Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 86, pp. 1–15.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXPONENTIAL STABILITY OF TRAVELING WAVES FOR
NON-MONOTONE DELAYED REACTION-DIFFUSION
EQUATIONS
YIXIN LIU, ZHIXIAN YU, JING XIA
Abstract. This article concerns the exponential stability of non-critical traveling waves (the wave speed is greater than the minimum speed) for nonmonotone time-delayed reaction-diffusion equations. With the help of the
weighted energy method, we prove that the non-critical travelling waves are
exponentially stable when the initial perturbation around the wave is small.
1. Introduction
In this article, we study the stability of traveling waves for the non-monotone
delayed reaction diffusion equation
∂υ(t, x)
∂ 2 υ(t, x)
=D
− d(υ(t, x)) + f (υ(t − r, x)),
∂t
∂x2
with the initial condition
(t, x) ∈ R+ × R
s ∈ [−r, 0], x ∈ R.
υ(s, x) = υ0 (s, x),
(1.1)
(1.2)
where D > 0, r ≥ 0 are constants. The nonlinear functions d(u) and f (u) satisfy
the following hypotheses:
(H1) d ∈ C 2 ([0, ∞], R), f ∈ C 2 ([0, ∞], R); there exist only two constant equilibria 0 and K > 0 such that f (0) = d(0), f (K) = d(K), d0 (0) − f 0 (0) < 0
and d0 (K) − f 0 (K) > 0.
(H2) There exists K ∗ ≥ K such that d(K ∗ ) ≥ max{f (υ)|0 ≤ υ ≤ K ∗ } and
d(υ) < d(K ∗ ) for all υ ∈ [0, K ∗ ), f 0 (0)υ ≥ f (υ) > 0, d(υ) ≥ d0 (0)υ and
f 0 (0)υ > d(υ) for all υ ∈ (0, K ∗ ].
(H3) d(υ) is strictly increasing on [0, K ∗ ] and d(υ) < f (υ) < 2d(K) − d(υ) for
υ ∈ [0, K), d(υ) > f (υ) > 2d(K) − d(υ) for υ ∈ (K, K ∗ ].
(H4) d0 (υ) ≥ d0 (0) and |f 0 (υ)| ≤ f 0 (0) for all υ ∈ (0, K ∗ ].
Equation (1.1) includes several practical models. Letting d(υ(t, x)) = δυ(t, x),
(1.1) reduces to the time-delayed reaction-diffusion equation
∂ 2 υ(t, x)
∂υ(t, x)
=D
− δυ(t, x) + f (υ(t − r, x)).
∂t
∂x2
2010 Mathematics Subject Classification. 35C07, 92D25, 35B35.
Key words and phrases. Stability; non-monotone; weighted energy method.
c
2016
Texas State University.
Submitted July 20, 2015. Published March 29, 2016.
1
(1.3)
2
Y. LIU, Z. YU, J. XIA
EJDE-2016/86
This model represents the single species population distribution such as the Australian blowfly [11, 12, 27]. Here υ(t, x) denotes the mature population of the
blowflies at location x and time t, D > 0 and δ > 0 are the diffusion coefficient
and death rate of the mature population, the time delay r > 0 is the time taken
from birth to maturity, and f (υ(t − r, x)) is the birth function. Especially, taking
f (υ) = pυe−aυ , p > 0, a > 0, (1.3) is a typical Nicholson’s blowflies model; i.e.,
∂ 2 υ(t, x)
∂υ(t, x)
− δυ(t, x) + pυ(t − r)e−aυ(t−r) .
(1.4)
=D
∂t
∂x2
When the birth function f is monotone, authors in [9, 13, 22, 23, 28, 29] investigated the existence of monotone traveling waves by using the monotone iteration
and fixed-points theorem with help of the upper-lower solutions. Schaaf [26] first
studied linear stability for the delayed reaction diffusion with the quasi-monotone
nonlinear terms, which includes (1.1), by using a spectral method. The authors in
[21] investigated the nonlinear stability of traveling waves by using the (technical)
weighted energy method. Then authors in [25] further employed its global stability
by using the weighted energy technique and the comparison principle. These results were then extended to more general delayed reaction diffusion equations with
the quasi-monotone nonlinearity in [15, 16, 30]. By using the Fourier transform,
Green’s function and the weighted energy method, the authors in [24, 25] showed
the global stability of critical traveling waves, which depends on the monotonicity
of both the equation and traveling waves.
However, because of the lack of monotonicity, the simple but useful methods
have failed. For this case [6, 7, 8, 17, 30, 10] show the existence of traveling waves
by developing different methods. Especially, the study on the stability of traveling
waves is quite limited. Wu, Zhao and Liu [34] first showed the stability of traveling waves with the large wave speed for (1.3) by using weighted energy method.
Recently, Lin et al [14] established the stability of traveling waves (including oscillating traveling waves) for (1.4) by using the weighted energy method and the
nonlinear Halanay inequality. Then Chern et al [3] followed the recent study [14]
and further answered all critical traveling waves for (1.4) are time-asymptotically
stable with the help of some new development.
To the best of our knowledge, the stability of traveling waves for the more general
non-monotone delayed reaction diffusion equation (1.1) is still not investigated. The
methods in [34, 3] can still be used owing to the boundedness of the solution with
the initial condition for the non-monotone delayed reaction diffusion equation (1.1),
which was proved in [32].
2. Preliminaries and main result
We first introduce some notation. Throughout this paper, C > 0 denotes a
generic constant, while Ci > 0 (i = 0, 1, 2 . . . ) represents a special constant. Letting
I be an interval, especially I = R, L2 (I) is the space of the square integrable
function on I, and H k (I)(k ≥ 0) is the Sobolev space of the L2 -function f (x)
di
2
defined on I whose derivatives dx
L2ω (I)
i f , i = 1, . . . , k, also belong to L (I).
2
represents the weighted L -space with the weight ω(x) > 0 and its norm is defined
by
Z
1/2
kf kL2ω =
ω(x)f 2 (x)dx
.
I
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EXPONENTIAL STABILITY OF TRAVELING WAVES
3
Hωk (I) is the weighted Sobolev space with the norm given by
k Z
X
1/2
di
kf kHωk =
ω(x)| i f (x)|2 dx
.
dx
i=0 I
Letting T > 0 and B space, we denote by C 0 ([0, T ]; B) the space of the B-valued
continuous functions on [0, T ], and L2 ([0, T ]; B) as the space of B-valued L2 -function
on [0, T ]. The corresponding spaces of the B-valued function on [0, ∞) are defined
similarly.
The traveling waves for (1.1) connecting 0 and K are the special solution to (1.1)
in the form of υ(t, x) = φ(x + ct), namely, φ satisfies
cφ0 (ξ) − Dφ00 (ξ) + d(φ(ξ)) − f (φ(ξ − cr)) = 0,
φ(−∞) = 0,
φ(+∞) = K.
(2.1)
(2.2)
If f 0 (0) > d0 (0), there exists a unique number c∗ > 0 such that for c > c∗ , the
characteristic equation ∆(c, λ) = 0 of linearized equation at 0 for (2.1) has two
positive roots λ1 = λ1 (c) > 0 and λ2 = λ2 (c) > 0, where
∆(c, λ) := cλ − Dλ2 + d0 (0) − e−λcr f 0 (0).
(2.3)
Moreover,
cλ − Dλ2 + d0 (0) > e−λcr f 0 (0),
for λ1 < λ < λ2 .
(2.4)
Let us recall the existence and uniqueness of traveling waves for (1.1) with the
non-monotone nonlinearity, (see [17]) and some related results can also be found
in [5, 33] and the boundedness of the solution with the initial condition for the
non-monotone delayed reaction diffusion equation (1.1), see [32].
Proposition 2.1. Assume that (H1)–(H3) hold, there exists a unique number c∗ >
0 such for every c > c∗ , Equation (1.1) has a unique (up to translation) traveling
wave solution φ(ξ) satisfying φ(−∞) = 0, φ(+∞) = K and 0 ≤ φ(ξ) ≤ K ∗ for all
ξ ∈ R.
Proposition 2.2. Assume that (H1)–(H4) hold and 0 ≤ υ0 (s, x) ≤ K ∗ for all
(s, x) ∈ [−r, 0] × R. Then the solution of Cauchy problem (1.1) and (1.2) satisfies
0 ≤ υ(t, x) ≤ K ∗
for all (t, x) ∈ [0, +∞) × R.
For λ1 < λ < λ2 and some number ξ∗ , define the weight function
(
e−2λ(ξ−ξ∗ ) , for ξ < ξ∗ ,
ω(ξ) =
1,
for ξ ≥ ξ∗ .
(2.5)
For a given weight function ω(ξ) and letting T ≥ 0, we define the solution spaces
as
X(−r, T ) = {u|u(t, ξ) ∈ C([−r, T ]; C(R) ∩ Hω1 (R))}
and
M (T )2 =
sup
t∈[−r,T ]
ku(t)k2C + ku(t)k2Hω1 .
In particular, when T = ∞, we can also define the solution space as X(−r, ∞) and
the norm of the solution space as M (∞). Now, we state the stability result for
(1.1).
4
Y. LIU, Z. YU, J. XIA
EJDE-2016/86
Theorem 2.3 (Stability). Assume that (H1)–(H4) hold and |f 0 (K)| is sufficiently
small. For any given traveling wave φ(ξ) of (1.4) with speed c > c∗ , if the initial
perturbation is small; i.e.,
Z 0
2
2
k(υ0 − φ)(s)k2Hω1 ds ≤ δ02 ,
max k(υ0 − φ)(s)kC + k(υ0 − φ)(0)kHω1 +
s∈[−r,0]
−r
then the unique solution υ(t, x) of (1.1) and (1.2) exists globally and satisfies
υ(t, x) − φ(x + ct) ∈ C([−r, ∞); C(R) ∩ Hω1 (R)),
−µt
sup |υ(t, x) − φ(x + ct)| ≤ Ce
, t>0
(2.6)
(2.7)
x∈R
for the constant C > 0 and µ > 0.
3. Proof of stability
To obtain the stability of non-monotone delayed reaction diffusion equations, we
need to give some results.
Lemma 3.1. Assume that (H1)–(H4) and φ is a traveling wave for (1.4). Then
there exists A > 0 such that
|φ0 (ξ)| ≤ A
and
lim φ0 (ξ) = 0
ξ→±∞
Proof. Letting
p
p
c2 + 4Dd0 (0)
c + c2 + 4Dd0 (0)
ρ1 =
and ρ2 =
,
2D
2D
it follows from (1.4) that
Z +∞
hZ ξ
i
1
ρ1 (ξ−s)
φ(ξ) =
e
H(φ)(s)ds +
eρ2 (ξ−s) H(φ)(s)ds ,
D(ρ2 − ρ1 ) −∞
ξ
c−
where H(φ)(s) = f (φ(s − cr)) + d0 (0)φ(ξ) − d(φ(ξ)). Differentiating the above
equation with respect to ξ, we obtain
Z +∞
hZ ξ
i
1
ρ1 (ξ−s)
0
ρ1 e
H(φ)(s)ds +
ρ2 eρ2 (ξ−s) H(φ)(s)ds .
φ (ξ) =
D(ρ2 − ρ1 ) −∞
ξ
(3.1)
q
Since ρ2 − ρ1 ≥ 2
d0 (0)
D ,
we obtain
|φ0 (ξ)| ≤ p
1
Dd0 (0)
max |H(φ)(s)| := A
s∈R
for all ξ ∈ R.
Finally, (3.1) and the L.Hopital’s rule imply that limξ→+∞ φ0 (ξ) = 0. the proof is
complete.
Lemma 3.2. Assume that f 0 (0) > d0 (0) and |f 0 (K)| is sufficiently small. Then,
for every c > c∗ , there exist ξ0 , ξ∗ ∈ R with ξ∗ > ξ0 such that
max{|f 0 (φ(ξ0 − cr))|, |f 0 (φ(ξ0 ))|} ≤
min{∆(c, λ) + e−λcr f 0 (0), d0 (0)}
,
cosh(λcr)
and for ξ ≥ ξ∗ − cr > ξ0 ,
max{|f 0 (φ(ξ − cr))|, |f 0 (φ(ξ))|} ≤ max{|f 0 (φ(ξ0 − cr))|, |f 0 (φ(ξ0 ))|}.
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EXPONENTIAL STABILITY OF TRAVELING WAVES
5
Since limξ→±∞ φ(ξ) = K and f 0 (K) is sufficiently small, the conclusion of the
above lemma obviously holds.
Letting u(t, ξ) := υ(t, x) − φ(ξ), ξ = x + ct, where φ(x + ct) is a given traveling
wave solution of (1.1), the Cauchy problem (1.1) and (1.2) can be reformulated as
ut (t, ξ) + cuξ (t, ξ) − Duξξ (t, ξ) = g(u(t − r, ξ − cr)) − p(u(t, ξ)),
(t, ξ) ∈ (0, +∞) × R,
u(s, ξ) = υ0 (s, ξ − cs) − φ(ξ) =: u0 (s, ξ),
(3.2)
(s, ξ) ∈ [−r, 0] × R,
where
g(u) = f (φ + u) − f (φ),
p(u) = d(φ + u) − d(φ).
By the iteration technique and the energy method (see [14, 18, 19] ), we can obtain
the existence of local solutions for (3.2).
Theorem 3.3. Assume that (H1)–(H4) hold. For any given traveling wave φ(ξ)
with c > c∗ , suppose u0 (s, ξ) ∈ X(−r, 0), and M (0) ≤ δ1 , where δ1 is a given
positive constant. Then there exists a small t0 = t0 (δ1 ) > 0 such that the local
solution u(t, ξ) of (3.2) uniquely exists for t ∈ [−r, t0 ] and satisfies u ∈ X(−r, t0 )
and M (t0 ) ≤ aM (0) for some constant a.
Proof. Let u(0) (t, ξ) := u0 (t, ξ) ∈ X(−r, 0) ⊆ X(−r, t0 ). Then define the iteration
u(n+1) = T (u(n) ) for n ≥ 0 by
∂u(n+1)
∂ 2 u(n+1)
∂u(n+1)
+c
−D
= g(u(n) (t − r, ξ − cr)) − p(u(n) (t, ξ)),
∂t
∂ξ
∂ξ 2
(n+1)
u
(3.3)
s ∈ [−r, 0], ξ ∈ R.
(s, ξ) = u0 (s, ξ),
Using Fourier transform, (3.3) can be written as
Z
Z tZ
(n+1)
u
(t, ξ) =
Γ(η, t)u0 (0, ξ − η)dη +
Γ(η, t − s)
R
0
R
(n)
× g(u (s − r, ξ − η + cr)) − p(u(n) (s, ξ − η)) dη dξ,
(3.4)
where Γ(η, t) is the heat kernel
Γ(η, t) = √
(η+ct)2
1
e− 4Dt .
4πDt
By applying regular energy estimates to both sides of (3.3), indicated as
Z tZ X
1
∂ξk ((3.3)) × ω(ξ)∂ξk u(n+1) dξ ds,
0
R
k=0
we can estimate
ku(n+1) (t)k2Hω1 +
Z
0
t
ku(n+1) (s)k2Hω1 ds
Z
≤ C ku0 (0)k2Hω1 +
0
−r
ku0 (s)k2Hω1 ds +
Z
t
0
(3.5)
ku(n) (s)k2Hω1 ds ,
t ∈ [0, t0 ]
for some positive constant C > 0. From (3.4) it follows that
ku(n+1) (t)kC ≤ Cku0 (0)kC + Ct0
sup
t∈[−r,t0 ]
ku(n) (t)kC ,
t ∈ [0, t0 ].
(3.6)
6
Y. LIU, Z. YU, J. XIA
EJDE-2016/86
According to (3.5) and (3.6), it holds that
Z
Mu(n+1) (t0 ) ≤ C max ku0 (s)k2C +ku0 (0)k2Hω1 +
s∈[−r,0]
Thus,
0
−r
when maxs∈[−r,0] ku0 (s)k2C
(n+1)
(n)
+
ku0 (0)k2H 1
ω
+
ku0 (s)k2Hω1 ds +Ct0 Mu(n) (t0 ).
R0
−r
ku0 (s)k2H 1 ds 1 with 0 <
ω
t0 1, u
= T (u ) defined in (3.3) is a contraction mapping from X(−r, t0 )
to X(−r, t0 ). Hence, by using Banach fixed point theorem, (3.2) admits a unique
local solution in X(−r, t0 ). This completes the proof.
A priori estimate. We rewrite (3.2) as
ut (t, ξ) + cuξ (t, ξ) − Duξξ (t, ξ) + d0 (φ(ξ))u(t, ξ) − f 0 (φ(ξ − cr))u(t − r, ξ − cr)
= G(u)(t, ξ) − E(u)(t, ξ),
(t, ξ) ∈ (0, +∞) × R,
u(s, ξ) = u0 (s, ξ),
(s, ξ) ∈ [−r, 0] × R,
(3.7)
where
G(u)(t, ξ) = f (u(t − r, ξ − cr) + φ(ξ − cr)) − f (φ(ξ − cr))
− f 0 (φ(ξ − cr))u(t − r, ξ − cr),
E(u)(t, ξ) = d(u(t, ξ) + φ(ξ)) − d(φ(ξ)) − d0 (φ(ξ))u(t, ξ).
(3.8)
(3.9)
Lemma 3.4. Let u(t, ξ) ∈ X(−r, T ). Then
Z t
Z
2
−2µ(t−s)
ku(t)kL2ω +
e
[Bη,µ,ω (ξ) − CM (t)]ω(ξ)u2 (s, ξ) dξ ds
0
−2µt
≤ Ce
ku0 (0)k2L2ω
Z
R
0
+
−r
(3.10)
ku0 (s)k2L2ω ds
,
where
ω(ξ + cr)
1
,
Bη,µ,ω (ξ) := Aη,ω (ξ) − 2µ − (e2µr − 1)|f 0 (φ(ξ))|
η
ω(ξ)
ω 0 (ξ)
D ω 0 (ξ) 2
Aη,ω (ξ) := − c
+ 2d0 (φ(ξ)) − (
) − η|f 0 (φ(ξ − cr))|
ω(ξ)
2 ω(ξ)
ω(ξ + cr)
1
,
− |f 0 (φ(ξ))|
η
ω(ξ)
(3.11)
(3.12)
and µ, η are positive constants.
Proof. Multiplying (3.7) by e2µt ω(ξ)u(t, ξ) with ξ ∈ R and 0 ≤ t ≤ T , we have
n1
o
n1
o
e2µt ωu2 + e2µt cωu2 − Dωuuξ + De2µt ωu2ξ + De2µt ω 0 uξ u
2
2
t
ξ
n c ω0
o
+ −
+ d0 (φ(ξ)) − µ e2µt ωu2 − e2µt ω(ξ)u(t, ξ)f 0 (φ(ξ − cr))u(t − r, ξ − cr)
2ω
= e2µt ω(ξ)u(t, ξ)[G(u) − E(u)].
(3.13)
By the Cauchy-Schwarz inequality,
ω 0 2
D
|De2µt ω 0 uξ u| ≤ De2µt ωu2ξ + e2µt
ωu2 ,
4
ω
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EXPONENTIAL STABILITY OF TRAVELING WAVES
we reduce (3.13) to
n1
e2µt ωu2
o
+ e2µt
n1
cωu2 − Dωuuξ
7
o
2
2
t
ξ
n c ω0
0 2 o
D ω
e2µt ωu2
+ d0 (φ(ξ)) − µ −
+ −
2ω
4 ω
− e2µt ω(ξ)u(t, ξ)f 0 (φ(ξ − cr))u(t − r, ξ − cr)
(3.14)
≤ e2µt ω(ξ)u(t, ξ)[G(u) − E(u)].
Integrating (3.14) over R × [0, t] with respect to ξ and t, we have
Z tZ
n
ω 0 (ξ)
D ω 0 (ξ) 2 o
e2µt ku(t)k2L2ω +
e2µs − c
+ 2d0 (φ(ξ)) − 2µ −
ω(ξ)
2 ω(ξ)
0
R
× ω(ξ)u2 (s, ξ) dξ ds
Z tZ
−2
e2µs ω(ξ)f 0 (φ(ξ − cr))u(s, ξ)u(s − r, ξ − cr) dξ ds
0
R
Z tZ
2
≤ ku0 (0)kL2ω + 2
e2µs ω(ξ)u(s, ξ)[G(u)(s, ξ) − E(u)(s, ξ)] dξ ds.
0
(3.15)
R
Since
Z tZ
2
e2µs ω(ξ)f 0 (φ(ξ − cr))u(s, ξ)u(s − r, ξ − cr) dξ ds
0
R
Z tZ
≤η
e2µs ω(ξ)|f 0 (φ(ξ − cr))|u2 (s, ξ) dξ ds
0
R
Z Z
1 t
e2µs ω(ξ)|f 0 (φ(ξ − cr))|u2 (s − r, ξ − cr) dξ ds
+
η 0 R
Z tZ
=η
e2µs ω(ξ)|f 0 (φ(ξ − cr))|u2 (s, ξ) dξ ds
0
R
Z t−r Z
1
e2µs ω(ξ + cr)|f 0 (φ(ξ))|u2 (s, ξ) dξ ds
+ e2µr
η
−r
R
Z tZ
≤η
e2µs ω(ξ)|f 0 (φ(ξ − cr))|u2 (s, ξ) dξ ds
0
R
Z Z
1 2µr t
e2µs ω(ξ + cr)|f 0 (φ(ξ))|u2 (s, ξ) dξ ds
+ e
η
0
R
Z Z
1 2µr 0
+ e
e2µs ω(ξ + cr)|f 0 (φ(ξ))|u20 (s, ξ) dξ ds,
η
−r R
(3.16)
Substituting (3.16) in (3.15), we have
Z tZ
2µt
2
e ku(t)kL2ω +
e2µs Bη,µ,ω (ξ)ω(ξ)u2 (s, ξ) dξ ds
0
≤
ku0 (0)k2L2ω
Z tZ
+2
0
e2µr
+
η
R
Z
0
−r
Z
e2µs ω(ξ + cr)|f 0 (φ(ξ)|u20 (s, ξ) dξ ds
R
e2µs ω(ξ)u(s, ξ)[G(u)(s, ξ) − E(u)(s, ξ)] dξ ds,
R
where Bη,µ,ω (ξ) is given by (3.11).
(3.17)
8
Y. LIU, Z. YU, J. XIA
EJDE-2016/86
By standard Sobolev’s embedding inequality H 1 (R) ,→ C 0 (R) and the embedding inequality Hω1 (R) ,→ H 1 (R) (since ω(ξ) ≥ 1, for all ξ ∈ R), we have, for all
ξ ∈ R and −r ≤ t ≤ T ,
|u(t, ξ)| ≤ sup |u(t, ξ)| ≤ σ0 ku(t, ·)kH 1 ≤ σ0 ku(t, ·)kHω1 ≤ σ0 M (t),
(3.18)
ξ∈R
where σ0 > 0 is the embedding constant. Since
|G(u)(t, ξ)| = |f (u(t − r, ξ − cr) + φ(ξ − cr)) − f (φ(ξ − cr))
− f 0 (φ(ξ − cr))u(t − r, ξ − cr)| ≤ C|u(t − r, ξ − cr)|2 ,
we obtain
Z tZ
e2µs ω(ξ)u(s, ξ)G(u)(s, ξ) dξ ds
2
0
R
Z tZ
≤ Cσ0 M (t)
e2µs ω(ξ)u2 (s − r, ξ − cr) dξ ds
0
R
Z t−r Z
= Cσ0 M (t)
e2µ(s+r) ω(ξ + cr)u2 (s, ξ) dξ ds
−r
≤ CM (t)e
2µr
R
nZ t Z
0
≤ CM (t)
Z
2
0
Z
o
e2µs ω(ξ + cr)u20 (s, ξ) dξ ds
ω(ξ)u (s, ξ) dξ ds +
−r
R
nZ t Z
0
e
2µs
e2µs ω(ξ)u2 (s, ξ) dξ ds +
Z
0
Z
−r
R
R
o
e2µs ω(ξ + cr)u20 (s, ξ) dξ ds .
R
(3.19)
On the other hand,
|E(u)(t, ξ)| = |d(u(t, ξ) + φ(ξ)) − d(φ(ξ)) − d0 (φ(ξ))u(t, ξ)| ≤ C|u(t, ξ)|2 ,
we can also obtain
Z tZ
Z tZ
2
e2µs ω(ξ)u(s, ξ)E(u)(s, ξ) dξ ds ≤ CM (t)
e2µs ω(ξ)u2 (s, ξ) dξ ds.
0
0
R
R
(3.20)
From (3.19), (3.20) and (3.17), we have
Z tZ
e2µt ku(t)k2L2ω +
e2µs [Bη,µ,ω (ξ) − CM (t)]ω(ξ)u2 (s, ξ) dξ ds
0
≤ ku0 (0)k2L2ω +
Z
R
0
Z
−r
≤ C ku0 (0)k2L2ω +
Z
R
0
−r
e2µs ω(ξ + cr)[CM (t) +
e2µr 0
|f (φ(ξ)|]u20 (s, ξ) dξ ds (3.21)
η
ku0 (s)k2L2ω ds ,
which immediately implies (3.10). This completes the proof.
Next we prove a key inequality.
Lemma 3.5. Letting η = e−λcr , there exists a unique number c∗ > 0, such for
every c > c∗ , there exists a constant C1 > 0 such that
Aη,ω (ξ) ≥ C1 > 0
for ξ ∈ R.
(3.22)
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EXPONENTIAL STABILITY OF TRAVELING WAVES
9
Proof. We distinguish three cases:
Case 1: For ξ < ξ∗ − cr, ω(ξ) = e−2λ(ξ−ξ∗ ) and ω(ξ + cr) = e−2λ(ξ−ξ∗ +cr)
ω 0 (ξ)
= −2λ,
ω(ξ)
ω(ξ + cr)
= e−2λcr .
ω(ξ)
By (H4) and (2.4), we can have
ω 0 (ξ)
D ω 0 (ξ) 2
+ 2d0 (φ(ξ)) −
− η|f 0 (φ(ξ − cr))|
ω(ξ)
2 ω(ξ)
ω(ξ + cr)
1
− |f 0 (φ(ξ))|
η
ω(ξ)
Aη,ω (ξ) := −c
(3.23)
= 2cλ − 2Dλ2 + 2d0 (φ(ξ)) − e−λcr |f 0 (φ(ξ − cr))| − e−λcr |f 0 (φ(ξ))|
≥ 2 cλ − Dλ2 + d0 (0) − e−λcr f 0 (0) =: C11 > 0.
Case 2: For ξ∗ − cr ≤ ξ ≤ ξ∗ , then ω(ξ) = e−2λ(ξ−ξ∗ ) and ω(ξ + cr) = 1, and by
Lemma 3.2, we have
Aη,ω (ξ)
D ω 0 (ξ) 2
ω 0 (ξ)
+ 2d0 (φ(ξ)) −
− η|f 0 (φ(ξ − cr))|
ω(ξ)
2 ω(ξ)
ω(ξ + cr)
1
− |f 0 (φ(ξ))|
η
ω(ξ)
:= −c
= 2cλ − 2Dλ2 + 2d0 (φ(ξ)) − η|f 0 (φ(ξ − cr))|
1
− |f 0 (φ(ξ))|e2λ(ξ−ξ∗ )
η
(3.24)
≥ 2(cλ − Dλ2 + d0 (0) − e−λcr f 0 (0)) + 2e−λcr f 0 (0) − η|f 0 (φ(ξ − cr))|
1
− |f 0 (φ(ξ))|
η
1
≥ 24(c, λ) + 2e−λcr f 0 (0) − (η + ) max{|f 0 (φ(ξ − cr))|, |f 0 (φ(ξ))|}
η
≥ 24(c, λ) + 2e−λcr f 0 (0) − 2 max{|f 0 (φ(ξ − cr))|, |f 0 (φ(ξ))|} cosh(λcr)
≥ 2 4(c, λ) + e−λcr f 0 (0) − max{|f 0 (φ(ξ0 − cr))|, |f 0 (φ(ξ0 ))|} cosh(λcr)
=: C12 > 0.
Case 3: For ξ ≥ ξ∗ , ω(ξ) = ω(ξ + cr) = 1, and by Lemma 3.2, we obtain
1
Aη,ω (ξ) : = 2d0 (φ(ξ)) − η|f 0 (φ(ξ − cr))| − |f 0 (φ(ξ))|
η
1
≥ 2 d0 (0) − (η + ) max{|f 0 (φ(ξ − cr))|, |f 0 (φ(ξ))|}
η
≥ 2 d0 (0) − max{|f 0 (φ(ξ0 − cr))|, |f 0 (φ(ξ0 ))|} cosh(λcr)
=: C13 > 0.
(3.25)
10
Y. LIU, Z. YU, J. XIA
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Combining (3.23)–(3.25), we obtain Aη,ω (ξ) ≥ C1 , where C1 = mini=1,2,3 {C1i } > 0.
This completes the proof.
Lemma 3.6. Let u(t, ξ) ∈ X(−r, T ). Then there exists a constant µ∗ > 0 such
that for 0 < µ < µ∗ , it holds
Z tZ
e−2µ(t−s) ku(s)k2L2ω ds
ku(t)k2L2ω +
0
R
(3.26)
Z 0
ku0 (s)k2L2ω ds ,
≤ Ce−2µt ku0 (0)k2L2ω +
−r
provided M (t) 1.
Proof. We distinguish three cases:
Case 1: For ξ < ξ∗ − cr, ω(ξ) = e−2λ(ξ−ξ∗ ) , ω(ξ + cr) = e−2λ(ξ−ξ∗ +cr) , and
according to Lemma 3.5,
1
ω(ξ + cr)
Bη,µ,ω (ξ) = Aη,ω (ξ) − 2µ − (e2µr − 1)|f 0 (φ(ξ))|
η
ω(ξ)
(3.27)
0
−λcr 2µr
≥ C1 − 2µ − f (0)e
(e
− 1)
=: C21 > 0 for 0 < µ < µ1 ,
where µ1 is the unique root of the equation
C1 − 2µ − f 0 (0)e−λcr (e2µr − 1) = 0.
Case 2: For ξ∗ − cr ≤ ξ ≤ ξ∗ , then ω(ξ) = e−2λ(ξ−ξ∗ ) and ω(ξ + cr) = 1,
1
ω(ξ + cr)
Bη,µ,ω (ξ) = Aη,ω (ξ) − 2µ − (e2µr − 1)|f 0 (φ(ξ))|
η
ω(ξ)
≥ C1 − 2µ − f 0 (0)eλcr (e2µr − 1)e2λ(ξ−ξ∗ )
0
λcr
≥ C1 − 2µ − f (0)e
=: C22 > 0
(e
2µr
(3.28)
− 1)
for 0 < µ < µ2
where µ2 is the unique root of the equation
C1 − 2µ − f 0 (0)eλcr (e2µr − 1) = 0.
Case 3: For ξ ≥ ξ∗ , ω(ξ) = ω(ξ + cr) = 1,
ω(ξ + cr)
1
Bη,µ,ω (ξ) = Aη,ω (ξ) − 2µ − (e2µr − 1)|f 0 (φ(ξ))|
η
ω(ξ)
≥ C1 − 2µ − f 0 (0)eλcr (e2µr − 1)
=: C22 > 0
(3.29)
for 0 < µ < µ2 .
Combining (3.27), (3.28) and (3.29), we obtain Bη,µ,ω (ξ) ≥ C2 , for 0 < µ < µ∗ ,
where C2 := min{C21 , C22 } and µ∗ = min{µ1 , µ2 }. It follows from (3.10) that
Z
Z t
e−2µ(t−s) [C2 − CM (t)]ω(ξ)u2 (s, ξ) dξ ds
ku(t)k2L2ω +
0
−2µt
≤ Ce
ku0 (0)k2L2ω
Z
R
0
+
−r
ku0 (s)k2L2ω dξ ds ,
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EXPONENTIAL STABILITY OF TRAVELING WAVES
11
which implies (3.26) by letting M (t) 1. This completes the proof.
Next, we shall establish the energy estimate for uξ , which is similar to (3.26).
Lemma 3.7. Let u(t, ξ) ∈ X(−r, T ). Then it holds
Z t
kuξ (t)k2L2ω +
e−2µ(t−s) kuξ (t)k2L2ω ds
0
≤ Ce−2µt ku0 (0)k2Hω1 +
Z
0
−r
(3.30)
ku0 (s)k2Hω1 ds .
provided M (t) 1.
Proof. Differentiating (3.2) with respect to ξ and multiplying the obtained equation
by e2µt ω(ξ)uξ (t, ξ), we have
n1
o
n1
o
e2µt ωu2ξ + e2µt cωu2ξ − Dωuξ uξξ + De2µt ωu2ξξ + De2µt ω 0 uξ uξξ
2
2
t
ξ
o
n c ω0
+ d0 (φ(ξ)) − µ e2µt ωu2ξ − e2µt ω(ξ)uξ (t, ξ)f 0 (φ(ξ − cr))
+ −
(3.31)
2ω
× uξ (t − r, ξ − cr)
= e2µt ω(ξ)uξ (t, ξ)[G2 (u) + G1 (u)] − e2µt ω(ξ)uξ (t, ξ)[E2 (u) + E1 (u)],
where
G1 (u)(t, ξ) = [f 0 (u(t − r, ξ − cr) + φ(ξ − cr)) − f 0 (φ(ξ − cr))]φ0 (ξ − cr),
G2 (u)(t, ξ) = [f 0 (u(t − r, ξ − cr) + φ(ξ − cr)) − f 0 (φ(ξ − cr))]uξ (t − r, ξ − cr),
E1 (u)(t, ξ) = [d0 (u(t, ξ) + φ(ξ)) − d0 (φ(ξ))]φ0 (ξ),
E2 (u)(t, ξ) = [d0 (u(t, ξ) + φ(ξ)) − d0 (φ(ξ))]uξ (t, ξ).
Using the Cauchy-Schwarz inequality
|De2µt ω 0 uξ uξξ | ≤ De2µt ωu2ξξ +
D 2µt ω 0 2 2
e
ωuξ ,
4
ω
it follows from (3.31) that
n1
o
n1
o
e2µt ωu2ξ + e2µt cωu2ξ − Dωuξ uξξ
2
2
t
ξ
n c ω0
ω 0 2 o
D
+ −
+ d0 (φ(ξ)) − µ −
e2µt ωu2ξ
2ω
4 ω
− e2µt ω(ξ)uξ (t, ξ)f 0 (φ(ξ − cr))uξ (t − r, ξ − cr)
(3.32)
≤ e2µt ω(ξ)uξ (t, ξ)[G2 (u) + G1 (u)] − e2µt ω(ξ)uξ (t, ξ)[E2 (u) + E1 (u)].
Integrating the above inequality over R × [0, t] with respect to ξ and t, we have
Z tZ
n
ω 0 (ξ)
D ω 0 (ξ) 2 o
e2µt kuξ (t)k2L2ω +
e2µs − c
+ 2d0 (φ(ξ)) − 2µ −
ω(ξ)
2 ω(ξ)
0
R
× ω(ξ)u2ξ (s, ξ) dξ ds
Z tZ
−2
e2µs ω(ξ)f 0 (φ(ξ − cr))uξ (s, ξ)uξ (s − r, ξ − cr) dξ ds
0
R
(3.33)
12
Y. LIU, Z. YU, J. XIA
EJDE-2016/86
Z tZ
e2µs ω(ξ)uξ (s, ξ)[G2 (u)(s, ξ) + G1 (u)(s, ξ)] dξ ds
≤ ku0 (0)k2Hω1 + 2
0
R
Z tZ
2µs
e ω(ξ)uξ (s, ξ)[E2 (u)(s, ξ) + E1 (u)(s, ξ)] dξ ds.
−2
0
R
By the Cauchy-Schwarz inequality, we have
Z tZ
e2µs ω(ξ)f 0 (φ(ξ − cr))uξ (s, ξ)uξ (s − r, ξ − cr) dξ ds
2
0
R
Z tZ
≤η
e2µs ω(ξ)|f 0 (φ(ξ − cr))|u2ξ (s, ξ) dξ ds
0
R
Z Z
1 2µr t
+ e
e2µs ω(ξ + cr)|f 0 (φ(ξ))|u2ξ (s, ξ) dξ ds
η
0
R
Z Z
1 2µr 0
e2µs ω(ξ + cr)|f 0 (φ(ξ))|u20ξ (s, ξ) dξ ds.
+ e
η
−r R
(3.34)
It follows from (3.33) and (3.34) that
Z tZ
2µt
2
e kuξ (t)kL2ω +
e2µs Bη,µ,ω (ξ)ω(ξ)u2ξ (s, ξ) dξ ds
0
≤ ku0 (0)k2Hω1
Z tZ
+2
e
0
2µs
Z
0
−r
Z
e2µs ω(ξ + cr)|f 0 (φ(ξ))|u20ξ (s, ξ) dξ ds
R
(3.35)
ω(ξ)uξ (s, ξ)[G2 (u)(s, ξ) + G1 (u)(s, ξ)] dξ ds
R
Z tZ
−2
0
R
e2µr
+
η
e2µs ω(ξ)uξ (s, ξ)[E2 (u)(s, ξ) + E1 (u)(s, ξ)] dξ ds.
R
Again, by the Taylor expansion,
|G2 (u)(t, ξ)| = [f 0 (u(t − r, ξ − cr) + φ(ξ − cr)) − f 0 (φ(ξ − cr))]uξ (t − r, ξ − cr)
≤ C|u(t − r, ξ − cr)|uξ (t − r, ξ − cr),
we can estimate the nonlinear term as
Z tZ
2
e2µs ω(ξ)uξ (s, ξ)G2 (u)(s, ξ) dξ ds
0
R
Z tZ
≤C
e2µs ω(ξ)uξ (s, ξ)|u(s − r, ξ − cr)|uξ (s − r, ξ − cr) dξ ds
0
R
Z tZ
≤ CM (t)
e2µs ω(ξ)u2ξ (s, ξ) dξ ds
0
+ CM (t)
R
Z 0
Z
+
−r
Z tZ
≤ CM (t)
0
Z
R
0
e2µ(s+r) ω(ξ + cr)u2ξ (s, ξ) dξ ds
R
e2µs ω(ξ)u2ξ (s, ξ) dξ ds
R
0 Z
+ CM (t)
−r
Z tZ R
e2µ(s+r) ω(ξ + cr)u20ξ (s, ξ) dξ ds,
(3.36)
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EXPONENTIAL STABILITY OF TRAVELING WAVES
13
and
Z tZ
e2µs ω(ξ)uξ (s, ξ)G1 (u)(s, ξ) dξ ds
2
0
R
Z tZ
e2µs ω(ξ)uξ (s, ξ)u(s − r, ξ − cr)φ0 (ξ − cr) dξ ds
≤ C
0
R
Z tZ
≤C
e2µs ω(ξ)uξ (s, ξ)|u(s − r, ξ − cr)| dξ ds
0
R
Z tZ
≤C
e2µs ω(ξ)u2ξ (s, ξ) dξ ds
0
+C
Z
R
0
−r
Z
+
Z tZ 0
R
(3.37)
e2µ(s+r) ω(ξ + cr)u2 (s, ξ) dξ ds.
R
Similarly, we obtain
Z tZ
Z tZ
2
e2µs ω(ξ)uξ (s, ξ)E2 (u)(s, ξ)) dξ ds ≤ CM (t)
e2µs ω(ξ)u2ξ (s, ξ) dξ ds
0
0
R
R
and
Z tZ
2
e2µs ω(ξ)uξ (s, ξ)E1 (u)(s, ξ)) dξ ds
0
R
Z tZ
≤ C
e2µs ω(ξ)uξ (s, ξ)u(s, ξ))φ0 (ξ − cr) dξ ds
0
R
Z tZ
≤C
e2µs ω(ξ)u2ξ (s, ξ) dξ ds
0
R
Z tZ
+C
e2µs ω(ξ)u2 (s, ξ) dξ ds.
0
(3.38)
R
It then follows from (3.36)-(3.38) and Lemma 3.6 that
Z t
Z
kuξ (t)k2L2ω +
e−2µ(t−s) kuξ (t)k2L2ω ds ≤ Ce−2µt ku0 (0)k2Hω1 +
0
−r
0
ku0 (s)k2Hω1 ds .
Combining (3.26) and (3.30), for some constant C, which is independent of T and
u(t, ξ), we have
Z 0
ku(t)k2Hω1 ≤ Ce−2µt ku0 (0)k2Hω1 +
ku0 (s)k2Hω1 ds , for all 0 ≤ t ≤ T.
−r
This completes the proof.
Proof of Theorem 2.3. It is based on the existence of a local solution and the estimate obtained above. The process is similar to the one in [20, 21], using the
continuity extension method, so we omit it.
Acknowledgments. The second author was supported by the National Natural
Science Foundation of China, by Shanghai Leading Academic Discipline Project
(No. XTKX2012), by Innovation Program of Shanghai Municipal Education Commission (No. 14YZ096), by the Hujiang Foundation of China (B14005). The
third author was supported by National Natural Science Foundation of China (No.
11301542).
14
Y. LIU, Z. YU, J. XIA
EJDE-2016/86
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Yixin Liu
College of Science, University of Shanghai for Science and Technology, Shanghai,
200093, China
E-mail address: 838080693@qq.com
Zhixian Yu (corresponding author)
College of Science, University of Shanghai for Science and Technology, Shanghai,
200093, China
E-mail address: zxyu@usst.edu.cn, zxyu0902@163.com
Jing Xia
Department of Fundamental Courses, Academy of Armored Force Engineering, Beijing
100072, China
E-mail address: xiajingcjh@163.com