε− + Υε(x+, x−),

where

Υε(x+, x−) = Σ|n|=2x
n1
+ xn2
− x

n3
+ xn4
− Φεn +Oε(|x+ + x−|3).



EJDE-2021/22 TRAVELING WAVES AND ASYMPTOTIC SPREADING 11

Furthermore, we obtain

dx+
dζ

= λε+x+ − (1 + αc − βc +
ε2

2
)cε+〈ψε+,L(x+Φε+ + x−Φε− + Υε(x+, x−))〉l

− (βc +
ε2

2
)cε+〈ψε+,J (x+Φε+ + x−Φε− + Υε(x+, x−))〉l,

dx−
dζ

= λε−x− − (1 + αc − βc +
ε2

2
)cε−〈ψε−,L(x+Φε+ + x−Φε− + Υε(x+, x−))〉l

− (βc +
ε2

2
)cε−〈ψε−,J (x+Φε+ + x−Φε− + Υε(x+, x−))〉l,

(3.7)
In addition, we have

DXΥε(Xc)
dXc

dζ
=N εΥε(Xc)−

(
1 + αc − βc +

ε2

2

)
(Gεc )⊥(L(Xc + Υε(Xc)))

−
(
βc +

ε2

2

)
(Gεc )⊥(J (Xc + Υε(Xc))),

where (Gεc )⊥ := id− Gεc . It follows from the definition of Ψε
±, N and J , that

〈Ψε
±,L(x+Φε+ + x−Φε− + Υε(x+, x−))〉l = 2l〈ψε±,L1(x+Φε+ + x−Φε− + Υε(x+, x−))〉

and

〈Ψε
±,J (x+Φε+ +x−Φε−+Υε(x+, x−))〉l = 2l〈ψε±,J1(x+Φε+ +x−Φε−+Υε(x+, x−))〉.

From the form of L1(X), J1(X) and Υε, we obtain

L1(x+Φε+ + x−Φε− + Υε(x+, x−)) = Oε(|x+ + x−|3),

J1(x+Φε+ + x−Φε− + Υε(x+, x−)) = Oε(|x+ + x−|3).

Because for each sub-system, namely the ones left after linearization is invariant, we
can work on each mode. Let Υε

n be the n-th mode of Υε and xc := x+φ
ε
+ + x−φ

ε
−,

then (3.7) can be written as

DxΥε
n(Xc)

dxc
dζ

=N εΥε(xc)−
(

1 + αc − βc +
ε2

2

)
(Gεc )⊥(L(Xc + Υε(Xc)))

−
(
βc +

ε2

2

)
(Gεc )⊥(J (Xc + Υε(Xc))).

(3.8)

Next we compute the value of Υε
n(Xc)(n = 0, 1, 2). When n = 0, we assume that

Υε
0(x+, x−) = Σ|n|=2x

n1
+ xn2
− x

n3
+ xn4
− γ

0
n(ε) +Oε(|x+ + x−|3), γ0n(ε) ∈ C6. (3.9)

Substituting (3.9) into (3.8), we obtain

Ξnγ
0
n(ε) = N ε

0 γ
0
n(ε)− (1 + αc − βc +

ε2

2
)Lε0,n − (βc +

ε2

2
)J ε0,n,

where

Ξn = Σ4
j=1λjnj , λ1,2 = λε±, λ3,4 = λ

ε

±,

Lε0,n = (0,L0,n, 0, 0, 0, 0)T , Jε0,n = (0,J0,n, 0, 0, 0, 0)T .

Therefore,

γ0n(ε) =
(

1 + αc − βc +
ε2

2

)
(N ε

0 − Ξnid)
−1 Lε0,n +

(
βc +

ε2

2

)
J ε0,n



12 B.-S. HAN, D.-Y. KONG, Q. SHI, F. WANG EJDE-2021/22

= (1 + αc − βc) (N 0
0 )−1L0

0,n + βc(N
0
0 )−1J 0

0,n +O(ε).

When n = 1, by using the same methods, we obtain

γ1n(ε) =
(

1 + αc − βc +
ε2

2

)
(N ε

1 − Ξnid)−1Lε1,n +
(
βc +

ε2

2

)
J ε1,n

=(1 + αc − βc)(N 0
1 )−1L0

1,n + βc(N 0
1 )−1J 0

1,n +O(ε).

Similarly, when n = 2, we have

γ2n(ε) =
(

1 + αc − βc +
ε2

2

)
(N ε

2 − Ξnid)−1Lε2,n +

(
βc +

ε2

2

)
J ε2,n

=(1 + αc − βc)(N 0
2 )−1L0

2,n + βc(N 0
1 )−1J 0

2,n +O(ε).

It is easy to see that, in the inner product

〈ψε±,L1(x+Φε+ + x−Φε− + Υε(x+, x−))〉, as ε→ 0.

About the cubic order, we see that(
1 + αc − βc +

ε2

2

)
cε+〈ψε+,L(x+Φε+ + x−Φε− + Υε(x+, x−))〉l

+
(
βc +

ε2

2

)
cε+〈ψε+,J (x+Φε+ + x−Φε− + Υε(x+, x−))〉l

=−
ςcε+2l

(1 + αc − βc)%0
(x+ + x−)|x+ + x−|2 +Oε(|x+ + x−|4),

(3.10)

and (
1 + αc − βc +

ε2

2

)
cε−〈ψε−,L(x+Φε+ + x−Φε− + Υε(x+, x−))〉l

+
(
βc +

ε2

2

)
〈ψε−,J (x+Φε+ + x−Φε− + Υε(x+, x−))〉l

=−
ςcε−2l

(1 + αc − βc)%0
(x+ + x−)|x+ + x−|2 +Oε(|x+ + x−|4),

(3.11)

where

%0 = −4σ2
c (1 + σ2

c )
( 3a2

(a2 + σ2
c )3
− 2

(1 + σ2
c )3

)
and ς is defined in (2.10). From (3.7), (3.10) and (3.11), we have

dx+
dζ

= λε+x+ +
ςcε+2l(1 + σ2

c )

(1 + αc − βc)%0
(x+ + x−)|x+ + x−|2 +Oε(|x+ + x−|4),

dx−
dζ

= λε−x+ +
ςcε−2l(1 + σ2

c )

(1 + αc − βc)%0
(x+ + x−)|x+ + x−|2 +Oε(|x+ + x−|4).

Through the three variable substitutions (similar to [13]),

Y = x+ + x−, Z = x+ − x−,
Y (t) = εu(εt), Z(t) = εv(εt),

u = q, v =
ϑ1√

ϑ21 − 4ϑ0ϑ2
q +

2ϑ2√
ϑ21 − 4ϑ0ϑ2

p,



EJDE-2021/22 TRAVELING WAVES AND ASYMPTOTIC SPREADING 13

we obtain the system

q′ = p+O(ε),

p′ =
1

ϑ2

(
− ϑ0q − ϑ1p+

ς(1 + σ2
c )ϑ2

(1 + αc − βc)σc
q|q|2

)
+O(ε),

(3.12)

which is equivalent to

q′′ +
ϑ1
ϑ2
q′ +

ϑ0
ϑ2
q − ς(1 + σ2

c )

(1 + αc − βc)ϑ0
q|q|2 = 0. (3.13)

To compare (3.13) and (3.2), let

− φ̂(σc)

2γ
=
ϑ0
ϑ2
,

s

γ
=
ϑ1
ϑ2
,

ς

γ
= − ς(1 + σ2

c )

(1 + αc − βc)%0
,

then system (3.12) is equivalent to (3.2), namely

q′ = p+O(ε),

p′ =
1

γ

(1

2
φ̂(σc)q − sp− ςq|q|2

)
+O(ε).

(3.14)

Next, analyzing the properties of (3.14), we prove Theorem 1.2, i.e. prove that
system (3.14) has heteroclinic orbits corresponding to the modulated traveling wave
solution.

Proof of Theorem 1.2. When ε = 0, system (3.14) can be written as

q′ = p,

p′ =
1

γ

(1

2
φ̂(σc)q − sp− ςq|q|2

)
.

(3.15)

Obviously, for every q on the circle |q| = φ̂(σc)
2ς , system (3.15) has a saddle connection

C0, which is tangent to the unstable direction at that point and connects the point
to the origin (q, p) = (0, 0), this property are similar to [4] and the proof is also
similar, we will not repeat it.

When ε > 0, (p, q) = (0, 0) has two eigenvalues ρ1,2 =
−s±
√
s2+2φ̂(σc)γ
2γ , in

addition ρ1,2 < 0 if s >
√
−2φ̂(σc)(1 + 1+αc−βc

2 φ̂′′(σc)). Thus, (0, 0) is stable node

for sufficiently small ε. To complete the proof of Theorem 1.2, we need another two
facts. One is that (3.14) has a circle of normally hyperbolic fixed points approaching

|q| =

√
φ̂(σc)
2ς , p = 0 as ε → 0, which is similar to [13, Lemma 4.3]. The other is

that (3.14) has a family of heteroclinic connections C0 (related to one another via
q → eiθq and p → eiθp) between the circle of fixed points and the origin, which is

given in [12, Lemma 4.2]. Thus, when s >
√
−2φ̂(σc)(1 + 1+αc−βc

2 φ̂′′(σc)), there

exists a ε0 > 0 such that for any ε ∈ (0, ε0) equation (1.1) has the modulated
traveling wave solutions of frequency σc, of the form

u(t, x) = u(x− εst, x) = Σn∈ZV
u
n (x− εst)e−inσcx

and having the boundary conditions at infinity

lim
ζ→−∞

u(ζ, x) = uε(x) ≈ 1 + ε

√
φ̂(σc)

2ς
cos(σcx), lim

ζ→+∞
u(ζ, x) = 1.



14 B.-S. HAN, D.-Y. KONG, Q. SHI, F. WANG EJDE-2021/22

All our results are still valid if σc is replaced by any σ, which satisfy σ = σc+ δ and(
1 + 1+αc−βc

2 φ̂′′(σc)
)
δ2 < − φ̂(σc)2 ε2. This completes the proof of Theorem 1.2. �

3.2. Numerical simulations. In section 2, we showed that equation (1.1) has a
periodic steady state under some conditions. Further, the existence of traveling
wave solutions connecting this periodic steady to the uniform steady u = 1 was
proved in subsection 3.1. In this subsection, we show a process of forming a steady
state around the uniform steady u = 1, by numerical simulation. We show that
equation (1.1) has traveling wave solutions connecting this periodic steady state to
the uniform steady u = 1. Here we only consider the special kernel function φ(x)
which defined by (2.1).

Similar to subsection 2.1, equation (1.1) can be replaced by (2.2), that is

ut = uxx + u(1 + αu− βu2 − (1 + α− β)(v + w)),

0 = vxx − a2v + 3a2u,

0 = wxx − w − 2u,

Before our numerical simulation, the initial value problem needs to be stated. We
define the initial value of u(t, x) as

u(0, x) =

{
1− τ sin bx, x < L0,

1, x > L0,
(3.16)

where τ, b, L0 are some constants. Since v(0, x) =
∫
R

3a
2 e
−a|x−y|u(y)dy, it follows

that

v(0, x) =

{
3− 3a

2(a2+b2)τ
[
2a sin bx− (a sin bL0 + b cos bL0)eax−aL0

]
, x < L0,

3− 3a
2(a2+b2)τe

(L0−x)a(a sin bL0 − b cos bL0), x > L0.

(3.17)
Since w(0, x) =

∫
R−e

−|x−y|u(y)dy, we have

w(0, x) =

{
−2 + 1

1+b2 τ
[
2 sin bx− (sin bL0 + b cos bL0)ex−L0

]
, x < L0,

−2 + 1
1+b2 τe

L0−x(sin bL0 − b cos bL0), x > L0.
(3.18)

In addition, the zero-flux boundary condition is considered here. Along with (3.16),
(3.17) and (3.18), the system (2.2) can be simulated through the pdepe package in
Matlab (see Figure 1).

Now, we explain our numerical results. Firstly, we see that the uniform steady
u = 1 will lose its stability as the value of α increasing. And then a periodic steady
state will occur (the theoretical analysis of this part is given in section 2.1, we do
not repeat the narrative). Secondly, Figure 2 shows a specific traveling wave which
connecting the uniform steady state u = 1 to a periodic steady state (it made a
perfect complement to the previous section). Lastly, we know that equation (1.1)
exists traveling wave solutions connecting 0 to 1 (or a periodic steady state), see

[21, Figure 3] when the kernel φ(x) = 1
2σ e
− |x|σ , σ > 0 and the initial condition is

u(x, 0) =

{
1, for x < L0,

0, for x ≥ L0.

However, here we know that equation (1.1) has traveling wave solutions connecting
1 to a periodic steady state when φ(x) = 3a

2 e
−a|x| − e−|x| and u(x, 0) id defined in



EJDE-2021/22 TRAVELING WAVES AND ASYMPTOTIC SPREADING 15

0

50

100

150

0

5

10

15

0

0.5

1

1.5

2

Distance x

α=1

Time t

S
p

e
c
ie

s
 u

0

50

100

150

0

5

10

15
0

0.5

1

1.5

2

Distance x

α=1.5

Time t

S
p

e
c
ie

s
 u

0

50

100

150

0

5

10

15

0

0.5

1

1.5

2

Distance x

α=1.7

Time t

S
p

e
c
ie

s
 u

0

50

100

150

0

5

10

15

0

0.5

1

1.5

2

Distance x

α=1.8

Time t

S
p

e
c
ie

s
 u

0
50

100
150

0

5

10

15

0

0.5

1

1.5

2

α=2

Distance x

Time t

S
p

e
c
ie

s
 u

0

50

100

150

0

5

10

15

0

1

2

3

4

α=2.5

Distance x

Time t

S
p

e
c
ie

s
 u

Figure 2. Time and space evolution for nonlocal equation (1.1)
with kernel φ(x) = 3a

2 e
−a|x| − e−|x|. The computational domain

is x ∈ [0, 150] and t ∈ [0, 15]. The corresponding parameters are
L0 = 40, β = 0.5, τ = 0.1, a = 0.7, b = 5, and α takes the values
of 1, 1.5, 1.7, 1.8, 2, 2.5.

(3.16). That is to say, the solution of equation (1.1) has a great relationship with
the form of the kernel function and the initial condition. Next, we consider the
influence of the initial conditions on the solution of equation (1.1).

4. Asymptotic rate of the Cauchy problem (1.4)

In this section, we study the asymptotic spreading speed for the solutions of the
Cauchy problem (1.4). Also we complete the proof of the Theorem 1.3. First, we
give a uniformly bounded of the solution u.

Lemma 4.1 ([10, Theorem 4.1]). There exists a positive constant C such that the
solution u(x, t) of the Cauchy problem (1.4) satisfies

u(x, t) ≤ C for (x, t) ∈ R× (0,∞).



16 B.-S. HAN, D.-Y. KONG, Q. SHI, F. WANG EJDE-2021/22

0 20 40 60 80

0

5

10

15

0

0.5

1

1.5

2

σ=10

Distance x

Time t

S
p

e
c
ie

s
 u

0 20 40 60 80
0

5

10

15

0

1

2

3

4

5 σ=10/3

Distance x

S
p

e
c
ie

s
 u

Time t

Figure 3. Time and space evolution for nonlocal equation (1.1)

with the kernel φ(x) = 1
2σ e
− |x|σ . The computational domain is

x ∈ [0, 80] and t ∈ [0, 15]. The corresponding parameters are
α = 0.9, β = 0.5 in the left figure, and α = 2, β = 0.4 in the right
figure.

Proof of Theorem 1.3(i). By contradictions, we assume the result is not true. Then,
for 0 ≤ c < 2, there exist two sequences (xn)n∈N in R and (tn)n∈N in (0,+∞) such
that

|xn| ≤ ctn, for all n ∈ N,
and

tn → +∞, u(tn, xn)→ 0 as n→ +∞. (4.1)

We define the shifting functions

un(t, x) = u(t+ tn, x+ xn), for all (t, x) ∈ (−tn,+∞)× R, n ∈ N.

It follows from Lemma 4.1 that (‖un‖L∞(−tn,+∞)×R)n∈N is bounded. Further, by

the standard parabolic estimating, we know that un converges in C1,2
loc (R × R),

extracting a subsequence and letting n→ +∞, we obtain u∞ satisfying

(u∞)t = (u∞)xx + u∞{1 + αu∞ − β(u∞)2 − (1 + α− β)(φ ∗ u∞)} in R× R,

and u∞ ≥ 0 in R × R, u∞(0, 0) = 0. By regarding 1 + αu∞ − β(u∞)2 − (1 + α −
β)(φ ∗ u∞) as a coefficient in L∞(R×R) and using the strong maximum principle
and the uniqueness of the solutions of the Cauchy problem (1.4), we know that
u∞(t, x) = 0 for all (x, t) ∈ R× R.

Further, we define

cn =
xn
tn
∈ [−c, c], (4.2)

and

vn(t, x) = un(t, x+ cnt) = u(t+ tn, x+ cn(t+ tn)) in (−tn,+∞)× R,

then vn(t, x) locally uniformly converge to 0 in R×R. Thus, combining the bound-
edness of (‖vn‖L∞((−tn,+∞)×R))n∈N, we know that φ∗vn also converges to 0 locally
uniformly in R× R.

Next we fix some parameters. Let δ > 0 satisfy

1− (1 + α− β)δ ≥ c2

4
+ δ, (4.3)



EJDE-2021/22 TRAVELING WAVES AND ASYMPTOTIC SPREADING 17

and R > 0 satisfy
π2

4R2
≤ δ. (4.4)

Without loss of generality, one can assume that tn > 1 for every n ∈ N. Because
φ ∗ vn → 0 uniformly locally in R×R as n→ +∞, for sufficiently large n(n ≥ N),
we can define

t∗n = inf {t ∈ [−tn + 1, 0]|φ ∗ vn ≤ δ for all (x, t) ∈ [−R,R]× [t, 0]} , n ≥ N,
where δ and R are respectively as in (4.3) and (4.4). From the definition of t∗n, we
can assume that t∗n < 0. By the continuity of φ ∗ vn in (−tn,+∞) × R and the
definition of t∗n, we know that for every n ≥ N ,

0 ≤ φ ∗ vn ≤ δ for all (x, t) ∈ [−R,R]× [t, 0]. (4.5)

On the other hand, u(1, ·) is continuous and combining with the strong maximum
principle, we know that u(1, ·) is positive. Then, there exist a ζ > 0 such that

u(1, x) ≥ ζ > 0 for all |x| ≤ R+ c. (4.6)

From (4.2) and (4.6), we know that, for n ≥ N ,

vn(−tn + 1, x) = u(1, x+ cn) ≥ ζ for all |x| ≤ R.
Therefore, we have the following dichotomy: either

t∗n > −tn + 1 and max
[−R,R]

(φ ∗ vn)(t∗n, .) = δ, (4.7)

or

t∗n = −tn + 1 and min
[−R,R]

vn(t∗n, .) > ζ. (4.8)

Next, we claim that there exists γ > 0 such that

min
[−R,R]

vn(t∗n, .) > γ > 0 for all n ≥ N, (4.9)

which is true when t∗n = −tn + 1 from (4.8). Assume (4.9) does not hold when
t∗n > −tn + 1, then, up to extraction a subsequence, there exist a sequence (zn)n≥N
in [−R,R] such that

vn(t∗n, zn)→ 0 and zn → z∞ ∈ [−R,R] as n→ +∞.
We define

wn(t, x) = vn(t+ t∗n, x) for all (t, x) ∈ (−tn − t∗n,+∞)× R,

for any n ≥ N . It is easy to see that vn satisfies

(vn)t = (vn)xx+cn(vn)x+vn{1+αvn−β(vn)2−(1+α−β)(φ∗vn)} in (−tn,+∞)×R.
(4.10)

Similarly, wn satisfies the same equation for (x, t) ∈ R× (−tn,+∞). Since

−tn − t∗n ≤ 1 for all n ≥ N, cn → c∞ as n→ +∞, wn ≥ 0

and (‖wn‖L∞(−tn−t∗n,+∞)×R)n≥N is bounded, combining the standard parabolic es-

timates, we know that wn converges in C1,2
loc ((−1,+∞)×R). Then, up to extraction

of a subsequence and let n→ +∞, we have w∞ satisfying

(w∞)t =(w∞)xx + c∞(w∞)x + w∞{1 + αw∞

− β(w∞)2 − (1 + α− β)(φ ∗ w∞)} in (−1,+∞)× R,



18 B.-S. HAN, D.-Y. KONG, Q. SHI, F. WANG EJDE-2021/22

and

w∞(t, x) ≥ 0 for all (t, x) ∈ (−1,+∞)× R and w∞(0, z∞) = 0.

The uniqueness of the solutions for the Cauchy problem and the maximum principle
imply that

w∞(t, x) = 0 for all (t, x) ∈ (−1,+∞)× R.
On the other hand, from wn → 0 as n→ +∞ locally uniformly in (−1,∞)×R and
the boundedness of (‖wn‖L∞((−1,∞)×R))n≥N it follows that φ∗wn → 0 as n→ +∞
locally uniformly in (−1,∞)× R. This implies

vn(t∗n, ·)→ 0 and (φ ∗ vn)(t∗n, ·)→ 0 locally uniformly in R as n→ +∞.

This is a contradiction to (4.7), so (4.9) is proved.
Now, from (4.5), (4.9) and (4.10), it is known that for every n ≥ N , −tn + 1 ≤

t∗n < 0 and vn ≥ 0, vn satisfies

(vn)t = (vn)xx + cn(vn)x + vn{1 + αvn − β(vn)2 − (1 + α− β)(φ ∗ vn)}
≥ (vn)xx + cn(vn)x + (1− (1 + α− β)δvn)

+ (vn)2(α− βvn) in [t∗n, 0]× [−R,R],

vn(t,±R) ≥ 0 for all t ∈ [t∗n, 0],

vn(t∗n, x) ≥ γ for all x ∈ [−R,R].

(4.11)

For n ≥ N , we define ϕn in [−R,R] as

ϕn(x) = νγe−
cnx
2 −

cR
2 cos

(πx
2R

)
,

where ν is sufficiently small and satisfying νγ < α
β and ν < 1. From (4.2)-(4.4), it

follows that ϕn(x) satisfies 0 ≤ ϕn(x) ≤ γ in [−R,R], ϕn(±R) = 0 and

ϕ′′n + cnϕ
′
n + (1− (1 + α− β)δ)ϕn + (ϕn)2(α− βϕn)

≥
(

1− (1 + α− β)δ − c2n
4
− π2

4R2

)
ϕn ≥ 0 in [−R,R].

That is to say, ϕn is a subsolution of (4.11). According to the maximum principle,
for all n ≥ N , we have

vn(x, t) ≥ ϕn(x) for all (t, x) ∈ [t∗n, 0]× [−R,R].

In particular,

u(tn, xn) = vn(0, 0) ≥ ϕn(0) = νγe−cR/2 for all n ≥ N.

However, (4.1) implies that u(tn, xn) → 0 as n → +∞ and νγe−cR/2 is a fixed
constant, which is a contradiction. Therefore, (i) holds. �

To complete the proof of Theorem 1.3, we need another Lemma.

Lemma 4.2. Assume c∗ is the minimal speed of

ut = uxx + u(1 + αu− βu2), (4.12)

then

lim
t→+∞

(
max
|x|≥c∗t

u(t, x) = 0
)
.



EJDE-2021/22 TRAVELING WAVES AND ASYMPTOTIC SPREADING 19

Since f(u) = u(1 + αu − βu2) satisfies the conditions [28, A1–A4]. By [28,
Theorems 2.17 and 4.3], it is easy to see that the Lemma 4.2 is true; we omit its
proof.

Proof of Theorem 1.3(ii). Since u0 is compactly supported, there exists R > 0 such
that u0(x) = 0 for a.e. |x| ≥ R. Combining u(x, t) ≥ 0 for all t > 0 with x ∈ R, we
have

u(t, x){1+αu(t, x)−βu2(t, x)−(1+α−β)(φ∗u)(t, x)} ≤ u(t, x)(1+αu(t, x)−βu2(t, x)),

for all t > 0 and x ∈ R. Let v(t, x) denote the solution of the Cauchy problem

vt = vxx + v(t, x)(1 + αv(t, x)− βv2(t, x)),

v(x, 0) = u0 for all x ∈ R.

By the maximum principle, we have

0 ≤ u(t, x) ≤ v(t, x) for all t > 0, x ∈ R.

On the other hand, [21, Lemma 2.1, Lemma 2.2] suggested that c∗ is the minimal

speed of (4.12) satisfying 2 ≤ c∗ ≤ 2
√

1 + α2

4β , especially, c∗ = 2 when α ≤
√
β/2.

Combining this with Lemma 4.2, we complete the proof. �

5. Appendix A

Proof of Lemma 2.2. Substituting v(t) = B(t)e + B(t)ē + Φ(B(t),B(t), ε, δ) in
equation (2.7) and comparing the coefficients of e, we obtain

dB

dt
=
(
− σ2 + (α− 2β)− (1 + α− β)φ̂(σ)

)
B +Oε,δ(B|B|2)

=
(
− (σc + δ)2 + (αc − 2βc)−

(
1 + αc − βc +

ε2

2

)
φ̂(σc + δ)

)
B +Oε,δ(B|B|2).

From the assumption on f(λ, σ, α, β) (i.e. the assumption (A1)(2)), we know that

−σ2
c − (1 + αc − βc)φ̂(σc) + (αc − 2βc) = 0,

−2σc − (1 + αc − βc)φ̂′(σc) = 0.

Thus

λ(ε, δ) :=− (σc + δ)2 + (αc − 2βc)− (1 + αc − βc)φ̂(σc + δ)− ε2

2
φ̂(σc + δ)

=−
(
(σc + δ)2 − σ2

c − 2δσc
)
− (1 + αc − βc)

×
(
φ̂(σc + δ)− φ̂(σc)− δφ̂′(σc)

)
− ε2

2
φ̂(σc + δ)

=
(
− 1− 1 + αc − βc

2
φ̂′′(σc)

)
δ2 − ε2

2
φ̂(σc) +O(|ε|2|δ|+ |δ|3),

as (ε, δ)→ (0, 0).
To obtain the coefficient ς in (2.9), let (ε, δ) = (0, 0) in (2.7) and we expect the

solutions have the Taylor expansion

v = B(t)e + B(t)ē + B2(t)e2,0 + B(t)B(t)e1,1 + B
2
(t)e0,2 +O(|B(t)|3). (5.1)



20 B.-S. HAN, D.-Y. KONG, Q. SHI, F. WANG EJDE-2021/22

Substituting (5.1) into (2.7) and comparing the coefficient of B2ei2x and BB, we
have

e2,0 =
αc − 3βc − (1 + αc − βc)φ̂(σc)

4σ2
c − (αc − 2βc) + (1 + αc − βc)φ̂(2σc)

ei2x + span(e, ē),

and

e1,1 =
2(1 + αc − βc)φ̂(σc)− 2(αc − 3βc)

αc − 2βc − (1 + αc − βc)φ̂(0)
+ span(e, ē).

It is easy to see that ς is the coefficient of the term B|BB| and only occurs in
the terms of (α− 3β)v2,−(1 + α− β)vφ ∗ v,−βv3. Thus

ς =− 〈(1 + αc − βc)(eφ ∗ e1,1 + ēφ ∗ e2,0 + e1,1φ ∗ e + e2,0φ ∗ ē), e〉
+ 〈2(αc − 3βc)(e · e1,1 + ē · e2,0, e)〉 − 〈3βce · e · ē, e〉·

(5.2)

Substituting e1,1 and ē2,0 in (5.2), we obtain the value of ς. This completes the
proof. �

6. Appendix B

Let ε = 0, then Nε
n can be rewrite as

N0
n =


0 1 0 0 0 0

n2σ2
c − (αc − 2βc) 2inσc ι 0 ι 0

0 0 0 1 0 0
−3a2 0 a2 + n2σ2

c 2inσc 0 0
0 0 0 0 0 1
2 0 0 0 1 + n2σ2

c 2inσc


It is easy to see that the characteristic polynomial of N0

n is

Gn(λ)

=
(
λ2 − 2inσcλ− n2σ2

c − σ2
c − 1− a2

+
(a2 + σ2

c )(1 + σ2
c )(a2 + (3a2 − 2)σ2

c )

3a2(1 + σ2
c )2 − 2(a2 + σ2

c )2

)
× (λ− i(n− 1)σc)

2
(λ− i(n+ 1)σc)

2
,

(6.1)
which implies

σ0
n =

{
i(n± 1)σc, inσc ±

√
1 + a2 + σ2

c −
(a2 + σ2

c )(1 + σ2
c )(a2 + (3a2 − 2)σ2

c )

3a2(1 + σ2
c )2 − 2(a2 + σ2

c )2

}
.

One can easily check that the algebraic multiplicity of each eigenvalue λ±,n =
i(n± 1)σc is two and the geometric multiplicity of λ±,n is one. Now, we study how
these eigenvalues are perturbed away when ε > 0. Here, we only consider λ−,1.
First, we write the characteristic polynomial of Nε

1 as

Gε1(λ) = G1(λ) + εDε
1(λ),

where G(λ) is given in (6.1) and

Dε
1(λ) = εϑ0 + ϑ1λ+O(ε|λ|+ |λ|2), as ε→ 0, |λ| → 0.



EJDE-2021/22 TRAVELING WAVES AND ASYMPTOTIC SPREADING 21

Here, we have

ϑ0 =
(
1− 2

3
a2
)
σ2
c −

1

2
a2, ϑ1 = (a2 + σ2

c )(1 + σ2
c )s.

Thus, we can seek λε−,1 of the form

λε−,1 = εχ+O(ε2),

where χ ∈ C. Since Gε1(λε−,1) = 0, it follows that χ satisfies the quadratic equation

ϑ2χ
2 + ϑ1χ+ ϑ0 = 0, (6.2)

where

ϑ2 = 4σ2
c

(
2σ2

c + 1 + a2 − (a2 + σ2
c )(1 + σ2

c )(a2 + (3a2 − 2)σ2
c )

3a2(1 + σ2
c )2 − 2(a2 + σ2

c )2

)
.

Then equation (6.2) has the roots

χ± =
−ϑ1 ±

√
ϑ21 − 4ϑ0ϑ2

2ϑ2
.

As a conclusion, the eigenvalue λ−,1 perturbs into two eigenvalues with asymptotic:

λε,±−,1 = εχ± +O(ε2), as ε→ 0.

Thus, there are only two center directions corresponding to the eigenvalues λε± of
Nε

1 .

7. Appendix C

Here the center manifold reduction is mainly to transform the infinite dimen-
sional system into a finite dimensional system. From Appendix B, we know that
Nε
n has eigenvalues λε± and

$±n := inσc ±

√
1 + a2 + σ2

c −
(a2 + σ2

c )(1 + σ2
c )(a2 + (3a2 − 2)σ2

c )

3a2(1 + σ2
c )2 − 2(a2 + σ2

c )2
.

We define E0 := ⊕∞n=0C6 and let X = (Xn,1, Xn,2, Xn,3, Xn,4, Xn,5,Xn,6) ∈ E0, n ≥
0. Let E denote the subset of E0, if X0,j , j = 1, 2, . . . , 6 are real. In addition, for
the functions of the form

V (ζ, x) = Σn∈ZVn(ζ)e−inσcx, Vn(ζ) = (V un (ζ), V vn (ζ), V wn (ζ))T ,

we define a one-to-one map

Iζ :
{

(V un , ∂zV
u
n , V

v
n , ∂zV

v
n , V

w
n , ∂zV

w
n )

T | for all n ∈ Z
}
→ E

for all ζ ∈ R as

Iζ(V )n = (V un , ∂zV
u
n |z=ζ , V vn , ∂zV vn |z=ζ , V wn , ∂zV wn |z=ζ)T ↔ Xn.

We note that V−n = V n, so Iζ(V )n ∈ C6, n > 0 and Iζ(V )0 ∈ R uniquely determine
by V . Following, we define an inner product about E0 as

〈X,Y 〉l =

∞∑
n=0

(1 + n2)l〈Xn, Yn〉C6 ,



22 B.-S. HAN, D.-Y. KONG, Q. SHI, F. WANG EJDE-2021/22

and define the Hilbert space H l
C = {X ∈ E0|〈X,Y 〉l < ∞}. The nonlinearity

L : E0 → E0 is defined as

Ln(X) =
(

0,
∑

p+q=n

(Xp,1Xq,3 +Xp,1Xq,5), 0, 0, 0, 0
)T
, n ≥ 0,

Jn(X) =
(

0,
∑

p+q+r=n

Xp,1Xq,1Xr,1, 0, 0, 0, 0
)T
, n ≥ 0.

Obviously, Ln(X) ∈ C1(H l
C(E0), H l

C(E0)) as long as l > 1/2. Furthermore, since
H l

C(E0) is a Banach algebra, we have the estimate

‖L(X)‖l ≥ C‖X‖2l , X ∈ H l
C(E0),

where C > 0 is a constant which is depends on l and the nonlinearity. Finally, we
also need a bilinear map L : E0 × E0 → E0 defined as

Ln(X,Y ) =
1

2

(
0,
∑

p+q=n

(Xp,1Yq,3 +Xp,1Yq,5 +Xq,3Yp,1 +Xq,5Yp,3)
)T
, n ≥ 0,

for any (X,Y ) ∈ E0 × E0. Thus, by using a center manifold reduction, we can
transform the system (2.10) into a finite dimensional system (3.6).

Acknowledgments. This work was supported by the NSF of China (11801470,
11701477), by the Fundamental Research Funds for the Central Universities
(2682018CX64), and by the Natural Science Foundation of Jiangsu Province, China
(Grant No. BK20190578)

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Bang-Sheng Han

School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 611756, China
Email address: hanbangsheng@swjtu.edu.cn

De-Yu Kong
School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 611756, China

Email address: KongDY@my.swjtu.edu.cn



24 B.-S. HAN, D.-Y. KONG, Q. SHI, F. WANG EJDE-2021/22

Qihong Shi

Department of Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730000,

China
Email address: shiqh@lut.edu.cn

Fan Wang (corresponding author)
School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 611756, China

Email address: wangf767@swjtu.edu.cn


	1. Introduction
	2. Existence of a periodic steady state
	2.1. Special kernel
	2.2. General kernel

	3. Traveling wave solutions connecting 1 to a periodic steady state
	3.1. Existence of traveling wave solutions
	3.2. Numerical simulations

	4. Asymptotic rate of the Cauchy problem (??)
	5. Appendix A
	6. Appendix B
	7. Appendix C
	Acknowledgments

	References