Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 179, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXPONENTIAL STABILITY OF TRAVELING FRONTS FOR A
2D LATTICE DELAYED DIFFERENTIAL EQUATION WITH
GLOBAL INTERACTION
SHI-LIANG WU, TIAN-TIAN LIU
Abstract. The purpose of this paper is to study traveling wave fronts of
a two-dimensional (2D) lattice delayed differential equation with global interaction. Applying the comparison principle combined with the technical
weighted-energy method, we prove that any given traveling wave front with
large speed is time-asymptotically stable when the initial perturbation around
the wave front need decay to zero exponentially as i cos θ + j sin θ → −∞,
where θ is the direction of propagation, but it can be allowed relatively large
in other locations. The result essentially extends the stability of traveling wave
fronts for local delayed lattice differential equations obtained by Cheng et al
[1] and Yu and Ruan [16].
1. Introduction
The purpose of this paper is to consider the exponential stability of traveling
wave fronts for a stage structured population model on a 2D spatial lattice. The
population model can be described by the delayed lattice differential equation with
global interaction (see Cheng et al [1] and Weng et al [13]):
dui,j (t)
= Dm [ui+1,j (t) + ui−1,j (t) + ui,j+1 (t) + ui,j−1 (t) − 4ui,j (t)] − dm ui,j (t)
dt
$ X
+ 2
βα (i − m)γα (j − n)b um,n (t − τ ) , i, j ∈ Z, t > 0,
4π
m,n∈Z
(1.1)
where Dm and dm represent the diffusion coefficient and the death rate of the
matured population, respectively, d(s) and D(s) are the death rate and
diffusion
Rτ
rate Rof the immature population, respectively, at age s ∈ (0, τ ), $ = e 0 d(s)ds and
τ
α = 0 D(s)ds represent the impact of the death rate for immature and the effect
of the dispersal rate of immature on the mature population, respectively, and
Z π
Z π
βα (l) = 2e−2α
cos(ls)e2α cos s ds, γα (l) = 2e−2α
cos(ls)e2α cos s ds, l ∈ Z.
0
0
2000 Mathematics Subject Classification. 35K57, 35R10, 35B40, 92D25.
Key words and phrases. Exponential stability; traveling wave front; global interaction;
lattice differential equation; comparison principle; weighted energy method.
c
2013
Texas State University - San Marcos.
Submitted July 1, 2013. Published August 4, 2013.
1
2
S.-L. WU, T.-T. LIU
EJDE-2013/179
The important feature of (1.1) is the reflection of the joint effect of the diffusion
dynamics and the nonlocal delayed effect. Under monstable and quasi-monotone
assumptions, the authors of [1, 13] established the existence of minimal wave speed
c∗ = c∗ (θ)(> 0), where θ ∈ [0, π2 ] is any fixed direction of propagation, and showed
that the minimal wave speed c∗ (θ) coincides with the spreading speed for any fixed
direction θ. Moreover, the effects of the maturation period τ and the direction of
propagation θ on the spreading speed were considered.
When D(a) = 0 for any 0 < a < τ (i.e. the immature population is non-mobile),
α = 0 and then β0 (0) = γ0 (0) = 2π and β0 (l) = γ0 (l) = 0 for any l ∈ Z \ {0}. In
this case, (1.1) reduces to the local delayed lattice differential equation
dui,j (t)
= Dm [ui+1,j (t) + ui−1,j (t) + ui,j+1 (t) + ui,j−1 (t) − 4ui,j (t)]
dt
− dm ui,j (t) + $b ui,j (t − τ ) .
(1.2)
Applying the weighted energy method, Cheng et al [2] proved the asymptotic stability of traveling wave fronts of (1.2). More precisely, they proved that, for the
Cauchy problem of (1.2) with initial data
ui,j (s) = u0i,j (s),
i, j ∈ Z, s ∈ [−τ, 0],
(1.3)
the traveling wave front φ(i cos θ + j sin θ + ct) of (1.2) connecting E + and E − with
large speed is time-asymptotically stable, when the initial perturbation around the
wave front (i.e. |u0i,j (s) − φ(i cos θ + j sin θ + cs)|) is sufficiently small in a weighted
norm. More recently, the authors of [16] further established the stability of traveling
wave fronts of (1.2) for relatively large initial perturbations by using the comparison
principle and the weighted-energy method. However, to the best of our knowledge,
there has been no results on the stability of traveling wave fronts for the delayed
lattice differential equation with global interaction.
The purpose of this paper is to consider the stability of traveling wave fronts of
(1.1). More precisely, we shall prove that any given traveling wave front φ(i cos θ +
j sin θ + ct) with large speed c (i.e. c satisfies (2.6) below) is time-asymptotically
stable when the initial perturbation around the wave front (i.e. |u0i,j (s) − φ(i cos θ +
j sin θ + cs)|) need to decay to zero exponentially as i cos θ + j sin θ → −∞, where
θ is the direction of propagation, but it can be allowed relatively large in other
locations (see Theorem 2.3). Here, we use an approach combining the comparison
principle and the weighted-energy method, which was developed by [5] to prove the
stability of traveling wave fronts of a Nicholson’s blowflies equation with diffusion.
This approach was further employed by many researchers to prove the stability of
traveling wave fronts of various reaction-diffusion equations with local or nonlocal
delays; see, e.g., [4, 6, 7, 8, 9, 14, 15].
Although the main idea and methods of the proof for our main theorem are
originally encouraged by [4, 6, 7, 8, 9, 14, 15, 16], we mention that difficulties and
challenge are existing for our arguments due to the convolution term. For example, in the construction of the weight function, we need to derive some important
estimations (see Lemma 2.2). In addition, the proof of the key inequality is more
technical (see Lemma 3.3). Similar to [1, 13], we make the following assumptions:
(A1) b ∈ C 2 ([0, K]), b(0) = 0, $b(K) = dm K, dm > $b0 (K), $b(u) > dm u for
u ∈ (0, K), where K is a positive constant;
(A2) b(u) ≤ b0 (0)u and b0 (u) ≥ 0 for u ∈ [0, K].
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STABILITY OF TRAVELING FRONTS
3
The rest of this paper is organized as follows. In Section 2, we first introduce
some known results on the existence of traveling wave fronts of (1.1), and then
present our stability results. The proofs of the main results are given in Section 3.
2
Notation. Throughout this paper, lw
denotes the weighted l2 space with weight
π
+
w(ξ) ∈ C(R, R ) and a fixed θ ∈ [0, 2 ]; that is,
n
o
X
2
2
lw
= ς = {ςi,j }i,j∈Z , ςi,j ∈ R :
w(i cos θ + j sin θ)ςi,j
<∞
i,j∈Z
with the norm
2 =
kςklw
hX
2
w(i cos θ + j sin θ)ςi,j
i1/2
.
i,j∈Z
2
In particular, if w ≡ 1, we denote lw
by l2 .
2. Preliminaries and main results
Throughout this article, a traveling wave solution connecting 0 and K refers to
a triplete (φ, c, θ), where φ = φ(·) : R → R is a function, c > 0 and θ ∈ [0, π/2] are
constants, such that ui,j (t) = φ(ξ), ξ = i cos θ + j sin θ + ct, is a solution of (1.1);
that is,
cφ0 (ξ) = Dm [φ(ξ + cos θ) + φ(ξ − cos θ) + φ(ξ + sin θ) + φ(ξ − sin θ) − 4φ(ξ)]
$ X
− dm φ(ξ) + 2
βα (m)γα (n)b φ(ξ − m cos θ − n sin θ − cτ )
4π
m,n∈Z
(2.1)
with the boundary conditions
φ(−∞) = 0,
φ(+∞) = K.
(2.2)
The constant θ represents the direction of the wave. We call c the wave speed and
φ the wave profile. Moreover, we say φ is a traveling (wave) front if φ(·) : R → R
is monotone.
It is clear that the characteristic function for (2.1) with respect to the trivial
equilibrium 0 can be represented by
∆(c, λ) = cλ − Dm eλ cos θ + e−λ cos θ + eλ sin θ + e−λ sin θ − 4 + dm
X
$
βα (m)γα (n)e−λ(m cos θ+n sin θ+cτ ) .
− 2 b0 (0)
4π
m,n∈Z
Properties of ∆(c, λ) and existence of traveling wave fronts of (1.1) were investigated
in [1, 13]. For the sake of completeness, we recall them as follows.
Proposition 2.1. Assume (A1)–(A2) hold. Then the following results hold:
(1) For each θ ∈ [0, π2 ], there exist λ∗ := λ∗ (θ) > 0 and c∗ := c∗ (θ) > 0 such
that
∂
∆(c∗ , λ∗ ) = 0 and
∆(c∗ , λ)
= 0.
∂λ
λ=λ∗
Furthermore, if c > c∗ (θ), then the equation ∆(c, λ) = 0 has two positive
real roots λ1 := λ1 (c, θ) and λ2 := λ2 (c, θ) with λ1 < λ∗ < λ2 .
(2) Fix θ ∈ [0, π/2]. Then, for every c ≥ c∗ (θ), (1.1) has a traveling wave front
φ(ξ) with direction θ and speed c.
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S.-L. WU, T.-T. LIU
EJDE-2013/179
For convenience, we denote
L1 = max b0 (u),
L2 =
u∈[0,K]
1 X
βα (m)γα (n) max 1, e−m cos θ−n sin θ .
2
4π
m,n∈Z
00
Note that if b (u) ≤ 0 for u ∈ [0, K], then L1 = b0 (0). Moreover, it is easy to see
that L2 = 1 when α = 0.
The following result plays an important role for constructing the weight function.
Lemma 2.2. Assume
1
dm > Dm e − 1 + $b0 (K)(1 + L2 ).
(2.3)
2
For any given traveling wave front φ(ξ) of (1.1) with direction θ ∈ [0, π2 ] and speed
c > c∗ (θ) obtained in Proposition 2.1, there exists ξ∗ > 0 such that for any ξ ≥ ξ∗ ,
X
$ 0
b φ(ξ))
βα (m)γα (n) max{1, e−m cos θ−n sin θ }
2
4π
m,n∈Z
$ X
βα (m)γα (n)b0 φ(ξ − m cos θ − n sin θ − cτ ))
+ 2
4π
m,n∈Z
≤ $b0 (K)(1 + L2 ) + ¯,
where ¯ = dm − Dm e − 1 − 21 $b0 (K)(1 + L2 ) > 0.
Proof. Since limξ→+∞ b0 φ(ξ)) = b0 (K), it suffices to show that
1 X
βα (m)γα (n)b0 φ(ξ − m cos θ − n sin θ − cτ )) = b0 (K).
lim
2
ξ→+∞ 4π
(2.4)
m,n∈Z
Given any > 0, since
1 X
1 X
βα (m) =
βα (n) = 1
2π
2π
m∈Z
n∈Z
(see [1, Lemma 2.1]), there exists M, N > 0 such that
X
X
π
βα (m),
γα (n) ≤
.
4L1
|m|≥M
0
|n|≥N
0
Noting that limξ→+∞ b φ(ξ)) = b (K), there exists ξ∗ > 0 such that for any ξ ≥
ξ∗ − M − N − cτ ,
|b0 φ(ξ)) − b0 (K)| < .
4
Then, for any ξ ≥ ξ∗ , we have
1 X
βα (m)γα (n)b0 φ(ξ − m cos θ − n sin θ − cτ ) − b0 (K)
2
4π
m,n∈Z
1 X
= 2
βα (m)γα (n) b0 φ(ξ − m cos θ − n sin θ − cτ ) − b0 (K) 4π
m,n∈Z
1
≤
4π 2
+
X
βα (m)γα (n)b0 φ(ξ − m cos θ − n sin θ − cτ ) − b0 (K)
|m|≥M,n∈Z
1
4π 2
X
|m|≤M,|n|≥N
βα (m)γα (n)b0 φ(ξ − m cos θ − n sin θ − cτ ) − b0 (K)
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+
1
4π 2
STABILITY OF TRAVELING FRONTS
X
5
βα (m)γα (n)b0 φ(ξ − m cos θ − n sin θ − cτ ) − b0 (K)
|m|≤M,|n|≤N
1 X
1 X
1
≤ 2L1
βα (m) + 2L1
γα (n) +
2π
2π
4 4π 2
|m|≥M
|n|≥N
X
βα (m)γα (n) < .
|m|≤M,|n|≤N
Thus, (2.4) holds. The proof is complete.
Based on the above lemma, we define the weight function w(ξ) as
(
e−(ξ−ξ∗ ) , for ξ < ξ∗ ,
w(ξ) =
1,
for ξ ≥ ξ∗ .
(2.5)
We can now state our main theorem.
Theorem 2.3. Assume (A1)–(A2) hold and b00 (u) ≤ 0 for u ∈ [0, K]. For any
given traveling wave front φ(ξ) of (1.1) with direction θ ∈ [0, π/2] and speed c
obtained in Proposition 2.1, if (2.3) holds,
c > max 2Dm e − 1 + b0 (0)$(1 + L2 ) − 2dm , c∗ (θ)
(2.6)
and the initial data satisfies 0 ≤ ui,j (s) ≤ K for (i, j, s) ∈ Z2 × [−τ, 0], and
2
{u0i,j (s) − φ(i cos θ + j sin θ + cs)}i,j∈Z ∈ C [−τ, 0], lw
,
then the unique solution ui,j (t) of the Cauchy problem (1.1) and (1.3) satisfies
0 ≤ ui,j (t) ≤ K (i, j, t) ∈ Z2 × [0, +∞),
2
{ui,j (t) − φ(i cos θ + j sin θ + ct)}i,j∈Z ∈ C [0, +∞), lw
,
and there exists positive number µ such that
sup ui,j (t) − φ(i cos θ + j sin θ + ct) ≤ C0 e−µt ,
t ≥ 0,
i,j∈Z
for some constant C0 > 0.
Remark 2.4. (i) Note that if Dm and b0 (K) are relatively small, then the technical
assumption (2.3) holds. As mentioned by Mei et al [6], the condition b0 (K) 1 is
natural, see e.g. [6, Remark 1].
(ii) From the condition
(2.6) and definitions of the weighted function w(ξ) and
2
the space C [−τ, 0], lw
, we see that the initial perturbation around the wave front
must converge to 0 exponentially as i cos θ + j sin θ → −∞ in the form
1
u0i,j (s) − φ(i cos θ + j sin θ + cs) = O(1)e− 2 |i cos θ+j sin θ| , s ∈ [−τ, 0].
Contrasting to [2], we do not require that the initial perturbation must be sufficiently small in a weighted norm.
(iii) Theorem 2.3 guarantees that any given traveling wave front of (1.1) with
large speed is time-asymptotically stable. However, we are unable to prove the
stability for any slower waves c > c∗ , particularly the case of critical waves with
c = c∗ . We leave this for future research.
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S.-L. WU, T.-T. LIU
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3. Proof of main results
In this section, we first state the existence of solutions of the Cauchy problem
(1.1) and (1.3) and establish the comparison principle. Then we prove our stability
results by using the comparison principle together with the weighted energy method.
In the sequel, we always assume that all the conditions in Theorem 2.3 hold.
Applying similar methods as in Cheng et al [2, Theorem 2.2], we obtain the
following existence result.
Lemma 3.1 (Existence). For any function u0 (s) = {u0i,j (s)}i,j∈Z ∈ C([−τ, 0], l∞ ),
equation (1.1) has a unique solution u(t) = {ui,j (t)}i,j∈Z ∈ C([−τ, +∞), l∞ ) with
u(s) = u0 (s) on [−τ, 0]. Furthermore, if
{u0i,j (s) − φ(i cos θ + j sin θ + cs)}i,j∈Z ∈ C [−τ, 0], l2 ,
then
{ui,j (t) − φ(i cos θ + j sin θ + ct)}i,j∈Z ∈ C [0, +∞), l2 .
Lemma 3.2 (Comparison Principle). Let {ui,j (t)}i,j∈Z and {ui,j (t)}i,j∈Z be the
solutions of (1.1) and (1.3) with initial data {u0i,j (s)}i,j∈Z and {u0i,j (s)}i,j∈Z , respectively. If
0 ≤ u0i,j (s) ≤ u0i,j (s) ≤ K
for i, j ∈ Z and s ∈ [−τ, 0],
then
0 ≤ ui,j (t) ≤ ui,j (t) ≤ K
for i, j ∈ Z and t ≥ 0.
Proof. Put wi,j (t) = ui,j (t) − ui,j (t) for i, j ∈ Z and t ≥ −τ . Direct computation
shows that
0
wi,j
(t) = Dm [wi+1,j (t) + wi−1,j (t) + wi,j+1 (t) + wi,j−1 (t) − 4wi,j (t)]
− dm wi,j (t) + hi,j (t),
where
hi,j (t) =
$ X
βα (i − m)γα (j − n) b um,n (t − τ ) − b um,n (t − τ ) .
2
4π
m,n∈Z
We claim that
wi,j (t) =
1 −dm t X
e
βDm t (i − k)γDm t (j − l)wk,l (0)
4π 2
k,l∈Z
Z
1 X t −dm (t−s)
+ 2
e
βDm (t−s) (i − k)γDm (t−s) (j − l)hk,l (s)ds.
4π
0
k,l∈Z
We note that this claim can be proved by using discrete Fourier transformation
as in Cheng et al. [2]. For the sake of completeness and reader’s convenience, we
provide its proof here. Note that the grid function wi,j (t) can be viewed as the
discrete spectral of a periodic function w(t,
b λ) by discrete Fourier transformation
(see Goldberg, 1965; Titchmarsh, 1962):
1 X −i(kλ1 +lλ2 )
e
wk,l (t),
w(t,
b λ) =
2π
k,l∈Z
Z π Z π
1
wk,l (t) =
ei(kλ1 +lλ2 ) w(t,
b λ)dλ1 dλ2 ,
2π −π −π
EJDE-2013/179
STABILITY OF TRAVELING FRONTS
7
where i is the imaginary unit and λ = (λ1 , λ2 ). Using discrete Fourier transformation, we obtain
∂
w(t,
b λ) = Dm [eiλ1 + e−iλ1 + eiλ2 + e−iλ2 − 4]w(t,
b λ) − dm w(t,
b λ) + b
h(t, λ)
∂t
h
i
λ1
λ2
= − 4Dm (sin2
+ sin2 ) + dm w(t,
b λ) + b
h(t, λ).
2
2
(3.1)
This equation can be solved as:
2 λ1
2 λ2
w(t,
b λ) = w(0,
b λ)e−4Dm t(sin 2 +sin 2 ) e−dm t
Z t
2 λ2
2 λ1
b
+
h(s, λ)e−4Dm (t−s)(sin 2 +sin 2 ) e−dm (t−s) ds.
0
Note that
w(0,
b λ) =
1 X −i(kλ1 +lλ2 )
e
wk,l (0),
2π
k,l∈Z
1 X −i(kλ1 +lλ2 )
b
h(s, λ) =
e
hk,l (s).
2π
k,l∈Z
Using the inverse discrete Fourier transformation, we obtain
Z Z
2
2 λ1
1 −dm t π π i(iλ1 +jλ2 )
wi,j (t) =
e
e
w(0,
b λ)e−4Dm t(sin 2 +sin
2π
−π −π
Z t
Z π Z π
1
−dm (t−s)
+
e
ei(iλ1 +jλ2 ) b
h(s, λ)
2π 0
−π −π
2 λ1
λ2
2
)
dλ1 dλ2
2 λ2
× e−4Dm (t−s)(sin 2 +sin 2 ) dλ1 dλ2 ds
Z π
2 λ1
1 −dm t X
=
e
wk,l (0)
ei(i−k)λ1 e−4Dm t sin 2 dλ1
4π 2
−π
k,l∈Z
Z π
Z
2 λ2
1 X t −dm (t−s)
×
ei(j−l)λ2 e−4Dm t sin 2 dλ2 + 2
e
hk,l (s)
4π
−π
k,l∈Z 0
Z π
Z π
2 λ1
2 λ2
×
ei(i−k)λ1 e−4Dm (t−s) sin 2 dλ1
ei(j−l)λ2 e−4Dm (t−s) sin 2 dλ2 ds
−π
−π
1 −dm t X
=
e
βDm t (i − k)γDm t (j − l)wk,l (0)
4π 2
k,l∈Z
Z
1 X t −dm (t−s)
+ 2
e
βDm (t−s) (i − k)γDm (t−s) (j − l)hk,l (s)ds.
4π
0
k,l∈Z
Since b0 (u) ≥ 0 for u ∈ [0, K] and 0 ≤ u0i,j (s) ≤ u0i,j (s) ≤ K for s ∈ [−τ, 0], we have
wi,j (t) ≥ 0 for i, j ∈ Z and t ∈ [0, τ ]. Inductively, we obtain that wi,j (t) ≥ 0 for
i, j ∈ Z and t ≥ 0, i.e. ui,j (t) ≤ ui,j (t) for i, j ∈ Z and t > 0. Similarly, we can
show that ui,j (t) ≥ 0 and ui,j (t) ≤ K for i, j ∈ Z and t > 0. This completes the
proof.
In what follows, we shall prove the stability theorem by means of the comparison
principle together with the weighed energy method.
We assume that the initial data {u0i,j (s)}i,j∈Z of (1.1) satisfying 0 ≤ u0i,j (s) ≤
K for i, j ∈ Z and s ∈ [−τ, 0], and {u0i,j (s) − φ(i cos θ + j sin θ + cs)}i,j∈Z ∈
8
S.-L. WU, T.-T. LIU
EJDE-2013/179
2
. Take
C [−τ, 0], lw
0
ϕ+
i,j (s) := max ui,j (s), φ(i cos θ + j sin θ + cs) ,
0
ϕ−
i,j (s) := min ui,j (s), φ(i cos θ + j sin θ + cs)
for i, j ∈ Z and s ∈ [−τ, 0]. Then,
2
{ϕ±
i,j (s) − φ(i cos θ + j sin θ + cs)}i,j∈Z ∈ C [−τ, 0], lw
and
0
0 ≤ ϕ−
i,j (s) ≤ ui,j (s),
φ(i cos θ + j sin θ + cs) ≤ ϕ+
i,j (s) ≤ K.
Let u± (t) = {u±
i,j (t)}i,j∈Z be the solutions of (1.1) with respect to the initial data
ϕ± (s) = {ϕ±
(s)}
i,j∈Z , i.e.
i,j
du±
i,j (t)
±
±
±
±
±
= Dm [u±
i+1,j (t) + ui−1,j (t) + ui,j+1 (t) + ui,j−1 (t) − 4ui,j (t)] − dm ui,j (t)
dt
$ X
βα (i − m)γα (j − n)b u±
i, j ∈ Z, t > 0,
+ 2
m,n (t − τ ) ,
4π
m,n∈Z
±
u±
i,j (s) = ϕi,j (s),
i, j ∈ Z, s ∈ [−τ, 0].
(3.2)
Applying the comparison principle, we have
0 ≤ u−
i,j (t) ≤ ui,j (t),
φ(i cos θ + j sin θ + ct) ≤ u+
i,j (t) ≤ K
for i, j ∈ Z, t > 0.
3.1. Weighted energy estimate. For convenience, we denote
Ui,j (t) = u+
i,j (t) − φ(i cos θ + j sin θ + ct),
ξi,j (t) = i cos θ + j sin θ + ct.
It is easy to verify that {Ui,j (t)}i,j∈Z satisfies
dUi,j (t)
= Dm [Ui+1,j (t) + Ui−1,j (t) + Ui,j+1 (t) + Ui,j−1 (t) − 4Ui,j (t)]
dt
$ X
+ 2
βα (m)γα (n)b0 φ(ξi−m,j−n (t − τ )) Ui−m,j−n (t − τ )
4π
(3.3)
m,n∈Z
− dm Ui,j (t) + Gi,j (t),
0
Ui,j (s) = ϕ+
i,j (s) − φ(ξi,j (s)) := Ui,j (s),
where i, j ∈ Z, t > 0, s ∈ [−τ, 0] and the nonlinear term Gi,j (t) is given by
$ X
Gi,j (t) =
βα (m)γα (n) b Ui−m,j−n (t − τ ) + φ(ξi−m,j−n (t − τ ))
2
4π
m,n∈Z
− b φ(ξi−m,j−n (t − τ )) − b0 φ(ξi−m,j−n (t − τ )) Ui−m,j−n (t − τ ) .
To obtain a weighted energy
estimate, we need the following key inequality. Take
C0 (µ) = min C1 (µ), C2 (µ) , where
C1 (µ) = c + 2dm − 2Dm (e − 1) − L1 $(1 + L2 ) − 2µ − $L1 L2 (e2µτ − 1),
1
C2 (µ) = dm − Dm (e − 1) − $b0 (K)(1 + L2 ) − 2µ − $L1 L2 (e2µτ − 1).
2
EJDE-2013/179
STABILITY OF TRAVELING FRONTS
9
Define
wξ (ξi,j (t))
+ 2(dm − µ) − Dm L(w)(ξi,j (t)) − 4
w(ξi,j (t))
X
$
w(ξi+m,j+n (t + τ ))
βα (m)γα (n)
− 2 e2µτ b0 φ(ξi,j (t)))
4π
w(ξi,j (t))
m,n∈Z
X
$
− 2
βα (m)γα (n)b0 φ(ξi−m,j−n (t − τ )) ,
4π
Bi,j (µ, t) = −c
(3.4)
m,n∈Z
where
L(w)(ξi,j (t)) =
w(ξi+1,j (t)) w(ξi−1,j (t)) w(ξi,j+1 (t)) w(ξi,j−1 (t))
+
+
+
.
w(ξi,j (t))
w(ξi,j (t))
w(ξi,j (t))
w(ξi,j (t))
Lemma 3.3 (Key inequality). Let w(ξ) be the weight function given in (2.5). Then
Bi,j (µ, t) ≥ C0 (µ) > 0,
for all i, j ∈ Z, t > 0, and 0 < µ < µ0 := min{µ1 , µ2 }, where µi is the unique
solution to the equation Ci (µ) = 0, i = 1, 2.
Proof. We distinguish two cases:
Case (i): ξi,j (t) < ξ∗ . In this case w(ξi,j (t)) = e−(ξi,j (t)−ξ∗ ) . Since w(ξ) is
non-increasing in R, we have
w(ξi+m,j+n (t + τ ))
w(ξi,j (t))
w(ξi+m,j+n (t))
≤
w(ξi,j (t))
(
(ξi,j (t)−ξ∗ )
e
≤ 1,
if ξi+m,j+n (t) ≥ ξ∗ ,
=
(ξi,j (t)−ξi+m,j+n (t))
−m cos θ−n sin θ
e
=e
, if ξi+m,j+n (t) < ξ∗ .
Hence,
w(ξi+m,j+n (t + τ ))
≤ max{1, e−m cos θ−n sin θ }
w(ξi,j (t))
for any m, n ∈ Z.
Similarly, it is easy to verify that
L(w)(ξi,j (t)) ≤
w(ξi−1,j (t)) w(ξi,j−1 (t))
+
+2
w(ξi,j (t))
w(ξi,j (t))
≤ max{1, ecos θ } + max{1, esin θ } + 2 ≤ 2(e + 1).
Thus, we have
Bi,j (µ, t)
≥ c + 2dm − 2Dm (e − 1) − 2µ − L1 $
X
$
− L1 2 e2µτ
βα (m)γα (n) max{1, e−m cos θ−n sin θ }
4π
m,n∈Z
= c + 2dm − 2Dm (e − 1) − L1 $(1 + L2 ) − 2µ − L1 L2 $(e2µτ − 1)
= C1 (µ) > 0
provided that (2.6) holds.
for 0 < µ < µ1 ,
10
S.-L. WU, T.-T. LIU
EJDE-2013/179
Case (ii): ξi,j (t) ≥ ξ∗ . In this case w(ξi,j (t)) = 1. Similarly, we can show that
w(ξi+m,j+n (t + τ ))
≤ max{1, e−m cos θ−n sin θ }
w(ξi,j (t))
for any m, n ∈ Z,
and
L(w)(ξi,j (t)) ≤ max{1, ecos θ } + max{1, esin θ } + 2 ≤ 2(e + 1).
Note that ξi−m,j−n (t − τ ) = ξi,j (t) − m cos θ − n sin θ − cτ . It follows from Lemma
2.2 that
Bi,j (µ, t) ≥ 2dm − 2Dm (e − 1)
X
$
βα (m)γα (n) max{1, e−m cos θ−n sin θ }
− 2 b0 φ(ξi,j (t)))
4π
m,n∈Z
$ X
− 2
βα (m)γα (n)b0 φ(ξi−m,j−n (t − τ )) − 2µ
4π
m,n∈Z
X
$ 2µτ
− 1)b0 φ(ξi,j (t)))
βα (m)γα (n) max{1, e−m cos θ−n sin θ }
− 2 (e
4π
m,n∈Z
≥ 2dm − 2Dm (e − 1) − $b0 (K)(1 + L2 ) − ¯ − 2µ
X
$
− 2 (e2µτ − 1)L1
βα (m)γα (n) max{1, e−m cos θ−n sin θ }
4π
m,n∈Z
1
= dm − Dm (e − 1) − $b0 (K)(1 + L2 ) − 2µ − L1 L2 $(e2µτ − 1)
2
= C2 (µ) > 0 for 0 < µ < µ2 .
Now, let 0 < µ < µ0 := min{µ1 , µ2 }, then Bi,j (µ, t) ≥ C0 (µ) > 0 for all i, j ∈ Z,
t > 0. This completes the proof.
Lemma 3.4 (Weighted energy estimate). There exists µ > 0 such that
1/2
Z 0
2 ≤
kU (t)klw
kU 0 (0)k2lw
kU 0 (s)k2lw
e−µt , t ≥ 0
2 + C2
2 ds
−τ
for some constant C2 > 0.
Proof. Multiplying (3.3) by e2µt w(ξi,j (t))Ui,j (t) for 0 < µ < µ0 , we have
cw
1
ξ
2
2
e2µt wUi,j
(t) + −
+ dm − µ e2µt wUi,j
(t)
2
2 w
t
− Dm e2µt w[Ui+1,j (t) + Ui−1,j (t) + Ui,j+1 (t) + Ui,j−1 (t) − 4Ui,j (t)]Ui,j (t)
$ X
− 2
βα (m)γα (n)b0 φ(ξi−m,j−n (t − τ )) Ui−m,j−n (t − τ )e2µt wUi,j (t)
4π
m,n∈Z
=e
2µt
wUi,j (t)Gi,j (t),
(3.5)
2
2
where w = w(ξi,j (t)). Noting that 2Ui±1,j±1 (t)Ui,j (t) ≤ Ui±1,j±1
(t) + Ui,j
(t), and
$ X
1 00 Gi,j (t) =
β
(m)γ
(n)
b θ1 Ui−m,j−n (t − τ )
α
α
4π 2
2
m,n∈Z
2
+ φ(ξi−m,j−n (t − τ )) Ui−m,j−n
(t − τ ) ≤ 0
EJDE-2013/179
STABILITY OF TRAVELING FRONTS
11
for all i, j ∈ Z, t > 0, where θ1 ∈ (0, 1), substituting these into (3.5), we obtain
wξ
2
2
e2µt wUi,j
(t) t + − c
+ 2dm − 2µ e2µt wUi,j
(t)
w
2
2
2
2
2
− Dm e2µt w[Ui+1,j
(t) + Ui−1,j
(t) + Ui,j+1
(t) + Ui,j−1
(t) − 4Ui,j
(t)]
X
$
0
βα (m)γα (n)b φ(ξi−m,j−n (t − τ )) Ui−m,j−n (t − τ )e2µt wUi,j (t)
− 2
2π
m,n∈Z
≤ 0.
(3.6)
Summing (3.6) about all i, j ∈ Z and integrating the inequality over [0, t], we have
Z t X X
$
βα (m)γα (n)b0 φ(ξi−m,j−n (s − τ ))
e2µt kU (t)k2lw
−
2
2
2π 0
i,j∈Z m,n∈Z
2µs
× Ui−m,j−n (s − τ )e w(ξi,j (s))Ui,j (s)ds
Z t X
wξ (ξi,j (t))
+
−c
+ 2(dm − µ) − Dm L(w)(ξi,j (s)) − 4
w(ξi,j (s))
0
(3.7)
i,j∈Z
×e
2µs
2
w(ξi,j (s))Ui,j
(s)ds
≤ kU 0 (0)k2L2w .
Using the inequality 2ab ≤ a2 + b2 and making changes of variables s − τ → s,
i − m → i, and j − n → j, we obtain
Z t X X
$
βα (m)γα (n)b0 φ(ξi−m,j−n (s − τ ))
2
2π 0
i,j∈Z m,n∈Z
× Ui−m,j−n (s − τ )e2µs w(ξi,j (s))Ui,j (s)ds
Z t X X
$
2
≤
βα (m)γα (n)b0 φ(ξi−m,j−n (s − τ )) e2µs w(ξi,j (s))Ui,j
(s)ds
2
4π 0
i,j∈Z m,n∈Z
Z t X X
$
βα (m)γα (n)
+ 2
4π 0
i,j∈Z m,n∈Z
2
× b0 φ(ξi−m,j−n (s − τ )) e2µs w(ξi,j (s))Ui−m,j−n
(s − τ )ds
Z t X X
$
2
≤
βα (m)γα (n)b0 φ(ξi−m,j−n (s − τ )) e2µs w(ξi,j (s))Ui,j
(s)ds
2
4π 0
i,j∈Z m,n∈Z
Z
$ 2µτ t X X
+ 2e
βα (m)γα (n)b0 φ(ξi,j (s)))
4π
0
i,j∈Z m,n∈Z
w(ξi+m,j+n (s + τ )) 2µs
2
×
e w(ξi,j (s))Ui,j
(s)ds
w(ξi,j (s))
Z 0 X X
$
+ 2 e2µτ
βα (m)γα (n)b0 φ(ξi,j (s)))
4π
−τ
i,j∈Z m,n∈Z
×
w(ξi+m,j+n (s + τ )) 2µs
2
e w(ξi,j (s))Ui,j
(s)ds
w(ξi,j (s))
(3.8)
12
S.-L. WU, T.-T. LIU
EJDE-2013/179
From the proof of Lemma 3.3, we see that
w(ξi+m,j+n (t + τ ))
≤ max{1, e−m cos θ−n sin θ }
w(ξi,j (t))
for any i, j, m, n ∈ Z.
Thus, we have
Z
w(ξi+m,j+n (s + τ ))
$ 2µτ 0 X X
e
βα (m)γα (n)b0 φ(ξi,j (s)))
2
4π
w(ξi,j (s))
−τ
i,j∈Z m,n∈Z
2
w(ξi,j (s))Ui,j
(s)ds
Z 0 X X
$
βα (m)γα (n) max{1, e−m cos θ−n sin θ }
L1 e2µτ
≤
4π 2
−τ
×e
2µs
(3.9)
i,j∈Z m,n∈Z
2
w(ξi,j (s))Ui,j
(s)ds
Z 0
= $L1 e2µτ L2
kU 0 (s)k2lw
2 ds.
−τ
Substituting (3.8) and (3.9) into (3.7), we have
Z tX
2µt
2
2
e kU (t)klw
Bi,j (µ, s)e2µs w(ξi,j (s))Ui,j
(s)ds
2 +
0
≤ kU
0
(0)k2lw
2
i,j
Z
(3.10)
0
kU
+ C2
−τ
0
(s)k2lw
2 ds,
where C2 = $L1 e2µ0 τ L2 > 0. It then follows from Lemma 3.3 that
Z 0
1/2
2 ≤
kU (t)klw
kU 0 (0)k2lw
+
C
kU 0 (s)k2lw
e−µt for t ≥ 0.
2
2 ds
2
−τ
This completes the proof.
3.2. Proof of Theorem 2.3. By Lemma 3.4 and the standard Sobolev’s embed2
,→ l2 for w(·) ≥ 1 defined as in (2.5), we obtain the
ding inequality l2 ,→ l∞ and lw
+
convergence of ui,j (t), that is there exists a constant µ01 > 0 such that
0
−µ1 t
sup |u+
,
i,j (t) − φ(i cos θ + j sin θ + ct)| ≤ C2 e
t ≥ 0,
i,j∈Z
for some constant C2 > 0.
−
Let Vi,j (t) = φ(i cos θ + j sin θ + ct) − u−
i,j (t). We can similarly prove that ui,j (t)
converges to φ(i cos θ + j sin θ + ct), i.e. there exists a constant µ02 > 0 such that
0
−µ2 t
sup |u−
,
i,j (t) − φ(i cos θ + j sin θ + ct)| ≤ C3 e
t ≥ 0,
i,j∈Z
for some constant C3 > 0.
+
Take µ0 = min{µ01 , µ02 }, Note that u−
i,j (t) ≤ ui,j (t) ≤ ui,j (t) for i, j ∈ Z, t ≥ 0.
Using the Squeeze Theorem, we can easily show that ui,j (t) converges to φ(i cos θ +
j sin θ + ct); that is,
0
sup |ui,j (t) − φ(i cos θ + j sin θ + ct)| ≤ C4 e−µ t ,
t ≥ 0,
i,j∈Z
for some constant C4 > 0. We now complete the proof of Theorem 2.3.
EJDE-2013/179
STABILITY OF TRAVELING FRONTS
13
Acknowledgments. The authors thank the anonymous referees for their valuable
comments and suggestions that help the improvement of the original manuscript.
Shi-Liang Wu was Supported by grant 2013JQ1012 from the NSF of Shaanxi
Province of China, and grant K5051370002 from the Fundamental Research Funds
for the Central Universities.
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Shi-Liang Wu
Department of Mathematics, Xidian University, Xi’an, Shaanxi 710071, China
E-mail address: slwu@xidian.edu.cn
Tian-Tian Liu
Department of Mathematics, Xidian University, Xi’an, Shaanxi 710071, China
E-mail address: ahtian4402@qq.com