Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 601325, 13 pages
http://dx.doi.org/10.1155/2013/601325
Research Article
Weakly Compact Uniform Attractor for the Nonautonomous
Long-Short Wave Equations
Hongyong Cui, Jie Xin, and Anran Li
School of Mathematics and Information, Ludong University, Yantai, 264025, China
Correspondence should be addressed to Jie Xin; fdxinjie@sina.com
Received 8 November 2012; Revised 25 January 2013; Accepted 27 January 2013
Academic Editor: Lucas Jódar
Copyright © 2013 Hongyong Cui et al. 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.
Solutions and weakly compact uniform attractor for the nonautonomous long-short wave equations with translation compact forces
were studied in a bounded domain. We first established the existence and the uniqueness of the solution to the system by using
Galerkin method and then obtained the uniform absorbing set and the weakly compact uniform attractor of the problem by applying
techniques of constructing skew product flow in the extended phase space.
1. Introduction
The long wave-short wave (LS) resonance equations arise in
the study of the interaction of the surface waves with both
gravity and capillary modes presence and also in the analysis
of internal waves, as well as Rossby wave [1]. In the plasma
physics they describe the resonance of the high-frequency
electron plasma oscillation and the associated low-frequency
ion density perturbation [2]. Benney [3] presents a general
theory for the interaction between the short wave and the
long wave.
Due to their rich physical and mathematical properties
the long wave-short wave resonance equations have drawn
much attention of many physicists and mathematicians.
For one-dimensional propagation of waves, there are many
studies on this interaction. Guo [4, 5] obtains the existence of
global solution for long-short wave equations and generalized
long-short wave equations, respectively. The existence of
global attractor was studied in [6–8]. The orbital stability of
solitary waves for this system has been studied in [9]. In [10],
Guo investigated the asymptotic behavior of solutions for
the long-short wave equations with zero order dissipation in
2
1
×𝐻per
. The approximation inertial manifolds for LS type
𝐻per
equations have been studied in [11]. The well posedness of
the Cauchy problem for the long wave-short wave resonance
equations was studied in [8, 12–17].
In this paper, nonautonomous LS equations with translation compact forces were studied. The essential difference
between nonautonomous systems and autonomous ones is
that the former get much influenced by the time-depended
external forces, which breaks semigroup property of the flow
or semiflow created by autonomous systems. Also, attractors
of nonautonomous systems are no longer invariable; they
change with the changing of the initial time. This makes
it impossible for us to consider nonautonomous systems
completely in the same way of autonomous ones. Fortunately,
Chepyzhov and Vishik [18, 19] developed techniques by
which skills in the study of autonomous systems can be used
in dealing with nonautonomous problems. Their central idea
is that constructing skew product flow in extended phase
space is obtained by
𝑆 (𝑡) (𝑢, 𝜎) = (𝑈𝜎 (𝑡, 0) 𝑢, 𝑇 (𝑡) 𝜎) ,
𝑡 ≥ 0, (𝑢, 𝜎) ∈ 𝐸 × ∑,
(1)
where {𝑈𝜎∈∑ (𝑡, 𝜏)} is a family of processes, {𝑇(𝑡)} is a translation semigroup, and the flow {𝑆(𝑡)} can be proved to be a
semigroup under some preconditions, such as the translation
identity and (𝐸 × ∑, 𝐸)-continuity of {𝑈𝜎 (𝑡, 𝜏)}, and more
importantly, the compactness of the symbol space ∑. By
this means, we can get the uniform attractor by projecting
the global attractor of {𝑆(𝑡)} to the phase space if the latter
2
Abstract and Applied Analysis
exists. We consider the following nonautonomous dissipative
generalized long-short wave equations:
𝑖𝑢𝑡 + 𝑢𝑥𝑥 − 𝑛𝑢 + 𝑖𝛼𝑢 + 𝑔 (|𝑢|2 ) 𝑢 = 𝑎 (𝑥, 𝑡) ,
(2)
𝑛𝑡 + 𝛽𝑛 + |𝑢|2𝑥 + 𝑓 (|𝑢|2 ) = 𝑏 (𝑥, 𝑡) ,
(3)
𝑛 (𝑥, 𝜏) = 𝑛𝜏 (𝑥) ,
(4)
and the boundary value conditions
𝑢 (𝑥, 𝑡) |𝜕Ω = 0,
𝑛 (𝑥, 𝑡) |𝜕Ω = 0,
(5)
where 𝑥 ∈ Ω = (−𝐷, 𝐷) ⊂ R. 𝑡 ≥ 𝜏 ∈ R+ . Nonautonomous
terms 𝑎(𝑥, 𝑡) and 𝑏(𝑥, 𝑡) are time-depended external forces,
which are supposed to be translation compact (cf. [18] or
Assumption 1). Nonlinear terms 𝑓(𝑠) and 𝑔(𝑠) are given
smooth and real, satisfying
󵄨
󵄨󵄨
𝑝/2
󵄨󵄨𝑓 (𝑠)󵄨󵄨󵄨 ⩽ 𝑐1 (𝑠 + 1) ,
󵄨󵄨 (𝑘) 󵄨󵄨
󵄨󵄨𝑓 (𝑠)󵄨󵄨 ≤ 𝑐3 ,
󵄨
󵄨
𝑡+1
󵄩󵄩 󵄩󵄩2
󵄩󵄩𝜑󵄩󵄩𝐿2 (R;𝑋) = sup ∫
𝑏
𝑡
𝑡∈R
with the initial conditions
𝑢 (𝑥, 𝜏) = 𝑢𝜏 (𝑥) ,
We denote all the translation compact functions in
𝐿2loc (R; 𝑋) by 𝐿2𝑐 (R; 𝑋), where 𝑋 is a Banach space. Apparently, 𝜑 ∈ 𝐿2𝑐 (R; 𝑋) implies that 𝜑 is translation bounded;
that is,
󵄨
󵄨󵄨
1/2
󵄨󵄨𝑔 (𝑠)󵄨󵄨󵄨 ⩽ 𝑐2 (𝑠 + 1) ,
󵄨󵄨 (𝑘) 󵄨󵄨
󵄨󵄨𝑔 (𝑠)󵄨󵄨 ≤ 𝑐4 ,
󵄨
󵄨
(6)
(7)
where 0 ≤ 𝑠 < ∞, 𝑝 < 2, 𝑘 = 1, 2, and constants 𝛼, 𝛽, and 𝑐𝑗
are given in R+ for 𝑗 = 1, 2, 3, 4.
Our aim here is, firstly, to get the unique existence
of solutions for problem (2)∼(5) and then to derive the
existence of weakly compact uniform attractor for it with the
above-mentioned method. Here, and throughout this paper,
uniform means uniform about symbols (𝜎) in symbol space
(∑) unless there is special explanation. In fact, it is the same
if we say uniform about the initial time, since the translation
identity and the (𝐸 × ∑, 𝐸)-continuity of {𝑈𝜎 (𝑡, 𝜏)} hold in
our case (cf. [20]).
Throughout this paper, we denote by ‖ ⋅ ‖ the norm of 𝐻 =
𝐿2 (Ω) with usual inner product (⋅, ⋅), denote by ‖ ⋅ ‖ the norm
of 𝐿𝑝 (Ω) for all 1 ≤ 𝑝 ≤ ∞ (‖ ⋅ ‖2 = ‖ ⋅ ‖), and denote
by ‖ ⋅ ‖𝐻𝑘 the norm of a usual Sobolev space 𝐻𝑘 (Ω) for all
1 ≤ 𝑘 ≤ ∞. And we denote different constants by a same
letter 𝐶, and 𝐶(⋅, ⋅) represents that the constant relies only on
the parameters appearing in the brackets.
This paper is organized as follows. In Section 2, we recall
some facts about the nonautonomous system. In Section 3, we
provide the uniform a priori estimates in time. In Section 4,
we obtain the unique existence of the solutions for problem
(2)∼(5) by Galerkin method. Section 5 contains the weakly
compact uniform attractor for the nonautonomous system
(2)∼(5), and in the proof of Theorem 13, the (𝐸0 × ∑, 𝐸0 )continuity of {𝑈𝜎∈∑ (𝑡, 𝜏)} is proved.
2. Preliminary Results
(8)
is called the hull of 𝜑 in I, denoted by H(𝜑). 𝜑 is translation
compact if H(𝜑) is compact in I.
(9)
Let 𝐸 be a Banach space, and let a family of two-parameter
mappings {𝑈(𝑡, 𝜏)} = {𝑈(𝑡, 𝜏) | 𝑡 ≥ 𝜏, 𝜏 ∈ R} act in 𝐸. We also
need the following definitions and lemma (cf. [19, 20]).
Definition 1. Let ∑ be a parameter set. {𝑈𝜎 (𝑡, 𝜏), 𝑡 ≥ 𝜏, 𝜏 ∈
R}, 𝜎 ∈ ∑ is said to be a family of processes in Banach space
𝐸, if for each 𝜎 ∈ ∑, {𝑈𝜎 (𝑡, 𝜏)} from 𝐸 to 𝐸 satisfies
𝑈𝜎 (𝑡, 𝑠) ∘ 𝑈𝜎 (𝑠, 𝜏) = 𝑈𝜎 (𝑡, 𝜏) ,
𝑈𝜎 (𝜏, 𝜏) = 𝐼,
∀𝑡 ≥ 𝑠 ≥ 𝜏, 𝜏 ∈ R,
𝐼 is the identity operator, 𝜏 ∈ R.
(10)
Definition 2. {𝑈𝜎∈∑ (𝑡, 𝜏)}, a family of processes in Banach
space 𝐸, is called (𝐸 × ∑, 𝐸)-continuous, if for all fixed 𝑇 and
𝜏, 𝑇 ≥ 𝜏, projection (𝑢𝜏 , 𝜎) → 𝑈𝜎 (𝑇, 𝜏)𝑢𝜏 is continuous
from 𝐸 × ∑ to 𝐸.
A set 𝐵0 ⊂ 𝐸 is said to be uniformly absorbing set for
the family of processes {𝑈𝜎∈∑ (𝑡, 𝜏)}, if for any 𝜏 ∈ R and 𝐵 ∈
B(𝐸) which denotes the set of all bounded subsets of 𝐸, there
exists 𝑡0 = 𝑡0 (𝜏, 𝐵) ≥ 𝜏, such that ⋃𝜎∈∑ 𝑈𝜎 (𝑡, 𝜏)𝐵 ⊆ 𝐵0 for all
𝑡 ≥ 𝑡0 . A set 𝑌 ⊂ 𝐸 is said to be uniformly attracting for the
family of process {𝑈𝜎 (𝑡, 𝜏)}, 𝜎 ∈ ∑ if for any fixed 𝜏 ∈ R and
every 𝐵 ∈ B(𝐸),
lim (sup dist𝐸 (𝑈𝜎 (𝑡, 𝜏) 𝐵, 𝑌)) = 0.
𝑡 → +∞
𝜎∈∑
(11)
Definition 3. A closed set A∑ ⊂ 𝐸 is called the uniform
attractor of the family of process {𝑈𝜎 (𝑡, 𝜏)}, 𝜎 ∈ ∑ if it is
uniformly attracting (attracting property), and it is contained
in any closed uniformly attracting set A󸀠 of the family of
process {𝑢𝜎 (𝑡, 𝜏)}, 𝜎 ∈ ∑ : A𝐸 ⊆ A󸀠 (minimality property).
Lemma 4. Let ∑ be a compact metric space, and suppose
{𝑇(ℎ) | ℎ ≥ 0} is a family of operators acting on ∑, satisfying
the following:
(i)
𝑇 (ℎ) ∑ = ∑,
∀ℎ ∈ R+ ;
(12)
(ii) translation identity:
𝑈𝜎 (𝑡 + ℎ, 𝜏 + ℎ) = 𝑈𝑇(ℎ)𝜎 (𝑡, 𝜏) ,
∀𝜎 ∈ ∑, 𝑡 ≥ 𝜏, 𝜏 ∈ R, ℎ ≥ 0,
Let I be a topological space, and let 𝜑(𝑠) ∈ I be a function.
The set
H (𝜑) = {𝜑(ℎ + 𝑠) | ℎ ∈ R}
󵄩󵄩 󵄩󵄩2
󵄩󵄩𝜑󵄩󵄩𝑋 𝑑𝑠 < ∞.
(13)
where 𝑈𝜎 (𝑇, 𝜏) is arbitrarily a process in compact metric space
𝐸. Moreover, if the family of processes {𝑈𝜎∈∑ (𝑇, 𝜏)} is (𝐸×∑, 𝐸)
continuous, and it has a uniform compact attracting set, then
the skew product flow corresponding to it has a global attractor
A on 𝐸 × ∑, and the projection of A on ∑, A∑ is the compact
uniform attractor of {𝑈𝜎∈∑ (𝑇, 𝜏)}.
Abstract and Applied Analysis
3
Remark 5. Assumption (13) holds if the system has a unique
solution.
For brevity, we rewrite system (2)∼(5) in the vector form
by introducing 𝑊(𝑥, 𝑡) = (𝑢(𝑥, 𝑡), 𝑛(𝑥, 𝑡)) and 𝑌(𝑥, 𝑡) =
(𝑎(𝑥, 𝑡), 𝑏(𝑥, 𝑡)). We denote by 𝐸0 = 𝐻2 ⋂ 𝐻01 (Ω) × 𝐻01 (Ω)
the space of vector functions 𝑊(𝑥, 𝑡) = (𝑢(𝑥, 𝑡), 𝑛(𝑥, 𝑡)) with
norm
‖𝑊‖𝐸0 = {‖𝑢‖2𝐻2 + ‖𝑛‖2𝐻1 }
1/2
.
1/2
.
(𝑖𝑢𝑡 + 𝑢𝑥𝑥 − 𝑛𝑢 + 𝑖𝛼𝑢 + 𝑔 (|𝑢|2 ) 𝑢, 𝑢) = (𝑎 (𝑥, 𝑡) , 𝑢) .
(20)
Taking the imaginary part of (20), we obtain that
1 𝑑
‖𝑢‖2 + 𝛼‖𝑢‖2 = Im (𝑎 (𝑥, 𝑡) , 𝑢) .
2 𝑑𝑡
(14)
Similarly, we denote by ∑0 the space of 𝑌(𝑥, 𝑡) with norm
‖𝑌‖∑0 = {‖𝑎‖2𝐻1 + ‖𝑏‖2𝐻1 }
Proof. Taking the inner product of (2) with 𝑢 in 𝐻 we get that
(21)
By Young inequality and Remark 6 we have
(15)
𝑑
1
1󵄩
󵄩2
‖𝑢‖2 + 𝛼‖𝑢‖2 ≤ ‖𝑎(𝑥, 𝑡)‖2𝐿2 (R;𝐻1 ) ≤ 󵄩󵄩󵄩𝑎0 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 (R;𝐻1 ) .
𝑏
𝑏
𝑑𝑡
𝛼
𝛼
(22)
Then system (2)∼(5) can be considered as
𝜕𝑡 𝑊 = 𝐴𝑊 + 𝜎 (𝑡) ,
𝑊|𝑡=𝜏 = (𝑢𝜏 , 𝑛𝜏 ) = 𝑊𝜏 ,
(16)
And then by Gronwall lemma we can complete the proof.
𝑊|𝜕Ω = 0,
where 𝜎(𝑡) = 𝑌(𝑥, 𝑡) is the symbol of (16).
Assumption 1. Assume that the symbol 𝜎(𝑡) comes from the
symbol space ∑ defined by
∑ = {𝑌0 (𝑥, 𝑠 + 𝑟) | 𝑟 ∈ R+ },
(17)
In the following, we denote that ∫ ⋅ 𝑑𝑥 = ∫Ω ⋅ 𝑑𝑥, which
will not cause confusions.
Lemma 9. Under assumptions of (6), (7) and Assumption 1, if
𝑊(𝜏) ∈ 𝐻1 × 𝐻, solutions of problem (2)∼(5) satisfy
where 𝑌0 = (𝑎0 (𝑥, 𝑡), 𝑏0 (𝑥, 𝑡)) ∈ 𝐿2𝑐 (R; 𝐸0 ) and the closure
is taken in the sense of local quadratic mean convergence
topology in the topological space 𝐿2loc (R; ∑0 ). Moreover, we
suppose that 𝑎0𝑡 (𝑥, 𝑡) ∈ 𝐿2𝑏 (R; 𝐻1 ).
where 𝐶2 = 𝐶(𝛼, 𝛽, 𝑓, 𝑔, 𝑌0 , 𝑎0𝑡 ) and 𝑡2 = 𝐶(𝛼, 𝛽, 𝑓, 𝑔, 𝑌0 , 𝑎0𝑡 ,
‖𝑊𝜏 ‖𝐻1 ×𝐻).
Remark 6. By the conception of translation compact/boundedness we remark that
Proof. Taking the inner product of (2) with 𝑢𝑡 in 𝐻 and taking
the real part, we get that
(i) ∀𝑌1 ∈ ∑, ‖𝑌1 ‖2𝐿2 (R;∑ ) ≤ ‖𝑌0 ‖2𝐿2 (R;∑ ) ;
0
𝑏
‖𝑊(𝑡)‖2𝐻1 ×𝐻 ≤ 𝐶2 ,
0
𝑏
(ii) 𝑇(𝑡) ∑ = ∑, ∀𝑡 ∈ R, where 𝑇(𝑡)𝜑(𝑠) = 𝜑(𝑠 + 𝑡) is an
translation operator.
−
1 𝑑 󵄩󵄩 󵄩󵄩2 1
𝑑 2
󵄩𝑢 󵄩 − ∫ 𝑛 |𝑢| 𝑑𝑥 + Re (𝑖𝛼𝑢, 𝑢𝑡 )
2 𝑑𝑡 󵄩 𝑥 󵄩
2
𝑑𝑡
1
𝑑
+ ∫ 𝑔 (|𝑢|2 ) |𝑢|2 𝑑𝑥 = Re (𝑎 (𝑥, 𝑡) , 𝑢𝑡 ) .
2
𝑑𝑡
3. Uniform a Priori Estimates in Time
In this section, we derive uniform a priori estimates in time
which enable us to show the existence of solutions and the
uniform attractor. First we recall the following interpolation
inequality (cf. [21]).
Lemma 7. Let 𝑗, 𝑚 ∈ N ∪ {0}, 𝑞, 𝑟 ∈ R+ , such that 0 ≤ 𝑗 < 𝑚,
1 ≤ 𝑞, 𝑟 ≤ ∞. Then one has
󵄩󵄩 𝑗 󵄩󵄩
󵄩󵄩𝐷 𝑢󵄩󵄩 ≤ 𝐶󵄩󵄩󵄩󵄩𝐷𝑚 𝑢󵄩󵄩󵄩󵄩𝑎𝑟 ‖𝑢‖1−𝑎
(18)
𝑞 ,
󵄩
󵄩𝑝
𝑚,𝑟
𝑞
1
for 𝑢 ∈ 𝑊 (Ω) ∩ 𝐿 (Ω), where Ω ⊂ R , 𝑗/𝑚 ≤ 𝑎 ≤ 1, and
1/𝑝 = 𝑗 + 𝑎(1/𝑟 − 𝑚) + 1 − 𝑎/𝑞.
Lemma 8. If 𝑢𝜏 (𝑥) ∈ 𝐿 (Ω) and 𝑌(𝑥, 𝑡) ∈ Σ, then for the
solutions of problem (2)∼(5), one has
∀𝑡 ≥ 𝑡1 ,
where 𝐶1 = 𝐶(𝛼, 𝑎0 ), 𝑡1 = 𝐶(𝛼, 𝑎0 , ‖𝑢𝜏 ‖).
(19)
(23)
(24)
By (3) we know that
𝑑
𝑑
∫ 𝑛 |𝑢|2 𝑑𝑥
𝑑𝑡
𝑑𝑡
=
𝑑
∫ 𝑛|𝑢|2 𝑑𝑥 − ∫ |𝑢|2 𝑛𝑡 𝑑𝑥
𝑑𝑡
=
𝑑
∫ 𝑛|𝑢|2 𝑑𝑥 + ∫ |𝑢|2 |𝑢|2𝑥 𝑑𝑥 + 𝛽 ∫ 𝑛|𝑢|2 𝑑𝑥
𝑑𝑡
2
2
2
+ ∫ 𝑓 (|𝑢| ) |𝑢| 𝑑𝑥 − ∫ 𝑏 (𝑥, 𝑡) |𝑢| 𝑑𝑥
2
‖𝑢 (𝑡)‖ ≤ 𝐶1 ,
∀𝑡 ≥ 𝑡2 ,
=
𝑑
∫ 𝑛|𝑢|2 𝑑𝑥 + 𝛽 ∫ 𝑛|𝑢|2 𝑑𝑥
𝑑𝑡
+ ∫ 𝑓 (|𝑢|2 ) |𝑢|2 𝑑𝑥 − ∫ 𝑏 (𝑥, 𝑡) |𝑢|2 𝑑𝑥,
(25)
4
Abstract and Applied Analysis
which shows that
1 𝑑 󵄩󵄩 󵄩󵄩2 1 𝑑
2
2
󵄩𝑢 󵄩 − ( ∫ 𝑛|𝑢| 𝑑𝑥 + 𝛽 ∫ 𝑛|𝑢| 𝑑𝑥
2 𝑑𝑡 󵄩 𝑥 󵄩
2 𝑑𝑡
−
+ ∫ 𝑓 (|𝑢|2 ) |𝑢|2 𝑑𝑥 − ∫ 𝑏 (𝑥, 𝑡) |𝑢|2 𝑑𝑥)
In the following, we denote by 𝐶 any constants depending
only on the data (𝛼, 𝛽, 𝑓, 𝑔), and 𝐶(⋅, ⋅) means it depends not
only on (𝛼, 𝛽, 𝑓, 𝑔) but also on parameters in the brackets.
∀𝜌 > 0, when 𝑡 is sufficiently large, by (6) and Lemmas 7 and
8, we have
󵄨
󵄨󵄨
󵄨󵄨− ∫ 𝑓 (|𝑢|2 ) |𝑢|2 𝑑𝑥󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨
󵄨
+
1 𝑑
∫ 𝐺 (|𝑢|2 ) 𝑑𝑥 + Re (𝑖𝛼𝑢, 𝑢𝑡 )
2 𝑑𝑡
−
𝑑
Re (𝑎 (𝑥, 𝑡) , 𝑢) + Re ∫ 𝑎𝑡 (𝑥, 𝑡) 𝑢 𝑑𝑥 = 0,
𝑑𝑡
≤ 𝐶 ∫ |𝑢|𝑝+2 𝑑𝑥 + 𝐶 ∫ |𝑢|2 𝑑𝑥
(26)
≤ 𝐶 ∫ (|𝑢|2𝑝 + |𝑢|4 ) 𝑑𝑥 + 𝐶‖𝑢‖2
where 𝐺(𝑠) is introduced by
󵄩 󵄩
≤ 𝐶‖𝑢‖44 + 𝐶 ≤ 𝐶 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 ‖𝑢‖3 + 𝐶
𝑠
𝐺 (𝑠) = ∫ 𝑔 (𝜉) 𝑑𝜉.
(27)
0
󵄩 󵄩2
≤ 𝜌󵄩󵄩󵄩𝑢x 󵄩󵄩󵄩 + 𝐶 (𝜌) ,
Taking the inner product of (2) with 𝑢 in 𝐻 and taking the
real part, we get that
󵄨
󵄨󵄨
󵄨󵄨∫ 𝑏 (𝑥, 𝑡) |𝑢|2 𝑑𝑥󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨
󵄨
󵄩 󵄩2
Re (𝑖𝑢𝑡 , 𝑢) − 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 − ∫ 𝑛|𝑢|2 𝑑𝑥
≤ ‖𝑏(𝑥, 𝑡)‖2𝐿2 (R;∑0 ) + ‖𝑢‖44
(28)
𝑏
+ ∫ 𝑔 (|𝑢|2 ) |𝑢|2 𝑑𝑥 − Re (𝑎 (𝑥, 𝑡) , 𝑢) = 0.
󵄩 󵄩2
󵄩 󵄩
≤ 󵄩󵄩󵄩𝑌0 󵄩󵄩󵄩𝐿2 (R;∑ ) + 𝐶 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩
0
𝑏
Multiply (28) by 𝛼, and add the resulting identity to (26) to
get
−
1 𝑑 󵄩󵄩 󵄩󵄩2 1 𝑑
∫ 𝑛|𝑢|2 𝑑𝑥
󵄩𝑢 󵄩 −
2 𝑑𝑡 󵄩 𝑥 󵄩
2 𝑑𝑡
󵄩 󵄩2
≤ 𝜌󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + 𝐶2 (𝜌) .
By (6) we deduce that
1
1
− (𝛽 + 2𝛼) ∫ 𝑛|𝑢|2 𝑑𝑥 − ∫ 𝑓 (|𝑢|2 ) |𝑢|2 𝑑𝑥
2
2
+
2
|𝐺 (𝑠)| ≤ 𝑐2 𝑠3/2 + 𝑐2 𝑠,
3
1
1 𝑑
󵄩 󵄩2
∫ 𝑏 (𝑥, 𝑡) |𝑢|2 𝑑𝑥 +
∫ 𝐺 (|𝑢|2 ) 𝑑𝑥 − 𝛼󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩
2
2 𝑑𝑡
+ 𝛼 ∫ 𝑔 (|𝑢|2 ) |𝑢|2 𝑑𝑥 − 𝛼 Re (𝑎 (𝑥, 𝑡) , 𝑢)
−
𝑑
Re (𝑎 (𝑥, 𝑡) , 𝑢) + Re ∫ 𝑎𝑡 (𝑥, 𝑡) 𝑢 𝑑𝑥 = 0.
𝑑𝑡
∀𝑠 ≥ 0.
(32)
And then
(29)
That is,
𝑑 󵄩󵄩 󵄩󵄩2
(󵄩𝑢 󵄩 +∫ 𝑛|𝑢|2 𝑑𝑥 − ∫ 𝐺 (|𝑢|2 ) 𝑑𝑥+2 Re ∫ 𝑎 (𝑥, 𝑡) 𝑢 𝑑𝑥)
𝑑𝑡 󵄩 𝑥 󵄩
󵄩 󵄩2
+ 𝛼 (󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + ∫ 𝑛|𝑢|2 𝑑𝑥 − ∫ 𝐺 (|𝑢|2 ) 𝑑𝑥
󵄨
󵄨󵄨
󵄨󵄨−𝛼 ∫ 𝐺 (|𝑢|2 ) 𝑑𝑥󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨
󵄨
Ω
≤ 𝐶 ∫ (|𝑢|3 + |𝑢|2 ) 𝑑𝑥 ≤ 𝐶‖𝑢‖33 + 𝐶‖𝑢‖2
(33)
󵄩 󵄩1/2
󵄩 󵄩2
≤ 𝐶󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 ‖𝑢‖5/2 + 𝐶 ≤ 𝜌󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + 𝐶3 (𝜌) ,
󵄨
󵄨󵄨
󵄨󵄨− (𝛼 + 𝛽) ∫ 𝑛|𝑢|2 𝑑𝑥󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨
󵄨
Ω
󵄩 󵄩2
+2 Re ∫ 𝑎 (𝑥, 𝑡) 𝑢 𝑑𝑥) + 𝛼󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩
≤ 𝜌‖𝑛‖2 + 𝐶 (𝜌) ‖𝑢‖44
(34)
󵄩 󵄩2
≤ 𝜌‖𝑛‖2 + 𝜌󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + 𝐶4 (𝜌) ,
= − ∫ 𝑓 (|𝑢|2 ) |𝑢|2 𝑑𝑥 + ∫ 𝑏 (𝑥, 𝑡) |𝑢|2 𝑑𝑥
2
(31)
󵄨
󵄨󵄨
󵄨󵄨2 Re ∫ 𝑎 (𝑥, 𝑡) 𝑢 𝑑𝑥󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
𝑡
󵄨
󵄨
Ω
2
− 𝛼 ∫ 𝐺 (|𝑢| ) 𝑑𝑥 − (𝛼 + 𝛽) ∫ 𝑛|𝑢| 𝑑𝑥
󵄩
󵄩2
≤ 󵄩󵄩󵄩𝑎𝑡 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 (R;Σ0 ) + ‖𝑢‖2
𝑐
+ 2 Re ∫ 𝑎𝑡 (𝑥, 𝑡) 𝑢 𝑑𝑥.
(30)
󵄩 󵄩2
≤ 𝐶 (󵄩󵄩󵄩𝑎0𝑡 󵄩󵄩󵄩𝐿2 (R;Σ0 ) , ‖𝑢‖2 ) .
𝑐
(35)
Abstract and Applied Analysis
5
By (30)∼(35) we get that
It comes from (38)∼(40) that
𝑑
𝑑
∫ 𝑖 (𝑢𝑢𝑥 − 𝑢𝑥 𝑢) 𝑑𝑥
‖𝑛‖2 +
𝑑𝑡
𝑑𝑡
𝑑 󵄩󵄩 󵄩󵄩2
(󵄩𝑢 󵄩 +∫ 𝑛|𝑢|2 𝑑𝑥 − ∫ 𝐺 (|𝑢|2 ) 𝑑𝑥+2 Re ∫ 𝑎 (𝑥, 𝑡) 𝑢 𝑑𝑥)
𝑑𝑡 󵄩 𝑥 󵄩
+ 2𝛽‖𝑛‖2 + 𝑖𝛼 ∫ (𝑢𝑢𝑥 − 𝑢𝑥 𝑢) 𝑑𝑥
󵄩 󵄩2
+ 𝛼 (󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + ∫ 𝑛|𝑢|2 𝑑𝑥 − ∫ 𝐺 (|𝑢|2 ) 𝑑𝑥
≤ 𝑖𝛼 ∫ (𝑢𝑢𝑥 − 𝑢𝑥 𝑢) 𝑑𝑥 − 4 Re ∫ 𝑖𝛼𝑢𝑢𝑥 𝑑𝑥
󵄩 󵄩2
+2 Re ∫ 𝑎 (𝑥, 𝑡) 𝑢 𝑑𝑥) + 𝛼󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩
+ 4 Re ∫ 𝑎 (𝑥, 𝑡) 𝑢𝑥 𝑑𝑥
󵄩 󵄩2
󵄩 󵄩2
≤ 𝜌‖𝑛‖2 + 4𝜌󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + 𝐶 (𝜌) + 𝐶 (󵄩󵄩󵄩𝑎0𝑡 󵄩󵄩󵄩𝐿2 (R;Σ0 ) , ‖𝑢‖2 ) .
𝑏
(36)
Similarly we can also deduce that
𝑑 󵄩󵄩 󵄩󵄩2
(󵄩𝑢 󵄩 +∫ 𝑛|𝑢|2 𝑑𝑥 − ∫ 𝐺 (|𝑢|2 ) 𝑑𝑥+2 Re ∫ 𝑎 (𝑥, 𝑡) 𝑢 𝑑𝑥)
𝑑𝑡 󵄩 𝑥 󵄩
(41)
− 2 ∫ 𝑓 (|𝑢|2 ) 𝑛 𝑑𝑥 + 2 ∫ 𝑏 (𝑥, 𝑡) 𝑛 𝑑𝑥.
Deal with the right hand side of inequality (41), by Lemmas 7
and 8,
󵄨
󵄨󵄨
󵄨󵄨𝑖𝛼 ∫ (𝑢𝑢 − 𝑢 𝑢) 𝑑𝑥󵄨󵄨󵄨 ≤ 2𝛼 ‖𝑢‖ 󵄩󵄩󵄩𝑢 󵄩󵄩󵄩 ≤ 𝜌 󵄩󵄩󵄩𝑢 󵄩󵄩󵄩 + 𝐶 (𝜌) ,
󵄨󵄨
󵄨󵄨
𝑥
𝑥
1
󵄩 𝑥󵄩
󵄩 𝑥󵄩
󵄨
󵄨
(42)
󵄩 󵄩2
+ 𝛽 (󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + ∫ 𝑛|𝑢|2 𝑑𝑥 − ∫ 𝐺 (|𝑢|2 ) 𝑑𝑥
󵄨
󵄨󵄨
󵄨󵄨−4 Re ∫ 𝑖𝛼𝑢𝑢 𝑑𝑥󵄨󵄨󵄨 ≤ 4𝛼 ‖𝑢‖ 󵄩󵄩󵄩𝑢 󵄩󵄩󵄩 ≤ 𝜌 󵄩󵄩󵄩𝑢 󵄩󵄩󵄩 + 𝐶 (𝜌) ,
󵄨󵄨
󵄨󵄨
𝑥
2
󵄩 𝑥󵄩
󵄩 𝑥󵄩
󵄨
󵄨
󵄩 󵄩2
+2 Re ∫ 𝑎 (𝑥, 𝑡) 𝑢 𝑑𝑥) + (2𝛼 − 𝛽) 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩
(43)
󵄨
󵄨󵄨
󵄨󵄨−2 ∫ 𝑓 (|𝑢|2 ) 𝑛 𝑑𝑥󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨
󵄨
󵄩 󵄩2
󵄩 󵄩2
≤ 𝜌‖𝑛‖2 + 4𝜌󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + 𝐶 (𝜌) + 𝐶 (󵄩󵄩󵄩𝑎0𝑡 󵄩󵄩󵄩𝐿2 (R;Σ0 ) , ‖𝑢‖2 ) .
𝑏
(37)
≤ 𝐶 ∫ |𝑢|𝑝 |𝑛| 𝑑𝑥 + 𝐶 ∫ |𝑛| 𝑑𝑥
1
1
≤ 𝜌‖𝑛‖2 + 𝐶 (𝜌) ∫ |𝑢|2𝑝 𝑑𝑥 + 𝜌‖𝑛‖2 + 𝐶 (𝜌)
2
2
󵄩 󵄩𝑝−1
≤ 𝜌‖𝑛‖2 + 𝐶 (𝜌) 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 ‖𝑢‖𝑝+1 + 𝐶 (𝜌)
󵄩 󵄩2
≤ 𝜌‖𝑛‖2 + 𝜌󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + 𝐶3 (𝜌) ,
Taking the inner product of (3) with 𝑛 in 𝐻, we see that
1 𝑑
‖𝑛‖2 + ∫ 𝑛|𝑢|2𝑥 𝑑𝑥 + 𝛽‖𝑛‖2
2 𝑑𝑡
(38)
2
+ ∫ 𝑓 (|𝑢| ) 𝑛 𝑑𝑥 − ∫ 𝑏 (𝑥, 𝑡) 𝑛 𝑑𝑥 = 0.
(44)
󵄨󵄨
󵄨
󵄨󵄨4 Re ∫ 𝑎 (𝑥, 𝑡) 𝑢 𝑑𝑥󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
𝑥
󵄨
󵄨
By (2) we get that
󵄩 󵄩
󵄩 󵄩
≤ 4‖𝑎 (𝑥, 𝑡)‖𝐿2𝑏 (R;Σ0 ) 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 ≤ 𝜌 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩
󵄩
󵄩
+ 𝐶4 (𝜌, 󵄩󵄩󵄩𝑎0 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 (R;Σ0 ) ) ,
∫ 𝑛|𝑢|2𝑥 𝑑𝑥 = ∫ 𝑛𝑢𝑥 𝑢 𝑑𝑥 + ∫ 𝑛𝑢𝑢𝑥 𝑑𝑥
= 𝑖 ∫ (𝑢𝑡 𝑢𝑥 − 𝑢𝑡 𝑢𝑥 ) 𝑑𝑥 + 2 Re ∫ 𝑖𝛼𝑢𝑢𝑥 𝑑𝑥
𝑏
󵄨
󵄨󵄨
󵄨󵄨2 ∫ 𝑏 (𝑥, 𝑡) 𝑛 𝑑𝑥󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨
󵄨
(39)
≤ 2‖𝑏 (𝑥, 𝑡)‖𝐿2𝑏 (R;Σ0 ) ‖𝑛‖ ≤ 𝜌‖𝑛‖2
− 2 Re ∫ 𝑏 (𝑥, 𝑡) 𝑢𝑥 𝑑𝑥,
𝑏
So
= 𝑖 ∫ (𝑢𝑡 𝑢𝑥 + 𝑢𝑢𝑥𝑡 − 𝑢𝑥𝑡 𝑢 − 𝑢𝑥 𝑢𝑡 ) 𝑑𝑥
(40)
= 2𝑖 ∫ (𝑢𝑡 𝑢𝑥 − 𝑢𝑡 𝑢𝑥 ) 𝑑𝑥.
(46)
󵄩
󵄩
+ 𝐶5 (𝜌, 󵄩󵄩󵄩𝑏0 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 (R;Σ0 ) ) .
𝑑
∫ (𝑖𝑢𝑢𝑥 − 𝑖𝑢𝑥 𝑢) 𝑑𝑥
𝑑𝑡
= 𝑖 ∫ (𝑢𝑡 𝑢𝑥 − 𝑢𝑥 𝑢𝑡 + 𝑢𝑡 𝑢𝑥 − 𝑢𝑥 𝑢𝑡 ) 𝑑𝑥
(45)
𝑑
(‖𝑛‖2 + 𝑖 ∫ (𝑢𝑢𝑥 − 𝑢𝑥 𝑢) 𝑑𝑥)
𝑑𝑡
+ 2𝛽‖𝑛‖2 + 𝑖𝛼 ∫ (𝑢𝑢𝑥 − 𝑢𝑥 𝑢) 𝑑𝑥
(47)
󵄩 󵄩2
󵄩
󵄩
≤ 2𝜌‖𝑛‖2 + 4𝜌󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + 𝐶 (𝜌, 󵄩󵄩󵄩𝑌0 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 (R;Σ0 ) ) .
𝑏
6
Abstract and Applied Analysis
where 𝐶(𝑅) = 𝐶(𝜌, ‖𝑌0 (𝑥, 𝑡)‖𝐿2𝑏 (R;Σ0 ) , ‖𝑎0𝑡 (𝑥, 𝑡)‖𝐿2𝑏 (R;Σ0 ) , 𝑅)
when ‖𝑊𝜏 ‖𝐻1 ×𝐻 ≤ 𝑅. Then by (52) we infer that
Analogously, we can also deduce that
𝑑
(‖𝑛‖2 + 𝑖 ∫ (𝑢𝑢𝑥 − 𝑢𝑥 𝑢) 𝑑𝑥)
𝑑𝑡
+ 2𝛽‖𝑛‖2 + 𝑖𝛽 ∫ (𝑢𝑢𝑥 − 𝑢𝑥 𝑢) 𝑑𝑥
(48)
𝐸 (𝑡) ≤ 𝐶 (𝑅) 𝑒−𝛾(𝑡−𝑡0 ) +
2𝐶0
≤
,
𝛾
󵄩 󵄩2
󵄩
󵄩
≤ 2𝜌‖𝑛‖2 + 4𝜌󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + 𝐶 (𝜌, 󵄩󵄩󵄩𝑌0 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 (R;Σ0 ) ) .
𝑏
∀𝑡 ≥ 𝑡0 ,
(55)
∀𝑡 ≥ 𝑡∗ ,
where 𝑡∗ = inf{𝑡 | 𝑡 ≥ 𝑡0 and 𝐶(𝑅)𝑒−𝛾(𝑡∗ −𝑡0 ) ≤ 𝐶0 /𝛾}. By (49),
(53), and (55) we infer that
Set 𝛾 = min{𝛼, 𝛽}, and
󵄩 󵄩2
𝐸 = 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + ‖𝑛‖2 + ∫ 𝑛|𝑢|2 𝑑𝑥
󵄩2
󵄩 󵄩2
󵄩󵄩
2
2
󵄩󵄩𝑢𝑥 (𝑡)󵄩󵄩󵄩 + ‖𝑛(𝑡)‖ ≤ 𝜌‖𝑛‖ + 𝜌󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + 𝐶0 .
2
− ∫ 𝐺 (|𝑢| ) 𝑑𝑥 + 2 Re ∫ 𝑎𝑢 𝑑𝑥
(49)
(56)
Choose 𝜌 = min{𝛼/8, 𝛽/3, 1/2}; then we have
󵄩󵄩 󵄩󵄩2
2
󵄩󵄩𝑢𝑥 󵄩󵄩 + ‖𝑛(𝑡)‖
+ 𝑖 ∫ (𝑢𝑢𝑥 − 𝑢𝑥 𝑢) 𝑑𝑥.
Then by (36), (47) and (37), (48) we can, respectively, get
𝑑
󵄩 󵄩2
𝐸 + 𝛼𝐸 + 𝛼󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + 𝛽‖𝑛‖2
𝑑𝑡
󵄩 󵄩2
󵄩
󵄩
≤ 8𝜌󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + 3𝜌‖𝑛‖2 + 𝐶 (𝜌, 󵄩󵄩󵄩Y0 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 (R;Σ0 ) ) ,
𝑏
𝑑
󵄩 󵄩2
𝐸 + 𝛽𝐸 + 𝛼󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + 𝛽‖𝑛‖2
𝑑𝑡
󵄩 󵄩2
󵄩
󵄩
≤ 8𝜌󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + 3𝜌‖𝑛‖2 + 𝐶 (𝜌, 󵄩󵄩󵄩𝑌0 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 (R;Σ0 ) ) ,
(50)
Similar to (33), (34), (45), and (42), for 𝑡 ≥ 𝑡0 we have
󵄨󵄨
󵄨󵄨∫ 𝑛|𝑢|2 𝑑𝑥 − ∫ 𝐺 (|𝑢|2 ) 𝑑𝑥
󵄨󵄨
󵄨
∀𝑡 ≥ 𝑡∗ ,
(57)
Lemma 10. Under assumptions of Lemma 9, if 𝑊(𝜏) ∈ 𝐸0 =
𝐻2 × 𝐻1 , solutions of problem (2)∼(5) satisfy
∀𝑡 ≥ 𝑡3 ,
(58)
where 𝐶2 = 𝐶(𝛼, 𝛽, 𝑓, 𝑔, 𝑌0 , 𝑎0𝑡 ) and 𝑡3 = 𝐶(𝛼, 𝛽, 𝑓, 𝑔, 𝑌0 , 𝑎0𝑡 ,
‖𝑊𝜏 ‖𝐸0 ).
Proof. Taking the real part of the inner product of (2) with
𝑢𝑥𝑥𝑡 in 𝐻, we have
𝑑 󵄩󵄩 󵄩󵄩2
󵄩𝑢 󵄩 − Re ∫ 𝑛𝑢𝑢𝑥𝑥𝑡 𝑑𝑥 + Re (𝑖𝛼𝑢, 𝑢𝑥𝑥𝑡 )
𝑑𝑡 󵄩 𝑥𝑥 󵄩
+ Re ∫ 𝑔 (|𝑢|2 ) 𝑢𝑢𝑥𝑥𝑡 𝑑𝑥
(59)
By (2) and (3), we have
(53)
− Re ∫ 𝑛𝑢𝑢𝑥𝑥𝑡 𝑑𝑥
=−
𝑏
And then
󵄨󵄨
󵄨󵄨 󵄩󵄩
󵄩󵄩2 󵄩󵄩
󵄩󵄩2
󵄨󵄨𝐸 (𝑡0 )󵄨󵄨 ≤ 󵄩󵄩𝑢𝑥 (𝑡0 )󵄩󵄩 + 󵄩󵄩𝑛(𝑡0 )󵄩󵄩
󵄨󵄨
󵄨
󵄨2
󵄨
󵄨2
+ 󵄨󵄨󵄨󵄨∫ 𝑛 (𝑡0 ) 󵄨󵄨󵄨𝑢 (𝑡0 )󵄨󵄨󵄨 𝑑𝑥 − ∫ 𝐺 (󵄨󵄨󵄨𝑢 (𝑡0 )󵄨󵄨󵄨 ) 𝑑𝑥
󵄨
󵄨󵄨
+𝑖 ∫ (𝑢 (𝑡0 ) 𝑢𝑥 (𝑡0 ) − 𝑢𝑥 (𝑡0 ) 𝑢 (𝑡0 )) 𝑑𝑥󵄨󵄨󵄨󵄨
󵄨
𝑏
− Re ∫ 𝑎 (𝑥, 𝑡) 𝑢𝑥𝑥𝑡 𝑑𝑥 = 0.
󵄩 󵄩2
󵄩
󵄩
≤ 𝜌‖𝑛‖2 + 𝜌󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + 𝐶 (𝜌, 󵄩󵄩󵄩𝑎0 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 (R;Σ0 ) ) .
+ 2 Re ∫ 𝑎 (𝑡0 ) 𝑢 (𝑡0 ) 𝑑𝑥
𝑏
‖𝑊(𝑡)‖2𝐸0 ≤ 𝐶2 ,
𝑏
󵄨󵄨
+2 Re ∫ 𝑎𝑢 𝑑𝑥 + 𝑖 ∫ (𝑢𝑢𝑥 − 𝑢𝑥 𝑢) 𝑑𝑥󵄨󵄨󵄨󵄨
󵄨
󵄩
󵄩
󵄩
󵄩
≤ 𝐶 (󵄩󵄩󵄩𝑌0 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 (R;Σ0 ) , 󵄩󵄩󵄩𝑎0𝑡 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 (R;Σ0 ) ) ,
which concludes the proof by using Lemma 8.
which shows that if we set 𝜌 ≤ min{𝛼/8, 𝛽/3}, we can deduce
that
𝑑
(51)
𝐸 + 𝛾𝐸 ≤ 𝐶0 , ∀𝑡 ≥ 𝑡0 ,
𝑑𝑡
where 𝐶0 = 𝐶(𝜌, ‖𝑌0 (𝑥, 𝑡)‖𝐿2𝑏 (R;Σ0 ) , ‖𝑎0𝑡 (𝑥, 𝑡)‖𝐿2𝑏 (R;Σ0 ) ). By
Gronwall lemma we see that
𝐶
𝐸 (𝑡) ≤ 𝐸 (𝑡0 ) 𝑒−𝛾(𝑡−𝑡0 ) + 0 , ∀𝑡 ≥ 𝑡0 .
(52)
𝛾
≤ 𝐶 (𝑅) ,
𝐶0
,
𝛾
𝑑
∫ Re (𝑛𝑢𝑢𝑥𝑥 ) 𝑑𝑥 + Re ∫ 𝑛𝑡 𝑢𝑢𝑥 𝑥𝑑𝑥
𝑑𝑡
+ Re ∫ 𝑛𝑢𝑡 𝑢𝑥𝑥 𝑑𝑥
=−
(54)
𝑑
∫ Re (𝑛𝑢𝑢𝑥𝑥 ) 𝑑𝑥
𝑑𝑡
− Re ∫ 𝑢𝑢𝑥𝑥 (|𝑢|2𝑥 + 𝛽𝑛 + 𝑓 (|𝑢|2 ) − 𝑏) 𝑑𝑥
+ Re ∫ 𝑛𝑢𝑥𝑥 (−𝑖𝑛𝑢 − 𝛼𝑢 + 𝑖𝑔 (|𝑢|2 ) 𝑢 − 𝑖𝑎) 𝑑𝑥.
(60)
Abstract and Applied Analysis
7
Since
We take care of terms in (66) as follows:
Re (𝑖𝛼𝑢, 𝑢𝑥𝑥𝑡 ) = Re ∫ 𝑖𝛼𝑢𝑢𝑡𝑥𝑥 𝑑𝑥 = − Re ∫ 𝑖𝛼𝑢𝑡 𝑢𝑥𝑥 𝑑𝑥, (61)
∫ 𝑔󸀠󸀠 (|𝑢|2 ) |𝑢|2𝑡 2 Re (𝑢𝑢𝑥 ) Re (𝑢𝑢𝑥 ) 𝑑𝑥
2
= 4 ∫ 𝑔󸀠󸀠 (|𝑢|2 ) (Re (𝑢𝑢𝑥 ))
we see that
󵄩 󵄩2
Re (𝑖𝛼𝑢, 𝑢𝑥𝑥𝑡 ) = 𝛼󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 − 𝛼 Re ∫ 𝑛𝑢𝑢𝑥𝑥 𝑑𝑥
+ 𝛼 Re ∫ 𝑔 (|𝑢|2 ) 𝑢𝑢𝑥𝑥 𝑑𝑥
× (Re (𝑖𝑢𝑥𝑥 𝑢) − 𝛼|𝑢|2 − Re (𝑖𝑎𝑢)) 𝑑𝑥,
(62)
= ∫ 𝑔󸀠 (|𝑢|2 ) 2 Re (𝑢𝑡 𝑢𝑥 ) Re (𝑢𝑢𝑥 ) 𝑑𝑥
− 𝛼 Re ∫ 𝑎𝑢𝑥𝑥 𝑑𝑥.
Multiplying (2) by 𝑢 and taking the real part, we find that
|𝑢|2𝑡 = 2 Re (𝑖𝑢𝑥𝑥 𝑢) − 2𝛼|𝑢|2 − 2 Re (𝑖𝑎𝑢) ,
∫ 𝑔󸀠 (|𝑢|2 ) 2 Re (𝑢𝑢𝑥 )𝑡 Re (𝑢𝑢𝑥 ) 𝑑𝑥
(63)
+ ∫ 𝑔󸀠 (|𝑢|2 ) 2 Re (𝑢𝑢𝑥𝑡 ) Re (𝑢𝑢𝑥 ) 𝑑𝑥
= 2 ∫ 𝑔󸀠 (|𝑢|2 ) Re (𝑢𝑢𝑥 )
therefore,
× Re (𝑖𝑢𝑥 (𝑢𝑥𝑥 − 𝑛𝑢 + 𝑖𝛼𝑢 + 𝑔 (|𝑢|2 ) 𝑢 − 𝑎)) 𝑑𝑥
Re ∫ 𝑔 (|𝑢|2 ) 𝑢𝑢𝑥𝑥𝑡 𝑑𝑥
+ ∫ 𝑔󸀠 (|𝑢|2 ) |𝑢|2𝑥 Re (𝑢𝑢𝑥𝑡 ) 𝑑𝑥,
= − ∫ 𝑔󸀠 (|𝑢|2 ) |𝑢|2𝑥 Re (𝑢𝑢𝑥𝑡 ) 𝑑𝑥
− ∫ 𝑔 (|𝑢|2 )
1 𝑑 󵄨󵄨 󵄨󵄨2
󵄨𝑢 󵄨 𝑑𝑥
2 𝑑𝑡 󵄨 𝑥 󵄨
= − ∫ 𝑔󸀠 (|𝑢|2 ) |𝑢|2𝑥 Re (𝑢𝑢𝑥𝑡 ) 𝑑𝑥
−
∫ 𝑔󸀠 (|𝑢|2 ) 2 Re (𝑢𝑢𝑥 ) Re (𝑢𝑡 𝑢𝑥 ) 𝑑𝑥
= 2 ∫ 𝑔󸀠 (|𝑢|2 ) Re (𝑢𝑢𝑥 )
(64)
1 𝑑
󵄨 󵄨2
∫ 𝑔 (|𝑢|2 ) 󵄨󵄨󵄨𝑢𝑥 󵄨󵄨󵄨 𝑑𝑥
2 𝑑𝑡
It follows from (66)∼(67) that
󵄨 󵄨2
+ ∫ 𝑔 (|𝑢| ) 󵄨󵄨󵄨𝑢𝑥 󵄨󵄨󵄨 (Re (𝑖𝑢𝑥𝑥 𝑢) − 𝛼|𝑢|2
󸀠
× Re (𝑖𝑢𝑥 (𝑢𝑥𝑥 − 𝑛𝑢 + 𝑖𝛼𝑢 + 𝑔 (|𝑢|2 ) 𝑢 − 𝑎)) 𝑑𝑥.
(67)
2
− ∫ 𝑔󸀠 (|𝑢|2 ) |𝑢|2𝑥 Re (𝑢𝑢𝑥𝑡 ) 𝑑𝑥
− Re (𝑖𝑎𝑢)) 𝑑𝑥.
=−
Now we deal with (64) to get (70). Due to equalities
2
+ 4 ∫ 𝑔󸀠󸀠 (|𝑢|2 ) (Re (𝑢𝑢𝑥 ))
|𝑢|2𝑥 = 2 Re (𝑢𝑢𝑥 ) ,
𝑑
Re (𝑢𝑢𝑥 ) = Re (𝑢𝑡 𝑢𝑥 ) + Re (𝑢𝑢𝑥𝑡 ) ,
𝑑𝑡
𝑑
∫ 𝑔󸀠 (|𝑢|2 ) 2 Re (𝑢𝑢𝑥 ) Re (𝑢𝑢𝑥 ) 𝑑𝑥
𝑑𝑡
(65)
× (Re (𝑖𝑢𝑥𝑥 𝑢) − 𝛼|𝑢|2 − Re (𝑖𝑎𝑢)) 𝑑𝑥
+ 4 ∫ 𝑔󸀠 (|𝑢|2 ) Re (𝑢𝑢𝑥 )
we deduce that
× Re (𝑖𝑢𝑥 (𝑢𝑥𝑥 − 𝑛𝑢 + 𝑖𝛼𝑢 + 𝑔 (|𝑢|2 ) 𝑢 − 𝑎)) 𝑑𝑥
∫ 𝑔󸀠 (|𝑢|2 ) |𝑢|2𝑥 Re (𝑢𝑢𝑥𝑡 ) 𝑑𝑥
=
+ ∫ 𝑔󸀠 (|𝑢|2 ) |𝑢|2𝑥 Re (𝑢𝑢𝑥𝑡 ) 𝑑𝑥.
𝑑
∫ 𝑔󸀠 (|𝑢|2 ) 2 Re (𝑢𝑢𝑥 ) Re (𝑢𝑢𝑥 ) 𝑑𝑥
𝑑𝑡
− ∫ 𝑔󸀠󸀠 (|𝑢|2 ) |𝑢|2𝑡 2 Re (𝑢𝑢𝑥 ) Re (𝑢𝑢x ) 𝑑𝑥
− ∫ 𝑔󸀠 (|𝑢|2 ) 2 Re (𝑢𝑢𝑥 )𝑡 Re (𝑢𝑢𝑥 ) 𝑑𝑥
− ∫ 𝑔󸀠 (|𝑢|2 ) 2 Re (𝑢𝑢𝑥 ) Re (𝑢𝑡 𝑢𝑥 ) 𝑑𝑥.
(68)
(66)
And then
− ∫ 𝑔󸀠 (|𝑢|2 ) |𝑢|2𝑥 Re (𝑢𝑢𝑥𝑡 ) 𝑑𝑥
=−
𝑑
∫ 𝑔󸀠 (|𝑢|2 ) Re (𝑢𝑢𝑥 ) Re (𝑢𝑢𝑥 ) 𝑑𝑥
𝑑𝑡
8
Abstract and Applied Analysis
2
+ 2 ∫ 𝑔󸀠󸀠 (|𝑢|2 ) (Re (𝑢𝑢𝑥 ))
+ 2 Re ∫ 𝑛𝑢𝑥𝑥 (−𝑖𝑛𝑢 − 𝛼𝑢 + 𝑖𝑔 (|𝑢|2 ) − 𝑖𝑎) 𝑑𝑥
× (Re (𝑖𝑢𝑥𝑥 𝑢) − 𝛼 |𝑢|2 − Re (𝑖𝑎𝑢)) 𝑑𝑥
+ 2𝛼 Re ∫ 𝑔 (|𝑢|2 ) 𝑢𝑢𝑥𝑥 𝑑𝑥
+ 2 ∫ 𝑔󸀠 (|𝑢|2 ) Re (𝑢𝑢𝑥 )
2
+ 4 ∫ 𝑔󸀠󸀠 (|𝑢|2 ) (Re (𝑢𝑢𝑥 ))
× Re (𝑖𝑢𝑥 (𝑢𝑥x − 𝑛𝑢 + 𝑖𝛼𝑢 + 𝑔 (|𝑢|2 ) 𝑢 − 𝑎)) 𝑑𝑥.
(69)
+ 4 ∫ 𝑔󸀠 (|𝑢|2 ) Re (𝑢𝑢𝑥 )
From (64) and (69) we have
× Re (𝑖𝑢𝑥 (𝑢𝑥𝑥 − 𝑛𝑢 + 𝑖𝛼𝑢 + 𝑔 (|𝑢|2 ) 𝑢 − 𝑎)) 𝑑𝑥
2
Re ∫ 𝑔 (|𝑢| ) 𝑢𝑢𝑥𝑥𝑡 𝑑𝑥
=−
× (Re (𝑖𝑢𝑥𝑥 𝑢) − 𝛼|𝑢|2 − Re (𝑖𝑎𝑢)) 𝑑𝑥
󵄨 󵄨2
+ 2 ∫ 𝑔󸀠 (|𝑢|2 ) 󵄨󵄨󵄨𝑢𝑥 󵄨󵄨󵄨 (Re (𝑖𝑢𝑥𝑥 𝑢) − 𝛼|𝑢|2 − Re (𝑖𝑎𝑢)) 𝑑𝑥
𝑑
∫ 𝑔󸀠 (|𝑢|2 ) Re (𝑢𝑢𝑥 ) Re (𝑢𝑢𝑥 ) 𝑑𝑥
𝑑𝑡
+ 2 Re ∫ 𝑎𝑡 𝑢𝑥𝑥 𝑑𝑥 = 0,
1 𝑑
󵄨 󵄨2
−
∫ 𝑔 (|𝑢|2 ) 󵄨󵄨󵄨𝑢𝑥 󵄨󵄨󵄨 𝑑𝑥
2 𝑑𝑡
+ 2 ∫ 𝑔󸀠󸀠 (|𝑢|2 ) (Re (𝑢𝑢𝑥 ))
(71)
2
where ∫ ⋅ = ∫ ⋅ 𝑑𝑥.
For later purpose, we let
× (Re (𝑖𝑢𝑥𝑥 𝑢) − 𝛼|𝑢|2 − Re (𝑖𝑎𝑢)) 𝑑𝑥
𝐹 (𝑢, 𝑛) = −2 Re ∫ 𝑛𝑢𝑢𝑥𝑥 𝑑𝑥
+ 2 ∫ 𝑔󸀠 (|𝑢|2 ) Re (𝑢𝑢𝑥 )
× Re (𝑖𝑢𝑥 (𝑢𝑥𝑥 − 𝑛𝑢 + 𝑖𝛼𝑢 + 𝑔 (|𝑢|2 ) 𝑢 − 𝑎)) 𝑑𝑥
󵄨 󵄨2
+ ∫ 𝑔󸀠 (|𝑢|2 ) 󵄨󵄨󵄨𝑢𝑥 󵄨󵄨󵄨
2
󵄨 󵄨2
− 2 ∫ 𝑔󸀠 (|𝑢|2 ) Re (𝑢𝑢𝑥 ) 𝑑𝑥 − ∫ 𝑔 (|𝑢|2 ) 󵄨󵄨󵄨𝑢𝑥 󵄨󵄨󵄨 𝑑𝑥
− 2 Re ∫ 𝑎𝑢𝑥𝑥 𝑑𝑥,
(72)
× (Re (𝑖𝑢𝑥𝑥 𝑢) − 𝛼|𝑢|2 − Re (𝑖𝑎𝑢)) 𝑑𝑥.
(70)
− 𝐺 (𝑢, 𝑛)
2
= 2𝛼 ∫ 𝑛𝑢𝑢𝑥𝑥 𝑑𝑥 + 4𝛼 ∫ 𝑔󸀠 (|𝑢|2 ) (Re (𝑢𝑢𝑥 )) 𝑑𝑥
By (59), (60), (62), and (70) we conclude that
𝑑 󵄩󵄩 󵄩󵄩2
2
(󵄩𝑢 󵄩 − 2 Re ∫ 𝑛𝑢𝑢𝑥𝑥 − 2 ∫ 𝑔󸀠 (|𝑢|2 ) (Re (𝑢𝑢𝑥 ))
𝑑𝑡 󵄩 𝑥𝑥 󵄩
− ∫ 𝑔 (|𝑢|2 ) |𝑢|2𝑥 − 2 Re ∫ 𝑎𝑢𝑥𝑥 )
+ 2𝛼 Re ∫ 𝑔 (|𝑢|2 ) 𝑢𝑢𝑥𝑥 𝑑𝑥
2
− 2 ∫ 𝑔󸀠 (|𝑢|2 ) (Re (𝑢𝑢𝑥 ))
2
+ 4 ∫ 𝑔󸀠󸀠 (|𝑢|2 ) (Re (𝑢𝑢𝑥 ))
󵄨 󵄨2
− ∫ 𝑔 (|𝑢| ) 󵄨󵄨󵄨𝑢𝑥 󵄨󵄨󵄨 − 2 Re ∫ 𝑎𝑢𝑥𝑥 )
2
2
2
+ 2𝛼 ∫ 𝑛𝑢𝑢𝑥𝑥 𝑑𝑥 + 4𝛼 ∫ 𝑔 (|𝑢| ) (Re (𝑢𝑢𝑥 )) 𝑑𝑥
󵄨 󵄨2
+ 2𝛼 ∫ 𝑔 (|𝑢|2 ) 󵄨󵄨󵄨𝑢𝑥 󵄨󵄨󵄨 𝑑𝑥 + 2𝛼 Re ∫ 𝑎𝑢𝑥𝑥 𝑑𝑥
−
2 Re ∫ 𝑢𝑢𝑥𝑥 (|𝑢|2𝑥
− 2 Re ∫ 𝑢𝑢𝑥𝑥 (|𝑢|2𝑥 + 𝛽𝑛 + 𝑓 (|𝑢|2 ) − 𝑏) 𝑑𝑥
+ 2 Re ∫ 𝑛𝑢𝑥𝑥 (−𝑖𝑛𝑢 − 𝛼𝑢 + 𝑖𝑔 (|𝑢|2 ) − 𝑖𝑎) 𝑑𝑥
󵄩 󵄩2
+ 2𝛼 (󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 − 2 Re ∫ 𝑛𝑢𝑢𝑥𝑥
󸀠
󵄨 󵄨2
+ 2𝛼 ∫ 𝑔 (|𝑢|2 ) 󵄨󵄨󵄨𝑢𝑥 󵄨󵄨󵄨 𝑑𝑥 + 2𝛼 Re ∫ 𝑎𝑢𝑥𝑥 𝑑𝑥
2
+ 𝛽𝑛 + 𝑓 (|𝑢| ) − 𝑏) 𝑑𝑥
× (Re (𝑖𝑢𝑥𝑥 𝑢) − 𝛼|𝑢|2 − Re (𝑖𝑎𝑢)) 𝑑𝑥
+ 4 ∫ 𝑔󸀠 (|𝑢|2 ) Re (𝑢𝑢𝑥 )
× Re (𝑖𝑢𝑥 (𝑢𝑥𝑥 − 𝑛𝑢 + 𝑖𝛼𝑢
+𝑔 (|𝑢|2 ) 𝑢 − 𝑎)) 𝑑𝑥
Abstract and Applied Analysis
9
󵄨 󵄨2
+ 2 ∫ 𝑔󸀠 (|𝑢|2 ) 󵄨󵄨󵄨𝑢𝑥 󵄨󵄨󵄨
by (78) we can deduce that
× (Re (𝑖𝑢𝑥𝑥 𝑢) − 𝛼|𝑢|2 − Re (𝑖𝑎𝑢)) 𝑑𝑥
𝑑 󵄩󵄩 󵄩󵄩2
󵄩𝑛 󵄩 + 4 ∫ Re (𝑢𝑢𝑥𝑥 𝑛𝑥 ) 𝑑𝑥
𝑑𝑡 󵄩 𝑥 󵄩
+ 2 Re ∫ 𝑎𝑡 𝑢𝑥𝑥 𝑑𝑥.
󵄩 󵄩2
󵄨 󵄨2
+ 4 ∫ 󵄨󵄨󵄨𝑢𝑥 󵄨󵄨󵄨 𝑛𝑥 𝑑𝑥 + 2𝛽󵄩󵄩󵄩𝑛𝑥 󵄩󵄩󵄩
(73)
(80)
󸀠
2
+ 2 ∫ 𝑓 (|𝑢| ) (𝑢𝑥 𝑢 + 𝑢𝑢𝑥 ) 𝑛𝑥 𝑑𝑥
Then from (71) we have
𝑑 󵄩󵄩 󵄩󵄩2
󵄩 󵄩2
(󵄩𝑢 󵄩 + 𝐹) + 2𝛼 (󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 + 𝐹) = 𝐺,
𝑑𝑡 󵄩 𝑥𝑥 󵄩
(74)
− 2 ∫ 𝑏𝑥 𝑛𝑥 𝑑𝑥 = 0.
or
𝑑 󵄩󵄩 󵄩󵄩2
󵄩 󵄩2
󵄩 󵄩2
(󵄩𝑢 󵄩 + 𝐹) + 𝛼 (󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 + 𝐹) + 𝛼󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 = 𝐺 − 𝛼𝐹.
𝑑𝑡 󵄩 𝑥𝑥 󵄩
(75)
By Lemma 9 and Agmon inequality we have
‖𝑢(𝑡)‖2𝐻1 + ‖𝑢(𝑡)‖2∞ + ‖𝑛(𝑡)‖2𝐻 ≤ 2𝐶2 ,
∀𝑡 ≥ 𝑡2 .
(76)
In the following, we denote by 𝐶 = 𝐶(𝛼, 𝛽, 𝑓, 𝑔, 𝑌0 , 𝑎0𝑡 ). By
Lemma 7 and (76) we estimate the size of |𝐺 − 𝛼𝐹| to get
𝑑 󵄩󵄩 󵄩󵄩2
󵄩 󵄩2
󵄩 󵄩2
(󵄩𝑢 󵄩 + 𝐹) + 𝛼 (󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 + 𝐹) + 𝛼󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
𝑑𝑡 󵄩 𝑥𝑥 󵄩
(82)
2
+ Re ∫ 𝑔 (|𝑢|
(77)
) |𝑢|2𝑥 𝑢𝑢𝑥𝑥 𝑑𝑥
+ Re ∫ 𝑔 (|𝑢|2 ) 𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥 − Re ∫ 𝑎𝑥 𝑢𝑥𝑥 𝑑𝑥 = 0.
Because of
󵄩 󵄩7/4
+ 𝐶‖𝑢‖𝐿∞ ‖𝑛‖ 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 ‖𝑢‖1/4 + 𝐶
󵄩 󵄩7/4
󵄩 󵄩 󵄩 󵄩1/2
≤ 𝐶 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑛𝑥 󵄩󵄩󵄩 + 𝐶󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 + 𝐶
𝑑
Re ∫ 𝑖𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥 = 2 Re ∫ 𝑖𝑢𝑡𝑥 𝑢𝑥𝑥 𝑑𝑥,
𝑑𝑡
𝛼 󵄩󵄩 󵄩󵄩2 𝛽 󵄩󵄩 󵄩󵄩2
󵄩𝑢 󵄩 + 󵄩󵄩𝑛𝑥 󵄩󵄩 + 𝐶.
2 󵄩 𝑥𝑥 󵄩
2
(78)
2
+ ∫ 𝑓 (|𝑢| ) 𝑛𝑥𝑥 𝑑𝑥 − ∫ 𝑏𝑛𝑥𝑥 𝑑𝑥 = 0.
1 𝑑
Re ∫ 𝑖𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥 − Re ∫ 𝑛𝑥 𝑢𝑢𝑥𝑥 𝑑𝑥
2 𝑑𝑡
− Re ∫ 𝑛𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥 + Re ∫ 𝑖𝛼𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥
+ Re ∫ 𝑔󸀠 (|𝑢|2 ) |𝑢|2𝑥 𝑢𝑢𝑥𝑥 𝑑𝑥
Since
∫ |𝑢|2𝑥 𝑛𝑥𝑥 𝑑𝑥 = 2 ∫ Re (𝑢𝑢𝑥 𝑛𝑥𝑥 ) 𝑑𝑥
󵄨 󵄨2
= −2 ∫ Re (𝑢𝑢𝑥𝑥 𝑛𝑥 + 󵄨󵄨󵄨𝑢𝑥 󵄨󵄨󵄨 𝑛𝑥 ) 𝑑𝑥,
(83)
it holds that
Taking the inner product of (3) with 𝑛𝑥𝑥 in 𝐻, we see that
1 𝑑 󵄩󵄩 󵄩󵄩2
󵄩 󵄩2
2
󵄩𝑛 󵄩 + ∫ |𝑢|𝑥 𝑛𝑥𝑥 𝑑𝑥 − 𝛽󵄩󵄩󵄩𝑛𝑥 󵄩󵄩󵄩
2 𝑑𝑡 󵄩 𝑥 󵄩
(81)
Taking the real part of the inner product to (81) with 𝑢𝑥𝑥 in
𝐻, we have
󸀠
󵄩 󵄩3/2 󵄩 󵄩3/2
󵄩 󵄩 󵄩 󵄩1/2
≤ 𝐶 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑛𝑥 󵄩󵄩󵄩 ‖𝑛‖3/2 + 𝐶󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩
−
+ 𝑔󸀠 (|𝑢|2 ) |𝑢|2𝑥 𝑢 + 𝑔 (|𝑢|2 ) 𝑢𝑥 − 𝑎𝑥 (𝑥, 𝑡) = 0.
− Re ∫ 𝑛𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥 + Re ∫ 𝑖𝛼𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥
󵄨 󵄨2
+ 𝐶 ∫ 󵄨󵄨󵄨𝑢𝑥 󵄨󵄨󵄨 |𝑛𝑢| 𝑑𝑥 + 𝐶
≤
𝑖𝑢𝑡𝑥 + 𝑢𝑥𝑥𝑥 − 𝑛𝑥 𝑢 − 𝑛𝑢𝑥 + 𝑖𝛼𝑢𝑥
Re ∫ 𝑖𝑢𝑡𝑥 𝑢𝑥𝑥 − Re ∫ 𝑛𝑥 𝑢𝑢𝑥𝑥 𝑑𝑥
󵄨 󵄨2 󵄨 󵄨
󵄨 󵄨
≤ 𝐶 ∫ |𝑛|2 󵄨󵄨󵄨𝑢𝑥𝑥 󵄨󵄨󵄨 𝑑𝑥 + 𝐶 ∫ 󵄨󵄨󵄨𝑢𝑥 󵄨󵄨󵄨 󵄨󵄨󵄨𝑢𝑥𝑥 󵄨󵄨󵄨 𝑑𝑥
󵄩 󵄩
󵄩 󵄩 󵄩 󵄩2
≤ 𝐶 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 ‖𝑛‖24 + 𝐶 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩4
󵄩 󵄩2
+ 𝐶‖𝑢‖𝐿∞ ‖𝑛‖ 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩4 + 𝐶
From (2) we know that
+ Re ∫ 𝑔 (|𝑢|2 ) 𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥
(79)
− Re ∫ 𝑎𝑥 𝑢𝑥𝑥 𝑑𝑥 = 0.
(84)
10
Abstract and Applied Analysis
By (84) and (80), we find that
By (77) and (87) we deduce that
𝑑 󵄩󵄩 󵄩󵄩2
𝑑
󵄨 󵄨2
󵄩 󵄩2
󵄩𝑛 󵄩 + 2 Re ∫ 𝑖𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥 + 4 ∫ 󵄨󵄨󵄨𝑢𝑥 󵄨󵄨󵄨 𝑛𝑥 𝑑𝑥 + 2𝛽󵄩󵄩󵄩𝑛𝑥 󵄩󵄩󵄩
𝑑𝑡 󵄩 𝑥 󵄩
𝑑𝑡
∀𝑡 ≥ 𝑡2 ,
(89)
which has the same form with (51) in the proof of Lemma 9.
Similar to the study of (51), we can derive that
+ 2 ∫ 𝑓󸀠 (|𝑢|2 ) |𝑢|2𝑥 𝑛𝑥 𝑑𝑥
− 2 ∫ 𝑏𝑥 𝑛𝑥 𝑑𝑥 − 4 Re ∫ 𝑛𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥
𝐸 (𝑡2 ) ≤ 𝐶 (𝑅2 ) ,
+ 4𝛼 Re ∫ 𝑖𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥 + 4 Re ∫ 𝑔󸀠 (|𝑢|2 ) |𝑢|2𝑥 𝑢𝑢𝑥𝑥 𝑑𝑥
𝐸 (𝑡) ≤
2𝐶
,
𝛾
∀𝑡 ≥ 𝑡2∗ ,
(90)
where 𝑡2∗ = inf{𝑡 | 𝑡 ≥ 𝑡2∗ , 𝐶(𝑅2 )𝑒−𝛾(𝑡2∗ −𝑡0 ) ≤ 𝐶0 /𝛾} and
𝐶(𝑅2 ) = 𝐶(𝛼, 𝛽, 𝑓, 𝑔, 𝑌0 , 𝑎0𝑡 , 𝑅2 ) when ‖𝑊𝜏 ‖𝐻2 ×𝐻1 ≤ 𝑅2 . By
(72) we deduce that
+ 4 Re ∫ 𝑔 (|𝑢|2 ) 𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥 − 4 Re ∫ 𝑎𝑥 𝑢𝑥𝑥 𝑑𝑥 = 0.
(85)
That is,
󵄨
󵄨󵄨
󵄨󵄨𝐹 + 2 Re ∫ 𝑖𝑢 𝑢 𝑑𝑥󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
𝑥 𝑥𝑥
󵄨
󵄨
󵄩 󵄩
󵄩 󵄩2
≤ 2‖𝑢‖∞ ‖𝑛‖ 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 + 𝐶‖𝑢‖2∞ 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩
𝑑 󵄩󵄩 󵄩󵄩2
(󵄩𝑛 󵄩 + 2 Re ∫ 𝑖𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥)
𝑑𝑡 󵄩 𝑥 󵄩
󵄩 󵄩2
󵄩 󵄩
+ ‖𝑢‖∞ 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 + ‖𝑎‖𝐿2𝑏 (R;∑0 ) 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
(91)
󵄩 󵄩󵄩 󵄩
+ 2 󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 + 𝐶
󵄩 󵄩2
+ 2𝛽 (󵄩󵄩󵄩𝑛𝑥 󵄩󵄩󵄩 + 2 Re ∫ 𝑖𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥)
1 󵄩 󵄩2
󵄩 󵄩
≤ 𝐶 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 + 𝐶 ≤ 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 + 𝐶,
2
󵄨 󵄨2
= 4𝛽 Re ∫ 𝑖𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥 − 4 ∫ 󵄨󵄨󵄨𝑢𝑥 󵄨󵄨󵄨 𝑛𝑥 𝑑𝑥
− 2 ∫ 𝑓󸀠 (|𝑢|2 ) |𝑢|2𝑥 𝑛𝑥 𝑑𝑥 + 2 ∫ 𝑏𝑥 𝑛𝑥 𝑑𝑥
𝑑
𝐸 + 𝛾𝐸 ≤ 𝐶,
𝑑𝑡
and then by (88), (90), and (91) we deduce that
(86)
󵄩󵄩 󵄩󵄩2
󵄩 󵄩2
󵄩󵄩𝑢𝑥𝑥 󵄩󵄩 + 2󵄩󵄩󵄩𝑛𝑥 󵄩󵄩󵄩 ≤ 𝐶,
∀𝑡 ≥ 𝑡2∗ ,
(92)
+ 4 Re ∫ 𝑛𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥 − 4𝛼 Re ∫ 𝑖𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥
which concludes the proof by Lemma 9.
− 4 Re ∫ 𝑔󸀠 (|𝑢|2 ) |𝑢|2𝑥 𝑢𝑢𝑥𝑥 𝑑𝑥
4. Solutions for (2)∼(5)
− 4 Re ∫ 𝑔 (|𝑢|2 ) 𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥 + 4 Re ∫ 𝑎𝑥 𝑢𝑥𝑥 𝑑𝑥.
Theorem 11. Under assumptions of Lemma 10, for each 𝑊𝜏 ∈
𝐸0 , system (2)∼(5) has a unique global solution 𝑊(𝑥, 𝑡) ∈
𝐿∞ (𝜏, 𝑇; 𝐸0 ), ∀𝑇 > 𝜏.
Similar to (77), we estimate each term in (86), and then we
get
𝑑 󵄩󵄩 󵄩󵄩2
(󵄩𝑛 󵄩 + 2 Re ∫ 𝑖𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥)
𝑑𝑡 󵄩 𝑥 󵄩
Step 1. The existence of the solution.
By Galërkin’s method, we apply the following approximate solution:
󵄩 󵄩2
󵄩 󵄩2
+ 𝛽 (󵄩󵄩󵄩𝑛𝑥 󵄩󵄩󵄩 + 2 Re ∫ 𝑖𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥) + 𝛽󵄩󵄩󵄩𝑛𝑥 󵄩󵄩󵄩
𝑙
𝑊𝑙 (𝑥, 𝑡) = ∑𝑤𝑗𝑙 (𝑡) 𝜂𝑗 (𝑥) ,
󵄩 󵄩2 󵄩 󵄩
󵄩 󵄩
≤ 2𝛽 Re ∫ 𝑖𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥 + 𝐶 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 + 𝐶󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩4 󵄩󵄩󵄩𝑛𝑥 󵄩󵄩󵄩
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩
+ 𝐶 󵄩󵄩󵄩𝑛𝑥 󵄩󵄩󵄩 + 𝐶󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩∞ ‖𝑛‖ 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
(87)
to approach the solution of the problem (2)∼(5), where {𝜂𝑗 }∞
𝑗=1
is a orthogonal basis of 𝐻(Ω) satisfying −Δ𝜂𝑗 = 𝜆 𝑗 𝜂𝑗 (𝑗 =
1, 2, . . .). And 𝑊𝑙 (𝑥, 𝑡) satisfies
󵄨 󵄨2
𝑙
(𝑖𝑢𝑡𝑙 + 𝑢𝑥𝑥
− 𝑛𝑙 𝑢𝑙 + 𝑖𝛼𝑢𝑙 + 𝑔 (󵄨󵄨󵄨󵄨𝑢𝑙 󵄨󵄨󵄨󵄨 ) 𝑢𝑙 − 𝑎, 𝜂𝑗 ) = 0,
󵄩 󵄩
󵄩 󵄩
󵄩 󵄩3/4
+ 𝐶 󵄩󵄩󵄩𝑛𝑥 󵄩󵄩󵄩 + 𝐶‖𝑢‖1/4 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 ‖𝑛‖ 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩
󵄨 󵄨2
󵄨 󵄨2
(𝑛𝑡𝑙 + 𝛽𝑛𝑙 + 󵄨󵄨󵄨󵄨𝑢𝑙 󵄨󵄨󵄨󵄨𝑥 + 𝑓 (󵄨󵄨󵄨󵄨𝑢𝑙 󵄨󵄨󵄨󵄨 ) − 𝑏, 𝜂𝑗 ) = 0,
𝛼 󵄩󵄩 󵄩󵄩2 𝛽 󵄩󵄩 󵄩󵄩2
󵄩𝑢 󵄩 + 󵄩󵄩𝑛𝑥 󵄩󵄩 + 𝐶.
2 󵄩 𝑥𝑥 󵄩
2
(𝑊𝑙 (𝑥, 𝜏) , 𝜂𝑗 ) = (𝑊𝜏 , 𝜂𝑗 ) ,
Let 𝛾 = min{𝛼, 𝛽}, and
󵄩 󵄩2 󵄩 󵄩2
𝐸 = 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑛𝑥 󵄩󵄩󵄩 + 𝐹 + 2 Re ∫ 𝑖𝑢𝑥 𝑢𝑥𝑥 𝑑𝑥.
(93)
𝑗=1
󵄩 󵄩3/2 󵄩 󵄩1/2 󵄩 󵄩
󵄩 󵄩
≤ 𝐶 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 + 𝐶󵄩󵄩󵄩𝑢𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑢𝑥𝑥 󵄩󵄩󵄩 󵄩󵄩󵄩𝑛𝑥 󵄩󵄩󵄩
≤
Proof. We prove this theorem briefly by two steps.
(88)
(94)
𝑊𝑙 |𝜕Ω = 0,
where 𝑗 = 1, 2, . . . , 𝑙. Then (94) becomes an initial boundary
value problem of ordinary differential equations. According
Abstract and Applied Analysis
11
to the standard existence theory for the ordinary differential
equations, there exists a unique solution of (94). Similar to
[4, 22], by the a priori estimates in Section 3 we know that
{𝑊𝑙 }∞
𝑙=1 converges (weakly star) to a 𝑊(𝑥, 𝑡) which solves (2)∼
(5).
Step 2. The uniqueness of the solution.
Suppose 𝑊1 , 𝑊2 are two solutions of the problem (2)∼(5).
Let 𝑊 = 𝑊1 − 𝑊2 , then 𝑊(𝑥, 𝑡) = (𝑢(𝑥, 𝑡), 𝑛(𝑥, 𝑡)) satisfies
𝑖𝑢𝑡 + 𝑢𝑥𝑥 − 𝑛1 𝑢1 + 𝑛2 𝑢2 + 𝑖𝛼𝑢
𝑊|𝑡=𝜏 = 0,
in 𝐸0 ,
(98)
where 𝑊𝜎𝑘 (𝑡1 ) = (𝑢𝑘 (𝑡1 ), 𝑛𝑘 (𝑡1 )) = 𝑈𝜎𝑘 (𝑡1 , 𝜏)𝑊𝜏𝑘 , 𝑊𝜎 (𝑡1 ) =
(𝑢(𝑡1 ), and 𝑛(𝑡1 )) = 𝑈𝜎 (𝑡1 , 𝜏)𝑊𝜏 .
From (97) and Theorem 11 we know that
󵄩󵄩
󵄩󵄩
(99)
󵄩󵄩𝑊𝜏𝑘 󵄩󵄩𝐸0 ≤ 𝐶,
󵄩
󵄩
sup 󵄩󵄩󵄩󵄩𝑊𝜎𝑘 (𝑡)󵄩󵄩󵄩󵄩𝐸 ≤ 𝐶.
(100)
0
𝑡∈[𝜏,𝑇]
‖V‖∞ ≤ 𝐶‖V‖𝐻1 .
(95)
We see that
󵄩󵄩
󵄩
󵄩󵄩𝑊𝜎𝑘 (𝑡)󵄩󵄩󵄩 ≤ 𝐶,
󵄩
󵄩∞
∀0 ≤ 𝑡 ≤ 𝑇.
(101)
(102)
Note that
𝑊|𝜕Ω = 0.
󵄨 󵄨2
𝑖𝑢𝑘𝑡 = −𝑢𝑘𝑥𝑥 + 𝑛𝑘 𝑢𝑘 − 𝑖𝛼𝑢𝑘 − 𝑔 (󵄨󵄨󵄨𝑢𝑘 󵄨󵄨󵄨 ) 𝑢𝑘 + 𝑎𝑘 (𝑥, 𝑡) ,
Similar to [4, 5, 22], we can deduce that ‖𝑊‖ = 0.
󵄨 󵄨2
󵄨 󵄨2
𝑛𝑘𝑡 = −󵄨󵄨󵄨𝑢𝑘 󵄨󵄨󵄨𝑥 − 𝛽𝑛𝑘 − 𝑓 (󵄨󵄨󵄨𝑢𝑘 󵄨󵄨󵄨 ) + 𝑏𝑘 (𝑥, 𝑡) ,
5. Uniform Absorbing Set and
Uniform Attractor
From Theorem 11 we know that {𝑈𝜎∈∑ (𝑡, 𝜏)}, the family of
processes corresponding to (2)∼(5), is well defined. And
assumption (13) is satisfied.
Theorem 12. Under assumptions of Theorem 11, {𝑈𝜎∈∑ (𝑡, 𝜏)}
possesses a bounded uniformly absorbing set 𝐵0 in 𝐸0 .
Proof. Let 𝐵0 = {𝑊 ∈ 𝐸0 | ‖𝑊‖2𝐸0 ≤ 𝐶(‖𝑊𝜏 ‖𝐸0 , ‖𝑌0 ‖𝐿2𝑐 (R;∑0 ) )}.
From Theorem 11 we know that 𝐵0 is a bounded absorbing set
of the process 𝑈𝜎=𝑌0 .
On the other hand, from Assumption 1 we know that
for each 𝑌 ∈ ∑, ‖𝑌‖2𝐿2 (R;∑ ) ≤ ‖𝑌0 ‖2𝐿2 (R;∑ ) holds. Thus, the
0
0
𝑏
𝑏
solution of our system satisfies
󵄩 󵄩
‖𝑊‖𝐸0 ≤ 𝐶 (󵄩󵄩󵄩𝑊𝜏 󵄩󵄩󵄩𝐸0 , ‖𝑌‖𝐿2𝑏 (R;∑0 ) )
󵄩 󵄩 󵄩 󵄩
≤ 𝐶 (󵄩󵄩󵄩𝑊𝜏 󵄩󵄩󵄩𝐸0 , 󵄩󵄩󵄩𝑌0 󵄩󵄩󵄩𝐿2 (R;∑ ) ) .
𝑏
(96)
0
Theorem 13. Under assumptions of Theorem 12, {𝑈𝜎∈∑ (𝑡, 𝜏)}
admits a weakly compact uniform attractor A∑ .
Proof. To prove the existence of weakly compact uniform
attractor in 𝐸0 , from Lemma 4 and Theorems 11 and 12,
the only thing we should do is to verify that {𝑈𝜎∈∑ (𝑡, 𝜏)} is
(𝐸×∑, 𝐸)-continuous. Through the following proof, ⇀ means
∗
weak converges, and ⇀ means ∗ weak converges.
For any fixed 𝑡1 ≥ 𝜏 ∈ R, let
in 𝐸0 × ∑ .
(97)
(103)
(104)
and 𝜎𝑘 = (𝑎𝑘 (𝑥, 𝑡), 𝑏𝑘 (𝑥, 𝑡)) ∈ ∑. By (100) and (102), we find
that 𝜕𝑡 𝑊𝜎𝑘 (𝑡) ∈ 𝐿∞ (𝜏, 𝑇; 𝐻) and
󵄩󵄩󵄩𝜕 𝑊 (𝑡)󵄩󵄩󵄩
(105)
󵄩󵄩 𝑡 𝜎𝑘 󵄩󵄩𝐿∞ (𝜏,𝑇;𝐻) ≤ 𝐶.
̃ ≜
Due to Theorem 11 and (105), we know that there exist 𝑊(𝑡)
∞
(̃
𝑢(𝑡), 𝑛̃(𝑡)) ∈ 𝐿 (𝜏, 𝑇; 𝐸0 ), and subsequences of {𝑊𝜎𝑘 (𝑡)},
which are still denoted by {𝑊𝜎𝑘 (𝑡)}, such that
∗
̃ (𝑡)
𝑊𝜎𝑘 (𝑡) ⇀ 𝑊
∗
̃ (𝑡)
𝜕𝑡 𝑊𝜎𝑘 (𝑡) ⇀ 𝜕𝑡 𝑊
in 𝐿∞ (𝜏, 𝑇; 𝐸0 ) ,
in 𝐿∞ (𝜏, 𝑇; 𝐻) .
(106)
(107)
Besides, for ∀𝑡1 ∈ [𝜏, 𝑇], by (100) we know that there exists
𝑊0 ≜ (𝑢0 , 𝑛0 ) ∈ 𝐸0 , such that
𝑊𝜎𝑘 (𝑡1 ) ⇀ 𝑊0
So the set 𝐵0 = {𝑊 ∈ 𝐸0 | ‖𝑊‖2𝐸0 ≤ 𝜌02 ≜
𝐶(‖𝑊𝜏 ‖𝐸0 , ‖𝑌0 ‖𝐿2𝑏 (R;∑0 ) )} is a bounded uniformly absorbing
set of {𝑈𝜎∈∑ (𝑡, 𝜏)}.
(𝑊𝜏𝑘 , 𝜎𝑘 ) ⇀ (𝑊𝜏 , 𝜎)
𝑊𝜎𝑘 (𝑡1 ) ⇀ 𝑊𝜎 (𝑡1 )
By Agmon inequality,
󵄨 󵄨2
󵄨 󵄨2
+ 𝑔 (󵄨󵄨󵄨𝑢1 󵄨󵄨󵄨 ) 𝑢1 − 𝑔 (󵄨󵄨󵄨𝑢2 󵄨󵄨󵄨 ) 𝑢2 = 0,
󵄨 󵄨2 󵄨 󵄨2
𝑛𝑡 + 𝛽𝑛 + 󵄨󵄨󵄨𝑢1 󵄨󵄨󵄨𝑥 − 󵄨󵄨󵄨𝑢2 󵄨󵄨󵄨𝑥
󵄨 󵄨
󵄨 󵄨2
+ 𝑓 (󵄨󵄨󵄨𝑢1 󵄨󵄨󵄨 ) − 𝑓 (󵄨󵄨󵄨󵄨𝑢22 󵄨󵄨󵄨󵄨) = 0,
We will complete the proof if we deduce that
in 𝐸0 .
(108)
By (106) and a compactness embedding theorem, we claim
that
̃ (𝑡)
𝑢𝑘 (𝑡) 󳨀→ 𝑢
strongly in 𝐿2 (0, 𝑇; 𝐻) .
(109)
̃ is a solution of
In the following, we shall show that 𝑊(𝑡)
the problem (2)∼(5).
For ∀V ∈ 𝐻, ∀𝜓 ∈ 𝐶0∞ (𝜏, 𝑇), by (103) we find that
𝑇
𝑇
𝜏
𝜏
∫ (𝑖𝑢𝑘𝑡 , 𝜓 (𝑡) V) 𝑑𝑡 + ∫ (𝑢𝑘𝑥𝑥 , 𝜓 (𝑡) V) 𝑑𝑡
𝑇
𝑇
𝜏
𝜏
− ∫ (𝑛𝑘 𝑢𝑘 , 𝜓 (𝑡) V) 𝑑𝑡 + ∫ (𝑖𝛼𝑢𝑘 , 𝜓 (𝑡) V) 𝑑𝑡
𝑇
󵄨 󵄨2
+ ∫ (𝑔 (󵄨󵄨󵄨𝑢𝑘 󵄨󵄨󵄨 ) 𝑢𝑘 , 𝜓 (𝑡) V) 𝑑𝑡
𝜏
𝑇
− ∫ (𝑎𝑘 (𝑥, 𝑡) , 𝜓 (𝑡) V) 𝑑𝑡 = 0.
𝜏
(110)
12
Abstract and Applied Analysis
Since
Assumption (97) implies that
𝑇
𝑇
𝑢𝑘 (𝜏) = 𝑢𝜏𝑘 ⇀ 𝑢𝜏
̃ , 𝜓 (𝑡) V) 𝑑𝑡
∫ (𝑛𝑘 𝑢𝑘 , 𝜓 (𝑡) V) 𝑑𝑡 − ∫ (̃𝑛𝑢
𝜏
𝜏
in 𝐻.
(117)
Then by (116) and (117), we have
𝑇
̃ ) 𝑛𝑘 , 𝜓 (𝑡) V) 𝑑𝑡
= ∫ ((𝑢𝑘 − 𝑢
𝜏
(111)
𝑇
𝑇
− ∫ (𝑖̃
𝑢𝑥𝑥 , V) 𝜓 (𝑡) 𝑑𝑡
𝑢, V) 𝜓󸀠 (𝑡) 𝑑𝑡 + ∫ (̃
𝜏
𝑇
𝑢 (𝑛𝑘 − 𝑛̃) , 𝜓 (𝑡) V) 𝑑𝑡,
+ ∫ (̃
𝜏
𝑇
𝑇
𝜏
𝜏
̃ , V) 𝜓 (𝑡) 𝑑𝑡 + ∫ (𝑖𝛼̃
− ∫ (̃𝑛𝑢
𝑢, V) 𝜓 (𝑡) 𝑑𝑡
𝜏
by (102), (109), and (106),
𝑇
𝑇
𝜏
𝜏
̃ , V) 𝜓 (𝑡) 𝑑𝑡 − ∫ (𝑎 (𝑥, 𝑡) , V) 𝜓 (𝑡) 𝑑𝑡
+ ∫ (𝑔 (|̃
𝑢|2 ) 𝑢
𝑇
̃ ) 𝑛𝑘 , 𝜓 (𝑡) V) 𝑑𝑡
∫ ((𝑢𝑘 − 𝑢
𝜏
󵄩
󵄩
󵄩
󵄩 󵄩
̃ 󵄩󵄩󵄩󵄩𝐿2 (0,𝑇;𝐻) 󳨀→ 0,
≤ sup 󵄩󵄩󵄩𝑛𝑘 (𝑡)󵄩󵄩󵄩∞ 󵄩󵄩󵄩𝜓(𝑡)V󵄩󵄩󵄩𝐿2 (0,𝑇;𝐻) 󵄩󵄩󵄩𝑢𝑘 − 𝑢
= 𝑖 (𝑢𝜏 , V) .
(118)
0≤𝑡≤𝑇
While from (115) we know that
𝑇
𝑢 (𝑛𝑘 − 𝑛̃) , 𝜓 (𝑡) V) 𝑑𝑡
∫ (̃
𝑇
𝑇
− ∫ (𝑖̃
𝑢𝑥𝑥 , V) 𝜓 (𝑡) 𝑑𝑡
𝑢, V) 𝜓󸀠 (𝑡) 𝑑𝑡 + ∫ (̃
𝜏
𝜏
𝑇
= ∫ ((𝑛𝑘 − 𝑛̃) , 𝜓 (𝑡) Ṽ
𝑢) 𝑑𝑡 󳨀→ 0.
𝜏
(112)
Then we have
𝑇
𝑇
𝜏
𝜏
(113)
By using the similar methods to the other terms of (110), we
have
𝑇
𝜏
𝜏
𝑇
𝜏
𝜏
𝜏
𝜏
𝑇
𝑇
𝜏
𝜏
= 𝑖 (̃
𝑢 (𝜏) , V) .
(119)
It come from (118) and (119) that
(𝑢𝜏 , V) = (̃
𝑢 (𝜏) , V) ,
𝑢𝑡 , V) 𝜓 (𝑡) 𝑑𝑡 + ∫ (̃
𝑢𝑥𝑥 , V) 𝜓 (𝑡) 𝑑𝑡
∫ (𝑖̃
𝑇
𝑇
̃ , V) 𝜓 (𝑡) 𝑑𝑡 + ∫ (𝑖𝛼̃
− ∫ (̃𝑛𝑢
𝑢, V) 𝜓 (𝑡) 𝑑𝑡
̃ , V) 𝜓 (𝑡) 𝑑𝑡 − ∫ (𝑎 (𝑥, 𝑡) , V) 𝜓 (𝑡) 𝑑𝑡
+ ∫ (𝑔 (|̃
𝑢|2 ) 𝑢
̃ , V) 𝜓 (𝑡) 𝑑𝑡.
∫ (𝑛𝑘 𝑢𝑘 , 𝜓 (𝑡) V) 𝑑𝑡 󳨀→ ∫ (̃𝑛𝑢
𝑇
𝜏
𝑇
(114)
𝑇
̃ , V) 𝜓 (𝑡) 𝑑𝑡
+ ∫ (𝑔 (|̃
𝑢|2 ) 𝑢
̃ (𝜏) = 𝑢𝜏 .
𝑢
̃ (𝑡) = 𝑢 (𝑡) .
𝑢
𝑇
with 𝜓(𝜏) = 0, 𝜓(𝑡1 ) = 1, then
For ∀V ∈ 𝐻, ∀𝜓 ∈
repeating the procedure of proofs of (116)∼(119), by (108) we
deduce that
𝜏
Therefore, we obtain
𝑇
− ∫ (𝑖𝑢𝑘 , V) 𝜓 (𝑡) 𝑑𝑡 + ∫ (𝑢𝑘𝑥𝑥 , V) 𝜓 (𝑡) 𝑑𝑡
𝑇
𝜏
𝜏
𝑢𝑘 (𝑡1 ) ⇀ 𝑢 (𝑡1 )
in 𝐻2 (Ω) .
(124)
in 𝐻1 (Ω) .
(125)
Similarly, we can also deduce that
By (124) and (125), we derive (98). We complete the proof of
Theorem 13.
− ∫ (𝑛𝑘 𝑢𝑘 , V) 𝜓 (𝑡) 𝑑𝑡 + ∫ (𝑖𝛼𝑢𝑘 , V) 𝜓 (𝑡) 𝑑𝑡
𝑇
(123)
It comes from (108), (122), and (123) that
𝑛𝑘 (𝑡1 ) ⇀ 𝑛 (𝑡1 )
𝜏
𝑇
̃ (𝑡1 ) .
𝑢0 = 𝑢
(115)
which shows that (̃
𝑢, 𝑛̃, 𝑎(𝑡)) satisfies (2).
For ∀V ∈ 𝐻, ∀𝜓 ∈ 𝐶0∞ (𝜏, 𝑇) with 𝜓(𝑇) = 0, 𝜓(𝜏) = 1, by
(103) we find that
𝜏
(122)
𝐶0∞ (𝜏, 𝑡1 ),
− ∫ (𝑎 (𝑥, 𝑡) , V) 𝜓 (𝑡) 𝑑𝑡 = 0.
̃ 𝑥𝑥 − 𝑢
̃ 𝑛̃ + 𝑖𝛼̃
̃ = 𝑎 (𝑥, 𝑡) ,
𝑖̃
𝑢𝑡 + 𝑢
𝑢 + 𝑔 (|̃
𝑢|2 ) 𝑢
(121)
By (115) and (121), we have
𝜏
󸀠
(120)
And then
̃ , V) 𝜓 (𝑡) 𝑑𝑡 + ∫ (𝑖𝛼̃
𝑢, V) 𝜓 (𝑡) 𝑑𝑡
− ∫ (̃𝑛𝑢
𝑇
∀V ∈ 𝐻.
𝑇
󵄨 󵄨2
+ ∫ (𝑔 (󵄨󵄨󵄨𝑢𝑘 󵄨󵄨󵄨 ) 𝑢𝑘 , V) 𝜓 (𝑡) 𝑑𝑡 − ∫ (𝑎𝑘 (𝑥, 𝑡) , V) 𝜓 (𝑡) 𝑑𝑡
𝜏
𝜏
= 𝑖 (𝑢𝑘 (𝜏) , V) .
(116)
Acknowledgments
The authors were supported by the NSF of China (nos.
11071162, 11271050), the Project of Shandong Province Higher
Educational Science and Technology Program (no. J10LA09).
Abstract and Applied Analysis
References
[1] R. H. J. Grimshaw, “The modulation of an internal gravity-wave
packet, and the resonance with the mean motion,” vol. 56, no. 3,
pp. 241–266, 1976/77.
[2] D. R. Nicholson and M. V. Goldman, “Damped nonlinear
Schrödinger equation,” The Physics of Fluids, vol. 19, no. 10, pp.
1621–1625, 1976.
[3] D. J. Benney, “A general theory for interactions between short
and long waves,” vol. 56, no. 1, pp. 81–94, 1976/77.
[4] B. L. Guo, “The global solution for one class of the system of LS
nonlinear wave interaction,” Journal of Mathematical Research
and Exposition, vol. 7, no. 1, pp. 69–76, 1987.
[5] B. Guo, “The periodic initial value problems and initial value
problems for one class of generalized long-short type equations,” Journal of Engineering Mathematics, vol. 8, pp. 47–53,
1991.
[6] X. Du and B. Guo, “The global attractor for LS type equation in
R1 ,” Acta Mathematicae Applicatae Sinica, vol. 28, pp. 723–734,
2005.
[7] R. Zhang, “Existence of global attractor for LS type equations,”
Journal of Mathematical Research and Exposition, vol. 26, no. 4,
pp. 708–714, 2006.
[8] Y. Li, “Long time behavior for the weakly damped driven longwave-short-wave resonance equations,” Journal of Differential
Equations, vol. 223, no. 2, pp. 261–289, 2006.
[9] B. Guo and L. Chen, “Orbital stability of solitary waves of
the long wave-short wave resonance equations,” Mathematical
Methods in the Applied Sciences, vol. 21, no. 10, pp. 883–894,
1998.
[10] B. Guo and B. Wang, “Attractors for the long-short wave
equations,” Journal of Partial Differential Equations, vol. 11, no.
4, pp. 361–383, 1998.
[11] B. Guo and B. Wang, “The global solution and its long time
behavior for a class of generalized LS type equations,” Progress
in Natural Science, vol. 6, no. 5, pp. 533–546, 1996.
[12] D. Bekiranov, T. Ogawa, and G. Ponce, “On the well-posedness
of Benney’s interaction equation of short and long waves,”
Advances in Differential Equations, vol. 1, no. 6, pp. 919–937,
1996.
[13] D. Bekiranov, T. Ogawa, and G. Ponce, “Interaction equations
for short and long dispersive waves,” Journal of Functional
Analysis, vol. 158, no. 2, pp. 357–388, 1998.
[14] Z. Huo and B. Guo, “Local well-posedness of interaction
equations for short and long dispersive waves,” Journal of Partial
Differential Equations, vol. 17, no. 2, pp. 137–151, 2004.
[15] M. Tsutsumi and S. Hatano, “Well-posedness of the Cauchy
problem for Benney’s first equations of long wave short wave
interactions,” Funkcialaj Ekvacioj, vol. 37, no. 2, pp. 289–316,
1994.
[16] M. Tsutsumi and S. Hatano, “Well-posedness of the Cauchy
problem for the long wave-short wave resonance equations,”
Nonlinear Analysis. Theory, Methods & Applications A: Theory
and Methods, vol. 22, no. 2, pp. 155–171, 1994.
[17] R. Zhang and B. Guo, “Global solution and its long time
behavior for the generalized long-short wave equations,” Journal
of Partial Differential Equations, vol. 18, no. 3, pp. 206–218, 2005.
[18] V. V. Chepyzhov and M. I. Vishik, “Non-autonomous evolutionary equations with translation-compact symbols and their
attractors,” Comptes Rendus de l’Académie des Sciences, vol. 321,
no. 2, pp. 153–158, 1995.
13
[19] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of
Mathematical Physics, vol. 49 of American Mathematical Society Colloquium Publications, American Mathematical Society,
Providence, RI, USA, 2002.
[20] V. V. Chepyzhov and M. I. Vishik, “Attractors of nonautonomous dynamical systems and their dimension,” Journal de
Mathématiques Pures et Appliquées, vol. 73, no. 3, pp. 279–333,
1994.
[21] J. Bergh and J. Laöfstraöm, Interpolation Spaces, SpringerVerlag, Berlin, Germany, 1976.
[22] J. Xin, B. Guo, Y. Han, and D. Huang, “The global solution of the
(2+1)-dimensional long wave-short wave resonance interaction
equation,” Journal of Mathematical Physics, vol. 49, no. 7, pp.
073504–07350413, 2008.
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