web.xidian.edu.cnweb.xidian.edu.cn/slwu/files/20120929_125230.pdf · Z. Angew. Math. Phys. 62 (2011), 377–397 c 2010 Springer Basel AG 0044-2275/11/030377-21 published online December

Z. Angew. Math. Phys. 62 (2011), 377–397c© 2010 Springer Basel AG0044-2275/11/030377-21published online December 21, 2010DOI 10.1007/s00033-010-0112-1

Zeitschrift fur angewandteMathematik und Physik ZAMP

Asymptotic stability of traveling waves for delayed reaction-diffusion equationswith crossing-monostability

Shi-Liang Wu, Hai-Qin Zhao and San-Yang Liu

Abstract. This paper is concerned with the traveling waves for a class of delayed reaction-diffusion equations with crossing-monostability. In the previous papers, we established the existence and uniqueness of traveling waves which may not bemonotone. However, the stability of such traveling waves remains open. In this paper, by means of the (technical) weightedenergy method, we prove that the traveling wave is exponentially stable, when the initial perturbation around the waveis relatively small in a weighted norm. As applications, we consider the delayed diffusive Nicholson’s blowflies equation inpopulation dynamics and Mackey–Glass model in physiology.

Mathematics Subject Classification (2000). 35K57 · 35R10 · 35B40 · 92D25.

Keywords. Asymptotic stability · Non-monotone traveling waves · Delayed reaction-diffusion equations ·Crossing-monostability · Weighted energy method.

1. Introduction

In this paper, which may be regarded as a sequel to [35,36], we are concerned with the traveling wavesfor the delayed reaction-diffusion equation with crossing-monostability (Gourley and Wu [9])

∂u(x, t)∂t

= D∂2u(x, t)

∂x2− du(x, t) + f (u(x, t − τ)), (1.1)

where x ∈ R, t > 0, D > 0, d > 0 and τ ≥ 0 are given constants, and f is Lipschitz continuous on anycompact interval, f(0) = 0 and f(K) = dK for some constant K > 0.

A special case of Eq. (1.1) which has been widely investigated in the literature is the following delayeddiffusive Nicholson’s blowflies equation [12,14,21,27]

∂N(x, t)∂t

= D∂2N(x, t)

∂x2− γN(x, t) + pN(x, t − τ)e−aN(x,t−τ), (1.2)

where N(x, t) denotes the mature population of the blowflies at location x and time t, D > 0 and γ > 0are the diffusion coefficient and death rate of the mature population, respectively, the delay τ ≥ 0 is thetime taken from birth to maturity and the remaining delayed term is adult recruitment.

Another special case of Eq. (1.1) is the delayed diffusive equation with Mackey-Glass type nonlinearity

∂w(x, t)∂t

= D∂2w(x, t)

∂x2− dw(x, t) +

αβmw(x, t − τ)βm + wm(x, t − τ)

, (1.3)

S.-L. Wu is Supported by the Fundamental Research Funds for the Central Universities JY10000970005 and the NSFof China 11026127. H.-Q.Zhao is Supported by the Talent Project of Xianyang Normal University 06XSYK247. S.-Y.Liu isSupported by the NSF of China 60674108.

378 S.-L. Wu, H.-Q. Zhao and S.-Y. Liu ZAMP

Fig. 1. f is non-decreasing on [0, K]

Fig. 2. f is non-monotone on [0, K]

where D, d, β, α,m are given positive constants. We refer to Mckey and Glass [18], Ruan [26] and Wuet al. [35] for more details on the Mckey-Glass models.

In recent years, traveling wave solutions of reaction-diffusion equations with delays have been widelyinvestigated due to the significant applications in several subjects. For Eq. (1.1), when the function f(·)is non-decreasing on the interval [0,K] (i.e., monostable case, see Fig. 1), the problem of traveling wavesis quite well understood since the whole interaction term is quasi-monotone and the solution semiflow ismonotone. Actually, in the monostable case, the existence, uniqueness and stability of monotone travel-ing waves have been established by many researchers for far more general reaction-diffusion systems withlocal or non-local delays, see eg., [3,10,12–16,21–24,27,29–34,37–40], and the recent surveys of Gourleyand Wu [9] and Gourley et al. [7].

If f(·) is not non-decreasing on [0,K] (i.e., crossing-monostable case [9], see Fig. 2), the travelingwave problem becomes harder, especially for the stability of traveling waves, due to the lack of quasi-monotonicity. Recently, there have been many efforts on the existence of traveling waves for the non-monotone delayed equations, see e.g., [5,6,25,28,30,34], in which these authors showed the existence for

Vol. 62 (2011) Asymptotic stability of traveling waves 379

sufficiently large wave speeds or small delays. Ma [17] further applied Schauder’s fixed-point theoremto a delayed non-local reaction-diffusion equation and obtained the existence of traveling waves for allvalues of the time delay. Inspired by Ma [17], Faria and Trofimchuk [6] and Huang [4], in [35], we provedthe existence of oscillatory waves, i.e., non-monotone traveling waves and periodic waves, for a class ofreaction-diffusion equations with non-local delay and crossing-monostability, which includes (1.1) as aparticular case.

It seems that little has been done for the uniqueness of traveling waves of such equations. One excep-tion is the work of Aguerrea et. al. [2], where the uniqueness (up to translations) of traveling waves of(1.1) for sufficiently large wave speeds were obtained, using the method of Lyapunov–Schmidt reduction.The other is the recent work by Wu and Liu [36]. In [36], we established the uniqueness of all the non-monotone traveling waves of (1.1) with speed c > c∗, where c∗ is the minimal wave speed, by using thetheory on non-trivial solutions of convolution equations. However, to the best of our knowledge, therehas been no results on the stability of the non-monotone traveling waves of delayed reaction-diffusionequations with crossing-monostability (Ma and Zou [16] and Gourley and Wu [9]).

The purpose of this paper is to address the stability problem of traveling waves of (1.1) in thecrossing-monostable case. More precisely, for the Cauchy problem to (1.1) with the initial data

u(x, s) = u0(x, s), x ∈ R, s ∈ [−τ, 0], (1.4)

that satisfies

u0(x, s) → E±(E− := 0, E+ := K), for s ∈ [−τ, 0] as x → ±∞,

we shall show that the global solution u(x, t) of (1.1) and (1.4) converges exponentially to a traveling waveφ(x+ct) (in time), when the initial perturbation around the wave, that is, |u0(x, s)−φ(x+cs)|(s ∈ [−τ, 0])is suitably small in a weighted norm. The exponential convergent rate will also be given (Theorems 2.4and 2.6).

We remark that the stability of traveling waves for delayed reaction-diffusion equations withmonostable nonlinearity has been extensively studied in the literature and many methods have beendeveloped for this issue. An effective method is the comparison principle combined with the squeez-ing technique, which has been used by many authors for various monostable equation, see e.g., [16,32].However, it seems difficult, if not impossible, to use the above method to the crossing-monostable non-linearity in (1.1) since the comparison theorems are not applicable for the equation. Another effectivemethod is the (technical) weighted energy method. This method was used by Mei et al. [21] for theNicholson’s blowflies Eq. (1.2) in the monostable case (i.e., 1 < p/γ ≤ e), and further employed by manyresearchers to prove the stability of monotone traveling waves of various monostable reaction-diffusionequations with delays, see, eg., [8,13,22,38] and the references therein. In the method, the key step is toestablish a priori estimate. Although the weighted energy method can only be used to prove the stabilityof traveling waves for small initial perturbations, the comparison principle is not needed in this method.

It is natural to ask if the method can be extended to the delayed reaction-diffusion equations withcrossing-monostable nonlinearity. We shall give an affirmative answer. More precisely, by adopting theweighted energy method, we shall prove the asymptotic stability of the non-monotone traveling waves of(1.1), including even the slower waves whose speeds are close to the minimal speed. In order to overcomethe difficulty of the energy estimates caused by the crossing-monostable nonlinearity and the non-mono-tonicity of the traveling waves, we introduce two ideal weight functions and carefully take the energyestimates. This is probably the first time the asymptotic stability of non-monotone traveling waves ofdelayed reaction-diffusion equations with crossing-monostability has been studied.

The rest of this paper is organized as follows. In Sect. 2, we first introduce some known results onthe existence and uniqueness of traveling waves. After defining two suitable weight functions, we stateour main results on the asymptotic stability of traveling waves of (1.1). The proofs of the main resultsare given in Sect. 3 by using the weighted energy method. The key step is to establish a priori estimate,and the weight functions play an important role in proving the stability results. In Sect. 4, we apply our


abstract results to the Nicholson’s blowflies Eq. (1.2) for the case where e < p/γ ≤ e2 and the diffusiveMackey-Glass model (1.3) for m > 1 and α

d > mm−1 .

Notations. Throughout this paper, C > 0 denotes a generic constant, while Ci > 0(i = 0, 1, 2, . . .) rep-resent specific constants. Let I be an interval, typically I = R. L2(I) is the space of the square integrablefunctions on I, and Hk(I)(k ≥ 0) is the Sobolev space of the L2-functions f(x) defined on the interval I

whose derivatives di

dxi f, i = 1, . . . , k, also belong to L2(I). L2w(I) represents the weighted L2-space with

the weight w(x) > 0 and its norm is defined by

‖f‖L2w

=

⎛⎝

∫

I

w(x)f2(x)dx

⎞⎠

1/2

.

Hkw(I) is the weighted Sobolev space with the norm

‖f‖Hkw

=

⎛⎝

k∑i=0

∫

I

w(x)∣∣∣∣

di

dxif(x)

∣∣∣∣2

dx

⎞⎠

1/2

.

Let T > 0 and let B be a Banach space, we denote by C0([0, T ];B) the space of the B-valued continuousfunctions on [0, T ], and L2([0, T ];B) as the space of B-valued L2-functions on [0, T ]. The correspondingspaces of the B-valued functions on [0,∞) are defined similarly.

2. Preliminaries and main results

A traveling wave solution of (1.1) connecting 0 and K always refers to a pair (φ, c), where φ = φ(·) is afunction on R and c > 0 is a constant, such that u(x, t) = φ(ξ), ξ = x + ct, is a solution of (1.1), that is,

cφ′(ξ) − Dφ′′(ξ) + dφ(ξ) − f (φ(ξ − cτ)) = 0, (2.1)

and

φ(−∞) = 0, φ(+∞) = K. (2.2)

For the sake of convenience, we first introduce the results of Wu et al. [35, Theorem 1.2] andWu and Liu [36, Theorem 2.2] on the existence and uniqueness of the non-monotone traveling wavesof (1.1) in the crossing-monostable case (see also Ma [17], Faria and Trofimchuk [6] and Trofimchuk et al.[28]). Assume that(A1) f ′(0) > d and there exists ν ∈ (0, 1] such that lim supu→0+ [f ′(0) − f(u)/u]u−ν < +∞;(A2) max{f ′(0)u, dK∗} ≥ f(u) > 0 for some K∗ ≥ K and for all u ∈ (0,K∗];(A3) du < f(u) < 2dK − du for u ∈ [K∗,K) and du > f(u) > 2dK − du for u ∈ (K,K∗], where

K∗ :=1d

infη∈(0,K∗]

{f(η) : f(η) ≤ dη}.

If f ′(0) > d, there exists a unique number c∗ ∈(0, 2

√D(f ′(0) − d)

)such that the characteristic

equation

�1(c, λ) := cλ − Dλ2 + d − f ′(0)e−λcτ = 0 (2.3)

has only one double real root λ∗ ∈(0, 2

√(f ′(0) − d)/D

). Moreover, for c > c∗, (2.3) has two positive

real roots λ1(c) and λ2(c) satisfy λ1(c) < λ∗ < λ2(c)

�1(c, λ) ={

< 0 for λ ∈ R\ (λ1(c), λ2(c)),> 0 for λ ∈ (λ1(c), λ2(c)).

(2.4)


Proposition 2.1. Assume that (A1)–(A3) hold. Then, for every c > c∗, Eq. (1.1) has a traveling wavesolution φ(ξ) satisfying φ(−∞) = 0, φ(+∞) = K and 0 ≤ φ(ξ) ≤ K∗ for all ξ ∈ R. Moreover,

(i) if f ∈ C2, f ′(K)τe(1+dτ) < −1, and the equilibria 0 and K of the equation

u′(t) = −du(t) + f (u(t − τ))

are hyperbolic and exponentially asymptotically stable, respectively, then the traveling profile φ(ξ)oscillates about K for sufficiently large wave speed c;

(ii) if |f(u) − f(v)| ≤ f ′(0)|u − v| for all u, v ∈ [0,K], then, for any c > c∗, the traveling wave solutionsof (1.1) with speed c are unique up to translation.

In what follows, we always assume that (A1)–(A3) hold and let φ(ξ) be a traveling wave of (1.1) withspeed c > c∗. Before stating our main results, we first give the result on the wave profiles.

Lemma 2.2.

|φ′(ξ)| ≤ 1√Dd

maxu∈[0,K∗]

f(u) for all ξ ∈ R and limξ→±∞

φ′(ξ) = 0.

Proof. Let

λ1 =c − √

c2 + 4Dd

2Dand λ2 =

c +√

c2 + 4Dd

2D.

Then, it follows from (2.1) that

φ(ξ) =1

D(λ2 − λ1)

⎡⎢⎣

ξ∫

−∞eλ1(ξ−s)H(φ)(s)ds +

+∞∫

ξ

eλ2(ξ−s)H(φ)(s)ds

⎤⎥⎦ ,

where H(φ)(s) = f(φ(s − cτ)). Differentiating the above equality with respect to ξ, we get

φ′(ξ) =1

D(λ2 − λ1)

⎡⎢⎣

ξ∫

−∞λ1e

λ1(ξ−s)H(φ)(s)ds +

+∞∫

ξ

λ2eλ2(ξ−s)H(φ)(s)ds

⎤⎥⎦ . (2.5)

Note that λ2 − λ1 ≥ 2√

dD and 0 ≤ φ(ξ) ≤ K∗ for all ξ ∈ R, we obtain

|φ′(ξ)| ≤ 1√Dd

maxu∈[0,K∗]

f(u) for all ξ ∈ R.

Finally, it follows from (2.5) and the L. Hopital’s rule that limξ→±∞ φ′(ξ)= 0. This completes the proof.�

Lemma 2.3. Assume that f ∈ C1 ([0,K∗], R) and d > |f ′(K)|. Let ε := d − |f ′(K)| ∈ (0, d), then thereexists ξ∗ ∈ R such that, for all ξ ≥ ξ∗,

|f ′ (φ(ξ − cτ))| + |f ′ (φ(ξ))| ≤ 2 |f ′(K)| + ε.

Proof. Note that limξ→+∞ φ(ξ) = K, f ∈ C1([0,K∗], R), and 0 ≤ φ(ξ) ≤ K∗ for ξ ∈ R. The assertion isa direct consequence of these observations. This completes the proof. �

Take

β =c

2Dand L= max

u∈[0,K∗]|f ′(u)|.

Define the weight function w1(ξ) as following:

w1(ξ) ={

e−β(ξ−ξ∗), for ξ < ξ∗,1, for ξ ≥ ξ∗.

(2.6)


Theorem 2.4. Let (A1)–(A3) hold. Assume that f ∈ C2 ([0,K∗], R) and d > |f ′(K)|. For any giventraveling wave φ(ξ) of (1.1) with speed c satisfying

c > c := 2√

2D(L − d), (2.7)

if the initial data satisfies

u0(x, s) − φ(x + cs) ∈ C0([−τ, 0];H1

w1(R)

),

then there exist positive constants δ0 = δ0(D, d, τ, f, c) and μ0 = μ0(D, d, τ, f, c), such that when

sups∈[−τ,0]

‖u0(·, s) − φ(· + cs)‖H1w1

≤ δ0,

the unique solution u(x, t) of the Cauchy problem (1.1) and (1.4) exists globally, and it satisfies

u(x, t) − φ(x + ct) ∈ C0([0,∞);H1

w1(R)

),

and

supx∈R

|u(x, t) − φ(x + ct)| ≤ Ce−μ0t, t ≥ 0,

for some constant C > 0. Moreover, if 0 ≤ u0(x, s) ≤ K∗ for all (x, s) ∈ R×[−τ, 0], then 0 ≤ u(x, t) ≤ K∗

for all (x, t) ∈ R × [0,+∞).

Note that c∗ ∈(0, 2

√D(f ′(0) − d)

)and L = maxu∈[0,K∗] |f ′(u)|, one can easily see that

c = 2√

2D(L − d) > c∗.

Theorem 2.4 shows that, for c > c, the traveling wave with a given speed c of (1.1) in the crossing-mono-stable case is exponentially stable.

As is well known, the stability of “slower” traveling waves (i.e., the wave speed satisfies c∗ < c ≤ c)is much more interesting and significant. To obtain the stability of slower traveling waves, we need theproperty of the critical waves, and more carefully select the weight function inspired by Gourley [8], seealso [23,24].

Note that �1(c, λ∗) > 0, for c > c∗, the following result holds.

Lemma 2.5. Assume that f ′(0) > d, f ∈ C1([0,K∗], R) and |f ′(K)| is sufficiently small. Then, for c > c∗,there exists ξ0, ξ

1∗ ∈ R with ξ1

∗ > ξ0 such that

max {|f ′(φ(ξ0 − cτ))| , |f ′ (φ(ξ0))|} ≤ min{�1(c, λ∗) + e−λ∗cτf ′(0), d

}cosh(λ∗cτ)

.

and for ξ ≥ ξ1∗ − cτ ,

max {|f ′(φ(ξ − cτ))| , |f ′ (φ(ξ))|} ≤ max {|f ′(φ(ξ0 − cτ))| , |f ′ (φ(ξ0))|}Now, we define another weight function w2(ξ) as following:

w2(ξ) ={

e−2λ∗(ξ−ξ1∗), for ξ < ξ1

∗ ,1, for ξ ≥ ξ1

∗ .(2.8)

Theorem 2.6. Let (A1)–(A3) hold. Assume that f ∈ C2([0,K∗], R), |f ′(K)| is sufficiently small, and|f ′(u)| ≤ f ′(0) for all u ∈ [0,K∗]. Then, for any given traveling wave φ(ξ) of (1.1) with speed c > c∗, ifthe initial data satisfies

u0(x, s) − φ(x + cs) ∈ C0([−τ, 0];H1

w2(R)

),

then there exist positive constants δ1 = δ1(D, d, τ, f, c) and μ1 = μ1(D, d, τ, f, c), such that when

sups∈[−τ,0]

‖u0(·, s) − φ(· + cs)‖H1w2

≤ δ1,


the unique solution u(x, t) of the Cauchy problem (1.1) and (1.4) exists globally, and it satisfies

u(x, t) − φ(x + ct) ∈ C0([0,∞);H1

w2(R)

),

and

supx∈R

|u(x, t) − φ(x + ct)| ≤ Ce−μ1t, t ≥ 0,

for some constant C > 0.

Remark 2.7. (i) Theorems 2.4 and 2.6 show that, for the crossing-monostable case, the traveling waveof (1.1) with speed c > c∗ is exponentially stable, when the initial perturbation around the wave,that is, |u0(x, s)−φ(x+ cs)|(s ∈ [−τ, 0]) is relatively small in some weighted norm. This is probablythe first time the asymptotic stability of traveling waves of delayed reaction-diffusion equations withcrossing-monostability has been studied.

However, the so-called large initial perturbation problem, that is, the wave stability for a largeinitial perturbation remans open. Another interesting and challenging problem is to address theexistence, uniqueness and stability of the critical wave of (1.1) in the crossing-monostable case.

(ii) The condition |f ′(K)| 1 (or |f ′(K)| < d) is crucial in our argument. We explain the conditionfor f(u) = pue−au (a > 0) with p/d > e. For this case, the function f is non-monotone and has asingle hump on the interval [0,K]. There exists a constant K ∈ (0,K) such that f(u) is strictlyincreasing for u ∈ (0,K) and strictly decreasing for u > K, and f ′(K) = 0 (see Fig. 2). Obviously,|f ′(K)| 1 is equivalent to K − K 1, i.e., K is close to K.

On the other hand, as showed in Al-Omari and Gourley [1], the dynamic behavior of (1.1)depend largely on the value of K − K. If K − K is suitable small, linear analysis predicts thatsolutions will tend to K, i.e., K is linearly stable, but oscillatory dynamics are possible for largerK − K.

We conjecture that, for the cross-monostable case, this condition is necessary for the stabilityof traveling waves.

3. Proof of the main results

This section is devoted to the proof of the stability results, i.e., Theorems 2.4 and 2.6. Our proof relies onthe weighted energy method. The existence and uniqueness of global solutions for the Cauchy problem(1.1) and (1.4) can be proved by the energy method reported in [21], see also a different method statedin Wu [33].

Lemma 3.1. (Boundedness) Assume that

0 ≤ u0(x, s) ≤ K∗ for all (x, s) ∈ R × [−τ, 0].

Then the solution of the Cauchy problem (1.1) and (1.4) satisfies

0 ≤ u(x, t) ≤ K∗ for all (x, t) ∈ R × [0,+∞).

Proof. Similar to Mei et al. [21, Theorem 2.1], one can easily show that u(x, t) ≥ 0 with (x, t) ∈ R ×[0,+∞). Now we are going to prove the case u(x, t) ≤ K∗ for all (x, t) ∈ R × [0,+∞).

Let v(x, t) = K∗ − u(x, t), then{

wt − Dwxx + dw = dK∗ − f(u(x, t − τ)), (x, t) ∈ R × [0,+∞),w(x, s) = K∗ − u0(x, s), (x, s) ∈ R × [−τ, 0].

Fixed t ∈ [0, τ ], we have u(x, t − τ) = u0(x, t − τ) for all x ∈ R. According to 0 ≤ u0(x, s) ≤K∗ for all (x, s) ∈ R × [−τ, 0], we then have 0 ≤ u(x, t − τ) = u0(x, t − τ) ≤ K∗ for (x, t) ∈ R × [0, τ ]. By


the assumption dK∗ ≥ f(u) for u ∈ [0,K∗], we obtain{

wt − Dwxx + dw ≥ 0, (x, t) ∈ R × [0, τ ],w(x, s) = K∗ − u0(x, s) ≥ 0, (x, s) ∈ R × [−τ, 0],

which guarantees, by the extreme value principle for parabolic partial differential equations, that

0 ≤ u(x, t) ≤ K∗ for all (x, t) ∈ R × [0, τ ].

Repeating this procedure, for t ∈ [nτ, (n + 1)τ ], where n ∈ N, we can prove

0 ≤ u(x, t) ≤ K∗ for all (x, t) ∈ R × [nτ, (n + 1)τ ],

which implies that

0 ≤ u(x, t) ≤ K∗ for all (x, t) ∈ R × [0,+∞),

and the assertion is proved. �

Set U(ξ, t) = u(x, t) − φ(ξ), ξ = x + ct. Then, the original problem (1.1) and (1.4) can be reformu-lated as ⎧⎨

⎩Ut(ξ, t) + cUξ(ξ, t) − DUξξ(ξ, t) + dU(ξ, t)

−f ′ (φ(ξ − cτ)) U(ξ − cτ, t − τ) = G(U)(ξ, t), (ξ, t) ∈ R × (0,+∞),U(ξ, s) = u0(ξ − cs, s) − φ(ξ) =: U0(ξ, s), (ξ, s) ∈ R × [−τ, 0].

(3.1)

The nonlinear term G(U)(ξ, t) is given by

G(U)(ξ, t) = f (U(ξ − cτ, t − τ) + φ(ξ − cτ))−f (φ(ξ − cτ)) − f ′ (φ(ξ − cτ)) U(ξ − cτ, t − τ), (3.2)

where

f ′ (φ(ξ − cτ)) :=df(u)du

∣∣u=φ(ξ−cτ) .

For a given weight function w(ξ) and any constants α ≥ 0 and T ≥ 0, we define the solution space by

X(α − τ, T + α) ={U

∣∣U(ξ, t) ∈ C0 ([α − τ, T + α] ;H1w(R)

)},

and

Mα(T ) = supt∈[α−τ,T+α]

‖U(t)‖H1w,

in particular, M(T ) = M0(T ) for α = 0. For simplicity, here and in what follows, we denote U(t) = U(·, t).The following local existence can be easily obtained by an elementary energy method. We omit the

proof.

Lemma 3.2. (Local existence) Consider the Cauchy problem with initial time α ≥ 0,⎧⎨⎩

Ut(ξ, t) + cUξ(ξ, t) − DUξξ(ξ, t) + dU(ξ, t) − f ′ (φ(ξ − cτ)) U(ξ − cτ, t − τ)= G(U)(ξ, t), (ξ, t) ∈ R × (α,+∞),

U(ξ, s) = u0(ξ − cs, s) − φ(ξ) =: Uα(ξ, s), (ξ, s) ∈ R × [α − τ, α].(3.3)

If Uα(ξ, s) ∈ C0([α − τ, α];H1

w(R)), and Mα(0) ≤ δ1 for a given positive constant δ1, then there exists a

small t0 = t0(δ1) > 0 such that U(ξ, t) ∈ X(α − τ, α + t0) and Mα(t0) ≤ √2(1 + τ)Mα(0).


3.1. Proof of Theorem 2.4

In this subsection, we prove Theorem 2.4. Obviously, it suffices to show that the following result forCauchy problem (3.1) holds.

Theorem 3.3. Let the assumptions of Theorem 2.4 be satisfied. For the given traveling wave φ(ξ) withspeed c satisfying (2.7), if

U0(ξ, s) ∈ C0([−τ, 0];H1

w1(R)

),

then there exist two positive constants δ0, μ0, such that when

sups∈[−τ,0]

‖U0(·, s)‖H1w1

≤ δ0,

the solution U(ξ, t) of the Cauchy problem (3.1) exists uniquely and globally, and satisfies

U(ξ, t) ∈ C0([0,∞);H1

w1(R)

),

and

supξ∈R

|U(ξ, t)| ≤ Ce−μ0t, t ≥ 0. (3.4)

We will prove Theorem 3.3 based on two propositions: the local existence (i.e., Lemma 3.2) and apriori estimate (Lemma 3.5), by the continuity argument (see [11,19,20]).

Throughout this subsection, we always assume that the conditions of Theorem 2.4 hold. Take C+(μ) =min

{C+

1 (μ), C+2 (μ)

}, where

C+1 (μ) =

c2

4D− 2(L − d) − L

(e2μτ − 1

) − 2μ,

C+2 (μ) = d − |f ′(K)| − L

(e2μτ − 1

) − 2μ.

Define

B+μ (ξ) = −c

w′1(ξ)

w1(ξ)− D

(w′

1(ξ)w1(ξ)

)2

+ 2d − 2μ

− |f ′(φ(ξ − cτ))| − e2μτ |f ′ (φ(ξ))| . (3.5)

Lemma 3.4. (Key inequalities) Let w1(ξ) be the weight function given in (2.6). If (2.7) holds, then

B+μ (ξ) ≥ C+(μ) > 0,

for all ξ ∈ R, and 0 < μ < μ+ := min{μ+1 , μ+

2 }, where μ+i is the unique solution to the equation C+

i (μ)= 0, i = 1, 2.

Proof. We distinguish among two cases:Case (i): For ξ < ξ∗, w1(ξ) = e−β(ξ−ξ∗). Note that 0 ≤ φ(ξ) ≤ K∗ for all ξ ∈ R, we have

B+μ (ξ) = cβ − Dβ2 + 2d − 2μ − |f ′(φ(ξ − cτ))| − e2μτ |f ′ (φ(ξ))|

≥ cβ − Dβ2 + 2d − 2μ − L(1 + e2μτ

)

≥ c2

4D+ 2d − 2μ − L

(1 + e2μτ

)

=c2

4D− (2L − 2d) − L

(e2μτ − 1

) − 2μ

= C+1 (μ) > 0 for 0 < μ < μ+

1 .


Case (i): For ξ ≥ ξ∗, w1(ξ) = 1, and by Lemma 2.3, we obtain

B+μ (ξ) = 2d − 2μ − |f ′(φ(ξ − cτ))| − e2μτ |f ′ (φ(ξ))|

= 2d − 2μ − |f ′(φ(ξ − cτ))| − |f ′ (φ(ξ))| − (e2μτ − 1

) |f ′ (φ(ξ))|≥ 2d − (2|f ′(K)| + ε) − L(e2μτ − 1) − 2μ

= d − |f ′(K)| − L(e2μτ − 1) − 2μ

= C+2 (μ) > 0 for 0 < μ < μ+

2 .

Thus, B+μ (ξ) ≥ C+(μ) > 0 for all ξ ∈ R and 0 < μ < min{μ+

1 , μ+2 }. This completes the proof.

�

Lemma 3.5. (A prior estimate) Assume that 0 ≤ u0(x, s) ≤ K∗ for all (x, s) ∈ R × [−τ, 0]. Let U(ξ, t) ∈X(−τ, T ) be a local solution of (3.1). Then there exist positive constants μ0, δ2 and C1 > 1 independentof T , such that M(T ) ≤ δ2 implies

‖U(t)‖2H1

w1≤ C1

⎛⎝‖U0(0)‖2

H1w1

+

0∫

−τ

‖U0(s)‖2H1

w1ds

⎞⎠ e−2μ0t, 0 ≤ t ≤ T. (3.6)

Proof. The idea of the proof is based on [21], see also [8,22]. From Lemma 3.1,

0 ≤ u(x, t) ≤ K∗ for all (x, t) ∈ R × [0, T ).

Take 0 < μ0 < min{μ+1 , μ+

2 }. Multiplying Eq. (3.1) by e2μ0tw1(ξ)U(ξ, t) for ξ ∈ R and 0 ≤ t ≤ T , wehave (

12e2μ0tw1U

2

)

t

+ e2μ0t( c

2w1U

2 − Dw1UUξ

)ξ

+Dw1U2ξ e2μ0t + Dw′

1UUξe2μ0t +

(− c

2w′

1

w1+ d − μ0

)e2μ0tw1U

2

−e2μ0tf ′ (φ(ξ − cτ)) Uw1U(ξ − cτ, t − τ)= e2μ0tw1UG(U), (3.7)

where φ = φ(ξ), w1 = w1(ξ), U = U(ξ, t), and G(U) = G(U)(ξ, t). Using the Cauchy-Schwarz inequality2ab ≤ a2 + b2, we have

|Dw′1UUξ| =

∣∣∣∣Dw1

(w′

1

w1U

)Uξ

∣∣∣∣ ≤ D

2w1U

2ξ +

D

2

(w′

1

w1

)2

w1U2.

Substituting it into (3.7) and integrating the resulting inequality over [0, t] × R, we have

e2μ0t‖U(t)‖2L2

w1+ D

t∫

0

e2μ0s‖Uξ(s)‖2L2

w1ds

+

t∫

0

∫

R

[−c

w′1(ξ)

w1(ξ)− D

(w′

1(ξ)w1(ξ)

)2

+ 2d − 2μ0

]e2μ0sw1(ξ)U2(ξ, s)dξds

−2

t∫

0

∫

R

e2μ0sw1(ξ)f ′ (φ(ξ − cτ)) U(ξ, s)U(ξ − cτ, s − τ)dξds

≤ ‖U0(0)‖2L2

w1+ 2

t∫

0

∫

R

e2μ0sw1(ξ)U(ξ, s)G(U)(ξ, s)dξds. (3.8)


Using the Cauchy-Schwarz inequality again and making the change of variables ξ−cτ → ξ and s−τ → s,we have

2

∣∣∣∣∣∣

t∫

0

∫

R

e2μ0sw1(ξ)f ′ (φ(ξ − cτ)) U(ξ, s)U(ξ − cτ, s − τ)dξds

∣∣∣∣∣∣

≤t∫

0

∫

R

e2μ0sw1(ξ) |f ′ (φ(ξ − cτ))| [U2(ξ, s) + U2(ξ − cτ, s − τ)]dξds

=

t∫

0

∫

R

e2μ0sw1(ξ) |f ′ (φ(ξ − cτ))| U2(ξ, s)dξds

+

t−τ∫

−τ

∫

R

e2μ0(s+τ) |f ′ (φ(ξ))| w1(ξ + cτ)U2(ξ, s)dξds

≤t∫

0

∫

R


+

0∫

−τ

∫

R

e2μ0sw1(ξ)[e2μ0τ |f ′ (φ(ξ))| w1(ξ + cτ)

w1(ξ)

]U2

0 (ξ, s)dξds

+

t∫

0

∫

R

e2μ0sw1(ξ)[e2μ0τ |f ′ (φ(ξ))| w1(ξ + cτ)

w1(ξ)

]U2(ξ, s)dξds

≤t∫

0

∫

R


+

0∫

−τ

∫

R

e2μ0sw1(ξ)Le2μ0τU20 (ξ, s)dξds

+

t∫

0

∫

R

e2μ0sw1(ξ)e2μ0τ |f ′ (φ(ξ))| U2(ξ, s)dξds. (3.9)

Substituting (3.9) into (3.8) yields

e2μ0t‖U(t)‖2L2

w1+

t∫

0

∫

R

B+μ0

(ξ)e2μ0sw1(ξ)U2(ξ, s)dξds

≤ ‖U0(0)‖2L2

w1+ Le2μ0τ

0∫

−τ

‖U0(s)‖2L2

w1ds

+2

t∫

0

∫

R



From Lemma 3.4, B+μ0

(ξ) ≥ C+(μ0) > 0 for all ξ ∈ R, then we can reduce (3.10) as

e2μ0t‖U(t)‖2L2

w1+ C+(μ0)

t∫

0

e2μ0s‖U(s)‖2L2

w1ds ≤ ‖U0(0)‖2

L2w1

+ Le2μ0τ

0∫

−τ

‖U0(s)‖2L2

w1ds

+2

t∫

0

∫

R


By the standard Sobolev’s embedding inequality H1(R) ↪→ C0(R) and the embedding inequalityH1

w1(R) ↪→ H1(R)( since w1(ξ) ≥ 1, for all ξ ∈ R), we have, for all ξ ∈ R and −τ ≤ t ≤ T ,

|U(ξ, t)| ≤ supξ∈R

|U(ξ, t)| ≤ σ0‖U(·, t)‖H1 ≤ σ0‖U(·, t)‖H1w1

≤ σ0M(t), (3.12)

where σ0 > 0 is the embedding constant.Take L2 := maxu∈[0,K∗] |f ′′(u)|. Using Taylor’s expansion, we get

|G(U)(ξ, t)| = |f (U(ξ − cτ, t − τ) + φ(ξ − cτ))−f (φ(ξ − cτ)) − f ′ (φ(ξ − cτ)) U(ξ − cτ, t − τ)|

=∣∣∣∣f ′′(η1)

2

∣∣∣∣U2(ξ − cτ, t − τ)

≤ L2

2U2(ξ − cτ, t − τ), (3.13)

for ξ ∈ R and 0 ≤ t ≤ T , where

η1 = (1 − θ1)U(ξ − cτ, t − τ) + φ(ξ − cτ)= (1 − θ1)u(ξ − ct, t) + θ1φ(ξ − cτ)∈ [0,K∗], θ1 ∈ (0, 1). (3.14)

Note that w1(ξ + cτ)/w1(ξ) ≤ 1 for all ξ ∈ R, by (3.12) and (3.13),

2

∣∣∣∣∣∣

t∫

0

∫

R

e2μ0sw1(ξ)U(ξ, s)G(U)(ξ, s)dξds

∣∣∣∣∣∣

≤ σ0L2M(t)

t∫

0

∫

R

e2μ0sw1(ξ)U2(ξ − cτ, s − τ)dξds

= σ0L2M(t)

t−τ∫

−τ

∫

R

e2μ0(s+τ)w1(ξ + cτ)U2(ξ, s)dξds

≤ σ0L2M(t)

⎧⎨⎩

t∫

0

∫

R

e2μ0(s+τ)w1(ξ)U2(ξ, s)w1(ξ + cτ)

w1(ξ)dξds

+

0∫

−τ

∫

R

e2μ0(s+τ)w1(ξ)U20 (ξ, s)

w1(ξ + cτ)w1(ξ)

dξds

⎫⎬⎭

≤ σ0L2e2μ0τM(t)

⎧⎨⎩

t∫

0

e2μ0s‖U(s)‖2L2

w1ds +

0∫

−τ

e2μ0s‖U0(s)‖2L2

w1ds

⎫⎬⎭ . (3.15)


Substituting (3.15) into (3.11), we obtain

e2μ0t‖U(t)‖2L2

w1+

[C+(μ0) − σ0L2e

2μ0τM(t)] t∫

0

e2μ0s‖U(s)‖2L2

w1ds

≤ ‖U0(0)‖2L2

w1+ e2μ0τ (L + σ0L2M(t))

0∫

−τ

‖U0(s)‖2L2

w1ds. (3.16)

Since C+(μ0) > 0, there exists a positive constant δ12 = δ1

2(D, d, τ, c, f) < K∗ such that

C+(μ0) − σ0L2e2μ0τδ1

2 > 0.

Thus, when M(T ) ≤ δ12 ,

C+(μ0) − σ0L2e2μ0τM(t) ≥ C+(μ0) − σ0L2e

2μ0τM(T )≥ C+(μ0) − σ0L2e

2μ0τδ12

> 0,

for all 0 ≤ t ≤ T , it follows that

‖U(t)‖2L2

w1≤ C2

⎛⎝‖U0(0)‖2

L2w1

+

0∫

−τ

‖U0(s)‖2L2

w1ds

⎞⎠ e−2μ0t, (3.17)

where C2 = max{1, e2μ0τ (L + σ0L2K

∗)}

> 1.Next, we shall establish the energy estimate for Uξ which is similar to (3.17). Differentiating (3.1)

with respect to ξ and then multiplying the resultant equation by e2μ0tw1(ξ)Uξ(ξ, t), we obtain(

12e2μ0tw1U

2ξ

)

t

+ e2μ0t( c

2w1U

2ξ − Dw1UξUξξ

)ξ

+Dw1U2ξξe

2μ0t + Dw′1UξUξξe

2μ0t +(

− c

2w′

1

w1+ d − μ0

)e2μ0tw1U

2ξ

−e2μ0tf ′ (φ(ξ − cτ)) Uξw1Uξ(ξ − cτ, t − τ)

= e2μ0tw1Uξ[G1(U) + G2(U)], (3.18)

where

G1(U)(ξ, t) = [f ′ (U(ξ − cτ, t − τ) + φ(ξ − cτ)) − f ′ (φ(ξ − cτ))] φ′(ξ − cτ)G2(U)(ξ, t) = [f ′ (U(ξ − cτ, t − τ) + φ(ξ − cτ)) − f ′ (φ(ξ − cτ))]

×Uξ(ξ − cτ, t − τ).

Integrating (3.18) over [0, t] × R, and carrying out similar steps to those led to (3.10), we get

e2μ0t‖Uξ(t)‖2L2

w1+

t∫

0

∫

R

B+μ0

(ξ)e2μ0sw1(ξ)U2ξ (ξ, s)dξds

≤ ‖U0(0)‖2H1

w1+ Le2μ0τ

0∫

−τ

‖U0(s)‖2H1

w1ds,

+2

t∫

0

∫

R

e2μ0sw1(ξ)Uξ(ξ, s) [G1(U)(ξ, s) + G2(U)(ξ, s)] dξds, (3.19)


Now we estimate the last two terms on the right-hand side of (3.19). Similar to (3.9), we have

2

∣∣∣∣∣∣

t∫

0

∫

R

e2μ0sw1(ξ)Uξ(ξ, s)G2(U)(ξ, s))dξds

∣∣∣∣∣∣

= 2

∣∣∣∣∣∣

t∫

0

∫

R

e2μ0sw1(ξ)Uξ(ξ, s)f ′′(η2)U(ξ − cτ, s − τ)Uξ(ξ − cτ, s − τ)dξds

∣∣∣∣∣∣

≤ 2σ0L2M(t)

t∫

0

∫

R

e2μ0sw1(ξ)Uξ(ξ, s)Uξ(ξ − cτ, s − τ)dξds

≤ σ0L2M(t)

t∫

0

∫

R

e2μ0sw1(ξ)U2ξ (ξ, s)dξds

+σ0L2M(t)

t∫

0

∫

R

e2μ0sw1(ξ)U2ξ (ξ − cτ, s − τ)dξds

≤ σ0L2M(t)

t∫

0

∫

R


+σ0L2M(t)

⎛⎝

0∫

−τ

∫

R

+

t∫

0

∫

R

⎞⎠ w1(ξ + cτ)

w1(ξ)e2μ0(s+τ)w1(ξ)U2

ξ (ξ, s)dξds

≤ σ0L2M(t)

⎛⎝(1 + e2μ0τ )

t∫

0


w1+ e2μ0τ

0∫

−τ

‖U0(s)‖2H1

w1ds

⎞⎠ (3.20)

and

2

∣∣∣∣∣∣

t∫

0

∫

R

e2μ0sw1(ξ)Uξ(ξ, s)G1(U)(ξ, s))dξds

∣∣∣∣∣∣

= 2

∣∣∣∣∣∣

t∫

0

∫

R

e2μ0sw1(ξ)Uξ(ξ, s)f ′′(η2)U(ξ − cτ, s − τ)φ′(ξ − cτ)dξds

∣∣∣∣∣∣

≤ 2L1

∣∣∣∣∣∣

t∫

0

∫

R

e2μ0sw1(ξ)Uξ(ξ, s)U(ξ − cτ, s − τ)dξds

∣∣∣∣∣∣

≤ 2L1η0

t∫

0

∫

R


+L1

2η0

t∫

−τ

∫

R

w1(ξ + cτ)w1(ξ)

e2μ0(s+τ)w1(ξ)U2(ξ, s)dξds


≤ 12C+(μ0)

t∫

0


w1ds

+2L2

1e2μ0τ

C+(μ0)

⎛⎝

0∫

−τ

‖U0(s)‖2H1

w1ds +

t∫

0

e2μ0s‖U(s)‖2L2

w1ds

⎞⎠ , (3.21)

where η0 = C+(μ0)/(4L1) > 0, η2 = (1 − θ2)U(ξ − cτ, s − τ) + φ(ξ − cτ) ∈ [0,K∗], θ2 ∈ (0, 1), and

L1 =1√Dd

maxu∈[0,K∗]

f(u) maxu∈[0,K∗]

|f ′′(u)| > 0.

It then follows from (3.19)–(3.21) and Lemma 3.4 that


w1+ C+(μ0)

t∫

0


w1ds

≤ e2μ0t‖Uξ(t)‖2L2

w1+

t∫

0

∫

R

B+μ0

(ξ)e2μ0sw1(ξ)U2ξ (ξ, s)dξds

≤ C3

t∫

0

‖U(s)‖2L2

w1ds +

(C4M(t) +

12C+(μ0)

) t∫

0


w1

+C5

⎛⎝‖U0(0)‖2

H1w1

+

0∫

−τ

‖U0(s)‖2H1

w1ds

⎞⎠ , (3.22)

provided that M(t) ≤ K∗, where C3 = 2L21e

2μ0τ/

C+(μ0), C4 = (1 + e2μ0τ )σ0L2, and C5 = 1 +(L + σ0

K∗L2 + 2L21

/C+(μ0)

)e2μ0τ .

Using the energy estimate (3.17), we get


w1+

(12C+(μ0) − C4M(t)

) t∫

0


w1ds

≤ C6

⎛⎝‖U0(0)‖2

H1w1

+

0∫

−τ

‖U0(s)‖2H1

w1ds

⎞⎠ , (3.23)

where C6 = C5 + C3C2/ (2μ0). Choose a positive constant δ2 < δ12 < K∗ such that 1

2C−(μ0) − C4δ2 > 0.Thus, when M(T ) ≤ δ2,

12C+(μ0) − C4M(t) ≥ 1

2C+(μ0) − C4M(T )

≥ 12C+(μ0) − C4δ2 > 0.

It then follows from (3.23) that, for all 0 ≤ t ≤ T ,

‖Uξ(t)‖2L2

w1≤ C6

⎛⎝‖U0(0)‖2

H1w1

+

0∫

−τ

‖U0(s)‖2H1

w1ds

⎞⎠ e−2μ0t. (3.24)

Obviously, μ0 depends only on the coefficients D, d, τ , the wave speed c, and the function f . Combin-ing (3.17) and (3.24), we finally have, for some absolute constant C1 = C2 +C6 > 1 which is independent


of T and U(ξ, t),

‖U(t)‖2H1

w1≤ C1

⎛⎝‖U0(0)‖2

H1w1

+

0∫

−τ

‖U0(s)‖2H1

w1ds

⎞⎠ e−2μ0t, for all 0 ≤ t ≤ T,

provided that M(T ) ≤ δ2. This completes the proof. �

Proof of Theorem 3.3. Based on Lemmas 3.2 and 3.5, the proof is similar to those of Mei and So[22, Theorem 3.1] and Mei et al. [21, Theorem 3.1], by using the continuity argument. We omit it here.

�

3.2. Proof of Theorem 2.6

The proof of Theorem 2.6 is similar to that of Theorem 2.4, we shall omit the details, and only give someimportant lemmas. In what follows, we assume that the conditions of Theorem 2.6 are satisfied.

Recall that L = maxu∈[0,K∗] |f ′(u)| and λ∗ ∈(λ1(c), 2

√(f ′(0) − d)/D

). Let C−(μ) = mini=1,2,3{

C−i (μ)

}, where

C−1 (μ) := 2�1(c, λ∗) − eλ∗cτ

(e2μτ − 1

)f ′(0) − 2μ,

C−2 (μ) := 2�1(c, λ∗) + 2e−λ∗cτf ′(0) − eλ∗cτ

(e2μτ − 1

)L − 2μ

−2max {|f ′(φ(ξ0 − cτ))| , |f ′ (φ(ξ0))|} cosh(λ∗cτ),C−

3 (μ) := 2d − 2max {|f ′(φ(ξ0 − cτ))| , |f ′ (φ(ξ0))|} cosh(λ∗cτ)

−eλ∗cτ(e2μτ − 1

)L − 2μ.

Define

B−μ (ξ) = −c

w′2(ξ)

w2(ξ)− D

2

(w′

2(ξ)w2(ξ)

)2

+ 2d − 2μ

−α |f ′(φ(ξ − cτ))| − 1α

e2μτ w2(ξ + cτ)w2(ξ)

|f ′ (φ(ξ))|,

where w2(ξ) be the weight function given in (2.8) and α is a positive number which will be chosen later.

Lemma 3.6. (Key Inequality) Let α = e−λ∗cτ . Then

B−μ (ξ) ≥ C−(μ) > 0,

for all ξ ∈ R and 0 < μ < μ− =: mini=1,2,3{μ−i }, where μ−

i > 0 are the unique solutions to the equationC−

i (μ) = 0, i = 1, 2, 3.

Proof. We distinguish among three cases:

Case (i): ξ < ξ1∗ − cτ . In this case w2(ξ) = e−2λ∗(ξ−ξ1

∗) and w2(ξ + cτ) = e−2λ∗(ξ−ξ1∗+cτ). Note that

0 ≤ φ(ξ) ≤ K∗ for ξ ∈ R and |f ′(u)| ≤ f ′(0) for all u ∈ [0,K∗], we have


B−μ (ξ) = 2cλ∗ − 2Dλ2

∗ + 2d − 2μ

−α |f ′(φ(ξ − cτ))| − 1α

e2μτe−2λ∗cτ |f ′ (φ(ξ))|

≥ 2cλ∗ − 2Dλ2∗ + 2d − 2μ − αf ′(0) − 1

αe−2λ∗cτf ′(0)

− 1α

(e2μτ − 1

)e−2λ∗cτf ′(0)

≥ 2�1(c, λ∗) − 1α

(e2μτ − 1

)f ′(0) − 2μ

= C−1 (μ) > 0 for 0 < μ < μ−

1 .

Case (ii): ξ1∗ − cτ ≤ ξ ≤ ξ1

∗ , then w2(ξ) = e−2λ∗(ξ−ξ1∗) and w2(ξ + cτ) = 1, and by Lemma 2.5, we obtain

B−μ (ξ) = 2cλ∗ − 2Dλ2

∗ + 2d − 2μ

−α |f ′(φ(ξ − cτ))| − 1α

e2μτe2λ∗(ξ−ξ1∗) |f ′ (φ(ξ))|

≥ 2�1(c, λ∗) + 2e−λ∗cτf ′(0) − α |f ′(φ(ξ − cτ))| − 1α

|f ′ (φ(ξ))|

− 1α

(e2μτ − 1

) |f ′ (φ(ξ))| − 2μ

≥ 2�1(c, λ∗) + 2e−λ∗cτf ′(0) −(

α +1α

)max {|f ′(φ(ξ − cτ))| , |f ′ (φ(ξ))|}

− 1α

(e2μτ − 1

) |f ′ (φ(ξ))| − 2μ

≥ 2�1(c, λ∗) + 2e−λ∗cτf ′(0) − 2max {|f ′(φ(ξ0 − cτ))| , |f ′ (φ(ξ0))|} cosh(λ∗cτ)

− 1α

(e2μτ − 1

)L − 2μ

= C−2 (μ) > 0 for 0 < μ < μ−

2 .

Case (iii): ξ ≥ ξ1∗ , then w2(ξ) = w2(ξ + cτ) = 1, and by Lemma 2.5, we get

B−μ (ξ) = 2d − 2μ − α |f ′(φ(ξ − cτ))| − 1

αe2μτ |f ′ (φ(ξ))|

= 2d − 2μ − α |f ′(φ(ξ − cτ))| − 1α

|f ′ (φ(ξ))| − 1α

(e2μτ − 1

) |f ′ (φ(ξ))|

≥ 2d − 2max {|f ′(φ(ξ0 − cτ))| , |f ′ (φ(ξ0))|} cosh(λ∗cτ) − 1α

(e2μτ − 1

)L − 2μ

= C−3 (μ) > 0 for 0 < μ < μ−

0 .

Thus, Bμ(ξ) ≥ C−(μ) > 0 for ξ ∈ R and 0 < μ < μ−. This completes the proof.�

Lemma 3.7. (A prior estimate) Assume that 0 ≤ u0(x, s) ≤ K∗ for all (x, s) ∈ R × [−τ, 0]. Let U(ξ, t) ∈X(−τ, T ) be a local solution of (3.1). Then there exist positive constants μ1, δ3 and C7 > 1 independentof T , such that M(T ) ≤ δ3 implies

‖U(t)‖2H1

w2≤ C7

⎛⎝‖U0(0)‖2

H1w2

+

0∫

−τ

‖U0(s)‖2H1

w2ds

⎞⎠ e−2μ1t, 0 ≤ t ≤ T.

Proof. The proof is similar to that of Lemma 3.5 and is omitted. �Theorem 2.6 follows from Lemmas 3.2 and 3.7.


4. Applications

In the previous sections, we prove the asymptotic stability of the traveling waves of (1.1) in the crossing-monostable case. We expect that our results are extended to the following delayed nonlocal reaction-dif-fusion equation with crossing-monostability

∂u

∂t= DΔu − du(x, t) +

∞∫

−∞h(y)f (u(y, t − τ)) dy, (4.1)

where x ∈ R, t > 0, D > 0, d > 0 and τ ≥ 0 are given constants, f is Lipschitz continuous on any com-pact interval, f(0) = 0 and f(K) = dK for some constant K > 0, and the kernel h(x) is any integrablenon-negative function satisfying h(−x) = h(x),

+∞∫

−∞h(y)dy = 1 and

+∞∫

−∞e−λyh(y)dy < +∞, for any λ > 0.

Now, we apply our abstract results developed in Sects. 2–3 to the delayed diffusive Nicholson’s blow-flies equation in population dynamics (see [12,14,21,27]) and Mackey-Glass model in physiology (seeMckey and Glass [18] and Ruan [26]).

Example 4.1. Consider the Nicholson’s blowflies Eq. (1.2), that is,

∂u

∂t= DΔu − γu(x, t) + pu(x, t − τ)e−au(x,t−τ), (4.2)

where γ > 0, p > 0, a > 0, D > 0 and τ ≥ 0 are constants. The model (4.2) has been studied by manyresearchers and many nice results on the existence, uniqueness and stability of monotone traveling waveshas been obtained for the monostable case where 1 < p/γ ≤ e, see [12,14,21,27,32,38]. For p/γ > e, i.e.,crossing-monostable case, the existence and uniqueness of traveling wave which may be non-monotonehave also been established in [36,35]. However, there has been no result on the stability of such travelingwaves for the non-monotone case.

Assume that e < p/γ ≤ e2. It is easy to see that (4.2) has two equilibria 0 and K := 1a ln p

γ . Letf(u) := pue−au, then

|f ′(K)| = (lnp

γ− 1)γ ≤ γ, K∗ =

p

aγe,

K∗ =1γ

f(K∗) =p2

aeγ2exp{− p

eγ}, L := max

u∈[0,K∗]|f ′(u)| = p = f ′(0).

Moreover, there exists a unique number c∗ ∈(0, 2

√D(p − γ)

)such that the equation

�2(c, λ) = cλ − Dλ2 + γ − pe−λcτ = 0. (4.3)

has only one double real root λ∗ ∈(0, 2

√(p − γ)/D

), and for c > c∗, (4.3) has two real roots λ1(c), λ2(c)

satisfy λ1(c) < λ∗ < λ2(c) and

�2(c, λ) ={

< 0, ∀λ ∈ R\ (λ1(c), λ2(c)),> 0, ∀λ ∈ (λ1(c), λ2(c)).

(4.4)

From Theorems 2.4 and 2.6, we have the following result.


Theorem 4.1. Assume that e < p/γ ≤ e2. Then the following statements hold:

(i) The traveling wave of (4.2) with speed

c > 2√

2D(p − γ) (> c∗),

is stable in the sense under Theorem 2.4;

(ii) If γ(ln pγ − 1) 1, then the traveling wave of (4.2) with speed c > c∗ is stable in the sense under

Theorem 2.6.

Remark 4.2. When p/γ ∈ (e, e2] is close to e, then the condition γ(ln pγ − 1) 1 holds.

Example 4.2. Consider the diffusive Mackey-Glass model

∂u


αβmu(x, t − τ)βm + um(x, t − τ)

, (4.5)

where D > 0, d > 0, β > 0, α > 0,m > 0 and τ ≥ 0 are given constants.

Let u = βu∗, Eq. (4.5) reduces to (for the sake of convenience, drop the star on u)

∂u


αu(x, t − τ)1 + um(x, t − τ)

. (4.6)

Denote f(u) = αu/ (1 + um), then

f ′(u) = α1 − (m − 1)um

(1 + um)2.

If 0 < m ≤ 1 and 1 < αd < ∞ or m > 1 and 1 < α

d ≤ mm−1 , then f ′(u) ≥ 0, u ∈ [0,K], where K =

(αd − 1

) 1m . In the two cases, system (4.6) is monostable, the existence, uniqueness and stability of mono-

tone traveling waves follow from the results in [32,38]. However, for the case where m > 1 and αd > m

m−1 ,few results on the existence, uniqueness and stability of traveling waves.

For mathematical simplicity, we assume that m = 2 and αd > 2. Then f(u) = αu/

(1+u2

)and K =(

αd − 1

) 12 . It is easy to verify that

|f ′(K)| =d2

α

(α

d− 2

)< d, K∗ =

1d

maxu∈[0,∞)

f(u) =α

2d,

K∗ =αK∗

d (1 + (K∗)2), L = max

u∈[0,K∗]|f ′(u)| = α = f ′(0).

Similar to Theorem 4.1, it follows from Theorems 2.4 and 2.6 that the following statements hold.

Theorem 4.3. Assume that m = 2 and αd > 2. Then

(i) The traveling wave of (4.5) with speed

c > 2√

2D(α − d) (> c∗),

is stable in the sense under Theorem 2.4;

(ii) If d2

α

(αd − 2

) 1, then the traveling wave of (4.5) with speed c > c∗ is stable in the sense underTheorem 2.6.


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Shi-Liang Wu and San-Yang LiuDepartment of Applied MathematicsXidian UniversityXi’anShaanxi 710071People’s Republic of Chinae-mail: [email protected]

Hai-Qin ZhaoDepartment of MathematicsXianyang Normal UniversityXianyangShaanxi 712000People’s Republic of China

(Received: April 15, 2009; revised: June 6, 2010)

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web.xidian.edu.cnweb.xidian.edu.cn/slwu/files/20120929_125230.pdf · Z. Angew. Math. Phys. 62 (2011), 377–397 c 2010 Springer Basel AG 0044-2275/11/030377-21 published online December