Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 390476, 8 pages
http://dx.doi.org/10.1155/2013/390476
Research Article
Exponential Attractor for Coupled Ginzburg-Landau
Equations Describing Bose-Einstein Condensates and Nonlinear
Optical Waveguides and Cavities
Gui Mu and Jun Liu
College of Mathematics and Information Science, Qujing Normal University, Qujing, Yunnan 655011, China
Correspondence should be addressed to Jun Liu; liujunxei@126.com
Received 4 February 2013; Accepted 6 March 2013
Academic Editor: de Dai
Copyright © 2013 G. Mu and J. Liu. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The existence of the exponential attractors for coupled Ginzburg-Landau equations describing Bose-Einstein condensates and
nonlinear optical waveguides and cavities with periodic initial boundary is obtained by showing Lipschitz continuity and the
squeezing property.
1. Introduction
Inertial set was introduced (see [1–5]) in order to overcome
some of the theoretical difficulties that are associated with
inertial manifolds. An inertial set, by definition, contains
the global attractor and attracts all trajectories at a uniform
exponential rate. Consequently, it contains the slow transients
as well as the global attractor. In the theory of dynamical
systems the slow transients correspond to slowly converging
stable manifolds that are in some sense close to central
manifolds. Numerical simulations of infinite dimensional
dynamical systems often capture both slow transients and
parts of the attractor. After a large but finite time the state
of the system obtained from the numerical calculation may
often be at a finite distance from the global attractor but at
an infinitesimal distance to the inertial set. In this sense, we
propose to call the inertial set an exponential attractor to be
consistent with the physical intuition [5].
An exponential attractor is an exponentially attracting
compact set with finite fractal dimension that is positively
invariant under the forward semiflow. The notion of exponential attractors was introduced by Eden et al. [3] and has
been shown to be one of the very important notions in the
study of long time behavior of solutions to nonlinear diffusion
equations [6]. The easiest way of obtaining an exponential
attractor is by taking the intersection of an absorbing set with
an inertial manifold.
In the area of hyperbolic evolutionary equations, the
existence of exponential attractors has been proved for many
equations. In this paper, we will prove the existence of exponential attractor for coupled Ginzburg-Landau equations
𝑖𝑢𝑡 + 𝛾2 Δ𝑢 + 𝑖𝛾𝑢 + (𝜎1 + 𝑖𝜎2 |𝑢|2 ) |𝑢|2 𝑢 + V = 0,
𝑖V𝑡 + 𝛾2 Δ𝑢 + (𝑖Γ − 𝜒) V + 𝑢 = 0,
(1)
with the periodic boundary conditions
𝑢 (𝑥, 𝑡) = 𝑢 (𝑥 + 𝐷, 𝑡) ,
V (𝑥, 𝑡) = V (𝑥 + 𝐷, 𝑡) ,
𝑥 ∈ 𝑅,
𝑡 > 0,
V (𝑥, 0) = V0 (𝑥) ,
𝑥 ∈ 𝑅.
(2)
and initial value
𝑢 (𝑥, 0) = 𝑢0 (𝑥) ,
(3)
Its physical realizations include systems from nonlinear
optics and a double-cigar-shaped Bose-Einstein condensate
with a negative scattering length. In particular, in the case of
the optical systems, 𝑢 and V are amplitudes of electromagnetic
waves in two cores of the system, the evolutional variable 𝑡 is
either time or propagation distance in the dual-core optical
fiber, and 𝑥 is the transverse coordinate in the cavity or the
reduced time in the application to the fibers [7].
This paper is organized as follows. In Section 2, we give
a description of preliminaries with existence of exponential
2
Abstract and Applied Analysis
attractor and the properties of solutions and bounded absorbing sets of (1). In Section 3, the existence of the exponential
attractor in 𝑉2 type exponential attractor is proved. In
Section 4, we give some conclusions for this paper.
2. Preliminaries
Let 𝑉1 , 𝑉2 be two Hilbert spaces, and let 𝑉2 be dense in 𝑉1 and
compactly imbedded into 𝑉1 . Let 𝑆(𝑡)𝑡≥0 be a continuous map
from 𝑉1 , 𝑉2 into itself. We study
𝑑𝑢
+ 𝐴𝑢 + 𝑔 (𝑢) = 𝑓 (𝑥) ,
𝑑𝑡
𝑡 > 0, 𝑥 ∈ Ω,
𝑢 (𝑥, 0) = 𝑢0 (𝑥) ,
Dirichlet problem or periodic boundary problem,
(4)
holds, then we also have
󵄩󵄩
󵄩
󵄩󵄩𝑆 (𝑡∗ ) 𝑢 − 𝑆 (𝑡∗ ) V󵄩󵄩󵄩V2 ≤ 𝛿‖𝑢 − V‖V2 ,
(13)
(5)
where 𝑄𝑁0 = 𝐼 − 𝑃𝑁0 .
(6)
Theorem 4 (see [3]). Suppose (4)–(6) satisfy the following
conditions.
where Ω is a bounded open set in 𝑅𝑛 , 𝜕Ω is smooth, and
𝐴 is a positive self-adjoint operator with a compact inverse.
Letting {𝑤𝑖 }∞
𝑖=1 denote the complete set of eigenvectors of 𝐴,
the corresponding eigenvalues are
0 ≤ 𝜆 1 < 𝜆 1 ⋅ ⋅ ⋅ 𝜆 𝑖 < ⋅ ⋅ ⋅ 󳨀→ +∞.
Definition 3. A continuous semigroup 𝑆(𝑡)𝑡≥0 is said to satisfy
the squeezing property on 𝐶 if there exists 𝑡∗ > 0 such that
𝑆(𝑡∗ ) satisfies the following.
For every 𝛿 ∈ (0, (1/8)), there exists an orthogonal
projection 𝑃𝑁0 of rank equal to 𝑁0 such that for every 𝑢 and
V in 𝐶 if
󵄩󵄩
󵄩
󵄩
󵄩
󵄩󵄩𝑃𝑁0 (𝑆 (𝑡∗ ) 𝑢 − 𝑆 (𝑡∗ ) V)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑄𝑁0 (𝑆 (𝑡∗ ) 𝑢 − 𝑆 (𝑡∗ ) V)󵄩󵄩󵄩
󵄩
󵄩V2 󵄩
󵄩V2
(12)
(7)
(1) There exist nonlinear semigroup 𝑆(𝑡)𝑡≥0 and a compact
attractor B.
(2) There exists a compact set C in 𝑉2 which is positively
invariant for 𝑆(𝑡)𝑡≥0 .
(3) 𝑆(𝑡)𝑡≥0 is Lipschitz continuous and is squeezing in 𝐶.
We assume that the nonlinear semigroup 𝑆(𝑡)𝑡≥0 defined
in (4)–(6) possesses a compact attractor B of (𝑉2 , 𝑉1 )-type;
namely, there exists a compact set B in 𝑉1 , and B attracts all
bounded subsets in 𝑉2 and is invariant under the action of
𝑆(𝑡)𝑡≥0 .
Let 𝐶 be a compact subset of 𝑉2 . 𝑆(𝑡)𝑡≥0 leaves the set 𝐶
invariant and set
Then (4)–(6) admit an exponential attractor 𝑀 in 𝑉2 for
𝑆(𝑡)𝑡≥0 and
B = ⋂ ⋃ 𝑆(𝑡)𝑡≥0 𝐶,
𝑀∗ = B ⋃ (⋃ ⋃ 𝑆(𝑡∗ ) (𝐸(𝑘) )) .
0≤𝑡≤𝑡∗
(14)
where
∞ ∞
(8)
𝑠≥0 𝑡≥𝑠
𝑗
(15)
𝑗=1 𝑘=1
that is, for 𝑆(𝑡)𝑡≥0 on 𝐶, B is the global attractor.
Moreover,
Definition 1. A compact set 𝑀 is called an exponential
attractors for 𝑆(𝑡)𝑡≥0 , 𝐶 if
(i) B ⊆ 𝑀 ⊆ 𝐶;
(iii) 𝑀 has finite fractal dimension 𝑑𝐹 < ∞;
(9)
where
󵄩
󵄩
distV2 (𝐴, 𝐷) = sup inf 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩𝑉2 .
𝑦∈𝐷
𝑥∈𝐴
dist𝑉2 (𝑆 (𝑡) 𝐵, 𝑀) ≤ 𝐶0 exp (−𝐶1 𝑡) ,
(16)
Proposition 5. There exists 𝑡0 (𝐵0 ) such that
(iv) There exist constants 𝑐1 and 𝑐2 such that
∀𝑡 > 0,
𝑑𝐹 (𝑀) ≤ 1 + 𝐶𝑁0 ,
where 𝑁0 , 𝐸(𝑘) are defined as in [4], 𝐶, 𝐶0 , 𝐶1 are the constants
independent of 𝐵, and 𝑡∗ is a positive constant.
(ii) 𝑆(𝑡)𝑀 ⊆ 𝑀, for every 𝑡 ≥ 0;
distV2 (𝑆 (𝑡) 𝐶, 𝑀) ≤ 𝑐1 exp (−𝑐2 𝑡) ,
𝑀 = ⋃ 𝑆 (𝑡) 𝑀∗ ,
(10)
𝐵∗ = ⋃ 𝑆 (𝑡) 𝐵0
0≤𝑡≤𝑡0
(17)
is a compact positively invariant set in 𝑉1 and is absorbing set
for all bounded subsets in 𝑉2 , where 𝐵0 is a closed absorbing set
in 𝑉2 for 𝑆(𝑡)𝑡≥0 .
(11)
Proposition 6. Let 𝐵0 , 𝐵1 be bounded and closed absorbing
sets for (4)–(6) in (𝑉2 , 𝑉1 ), respectively. Then there exists
a compact attractor 𝐴∗ of (𝑉2 , 𝑉1 )-type. For the proof of
Proposition 5 and Proposition 6, we refer the reader to [5].
for every 𝑢, V ∈ 𝐶, then we say 𝑆(𝑡) is Lipschitz continuous in
𝐶 and 𝑙(𝑡) is Lipschitz constant for 𝑆(𝑡) in 𝐶.
Denoting by | ⋅ |𝐿𝑝 the norm in 𝐿𝑝 (0, 𝐿), 1 ≤ 𝑝 ≤ ∞, for
simplicity, we denote by | ⋅ |0 and | ⋅ |∞ the norm in the case
Definition 2. If there exists a bounded function 𝑙(𝑡) independent 𝑢 and V such that
‖𝑆 (𝑡) 𝑢 − 𝑆 (𝑡) V‖𝑉2 ≤ 𝑙 (𝑡) ‖𝑢 − V‖𝑉2 ,
Abstract and Applied Analysis
3
𝑝 = 2 and 𝑝 = ∞, respectively. Suppose that 𝐻 = 𝐿2 (0, 𝐿),
𝐸𝑖 = 𝐻𝑖 (0, 𝐿) × 𝐻𝑖 (0, 𝐿) (𝑖 = 1, 2), where 𝐻𝑖 (0, 𝐿) is a Hilbert
space for the scalar product
𝑖
𝜕
𝐷=
.
𝜕𝑥
((⋅, ⋅))𝐻𝑖 = (⋅, ⋅) + ∑ (𝐷𝑗 ⋅, 𝐷𝑗 ⋅) ,
𝑗=1
(18)
3. Exponential Attractor in 𝑉2 for Problem
(1)-(2)
In this section, we show the existence of the exponential
attractor in 𝑉2 for problem (1)-(2). In order to prove the
Lipschitz continuity and the squeezing property, we need to
extend Hölder inequality
The norm of 𝐸𝑖 is defined by ‖(𝑢, V)‖2𝐸𝑖 = ‖𝑢‖2𝐻𝑖 + ‖V‖2𝐻𝑖 .
We now establish some time-uniform a priori estimates
on (𝑢, V) in 𝐸1 and 𝐸2 , respectively.
Lemma 7. Assume that (𝑢0 , V0 ) ∈ 𝐸1 ; then
󵄩
󵄩2
‖(𝑢, V)‖2𝐸1 ≤ 𝑐󵄩󵄩󵄩(𝑢0 , V0 )󵄩󵄩󵄩𝐸1 𝑒−𝛿1 𝑡 + 𝑐1 .
(19)
𝑘
󵄩 󵄩
󵄨
󵄨
∫ 󵄨󵄨󵄨𝑢 (𝑥) 𝑢2 (𝑥) ⋅ ⋅ ⋅ 𝑢𝑘 (𝑥)󵄨󵄨󵄨 𝑑𝑥 ≤ ∏󵄩󵄩󵄩󵄩𝑢𝑗 󵄩󵄩󵄩󵄩𝐿𝑝𝑗 ,
Ω
where ∑𝑘𝑗=1 (1/𝑝𝑗 ) = 1, 𝑝𝑗 > 1 and Gagliardo-Nirenberg (GN) inequality
󵄩󵄩 𝑗 󵄩󵄩
󵄩󵄩∇ 𝑢󵄩󵄩 ≤ 𝑐󵄩󵄩󵄩󵄩∇𝑚 𝑢󵄩󵄩󵄩󵄩𝑎𝑟 ‖𝑢‖1−𝑎
𝑞 ,
󵄩𝑝
󵄩
Thus there exists 𝑡1 = 𝑡1 (𝑅) > 0 such that
‖(𝑢, V)‖2𝐸1 ≤ 𝑐2 ,
𝑡 ≥ 𝑡1 ,
(20)
1 𝑗
1 𝑚
1−𝑎
= + 𝑎( − ) +
,
𝑝 𝑛
𝑟 𝑛
𝑞
Lemma 8. Assume that (𝑢0 , V0 ) ∈ 𝐸2 ; then
1 ≤ 𝑞,
(21)
Thus there exists 𝑡2 = 𝑡2 (𝑅) > 0 such that
‖(𝑢, V)‖2𝐸2 ≤ 𝑐4 ,
𝑡 ≥ 𝑡2 ,
(22)
Theorem 9. Assume that all the parameters of (1) are positive.
For (𝑢0 , V0 ) given in 𝐸𝑖 (𝑖 = 1, 2), there exists a unique solution
(𝑢, V) ∈ 𝐿∞ (𝑅+ , 𝐸𝑖 ) .
(23)
And also
∀ (𝑢0 , V0 ) ∈ 𝐸1 .
(24)
Furthermore, the solution operator of the system is a continuous
semigroup 𝑆(𝑡) on 𝐸1 which possesses bounded absorbing sets
𝐵𝑖 ⊂ 𝐸𝑖 , for 𝑖 = 1, 2.
𝑎𝑏 ≤
𝐵1 = {(𝑢, V) ∈ 𝐸1 , ‖𝑢‖𝐻1 + ‖V‖𝐻1 ≤ 𝑘} ,
𝐵2 = {(𝑢, V) ∈ 𝐸2 , ‖𝑢‖𝐻2 + ‖V‖𝐻2 ≤ 𝑘}
(25)
are bounded absorbing sets for 𝑆(𝑡) in 𝐸1 and 𝐸2 , respectively:
Let
𝑉2 = 𝐸2 ,
0 ≤ 𝑗 < 𝑚,
𝐵 = ⋃ 𝑆 (𝑡) 𝐵2 ,
𝑡≥0
(26)
then 𝐵 is a compact invariant subset in 𝑉2 ; we know that
semigroup 𝑆(𝑡) defined by problem (31)–(34) possesses a 𝑉2 type compact attractor. According to Theorem 4, we need
only to show the Lipschitz continuity and the squeezing
property of the dynamical system 𝑆(𝑡) in 𝐵, respectively. That
is what we proceed to do in the following sections.
𝜀 𝑝 1 (−𝑞/𝑏) 𝑞
𝑏,
𝑎 + 𝜀
𝑝
𝑞
𝑞 < ∞,
(29)
𝑎, 𝑏, 𝜀 > 0, 1 < 𝑝,
(30)
1 1
+ = 1.
𝑝 𝑞
Theorem 10. Assume 𝑤1 (𝑡) = (𝑢1 (𝑡), V1 (𝑡)), and 𝑤2 (𝑡) =
(𝑢2 (𝑡), V2 (𝑡)) are two solutions of problem (1)-(2) with initial
values 𝑤10 = (𝑢10 , V10 ), 𝑤20 = (𝑢20 , V20 ) ∈ 𝐵 = 𝐻2 × 𝐻2 ; then
one has
󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑤1 (𝑡) − 𝑤2 (𝑡)󵄩󵄩󵄩𝑉2 ≤ exp (2𝐶0 𝑡) 󵄩󵄩󵄩𝑤10 − 𝑤20 󵄩󵄩󵄩𝑉2 .
(31)
Proof. Letting ℎ(𝑡) = 𝑢1 (𝑡) − 𝑢2 (𝑡), 𝑔(𝑡) = V1 (𝑡) − V2 (𝑡), from
(1)-(2), we have
Thus, we observe that Lemmas 7 and 8 show that there
exists constant 𝑘 depending only on the data that the balls
𝑉1 = 𝐸1 ,
𝑟 ≤ ∞,
𝑗
≤ 𝑎 ≤ 1,
𝑚
and the Young’s inequality
whenever ‖(𝑢0 , V0 )‖𝐸2 ≤ 𝑅.
(𝑢, V) ∈ C (𝑅+ , 𝐸1 ) ,
(28)
where
whenever ‖(𝑢0 , V0 )‖𝐸1 ≤ 𝑅.
󵄩
󵄩2
‖(𝑢, V)‖2𝐸2 ≤ 𝑐󵄩󵄩󵄩(𝑢0 , V0 )󵄩󵄩󵄩𝐸2 𝑒−𝛿2 𝑡 + 𝑐3 .
(27)
𝑗=1
𝑖ℎ𝑡 + 𝛾2 Δℎ + 𝑖𝛾ℎ + 𝑓 (𝑢1 , 𝑢2 ) + 𝑔 = 0,
(32)
𝑖𝑔𝑡 + 𝛾2 Δ𝑔 + (𝑖Γ − 𝜒) 𝑔 + ℎ = 0,
(33)
with periodic initial value
ℎ (𝑥, 𝑡) = ℎ (𝑥 + 𝐷, 𝑡) ,
𝑔 (𝑥, 𝑡) = 𝑔 (𝑥 + 𝐷, 𝑡) ,
𝑥 ∈ 𝑅,
ℎ (𝑥, 0) = 𝑢10 (𝑥) − 𝑢20 (𝑥) ,
𝑡 > 0,
(34)
𝑔 (𝑥, 0) = V10 (𝑥) − V20 (𝑥) ,
𝑥 ∈ 𝑅,
(35)
where
󵄨 󵄨2
󵄨 󵄨2
󵄨 󵄨4
󵄨 󵄨4
𝑓 (𝑢1 , 𝑢2 ) = 𝜎1 (󵄨󵄨󵄨𝑢1 󵄨󵄨󵄨 𝑢1 − 󵄨󵄨󵄨𝑢2 󵄨󵄨󵄨 𝑢2 ) + 𝑖𝜎2 (󵄨󵄨󵄨𝑢1 󵄨󵄨󵄨 𝑢1 − 󵄨󵄨󵄨𝑢2 󵄨󵄨󵄨 𝑢2 ) .
(36)
4
Abstract and Applied Analysis
Taking 𝜙1 (𝑢) = |𝑢|2 and 𝜙2 (𝑢) = |𝑢|4 , then we get
󵄨 󵄨2 󵄨 󵄨2
𝜙1󸀠 (𝜉) ℎ = 󵄨󵄨󵄨𝑢1 󵄨󵄨󵄨 − 󵄨󵄨󵄨𝑢2 󵄨󵄨󵄨 ,
󵄨 󵄨4 󵄨 󵄨4
𝜙2󸀠 (𝜂) ℎ = 󵄨󵄨󵄨𝑢1 󵄨󵄨󵄨 − 󵄨󵄨󵄨𝑢2 󵄨󵄨󵄨 .
by using the extend Hölder inequality, we can obtain
(37)
󵄨
󵄨󵄨
󵄨󵄨Im ∫ 𝑔ℎ 𝑑𝑥󵄨󵄨󵄨 ≤ 1 (󵄩󵄩󵄩𝑔󵄩󵄩󵄩2 + ‖ℎ‖2 ) ,
󵄨󵄨
󵄨󵄨
󵄨 2 󵄩 󵄩
󵄨
Ω
(38)
󵄨
󵄨󵄨
󵄨󵄨𝜎 Im ∫ 𝑢 𝜙󸀠 (𝜉) |ℎ|2 𝑑𝑥󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨𝜎 󵄨󵄨󵄨 ∫ 󵄨󵄨󵄨𝑢 󵄨󵄨󵄨 𝜙󸀠 (𝜉) |ℎ|2 𝑑𝑥
󵄨󵄨 󵄨 1 󵄨 󵄨 2 󵄨 1
󵄨󵄨 1
2 1
󵄨
󵄨
Ω
Ω
Substituting (37) and (38) into (36), we get
󵄨 󵄨2
󵄨 󵄨2
󵄨 󵄨2
󵄨 󵄨2
𝑓 (𝑢1 , 𝑢2 ) = 𝜎1 (󵄨󵄨󵄨𝑢1 󵄨󵄨󵄨 𝑢1 − 󵄨󵄨󵄨𝑢1 󵄨󵄨󵄨 𝑢2 + 󵄨󵄨󵄨𝑢1 󵄨󵄨󵄨 𝑢2 − 󵄨󵄨󵄨𝑢2 󵄨󵄨󵄨 𝑢2 )
󵄩
󵄩 󵄩 󵄩
󵄨 󵄨
≤ 󵄨󵄨󵄨𝜎1 󵄨󵄨󵄨 ‖ℎ‖2 󵄩󵄩󵄩𝑢2 󵄩󵄩󵄩∞ 󵄩󵄩󵄩󵄩𝜙1󸀠 (𝜉)󵄩󵄩󵄩󵄩∞
󵄨 󵄨4
󵄨 󵄨4
󵄨 󵄨4
󵄨 󵄨4
+ 𝑖𝜎2 (󵄨󵄨󵄨𝑢1 󵄨󵄨󵄨 𝑢1 − 󵄨󵄨󵄨𝑢1 󵄨󵄨󵄨 𝑢1 + 󵄨󵄨󵄨𝑢1 󵄨󵄨󵄨 𝑢2 − 󵄨󵄨󵄨𝑢2 󵄨󵄨󵄨 𝑢2 )
≤ 𝐶‖ℎ‖2 ,
= 𝜎1 ℎ (𝜙1 (𝑢1 ) + 𝑢2 𝜙1󸀠 (𝜉))
+ 𝑖𝜎2 ℎ (𝜙2 (𝑢1 ) + 𝑢2 𝜙2󸀠 (𝜂)) .
(39)
󵄨󵄨
󵄨
󵄨󵄨𝜎 Re ∫ 𝑢 𝜙󸀠 (𝜂) |ℎ|2 𝑑𝑥󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨𝜎 󵄨󵄨󵄨 ∫ 󵄨󵄨󵄨𝑢 󵄨󵄨󵄨 𝜙󸀠 (𝜉) |ℎ|2 𝑑𝑥
󵄨󵄨 2
󵄨󵄨 󵄨 2 󵄨 󵄨 2 󵄨 2
2 2
󵄨
󵄨
Ω
Ω
󵄩
󵄩 󵄩 󵄩
󵄨 󵄨
≤ 󵄨󵄨󵄨𝜎2 󵄨󵄨󵄨 ‖ℎ‖2 󵄩󵄩󵄩𝑢2 󵄩󵄩󵄩∞ 󵄩󵄩󵄩󵄩𝜙2󸀠 (𝜂)󵄩󵄩󵄩󵄩∞
Substituting (39) into (32), we obtain
𝑖ℎ𝑡 + 𝛾2 Δℎ + 𝑖𝛾ℎ + 𝜎1 ℎ (𝜙1 (𝑢1 ) + 𝑢2 𝜙1󸀠 (𝜉))
+ 𝑖𝜎2 ℎ (𝜙2 (𝑢1 ) + 𝑢2 𝜙2󸀠 (𝜂)) + 𝑔 = 0,
𝑖𝑔𝑡 + 𝛾2 Δ𝑔 + (𝑖Γ − 𝜒) 𝑔 + ℎ = 0.
≤ 𝐶‖ℎ‖2 .
(40)
(41)
To prove the Theorem 4, we take the following four steps.
Step 1. Taking the inner product of (40) with ℎ and (41) with
𝑔, respectively, we have
Combining (46) and (47), then we infer that
1 𝑑
󵄩 󵄩2
󵄩 󵄩2
(‖ℎ‖2 + 󵄩󵄩󵄩𝑔󵄩󵄩󵄩 ) + 𝛾‖ℎ‖2 + 𝛾󵄩󵄩󵄩𝑔󵄩󵄩󵄩
2 𝑑𝑡
󵄩 󵄩2
+ 𝜎2 ∫ 𝜙2 (𝑢1 ) |ℎ|2 𝑑𝑥 ≤ 𝐶‖ℎ‖2 + 󵄩󵄩󵄩𝑔󵄩󵄩󵄩 .
(𝑖ℎ𝑡 , ℎ) + (𝛾2 Δℎ, ℎ) + (𝑖𝛾ℎ, ℎ)
𝑢2 𝜙2󸀠
(42)
(𝜂)) , ℎ) + (𝑔, ℎ) = 0,
(𝑖𝑔𝑡 , 𝑔) + (𝛾2 Δ𝑔, 𝑔) + ((𝑖Γ − 𝜒) 𝑔, 𝑔) + (ℎ, 𝑔) = 0,
(43)
using
𝑑
𝑑
∫ |𝑢|2 𝑑𝑥 =
∫ 𝑢𝑢 𝑑𝑥 = ∫ (𝑢𝑡 𝑢 + 𝑢𝑢𝑡 ) 𝑑𝑥
𝑑𝑡 Ω
𝑑𝑡 Ω
Ω
Step 2. Taking the inner product of (40) with −ℎ𝑥𝑥 and (41)
with −𝑔𝑥𝑥 , respectively, we have
(𝑖ℎ𝑡 , ℎ𝑥𝑥 ) + (𝛾2 Δℎ, −ℎ𝑥𝑥 ) + (𝑖𝛾ℎ, −ℎ𝑥𝑥 )
+ (𝜎1 ℎ (𝜙1 (𝑢1 ) + 𝑢2 𝜙1󸀠 (𝜉)) , −ℎ𝑥𝑥 )
+ (𝑖𝜎2 ℎ (𝜙2 (𝑢1 ) + 𝑢2 𝜙2󸀠 (𝜂)) , −ℎ𝑥𝑥 )
(44)
= 2 Re ∫ 𝑢𝑡 𝑢 𝑑𝑥.
Ω
Ω
1 𝑑
∫ |𝑢|2 𝑑𝑥,
2 𝑑𝑡 Ω
󵄩 󵄩2
(𝛾2 Δℎ, ℎ) = −𝛾2 󵄩󵄩󵄩ℎ𝑥 󵄩󵄩󵄩 ,
(𝑖𝑔𝑡 , −𝑔𝑥𝑥 ) + (𝛾2 Δ𝑔, −𝑔𝑥𝑥 ) + ((𝑖Γ − 𝜒) 𝑔, −𝑔𝑥𝑥 )
(45)
1 𝑑
‖ℎ‖2 + 𝛾‖ℎ‖2 + 𝜎1 Im ∫ 𝑢2 𝜙1󸀠 (𝜉) |ℎ|2 𝑑𝑥
2 𝑑𝑡
Ω
2
+ 𝜎2 Im ∫ 𝜙2 (𝑢1 ) |ℎ| 𝑑𝑥
Ω
Note that
(𝑖ℎ𝑡 , −ℎ𝑥𝑥 ) = 𝑖 ∫ ℎ𝑥𝑡 ℎ𝑥 𝑑𝑥,
(46)
Ω
+ 𝜎2 Re ∫
+ (ℎ, −𝑔𝑥𝑥 ) = 0.
(𝑖𝛾ℎ, ℎ) = 𝑖𝛾‖ℎ‖2 ,
then taking the imaginary part of (42) and (43), respectively,
𝜙2󸀠
(50)
+ (𝑔, −ℎ𝑥𝑥 ) = 0,
Thus,
Im (𝑖 ∫ 𝑢𝑡 𝑢 𝑑𝑥) =
(49)
Ω
+ (𝜎1 ℎ (𝜙1 (𝑢1 ) + 𝑢2 𝜙1󸀠 (𝜉)) , ℎ)
+ (𝑖𝜎2 ℎ (𝜙2 (𝑢1 ) +
(48)
Ω
(𝑔, −ℎ𝑥𝑥 ) = ∫ 𝑔𝑥 ℎ𝑥 𝑑𝑥,
Ω
2
(𝜂) |ℎ| 𝑑𝑥 + Im ∫ 𝑔ℎ 𝑑𝑥 = 0,
Ω
1 𝑑 󵄩󵄩 󵄩󵄩2
󵄩 󵄩2
󵄩𝑔󵄩 + Γ󵄩󵄩󵄩𝑔󵄩󵄩󵄩 + Im ∫ ℎ𝑔 𝑑𝑥 = 0,
2 𝑑𝑡 󵄩 󵄩
Ω
(𝛾2 Δℎ, −ℎ𝑥𝑥 ) = ‖Δℎ‖2 ,
(47)
󵄩 󵄩2
(𝑖𝛾ℎ, −ℎ𝑥𝑥 ) = 𝑖𝛾󵄩󵄩󵄩ℎ𝑥 󵄩󵄩󵄩 ,
(51)
Abstract and Applied Analysis
5
(𝜎1 ℎ (𝜙1 (𝑢1 ) + 𝑢2 𝜙1󸀠 (𝜉)) , −ℎ𝑥𝑥 )
Combining (53) and (54), one can obtain
󵄨 󵄨2
= 𝜎1 ∫ [󵄨󵄨󵄨ℎ𝑥 󵄨󵄨󵄨 (𝜙1 (𝑢1 ) + 𝑢2 𝜙1󸀠 (𝜉))
Ω
1 𝑑 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2
󵄩 󵄩2
󵄩 󵄩2
(󵄩ℎ 󵄩 + 󵄩𝑔 󵄩 ) + 𝛾󵄩󵄩󵄩ℎ𝑥 󵄩󵄩󵄩 + 𝛾󵄩󵄩󵄩𝑔𝑥 󵄩󵄩󵄩
2 𝑑𝑡 󵄩 𝑥 󵄩 󵄩 𝑥 󵄩
+ ℎℎ𝑥 (𝜙1 (𝑢1 ) + 𝑢2 𝜙1󸀠 (𝜉))𝑥 ] 𝑑𝑥,
󵄨 󵄨2
󵄩 󵄩2 󵄩 󵄩2
+ 𝜎2 ∫ 𝜙2 (𝑢1 ) 󵄨󵄨󵄨ℎ𝑥 󵄨󵄨󵄨 𝑑𝑥 ≤ 𝐶󵄩󵄩󵄩ℎ𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑔𝑥 󵄩󵄩󵄩 + 𝑐‖ℎ‖2 .
(𝑖𝜎2 ℎ (𝜙2 (𝑢1 ) + 𝑢2 𝜙2󸀠 (𝜂)) , −ℎ𝑥𝑥 )
Ω
󵄨 󵄨2
= 𝑖𝜎2 ∫ [󵄨󵄨󵄨ℎ𝑥 󵄨󵄨󵄨 (𝜙2 (𝑢1 ) + 𝑢2 𝜙2󸀠 (𝜂))
Ω
Step 3. Taking the inner product of (40) with ℎ𝑥𝑥𝑥𝑥 and (41)
with 𝑔𝑥𝑥𝑥𝑥 , respectively, we have
+ ℎℎ𝑥 (𝜙2 (𝑢1 ) + 𝑢2 𝜙2󸀠 (𝜂))𝑥 ] 𝑑𝑥,
(52)
(𝑖ℎ𝑡 , ℎ𝑥𝑥𝑥𝑥 ) + (𝛾2 Δℎ, ℎ𝑥𝑥𝑥𝑥 ) + (𝑖𝛾ℎ, ℎ𝑥𝑥𝑥𝑥 )
+ (𝜎1 ℎ (𝜙1 (𝑢1 ) + 𝑢2 𝜙1󸀠 (𝜉)) , ℎ𝑥𝑥𝑥𝑥 )
then taking the imaginary part of (50) and (51), respectively,
+ (𝑖𝜎2 ℎ (𝜙2 (𝑢1 ) + 𝑢2 𝜙2󸀠 (𝜂)) , ℎ𝑥𝑥𝑥𝑥 )
1 𝑑 󵄩󵄩 󵄩󵄩2
󵄩 󵄩2
󵄩ℎ 󵄩 + 𝛾󵄩󵄩󵄩ℎ𝑥 󵄩󵄩󵄩
2 𝑑𝑡 󵄩 𝑥 󵄩
(57)
+ (𝑔, ℎ𝑥𝑥𝑥𝑥 ) = 0,
󵄨 󵄨2
+ 𝜎1 Im ∫ (𝑢2 𝜙1󸀠 (𝜉) 󵄨󵄨󵄨ℎ𝑥 󵄨󵄨󵄨
(𝑖𝑔𝑡 , 𝑔𝑥𝑥𝑥𝑥 ) + (𝛾2 Δ𝑔, 𝑔𝑥𝑥𝑥𝑥 ) + ((𝑖Γ − 𝜒) 𝑔, 𝑔𝑥𝑥𝑥𝑥 )
Ω
+ ℎℎ𝑥 (𝜙1 (𝑢1 ) +
(56)
𝑢2 𝜙1󸀠
(𝜉))x ) 𝑑𝑥
+ (ℎ, 𝑔𝑥𝑥𝑥𝑥 ) = 0,
󵄨 󵄨2
+ 𝜎2 Im ∫ 𝜙2 (𝑢1 ) 󵄨󵄨󵄨ℎ𝑥 󵄨󵄨󵄨 𝑑𝑥
(53)
Ω
using
󵄨 󵄨2
+ 𝜎2 Re ∫ (𝑢2 𝜙2󸀠 (𝜂) 󵄨󵄨󵄨ℎ𝑥 󵄨󵄨󵄨
(𝑖ℎ𝑡 , ℎ𝑥𝑥𝑥𝑥 ) = 𝑖 ∫ ℎ𝑥𝑥𝑡 ℎ𝑥𝑥 𝑑𝑥,
Ω
Ω
+ ℎℎ𝑥 (𝜙2 (𝑢1 ) + 𝑢2 𝜙2󸀠 (𝜂))𝑥 ) 𝑑𝑥
(𝑔, ℎ𝑥𝑥𝑥𝑥 ) = ∫ 𝑔𝑥𝑥 ℎ𝑥𝑥 𝑑𝑥,
Ω
+ Im ∫ 𝑔𝑥 ℎ𝑥 𝑑𝑥 = 0,
󵄩2
󵄩
(𝛾2 Δℎ, ℎ𝑥𝑥𝑥𝑥 ) = 󵄩󵄩󵄩ℎ𝑥𝑥𝑥 󵄩󵄩󵄩 ,
Ω
1 𝑑 󵄩󵄩 󵄩󵄩2
󵄩 󵄩2
󵄩𝑔󵄩 + Γ󵄩󵄩󵄩𝑔󵄩󵄩󵄩 + Im ∫ ℎ𝑔 𝑑𝑥 = 0.
2 𝑑𝑡 󵄩 󵄩
Ω
󵄩 󵄩2
(𝑖𝛾ℎ, ℎ𝑥𝑥𝑥𝑥 ) = 𝑖𝛾󵄩󵄩󵄩ℎ𝑥𝑥 󵄩󵄩󵄩 ,
(54)
(𝜎1 ℎ (𝜙1 (𝑢1 ) + 𝑢2 𝜙1󸀠 (𝜉)) , ℎ𝑥𝑥𝑥𝑥 )
Note the following inequalities:
󵄨󵄨
󵄨
󵄨󵄨𝜎 Im ∫ (𝑢 𝜙󸀠 (𝜉) 󵄨󵄨󵄨ℎ 󵄨󵄨󵄨2 + ℎℎ (𝜙 (𝑢 ) + 𝑢 𝜙󸀠 (𝜉)) ) 𝑑𝑥󵄨󵄨󵄨
󵄨󵄨 1
󵄨󵄨
2 1
𝑥󵄨
𝑥 1
1
2 1
󵄨
𝑥
󵄨
󵄨
Ω
= 𝜎1 ((ℎ (𝜙1 (𝑢1 ) + 𝑢2 𝜙1󸀠 (𝜉)))𝑥𝑥 , ℎ𝑥𝑥 )
= 𝜎1 (ℎ𝑥𝑥 𝜙1 (𝑢1 ) + 𝜓1 , ℎ𝑥𝑥 ) ,
󵄨 󵄨 󵄨2
󵄨 󵄨
󵄨 󵄨󵄨
≤ 󵄨󵄨󵄨𝜎1 󵄨󵄨󵄨 Im ∫ (󵄨󵄨󵄨𝑢2 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝜙1󸀠 (𝜉)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨ℎ𝑥 󵄨󵄨󵄨
Ω
󵄨
󵄨 󵄨 󵄨
󵄨 󵄨 󵄨󵄨
+ |ℎ| 󵄨󵄨󵄨󵄨ℎ𝑥 󵄨󵄨󵄨󵄨 (󵄨󵄨󵄨𝜙1 (𝑢1 )𝑥 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑢2𝑥 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝜙1󸀠 (𝜉)󵄨󵄨󵄨󵄨
󵄨
󵄨 󵄨󵄨
+ 󵄨󵄨󵄨𝑢2 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝜙1󸀠 (𝜉)𝑥 󵄨󵄨󵄨󵄨)) 𝑑𝑥
󵄩 󵄩 󵄩
󵄩
󵄩 󵄩2 󵄨 󵄨
󵄩
󵄩 󵄩 󵄩
≤ 𝐶󵄩󵄩󵄩ℎ𝑥 󵄩󵄩󵄩 + 󵄨󵄨󵄨𝜎1 󵄨󵄨󵄨 ‖ℎ‖ 󵄩󵄩󵄩󵄩ℎ𝑥 󵄩󵄩󵄩󵄩 (󵄩󵄩󵄩𝜙1 (𝑢1 )𝑥 󵄩󵄩󵄩∞ + 󵄩󵄩󵄩𝑢2𝑥 󵄩󵄩󵄩∞ 󵄩󵄩󵄩󵄩𝜙1󸀠 (𝜉)󵄩󵄩󵄩󵄩∞
󵄩
󵄩 󵄩 󵄩
+ 󵄩󵄩󵄩𝑢2 󵄩󵄩󵄩∞ 󵄩󵄩󵄩󵄩𝜙1󸀠 (𝜉)𝑥 󵄩󵄩󵄩󵄩∞ ) ,
󵄩 󵄩2
≤ 𝐶󵄩󵄩󵄩ℎ𝑥 󵄩󵄩󵄩 + 𝑐‖ℎ‖2 ,
(𝑖𝜎2 ℎ (𝜙2 (𝑢1 ) + 𝑢2 𝜙2󸀠 (𝜂)) , ℎ𝑥𝑥𝑥𝑥 )
= 𝑖𝜎2 ((ℎ (𝜙2 (𝑢1 ) + 𝑢2 𝜙2󸀠 (𝜂)))𝑥𝑥 , ℎ𝑥𝑥 )
= 𝑖𝜎2 (ℎ𝑥𝑥 𝜙2 (𝑢1 ) + 𝜓2 , ℎ𝑥𝑥 ) ,
where
𝜓1 = ℎ𝑥𝑥 𝑢2 𝜙1󸀠 (𝜉)
󵄨 󵄨2
𝜎2 Re ∫ (𝑢2 𝜙2󸀠 (𝜂) 󵄨󵄨󵄨ℎ𝑥 󵄨󵄨󵄨 + ℎℎ𝑥 (𝜙2 (𝑢1 ) + 𝑢2 𝜙2󸀠 (𝜂))𝑥 ) 𝑑𝑥
+ 2ℎ𝑥 (𝜙1 (𝑢1 )𝑥 + 𝑢2𝑥 𝜙1󸀠 (𝜉) + 𝑢2 𝜙1󸀠 (𝜉)𝑥 )
󵄩 󵄩2
≤ 𝐶󵄩󵄩󵄩ℎ𝑥 󵄩󵄩󵄩 + 𝑐‖ℎ‖2 ,
+ ℎ (𝜙1 (𝑢1 )𝑥𝑥 + 𝑢2𝑥𝑥 𝜙1󸀠 (𝜉) + 2𝑢2𝑥 𝜙1󸀠 (𝜉)𝑥
Ω
(55)
+ 𝑢2 𝜙1󸀠 (𝜉)𝑥𝑥 ) ,
(58)
6
Abstract and Applied Analysis
𝜓2 = ℎ𝑥𝑥 𝑢2 𝜙2󸀠 (𝜂)
that is,
󵄩
󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑤1 (𝑡) − 𝑤2 (𝑡)󵄩󵄩󵄩𝑉2 ≤ exp (2𝐶0 𝑡) 󵄩󵄩󵄩𝑤10 − 𝑤20 󵄩󵄩󵄩𝑉2 .
+ 2ℎ𝑥 (𝜙2 (𝑢1 )𝑥 + 𝑢2𝑥 𝜙2󸀠 (𝜂) + 𝑢2 𝜙2󸀠 (𝜂)𝑥 )
+ ℎ (𝜙2 (𝑢1 )𝑥𝑥 + 𝑢2𝑥𝑥 𝜙2󸀠 (𝜂) + 2𝑢2𝑥 𝜙2󸀠 (𝜂)𝑥
+ 𝑢2 𝜙2󸀠 (𝜂)𝑥𝑥 ) ,
(59)
Meanwhile, it indicates that the Lipschitz constant 𝑙(𝑡) ≤
exp(2𝐶0 𝑡). This completes the proof.
Now, we intend to show the squeezing property for
semigroup 𝑆(𝑡). To this end, we introduce the operator 𝐴 =
−(𝜕/𝜕𝑥2 ) from 𝐷(𝐴) to 𝐻 with domain
𝐷 (𝐴) = {𝑢 ∈ 𝐻2 (Ω)} .
then taking the imaginary part of (50) and (51), respectively,
1 𝑑 󵄩󵄩 󵄩󵄩2
󵄩 󵄩2
󵄩ℎ 󵄩 + 𝛾󵄩󵄩󵄩ℎ𝑥𝑥 󵄩󵄩󵄩 + 𝜎1 Im (𝜓1 , ℎ𝑥𝑥 )
2 𝑑𝑡 󵄩 𝑥𝑥 󵄩
+ 𝜎2 (ℎ𝑥𝑥 𝜙2 (𝑢1 ) , ℎ𝑥𝑥 ) + 𝜎2 Re (𝜓2 , ℎ𝑥𝑥 ) (60)
Ω
1 𝑑 󵄩󵄩 󵄩󵄩2
󵄩 󵄩2
󵄩𝑔 󵄩 + Γ󵄩󵄩󵄩𝑔𝑥𝑥 󵄩󵄩󵄩 + Im ∫ ℎ𝑥𝑥 𝑔𝑥𝑥 𝑑𝑥 = 0.
2 𝑑𝑡 󵄩 𝑥𝑥 󵄩
Ω
0 ≤ 𝜆 1 < 𝜆 1 ⋅ ⋅ ⋅ 𝜆 𝑖 < ⋅ ⋅ ⋅ 󳨀→ +∞,
(61)
󵄨󵄨
󵄨
󵄨󵄨𝜎2 Re (𝜓2 , ℎ𝑥𝑥 )󵄨󵄨󵄨 ≤ 𝐶‖ℎ‖2𝐻2 .
󵄨
󵄨
(62)
Combining (60) and (61), one can obtain
1 𝑑 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2
󵄩 󵄩2
󵄩 󵄩2
(󵄩󵄩ℎ𝑥𝑥 󵄩󵄩 + 󵄩󵄩𝑔𝑥𝑥 󵄩󵄩 ) + 𝛾󵄩󵄩󵄩ℎ𝑥𝑥 󵄩󵄩󵄩 + Γ󵄩󵄩󵄩𝑔𝑥𝑥 󵄩󵄩󵄩
2 𝑑𝑡
󵄨 󵄨2
󵄩 󵄩2
+ 𝜎2 ∫ 𝜙2 (𝑢1 ) 󵄨󵄨󵄨ℎ𝑥𝑥 󵄨󵄨󵄨 𝑑𝑥 ≤ 𝐶‖ℎ‖2𝐻2 + 󵄩󵄩󵄩𝑔𝑥𝑥 󵄩󵄩󵄩 .
Ω
(63)
(71)
Decompose ℎ, 𝑔 as
𝑔 = 𝑃𝑔 + 𝑄𝑔.
𝑖𝑄ℎ𝑡 + 𝛾2 Δ𝑄ℎ + 𝑖𝛾ℎ + 𝑄𝑓 (𝑢1 , 𝑢2 ) + 𝑄𝑔 = 0,
𝑖𝑄𝑔𝑡 + 𝛾2 Δ𝑄𝑔 + (𝑖Γ − 𝜒) 𝑄𝑔 + 𝑄ℎ = 0.
󵄨 󵄨2 󵄨 󵄨2
+ 𝜎2 ∫ 𝜙2 (𝑢1 ) (|ℎ| + 󵄨󵄨󵄨ℎ𝑥 󵄨󵄨󵄨 + 󵄨󵄨󵄨ℎ𝑥𝑥 󵄨󵄨󵄨 ) 𝑑𝑥
Ω
2
󵄩 󵄩2
󵄩 󵄩 2 󵄩 󵄩2 󵄩 󵄩2
≤ 𝐶 (‖ℎ‖2 + 󵄩󵄩󵄩ℎ𝑥 󵄩󵄩󵄩 + ‖ℎ‖2𝐻2 ) + 󵄩󵄩󵄩𝑔󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑔𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑔𝑥𝑥 󵄩󵄩󵄩 .
(64)
Taking 𝜇 = min(Γ, 𝛾),𝐶0 = max(𝐶, 1) and noting that
(72)
(65)
(73)
(74)
Take the inner product of (73) with 𝑄ℎ and (74) with 𝑄𝑔,
respectively. Then like Step 1, we can get
1 𝑑
󵄩 󵄩2
󵄩 󵄩2
(‖𝑄ℎ‖2 + 󵄩󵄩󵄩𝑄𝑔󵄩󵄩󵄩 ) + 𝛾‖𝑄ℎ‖2 + Γ󵄩󵄩󵄩𝑄𝑔󵄩󵄩󵄩
2 𝑑𝑡
+ 𝜎2 ∫ 𝑄𝜙2 (𝑢1 ) |𝑄ℎ|2 𝑑𝑥
so (64) can be reduced to
1 𝑑
󵄩 󵄩2
󵄩 󵄩2
(‖ℎ‖2𝐻2 + 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐻2 ) + 𝜇 (‖ℎ‖2𝐻2 + 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐻2 )
2 𝑑𝑡
󵄩 󵄩2
≤ 𝐶0 (‖ℎ‖2𝐻2 + 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐻2 ) .
󵄩󵄩 (1/2) 󵄩󵄩
󵄩󵄩𝐴
𝑢󵄩󵄩󵄩 ≥ 𝜆(1/2)
𝑢 ∈ 𝑄𝑁𝐻,
𝑁+1 ,
󵄩
󵄩󵄩
󵄩
󵄩󵄩𝑄𝑁𝑢󵄩󵄩󵄩 ≤ ‖𝑢‖ , 𝑢 ∈ 𝐻,
󵄩󵄩
󵄩 󵄩
󵄩
󵄩󵄩𝐴𝑄𝑁𝑢󵄩󵄩󵄩 = 󵄩󵄩󵄩𝑄𝑁𝐴𝑢󵄩󵄩󵄩 ≤ ‖𝐴𝑢‖ , 𝑢 ∈ 𝐷 (𝐴) .
Applying 𝑄 to (32) and (33) we find that
1 𝑑
󵄩 󵄩2
󵄩 󵄩2
(‖ℎ‖2𝐻2 + 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐻2 ) + 𝛾‖ℎ‖2𝐻2 + 𝛾󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐻2
2 𝑑𝑡
Ω
(70)
󵄩󵄩 (1/2) 󵄩󵄩 󵄩󵄩󵄩󵄩 𝜕𝑢 󵄩󵄩󵄩󵄩
󵄩󵄩𝐴
𝑢󵄩󵄩󵄩 = 󵄩󵄩 󵄩󵄩 ,
󵄩
󵄩󵄩 𝜕𝑥 󵄩󵄩
ℎ = 𝑃ℎ + 𝑄ℎ,
Step 4. Combining (49), (56) and (63), we get
󵄨 󵄨2 󵄨 󵄨2
𝜎2 ∫ (|ℎ|2 + 󵄨󵄨󵄨ℎ𝑥 󵄨󵄨󵄨 + 󵄨󵄨󵄨ℎ𝑥𝑥 󵄨󵄨󵄨 ) 𝑑𝑥 ≥ 0,
when 𝑖 󳨀→ ∞.
For all 𝑁 denote by 𝑃 = 𝑃𝑛 : 𝐻 → span{𝑤1 , 𝑤2 , . . . , 𝑤𝑛 } the
projector 𝑄 = 𝑄𝑁 = 𝐼 − 𝑃𝑁. In the following, we will use
Note the following inequalities:
󵄨󵄨
󵄨
󵄨󵄨Im (𝜓1 , ℎ𝑥𝑥 )󵄨󵄨󵄨 ≤ 𝐶‖ℎ‖2𝐻2 ,
󵄨
󵄨
(69)
Obviously, 𝐴 is an unbounded self-adjoint positive operator and the inverse 𝐴−1 is compact. Thus, there exists an
orthonormal basis {𝑤𝑖 }∞
𝑖=1 𝑖 = 1 of 𝐻 consisting of eigenvectors of 𝐴 such that
𝐴𝑤𝑖 = 𝜆 𝑖 𝑤𝑖 ,
+ Im ∫ 𝑔𝑥𝑥 ℎ𝑥𝑥 𝑑𝑥 = 0,
(68)
(75)
Ω
󵄩 󵄩2
≤ 𝐶‖𝑄ℎ‖2 + 󵄩󵄩󵄩𝑄𝑔󵄩󵄩󵄩 .
Take the inner product of (73) with −𝑄ℎ𝑥𝑥 and (74) with
−𝑄𝑔𝑥𝑥 , respectively. Then like Step 2, we can get
(66)
By Gronwall’s inequality
󵄩2
󵄩 󵄩2
󵄩
‖ℎ‖2𝐻2 + 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐻2 ≤ exp (2𝐶0 𝑡) (‖ℎ (0)‖2𝐻2 + 󵄩󵄩󵄩𝑔 (0)󵄩󵄩󵄩𝐻2 ) , (67)
1 𝑑 󵄩󵄩
󵄩2 󵄩
󵄩2
󵄩2
󵄩2
󵄩
󵄩
(󵄩󵄩𝑄ℎ𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑄𝑔𝑥 󵄩󵄩󵄩 ) + 𝛾󵄩󵄩󵄩𝑄ℎ𝑥 󵄩󵄩󵄩 + Γ󵄩󵄩󵄩𝑄𝑔𝑥 󵄩󵄩󵄩
2 𝑑𝑡
󵄨2
󵄨
+ 𝜎2 ∫ 𝑄𝜙2 (𝑢1 ) 󵄨󵄨󵄨𝑄ℎ𝑥 󵄨󵄨󵄨 𝑑𝑥
Ω
󵄩
󵄩2 󵄩
󵄩2
≤ 𝐶󵄩󵄩󵄩𝑄ℎ𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑄𝑔𝑥 󵄩󵄩󵄩 + 𝑐‖𝑄ℎ‖2 .
(76)
Abstract and Applied Analysis
7
Take the inner product of (73) with 𝑄ℎ𝑥𝑥𝑥𝑥 and (74) with
𝑄𝑔𝑥𝑥𝑥𝑥 , respectively. Then like Step 3, we can get
1 𝑑 󵄩󵄩
󵄩2 󵄩
󵄩2
󵄩2
󵄩2
󵄩
󵄩
(󵄩𝑄ℎ 󵄩󵄩 + 󵄩󵄩𝑄𝑔 󵄩󵄩 ) + 𝛾󵄩󵄩󵄩𝑄ℎ𝑥𝑥 󵄩󵄩󵄩 + Γ󵄩󵄩󵄩𝑄𝑔𝑥𝑥 󵄩󵄩󵄩
2 𝑑𝑡 󵄩 𝑥𝑥 󵄩 󵄩 𝑥𝑥 󵄩
󵄨2
󵄨
+ 𝜎2 ∫ 𝑄𝜙2 (𝑢1 ) 󵄨󵄨󵄨𝑄ℎ𝑥𝑥 󵄨󵄨󵄨 𝑑𝑥
Ω
(77)
󵄩2
󵄩
≤ 𝐶‖𝑄ℎ‖2𝐻2 + 󵄩󵄩󵄩𝑄𝑔𝑥𝑥 󵄩󵄩󵄩 .
From (82) and (84), we obtain
1
󵄩2
󵄩 󵄩2
󵄩
(‖ℎ (0)‖2𝐻2 + 󵄩󵄩󵄩𝑔 (0)󵄩󵄩󵄩𝐻2 ) .
‖𝑄ℎ‖2𝐻2 + 󵄩󵄩󵄩𝑄𝑔󵄩󵄩󵄩𝐻2 ≤
128
This shows that when 𝑡∗ > 0 is fixed, Lipschitz constant for
𝑆(𝑡) in 𝐵 is equal to exp(2𝐶0 𝑡∗ ) and 𝑁 satisfies
𝜆 𝑁+1 ≥ 256𝐶 exp (2𝐶0 𝑡∗ ) .
(86)
‖𝑄𝑤‖𝑉2 ≤ ‖𝑄𝑤 (0)‖𝑉2 .
(87)
We have
Combining (75), (76), and (77), we get
1 𝑑
󵄩 󵄩2
󵄩 󵄩2
(‖𝑄ℎ‖2𝐻2 + 󵄩󵄩󵄩𝑄𝑔󵄩󵄩󵄩𝐻2 ) + 𝜇 (‖𝑄ℎ‖2𝐻2 + 󵄩󵄩󵄩𝑄𝑔󵄩󵄩󵄩𝐻2 )
2 𝑑𝑡
󵄩 󵄩2
≤ 𝐶0 (‖𝑄ℎ‖2𝐻2 + 󵄩󵄩󵄩𝑄𝑔󵄩󵄩󵄩𝐻2 ) .
(78)
Using the G-N inequality
󵄩󵄩 󵄩󵄩2
󵄩 󵄩 1
󵄩 󵄩2
2
󵄩󵄩𝑢𝑥 󵄩󵄩 ≤ ‖𝑢‖ 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 ≤ (‖𝑢‖ + 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 ) ,
2
So when
󵄩󵄩
󵄩
󵄩
󵄩
󵄩󵄩𝑄𝑤 (𝑡∗ )󵄩󵄩󵄩𝑉2 > 󵄩󵄩󵄩𝑃𝑤 (𝑡∗ )󵄩󵄩󵄩𝑉2 ,
󵄩󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
󵄩󵄩𝑤 (𝑡∗ )󵄩󵄩󵄩𝑉2 = 󵄩󵄩󵄩𝑄𝑤 (𝑡∗ )󵄩󵄩󵄩𝑉2 + 󵄩󵄩󵄩𝑃𝑤 (𝑡∗ )󵄩󵄩󵄩𝑉2
1
󵄩
󵄩
< 2󵄩󵄩󵄩𝑄𝑤 (𝑡∗ )󵄩󵄩󵄩𝑉2 ≤ ‖𝑄𝑤 (0)‖𝑉2
64
(79)
from (78), we have
≤
1 𝑑
󵄩 󵄩2
󵄩 󵄩2
(‖𝑄ℎ‖2𝐻2 + 󵄩󵄩󵄩𝑄𝑔󵄩󵄩󵄩𝐻2 ) + 𝜇 (‖𝑄ℎ‖2𝐻2 + 󵄩󵄩󵄩𝑄𝑔󵄩󵄩󵄩𝐻2 )
2 𝑑𝑡
≤
3𝐶0
󵄩 󵄩 󵄩 󵄩
󵄩
󵄩
(‖𝑄ℎ‖ + 󵄩󵄩󵄩𝑄ℎ𝑥𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑄𝑔󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑄𝑔𝑥𝑥 󵄩󵄩󵄩)
2
≤
3𝐶0 −1 󵄩󵄩
󵄩 󵄩
󵄩
(󵄩𝑄ℎ 󵄩󵄩 + 󵄩󵄩𝑄𝑔 󵄩󵄩)
𝜆
2 𝑁+1 󵄩 𝑥𝑥 󵄩 󵄩 𝑥𝑥 󵄩
(80)
𝑑𝐹 (𝑀) ≤ 1 + 𝐶𝑁0 ,
dist𝑉2 (𝑆 (𝑡) 𝐵, 𝑀) ≤ 𝐶0 exp (−𝐶1 𝑡) ,
󵄩2
󵄩󵄩
󵄩󵄩𝑔 (0)󵄩󵄩󵄩𝐻2 ) exp (−2𝜇𝑡)
󵄩󵄩2
󵄩󵄩
2
+ 𝐶𝜆−1
𝑁+1 exp (2𝐶0 𝑡) (‖ℎ (0)‖𝐻2 + 󵄩
󵄩𝑔 (0)󵄩󵄩𝐻2 )
(81)
≤ [exp (−2𝜇𝑡) + 𝐶𝜆−1
𝑁+1 exp (2𝐶0 𝑡)]
Letting 𝑡∗ > 0 be fixed we take 𝑤(𝑡) = 𝑤1 (𝑡) − 𝑤2 (𝑡) =
(ℎ(𝑡), 𝑔(𝑡)) and assume that
1
.
256
(82)
Then we choose 𝑁 large enough so that
where 𝐶0 , 𝐶1 , 𝐶 are constants independent of the solution of
the equation.
In this paper, we have studied the coupled Ginzburg-Landau
equations which describe Bose-Einstein condensates and
nonlinear optical waveguides and cavities with periodic
initial boundary; the existence of the exponential attractors is
obtained by showing Lipschitz continuity and the squeezing
property. For exponential attractor, 𝑁 is only large enough
such that
𝜆 𝑁+1 ≥ 256𝐶 exp (2𝐶0 𝑡∗ ) .
1
,
256
(83)
𝜆 𝑁+1 ≥ 256𝐶 exp (2𝐶0 𝑡) .
(84)
𝐶𝜆−1
𝑁+1 exp (2𝐶0 𝑡) ≤
(90)
4. Conclusions
󵄩2
󵄩
+ 󵄩󵄩󵄩𝑔 (0)󵄩󵄩󵄩𝐻2 ) .
exp (−2𝜇𝑡∗ ) ≤
(89)
Then for the nonlinear semigroup 𝑆(𝑡) defined in (4) and (5),
𝑆(𝑡)𝑡≤0 ; 𝐵 admits an exponential attractor 𝑀 in 𝑉2 and
󵄩 󵄩2
‖𝑄ℎ‖2𝐻2 + 󵄩󵄩󵄩𝑄𝑔󵄩󵄩󵄩𝐻2
×
Theorem 11. The semigroup 𝑆(𝑡) associated with problem (1)(2) is squeezing in 𝐵. Now we conclude this section by giving
our main result.
𝜆 𝑁+1 ≥ 256𝐶 exp (2𝐶0 𝑡∗ ) .
By Gronwall lemma we get
(‖ℎ (0)‖2𝐻2
1
‖𝑤 (0)‖𝑉2 .
64
Theorem 12. Suppose that problem (1)-(2) satisfies Theorem 9;
there exist 𝑡∗ ≥ (1/2𝜇) ln(256) and N large enough such that
3𝐶0 −1
󵄩2
󵄩
exp (2𝐶0 𝑡) (‖ℎ (0)‖2𝐻2 + 󵄩󵄩󵄩𝑔 (0)󵄩󵄩󵄩𝐻2 ) .
𝜆
2 𝑁+1
≤ (‖ℎ (0)‖2𝐻2 +
(88)
This completes the proof of Theorem 4.
3𝐶0 −1
󵄩 󵄩2
≤
𝜆 𝑁+1 (‖𝑄ℎ‖2𝐻2 + 󵄩󵄩󵄩𝑄𝑔󵄩󵄩󵄩𝐻2 )
2
≤
(85)
(91)
Acknowledgment
that is,
This work was supported by Chinese Natural Science Foundation Grant no. 11061028.
8
References
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