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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 ≤ C for 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)
Page 11
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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)
Page 14
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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,
Page 15
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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
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NONLOCAL TIME-DELAYED REACTION-DIFFUSION EQUATIONS
2777
Splitting each integral on the above inequality into two parts due to R = (−∞,x0] ∪
[x0,∞), we get
?∞
?t
≤ ?v0(0)?L1(R)− J(t),
where
?x0
?t
In view of Lemma 3.1 and w1(ξ) ≥ 1 in (−∞,x0], we obtain
?x0
which, together with the boundedness of d?(φ(ξ)) and b?(φ(ξ)), implies that
eμ1t
x0
v(t,ξ)dξ +
?t
?∞
0
eμ1s
?∞
x0
d?(φ(ξ))v(t,ξ)dξds
− μ1
0
eμ1s
x0
v(s,ξ)dξds − εeμ1τ
?t−τ
0
?∞
x0
eμ1sb?(φ(ξ))v(s,ξ)dξds
(3.41)
J(t) : = eμ1t
−∞
v(t,ξ)dξ +
?t
0
eμ1s
?x0
−∞
d?(φ(ξ))v(t,ξ)dξds
−μ1
0
eμ1s
?x0
−∞
v(s,ξ)dξds − εeμ1τ
?t−τ
−τ
?x0
−∞
eμ1sb?(φ(ξ))v(s,ξ)dξds.
eμ1t
−∞
v(t,ξ)dξ ≤ C and
?t
0
eμ1s
?x0
−∞
v(t,ξ)dξds ≤ C,
(3.42)
|J(t)| ≤ C for t ≥ 0.
Applying (3.42) to (3.41), we then get
(3.43) eμ1t
?∞
x0
v(t,ξ)dξ+
?t
0
eμ1s
?∞
x0
[d?(φ(ξ))−εeμ1τb?(φ(ξ))−μ1]v(s,ξ)dξds ≤ C.
Since u+is stable (see (H2)), i.e., d?(u+) − εb?(u+) > 0, there exists a small μ1> 0
so that
d?(u+) − εeμ1τb?(u+) − μ1> 0.
By the continuity of limξ→+∞φ(ξ) = v+, it then follows that for all x0? 1, we have
(3.44)d?(φ(ξ)) − εb?(φ(ξ)) − μ1≥ 0, ξ ∈ [x0,∞).
Applying (3.44) to (3.43), we see that
?∞
Based on this inequality, as shown in Lemma 3.4, we can similarly prove the
following convergence in [x0,∞):
3√4?v(t)?
x0
v(t,ξ)dξ ≤ Ce−μ1t.
?v(t)?L∞[x0,∞)≤
1
3
L1[x0,∞)?vξ(t)?
2
3
L2[x0,∞)≤ Ce−μ1t
3.
This completes the proof.
Combining Lemmas 3.4 and 3.5, we have the following result.
Page 17
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2778
MING MEI, CHUNHUA OU, AND XIAO-QIANG ZHAO
Lemma 3.6. For any c > c∗, there holds
sup
x∈R|U+(t,x) − φ(x + ct)| = ?v(t)?L∞(R)≤ Ce−μ1t/3,t ≥ 0.
Next, we are going to prove the algebraic stability for the critical traveling wave
with c = c∗. In this case, we have λ = λ1= λ∗. Using the linearization of (3.3) at 0,
we can rewrite (3.3) as
∂v
∂t+ c∗∂v
∂ξ− D∂2v
?
∂ξ2+ d?(0)v
− εb?(0)
R
fα(y)v(t − τ,ξ − y − c∗τ)dy
?
fα(y)[b?(φ(ξ − y − c∗τ)) − b?(0)]v(t − τ,ξ − y − c∗τ)dy.
= −Q1(t,ξ) + ε
?
R
fα(y)Q2(t − τ,ξ − y − c∗τ)dy + [d?(0) − d?(φ(ξ))]v
+ ε
R
From (H2) and (H3), we see that
−Q1(t,ξ) ≤ 0,Q2(t − τ,ξ − y − cτ) ≤ 0,
and
d?(0) − d?(φ(ξ)) ≤ 0,b?(φ(ξ − y − c∗τ) − b?(0) ≤ 0.
It then follows that
∂v
∂t+ c∗∂v
∂ξ− D∂2v
∂ξ2+ d?(0)v − εb?(0)
?
R
fα(y)v(t − τ,ξ − y − c∗τ)dy ≤ 0.
Let ¯ v(t,ξ) be the solution of the following equation with the same initial data
v0(s,ξ):
(3.45)
⎧
⎪
By the comparison principle, we have
⎪
⎪
⎪
⎩
⎨
∂¯ v
∂t+ c∗∂¯ v
∂ξ− D∂2¯ v
∂ξ2+ d?(0)¯ v − εb?(0)
?
R
fα(y)¯ v(t − τ,ξ − y − c∗τ)dy = 0,
(t,ξ) ∈ R+× R,
¯ v(s,ξ) = v0(s,ξ),s ∈ [−τ,0],x ∈ R.
v(t,ξ) ≤ ¯ v(t,ξ)for (t,ξ) ∈ R+× R.
Let
(3.46)˜ v(t,ξ) := w1(ξ)¯ v(t,ξ).
From (3.45), we see that ˜ v(t,ξ) satisfies
∂˜ v
∂t+ k1∂˜ v
∂ξ− D∂2˜ v
∂ξ2+ k2˜ v = εb?(0)
?
R
fα(y)e−λ∗(y+c∗τ)˜ v(t − τ,ξ − y − c∗τ)dy,
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NONLOCAL TIME-DELAYED REACTION-DIFFUSION EQUATIONS
2779
where
k1:= c∗− 2Dλ∗ and k2:= c∗λ∗− Dλ2
∗+ d?(0) > 0.
Furthermore, letting
(3.47)ˆ v(t,ξ) := ek2t˜ v(t,ξ),
we have
∂ˆ v
∂t+ k1∂ˆ v
∂ξ− D∂2ˆ v
∂ξ2= εb?(0)ek2τ
?
R
fα(y)e−λ∗(y+c∗τ)ˆ v(t − τ,ξ − y − c∗τ)dy,
which is equivalent to
ˆ v(t,ξ) =
?
+ εb?(0)ek2τ
R
G(t,ξ − ζ)ˆ v0(0,ζ)dζ
?t
×
R
0
?
R
G(t − s,ξ − ζ)
?
fα(y)e−λ∗(y+c∗τ)ˆ v(s − τ,ζ − y − c∗τ)dydζds, (3.48)
where the Green function G(t,ξ − ζ) is defined as
G(t,ξ − ζ) =
1
√4πDt
e−(ξ−ζ+k1t)2
4Dt
.
Lemma 3.7. It holds that
(3.49)
?ˆ v(t)?L∞(R)≤ C(1 + t)−1
2ek2t, t > 0.
Proof. Note that
εb?(0)
?
R
fα(y)e−λ∗(y+c∗τ)dy = εb?(0)eαλ2
∗−c∗λ∗τ= k2
and
(3.50)0 < G(t,ξ) ≤ (4πD)−1
2t−1
2
and
?
R
G(t,ξ)dξ = 1.
If
(3.51)
?ˆ v(t − τ)?L∞ ≤ C(θ + t − τ)−1
2ek2(t−τ),t ≥ 0
where we take θ > 2τ in order to avoid the singularity, then (3.48) together with
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2780
MING MEI, CHUNHUA OU, AND XIAO-QIANG ZHAO
(3.50) implies that for all t > 0,
?ˆ v(t)?L∞(R)≤ t−1
2?ˆ v0?L1(R)
+ εb?(0)ek2τ
?t
×
0
?ˆ v(s − τ)?L∞(R)
?
?t
(θ + s − τ)−1
?
??
?−1
?
R
G(t − s,ξ − ζ)
R
fα(y)e−λ∗(y+c∗τ)dydζds
≤ C
?
?
?
?
t−1
2+ k2ek2τ
0
(θ + s − τ)−1
2ek2(s−τ)ds
?
= Ct−1
2+ k2
?
t
2
0
2ek2sds + k2
?t
?
?1
t
2
(θ + s − τ)−1
2ek2sds
?
≤ Ct−1
2+ k2e
k2t
2
t
2
0
(θ + s − τ)−1
2ds + k2
θ +t
2− τ
?−1
2?t
t
2
ek2sds
?
= Ct−1
2+ 2k2e
k2t
2
θ +t
2− τ
?1
2
−
?
?
θ − τ
2
?
+
?
θ +t
2− τ
2
[ek2t− ek2t
2]
≤ C
?
t−1
2+ e
k2t
2 (θ + t − τ)
1
2+ (θ + t − τ)−1
2ek2t?
. (3.52)
Since ?ˆ v(0)?L∞(R)≤ C, ˆ v(t,ξ) has no singularity for t around 0, and hence the first
term t−1
way in the heat equation). It then follows from (3.52) that
?
≤ C(θ + t)−1
≤ C(θ + t)−1
Here we have used the fact that
2 on the last line of (3.52) could be replaced by (θ+t)−1
2 (this is the standard
?ˆ v(t)?L∞(R)≤ C(θ + t)−1
2+ e
2ek2t?
2ek2t.
k2t
2 (θ + t − τ)
e−k2t+ e−k2t
1
2+ (θ + t − τ)−1
2ek2t?
2 (θ + t − τ) + 1
?
(3.53)
e−k2t
2(θ + t − τ) ≤ C, ∀t ≥ 0.
It is easy to check that
?ˆ v(t − τ)?L∞(R)= ?ˆ v0(t − τ)?L∞(R)≤ C(θ + t)−1
namely, (3.51) holds for all t ∈ [0,τ]. It then follows from (3.53) that
(3.54)
?ˆ v(t)?L∞(R)≤ C(1 + t)−1
Next, for any t ∈ [τ,2τ], i.e., t − τ ∈ [0,τ], (3.54) immediately implies that
?ˆ v(t − τ)?L∞(R)≤ C(1 + t)−1
and hence (3.51) holds for all t ∈ [τ,2τ]. Thus, we can apply (3.53) to get
(3.55)
?ˆ v(t)?L∞(R)≤ C(1 + t)−1
2ek2(t−τ), ∀t ∈ [0,τ],
2ek2t, ∀t ∈ [0,τ].
2ek2(t−τ), ∀t ∈ [τ,2τ],
2ek2t, ∀t ∈ [τ,2τ].
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NONLOCAL TIME-DELAYED REACTION-DIFFUSION EQUATIONS
2781
For t ∈ [nτ,(n + 1)τ], by repeating this procedure, we then obtain
?ˆ v(t)?L∞ ≤ C(1 + t)−1
Combining (3.54), (3.55), and (3.56), we see that (3.49) holds for all t ≥ 0.
As a consequence of (3.46), (3.47), and (3.49), we have the following result on the
algebraic decay for ¯ v(t,ξ).
Lemma 3.8. For c = c∗, it holds that
(3.56)
2ek2t, ∀t ∈ [nτ,(n + 1)τ].
?¯ v(t)?L∞
w1(R)≤ C(1 + t)−1
2,
∀t > 0.
Since v(t,ξ) ≤ ¯ v(t,ξ), we immediately obtain the following decay for v in the case
of c = c∗.
Lemma 3.9. For c = c∗, it holds that
?v(t)?L∞
w1(R)≤ C(1 + t)−1
2, ∀t > 0
and
(3.57)
?v(t)?L∞(−∞,x0]≤ C(1 + t)−1
2, ∀t > 0,
due to w1(ξ) ≥ 1 for ξ ∈ (−∞,x0].
On the other hand, v possesses the exponential decay for ξ ∈ [x0,∞) (because
v = u+is stable):
(3.58)lim
ξ→∞v(t,ξ) ≤ Ce−μ2t,
∀t > 0
for some μ2satisfying
0 < μ2< d?(u+) − εb?(u+).
Thus, (3.57) and (3.58) lead to the following algebraic decay property.
Lemma 3.10. For c = c∗, there holds
x∈R|U+(t,x) − φ(x + c∗t)| = ?¯ v(t)?L∞(R)≤ C(1 + t)−1
Combining Lemmas 3.6 and 3.10, we have the following result.
Lemma 3.11. There hold the exponential decay
sup
2,
∀t > 0.
sup
x∈R|U+(t,x) − φ(x + ct)| ≤ Ce−μt,
and the algebraic decay
∀t ≥ 0, for c > c∗,
sup
x∈R|U+(t,x) − φ(x + c∗t)| ≤ C(1 + t)−1
2,
∀t ≥ 0, for c = c∗.
Step 2 (the convergence of U−(t,x) to φ(x + ct)). For any given c ≥ c∗, let
ξ = x + ct and
v(t,ξ) = φ(x + ct) − U−(t,x),
As in Step 1, we can similarly prove that U−(t,x) converges to φ(x + ct) as follows.
v0(s,ξ) = φ(x + cs) − U−
0(s,x).
Page 21
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2782
MING MEI, CHUNHUA OU, AND XIAO-QIANG ZHAO
Lemma 3.12. There hold the exponential decay
sup
x∈R|U−(t,x) − φ(x + ct)| ≤ Ce−μt,
and the algebraic decay
∀t ≥ 0, for c > c∗,
sup
x∈R|U−(t,x) − φ(x + c∗t)| ≤ C(1 + t)−1
2,
∀t ≥ 0, for c = c∗.
Step 3 (the convergence of u(t,x) to φ(x + ct)). Finally, we prove that u(t,x)
converges to φ(x + ct) as follows.
Lemma 3.13. There hold the exponential decay
sup
x∈R|U−(t,x) − φ(x + ct)| ≤ Ce−μt, ∀t ≥ 0, for c > c∗,
and the algebraic decay
sup
x∈R|U−(t,x) − φ(x + c∗t)| ≤ C(1 + t)−1
Proof. Since the initial data satisfy U−
son principle implies that
2, ∀t ≥ 0, for c = c∗.
0(s,x) ≤ u0(s,x) ≤ U+
0(s,x), the compari-
U−(t,x) ≤ u(t,x) ≤ U+(t,x),
∀(t,x) ∈ R+× R.
Thanks to Lemmas 3.11 and 3.12, we have the following convergence results:
sup
x∈R|U±(t,x) − φ(x + ct)| ≤ Ce−μt, ∀t ≥ 0, for c > c∗,
and
sup
x∈R|U±(t,x) − φ(x + c∗t)| ≤ C(1 + t)−1
By these inequalities and the squeezing argument, it then follows that
2, ∀t ≥ 0, for c = c∗.
sup
x∈R|u(t,x) − φ(x + ct)| ≤ Ce−μt, ∀t ≥ 0, for c > c∗,
and
sup
x∈R|u(t,x) − φ(x + c∗t)| ≤ C(1 + t)−1
This completes the proof.
2, ∀t ≥ 0, for c = c∗.
4. Applications. In this section, we apply Theorem 2.2 to the monostable evo-
lution equations mentioned in (1.3)–(1.7) to obtain the global stability of all traveling
waves, including the critical one.
4.1. Nonlocal Nicholson’s blowflies equation. Consider the nonlocal Nichol-
son’s blowflies equation
⎧
⎩
(4.1)
⎨
∂u
∂t− D∂2u
u(s,x) = u0(s,x), s ∈ [−τ,0], x ∈ R.
∂x2+ δu = εp
?
R
fα(y)u(t − τ,x − y)e−au(t−τ,x−y)dy,
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NONLOCAL TIME-DELAYED REACTION-DIFFUSION EQUATIONS
2783
Here the death rate function is d(u) = δu and the birth rate is b(u) = pue−auwith
δ > 0, p > 0, and a > 0. The constant equilibria of (4.1) is
u−= 0 and u+=1
alnεp
δ.
For 1 <εp
where the condition d?(u+)2> ε2b?(0)b?(u+) is equivalent to
automatically holds when
δ≈ e. From Theorem 2.2, we immediately obtain the
following result.
Theorem 4.1. Let 1 <
δ≤ e and
φ(x + ct) of (4.1) with c ≥ c∗and φ(±∞) = u±, if the initial data satisfy
0 = u−≤ u0(s,x) ≤ u+,
and the initial perturbation u0(s,x)−φ(x+cs) is in C([−τ,0];L1
the solution of (4.1) converges to the traveling wave φ(x + ct) in the sense that
δ≤ e, it can be easily checked that the conditions (H1)–(H3) are satisfied,
δ
εp> 1 − lnεp
δ, which
εp
εp
δ
εp> 1 − lnεp
δ. For a given traveling wave
for (s,x) ∈ [−τ,0] × R,
w(R)∩H1(R)), then
sup
x∈R|u(t,x) − φ(x + ct)| ≤ Ce−μt, ∀t ≥ 0, for c > c∗ and μ = μ(c) > 0,
and
sup
x∈R|u(t,x) − φ(x + c∗t)| ≤ C(1 + t)−1
2, ∀t ≥ 0, for c = c∗.
Remark 3. The stability of traveling wavefronts of (4.1) was studied earlier in [28]
for fast waves and then improved in [26] for all slow waves under the condition that
α ? 1 if c ≈ c∗. Here we obtain the stability for all waves, including the critical wave,
without any restriction on the delay time τ and α (the total diffusion of immature
population).
4.2. Local Nicholson’s blowflies equation. Let α → 0. We then reduce
the nonlocal Nicholson’s blowflies equation (4.1) to the following local Nicholson’s
blowflies equation:
⎧
⎩
By a similar calculation as in section 3, we find that the condition
which is needed in Theorem 4.1, can be removed. Our new stability is as follows.
Theorem 4.2. Let 1 <
δ≤ e. For a given traveling wave φ(x + ct) of (4.2)
with c ≥ c∗and φ(±∞) = u±, if the initial data satisfy
0 = u−≤ u0(s,x) ≤ u+,
and the initial perturbation u0(s,x)−φ(x+cs) is in C([−τ,0];L1
the solution of (4.2) converges to the traveling wave φ(x + ct) in the sense that
(4.2)
⎨
∂u
∂t− D∂2u
u(s,x) = u0(s,x), s ∈ [−τ,0], x ∈ R.
∂x2+ δu = εpu(t − τ,x)e−au(t−τ,x), t > 0, x ∈ R,
δ
εp> 1−lnεp
δ,
εp
for (s,x) ∈ [−τ,0] × R,
w(R)∩H1(R)), then
sup
x∈R|u(t,x) − φ(x + ct)| ≤ Ce−μt, ∀t ≥ 0, for c > c∗ and μ = μ(c) > 0,
and
sup
x∈R|u(t,x) − φ(x + c∗t)| ≤ C(1 + t)−1
2, ∀t ≥ 0, for c = c∗.
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2784
MING MEI, CHUNHUA OU, AND XIAO-QIANG ZHAO
Remark 4. For the local time-delayed reaction-diffusion equation (4.2), it was
showed that fast waves are locally stable with an exponential decay in [29] and globally
stable in [23]. But the stability for those slow waves (except for the critical wave)
holds only when the delay time τ ? 1. Recently, the global stability of all noncritical
traveling waves was proved in [26] regardless of the magnitude of time delay. The
stability result presented in Theorem 4.2 for the critical wave improves these earlier
works in [23, 26, 29].
4.3. A nonlocal population model with age structure. Letting d(u) = δu2
with δ > 0 and ε = 1,b(u) = pe−γτu with p > 0 and γ > 0, we then reduce (1.1) to
the following age-structured population model which was first derived in [3]:
⎧
⎩
It is clear that the constant equilibria of (4.3) are u−= 0 and u+=p
conditions (H1)–(H3) are satisfied automatically. The following result is a straight-
forward consequence of Theorem 2.2.
Theorem 4.3. For a given traveling wave φ(x + ct) of (4.3) with c ≥ c∗ and
φ(±∞) = u±, if the initial data satisfy
0 = u−≤ u0(s,x) ≤ u+,
and the initial perturbation u0(s,x)−φ(x+cs) is in C([−τ,0];L1
the solution of (4.3) converges to the traveling wave φ(x + ct) in the sense that
(4.3)
⎨
∂u
∂t− D∂2u
u(s,x) = u0(s,x), s ∈ [−τ,0], x ∈ R.
∂x2+ δu2= pe−γτ
?∞
−∞
fα(y)u(t − τ,x − y)dy,
δe−γτ, and the
for (s,x) ∈ [−τ,0] × R,
w(R)∩H1(R)), then
sup
x∈R|u(t,x) − φ(x + ct)| ≤ Ce−μt, ∀t ≥ 0, for c > c∗ and μ = μ(c) > 0,
and
sup
x∈R|u(t,x) − φ(x + c∗t)| ≤ C(1 + t)−1
2, ∀t ≥ 0, for c = c∗.
4.4. A local population model with age-structure. Letting α → 0 in (4.3),
we obtain the following local population model with age-structure:
⎧
⎩
Here u−= 0 and u+=p
traveling waves, including the critical wave.
Theorem 4.4. For a given traveling wave φ(x + ct) of (4.4) with c ≥ c∗ and
φ(±∞) = u±, if the initial data satisfy
0 = u−≤ u0(s,x) ≤ u+,
and the initial perturbation u0(s,x)−φ(x+cs) is in C([−τ,0];L1
the solution of (4.4) converges to the traveling wave φ(x + ct) in the sense that
(4.4)
⎨
∂u
∂t− D∂2u
u(s,x) = u0(s,x), s ∈ [−τ,0], x ∈ R.
∂x2+ δu2= pe−γτu(t − τ,x), t > 0,x ∈ R,
δe−γτ. Now we give a complete answer to the stability for all
for (s,x) ∈ [−τ,0] × R,
w(R)∩H1(R)), then
sup
x∈R|u(t,x) − φ(x + ct)| ≤ Ce−μt, ∀t ≥ 0, for c > c∗ and μ = μ(c) > 0,
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NONLOCAL TIME-DELAYED REACTION-DIFFUSION EQUATIONS
2785
and
sup
x∈R|u(t,x) − φ(x + c∗t)| ≤ C(1 + t)−1
2, ∀t ≥ 0, for c = c∗.
Remark 5. The linear stability of all slow waves (except for the critical wave)
was given in [10] in the case where the time delay τ is sufficiently small. The global
nonlinear stability of noncritical waves was proved in [18] still with τ ? 1. Such a
restriction of smallness for the time delay was further removed in [30]. However, the
stability of the critical wave was not addressed in [10, 18, 30] because the weighted
L2-energy method cannot apply to this case. The stability of the critical wave in
Theorem 4.4 complements the existing results in [10, 18, 30].
4.5. Fisher–KPP equation. Taking τ = 0, D = δ = p = 1 in (4.4), we then
get the following well-known Fisher–KPP equation:
⎧
⎩
Here u−= 0, u+= 1, c∗= 2, λ∗ =
c > c∗= 2. Let the weight function w(x) be defined as in (2.2). We further choose a
large number x0such that
(4.5)
⎨
∂u
∂t−∂2u
u(0,x) = u0(x),
∂x2= u(1 − u), t > 0,x ∈ R,
x ∈ R.
c∗
2= 1, λ1=
c−√c2−4
2
, and λ2=
c+√c2−4
2
for
4φ(x0) > η +1
η.
Here φ(x + ct) is the given traveling wave, and η is a positive constant satisfying
2 −√3 < η < 2 +√3.
Thus, Theorem 2.2 implies the following result.
Theorem 4.5.For a given traveling wave solution φ(x + ct) of the Fisher–
KPP equation (4.5) with c ≥ c∗ = 2 and φ(±∞) = u±, if the initial data satisfy
0 ≤ u0(x) ≤ 1 and the initial perturbation u0(x) − φ(x) is in L1
the solution of (4.5) converges to the traveling wave φ(x + ct) in the sense that
w(R) ∩ H1(R), then
sup
x∈R|u(t,x) − φ(x + ct)| ≤ Ce−μt, ∀t ≥ 0, for c > c∗ and μ = μ(c) > 0,
and
sup
x∈R|u(t,x) − φ(x + c∗t)| ≤ C(1 + t)−1
2, ∀t ≥ 0, for c = c∗.
Remark 6. It is interesting to compare Theorem 4.5 with the classical results
for the Fisher–KPP equation. In [35], Sattinger proved the exponential stability for
all noncritical waves by the spectral analysis method, but the stability of the critical
wave was left open. The authors of [31, 15, 9] improved Sattinger’s result and proved
the stability for all waves including the critical wave. For the critical wave φ(x+c∗t),
when the initial perturbation around the wave φ(x + c∗t) decays as
|u0(x) − φ(x)| = O(1)e−c∗|x|/2as x → −∞,
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2786
MING MEI, CHUNHUA OU, AND XIAO-QIANG ZHAO
Moet [31] applied the maximum principle to prove
?u(t,x) − φ(x + c∗t)?L∞ = O(1)t−1/2,
and Kirchgassner [15] used the spectral analysis method to obtain
?u(t,x) − φ(x + c∗t)?L∞ = O(1)t−1/4,
which was then improved by Gallay [9], using the renormalization group method, as
?u(t,x) − φ(x + c∗t)?L∞ = O(1)t−3/2.
But the initial perturbation around the critical wave needs to be is much faster than
what we assumed, because his weight function is chosen as
?
(1 + x)3
w(x) =
e−β|x|
for x ≤ 0,
for x > 0.
It is also easy to see that the exponential stability of noncritical waves in [35] and
the algebraic stability of the critical wave in [31] are the consequences of our main
result. For the critical wave case, our obtained decay rate is faster than that in [15],
but slower than that in [9].
5. A generalization. We consider a more general time-delayed reaction-diffusion
equation
⎧
⎪
Here the death rate function d(u), the birth rate function b(u), the nonlinear function
F(u), and the kernel g(x) satisfy the following conditions:
(H1) There exist u−= 0 and u+> 0 such that d(0) = b(0) = F(0) = 0, d(u+) =
F(b(u+)), d ∈ C2[0,u+], b ∈ C2[0,u+], F ∈ C2[0,b(u+)];
(H2) F?(0)b?(0) > 0, F?(0)b?(0) > d?(0), F?(b(u+))b?(u+) < d?(u+), and d?(u+)2>
F?(0)2b?(0)b?(u+);
(H3) For 0 ≤ u ≤ u+, d?(u) ≥ 0, b?(u) ≥ 0, d??(u) ≥ 0, b??(u) ≤ 0 but at least one
of d??(u) and |b??(u)| is strictly greater than 0;
(H4) F ∈ C2[0,b(u+)], F?(u) ≥ 0, and F??(u) ≤ 0 for u ∈ [0,b(u+)];
(H5) The kernel g(x) is any integrable nonnegative function satisfying g(−x) =
g(x) and
R
By the theory of spreading speeds and traveling waves developed in [21, 22] for
monotone semiflows, we have the following result.
Lemma 5.1 (existence of traveling waves). Under the conditions (H1)–(H5),
there exist a minimum speed (also called the critical wave speed) c∗> 0 and a corre-
sponding number λ∗= λ(c∗) > 0 satisfying
Fc∗(λ∗) = Gc∗(λ∗),
where
?
(5.1)
⎪
⎪
⎪
⎩
⎨
∂u
∂t− D∂2u
∂x2+ d(u) = F
??
R
g(y)b(u(t − τ,x − y))dy
?
, t > 0,x ∈ R,
u(s,x) = u0(s,x), s ∈ [−τ,0], x ∈ R.
?
g(y)dy = 1.
F?
c∗(λ∗) = G?
c∗(λ∗),
Fc(λ) = F?(0)b?(0)
R
e−λ(y+cτ)g(y)dy,
Gc(λ) = cλ − Dλ2+ d?(0),
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NONLOCAL TIME-DELAYED REACTION-DIFFUSION EQUATIONS
2787
and (c∗,λ∗) is the tangent point of Fc(λ) and Gc(λ), namely,
c∗λ∗− Dλ2
∗+ d?(0) = F?(0)b?(0)
?
R
?
e−λ∗(y+c∗τ)g(y)dy,
c∗− 2Dλ∗= −F?(0)b?(0)
R
(y + c∗τ)e−λ∗(y+c∗τ)g(y)dy,
such that for any c ≥ c∗, the traveling wavefront φ(x + ct) of (5.1) connecting u±
exists, and for any c < c∗, no traveling wave φ(x+ct) exists. When c > c∗, there exist
two numbers λ1> 0 and λ2 > 0, as the solutions to the equation Fc(λi) = Gc(λi),
such that Fc(λ) < Gc(λ) for λ1< λ < λ2, and λ1< λ∗< λ2. When c = c∗, we have
λ1= λ∗= λ2.
Let η1> 0 be a number such that
0 <d?(u+) −?d?(u+)2− F?(0)2b?(0)b?(u+)
<d?(u+) +?d?(u+)2− F?(0)2b?(0)b?(u+)
F?(0)b?(0)
< η1
F?(0)b?(0)
,
and let x1be sufficiently large so that
2d?(φ(x1)) −1
η1F?(0)b?(φ(x1)) − η1F?(0)b?(0) > 0.
For any given c ≥ c∗, we define the weight function as follows:
(5.2)w2(x) =
?
e−λ(x−x1)
1
for x ≤ x1,
for x > x1,
where λ is any fixed number in (λ1,λ∗] when c > c∗, but λ = λ∗when c = c∗.
By similar arguments as in section 3, we can prove the following result on the
global stability of all traveling wavefronts, including the critical traveling wavefront.
Theorem 5.2 (global stability). Let the hypotheses (H1)–(H5) hold. For a
traveling wave φ(x + ct) of (5.1) with c ≥ c∗, if the initial data satisfy
0 = u−≤ u0(s,x) ≤ u+, for (s,x) ∈ [−τ,0] × R,
and the initial perturbation u0(s,x)−φ(x+cs) is in C([−τ,0];L1
the solution of (5.1) uniquely exists and satisfies
w2(R)∩H1(R)), then
0 = u−≤ u(t,x) ≤ u+,for (t,x) ∈ R+× R,
and
u(t,x) − φ(x + ct) ∈ C([0,∞)L1
w2(R) ∩ H1(R)),
and, in particular, the solution u(t,x) converges to the traveling wave φ(x+ct) in the
sense that
sup
x∈R|u(t,x) − φ(x + ct)| ≤ Ce−ˆ μt, ∀t ≥ 0, for c > c∗and ˆ μ = ˆ μ(c) > 0,
Page 27
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2788
MING MEI, CHUNHUA OU, AND XIAO-QIANG ZHAO
and
sup
x∈R|u(t,x) − φ(x + ct)| ≤ C(t + 1)−1
2, ∀t ≥ 0, for c = c∗.
If we replace the condition (H4) with the following weaker one
(H4?) F ∈ C2[0,b(u+)] with F?(u) ≥ 0, ∀u ∈ [0,b(u+)],
then we can prove the following local stability of traveling waves for a class of nonlocal
time-delayed reaction-diffusion equations.
Theorem 5.3 (local stability). Let the hypotheses (H1)–(H3), (H4?), and (H5)
hold. For a traveling wave φ(x+ct) of (5.1) with c ≥ c∗, if the the initial perturbation
u0(s,x) − φ(x + cs) is in C([−τ,0];L1
w2(R) ∩ H1(R)) and
?u0(s,·) − φ(· + cs)?L1
w2+ ?u0(s,·) − φ(· + cs)?H1 ? 1, s ∈ [−τ,0],
then the solution of (5.1) uniquely exists and satisfies
u(t,x) − φ(x + ct) ∈ C([0,∞)L1
w2(R) ∩ H1(R)),
and, in particular, the solution u(t,x) converges to the traveling wave φ(x+ct) in the
sense that
sup
x∈R|u(t,x) − φ(x + ct)| ≤ Ce−ˆ μt, ∀t ≥ 0, for c > c∗and ˆ μ = ˆ μ(c) > 0,
and
sup
x∈R|u(t,x) − φ(x + ct)| ≤ C(t + 1)−1
2, ∀t ≥ 0, for c = c∗.
As a final remark, we consider a nonlocal vector disease model
⎧
⎪
which is a generalized version of the model presented in [34]. Under appropriate
assumptions, the spreading speed and its coincidence with the minimal wave speed
can be established for system (5.3) in the same way as in [48, 49]. In the case where
F(0) = 0, F?(0) > 0, and h(0) > 0, e.g., F(u) = u and h(u) = 1−u as in [48, 49], the
linearization of (5.3) at u = 0 is
(5.3)
⎪
⎪
⎪
⎩
⎨
∂u
∂t− D∂2u
∂x2+ d(u) = F
?
h(u(t,x))
?
R
g(y)u(t − τ,x − y)dy
?
,
u(s,x) = u0(s,x), s ∈ [−τ,0], x ∈ R,
∂u
∂t− D∂2u
∂x2+ d?(0)u = F?(0)h(0)
?
R
g(y)u(t − τ,x − y)dy,
which is essentially the same as the linearized equation of (1.1) at u = 0. It then
follows that we can use the similar arguments as in this paper to obtain sufficient
conditions for the exponential stability of noncritical traveling waves and the algebraic
stability of the critical traveling wave for model system (5.3).
Page 28
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NONLOCAL TIME-DELAYED REACTION-DIFFUSION EQUATIONS
2789
Acknowledgments. We are grateful to Dr. Sergei Trofimchuk and two anony-
mous referees for careful reading and valuable comments which led to an important
improvement of our original manuscript.
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