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SIAM J. MATH. ANAL.

Vol. 42, No. 6, pp. 2762–2790

c ? 2010 Society for Industrial and Applied Mathematics

GLOBAL STABILITY OF MONOSTABLE TRAVELING WAVES FOR

NONLOCAL TIME-DELAYED REACTION-DIFFUSION EQUATIONS∗

MING MEI†, CHUNHUA OU‡, AND XIAO-QIANG ZHAO‡

Abstract. For a class of nonlocal time-delayed reaction-diffusion equations, we prove that all

noncritical wavefronts are globally exponentially stable, and critical wavefronts are globally alge-

braically stable when the initial perturbations around the wavefront decay to zero exponentially

near the negative infinity regardless of the magnitude of time delay. This work also improves and

develops the existing stability results for local and nonlocal reaction-diffusion equations with delays.

Our approach is based on the combination of the weighted energy method and the Green function

technique.

Key words. nonlocal reaction-diffusion equations, time delays, traveling waves, global stability,

the Fisher–KPP equation, L1-weighted energy, Green functions

AMS subject classifications. 35K57, 34K20, 92D25

DOI. 10.1137/090776342

1. Introduction. Regarding the spatial dynamics of a single-species population

with age-structure and spatial diffusion such as the Australian blowflies population

distribution, there is a class of time-delayed reaction-diffusion equations with nonlocal

nonlinearity (see, e.g., [6, 12, 32, 40, 41])

?∞

with the initial data

(1.1)

∂u

∂t− D∂2u

∂x2+ d(u(t,x)) = ε

−∞

fα(y)b(u(t − τ,x − y))dy, t > 0, x ∈ R,

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

Here u(t,x) denotes the total mature population of the species (with age greater than

the maturation age τ > 0) at time t and position x, D > 0 is the spatial diffusion rate

for the mature population, α > 0 is the total amount of diffusion for the immature

species and satisfies α ≤ τD, ε > 0 is the survival rate of the species in time τ period

and represents the impact of the death rate of the immature population, and fα(y) is

the heat kernel in the form of

fα(y) =

1

√4παe−y2/4α

with

?∞

−∞

fα(y)dy = 1.

The nonlinear functions d(u) and b(u) denote the death and birth rates of the mature

population, respectively, and satisfy the following hypotheses:

∗Received by the editors November 6, 2009; accepted for publication (in revised form) August 25,

2010; published electronically November 9, 2010.

http://www.siam.org/journals/sima/42-6/77634.html

†Department of Mathematics, Champlain College, Saint-Lambert, Quebec J4P 3P2, Canada.

Current address: Department of Mathematics and Statistics, McGill University, Montreal, Quebec

H3A 2K6, Canada (ming.mei@mcgill.ca). This author was supported in part by the NSERC of

Canada.

‡Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s,

NL A1C 5S7, Canada (ou@mun.ca, zhao@mun.ca). The first author was supported in part by the

NSERC of Canada and the IRIF grant of Newfoundland and Labrador Province. The second author

was supported in part by the NSERC of Canada and the MITACS of Canada.

2762

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NONLOCAL TIME-DELAYED REACTION-DIFFUSION EQUATIONS

2763

(H1) There exist u−= 0 and u+> 0 such that d(0) = b(0) = 0, d(u+) = εb(u+),

and d(u),b(u) ∈ C2[0,u+];

(H2) εb?(0) > d?(0) ≥ 0 and 0 ≤ εb?(u+) < d?(u+), and d?(u+)2> ε2b?(0)b?(u+);

(H3) For 0 ≤ u ≤ u+, d?(u) ≥ 0, b?(u) ≥ 0, d??(u) ≥ 0, b??(u) ≤ 0, but either

d??(u) > 0 or |b??(u)| > 0.

The equation (1.1) includes a lot of evolution equations for the single species

population with an age structure. For example, by taking the death rate function as

d(u) = δu with a positive coefficient δ > 0, (1.1) reduces to the following nonlocal

Nicholson’s blowflies population model (see, e.g., [12, 19, 20, 27, 28, 32, 40, 41, 45]):

?∞

where the birth rate function b(u) is usually taken as

(1.3)

∂u

∂t− D∂2u

∂x2+ δu = ε

−∞

fα(y)b(u(t − τ,x − y))dy, t > 0, x ∈ R,

b1(u) = pue−auq,b2(u) =

pu

1 + auq, p > 0, a > 0, q > 0.

In particular, when q = 1, b1(u) is just the so-called Nicholson’s birth rate function.

If we further assume that the immature species is almost nonmobile, i.e., the

impact factor α of spatial diffusion for the immature population is sufficiently close

to zero, by using the property of the heat kernel fα(y) =

?∞

we then obtain the following local Nicholson’s blowflies equation (see, e.g., [12, 13, 23,

26, 29, 39]):

1

√4παe−y2/4α,

b(u(t − τ,x)) = lim

α→0+

−∞

fα(y)b(u(t − τ,x − y))dy,

(1.4)

∂u

∂t− D∂2u

∂x2+ δu = εb(u(t − τ,x)), t > 0, x ∈ R.

On the other hand, if we take

d(u) = δu2, δ > 0 and εb(u) = pe−γτu, p > 0, γ > 0,

then (1.1) reduces to the following nonlocal age-structured population model (see, e.g.,

[1, 2, 3, 6, 11, 12, 32, 41, 44])

?∞

and by taking α → 0+in (1.5) for consideration of the nonmobile immature popula-

tion, we further derive the local age-structured population model (see, e.g., [2, 10, 11,

12, 18, 25, 30])

(1.5)

∂u

∂t− D∂2u

∂x2+ δu2= pe−γτ

−∞

fα(y)u(t − τ,x − y)dy, t > 0, x ∈ R,

(1.6)

∂u

∂t− D∂2u

∂x2+ δu2= pe−γτu(t − τ,x), t > 0, x ∈ R.

In particular, if we consider the case without time delay, i.e., τ = 0, and simply

take D = δ = p = 1 in (1.6), we get the well-known Fisher-KPP equation (c.f.

[8, 9, 15, 16, 31, 35, 43, 47])

(1.7)

∂u

∂t−∂2u

∂x2= u(1 − u), t > 0, x ∈ R.

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2764

MING MEI, CHUNHUA OU, AND XIAO-QIANG ZHAO

From (H1), it can be verified that both u−= 0 and u+> 0 are constant equilibria

of (1.1), and from (H2) we see that u− = 0 is unstable and u+ is stable for the

spatially homogeneous equation associated with (1.1). (H3) implies that, in [u−,u+]

under consideration, both the birth rate function b(u) and the death rate function

d(u) are nondecreasing, and b(u) is concave downward and d(u) is concave upward.

These characters are summarized from those typical examples in (1.3)–(1.6).

A traveling wavefront of (1.1) is a special solution in the form of u(t,x) = φ(x+ct)

with φ(±∞) = u±, where c is the wave speed. The main purpose of this paper is

to study the global asymptotic stability of traveling wavefronts φ(x + ct) of (1.1),

including the case of the critical wave φ(x + c∗t). Here the number c∗is called the

critical speed (or the minimum speed) in the sense that a traveling wave φ(x + ct)

exists if c ≥ c∗, while no traveling wave φ(x + ct) exists if c < c∗.

There have been extensive investigations on the stability of traveling waves for

reaction-diffusion equations without time delay; see, e.g., [5, 7, 9, 14, 15, 24, 31, 35,

37, 42, 46], the monographs [4, 43], the survey paper [47], and the references therein.

Regarding time-delayed reaction-diffusion equations such as those in [6, 12, 21, 22,

29, 32, 33, 36, 40, 41], the study of stability of traveling waves is quite limited. The

first study on the linearized stability was given by Schaaf [36] via a spectral analysis.

When the spatially homogeneous equation possesses two stable constant equilibria

(i.e., the bistable case), the stability of bistable traveling waves for local equations

was obtained by Smith and Zhao [38] via the upper-lower solutions method coupled

with the squeezing technique; see also the recent contribution by Wang, Li, and Ruan

[44] for nonlocal equations. In the monostable case (i.e., one equilibrium is stable, but

the other is unstable), the study of the stability of traveling waves is much harder,

due to the difficulty caused by the unstable equilibrium. The first work related to

this case for the local Nicholson’s blowflies equation was given by Mei et al. [29] via

the weighted L2-energy method, where the fast waves (i.e., the wave speeds are large)

were proved to be locally stable (i.e., the initial perturbation around the waves must

be small enough). Later on, a similar result for the nonlocal Nicholson’s blowflies

equation was obtained by Mei and So [28]. Furthermore, the global stability for all

waves, including those slow waves (but except for the critical one), was proved by

Mei et al. in [26] for the local equation and in [27] for the nonlocal equation via a

development of the ideas in [23]. Note that the nonlocal Nicholson’s blowflies equation

was considered in [27] under the condition that the total diffusion for the immature

population, α, is sufficiently small when the wave speed c is sufficiently close to the

critical wave speed c∗. This condition is acceptable but still a bit stiff because when

α ? 1, the nonlocal birth-rate term can be regarded as a small perturbation of the

corresponding local birth-rate term, which implies that the nonlocal equation is just

a small perturbation of the local equation for α ? 1. For the monostable equation

with age-structure, the linear stability for all slow waves (except for the critical one)

was studied by Gourley [10] when the time delay τ ? 1. Further, these waves were

proved to be nonlinearly stable, also globally stable by Li, Mei, and Wong [18] but

still with a small τ. Recently, such a smallness on time delay τ was removed by Mei

and Wong [30].

The stability of the critical traveling wave solutions to either local or nonlocal

time-delayed equations has been a challenging open problem. It is well known that

the stability of the critical waves is very important in the study of biological invasions.

This is because the critical wave speed is also the spreading speed for all solutions with

initial data having compact supports (see, e.g., [41, 21, 22] and the references therein).

Since the traditional methods, including the weighted L2-energy method, the upper-

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NONLOCAL TIME-DELAYED REACTION-DIFFUSION EQUATIONS

2765

lower solution method, as well as the spectral analysis approach, may not be used to

prove the stability of the critical traveling wavefronts for these generalized nonlocal

time-delayed reaction-diffusion equations, we need to look for a new strategy to attack

the problem. By a profound observation on the standing equation and using the

concavity of the nonlinear birth and death rate functions, we first establish a weighted

L1-energy estimate of solutions and then obtain the desired L2-energy estimate as

well as the exponential convergence rate to the noncritical wave by the ordinary

weighted energy method. When the wave is critical, the convergence rate to the

wave is proved to be algebraic by the Green function method. These stability results

improve and develop the existing works on monostable waves. As the applications

of our main result, we obtain the global and exponential stability of all noncritical

traveling waves and the algebraic stability of the critical wave for the local/nonlocal

Nicholson’s blowflies equations and the local/nonlocal population equations with age-

structure. In particular, the classical stability results for the Fisher–KPP equation,

for example, the exponential stability of all noncritical waves given by Sattinger [35]

and the algebraic stability of the critical waves shown by Moet [31], Kirchgassner [15],

and Gallay [9] are consequences of our main theorem.

The rest of this paper is organized as follows. In section 2, we introduce some

necessary notations and present the main results on the exsitence and nonlinear sta-

bility of traveling wavefronts. In section 3, we build up some energy estimates in

the weighted L1space, then establish the energy estimates in H1, and further prove

the global asymptotic stability result with a time-exponential decay for the non-

critical traveling waves and a time-algebraic decay for the critical traveling wave,

respectively. Section 4 is devoted to the application of our main result to the afore-

mentioned evolution equations, including the Fisher–KPP equation. In section 5, we

present a generalization of our stability result to a larger class of nonlocal time-delayed

reaction-diffusion equations and give a remark on a time-delayed integro-differential

vector disease model.

2. Main results. Throughout this paper, C > 0 denotes a generic constant,

while Ci > 0 (i = 0,1,2,...) represents a specific constant. Let I be an interval,

typically I = R. Lp(I) (p ≥ 1) is the Lebesque space of the integrable functions

defined on I, Wk,p(I) (k ≥ 0,p ≥ 1) is the Sobolev space of the Lp-functions f(x)

defined on the interval I whose derivatives

and in particular we denote Wk,2(I) as Hk(I). Further, Lp

Lp-space for a weight function w(x) > 0 with the norm defined as

di

dxif (i = 1,...,k) also belong to Lp(I),

w(I) denotes the weighted

?f?Lp

w=

??

I

w(x)|f(x)|pdx

?1/p

,

Wk,p

w(I) is the weighted Sobolev space with the norm given by

?f?Wk,p

w

=

?

k

?

i=0

?

I

w(x)

????

di

dxif(x)

????

p

dx

?1/p

,

and Hk

w(I) is defined with the norm

?f?Hk

w=

?

k

?

i=0

?

I

w(x)

????

di

dxif(x)

????

2

dx

?1/2

.

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2766

MING MEI, CHUNHUA OU, AND XIAO-QIANG ZHAO

Let T > 0 be a number and B be a Banach space. We denote by C0([0,T],B) the

space of the B-valued continuous functions on [0,T], L2([0,T],B) as the space of the

B-valued L2-functions on [0,T]. The corresponding spaces of the B-valued functions

on [0,∞) are defined similarly.

Recall that a traveling wavefront to (1.1) connecting u±is a solution in the form

of φ(x + ct) with a speed c. Namely, the function φ satisfies the following differential

equation:

⎧

⎩

where?=

Note that the existence of monotone traveling wavefronts of (1.1) can be proved by

the method of upper-lower solutions in a similar way as in [36, 40, 39, 41, 20]. However,

the nonexistence of traveling wavefronts may not be obtained by the linearization of

(2.1) at its zero solution since (2.1) is a mixed-type functional differential equation.

It is easy to see that (1.1) generates a monotone semiflow on C(R,[0,u+]) equipped

with the compact open topology. Consequently, the abstract results in [21, 22] imply

the following result on the existence of the minimum (critical) wave speed.

Lemma 2.1 (existence of traveling waves).

there exist a minimum wave speed (also called the critical wave speed) c∗> 0 and a

corresponding number λ∗= λ(c∗) > 0 satisfying

Fc∗(λ∗) = Gc∗(λ∗),

where

(2.1)

⎨

cφ?− Dφ??+ d(φ) = ε

φ(±∞) = u±,

?∞

−∞

fα(y)b(φ(ξ − y − cτ))dy,

d

dξ, ξ = x + ct.

Under the conditions (H1)–(H3),

F?

c∗(λ∗) = G?

c∗(λ∗),

Fc(λ) = εb?(0)eαλ2−λcτ,Gc(λ) = cλ − Dλ2+ d?(0),

and (c∗,λ∗) is the tangent point of Fc(λ) and Gc(λ), namely,

εb?(0)eαλ2

εb?(0)(2αλ∗− c∗τ)eαλ2

such that for any c ≥ c∗, a monotone traveling wavefront φ(x+ct) of (2.1) connecting

u±exists, and for any c < c∗, no traveling wave φ(x+ ct) exists. When c > c∗, there

exist two numbers depending on c: λ1= λ1(c) > 0 and λ2= λ2(c) > 0 as the solutions

to the equation Fc(λi) = Gc(λi), i.e.,

∗−λ∗c∗τ= c∗λ∗− Dλ2

∗−λ∗c∗τ= c∗− 2Dλ∗,

∗+ d?(0),

εb?(0)eαλ2

i−λicτ= cλi− Dλ2

i+ d?(0),i = 1,2,

such that

Fc(λ) < Gc(λ) for λ1< λ < λ2,

and particularly

Fc(λ∗) < Gc(λ∗) with λ1< λ∗< λ2.

When c = c∗, it holds that

Fc∗(λ∗) = Gc∗(λ∗) with λ1= λ∗= λ2.

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NONLOCAL TIME-DELAYED REACTION-DIFFUSION EQUATIONS

2767

Now we are going to define a weight function. Let x0be sufficiently large so that

2d?(φ(x0)) −ε

ηb?(φ(x0)) − εηb?(0) > 0,

where η > 0 is taken as

0 <d?(u+) −?d?(u+)2− ε2b?(0)b?(u+)

For the choices of x0and η, we refer the details to section 3 (see (3.29)–(3.32) below).

For any given c ≥ c∗, we define

?

1

εb?(0)

< η <d?(u+) +?d?(u+)2− ε2b?(0)b?(u+)

εb?(0)

.

(2.2)w(x) =

e−λ(x−x0)

for x ≤ x0,

for x > x0,

where λ is any fixed number in (λ1,λ∗] when c > c∗, but λ = λ∗when c = c∗. It is

easy to see that w(ξ) ≥ 1 for all ξ ∈ R and w(−∞) = ∞.

Theorem 2.2 (stability of traveling waves). Let d(u) and b(u) satisfy (H1)–(H3).

For a given traveling wave φ(x + ct) of (1.1) with c ≥ c∗ and φ(±∞) = u±, if the

initial data satisfies

0 = u−≤ u0(s,x) ≤ u+∀(s,x) ∈ [−τ,0] × R,

and the initial perturbation u0(s,x)−φ(x+cs) is in C([−τ,0],L1

the solution of (1.1) and (1.2) uniquely exists and satisfies

w(R)∩H1(R)), then

0 = u−≤ u(t,x) ≤ u+∀(t,x) ∈ R+× R,

u(t,x) − φ(x + ct) ∈ C([0,∞),L1

When c > c∗, the solution u(t,x) converges to the noncritical traveling wave φ(x+ct)

exponentially,

w(R) ∩ H1(R)).

sup

x∈R|u(t,x) − φ(x + ct)| ≤ Ce−μt, t > 0,

for a positive constant μ = μ1/3, where μ1= μ1(c,λ) > 0 for λ ∈ (λ1,λ∗] satisfies

(2.3)Gc(λ) − Fc(λ) − μ1− Fc(λ)(eμ1τ− 1) ≥ 0

and

(2.4)d?(u+) − εeμ1τb?(u+) − μ1> 0.

When c = c∗, the solution u(t,x) converges to the critical traveling wave φ(x + c∗t)

algebraically,

sup

x∈R|u(t,x) − φ(x + c∗t)| ≤ Ct−1

2, t > 0.

Remark 1. Theorem 2.2, as applied to monostable evolution equations (1.3)–

(1.6), implies the global stability of the critical wave φ(x + c∗t), which was left open

in the earlier works [10, 18, 23, 28, 29, 26, 27, 30].

Remark 2. For the Fisher–KPP equation (1.7), Theorem 2.2 also provides the

stability of all traveling wavefronts including the critical one, with a time-exponential

decay to the noncritical waves and a time-algebraic decay to the critical wave, which

are the same as those in [9, 15, 31, 35]. For more details, we refer to section 4.

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2768

MING MEI, CHUNHUA OU, AND XIAO-QIANG ZHAO

3. Proof of the global stability. The existence and uniqueness of the solution

to (1.1) and (1.2) can been proved via the standard energy method and continuity-

extension method (cf., [29, 28]) or the theory of abstract functional differential equa-

tions [17], and we omit the details here. The main target in this section is to prove the

stability for all noncritical traveling waves to (1.1) with an exponential convergence

rate and, in particular, the stability for the critical traveling wave with an algebraic

convergence rate. As in [18, 23, 26, 27], we will use the comparison principle and the

weighted energy method to prove the exponential stability for the noncritical waves

in Theorem 2.2, and use the Green function method to prove the algebraic stability

for the critical waves in Theorem 2.2. As usual, the crucial step is to establish the

L2-energy estimate for the solution in a suitable weighted Sobolev space H1

ever, such a weighted L2-energy method cannot be directly applied to the case of

the critical wavefront. Here, we develop a new strategy. Instead of the weighted

L2-energy estimate, we first establish a weighted L1-energy estimate by selecting a

suitable weight function and carefully treating each term. Then using this crucial

L1-estimate, we further obtain the desired L2-energy estimate.

Let c ≥ c∗and the initial data u0(s,x) be such that 0 = u−≤ u0(s,x) ≤ u+for

(s,x) ∈ [−τ,0] × R, and define

?

U−

w. How-

U+

0(s,x) = max{u0(s,x), φ(x + cs)},

0(s,x) = min{u0(s,x), φ(x + cs)},

∀(s,x) ∈ [−τ,0] × R,

which implies

0 = u−≤ U−

0 = u−≤ U−

0(s,x) ≤ u0(s,x) ≤ U+

0(s,x) ≤ φ(x + cs) ≤ U+

0(s,x) ≤ u+∀(s,x) ∈ [−τ,0] × R,

0(s,x) ≤ u+∀(s,x) ∈ [−τ,0] × R.

Let U+(t,x) and U−(t,x) be the corresponding solutions of (1.1) with the initial data

U+

0(s,x), respectively, that is,

?

U±(s,x) = U±

0(s,x),

0(s,x) and U−

∂U±

∂t

− D∂2U±

∂x2

+ d(U±) = ε

R

fα(y)b(U±(t − τ,x − y))dy,

x ∈ R, s ∈ [−τ,0].

By similar arguments as in [23, 26, 27, 18] or the abstract results in [17], it easily

follows that (1.1) admits the comparison principle. Thus, we have

u−≤ U−(t,x) ≤ u(t,x) ≤ U+(t,x) ≤ u+∀(t,x) ∈ R+× R,

u−≤ U−(t,x) ≤ φ(x + ct) ≤ U+(t,x) ≤ u+∀(t,x) ∈ R+× R.

(3.1)

(3.2)

In what follows, we are going to complete the proof for the stability in three steps.

Step 1 (the convergence of U+(t,x) to φ(x + ct)). For any given c ≥ c∗, let

ξ := x + ct and

v(t,ξ) := U+(t,x) − φ(x + ct),v0(s,ξ) := U+

0(s,x) − φ(x + cs).

It follows from (3.1) and (3.2) that

v(t,ξ) ≥ 0,v0(s,ξ) ≥ 0.

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NONLOCAL TIME-DELAYED REACTION-DIFFUSION EQUATIONS

2769

We see from (1.1) that v(t,ξ) satisfies (by linearizing it at 0)

∂v

∂t+ c∂v

∂ξ− D∂2v

?

∂ξ2+ d?(0)v

−εb?(0)

R

fα(y)v(t − τ,ξ − y − cτ)dy

?

fα(y)[b?(φ(ξ − y − cτ) − b?(0)]v(t − τ,ξ − y − cτ)dy

=: I1(t,ξ) + I2(t,ξ) + I3(t,ξ) + I4(t,ξ),

= −Q1(t,ξ) + ε

?

R

fα(y)Q2(t − τ,ξ − y − cτ)dy + [d?(0) − d?(φ(ξ))]v

+ε

R

(3.3)

with the initial data

(3.4)v(s,ξ) = v0(s,ξ), s ∈ [−τ,0],

where

(3.5)Q1(t,ξ) = d(φ + v) − d(φ) − d?(φ)v

with φ = φ(ξ) and v = v(t,ξ), and

(3.6)Q2(t − τ,ξ − y − cτ) = b(φ + v) − b(φ) − b?(φ)v

with φ = φ(ξ −y −cτ) and v = v(t−τ,ξ −y −cτ). Here Ii(t,ξ),i = 1,2,3,4, denotes

the ith term in the right-side of line above (3.3).

Lemma 3.1. It holds that

?t−τ

and

?t−τ

where w1(ξ) = e−λ(ξ−x0), λ is chosen as in (2.2), and μ1> 0 is the small constant

given in (2.3) and (2.4) for c > c∗.

Proof. In order to establish the energy estimate (3.7), technically we need the

good enough regularity for the solution of (3.3) and (3.4). To do it, the usual approach

is via the mollification. Now let us mollify the initial data as

?

where J¯ ?(ξ) is the mollifier. Let v¯ ?(t,ξ) be the solution to (3.3) with the above mollified

initial data. We then have

(3.7)

?v(t)?L1

w1(R)+

0

e−μ1(t−s)?v(s)?2

L2

w1(R)ds ≤ Ce−μ1t

for c > c∗,

(3.8)

?v(t)?L1

w1(R)+

0

?v(s)?2

L2

w1(R)ds ≤ Cfor c = c∗,

v0¯ ?(s,ξ) = (J¯ ?∗ v0)(s,ξ) =

R

J¯ ?(ξ − y)v0(s,y)dy ∈ C0([−τ,0],W2,1

w(R) ∩ H2(R)),

(3.9)v¯ ?(t,ξ) ∈ C0([0,∞),W2,1

w(R) ∩ H2(R)).

To show (3.9), we first prove the local existence for the solution in the designed

solution space within [0,t0] for some t0> 0. Then, by Zorn’s lemma (for example,

see [50]), the solution either globally exists in the given solution space or blows up

Page 9

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2770

MING MEI, CHUNHUA OU, AND XIAO-QIANG ZHAO

at a finite time in the norm of the given space. We further show that, by using the

energy method, for any time T > 0, the solution within the designed space (3.9) is

bounded by a constant depending on T and doesn’t blow up. Consequently, we obtain

the global existence in the given solution space. Here we omit the detail of the proof

since it is rather standard.

Next we are going to derive (3.7) and (3.8) for all t > 0. Multiplying (3.3) by

w1(ξ)eμ1t, where μ1> 0 is given in (2.3) (we will show how to determine it later), we

have

∂

∂t(eμ1tw1v¯ ?) + eμ1t∂

+eμ1t?

−εb?(0)w1eμ1t(ξ)

= eμ1tw1(ξ)[I1(t,ξ) + I2(t,ξ) + I3(t,ξ) + I4(t,ξ)].

∂ξ

?

1+ d?(0)w1

cw1v¯ ?− Dw1v¯ ?ξ+ Dw?

?

fα(y)v¯ ?(t − τ,ξ − y − cτ)dy

1v¯ ?

?

− μ1eμ1tw1v¯ ?

(3.10)

− cw?

1− Dw??

v¯ ?

?

R

Integrating the above equation over R × [0,t] with respect to ξ and t gives

?

?t

−εb?(0)

0

R

?t

eμ1t

R

w1(ξ)v¯ ?(t,ξ)dξ

?

?t

= ?v0¯ ?(0)?L1

+

0

R

eμ1s?

− cw?

1(ξ) − Dw??

??

eμ1sw1(ξ)(I1+ I2+ I3+ I4)dξds.

1(ξ) + d?(0)w1(ξ) − μ1w1(ξ)

?

dξds

v¯ ?(s,ξ)dξds

?

eμ1sw1(ξ)

?

R

fα(y)v¯ ?(s − τ,ξ − y − cτ)dy

?

w1+

0

R

(3.11)

Here we have used (3.9) to ensure that the integral of the second term in (3.10) is

zero. By applying Taylor’s expansion to (3.5) and (3.6) and noting (H3), we have

Q1(t,ξ) = d(φ + v¯ ?) − d(φ) − d?(φ)v¯ ?= d??(¯φ1)v2

Q2(t − τ,ξ − cτ) = b(φ + v¯ ?) − b(φ) − b?(φ)v¯ ?= b??(¯φ2)v2

¯ ?≥ C1v2

¯ ?≤ −C2v2

¯ ?,

¯ ?,

for some¯φ1,¯φ2 ∈ [0,φ + v¯ ?] and nonnegative constants Ci ≥ 0 (i = 1,2) with

C1+ C2> 0, namely, at least one of C1and C2is positive (see (H3)), which implies

?t

?t

0

?

?

R

eμ1sw1(ξ)I1(s,ξ)dξds ≤ −C1

?t

?t

fα(y) × v2

0

eμ1s

?

R

?

w1(ξ)v2

¯ ?(s,ξ)dξds,(3.12)

0

R

eμ1sw1(ξ)I2(s,ξ)dξds ≤ −εC2

0

eμ1s

R

w1(ξ)

??

R

¯ ?(s − τ,ξ − y − cτ)dy

?

dξds.(3.13)

Notice from (H3) that d?(u) is increasing and b?(u) is decreasing, which implies

d?(0) − d?(φ) ≤ 0 and b?(φ) − b?(0) ≤ 0 for φ ≥ 0,

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NONLOCAL TIME-DELAYED REACTION-DIFFUSION EQUATIONS

2771

namely,

I3(t,ξ) ≤ 0 and I4(t,ξ) ≤ 0.

Thus, we have

(3.14)

?t

0

?

R

eμ1sw1(ξ)[I3(s,ξ) + I4(s,ξ)]dξds ≤ 0.

Applying (3.12), (3.13), and (3.14) to (3.11) gives

eμ1t?v¯ ?(s)?L1

?t

w1(R)

+

0

?

R

eμ1s{−cw?

?t

?t

?t

1(ξ) − Dw??

1(ξ) + d?(0)w1(ξ) − μ1w1(ξ)}v¯ ?(s,ξ)dξds

− εb?(0)

0

?

R

eμ1sw1(ξ)

??

R

fα(y)v¯ ?(s − τ,ξ − y − cτ)dy

?

dξds

+ C1

0

?

R

?

eμ1sw1(ξ)v2

¯ ?(s,ξ)dξds

+ εC2

0

R

eμ1sw1(ξ)

??

R

fα(y)v2

¯ ?(s − τ,ξ − y − cτ)dy

?

dξds

≤ ?v0¯ ?(0)?L1

w1(R).(3.15)

By changing variables y → y, ξ − y − cτ → ξ, s − τ → s, and using the fact

?

R

fα(y)w1(ξ + y + cτ)

w1(ξ)

dy = eαλ2−λcτ,

we obtain

εb?(0)

?t

0

?

R

eμ1sw1(ξ)

??

R

fα(y)v¯ ?(s − τ,ξ − y − cτ)dy

??

??

??

?

?0

?

dξds

= εb?(0)

?t−τ

?t−τ

?0

−τ

?

?

?

R

eμ1(s+τ)

R

w1(ξ + y + cτ)fα(y)dy

?

?

v¯ ?(s,ξ)dξds

= εb?(0)

0

R

eμ1(s+τ)

R

w1(ξ + y + cτ)

w1(ξ)

fα(y)dyw1(ξ)v¯ ?(s,ξ)dξds

+ εb?(0)

−τ

R

eμ1(s+τ)

R

w1(ξ + y + cτ)

w1(ξ)

fα(y)dy

?

w1(ξ)v0¯ ?(s,ξ)dξds

≤ εb?(0)eαλ2−λcτeμ1τ

?t

0

R

eμ1sw1(ξ)v¯ ?(s,ξ)dξds

+ εb?(0)eαλ2−λcτeμ1τ

−τ

eμ1s?v0¯ ?(s)?L1

w1(R)ds(3.16)

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2772

MING MEI, CHUNHUA OU, AND XIAO-QIANG ZHAO

and

εC2

?t

0

?

R

eμ1sw1(ξ)

?t−τ

?t−τ

?0

≥ εC2eαλ2−λcτeμ1τ

??

R

fα(y)v2

?

??

??

eμ1s?v¯ ?(s)?2

¯ ?(s − τ,ξ − y − cτ)dy

?

dξds

= εC2

−τ

?

?

?

R

eμ1(s+τ)

R

w1(ξ + y + cτ)fα(y)v2

¯ ?(s,ξ)dydξds

?

?

= εC2

0

R

eμ1(s+τ)

R

w1(ξ + y + cτ)

w1(ξ)

w1(ξ + y + cτ)

w1(ξ)

fα(y)dyw1(ξ)v2

¯ ?(s,ξ)dydξds

+ εC2

−τ

R

eμ1(s+τ)

R

fα(y)dyw1(ξ)v2

0¯ ?(s,ξ)dydξds

?t−τ

0

L2

w1(R)ds.(3.17)

Substituting (3.17) and (3.16) to (3.15), we then get

eμ1t?v¯ ?(t)?L1

w1(R)+

?t

?t

0

?

R

eμ1s¯A(c,μ1,ξ)w1(ξ)v¯ ?(s,ξ)dξds

+ C1

?

0

eμ1s?v¯ ?(s)?2

L2

?0

w1(R)ds + εC2eαλ2−λcτeμ1τ

?t−τ

0

eμ1s?v¯ ?(s)?2

L2

w1(R)ds

≤ C

?v0¯ ?(0)?L1

w1(R)+

−τ

?v0¯ ?(s)?L1

w1(R)ds

?

, (3.18)

where

¯A(c,μ1,ξ) := A(c,ξ) − μ1− εb?(0)eαλ2−λcτ[eμ1τ− 1]

and

A(c,ξ) := −cw?

1(ξ)

w1(ξ)− Dw??

1(ξ)

w1(ξ)+ d?(0) − εb?(0)eαλ2−λcτ.

Using the facts that cλ−Dλ2+d?(0) > ( or =)εb?(0)eαλ2−λcτfor c > ( or =)c∗and

that w1(ξ) = e−λ(ξ−x0), we further obtain

A(c,ξ) = cλ − Dλ2+ d?(0) − εb?(0)eαλ2−λcτ

?

Gc∗(λ∗) − Fc∗(λ∗) = 0

=

Gc(λ) − Fc(λ) > 0for c > c∗, λ1< λ ≤ λ∗,

for c = c∗, λ1= λ = λ∗.

Thus, when c > c∗, we choose a small μ1> 0 such that

¯A(c,μ1,ξ) = A(c,ξ) − μ1− εb?(0)eαλ2−λcτ[eμ1τ− 1]

= Gc(λ) − Fc(λ) − μ1− Fc(λ)(eμ1τ− 1)

≥ 0,(3.19)

and when c = c∗, we can take only μ1= 0 such that

(3.20)

¯A(c∗,0,ξ) = A(c∗,ξ) ≥ 0.

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NONLOCAL TIME-DELAYED REACTION-DIFFUSION EQUATIONS

2773

Applying (3.19) for c > c∗ and (3.20) for c = c∗ to (3.18) and noting that

C1+C2> 0, namely, at least one of them is positive, we then establish the following

key energy estimate:

?t−τ

and

?t−τ

Letting ¯ ? → 0 in (3.21) and (3.22), we finally arrive at

?t−τ

and

?t−τ

This proves (3.7) and (3.8).

Lemma 3.2. For any c ≥ c∗, it holds that

?t

(3.21)eμ1t?v¯ ?(t)?L1

w1(R)+

0

eμ1s?v¯ ?(s)?2

L2

w1(R)ds ≤ C for c > c∗

(3.22)

?v¯ ?(t)?L1

w1(R)+

0

?v¯ ?(s)?2

L2

w1(R)ds ≤ C for c = c∗.

?v(t)?L1

w1(R)+

0

e−μ1(t−s)?v(s)?2

L2

w1(R)ds ≤ Ce−μ1tfor c > c∗

?v(t)?L1

w1(R)+

0

?v(s)?2

L2

w1(R)ds ≤ C for c = c∗.

(3.23)

?v(t)?2

L2(R)+

0

?v(s)?2

H1(R)ds ≤ C,t ≥ 0.

Proof. Since w1(ξ) = e−λ(ξ−x0)≥ 1 for ξ ∈ (−∞,x0], (3.7) and (3.8) guarantee

that for any c ≥ c∗,

?x0

and in particular by taking t = ∞, we have

?∞

Although we cannot directly work on the original equations (3.3) and (3.4) due

to the lack of regularity for the solution as illustrated in the proof of Lemma 3.1,

we can get a mollified solution first and then take the limit to get the corresponding

energy estimate for the original solution v(t,ξ). Therefore, for the sake of simplicity,

we formally use v(t,ξ) to establish the desired energy estimates in what follows.

Let us multiply (3.3) by v(t,ξ) and integrate it over R × [0,t] with respect to ξ

and t. Then we have

?t

− 2ε

0

R

?t

?t

−∞

v(t,ξ)dξ +

?t−τ

0

?x0

−∞

v2(s,ξ)dξds ≤ C for all t ≥ 0,

(3.24)

0

?x0

−∞

v2(s,ξ)dξds ≤ C.

?v(t)?2

L2(R)+ 2D

?t

0

?vξ(s)?2

L2(R)ds + 2

?t

0

?

R

d?(φ(ξ))v2(s,ξ)dξds

??

R

fα(y)b?(φ(ξ − y − cτ))v(s − τ,ξ − y − cτ)v(s,ξ)dydξds

?

fα(y)v(s,ξ)Q2(s − τ,ξ − y − cτ)dydξds.

= ?v0(0)?2

L2(R)− 2

?

0

R

v(s,ξ)Q1(s,ξ)dξds

+ 2ε

0

R

?

R

(3.25)

Page 13

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2774

MING MEI, CHUNHUA OU, AND XIAO-QIANG ZHAO

Using the Cauchy inequality |ab| ≤η

we obtain

?t

≤ε

η

0

?t

By changing variables y → y, ξ − y − cτ → ξ, s − τ → s, we have

ε

η

0

R

=ε

η

−τ

=ε

η

0

R

≤ C +ε

η

0

−∞

≤ C +ε

0x0

Here we have used (3.24) and the fact that b?(φ(ξ)) is decreasing. Similarly, we can

obtain

?t

≤ εηb?(0)

0

R

?t

≤ C + εηb?(0)

0x0

Substituting (3.27) and (3.28) to (3.26) yields

?t

≤ C +ε

0x0

By applying (3.29) to (3.25) and noting that v(t,ξ) ≥ 0, Q1(t,ξ) ≥ 0, and Q2(t −

τ,ξ − y − cτ) ≤ 0, we then obtain

?t

?t

≤ C.

2a2+1

2ηb2for η > 0, which will be specified later,

2ε

0

?

R

?

R

fα(y)b?(φ(ξ − y − cτ))v(s − τ,ξ − y − cτ)v(s,ξ)dydξds

?

?

?t

?

RR

fα(y)b?(φ(ξ − y − cτ))v2(s − τ,ξ − y − cτ)dydξds

?

+ εη

0

RR

fα(y)b?(φ(ξ − y − cτ))v2(s,ξ)dydξds.(3.26)

?t

??

R

fα(y)b?(φ(ξ − y − cτ))v2(s − τ,ξ − y − cτ)dydξds

?

?

?t−τ

ηb?(φ(x0))

?t−τ

?t−τ

R

??

b?(φ(ξ))v2(s,ξ)dξds +ε

R

fα(y)dy

?

b?(φ(ξ))v2(s,ξ)dξds

η

?0

−τ

?

R

b?(φ(ξ))v2

0(s,ξ)dξds

?∞

?x0

b?(φ(ξ))v2(s,ξ)dξds +ε

η

?t−τ

0x0

b?(φ(ξ))v2(s,ξ)dξds

?t

?∞

v2(s,ξ)dξds.(3.27)

εη

0

?

R

?

?t

R

fα(y)b?(φ(ξ − y − cτ))v2(s,ξ)dydξds

?

?x0

?t

??

R

fα(y)dy

?

v2(s,ξ)dξds

= εηb?(0)

0

−∞

v2(s,ξ)dξds + εηb?(0)

?t

0

?∞

x0

v2(s,ξ)dξds

?∞

v2(s,ξ)dξds. (3.28)

2ε

0

?

R

?

R

fα(y)b?(φ(ξ − y − cτ))v(s − τ,ξ − y − cτ)v(s,ξ)dydξds

?t

ηb?(φ(x0))

?∞

v2(s,ξ)dξds + εηb?(0)

?t

0

?∞

x0

v2(s,ξ)dξds.(3.29)

?v(t)?2

L2(R)+ 2D

?∞

0

?vξ(s)?2

L2(R)ds + 2

?t

0

?x0

−∞

d?(φ(ξ))v2(s,ξ)dξds

+

0x0

?

2d?(φ(ξ)) −ε

ηb?(φ(x0)) − εηb?(0)

?

v2(s,ξ)dξds

(3.30)

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NONLOCAL TIME-DELAYED REACTION-DIFFUSION EQUATIONS

2775

Since d?(u+)2> ε2b?(0)b?(u+) (see (H2)), we can choose η > 0 such that

0 <d?(u+) −?d?(u+)2− ε2b?(0)b?(u+)

εb?(0)

< η <d?(u+) +?d?(u+)2− ε2b?(0)b?(u+)

εb?(0)

.

It then follows that

(3.31)2d?(u+) −ε

ηb?(u+) − εηb?(0) > 0.

Thus, choosing x0sufficiently large such that |φ(x0) − u+| ? 1, we reach

C3:= 2d?(φ(x0)) −ε

(3.32)

ηb?(φ(x0)) − εηb?(0) > 0.

Since d?(u+) ≥ d?(φ(ξ) ≥ d?(φ(x0)) for ξ ∈ [x0,∞) (from (H3), d?(φ(ξ)) is increasing),

we see that (3.32) implies

(3.33)2d?(φ(ξ)) −ε

ηb?(φ(x0)) − εηb?(0) ≥ C3> 0, ξ ∈ [x0,∞).

Applying (3.33) to (3.30) and adding it with (3.24), that is, C3

C, we further obtain

?t

This proves (3.23).

Next we derive the L2-energy estimate for vξ(t,ξ). Let us differentiate (3.3) with

respect to ξ and multiply the resulting equation by vξ(t,ξ) and then integrate it over

R× [0,t] with respect to ξ and t. By using the key estimates (3.23), we can similarly

obtain the following high order estimate. The detail of proof is omitted.

Lemma 3.3. For any c ≥ c∗, it holds that

?t

Based on the above lemmas, we can prove the following two convergence results.

One is the exponential stability for the noncritical traveling waves with c > c∗, and

the other one is the algebraic stability for the critical traveling wave with c = c∗. We

first prove the exponential stability.

Lemma 3.4. For any c > c∗, there holds

?v(t)?L∞(−∞,x0]≤ Ce−μ1t/3,

Proof. Let I = (−∞,x0]. Then we have

?x0

and

?ξ

?t

0

?x0

−∞v2(s,ξ)dξds ≤

?v(t)?2

L2(R)+ 2D

0

?vξ(s)?2

L2(R)ds + C3

?t

0

?v(s)?2

L2(R)ds ≤ C.

(3.34)

?vξ(t)?2

L2(R)+

0

?vξ(s)?2

H1(R)ds ≤ C,t ≥ 0.

t ≥ 0.

(3.35)

?v(t)?2

L2(I)=

−∞

|v(ξ,t)|2dξ ≤ ?v(t)?L∞(I)?v(t)?L1(I),

v2(ξ,t) =

−∞

∂ξ(v2)dξ = 2

?ξ

−∞

v(ξ,t)vξ(ξ,t)dξ,

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2776

MING MEI, CHUNHUA OU, AND XIAO-QIANG ZHAO

which, by the H¨ older inequality, implies the following Sobolev inequality:

(3.36)

?v(t)?2

L∞(I)≤ 2?v(t)?L2(I)?vξ(t)?L2(I).

Combining (3.35) and (3.36), we obtain

(3.37)

?v(t)?L∞(I)≤

3√4?v(t)?

1

3

L1(I)?vξ(t)?

2

3

L2(I).

In view of ?vξ(t)?L2(I)≤ C from (3.34), w1(ξ) = e−λ(ξ−x0)≥ 1 for ξ ∈ I = (−∞,x0],

and (3.7), it follows that

(3.38)

?v(t)?L1(I)≤ ?v(t)?L1

w1(I)≤ Ce−μ1t.

Thus, (3.37) and (3.38) immediately yield

?v(t)?L∞(I)≤ Ce−μ1t/3,t ≥ 0,

and

v(t,x0) ≤ Ce−μ1t/3,t ≥ 0.

This completes the proof.

Now we are going to prove the exponential stability for noncritical traveling waves

in [x0,∞).

Lemma 3.5. For any c > c∗, there holds

?v(t)?L∞[x0,∞)≤ Ce−μ1t/3,

Proof. Multiplying (3.3) by eμ1tand integrating it with respect to (ξ,t) over

R × [0,t], and noting that −Q1≤ 0 and Q2≤ 0, we have

?t

− ε

0

R

≤ ?v0(0)?L1(R).

As shown before, by the change of variables ξ −y −cτ → ξ and s−τ → s, and using

the fact?

ε

0

R

?t−τ

= εeμ1τ

−τ

Substituting (3.40) into (3.39), we obtain

?t

− μ1

0

≤ ?v0(0)?L1(R).

t ≥ 0.

eμ1t?v(t)?L1(R)+

?t

(3.39)

0

eμ1s

?

R

d?(φ(ξ))v(t,ξ)dξds − μ1

?t

0

eμ1s?v(s)?L1(R)ds

??

R

eμ1sfα(y)b?(φ(ξ − y − cτ))v(s − τ,ξ − y − cτ)dydξds

Rfα(y)dy = 1, we have

?t

?

?t−τ

??

R

eμ1sfα(y)b?(φ(ξ − y − cτ))v(t − τ,ξ − y − cτ)dydξds

?

?

= ε

−τ

R

eμ1τfα(y)dy

R

eμ1sb?(φ(ξ))v(s,ξ)dξds

R

eμ1sb?(φ(ξ))v(s,ξ)dξds. (3.40)

eμ1t?v(t)?L1(R)+

0

eμ1s

?

R

d?(φ(ξ))v(t,ξ)dξds

?t

eμ1s?v(s)?L1(R)ds − εeμ1τ

?t−τ

−τ

?

R

eμ1sb?(φ(ξ))v(s,ξ)dξds