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Applied Mathematics Letters 22 (2009) 341–346

Contents lists available at ScienceDirect

Applied Mathematics Letters

journal homepage: www.elsevier.com/locate/aml

Travelling wavefronts of Belousov–Zhabotinskii system with diffusion

and delay$

Guo Lin∗, Wan-Tong Li

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China

a r t i c l ei n f o

Article history:

Received 17 January 2007

Received in revised form 1 April 2008

Accepted 25 April 2008

Keywords:

Travelling wavefront

Minimal wave speed

Asymptotic spreading

Comparison principle

Analytic semigroup

a b s t r a c t

This paper is concerned with the existence, nonexistence and minimal wave speed of the

travelling wavefronts of Belousov–Zhabotinskii system with diffusion and delay.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

In 1959, Belousov [2] proposed the so-called Belousov–Zhabotinskii system to model the chemical reaction, and one of

its simplified models takes the form as follows

and bromide ion, respectively, ∆ is the Laplacian operator on R. Model (1.1), in fact, was also derived in biochemical and

biological fields, see [10,11,21,22]. Recalling the chemical and biological backgrounds of (1.1), the following asymptotical

boundary conditions were proposed [5,6,17,20]

?

∂U(x,t)

∂t

∂V(x,t)

∂t

= ∆U(x,t) + U(x,t)[1 − U(x,t) − rV(x,t)],

= ∆V(x,t) − bU(x,t)V(x,t),

where x ∈ R,t > 0,r ∈ (0,1),b is a positive constant, and U,V ∈ R correspond to the concentration of bromic acid

(1.1)

lim

lim

x→−∞U(x,t) = 0,

x→∞U(x,t) = 1,

On the dynamics of (1.1) and (1.2), travelling wavefront, which takes the form of(U(x,t),V(x,t)) = (ρ(x+ct),?(x+ct))

for some wave speed c > 0 and monotone wave profile function (ρ,?), attracted much attention, see Murray [12],

Troy [17], Ye and Wang [20] and the references cited therein. Moreover, from the viewpoint of the chemical reaction, the

travelling wavefronts of (1.1) and (1.2) have significant sense, namely, the waves move from a region of higher bromic

lim

x→−∞V(x,t) = 1,

lim

x→∞V(x,t) = 0.

(1.2)

$Supported by NSFC (No. 10571078), NSF of Gansu Province of China (No. 3ZS061-A25-001).

∗Corresponding author.

E-mail address: ling02@lzu.cn (G. Lin).

0893-9659/$ – see front matter © 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.aml.2008.04.006

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G. Lin, W.-T. Li / Applied Mathematics Letters 22 (2009) 341–346

acid concentration to one of lower bromic acid concentration as it reduces the level of bromic ion (we can refer to Wu and

Zou [19]).

It is well known that time delay seems to be inevitable in many evolutionary processes, e.g., biological science [4],

therefore the time delay was incorporated into (1.1) by Wu and Zou [19], which takes the form as follows

wavefronts,see,forexample,Ma[8],andWuandZou[19].Inparticular,Ma[8]provedtheexistenceoftravellingwavefronts

of (1.3) with (1.2) by the upper and lower solution and Schauder’s fixed point theorem if the wave speed c > 2√

for the case of c ≤ 2√

this paper.

We first change the variables such that u = U and v = 1 − V, then (1.3) reduces to

lim limlim

∂U(x,t)

∂t

∂V(x,t)

∂t

= ∆U(x,t) + U(x,t)[1 − U(x,t) − rV(x,t − τ)],

= ∆V(x,t) − bU(x,t)V(x,t),

where τ ≥ 0 denotes a time delay. For model (1.3), some results have been established for the existence of travelling

(1.3)

1 − r. But

1 − r, the existence of travelling wavefronts of (1.3) remains open. This constitutes the purpose of

∂u(x,t)

∂t

∂v(x,t)

∂t

= ∆u(x,t) + u(x,t)[1 − r − u(x,t) + rv(x,t − τ)],

= ∆v(x,t) + bu(x,t)[1 − v(x,t)],

and we are interested in the following asymptotic boundary conditions (see (1.2))

(1.4)

x→−∞u(x,t) =

Let (u(x,t),v(x,t)) = (φ(x + ct),ψ(x + ct)) be the travelling wavefront of (1.4) and denote x + ct by ξ, then

(φ(ξ),ψ(ξ)),ξ ∈ R must satisfy

?

and the corresponding asymptotic boundary conditions as follows (see (1.5))

x→−∞v(x,t) = 0,

x→∞u(x,t) = lim

x→∞v(x,t) = 1.

(1.5)

cφ?(ξ) = φ??(ξ) + φ(ξ)[1 − r − φ(ξ) + rψ(ξ − cτ)],

cψ?(ξ) = ψ??(ξ) + bφ(ξ)[1 − ψ(ξ)],

(1.6)

lim

ξ→−∞φ(ξ) =

By the above notations, our main concern in this paper is to investigate the monotone nondecreasing solutions of (1.6)

and (1.7). In Section 2, we prove the existence of travelling wavefronts if c ≥ 2√

an approximation argument used in [3,16]. In Section 3, the nonexistence and minimal wave speed of (1.6) and (1.7)

will be proved by the theory of asymptotic spreading [7,16] and comparison principle for partial functional differential

equations [9]. This is probably the first time that the nonexistence of travelling wavefronts of (1.3) has been reported, even

for the case of τ = 0.

2. Existence of travelling wavefronts

lim

ξ→−∞ψ(ξ) = 0,

lim

ξ→∞φ(ξ) = lim

ξ→∞ψ(ξ) = 1.

(1.7)

1 − r by the method of Ma [8] and

In this section, we shall investigate the existence of monotone solution of (1.6) and (1.7). Throughout this paper, X will

be defined by

X = C(R,R2) =?

?

Then (1.6) can be rewritten as

?

For c > 0, define constants as follows

c −√

22

c −√

22

u(x)|u(x) : R → R2is uniformly continuous and bounded?,

H1(φ,ψ)(ξ) = 2φ(ξ) + φ(ξ)[1 − r − φ(ξ) + rψ(ξ − cτ)],

H2(φ,ψ)(ξ) = bψ(ξ) + bφ(ξ)[1 − ψ(ξ)].

which is a Banach space with the super norm. For (φ,ψ) ∈ X, denote (H1,H2) as follows

cφ?(ξ) = φ??(ξ) − 2φ(ξ) + H1(φ,ψ)(ξ),

cψ?(ξ) = ψ??(ξ) − bψ(ξ) + H2(φ,ψ)(ξ).

γ1(c) =

c2+ 8

,γ2(c) =

c +√

c +√

c2+ 8

,

γ3(c) =

c2+ 4b

,γ4(c) =

c2+ 4b

.

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Moreover, for (φ,ψ) ∈ X, we denote F = (F1,F2) by

F1(φ,ψ)(ξ) =

γ2(c) − γ1(c)

F2(φ,ψ)(ξ) =

γ4(c) − γ3(c)

which is similar to that in [8,19]. Then, it is sufficient to consider the fixed point of the operator F in the space X.

Furthermore, define constants as follows

c∗= 2√

2

The following existence result is established in Ma [8].

1

??ξ

??ξ

−∞

eγ1(c)(ξ−s)+

?∞

?∞

ξ

eγ2(c)(ξ−s)

?

?

H1(φ,ψ)(s)ds,

1

−∞

eγ3(c)(ξ−s)+

ξ

eγ4(c)(ξ−s)

H2(φ,ψ)(s)ds,

1 − r,λ∗(c) =

c −

?

c2− 4(1 − r)

.

Theorem 2.1. Let c > c∗be true such that be−λ∗(c)cτ≤ 1 − r. Then (1.6) and (1.7) have a monotone solution (φ(ξ),ψ(ξ)),

which is a travelling wavefront of (1.4) and (1.5).

In addition, from the definition of F, the following result is clear.

Lemma 2.2. For any c ∈ (c∗,c∗+ 1), (φ,ψ) formulated by Theorem 2.1 are equicontinuous.

Theorem 2.3. Assume that c = c∗and be−λ∗(c)cτ< 1 − r. Then (1.6) has a monotone solution (φ(ξ),ψ(ξ)) such that

lim

lim

ξ→∞(φ(ξ),ψ(ξ)) = (1,1),

for some constant α ∈ [0,1].

Proof. We prove this by an approximation method used in [3,16]. Since be−λ∗(c∗)c∗τ< 1 − r, there exists a δ1> 0 such

that be−λ∗(c)cτ≤ 1 − r for c ∈ (c∗,c∗+ δ1). Let δ = min{δ1,1}. Choose a sequence {cn}∞

cn→ c∗as n → ∞. Then Theorem 2.1 implies that (1.6) and (1.7) have a monotone solution(φn(ξ),ψn(ξ)) with c = cn, so

(φn(ξ),ψn(ξ)) is a fixed point of the operator F withγi(c) = γi(cn). Since such a travelling wavefront is invariant under the

sense of phase shift, then we can assume that φn(0) =

equicontinuous.

By Ascoli–Arzela lemma and a nested subsequence argument, there exists a subsequence of (φn(ξ),ψn(ξ)) which

converges uniformly on every compact subset of R, and hence pointwise on R to a vector function (φ(ξ),ψ(ξ)) ∈ X.

According to the Lebesgue’s dominant convergence theorem, (φ(ξ),ψ(ξ)) is a fixed point of the operator F with γi(c) =

γi(c∗). Hence,(φ(ξ),ψ(ξ)) satisfies (1.6) with c = c∗. Moreover, the monotonicity of(φn(ξ),ψn(ξ)) andφn(0) =

that (φ(ξ),ψ(ξ)) is nondecreasing and φ(0) =

We now consider the asymptotic behavior of (φ(ξ),ψ(ξ)). In fact, since (φ(ξ),ψ(ξ)) is nondecreasing and bounded,

then

ξ→±∞(φ??(ξ),ψ??(ξ)) =

Combining this with (1.6), then there exists constants φ±,ψ±such that

lim

(0,0) ≤ (φ±,ψ±) ≤ (1,1).

ξ→−∞(φ(ξ),ψ(ξ)) = (0,α)

(2.1)

n=1with cn ∈ (c∗,c∗+ δ) and

1

2for all n. Furthermore, Lemma 2.2 indicates that (φn(ξ),ψn(ξ)) is

1

2implies

1

2.

limlim

ξ→±∞(φ?(ξ),ψ?(ξ)) = (0,0).

ξ→±∞(φ(ξ),ψ(ξ)) = (φ±,ψ±),

Then φ(0) =

φ±(1 − r − φ±+ rψ±) = 0,

We now prove (2.1) in three cases.

(i) Since φ+∈ [1

(ii) If φ−= 0, then there exists some α ∈ [0,1] such that ψ−= α.

(iii) If φ−∈ (0,1

1, which is a contradiction.

It is clear that (i)–(iii) imply (2.1). The proof is complete.

1

2implies that φ−∈ [0,1

2],φ+∈ [1

φ±(1 − ψ±) = 0.

2,1] and the following equalities

(2.2)

2,1], then ψ+= 1 and φ+= 1 are obvious by (2.2).

2], then φ−(1 − ψ−) = 0 indicates that ψ−= 1 while φ−(1 − r − φ−+ rψ−) = 0 means that φ−= 0 or

?

Remark 2.4. In Theorem 2.3, we can only prove a weaker asymptotic boundary condition (2.1) than that of (1.7) since(0,α)

is the equilibrium of (1.6) for any α ∈ R. We conjecture α = 0 in (2.1), and we shall further investigate the problem in our

forthcoming research.

Remark 2.5. Although we prove Theorem 2.3 by a method similar to that of [7], their results cannot be applied directly

since (1.6) has infinite constant equilibrium states.

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3. Nonexistence and minimal wave speed

In this section, we will prove that (1.6) and (1.7) have no positive solution (do not require the monotonicity) in the sense

of functional if c < c∗holds.

We first consider the classical Fisher equation

Y = C(R,R) = {u(x) : u is uniformly continuous and bounded for all x ∈ R},

then it is clear that Y is a Banach space with the super norm. By the theory of asymptotic spreading of reaction-diffusion

equation, which was earlier proposed in Aronson and Weinberger [1] and recently developed by [7,16,18], the following

result is true.

∂w(x,t)

∂t

w(x,0) = w(x),

= ∆w(x,t) + dw(x,t)

?

1 −w(x,t)

K

?

,

(3.1)

where all the constants are positive and w(x) ∈ Y which is defined by

Lemma 3.1. Assume that w(x) ≥ 0 for x ∈ R, w(x) = 0 outside a bounded interval of R and 0 < w(x) < K on a nonempty

subset of R. Let w(x,t) be defined by (3.1), then

(i) limt→∞sup|x|>ctw(x,t) = 0 holds for any given c > 2√

(ii) limt→∞inf|x|<ctw(x,t) = K holds for any given c ∈

Moreover, for t ≥ 0 and w(x) ∈ Y, define T(t) as follows

T(t)w(x) =

4πt

R

then T(t) : Y → Y is an analytic semigroup for t ≥ 0 [13,15]. Thus the following result is obvious by the theory of abstract

functional differential equations [9] (we also refer to Smith and Zhao [14] for the delayed reaction-diffusion equation on R).

d;

?

0,2√

d

?

.

1

√

?

e−(x−y)2

4t w(y)dy,

Lemma 3.2. Assume that w(x,t) ∈ Y for all t ∈ [0,t?). If w(x,0) ≥ w(x) and

?t

for any 0 ≤ s < t < t?. Then w(x,t) ≥ w(x,t) holds for all (x,t) ∈ R × [0,t?).

We now consider the initial value problem

mild solution of (3.2) in the space X is formulated as follows.

w(x,t) ≥ T(t)w(x,s) +

s

T(t − θ)

?

dw(x,θ)

?

1 −w(x,θ)

K

??

dθ,

x ∈ R,

∂u(x,t)

∂t

∂v(x,t)

∂t

u(x,0) = u(x),v(x,s) = z(x,s),

with (u(·),z(·,s)) ∈ X for all s ∈ [−τ,0]. By the theory of the abstract functional differential equation [9], the existence of

= ∆u(x,t) + u(x,t)[1 − r − u(x,t) + rv(x,t − τ)],

= ∆v(x,t) + bu(x,t)[1 − v(x,t)],

x ∈ R, s ∈ [−τ,0],

x ∈ R, t > 0,

x ∈ R, t > 0,

(3.2)

Lemma 3.3. Assume that 0 ≤ u(x),z(x,s) ≤ 1 for any x ∈ R,s ∈ [−τ,0]. Then (3.2) has a unique mild solution

(u(x,t),v(x,t)) defined for all (x,t) ∈ R × (0,∞). Moreover, 0 ≤ u(x,t),v(x,t) ≤ 1 for all (x,t) ∈ R × (0,∞) and

takes the form as follows

Lemma 3.4. Assume that the initial value of (3.2) satisfies

(I) 0 < u(x),z(x,s) < 1 if x ∈ (−1,1),s ∈ [−τ,0],

(II) u(x) = z(x,s) = 0 if |x| ≥ 1,s ∈ [−τ,0].

Then limt→∞inf|x|<ctu(x,t) ≥ 1 − r holds for any given c ∈ (0,c∗).

u(x,t) = T(t)u(x) +

?t

0

T(t − s){u(x,s)[1 − r − u(x,s) + rv(x,s − τ)]}ds,

?t

v(x,t) = T(t)z(x,0) +

0

T(t − s){bu(x,s)[1 − v(x,s)]}ds.

(3.3)

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Note that T(t)u(x) ≥ 0 if u(x) ≥ 0, then (3.3) and Lemma 3.3 imply that

?t

for all (x,t) ∈ R × (0,∞). Therefore, Lemma 3.4 is a direct consequence of Lemmas 3.1–3.3, so we omit its proof here (we

can also refer to Smith and Zhao [14]).

u(x,t) ≥ T(t)u(x) +

0

T(t − s){u(x,s)[1 − r − u(x,s)]}ds

Lemma 3.5. Assume that (u1(·),z1(·,s)) ∈ X for s ∈ [−τ,0] and

0 ≤ u1(x) ≤ u(x) ≤ 1,

for all x ∈ R,s ∈ [−τ,0]. Let (u1(x,t),v1(x,t)) and (u(x,t),v(x,t)) be the mild solutions of (3.2) with initial values (u1,z1)

and (u,z), respectively. Then

0 ≤ z1(x,s) ≤ z(x,s) ≤ 1

0 ≤ u1(x,t) ≤ u(x,t) ≤ 1,

0 ≤ v1(x,t) ≤ v(x,t) ≤ 1,(x,t) ∈ R × (0,∞).

Lemma 3.5 is clear by Martin and Smith [9], and the proof is omitted here.

Theorem 3.6. Assume that c < c∗holds. Then (1.6) and (1.7) have no monotone solution.

Proof. We argue it by contradiction, were the stated conclusion false, then there exists a constant c1 ∈ (0,c∗) such that

thereexists(φ(ξ),ψ(ξ))satisfying(1.6)and(1.7)withc = c1andξ = x+c1t.Notethatthetravellingwavefrontisinvariant

in the sense of phase shift, so (φ(ξ + h),ψ(ξ + h)) also satisfies (1.6) and (1.7) with c = c1and ξ = x + c1t. Assume that

the initial value of (3.2) satisfies the conditions (I)–(II) in Lemma 3.4, then we can always choose h > 0 sufficiently large

such that

0 ≤ u(x) ≤ φ(x + h) ≤ 1,

hold for all x ∈ R,s ∈ [−τ,0]. Let c =

0 ≤ u(x,t) ≤ φ(x + c1t + h) ≤ 1,0 ≤ v(x,t) ≤ ψ(x + c1t + h) ≤ 1

for all (x,t) ∈ R×[0,∞), which further implies a contradiction between Lemma 3.4 and (1.7) as x+ ct → −∞. The proof

is complete.

?

0 ≤ z(x,s) ≤ ψ(x + c1s + h) ≤ 1

c1+c∗

2

in Lemma 3.4. Then the comparison principle (Lemma 3.5) indicates that

Theorem 3.7. Assume that c < c∗holds. Then (1.6) and (1.7) have no positive solution.

The proof of Theorem 3.7 is similar to that of Theorem 3.6, so we omit it here.

Remark 3.8. The results in Liang and Zhao [7] cannot be applied to consider the nonexistence of travelling wavefront

directly, the reason is similar to that of Remark 2.5. However, the proof in this paper is motivated by Liang and Zhao [7]

and Thieme and Zhao [16].

Remark 3.9. What we have done implies that c∗is the minimal wave speed of system (1.4), which is under the sense that

(1.6) and (1.7) have no nontrivial positive solution if c < c∗while they have a nontrivial monotone solution if c ≥ c∗.

Acknowledgements

The authors are grateful to two anonymous referees for their helpful suggestions which led to an improvement of our

original manuscript.

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