Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 112, pp. 1–29.

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

PYRAMIDAL TRAVELING FRONTS IN THE

BELOUSOV-ZHABOTINSKII REACTION-DIFFUSION

SYSTEMS IN R3

LUYI MA, HONG-TAO NIU, ZHI-CHENG WANG

Abstract. In this article, we consider a diffusion system with the Belousov-

Zhabotinskii (BZ for short) chemical reaction. The existence and stability
of V-shaped traveling fronts for the BZ system in R2 had been proved in

our previous papers [30, 31]. Here we establish the existence and stability of

pyramidal traveling fronts for the BZ system in R3.

1. Introduction

Consider the reaction-diffusion system

ut(x, t) = ∆u(x, t) + u(x, t)(1− u(x, t)− rv(x, t)),

vt(x, t) = ∆v(x, t)− bu(x, t)v(x, t),
(1.1)

where r, b > 0 are positive parameters and u, v correspond to the concentrations
of the bromous acid and bromide ion respectively. (1.1) is called the BZ system,
which stems from a typical chemical oscillating reaction. The possible existence
of such chemical oscillation was predicted by Turing [39] through the method of
mathematical calculation, and the chemical phenomenon was observed by Belousov
[1]. When the concentrations of the reactants change orderly along with time and
space, chemical waves appear [47]. To investigate the mechanism of the BZ reaction,
Field and his coworkers formulated a complex model [8] and then simplified it [9].
Later, the simplified model was nondimensionalized by Murray [25, 26] to be (1.1).
It is found that the front solution of (1.1) is an appropriate mathematical tool to
describe the planar waves [47].

After that, mathematical studies on system (1.1), a lot of progress has been made,
mainly including the existence of 1-D traveling wave fronts, admissible traveling
speeds and the asymptotic behavior of traveling wave fronts [12, 20, 21, 22, 37, 38,
41]. In fact, studies of traveling wave solutions on reaction-diffusion equations

ut(x, t) = ∆u(x, t) + f(u(x, t)), x ∈ RN , t > 0,

which can be originated from the pioneer work of Fisher [6], have attracted a lot of
attention [4, 10, 11, 23, 24, 42], which mainly focus on 1-D traveling wave solutions
and planar traveling wave solutions in RN (N ≥ 2). The gradually mature theory of

2010 Mathematics Subject Classification. 35K40, 35K57, 35C07.
Key words and phrases. Belousov-Zhabotinskii reaction; pyramidal traveling fronts;

supersolution; subsolution.
c©2020 Texas State University.

Submitted April 3, 2020. Published November 12, 2020.

1



2 L. MA, H.-T. NIU, Z.-C. WANG EJDE-2020/112

1-D traveling wave solutions promotes the research on multidimensional traveling
wave solutions, which are proper to describe the traveling wave phenomena in multi-
dimensional space, see [2, 3, 13, 14, 15, 28, 29, 34, 35, 44] for the scalar equation
and [5, 16, 17, 18, 19, 27, 36, 43, 45, 46] for the reaction diffusion system.

For the BZ reaction, along with the development of research, nonplanar chemical
waves were also observed. In 1995, stable V-shaped chemical waves were observed in
the BZ reaction [40], for which we have already made rigorous mathematical proofs
[30, 31]. However, mathematical studies of multidimensional nonplanar traveling
waves on the BZ system (1.1) are still very few.

In this article, we continue to study the pyramidal traveling fronts of (1.1) and
expect to give some theoretical implications to the observation of new nonplanar
chemical waves in the experiment. Precisely, we go on studying (1.1) under the
case r > 1, which means that (1.1) is bistable. We still want to emphasize that the
case ‘bistable’ for the BZ system is different from those in Ni and Taniguchi [27]
and Wang [43] since it is degenerate at one of its the equilibria.

Now we set u1(x, t) = u(x, t), u2(x, t) = 1 − v(x, t), u = (u1, u2), then system
(1.1) can be rewritten as

ut = ∆u + F(u), (1.2)

where F(u) = (f1(u), f2(u)) = (u1(1 − r − u1 + ru2), bu1(1 − u2)). Under the
condition r > 1, [37, Theorem 9] tells that system (1.2) admits a unique positive
traveling front U(ξ) = (U1(ξ), U2(ξ)) satisfying

U ′′1 (ξ)− cU ′1(ξ) + f1(U(ξ)) = 0,

U ′′2 (ξ)− cU ′2(ξ) + f2(U(ξ)) = 0
(1.3)

with

U(−∞) = (0, 0), U(+∞) = (1, 1), 0 < U1(ξ) < U2(ξ) < 1, ∀ξ ∈ R,

where c ∈
(
b/(2

√
(r + b)[min(1, b)(r + b)− 0.5b]), 2

√
min(1, b)

)
is the wave speed,

see [37, Theorem 6, Theorem 9 and Proposition 10].
Denote x′ = (x1, x2), x = (x′, x3). Without loss of generality, we assume that

the traveling solutions travel towards the −x3 direction with speed s, then they
have the form u(x, t) = v(x′, x3 + st) and satisfy

vt = ∆v − svx3
+ F(v), x ∈ R3, t > 0, (1.4)

v|t=0 = v0(x), x ∈ R3. (1.5)

We aim to find a nontrivial steady state V(x) of the system

−∆V + sVx3
− F(V) = 0, x ∈ R3. (1.6)

Since the acceleration effect of the curvature, it is natural to take s > c. Fix s and
set

m∗ =
√
s2 − c2

/
c.

Now we introduce the definition of pyramid. Let n ≥ 3 be a given integer. As-
sume {(Aj , Bj)}nj=1 is a set of unit vectors, i.e. A2

j +B2
j = 1 for all j ∈ {1, 2, . . . , n},

and satisfies

AjBj+1 −Aj+1Bj > 0, 1 ≤ j ≤ n− 1; AnB1 −A1Bn > 0.



EJDE-2020/112 PYRAMIDAL TRAVELING FRONTS 3

We also assume that (Aj , Bj) 6= (Ai, Bi) if i 6= j. Now (−m∗Aj ,−m∗Bj , 1) is a
normal vector of the plane {x ∈ R3 : −x3 = m∗(Ajx1 +Bjx2)}. We put

hj(x
′) = m∗(Ajx1 +Bjx2),

h(x′) = max
1≤j≤n

hj(x
′) = m∗ max

1≤j≤n
(Ajx1 +Bjx2).

Then −x3 = h(x′) represents a pyramid in R3. Set Ωj = {x′ ∈ R2 : h(x′) =
hj(x

′)}, j ∈ {1, 2, . . . n}. Then R2 = ∪nj=1Ωj . Denote

E := ∪nj=1∂Ωj ⊂ R2.

Now the lateral surfaces of a pyramid are Sj = {x ∈ R3 : −x3 = hj(x
′), x′ ∈ Ωj}

for j = 1, 2, . . . , n. We put

Γj =

{
Sj ∩ Sj+1 if 1 ≤ j ≤ n− 1,

Sn ∩ S1 if j = n.

Then Γj represents an edge of the pyramid and Γ = ∪nj=1Γj represents the set of
all edges. Denote

v−(x) = U
( c
s

(x3 + h(x′))
)

= max
1≤j≤n

U
( c
s

(x3 + hj(x
′))
)
. (1.7)

Since U(x) is a planar traveling wave of (1.2), it is easy to see that v−(x) is
combined with several such planar traveling waves and thus becomes a nonplanar
traveling wave with pyramidal level sets. We also define

D(γ) := {x ∈ R3 : dist(x,Γ) > γ}, ∀γ > 0.

We now define the a relation of order in R3. We say that x < y (resp. x ≤ y)
An interval [x1,x2] ⊂ R3 denotes the set of x ∈ R3 with x1 ≤ x ≤ x2. Throughout
this paper, we denote 0 = (0, 0) and 1 = (1, 1). The following theorem is the main
assertion in this paper.

Theorem 1.1. Assume that r > 1 and b > 0. Then for each s > c, there exists a
solution V(x) = (V1(x), V2(x)) to (1.6) with

v−(x) < V(x) < 1 in R3

and

lim
γ→∞

sup
x∈D(γ)

|Vi(x)− v−i (x)|(
v−2 (x)

)βi = 0, i = 1, 2.

Furthermore, for any u0(x) ∈ C(R3,R2) with u0(x) ∈ [0,1] for x ∈ R3 and

lim
γ→∞

sup
x∈D(γ)

|Vi(x)− u0,i(x)|
(v−2 (x))βi

= 0, i = 1, 2,

v−(x) ≤ u0(x), x ∈ R3,

(1.8)

the solution u(x, t; u0) of (1.2) with initial data u0 satisfies

lim
t→∞

∥∥ui(·, ·, ·, t; u0)− Vi(·, ·, ·+ st)

(v−2 (·))βi
∥∥
L∞(R3)

= 0, i = 1, 2.

Here, 0 < β2 < β1 < β∗ are arbitrary (see (3.4) for β∗).



4 L. MA, H.-T. NIU, Z.-C. WANG EJDE-2020/112

To use the comparison argument, we consider a modified system

ṽt = ∆ṽ − sṽx3 + F̃(ṽ), x ∈ R3, t > 0, (1.9)

ṽ|t=0 = v0(x, z), x ∈ R3, (1.10)

where F̃(u) = (f̃1(u), f̃2(u)) = F(u) + G(u), G(u) = (g1(u), g2(u)) with

g1(u) = 0, g2(u) = b(u1 − 1) max{0, u2 − 1}.

It is clear that both F(u) and F̃(u) are Lipschitz continuous in R2. Obviously,

∂u2
f1 ≥ 0, ∂u1

f̃2 ≥ 0 if (u1, u2) ∈ [0,+∞)× [0,+∞).

Then the comparison principle (see [32]) gives

ṽ(x, t; v1
0) ≤ ṽ(x, t; v2

0), ∀x ∈ R3, t ≥ 0

if 0 ≤ v1
0(x) ≤ v2

0(x) in R3, where ṽ(x, t; v0) denotes the solution of (1.9) and
(1.10). In particular, it holds

ṽ(x, t; v0) ∈ [0,1] if v0(x) ∈ [0,1], ∀x ∈ R3,

which implies that the interval [0,1] is invariant for the solution of (1.9) and (1.10).
Thus, for v0(x) ∈ [0,1], the solution ṽ(x, t; v0) of (1.9) and (1.10) is also the so-
lution of (1.4) and (1.5), namely, ṽ(x, t; v0) ≡ v(x, t; v0), where v(x, t; v0) denotes
the solution of (1.4) and (1.5).

For each unit vector (Aj , Bj), (1.6) admits a solution U( cs (x3 + hj(x
′))), which

is called a planar wave. It follows that the function v−(x) defined by (1.7) is a
subsolution of (1.9), and obviously v−z = c

sU
′( cs (x3 +h(x′))) > 0. Throughout this

paper, we define the operator L̃ by

L̃[v] := vt −∆v + svx3
− F̃(v).

The remainder of this paper is organized as follows: in Section 2 we give some
notation and known results. In Section 3 we prove the existence result of pyramidal
fronts by constructing an appropriate supersolution. And in Section 4 we prove the
asymptotic stability of the pyramidal traveling fronts constructed in Section 3.

2. Preliminaries

In this section, we give some notation and known results. By [30, Lemma 1.1]
or [37, Lemma 13], we have

lim
ξ→−∞

U ′2(ξ)

U2(ξ)
= λ2 = c.

Thus we can define

N1 := sup
x∈R

∣∣U ′2(x)

U2(x)

∣∣, N2 := sup
x∈R

∣∣U ′′2 (x)

U2(x)

∣∣. (2.1)

[30, Lemma 1.1] also implies that there exist two positive constants L1 < 1 and
L2 > 1 such that

L1e
max{λ1,2λ2}ξ < U1(ξ), U ′1(ξ) < L2e

min{λ1,2λ2}ξ, ξ < 0. (2.2)

L1e
λ1ξ < U2(ξ), U ′2(ξ) < L2e

λ2ξ, ξ < 0. (2.3)



EJDE-2020/112 PYRAMIDAL TRAVELING FRONTS 5

Using (1.2), the derivative matrix of F is

DF(u) = (fij(u))2×2 =

(
1− r − 2u1 + ru2 ru1

b(1− u2) −bu1

)
where fij(u) = ∂fi(u)

∂uj
. Because of r > 1, we can find two positive numbers p1, p2

that satisfy p1 > p2 ≥ 1 and p1

p2
> r. Let p = (p1, p2)T , where T means the

transpose. We have

q = (q1, q2)T := DF(1) · p < 0,

Fix an appropriate ε1 ∈ (0, 1) such that

DF(u) · p <
1

2
q, (1− ε1)1 ≤ u ≤ (1 + ε1)1. (2.4)

Now we introduce a mollified pyramid, see [34]. Let ρ̃(r) ∈ C∞([0,∞)) be a
function with the following properties:

ρ̃(r) > 0, ρ̃′(r) ≤ 0 for r ≥ 0,

ρ̃(r) ≡ 1 if 0 ≤ r ≤ 1,

ρ̃(r) ≡ e−r if r > 0 is large enough,

2π

∫ ∞
0

rρ̃(r)dr = 1.

Then ρ(x′) := ρ̃(|x′|) belongs to C∞(R2) and satisfies
∫
R2 ρ(x′)dx′ = 1. For a

pyramid −x3 = h(x′) we define its corresponding mollified pyramid −x3 = ϕ(x′),
where

ϕ(x′) =

∫
R2

ρ(x′ − y′)dy′ =

∫
R2

ρ(y′)h(x′ − y′)dy′. (2.5)

We set (aj , bj) = m∗(Aj , Bj). Then (aj , bj) ∈ R2 satisfies

s√
1 + a2

j + b2j

= c, for j = 1, 2, . . . n.

We put

S(x′) :=
s√

1 + |∇ϕ(x′)|2
− c, (2.6)

where ∇ϕ = (ϕx1 , ϕx2). The following two lemmas come from Taniguchi [34].

Lemma 2.1. Let ϕ and S be as in (2.5) and (2.6), respectively. Then

h(x′) < ϕ(x′) ≤ h(x′) + 2πm∗

∫ ∞
0

r2ρ̃(r)dr, |∇ϕ(x′)| < m∗,

0 < S(x′) ≤ s− c

for all x′ ∈ R2. In particular,

lim
λ→∞

sup{S(x′)|x′ ∈ R2,dist(x′, E) ≥ λ} = 0,

lim
λ→∞

sup{ϕ(x′)− h(x′)|x′ ∈ R2,dist(x′, E) ≥ λ} = 0

and there exists positive constants ν1, ν2 so that

0 < ν1 ≤
ϕ(x′)− h(x′)

S(x′)
≤ ν2, ∀x′ ∈ R2.



6 L. MA, H.-T. NIU, Z.-C. WANG EJDE-2020/112

Lemma 2.2. For all integers i1 ≥ 0, i2 ≥ 0, one has

C1 := sup
x′∈R2

|Di1
x1
Di2
x2
ϕ(x′)| < +∞,

and furthermore, for 2 ≤ i1 + i2 ≤ 3 one also has

C2 := sup
x′∈R2

|Di1
x1
Di2
x2
ϕ(x′)|

S(x′)
< +∞ .

3. Existence of pyramidal traveling fronts

Set z′ = αx′, z3 = αx3 and z = αx. Define

ς(x) =
x3 + ϕ(z′)/α√
1 + |∇ϕ(z′)|2

, η(x) =
c

s
(x3 + ϕ(z′)/α).

Since 1 ≤
√

1 + |∇ϕ|2 < s/c, we have

s

c
η(x) < ς(x) < η(x), if ς(x) < 0, (3.1)

η(x, z) < ς(x) <
s

c
η(x), if ς(x) > 0. (3.2)

Now we fix a function ω(x) ∈ C∞(R) with

ω(x) = 1, if x ≤ −1,

0 < ω(x) < 1, −1 < ω′(x) < 0, if − 1 < x < 1,

ω(x) = 0, if x ≥ 1.

(3.3)

In this section, we denote β := (β1, β2) and make it satisfies

0 < β2 < β1 < β∗ :=
λ2

λ1
, (3.4)

see [30, Lemmas 1.1 and 1.4] for λ1 and λ2. In the proof of the following lemma,
we denote

Π1(x) := x2 − cx+ 1− r, Π2(x) := x2 − cx.
Obviously, β∗ < 1 and Πi(βiλ2) < 0 for i = 1, 2.

Lemma 3.1. There exist a positive constant ε+
0 (β) < 1 and a positive function

α+
0 (ε,β) such that, for all 0 < ε < ε+

0 (β) and 0 < α < α+
0 (ε,β),

v+(x; ε,β, α) = U(ς(x)) + εS(z′)
(
(1− ω(η(x))) p + ω(η(x))Uβ(η(x))

)
is a supersolution to (1.9), where Uβ(ξ) :=

(
Uβ1

2 (ξ), Uβ2

2 (ξ)
)
. Furthermore,

lim
γ→+∞

sup
x∈D(γ)

|v+
i (x; ε,β, α)− v−i (x)|

(v−2 (x))βi
≤ 2ε, i = 1, 2, (3.5)

v−(x) < v+(x; ε,β, α), x ∈ R3, (3.6)

∂x3v
+(x; ε,β, α) > 0, x ∈ R3. (3.7)

Proof. For the sake of convenience, we denote ς(x), η(x) by ς and η, respectively.
We also denote v+(x; ε,β, α) by v+(x) for simplicity. By a direct computation, we
have

ςx1
= −αϕz1ϕz1z1 + ϕz2ϕz2z1

1 + |∇ϕ(z′)|2
ς +

ϕz1√
1 + |∇ϕ(z′)|2

, ηx1
=
c

s
ϕz1 ,



EJDE-2020/112 PYRAMIDAL TRAVELING FRONTS 7

ςx2
= −αϕz1ϕz1z2 + ϕz2ϕz2z2

1 + |∇ϕ(z′)|2
ς +

ϕz2√
1 + |∇ϕ(z′)|2

,

ηx2
=
c

s
ϕz2 , ςx3

=
1√

1 + |∇ϕ(z′)|2
, ηx3

=
c

s
,

ηx1x1
= α

c

s
ϕz1z1 , ηx2x2

= α
c

s
ϕz2z2 , ηx3x3

= 0, ςx3x3
= 0,

ςx1x1
= α2

[3(ϕz1ϕz1z1 + ϕz2ϕz2z1)2

(1 + |∇ϕ(z′)|2)2
−
ϕ2
z1z1 + ϕz1ϕz1z1z1 + ϕ2

z2z1 + ϕz2ϕz2z1z1
1 + |∇ϕ(z′)|2

]
ς

+ α
ϕz1z1(1− ϕ2

z1 + ϕ2
z2)− 2ϕz1ϕz2ϕz1z2

(1 + |∇ϕ(z′)|2)3/2
,

ςx2x2 = α2
[3(ϕz1ϕz1z2 + ϕz2ϕz2z2)2

(1 + |∇ϕ(z′)|2)2
−
ϕ2
z1z2 + ϕz1ϕz1z2z2 + ϕ2

z2z2 + ϕz2ϕz2z2z2
1 + |∇ϕ(z′)|2

]
ς

+ α
ϕz2z2(1 + ϕ2

z1 − ϕ
2
z2)− 2ϕz1ϕz2ϕz1z2

(1 + |∇ϕ(z′)|2)3/2
.

It is easy to check that ∂x3
v+(x) > 0 holds according to the definition of v+(x).

To prove that v+(x) is a supersolution, it suffices to verify that

L̃[v+(x)] = −∆v+(x) + sv+
x3

(x)− F̃(v+(x)) ≥ 0.

Throughout the proof, we assume that α < ε. From direct computations and (1.3),
we have

L̃[v+(x)]i =
(

1−
3∑
j=1

ς2xj

)
U ′′i (ς)−

( 3∑
j=1

ςxjxj

)
U ′i(ς)

− εα2
( 2∑
j=1

Szjzj

)[
(1− ω(η))pi + ω(η)Uβi2 (η)

]
− 2εα

( 2∑
j=1

Szjηxj

)[
ω′(η)(−pi + Uβi2 (η)) + ω(η)βiU

βi−1
2 (η)U ′2(η)

]
− εS(z′)

{[
ω′′(η)

3∑
j=1

η2
xj + ω′(η)

2∑
j=1

ηxjxj − cω′(η)
](
− pi + Uβi2 (η)

)
− 2βiω

′(η)U ′2(η)Uβi−1
2 (η)

3∑
j=1

η2
xj

+ ω(η)
[
βi(βi − 1)Uβi−2

2 (η)(U ′2(η))2
3∑
j=1

η2
xj

+ βiU
βi−1
2 (η)U ′2(η)

3∑
j=1

ηxjxj

+ βiU
βi−1
2 (η)U ′′2 (η)

3∑
j=1

η2
xj − cβiU

βi−1
2 (η)U ′2(η)

]}
+
( s√

1 + |∇ϕ(z′)|2
− c
)
U ′i(ς)− f̃i(v+(x)) + fi(U(ς)).



8 L. MA, H.-T. NIU, Z.-C. WANG EJDE-2020/112

Let

A1 := sup
ξ∈R

∣∣∑2
j=1 Szjzj (z

′)

S(z′)

∣∣, A2 := sup
ξ∈R

∣∣∑2
j=1 Szj (z

′)ηxj

S(z′)

∣∣.
By Lemmas 2.1-2.2, we know that 0 ≤ A1, A2 < +∞ are well defined. Lemma 2.1
and Lemma 2.2 also imply that there exist positive constants A3, A4, A5 and A6

such that ∣∣1− 3∑
j=1

ς2xj
∣∣ ≤ α(A3|ς|+A4ς

2)S(z′) ≤ ε
(
A3|ς|+A4ς

2
)
S(z′), (3.8)

∣∣ 2∑
j=1

ςxjxj
∣∣ ≤ α(A5 +A6|ς|)S(z′) ≤ ε(A5 +A6|ς|)S(z′), (3.9)

∣∣ 2∑
j=1

ηxjxj
∣∣ ≤ ∣∣αc

s

2∑
j=1

ϕzjzj
∣∣ ≤ 2αC1. (3.10)

Next, we consider three cases.

Case 1: ς < −X ′ for some X ′ > 0 large enough. Recalling (3.1), ς < η holds
in this case. Assume that ε ≤ 1

2(s−c) . And without loss of generality, suppose

that η ≤ −X∗ < −1, where X∗ > 0 is a positive constant such that U2(η) ≤ 1
2 if

η ≤ −X∗. Under these conditions, we know

v+
2 (x) ≤ 1

2
+ εS(z′) < 1 if η < −X∗,

which implies that f̃2(v+(x)) = f2(v+(x)). Then by (3.3) we have

L̃[v+(x)]i

=
(

1−
3∑
j=1

ς2xj

)
U ′′i (ς)−

( 3∑
j=1

ςxjxj

)
U ′i(ς)

− εS(z′)Uβi2 (η)
{α2

∑2
j=1 Szjzj

S(z′)
+
(

2αβi

∑2
j=1 Szjηxj

S(z′)
+ βi

2∑
j=1

ηxjxj

)U ′2(η)

U2(η)

+ βi(βi − 1)
(U ′2(η)

U2(η)

)2 3∑
j=1

η2
xj + βi

U ′′2 (η)

U2(η)

3∑
j=1

η2
xj − cβi

U ′2(η)

U2(η)

}
+
( s√

1 + |∇ϕ(z′)|2
− c
)
U ′i(ς)− fi(v+(x)) + fi(U(ς)).

Recall (3.1) and the monotonicity of the wave profile U. Then by (3.8)-(3.10) we
have

L̃[v+(x)]i

≥ −εS(z′)Uβi2 (η)(A3|ς|+A4ς
2)
|U ′′i (ς)|
Uβi2 (ς)

− εS(z′)Uβi2 (η)(A5 +A6|ς|)
U ′i(ς)

Uβi2 (ς)

− εS(z′)Uβi2 (η)
{
α2A1 + 2αA2

U ′2(η)

U2(η)
+ 2αC1

U ′2(η)

U2(η)

+ βi
( c
s

)2∣∣∣− (U ′2(η)

U2(η)

)2
+
U ′′2 (η)

U2(η)

∣∣∣(1 + |∇ϕ(z′)|2
)



EJDE-2020/112 PYRAMIDAL TRAVELING FRONTS 9

+ β2
i

( c
s

)2(U ′2(η)

Ui(η)

)2(
1 + |∇ϕ(z′)|2

)
− β2

i

(U ′2(η)

U2(η)

)2

+ β2
i

(U ′2(η)

U2(η)

)2

− cβi
U ′2(η)

U2(η)

}
+
( s√

1 + |∇ϕ(z′)|2
− c
)
U ′i(ς) + fi(U(ς))− fi(v+(x)).

Since λ2 = limx→−∞
U ′2(x)
U2(x) and λ2

2 = limx→−∞
U ′′2 (x)
U2(x) , we have

−
(U ′2(x)

U2(x)

)2

+
U ′′2 (x)

U2(x)
→ 0,

β2
1

(U ′2(x)

U2(x)

)2

− cβ1
U ′2(x)

U2(x)
+ 1− r → Π1(β1λ2) < 0,

β2
2

(U ′2(x)

U2(x)

)2

− cβ2
U ′2(x)

U2(x)
→ Π2(β2λ2) < 0

as x→ −∞. Thus there exists X1 > 0 large enough such that

U ′2(x)

U2(x)
<

3

2
λ2,

∣∣− (U ′2(x)

U2(x)

)2

+
U ′′2 (x)

U2(x)

∣∣ < − 1

16
Πi(βiλ2),

β2
1

(U ′2(x)

U2(x)

)2

− cβ1
U ′2(x)

U2(x)
+ 1− r < 1

2
Π1(β1λ2),

β2
2

(U ′2(x)

U2(x)

)2

− cβ2
U ′2(x)

U2(x)
<

1

2
Π2(β2λ2)

for any x < −X1. For the above positive constants A3, A4, A5 and A6, it follows
from [30, Lemma 1.1] that there exists X2 > 0 large enough such that

(A3|x|+A4x
2)
|U ′′i (x)|
Uβi2 (x)

< − 1

16
Πi(βiλ2),

(A5 +A6|x|)
|U ′i(x)|
Uβi2 (x)

< − 1

16
Πi(βiλ2)

for any x < −X2. Also there exists α1 ∈ (0, β∗) small enough such that

α2A1 + 3αA2λ2 + 3αC1λ2 < −
1

16
Πi(βiλ2), ∀α ∈ (0, α1), i = 1, 2.

For the reaction term fi, we have

fi
(
v+(x)

)
− fi

(
U(ς)

)
=
( 2∑
j=1

fij
(
θiv

+(x) + (1− θi)U(ς)
)
U
βj
2 (η)

)
εS(z′)

=
( 2∑
j=1

fij
(
U(ς) + εθiS(z′)Uβ(η)

)
U
βj
2 (η)

)
εS(z′),

where θi ∈ (0, 1), i = 1, 2. If i = 1, then 0 < U1(ξ) < U2(ξ) < 1 yields

f12

(
U(ς) + εθ1S(z′)Uβ(η)

)
= r

(
U1(ς) + εθ1S(z′)Uβ1

2 (η)
)
≤ r(1 + εs)Uβ1

2 (η).

It follows that

f1(v+(x))− f1(U(ς))



10 L. MA, H.-T. NIU, Z.-C. WANG EJDE-2020/112

≤
(
f11

(
U(ς) + εθ1S(z′)Uβ(η)

)
Uβ1

2 (η) + r(1 + εs)Uβ1

2 (η)Uβ2

2 (η)
)
εS(z′)

≤
(
f11

(
U(ς) + εθ1S(z′)Uβ(η)

)
+ r(1 + εs)Uβ2

2 (η)
)
εS(z′)Uβ1

2 (η),

and f11

(
U(x) + εθ1S(z′)Uβ(y)

)
+ r(1 + εs)Uβ2

2 (y)→ 1− r as x, y → −∞. If i = 2,
then

f2

(
v+(x)

)
− f2

(
U(ς)

)
=
( 2∑
j=1

f2j

(
U(ς) + εθ2S(z′)Uβ(η)

)
U
βj
2 (η)

)
εS(z′)

=
( 2∑
j=1

f2j

(
U(ς) + εθ2S(z′)Uβ(η)

)
(U2(η))βj−β2

)
εS(z′)Uβ2

2 (η).

Note that (U2(x))β1−β2 → 0 as x→ −∞, also we have

2∑
j=1

f2j

(
U(x) + εθ2S(z′)Uβ(y)

)
(U2(y))βj−β2 → 0 as x, y → −∞.

Then we know that there exists X3 > 0 large enough such that

f11

(
U(x) + εθ1S(z′)Uβ(y)

)
+ r(1 + εs)Uβ2

2 (y)− (1− r) < − 1

16
Π1(β1λ2),

2∑
j=1

f2j

(
U(x) + εθ2S(z′)Uβ(y)

)
(U2(y))βj−β2 < − 1

16
Π1(β1λ2)

for x, y < −X3.
Let X ′ = max{ scX

∗, X1, X2, X3}, then for ς < −X ′ we have

L̃[v+(x)]i ≥ −Uβi2 (η)εS(z′)
(
A3|ς|+A4ς

2
) |U ′′i (ς)|
Uβi2 (ς)

− Uβii (η)εS(z′) (A5 +A6|ς|)
U ′i(ς)

Uβi2 (ς)

− εS(z′)Uβi2 (η)
{
α2A1 + 3αA2λ2 + 3αC1λ2

+ βi

∣∣∣− (U ′2(η)

U2(η)

)2

+
U ′′2 (η)

U2(η)

∣∣∣+ β2
i

(U ′2(η)

U2(η)

)2

− β2
i

(U ′2(η)

U2(η)

)2

+ β2
i

(U ′2(η)

U2(η)

)2

− cβi
U ′2(η)

U2(η)

}
− εS(z′)

( 2∑
j=1

fij
(
U(ς) + εθiS(z′)Uβ(η)

)
U
βj
2 (η)

)
≥ εS(z′)Uβi2 (η)

( 1

16
Πi(βiλ2) +

1

16
Πi(βiλ2) +

1

16
Πi(βiλ2)

+
1

16
Πi(βiλ2)− 1

2
Πi(βiλ2) +

1

16
Πi(βiλ2)

)
> 0.



EJDE-2020/112 PYRAMIDAL TRAVELING FRONTS 11

Case 2: ς > X ′′ for some X ′′ > 0 large enough. Without loss of generality, suppose
that η > 1. By [30, Lemma 1.4], we can take X ′1 > 0 large enough such that

(A3|x|+ αA4x
2)|U ′′i (x)| < −qi

8
and (A5 +A6|x|)|U ′i(x)| < −qi

8

for all x > X ′1 and i = 1, 2.
Fix a constant α2 ∈ (0, β∗) such that α2A1p1 < mini=1,2

{
− qi8

}
for any α in

(0, α2). For the reaction term fi, we have

fi(v
+(x))− fi(U(ς)) =

( 2∑
j=1

fij
(
U(ς) + θiεS(z′)p

)
pj

)
εS(z′), i = 1, 2.

Since Ui(x) → 1 as x → +∞ for i = 1, 2, there exists X ′2 > 0 large enough such
that for all ε ∈ (0, ε1

p1(s−c) ) (see (2.4) for ε1), it holds

1− ε1 < Ui(x) + εθiS(z′)pi < 1 + ε1, x > X ′2, i = 1, 2,

and then
2∑
j=1

fij
(
U(x) + εθiS(z′)p

)
pj <

1

2
qi, x > X ′2, i = 1, 2.

According to the definition of f̃2, we know

f̃2(v+(x)) ≤ f2(v+(x)) + b(εS(z′))2p1p2.

Take X ′′ = max{X ′1, X ′2, 1}, then for ς > X ′′, we have

L̃[v+(x)]i

=
(

1−
3∑
j=1

ς2xj

)
U ′′i (ς)−

( 3∑
j=1

ςxjxj

)
U ′i(ς)− εα2S(z′)

∑2
j=1 Szjηxj

S(z′)
pi

+
( s√

1 + |∇ϕ(z′)|2
− c
)
U ′i(ς)− f̃i(v+(x)) + fi(U(ς))

≥ −εS(z′)(A3|ς|+A4|ς|2)|U ′′i (ς)|
− εS(z′)(A5 +A6|ς|)|U ′i(ς)| − εS(z′)α2A1p1

− εS(z′)
( 2∑
j=1

fij (U(ς) + θiεS(z′)p) pj

)
− b(εS(z′))2p1p2

≥ εS(z′)
(qi

8
+
qi
8

+
qi
8
− qi

2
− bεp1p2(s− c)

)
> 0

provided that ε < mini=1,2{− qi
8bp1p2(s−c)}.

Case 3: −X ′ ≤ ς ≤ X ′′. Define u∗ := min−X′≤x≤X′′ mini=1,2 U
′
i(x) and

Mij := sup
u∈[−ε11,(1+ε1)1]

fij(u), M0 := sup
1≤i,j≤2

Mij ,

M1 := sup
x∈R, i=1,2

|U ′i(x)|, M2 := sup
x∈R, i=1,2

|x||U ′i(x)|,

M3 := sup
x∈R, i=1,2

|x||U ′′i (x)|, M4 := sup
x∈R, i=1,2

x2|U ′′i (x)|.

We have

L̃[v+(x)]i ≥ −αS(z′)(A3|ς|+ αA4|ς|2)|U ′′i (ς)| − εS(z′)(A5 +A6|ς|)|U ′i(ς)|



12 L. MA, H.-T. NIU, Z.-C. WANG EJDE-2020/112

− εS(z′)α2A1pi − 2εαS(z′)A2(pi +N1)

− εS(z′) {(|ω′′(η)|+ α |∆z′ϕ(z′)|) pi +N1 |∆z′ϕ(z′)|+N2}
+ S(z′)u∗ − b(εS(z′))2p1p2

− εS(z′)
( 2∑
j=1

fij
(
U(ς) + εθi

(
ω(η)p + (1− ω(η))Uβ(η)

)) )
×
(
ω(η)pj + (1− ω(η))U

βj
2 (η)

)
≥ S(z′)

{
− αA3M3 − αA4M4 − αA5M1 − αA6M2

− αA1p1 − 2αA2(p1 +N1)− εA+ u∗ − bε(s− c)p1p2 − 2εM0p1

}
,

where A :=
(

supx∈R |ω′′(x)|+supz′∈R2 |∆z′ϕ(z′)|
)
p1+(supz′∈R2 |∆z′ϕ(z′)|)N1+N2.

See (2.1) for N1 and N2. Let

α < α3 :=
u∗

2[A3M3 +A4M4 +A5M1 +A6M2 +A1p1 + 2A2(N1 + p1)]
,

ε < ε2 :=
u∗

2[A+ 2M0p1 + b(s− c)p1p2]
,

then L̃[v+(x)]i > 0, i = 1, 2. Combining the three cases above, we have proved
that v+(x) is a supersolution to (1.9).

Next we prove that v−(x) < v+(x). Let

ξ(x) =
c

s
(x3 + h(x′)), ν(x) =

1√
1 + |∇ϕ(z′)|2

(x3 + h(x′)),

and recall that

η(x) =
c

s
(x3 + ϕ(z′)/α), ς(x) =

1√
1 + |∇ϕ(z′)|2

(x3 + ϕ(z′)/α).

If ς(x) ≥ ξ(x), then it is obvious that v−(x) < v+(x) since Ui(y)(i = 1, 2) are
monotone increasing in y. Thus we need only consider the case ς(x) < ξ(x). It
follows from the definitions of ς(x) and ξ(x) that

ς(x)− ξ(x) =
( 1√

1 + |∇ϕ(z′)|2
− c

s

)
(x3 + h(x′)) +

ϕ(z′)/α− h(x′)√
1 + |∇ϕ(z′)|2

=
1

s
S(z′)(x3 + h(x′)) +

ϕ(z′)/α− h(x′)√
1 + |∇ϕ(z′)|2

< 0.

Since 1√
1+|∇ϕ(z′)|2

− c
s > 0 and ν1 ≤ ϕ(z′)−h(z′)

S(z′) ≤ ν2, we have

x3 + h(x′) < −sϕ(z′)/α− h(x′)√
1 + |∇ϕ(z′)|2

1

S(z′)
≤ −cν1

α
< 0,

which implies that ς(x) < ξ(x) ≤ c
s (− cν1

α ) = − c
2ν1

αs < 0 and s
cη(x) < ς(x) < η(x) <

0. Then we have

v+
i (x)− v−i (x) ≥ Ui(ν(x)) + εS(z′)Uβi2 (η(x))− Ui(ξ(x))

=
( 1√

1 + |∇ϕ(z′)|2
− c

s

)
(x3 + h(x′))U ′i

(
θiν(x) + (1− θi)ξ(x)

)



EJDE-2020/112 PYRAMIDAL TRAVELING FRONTS 13

+ εS(z′)Uβi2 (η(x))

=
1

s
S(z′)ξ(x)U ′i (θiν(x) + (1− θi)ξ(x)) + εS(z′)Uβi2 (η(x))

for some θi ∈ (0, 1), i = 1, 2. Since x3 + h(x′) < 0, we have ν(x) < ς(x) < ξ(x) <
η(x) < 0, and hence

U ′i(θiν(x) + (1− θi)ξ(x)) ≤

{
L2e

min{λ1,2λ2}ξ(x), i = 1,

L2e
λ2ξ(x), i = 2,

Uβi2 (η(x)) ≥ Lβi1 e
λ1βiη(x) ≥ L1e

λ1βiξ(x), i = 1, 2.

Then for i = 2, we have

v+
2 (x)− v−2 (x) ≥ 1

s
S(z′)ξ(x)U ′2 (θ2ν(x) + (1− θ2)ξ(x)) + εS(z′)Uβ2

2 (η(x))

≥ S(z′)
(L2

s
ξ(x)eλ2ξ(x) + εL1e

λ1β2ξ(x)
)

≥ S(z′)eλ1β2ξ(x)
(L2

s
ξ(x)e(λ2−λ1β2)ξ(x) + εL1

)
≥ S(z′)eλ1β2ξ(x)

( L2

s(λ2 − λ1β2)2ξ(x)
sup
ω>0

ω2e−ω + εL1

)
≥ S(z′)eλ1β2ξ(x)

(
− 4L2α

c2e2(λ2 − λ1β2)2ν1
+ εL1

)
> 0,

provided that

α < α4 := min
{εL1c

2e2(λ2 − β2λ1)2ν1

4L2
,

εL1c
2e2 (min{λ1, 2λ2} − β1λ1)

2
ν1

4L2

}
.

A similar argument will lead to v+
1 (x)− v−1 (x) > 0 for the above α and we omit it.

Thus we have that v+(x) > v−(x) for all x ∈ R3.
Next, we prove (3.5). By Lemma 2.1 we know that for each fixed α, there exits

a positive constant mα = 1
α2πm∗

∫∞
0
r2ρ̃(r)dr such that

ξ(x) ≤ η(x) ≤ ξ(x) +mα, ∀x ∈ R3. (3.11)

Recall N1 := supx∈R

∣∣∣U ′2(x)
U2(x)

∣∣∣. Since U2(x + y)e−N1y is decreasing in y, we know

U2(x+ y)e−N1y ≤ U2(x) for any y ≥ 0. Using this fact and (3.11), we can get

U2(ξ(x)) ≤ U2(η(x)) ≤ U2(ξ(x))eN1mα ,∀x ∈ R3.

In other words,

1 ≤ U2(η(x))

U2(ξ(x))
≤ eN1mα , ∀x ∈ R3. (3.12)

According to the definition of v+(x) and v−(x), and using (3.12), it is sufficient to
prove that

lim
γ→+∞

sup
x∈D(γ)

|Ui(ς(x))− Ui(ξ(x))|
(U2(η(x)))βi

= 0, i = 1, 2.

We consider three cases.

Case 1: ξ(x) = c
s (x3 + h(x′)) → +∞. Then we have 0 < ξ(x) < ς(x) and

η(x) < ς(x) < s
cη(x), which implies that

|Ui(ς(x))− Ui(ξ(x))|
(U2(η(x)))βi

→ 0 as ξ(x)→ +∞, i = 1, 2.



14 L. MA, H.-T. NIU, Z.-C. WANG EJDE-2020/112

Case 2: ξ(x) = c
s (x3 + h(x′))→ −∞. We have

ς(x) =
1√

1 + |∇ϕ(αx′)|2
s

c
ξ(x) +

ϕ(αx′)/α− h(x′)√
1 + |∇ϕ(αx′)|2

.

Since 0 < ϕ(αx′)/α− h(x′) ≤ mα and c
s <

1√
1+|∇ϕ(αx′)|2

≤ 1, it follows that

s

c
ξ(x) ≤ ς(x) ≤ ξ(x) +mα, (3.13)

which implies ς(x)→ −∞ as ξ(x)→ −∞ and vice versa. Thus, we know

ξ(x) ≤ η(x) ≤ c

s
ς(x) <

c

s
η(x) < 0.

Using [30, Lemma 1.1], we have

U ′1(ξ)

U2(ξ)
=

λ1e
λ1ξ +O(e(2λ2−σ)ξ)

Aλ2eλ2ξ +O(e(λ1−σ)ξ)

=
λ1e

(λ1−λ2)ξ +O(e(λ2−σ)ξ)

Aλ2 +O(eλ1−λ2−σ)ξ)
→ 0 as ξ → −∞.

Thus we know C := supξ≤0,i=1,2
U ′i(ξ)
U2(ξ) < +∞. Then by (3.11), (3.13) and (2.3), we

have

|Ui(ς(x))− Ui(ξ(x))|
(U2(η(x)))βi

≤ U ′i(θiς(x)) + (1− θi)ξ(x))

(U2(η(x)))
βi

|ς(x)− ξ(x)|

≤ U ′i(θiς(x)) + (1− θi)ξ(x))

(U2(η(x)))βi

(s
c
|η(x)|+ |η(x)|+mα

)
≤ U ′i(θiς(x)) + (1− θi)ξ(x))

U2(θiς(x)) + (1− θi)ξ(x))

U2(η(x))

(U2(η(x)))βi

((s
c

+ 1
)
|η(x)|+mα

)
≤ CL2e

(1−βi)λ2η(x)
((s

c
+ 1
)
|η(x)|+mα

)
→ 0 as η(x)→ −∞.

And notice that η(x)→ −∞ is equivalent to ξ(x)→ −∞ by (3.11).

Case 3: ξ(x) = c
s (x3 + h(x′)) is bounded. It is obvious by (3.11) that η(x) is

bounded in this case. Suppose R0 > 0 is a constant such that |ξ(x)| ≤ R0. For
each γ > 2R0 and x ∈ D(γ), there must hold dist(x′, E) > γ and thus

lim
γ→+∞

sup
dist(x′,E)>γ

(ϕ(x′)− h(x′)) = 0, lim
γ→+∞

sup
dist(x′,E)>γ

S(x′) = 0,

and hence

lim
γ→+∞

sup
x∈D(γ)

|Ui(ς(x))− Ui(ξ(x))|
(U2(η(x)))βi

= 0, i = 1, 2.

Finally, let α+
0 (ε,β) = min{ε, α1, α2, α3, α4} and

ε+
0 (β) = min

{ ε1

p1(s− c)
, min
i=1,2
{− qi

8(s− c)bp1p2
}, ε2

}
.

The proof is complete. �



EJDE-2020/112 PYRAMIDAL TRAVELING FRONTS 15

Now we give the existence result for traveling curved fronts.

Theorem 3.2. For each s > c, (1.2) admits a pyramidal traveling front u(x, t) =
V(x′, x3 + st). V(x) satisfies (1.6) with ∂x3V(x) > 0 and

v−(x) < V(x) < v+(x; ε,β, α), ∀x ∈ R3,

where 0 < ε < ε+
0 (β) and 0 < α < α+

0 (ε,β). Furthermore, we have

lim
γ→+∞

sup
x∈D(γ)

|Vi(x)− v−i (x)|
(v−2 (x))βi

= 0, i = 1, 2. (3.14)

Proof. According to the parabolic estimates, we know that there exists a constant
C > 0 such that the solution v(x; t) of (1.4) and (1.5) with v0(x) ∈ [0,1] satisfies

‖v(·, t; v0)‖C3(R3) < C, ∀ t ≥ 1.

Since v− is a subsolution, we have v(x, t1; v−) < v(x, t2; v−) for all x ∈ R3 and
0 < t1 < t2, see [33] for more details. Thus,

V(x) := lim
t→+∞

v(x, t; v−), x ∈ R3, (3.15)

is well defined and independent of ε, α, and β. It follows that v(·, t; v−) converges
monotonically to V(·) under the norm ‖ · ‖C2

loc(R3), namely,

lim
t→+∞

‖v(·, t; v−)−V(·, ·)‖C2
loc(R3) = 0.

By the comparison principle, we know v−(x) < V(x) < v+(x; ε,β, α). And the
proof of (3.14) is similar to that of [44]. In view of the monotonicity of v−(x) on
the variable x3, we come to the conclusion that ∂x3V(x) ≥ 0 for all x ∈ R3. Then
the strong maximum principle implies the strict inequality. �

4. Stability of traveling curved fronts

This section discusses the stability of the pyramidal traveling fronts constructed
in Section 3 by improving the arguments of Taniguchi [35] and Wang [43, 45].
Consider the Cauchy problem

∂

∂t
w̃(ξ, η, t)−∆w̃(ξ, η, t) + s̄

∂

∂η
w̃(ξ, η, t)− F(w̃) = 0,

w̃(ξ, η, 0) = w̃0(ξ, η),
(4.1)

where w̃(ξ, η, t) = (w̃1(ξ, η, t), w̃2(ξ, η, t)) and (ξ, η) ∈ R2, t > 0. The following
theorem is established in [30, 31].

Theorem 4.1. Assume b > 0 and r > 1. Then for each s̄ > c, there exists a steady
state Φ(ξ, η; s̄) of (4.1) satisfying Φ(ξ, η; s̄) > ṽ−(ξ, η) and

lim
R→∞

sup
ξ2+η2>R2

∣∣Φi(ξ, η)− ṽ−i (ξ, η)

(ṽ−2 (ξ, η))βi

∣∣ = 0, i = 1, 2,

where

ṽ−(ξ, η) = U
( c
s̄

(
η +

√
s̄2 − c2
c

|ξ|
))
,



16 L. MA, H.-T. NIU, Z.-C. WANG EJDE-2020/112

and βi(i = 1, 2) is defined as in (3.4). Furthermore, for any w̃0(ξ, η) ∈ [0,1] with
w̃0 ∈ C(R2,R2) and

lim
R→∞

sup
ξ2+η2>R2

∣∣ ṽ−i (ξ, η)− w̃0i(ξ, η)

(ṽ−2 (ξ, η))βi

∣∣ = 0, i = 1, 2,

the solution w̃(ξ, η, t; w̃0) of (4.1) with initial data w̃0 satisfies

lim
t→∞

∥∥wi(·, ·, t; w̃0)− Φi(·, ·+ st)

(ṽ−2 (·, ·))βi
∥∥
L∞(R2)

= 0, i = 1, 2.

For any subset D ∈ R3, we denote the characteristic function of D by χD, namely,

χD(x) =

{
1, x ∈ D,
0, x ∈ Dc,

where Dc denotes the complementary set of D. Let hij(x, t) ∈ C(R3×R) (i, j = 1, 2)
be given continuous functions satisfying

0 ≤ hij(x, t) ≤Mij , i 6= j; sup
x∈R3,t>0

|hii(x, t)| ≤Mii,

where Mij ∈ R (i, j = 1, 2) are constants. Now consider the linear system

Lt[wi(x, t)]−
2∑
j=1

hij(x, t)wj(x, t) = 0, x ∈ R3, t > 0,

wi(x, 0) = wi,0(x), x ∈ R3, i = 1, 2,

(4.2)

where Lt := ∂
∂t −

∑3
k=1

∂2

∂x2
k

+ s ∂
∂x3

is a linear operator. We have the following

result.

Lemma 4.2. Let w(x, t) = (w1(x, t), w2(x, t)) be a solution of (4.2). Then there

exist positive constants Ã, B̃ and λ0 such that

sup
i=1,2

sup
x∈D(2γ)

|wi(x, t)|
(v−2 (x))βi

≤ 6eλ0t
Ãπ

B̃

∫ +∞

√
3γ

3t −
N1
2B̃

e−B̃r
2

drmax
i=1,2

sup
x∈D(γ)c

|wi,0(x)|
(v−2 (x))βi

+ eλ0tÃ
( π
B̃

)3/2

max
i=1,2

sup
x∈D(γ)

|wi,0(x)|
(v−2 (x))βi

, ∀ t > 0

(4.3)

for any γ > 0, where D(γ)c is the complementary set of D(γ). Moreover, we have

sup
i=1,2

sup
x∈R3

|wi(x, t)|
(v−2 (x))βi

≤ eλ0tÃ
( π
B̃

)3/2

max
i=1,2

sup
x∈R3

|wi,0(x)|
(v−2 (x))βi

, ∀t > 0. (4.4)

Proof. Let ŵ(x, t) = e−λ
′
0tw(x, t), where λ′0 =

∑2
i=1Mii. Then ŵ(x, t) satisfies

Ltŵi(x, t) + (λ′0 − hii(x, t))ŵi(x, t)− hij(x, t)ŵj(x, t) = 0, x ∈ R3, t > 0, j 6= i,

ŵi,0(x) = wi,0(x), x ∈ R3.

Now consider a linear system with constant coefficients

Ltw̃i(x, t) + (λ′0 −Mii)w̃i(x, t)−Mijw̃j(x, t) = 0, x ∈ R3, t > 0, j 6= i,

w̃i,0(x) = |wi,0(x)|, x ∈ R3.
(4.5)

A similar discussion as [45, Lemma 4.2] implies that

|ŵi(x, t)| ≤ w̃i(x, t), x ∈ R3, t > 0, i = 1, 2.



EJDE-2020/112 PYRAMIDAL TRAVELING FRONTS 17

By [7, Theorems 2 and 3, Chapter 9], we know that there exists a 2 × 2 matrix
function Γ(x,y, t, s) = Γ(x− y, t− s) such that

w̃(x, t) =

∫
R3

Γ(x− y, t) · w̃0(y)dy,

and there exist positive numbers à ≥ 1 and B̃ ≤ 1 such that

|Γ(x− y, t− s)| =
2∑

i,j=1

|Γij | ≤ Ã(t− s)−3/2 exp{−B̃ |x− y|2

t− s
}

for any 0 ≤ s < t ≤ 2. By the uniqueness of solutions, the solution of (4.5) can be
written as

w̃(x, t) =

∫
R3

Γ(x− y1, 1)dy1

∫
R3

Γ(y1 − y2, 1)dy2 · · ·∫
R3

Γ(yk−1 − yk, 1)dyk

∫
R3

Γ(yk − y, t− k) · w̃0(y)dy

for any t > 0, where k = max{[t − 1], 0} and [t] represents the largest integer no
more than t. Therefore

w̃i(x, t)

(v−2 (x))βi
=

∫
R3

Γi(x− y1, 1)dy1

∫
R3

Γ(y1 − y2, 1)dy2 · · ·
∫
R3

Γ(yk−1 − yk, 1)dyk∫
R3

Γ(yk − y, t− k) ·
(
χD(γ)c(y)

w̃0(y)

(v−2 (x))βi
+ χD(γ)(y)

w̃0(y)

(v−2 (x))βi

)
dy

=

∫
R3

Γ(x− y1, 1)dy1

∫
R3

Γ(y1 − y2, 1)dy2 · · ·
∫
R3

Γ(yk−1 − yk, 1)dyk∫
R3

Γ(yk − y, t− k) · χD(γ)c(y)
w̃0(y)

(v−2 (x))βi
dy

+

∫
R3

Γ(x− y1, 1)dy1

∫
R3

Γ(y1 − y2, 1)dy2 · · ·
∫
R3

Γ(yk−1 − yk, 1)dyk∫
R3

Γ(yk − y, t− k) · χD(γ)(y)
w̃0(y)

(v−2 (x))βi
dy

=: I + II

for any t > 0 and i = 1, 2. Then by a same computation as in [45, Lemma 4.2], we
have

|I| ≤ Ãk+1(1 + t)3/2
(∫

R3

e−
B̃|z|2

2 dz
)k ∫

R3

t−3/2e−B̃
|x−y|2

t χD(γ)c(y)
|w̃0(y)|

(v−2 (x))βi
dy

≤ Ãk+1e2t
(2π

B̃

)3k/2
∫
D(γ)c

t−3/2e−B̃
|y|2
t

(v−2 (x− y))βi

(v−2 (x))βi
|w̃0(x− y)|

(v−2 (x− y))βi
dy.

Moreover, since U2(x+y)e−N1y is decreasing in y, we know U2(x+y)e−N1y ≤ U2(x)
for any y ≥ 0. With the help of this fact, we have

v−2 (x− y)

v−2 (x)
=
U2

(
c
s (x3 − y3 + h(x′ − y′))

)
U2

(
c
s (x3 + h(x′))

)
≤
U2

(
c
s

(
x3 + h(x′) + max{1,m∗}Σ3

i=1|yi|
))

U2

(
c
s (x3 + h(x′))

)
≤ eN1

c
s max{1,m∗}Σ3

i=1|yi|



18 L. MA, H.-T. NIU, Z.-C. WANG EJDE-2020/112

≤ eN1Σ3
i=1|yi|.

Note that if x ∈ D(2γ), then γ ≤ |dist(x,Γ)−dist(y,Γ)| ≤ |x−y| for all y ∈ D(γ)c.
Therefore,

|I| ≤ Ck,tt−3/2

∫
B(x,γ)c

e−B̃
|x−y|2

t eN1
∑3
i=1 |xi−yi|dy max

i=1,2
sup

y∈D(γ)c

|w̃i,0(y)|
(v−2 (y))βi

= Ck,tt
−3/2e

3tN2
1

4B̃

∫
B(x,γ)c

e−
B̃
t

∑3
i=1(|yi−xi|−

tN1
2B̃

)
2

dy max
i=1,2

sup
y∈D(γ)c

|w̃i,0(y)|
(v−2 (y))βi

≤ Ck,tt−3/2e
3tN2

1
4B̃ 3

∫
y∈R3,|y1|≥

√
3

3 γ− tN1
2B̃

e−
B̃
t |y|

2

dy max
i=1,2

sup
y∈D(γ)c

|w̃i,0(y)|
(v−2 (y))βi

= 6Ck,te
3tN2

1
4B̃

Ãπ

B̃

∫ +∞

√
3γ

3t −
N1
2B̃

e−B̃r
2

dr · max
i=1,2

sup
y∈D(γ)c

|w̃i,0(y)|
(v−2 (y))βi

,

where Ck,t = Ãk+1e2t
(

2π

B̃

)3k/2
. Similarly,

|II| ≤ Ãk+1(1 + t)3/2
(∫

R3

e−
B̃|z|2

2 dz
)k

×
∫
R3

t−3/2e−B̃
|x−y|2

t eN1
∑3
i=1 |xi−yi|χD(γ)(y)

|w̃0(y)|
(v−2 (x))βi

dy

≤ Ck,tt−3/2e
3tN2

1
4B̃

∫
R3

e−B̃
|x−y|2

t χD(γ)(y)dy max
i=1,2

sup
y∈D(γ)

|w̃i,0(y)|
(v−2 (y))βi

≤ Ck,t
(
π

B̃

)3/2

e
3tN2

1
4B̃ max

i=1,2
sup

y∈D(γ)

|w̃i,0(y)|
(v−2 (y))βi

.

Note that 0 < t − k < 2 and let λ0 := λ′0 + 2 + ln
( 2
√

2π
√
πÃ

B̃
√
B̃

)
+

3N2
1

4B̃
, then (4.3)

follows. To prove the inequality (4.4), we just need to replace D(γ) by R3 in the
term II. Then the proof is complete. �

By a proper coordinate change, we show that the pyramidal traveling front V(x)
converges to a two dimensional V-form front on edges of the pyramid at infinity.
For each positive integer j ∈ {1, 2, . . . , n}, we consider a plane perpendicular to an
edge Γj = Sj∩Sj+1. Then the cross section of −x3 = max{hj(x′), hj+1(x′)} in this
plane is V-shaped. Let Vj be the two dimensional V-form front as in Theorem 4.1
corresponding to the cross section of −x3 = max{hj(x′), hj+1(x′)}. We now make
some preparations before giving the formulation of Vj .

It is easy to see that the expression of Γj is

x1

Bj −Bj+1
=

x2

Aj+1 −Aj
=

x3

m∗(AjBj+1 −Aj+1Bj)
, x3 < 0.

Define

pj := AjBj+1 −Aj+1Bj > 0, 1 ≤ j ≤ n;

qj :=
√

(Aj −Aj+1)2 + (Bj −Bj+1)2 > 0, 1 ≤ j ≤ n.



EJDE-2020/112 PYRAMIDAL TRAVELING FRONTS 19

Then the direction of Γj is given by

1√
m2
∗p

2
j + q2

j

(
Bj+1 −Bj , Aj −Aj+1,m∗(Aj+1Bj −AjBj+1)

)
and the traveling direction of the two dimensional V-form wave Vj is perpendicular
to the direction of Γj and given by

1

qj
√
m2
∗p

2
j + q2

j

(
m∗(Bj+1 −Bj)pj ,m∗(Aj −Aj+1)pj , q

2
j

)
.

Let sj be the speed of Vj and 2θj ∈ (0, π) be the angle between Sj and Sj+1. It is
not difficult to see that

sj sin θj = c, sin θj =

√
m2
∗p

2
j + q2

j

qj
√

1 +m2
∗
, sj =

sqj√
m2
∗p

2
j + q2

j

.

The speed of Vj toward the x3-axis equals

sj

√
m2
∗p

2
j + q2

j /qj = c
√

1 +m2
∗ = s,

which coincides with the speed of V. Now we define a matrix

Rj =


Aj+1−Aj

qj

m∗(Bj+1−Bj)pj
qj
√
m2
∗p

2
j+q

2
j

Bj+1−Bj√
m2
∗p

2
j+q

2
j

Bj+1−Bj
qj

m∗(Aj−Aj+1)pj

qj
√
m2
∗p

2
j+q

2
j

Aj−Aj+1√
m2
∗p

2
j+q

2
j

0
qj√

m2
∗p

2
j+q

2
j

− m∗pj√
m2
∗p

2
j+q

2
j


and make the following coordinate transformation:x1

x2

x3

 = Rj

ξη
ζ

 or

ξη
ζ

 = RT
j

x1

x2

x3

 .

Define Vj(x) := Φ(ξ, η; sj). Since Rj is an orthogonal matrix, the graph of Vj(x)
is the same as that of Φ(ξ, η; sj) except the position in the space. Direct calculations
show that Vj(x) satisfies (1.6) with speed sj for each j ∈ {1, 2, . . . , n}. Thus we
call Vj a planar V-form front corresponding to the edge Γj . Set

Qj := {x ∈ R3 : dist(x,Γ) = dist(x,Γj)}.

Then we have R3 = ∪nj=1Qj . Define V̂(x) = max1≤j≤n Vj(x). From the mono-

tonicity of Vj in x3, V̂(x) is strictly monotone in x3. In addition, V̂(x) has the
following properties.

Lemma 4.3. V̂(x) satisfies v−(x) < V̂(x) < V(x) for x ∈ R3 and

lim
γ→∞

sup
x∈D(γ)

|V̂i(x)− v−i (x)|
(v−2 (x))βi

= 0. (4.6)

Proof. By Theorem 4.1 we have

max
{
U
( c
s

(x3 + hj(x
′))
)
,U
( c
s

(x3 + hj+1(x′))
)}

< Vj(x), x ∈ R3.



20 L. MA, H.-T. NIU, Z.-C. WANG EJDE-2020/112

It follows that v−(x) = U
(
c
s (x3 + h(x′))

)
< V̂(x) for x ∈ R3. Moreover, taking

the left and right sides of the inequality

max
{

U
( c
s

(x3 + hj(x
′))
)
,U
( c
s

(x3 + hj+1(x′))
)}

< v−(x)

as initial values of (1.6) respectively, we obtain Vj(x) ≤ V(x) for x ∈ R3. Then

we have V̂(x) ≤ V(x) for x ∈ R3. Finally, (4.6) follows from (3.14). �

Let v(x, t; v0) be the solution of (1.4) with initial value v0 ∈ [0,1] which satisfies
(1.8). By Lemma 4.2 we have

max
i=1,2

sup
x∈D(γ)

|vi(x, t; v0)− Vi(x)|
(v−2 (x))βi

≤ 6eλ0t
πÃ

B̃

∫ +∞

√
3γ

3t −
N1
2B̃

e−B̃r
2

drmax
i=1,2

sup
x∈D(γ)c

|Vi(x)− vi,0(x)|
(v−2 (x))βi

+ eλ0tÃ
( π
B̃

)3/2

max
i=1,2

sup
x∈D(γ)

|Vi(x)− vi,0(x)|
(v−2 (x))βi

for any γ > 0 and t > 0. It follows that

lim
γ→+∞

sup
x∈D(γ)

|vi(x, t; v0)− Vi(x)|
(v−2 (x))βi

= 0,

which implies that

lim
γ→+∞

sup
x∈D(γ),x∈Qj

|vi(x, t; v0)− V ji (x)|
(v−2 (x))βi

= 0, (4.7)

lim
γ→+∞

sup
x∈D(γ)

|vi(x, t; v0)− V̂i(x)|
(v−2 (x))βi

= 0 (4.8)

for any fixed t > 0. Using Theorem 4.1 and (4.7), we obtain the following re-
sult through a similar discussion as Wang et al. [45, Proposition 4.5] or a slight
modification of the proof in [3, Proposition 1].

Proposition 4.4. Assume that v0 ∈ [0,1] satisfies (1.8). For any given ε > 0,
one can choose a T ∗ > 0 large enough such that

lim
R→∞

max
1≤j≤n

sup
|x|≥R,x∈Qj

|vi(x, t; v0)− V ji (x)|
(v−2 (x))βi

< ε (4.9)

for any fixed t ≥ T ∗.

Lemma 4.5. Assume that v0 ∈ [0,1] satisfies (1.8). Let V be defined by (3.15),
then for any given ε > 0 one can choose a T ∗ > 0 large enough such that

lim
R→∞

max
i=1,2

sup
|x|≥R

|vi(x, t; v0)− Vi(x)|
(v−2 (x))βi

< ε (4.10)

for any fixed t ≥ T ∗. In particular, one has

lim
R→∞

max
i=1,2

sup
|x|≥R

|V̂i(x)− Vi(x)|
(v−2 (x))βi

= 0. (4.11)



EJDE-2020/112 PYRAMIDAL TRAVELING FRONTS 21

Proof. For any ε > 0, by taking v0 = V in Proposition 4.4 we have

lim
R→∞

max
1≤j≤n

sup
|x|≥R,x∈Qj

|Vi(x)− V ji (x)|
(v−2 (x))βi

< ε.

Thus because of the arbitrariness of ε, we have

lim
R→∞

max
1≤j≤n

sup
|x|≥R,x∈Qj

|Vi(x)− V ji (x)|
(v−2 (x))βi

= 0. (4.12)

Then (4.11) follows from (4.12). And (4.10) follows from (4.12) and (4.9). The
proof is complete. �

Equality (4.11) shows that a pyramidal traveling front V converges to two di-
mensional V-form fronts Φ near edges. By a similar proof to that of Wang [45], we
can get the following lemma.

Lemma 4.6. Let V be defined by (3.15). Then for any δ ∈ (0, 1) we have

min
i=1,2

inf
V ji (x)∈[δ,1−δ]

∂

∂x3
V ji (x) > 0, j = 1, 2, 3, . . . , n;

min
i=1,2

inf
Vi(x)∈[δ,1−δ]

∂

∂x3
Vi(x) > 0.

Inequality (3.7) and Lemma 4.6 yield that for any M > 0, there exists a positive
C∗ such that

min
i=1,2

inf
| cs (x3+h(x′))|≤M

∂

∂x3
Vi(x) ≥ C∗, min

i=1,2
inf

|ς(x)|≤M

∂

∂x3
v+
i (x) ≥ C∗.

Then we can construct the following two supersolutions.

Lemma 4.7. Let V be as in (3.15). For any 0 < ϑ < min
{

mini=1,2

(
−

∏
i(βiλ2)

24λ2C1

)
, 1
}

,

there exists a positive constant κ small enough and a positive constant ρ = ρ(κ)
sufficiently large such that, for any δ ∈ (0, δ0) where

δ0 := min
{ 1

2(s− c+ 1)
,
ε1

p1
, min
i=1,2
{− qi

8bp1p2
}
}
,

the function

W+(x, t; δ) = V(x′, x3 + ξ + ρδ(1− e−κt)) + δe−κt
(
ω(θ)p + (1− ω(θ))Uβ(θ)

)
is a supersolution to (1.9), where θ = c

s (x3 +ξ+ρδ(1−e−κt)+ϕ(ϑx′)/ϑ) and ξ ∈ R
is a constant. See (2.4) for ε1.

Proof. By a direct computation, we have

L̃[W+]i

= ρδκe−κt∂x3Vi − δκe−κt[ω(θ)pi + (1− ω(θ))Uβi2 (θ)]

+
c

s
ρδ2κe−2κt

[(
pi − Uβi2 (θ)

)
ω′(θ) + βi(1− ω(θ))Uβi−1

2 (θ)U ′2(θ)
]

− δe−κt
{
ω′′(θ)

(
pi − Uβi2 (θ)

) 3∑
j=1

θ2
xj + ω′(θ)

(
pi − Uβi2 (θ)

) 3∑
j=1

θxjxj

− cω′(θ)
(
pi − Uβi2 (θ)

)
− 2βiω

′(θ)Uβi−1
2 (θ)U ′2(θ)

3∑
j=1

θ2
xj



22 L. MA, H.-T. NIU, Z.-C. WANG EJDE-2020/112

+ (1− ω(θ))
[
βi(βi − 1)Uβi−2

2 (θ)(U ′2(θ))2
3∑
j=1

θ2
xj + βiU

βi−1
2 (θ)U ′′2 (θ)

3∑
j=1

θ2
xj

+ βiU
βi−1
2 (θ)U ′2(θ)

3∑
j=1

θxjxj − cβiU
βi−1
2 (θ)U ′2(θ)

]}
− fi(W+)− gi(W+) + fi(V).

Then the rest of proof is almost the same as that of [30, Lemma 4.2], we omit it. �

By a similar argument to that of Lemma 4.7, we obtain the following lemma.

Lemma 4.8. There exists a positive κ constant small enough and a positive con-
stant ρ = ρ(κ) sufficiently large such that, for any δ ∈ (0, δ0) (δ0 is given in Lemma

4.7), ξ ∈ R and 0 < α < min
{
α+

0 (ε,β),mini=1,2

(
−

∏
i(βiλ2)

24C1λ2

)}
, the function

w+(x, t; δ) = v+(x′, x3 +ξ+ρδ(1−e−κt); ε,β, α)+δe−κt
(
ω(θ̂)p+

(
1−ω(θ̂)

)
Uβ(θ̂)

)
is a supersolution to (1.9), where θ̂ = c

s (x3 + ξ + ρδ(1 − e−κt) + ϕ(αx′)/α) and
ξ ∈ R is a constant.

Lemma 4.9. Let V be defined by (3.15). Then it satisfies

lim
R→∞

sup
|x3+h(x′)|≥R

∂x3
Vi(x)

(v−2 (x))βi
= 0, i = 1, 2. (4.13)

Proof. We split up the proof into two steps.

Step 1. We prove that

lim
R→∞

sup
x3+h(x′)≥R

∂x3
Vi(x)

(v−2 (x))βi
= 0, i = 1, 2.

It is sufficient to prove that limR→∞ supx3+h(x′)≥R ∂x3
Vi(x) = 0, since

lim
R→∞

sup
x3+h(x′)≥R

v−2 (x) = lim
R→∞

sup
x3+h(x′)≥R

U2

( c
s

(x3 + h(x′))
)

= 1.

Obviously, the assumption x3 + h(x′)→∞ implies that dist(x,Γ)→∞. It follows
that

lim
R→∞

sup
x3+h(x′)≥R

|V(x)− 1| = 0,

and thus limR→∞ supx3+h(x′)≥R |F(V(x))| = 0. Applying the interior Schauder’s
estimate to

−∆Vi + s∂x3
Vi = fi(V) in B(x0, 2), ∀x0 ∈ R3, i = 1, 2,

we have

lim
R→∞

sup
i=1,2
{‖Vi‖W 2,p(B(x0,1))

∣∣x0 ∈ R3, |x0
3 + h(x′0)| ≥ R} = 0

for p > 3. Therefore

lim
R→∞

sup
x3+h(x′)≥R

∂x3Vi(x) = 0, i = 1, 2.



EJDE-2020/112 PYRAMIDAL TRAVELING FRONTS 23

Step 2. We prove that

lim
R→∞

sup
x3+h(x′)≤−R

∂x3
Vi(x)

(v−2 (x))βi
= 0, i = 1, 2.

Note that the set {x ∈ R3|x3 + h(x′) ≤ −R} ⊆ D(R) = {x ∈ R3|dist(x,Γ) ≥ R}.
It follows from (3.14) that for i = 1, 2,

lim
R→∞

sup
x3+h(x′)≤−R

∣∣Vi(x)− v−i (x)
∣∣

(v−2 (x))βi
≤ lim
R→+∞

sup
x∈D(R)

|Vi(x)− v−i (x)|
(v−2 (x))βi

= 0.

Then a similar discussion as that in [30, Lemma 4.6] shows that (4.13) holds for
the case x3 + h(x′)→ −∞. The proof is complete. �

To prove the stability result, we also need the following lemma, which can be
referred to [45, Lemma 4.8].

Lemma 4.10. Assume that δ ∈ (0, ε1), where ε1 is in (2.4). For any x ∈ R3 with

δ ≤ V̂i(x) = max
1≤j≤n

V ji (x) ≤ 1− δ,

we have

inf
0<%<%0

V̂i(x
′, x3 + %)− V̂i(x)

%
≥ min

1≤j≤n
min
i=1,2

inf
δ
2≤V

j
i (x)≤1− δ2

∂

∂x3
V ji (x) > 0,

where %0 is a positive constant depending only on δ.

Let v+(x; ε,β, α) be as in Lemma 3.1. Define

V∗(x) := lim
t→∞

ṽ(x, t; v+), x ∈ R3.

Since v+ is a supersolution, V∗(x) is well defined and satisfies

−∆V∗ + sV∗x3
− F̃(V∗) = 0,

and it may depend on ε, α and β. By the comparison principle, we have

v−(x) < V̂(x) < V(x) ≤ V∗(x) < v+(x; ε,β, α), x ∈ R3.

From (3.5), we have

lim
γ→+∞

sup
x∈D(γ)

|V ∗i (x)− v−i (x)|
(v−2 (x))βi

= 0, i = 1, 2.

Then applying Proposition 4.4 to V∗ we obtain

lim
R→∞

max
i=1,2

sup
|x|≥R

|V̂i(x)− V ∗i (x)|
(v−2 (x))βi

= 0. (4.14)

It follows immediately that

lim
R→∞

max
i=1,2

sup
|x|≥R

|Vi(x)− V ∗i (x)|
(v−2 (x))βi

= 0. (4.15)

The next lemma says that V∗(x) is independent of ε, α and β.

Lemma 4.11. One has V(x) = V∗(x) in R3.



24 L. MA, H.-T. NIU, Z.-C. WANG EJDE-2020/112

Proof. Assume that V(x) 6≡ V∗(x). We take δ ∈ ( δ02 , δ0), where δ0 is given in
Lemma 4.7. By the definition of V∗ and (4.15), there exists λ > 0 sufficiently large
such that

V∗(x) ≤ V(x′, x3 + λ) + δ(v−2 (θ))βi

≤ V(x′, x3 + λ) + δ
(
ω(θ)p + (1− ω(θ))Uβ(θ)

)
,

where θ = c
s (x3 + λ+ ϕ(ϑx′)/ϑ). Then the comparison principle implies that

V∗(x) ≤W+(x′, x3 + λ, t; δ), x ∈ R3, t > 0.

Letting t→∞, we obtain

V∗(x) ≤ V(x′, x3 + λ+ ρδ), x ∈ R3.

Define
Λ := inf{λ > 0 : V∗(x) ≤ V(x′, x3 + λ), x ∈ R3}.

Then Λ ≥ 0, and V∗(x) ≤ V(x′, x3 + Λ) for all x ∈ R3. The assumption V(x) 6≡
V∗(x) implies that Λ 6= 0. Thus the strong maximum principle yields that either
V∗(x) ≡ V(x′, x3 + Λ) or V∗(x) < V(x′, x3 + Λ). We assert that the former
case is impossible. To see this, we choose a sequence {x′m}m∈N ⊆ R2 satisfying
h(x′m)→∞ and dist(x′m, E)→∞. Then by the fact that v− < V ≤ V∗ < v+, we
have

lim
m→∞

V∗(x′m,−h(x′m)) = U(0), lim inf
m→∞

V(x′m,−h(x′m) + Λ) ≥ U
( c
s

Λ
)
,

which contradicts V∗(x) ≡ V(x′, x3 + Λ). Now we assume that

V∗(x) < V(x′, x3 + Λ), x ∈ R3.

By Lemma 4.9, for any fixed ρ > 0 defined in Lemma 4.7, we can take a R∗ =
R∗(ρ) > 0 large enough such that

sup
|x3+h(x′)|≥R∗−Λ

2

∂x3
Vi(x)

(v−2 (x))βi
≤ 1

3ρ
, i = 1, 2.

Define
D := {x ∈ R3 : |x3 + h(x′)| ≤ R∗}.

Choose a constant σ satisfying 0 < σ < { δ02 ,
Λ
4ρ ,

ln 3
2

2N1ρ
}, where N1 = supx∈R3

U ′2(x)
U2(x) .

Using Lemma 4.10 for x ∈ D, we have

V̂i(x
′, x3 + Λ)− V̂i(x) ≥ min {%0,Λ} min

1≤j≤n
min
i=1,2

inf
δ∗
2 ≤V

j
i (x)≤1− δ∗2

∂

∂x3
V ji (x) > 0,

where

δ∗ = min
i=1,2

min
{δ0

2
, 1− max

1≤j≤n
sup
x∈D

V ji (x′, x3 + Λ), min
1≤j≤n

inf
x∈D

V ji (x))
}
,

and %0 is defined in Lemma 4.10 associated with δ∗. Thus, for x ∈ D, it follows
that

lim
R→∞

inf
|x|>R,x∈D

(Vi(x
′, x3 + Λ)− V ∗i (x))

≥ lim
R→∞

inf
|x|>R,x∈D

(
V̂i(x

′, x3 + Λ)− V ∗i (x)
)

≥ lim
R→∞

inf
|x|>R,x∈D

(
V̂i(x

′, x3 + Λ)− V̂i(x)
)

+ lim
R→∞

inf
|x|>R,x∈D

(
V̂i(x)− V ∗i (x)

)



EJDE-2020/112 PYRAMIDAL TRAVELING FRONTS 25

> 0,

since limR→∞ sup|x|>R,x∈D
(
V ∗i (x) − V̂i(x)

)
= 0 by (4.14). Thus we can choose a

small σ such that

Vi(x
′, x3 + Λ− 2ρσ) > V ∗i (x) in D.

In the domain R3 \ D, we have

Vi(x
′, x3 + Λ− 2ρσ)− Vi(x′, x3 + Λ)

Uβi2

(
c
s (x3 + Λ− 2ρσ + ϕ(αx′)/α)

)
≥ Vi(x

′, x3 + Λ− 2ρσ)− Vi(x′, x3 + Λ)(
v−2 (x′, x3 + Λ− 2ρσ)

)βi
≥
−2ρσ

∫ 1

0
∂x3

Vi(x
′, x3 + Λ− 2ρστ)dτ(

v−2 (x′, x3 + Λ− 2ρσ)
)βi

= −2ρσ
( v−2 (x′, x3 + Λ)

v−2 (x′, x3 + Λ− 2ρσ)

)βi ∫ 1

0

∂x3Vi(x
′, x3 + Λ− 2ρστ)

(v−2 (x′, x3 + Λ))βi
dτ

≥ −2ρσe2N1ρσβi

∫ 1

0

∂x3
Vi(x

′, x3 + Λ− 2ρστ)(
v−2 (x′, x3 + Λ− 2ρστ)

)βi dτ
≥ −2ρσ

3

2

1

3ρ
= −σ.

In other other words,

Vi(x
′, x3 + Λ) ≤ Vi(x′, x3 + Λ− 2ρσ) + σUβi2

( c
s

(x3 + Λ− 2ρσ + ϕ(αx′)/α)
)
.

Combining the above two cases, we obtain

V ∗i (x) ≤ Vi(x′, x3 + Λ− 2ρσ) + σUβi2

( c
s

(x3 + Λ− 2ρσ + ϕ(αx′)/α)
)
,

for all x ∈ R3. Then Lemma 4.7 and the comparison principle yield

V ∗i (x) ≤W+
i (x′, x3 + Λ− 2ρσ, t;σ), x ∈ R3, t > 0.

Letting t→∞, we have

V ∗i (x) ≤ Vi(x′, x3 + Λ− 2ρσ), x ∈ R3.

This contradicts the definition of Λ. Thus Λ ≡ 0. The proof is complete. �

Theorem 4.12. Assume b > 0 and r > 1. Fix a couple of β1, β2 ∈ (0, β∗) with
β2 < β1. Assume that v0 ∈ C(R3,R2) with v0 ∈ [0,1], v0(x) ≥ v−(x) for x ∈ R3

and

lim
γ→∞

sup
x∈D(γ)

|vi,0(x)− Vi,0(x)|
(v2(x))βi

= 0, i = 1, 2.

Then the solution of (1.2) with initial value v0 satisfies

lim
t→∞

∥∥vi(·, t; v0)− Vi(·)
(v2(x))βi

∥∥
L∞(R3)

= 0, i = 1, 2. (4.16)

Proof. Under the condition v0 ∈ [0,1], the solution ṽ(x, t; v0) of (1.9) and (1.10) is
also the solution of (1.4) and (1.5), namely, ṽ(x, t; v0) ≡ v(x, t; v0). For a random



26 L. MA, H.-T. NIU, Z.-C. WANG EJDE-2020/112

m > 1 and any given ε > 0, by Proposition 4.4, Lemma 4.5 and Lemma 4.11, we
can choose a positive constant R0 such that

Vi(x)− ε

m
(v−2 (x))βi ≤ ṽi(x, t; v−) ≤ Vi(x), |x| ≥ R0, i = 1, 2;

Vi(x) ≤ ṽi(x, t; v+) ≤ Vi(x) +
ε

m
(v−2 (x))βi , |x| ≥ R0, i = 1, 2,

for any fixed t ≥ T ∗. On the domain BR0(0) = {x||x| ≤ R0}, applying the interior
Shauder estimate to ṽi(x, t; v

−)− Vi(x) and ṽi(x, t; v
+)− Vi(x), and noticing that

v− is bounded on BR0
(0), we have

lim
t→∞

sup
x∈BR0

(0)

∣∣ ṽi(·, t; v−)− Vi(·)
(v2(x))βi

∣∣ = 0, lim
t→∞

sup
x∈BR0

(0)

∣∣ ṽi(·, t; v+)− Vi(·)
(v2(x))βi

∣∣ = 0

for i = 1, 2. Since m > 1 can be taken arbitrarily large, we have

lim
t→∞

∥∥ ṽi(·, t; v−)− Vi(·)
(v2(x))βi

∥∥
L∞(R3)

= 0, lim
t→∞

∥∥ ṽi(·, t; v+)− Vi(·)
(v2(x))βi

∥∥
L∞(R3)

= 0,

for i = 1, 2. Let δ ∈ (0, δ02 ) and ε < min{ε+
0 (β), δ04s}. Taking α ∈ (0, α+(ε,β)) small

enough and using Lemma 4.5, we have

vi(x, T
∗; v0) ≤ Vi(x) + δ(v−2 (x))βi ≤ v+

i (x) + δ(v−2 (x))βi .

A similar discussion as in [35] shows that

lim
t→∞

∥∥vi(x, t; v−)− Vi(x)

(v2(x))βi

∥∥
L∞(R3)

= 0,

lim
t→∞

∥∥vi(x, t; v0)− Vi(x)

(v2(x))βi

∥∥
L∞(R3)

= 0

for i = 1, 2. Take t̂ large enough such that

vi(x, t̂; v
−) ≤ vi(x, t̂; v+) < Vi(x) + δ(v−2 (x))βi , t ≥ t̂, i = 1, 2. (4.17)

Since w+(x, t; δ) is a supersolution, there exists a t̃ > 0 such that

vi(x, t+ T ∗ + 1; v0) ≤ Vi(x′, x3 + ρδ) + δe−λ0 t̂(v−2 (x))βi , t ≥ t̃, i = 1, 2.

Denote v+,δ(x) = v+(x′, x3 + ρδ). Then Lemma 4.2 implies that

vi(x, t̃+ T ∗ + t̂+ 1; v0)− v+,δ
i (x) ≤ δ(v−2 (x))βi , i = 1, 2.

Then by (4.17), we have

vi(x, t̃+ T ∗ + t̂+ 1; v0) ≤ Vi(x′, x3 + ρδ) + 2δ(v−2 (x))βi .

Thus it follows from Lemma 4.7 that

v(x, t+ t̃+ T ∗ + t̂+ 1; v0) ≤W+(x′, x3 + ρδ, t; 2δ).

Therefore, by letting t→∞ we have

Vi(x) ≤ vi(x, t; v0) ≤ Vi(x′, x3 + ρδ + 2ρδ) + 2δ(v−2 (x))βi ,

0 ≤ vi(x, t; v0)− Vi(x) ≤ Vi(x′, x3 + ρδ + 2ρδ)− Vi(x) + 2δ(v−2 (x))βi

≤ (v−2 (x))βi
( ∂x3Vi(x

′, x3 + 3ρδτ)

(v−2 (x′, x3 + 3ρδτ))βi
(v−2 (x′, x3 + 3ρδτ))βi

(v−2 (x))βi
+ 2δ

)
≤ (v−2 (x))βi

(
M∗e

c
sN13ρδ3ρ+ 2

)
δ,



EJDE-2020/112 PYRAMIDAL TRAVELING FRONTS 27

where τ ∈ (0, 1) and

M∗ = max
x∈R3

∂x3
Vi(x)

(v−2 (x))βi
.

Because of the arbitrariness of δ, (4.16) follows. Thus the proof is complete. �

Acknowledgements. This work is supported by the NNSF of China (11901330,
12071193).

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EJDE-2020/112 PYRAMIDAL TRAVELING FRONTS 29

Luyi Ma

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000,

China
Email address: maly16@lzu.edu.cn

Hong-Tao Niu (corresponding author)
School of Mathematics and Statistics, Xuzhou Institute of Technology, Xuzhou, Jiangsu

221000, China

Email address: aniuht@163.com

Zhi-Cheng Wang

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000,

China
Email address: wangzhch@lzu.edu.cn


	1. Introduction
	2. Preliminaries
	3. Existence of pyramidal traveling fronts
	4. Stability of traveling curved fronts
	Acknowledgements

	References