Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 492847, 17 pages
http://dx.doi.org/10.1155/2013/492847
Research Article
Uniformly Random Attractor for the Three-Dimensional
Stochastic Nonautonomous Camassa-Holm Equations
Zhehao Huang and Zhengrong Liu
Department of Mathematics, South China University of Technology, Guangzhou 510640, China
Correspondence should be addressed to Zhengrong Liu; [email protected]
Received 25 November 2012; Revised 3 July 2013; Accepted 17 July 2013
Academic Editor: Ziemowit Popowicz
Copyright © 2013 Z. Huang and Z. 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.
We consider the uniformly random attractor for the three-dimensional stochastic nonautonomous Camassa-Holm equations in
the periodic box [0, 𝑙]3 in this paper. We associate with the concepts of uniform attractor and random attractor and produce the
concept of uniformly random attractor for a process. Then we establish the existence of the uniformly random attractor in 𝐷(𝐴1/2 )
and 𝐷(𝐴) for the equations.
∇ ⋅ 𝑢 = 0,
1. Introduction
𝑢 (𝑥, 0) = 𝑢0 (𝑥) .
The Camassa-Holm equation
𝑢𝑡 − 𝑢𝑥𝑥𝑡 + 3𝑢𝑢𝑥 − 2𝑢𝑥 𝑢𝑥𝑥 − 𝑢𝑢𝑥𝑥𝑥 = 0
models the unidirectional propagation of shallow water
waves over a flat bottom [1–4]. It has been paid a large number
of attentions due to its rich nonlinear phenomenology. It is
completely integrable [1], and it has stable solutions [5]. It
possesses the peakons 𝑢 = 𝑒|𝑥−𝑐𝑡| which has been proved
stable [6, 7]. It has been shown that (1) is locally well-posed
for initial data 𝑢0 ∈ 𝐻𝑠 (R) (𝑠 > 3/2) [8, 9]. There are a rich
variety of global solutions and blow-up solutions obtained in
[8–10]. The global existence of weak solutions, conservative
solutions, and disispative solutions was investigated in [6, 11,
12].
Following the Camassa-Holm (1), some generalized types
of the equation have been deeply considered by many authors,
for instance [13–18]. The authors in [19] considered the threedimensional Camassa-Holm equations subject to periodic
boundary conditions:
𝜕 2
(𝛼 𝑢 − 𝛼12 Δ𝑢) − ]Δ (𝛼02 𝑢 − 𝛼12 Δ𝑢) − 𝑢
𝜕𝑡 0
× (∇ × (𝛼02 𝑢 − 𝛼12 Δ𝑢)) +
1
∇𝑝 = 𝑓 (𝑥) ,
𝜌0
(2)
(1)
They established the global regularity of solutions of the
equation and provided the estimates for the Hausdorff and
fractal dimensions of the global attractor.
In [20] the authors analyzed the effects produced by
stochastic perturbations in the deterministic version of
the three-dimensional Lagrangian averaged Navier-Stokes-𝛼
model:
𝜕
(𝑢 − 𝛼Δ𝑢) + ] (𝐴𝑢 − 𝛼Δ (𝐴𝑢))
𝜕𝑡
+ (𝑢 ⋅ ∇) (𝑢 − 𝛼Δ𝑢) − 𝛼∇𝑢∗ ⋅ Δ𝑢 + ∇𝑝
= 𝐹 (𝑡, 𝑢) + 𝐺 (𝑡, 𝑢) 𝑊̇ (𝑡) ,
∇ ⋅ 𝑢 = 0,
𝑢 = 0,
(3)
in 𝐷 × (0, +∞) ,
𝐴𝑢 = 0,
on 𝜕𝐷 × (0, +∞) ,
𝑢 (0) = 𝑢0 ,
that is, the persistence of exponential stability as well as
possible stabilization effects produced by the noise.
2
Abstract and Applied Analysis
In [21] the authors proved the existence of the pullback
and forward attractors for three-dimensional Lagrangian
averaged Navier-Stokes-𝛼 model with delay:
𝜕
(𝑢 − 𝛼Δ𝑢) + ] (𝐴𝑢 − 𝛼Δ (𝐴𝑢))
𝜕𝑡
= 𝑓 (𝑡) + 𝐹 (𝑡, 𝑢𝑡 ) ,
𝑢 = 0,
𝑢 (𝑡) = 𝜙 (𝑡 − 𝜏) ,
(4)
on 𝜕𝐷 × (𝜏, +∞) ,
in (𝜏 − ℎ, 𝜏) ,
In [22] the author investigated the existence of finite
dimensional uniform attractor for three-dimensional nonautonomous Camassa-Holm equations with periodic boundary
conditions:
𝜕 2
(𝛼 𝑢 − 𝛼12 Δ𝑢) − ]Δ (𝛼02 𝑢 − 𝛼12 Δ𝑢)
𝜕𝑡 0
− 𝑢 × (∇ × (𝛼02 𝑢 − 𝛼12 Δ𝑢)) +
1
∇𝑝 = 𝑓 (𝑥, 𝑡) ,
𝜌0
(5)
∇ ⋅ 𝑢 = 0,
𝑢 (𝑥, 𝜏) = 𝑢𝜏 (𝑥) .
In [23, 24] the author studied the existence of uniform
attractor and convergence of the attractor as 𝜀 → 0+ for
a nonautonomous three-dimensional Lagrangian averaged
Navier-Stokes-𝛼 model with singularly oscillating external
force:
𝜕
(𝑢 − 𝛼Δ𝑢) + ] (𝐴𝑢 − 𝛼Δ (𝐴𝑢))
𝜕𝑡
+ (𝑢 ⋅ ∇) (𝑢 − 𝛼Δ𝑢) − 𝛼∇𝑢∗ ⋅ Δ𝑢 + ∇𝑝 = 𝑔𝜀 ,
𝑢 = 0,
in 𝐷 × (𝜏, +∞) ,
𝐴𝑢 = 0,
(8)
∇ ⋅ 𝑢 = 0,
𝑢 (𝑥, 𝜏) = 𝑢𝜏 (𝑥) ,
𝑢 (𝜏) = 𝑢0 .
∇ ⋅ 𝑢 = 0,
1
∇𝑝
𝜌0
= 𝑓 (𝑥, 𝑡) + 𝑄𝑊𝑡 ,
in 𝐷 × (𝜏, +∞) ,
𝐴𝑢 = 0,
𝜕 2
(𝛼 𝑢 − 𝛼12 Δ𝑢) − ]Δ (𝛼02 𝑢 − 𝛼12 Δ𝑢)
𝜕𝑡 0
− 𝑢 × (∇ × (𝛼02 𝑢 − 𝛼12 Δ𝑢)) +
+ (𝑢 ⋅ ∇) (𝑢 − 𝛼Δ𝑢) − 𝛼∇𝑢∗ ⋅ Δ𝑢 + ∇𝑝
∇ ⋅ 𝑢 = 0,
stochastic nonautonomous Camassa-Holm equation in the
periodic box [0, 𝑙]3 :
(6)
on 𝜕𝐷 × (𝜏, +∞) ,
𝑢 (𝜏) = 𝑢𝜏 ,
where
𝑔𝜀 (𝑥, 𝑦, 𝑧, 𝑡)
{𝑔 (𝑥, 𝑦, 𝑧, 𝑡) + 1 𝑔1 (𝑥, 𝑦, 𝑧, 𝑡) , 𝜀 ∈ (0, 1] , 𝜌 ∈ [0, 1) ,
={ 0
𝜀𝜌
𝜀 = 0.
{𝑔0 (𝑥, 𝑦, 𝑧, 𝑡) ,
(7)
Motivated by all their works, we initial our work to
investigate the equations perturbed by an additive noise. We
consider the following viscous version of three-dimensional
where 𝑝/𝜌0 = 𝜋/𝜌0 + 𝛼02 |𝑢|2 − 𝛼12 (𝑢 ⋅ Δ𝑢) is the modified
pressure, while 𝜋 is the pressure, ] > 0 is the constant
viscosity, and 𝜌0 > 0 is a constant density. The function
𝑓 is a given body forcing, and 𝛼0 > 0, 𝛼1 ≥ 0 are
scale parameters. 𝑊(𝑡) are two-sided real-valued Wiener
processes on a probability space which will be specified later.
𝑄 : R𝑛 → 𝐿2 ([0, 𝑙]3 )3 is a bounded linear operator. Also
observing that at the limiting case 𝛼0 = 1, 𝛼1 = 0, we obtain
the three-dimensional stochastic Navier-Stokes with periodic
boundary conditions.
Attractor is an important concept to describe the longtime behavior of solutions for a system in mathematical
physics [25–27]. The notion of uniform attractor parallelling
to that of the global autonomous systems has been systematically considered in [26]. In the approach presented in [27],
to construct the uniform attractor, instead of the associated
process {U𝜎 (𝑡, 𝜏), 𝑡 ≥ 𝜏, 𝜏 ∈ R}, one should consider a family
of processes {U𝜎 (𝑡, 𝑠)}, 𝜎 ∈ Σ, in some Banach space 𝐸 where
the functional parameter 𝜎0 (𝑠), 𝑠 ∈ R, is called the symbol
and Σ is the symbol space including 𝜎0 (𝑠). The approach
implies that the structure of uniform attractor is described
by the representation as a union of sections of all kernels of
the family of processes. The kernel is the set of all complete
trajectories of a process.
While in the real world, a system is usually uncertain due
to some external noise, which is random. The random effects
are considered not only as compensations for the defects in
some deterministic models but rather essential phenomena
[28–32]. In order to capture the essential dynamics of random
dynamical systems with wide fluctuations, the concept of
pullback random attractor was introduced in [29, 33, 34], as
an extension to stochastic systems of the theory of attractors
for deterministic systems in [25, 27, 35–43]. A pullback
random attractor A(𝜔) which can be constructed by a closed
random absorbing set K(𝜔) for an asymptotically compact
stochastic mapping S(𝑡, 𝑠, 𝜔) is given by
A (𝜔) = ⋂ ⋃S(𝑡, 𝜏, 𝜃𝜏−𝑡 𝜔, K(𝜃𝜏−𝑡 𝜔)),
𝑠≥𝜏 𝑡≥𝑠
(9)
where 𝜃𝑡 is the metric process on probability space. The
existence of random attractors for stochastic dynamical
systems has been investigated extensively by many authors
[29, 33, 34, 44–49]. In our paper, we associate with the
concepts of uniform attractor and random attractor together
Abstract and Applied Analysis
3
and give the concept of uniformly random attractor. Then we
consider (8) in an appropriate space and show that there is a
uniformly (with respect to 𝑓) random attractor A(𝜔) which
all solutions approach as 𝑡 → ∞. To our best knowledge,
the long-time dynamical behavior of the three-dimensional
stochastic nonautonomous Camassa-Holm equations has not
been discussed, and we believe that it is a significant work to
obtain a uniformly (with respect to 𝑓) random attractor for
the system.
The paper is organized as follows. In Section 2, we
present the abstract results describing the uniformly random
attractor and some relevant definitions. In Section 3, we give
some functional settings which are foundations for us to
obtain the existence of uniformly (with respect to 𝑓) random
attractor of (8). In Section 4, we convert (8) with an additive
noise to deterministic equations with random parameters
and define a process corresponding to the equations. In
Section 5, we obtain the existence of uniformly (with respect
to 𝑓) random attractors for (8) on the basis of the above
preparations.
Definition 5. A family of processes U𝜎 (𝑡, 𝜏, 𝜔), 𝜎 ∈ Σ, acting
in 𝐸 is said to be (𝐸×Σ, 𝐸)-continuous, if, for P a.e. 𝜔 ∈ Ω and
fixed 𝑡, 𝜏, 𝑡 ≥ 𝜏, the random mapping (𝑢, 𝜎) 󳨃→ U𝜎 (𝑡, 𝜏, 𝜔)𝑢
is continuous from 𝐸 × Σ into 𝐸.
2. Abstract Results
is said to be the random kernel section at time 𝑡 = 𝑠, 𝑠 ∈ R.
In this section, we associate with the concepts of uniform
attractor and random attractor and obtain the notion of uniformly random attractor. Let (𝐸, ‖ ⋅ ‖𝐸 ) be a separable Hilbert
space with the Borel 𝜎-algebra B(𝐸), and let (Ω, F, P) be a
probability space.
Let 𝐵𝑡 (𝜔) = ⋃𝜎∈Σ ⋃𝑠≥𝑡 U𝜎 (𝑠, 𝑡, 𝜃𝑡−𝑠 𝜔)𝐵(𝜃𝑡−𝑠 𝜔), where
𝐵(𝜔) is a random set. Denote that 𝐵(𝜔) is closure of the set
𝐵(𝜔) and R𝜏 = {𝑡 ∈ R | 𝑡 ≥ 𝜏}.
Definition 1. (𝜃𝑡 )𝑡∈R is called a measurable flow on probability
space (Ω, F) if 𝜃 : R × Ω → Ω is (R × F, F)-measurable, 𝜃0
is the identity on Ω, 𝜃𝑠+𝑡 = 𝜃𝑡 ∘ 𝜃𝑠 for all 𝑠, 𝑡 ∈ R, and 𝜃𝑡 P = P
for all 𝑡 ∈ R.
Definition 2. A random bounded set {𝐵(𝜔)}𝜔∈Ω of 𝐸, is called
tempered with respect to (𝜃𝑡 )𝑡∈R if for P a.e. 𝜔 ∈ Ω,
lim 𝑒−𝛽𝑡 𝑑 (𝐵 (𝜃−𝑡 𝜔)) = 0 ∀𝛽 > 0,
𝑡→∞
(10)
{U (𝑡, 𝜏, 𝜔)} 𝑢 (𝜏, 𝜔) = 𝑢 (𝑡, 𝜔) ,
∀𝑡 ≥ 𝜏, 𝑡, 𝜏 ∈ R.
Definition 3. A random set {K(𝜔)}𝜔∈Ω is called an absorbing
set of a stochastic mapping S(𝑡, 𝑠, 𝜔) in 𝐸 if for every random
bounded set 𝐵 and P a.e. 𝜔 ∈ Ω, there exists 𝑡𝐵 (𝜔) > 0 such
that
S (𝑡, 𝑠, 𝜃𝑠−𝑡 𝜔) 𝐵 (𝜃𝑠−𝑡 (𝜔)) ⊆ K (𝜔)
∀𝑡 ≥ 𝑡𝐵 (𝜔) .
(11)
Let {U(𝑡, 𝜏, 𝜔)} = {U(𝑡, 𝜏, 𝜔), 𝑡 ≥ 𝜏, 𝑡, 𝜏 ∈ R, 𝜔 ∈ Ω} be a
three-parameter family of mappings acting on 𝐸:
𝑡 ≥ 𝜏, 𝑡, 𝜏 ∈ R, 𝜔 ∈ Ω.
(12)
Definition 4. A three-parameter family of random mappings
{U(𝑡, 𝜏, 𝜔)} is said to be a process in 𝐸 if it is (B(R+ ) ×
B(R+ ) × F × B(𝐸), B(𝐸))-measurable and for P a.e. 𝜔 ∈ Ω
it satisfies
U (𝑡, 𝑠, 𝜃𝑠−𝜏 𝜔) ∘ U (𝑠, 𝜏, 𝜔) = U (𝑡, 𝜏, 𝜔) ,
∀𝑡 ≥ 𝑠 ≥ 𝜏, 𝜏 ∈ R,
U (𝜏, 𝜏, 𝜔) = 𝐼𝑑,
𝜏 ∈ R.
(13)
(14)
Definition 7. The random kernel N(𝜔) of the process
{U(𝑡, 𝜏, 𝜔)} consists of all bounded complete trajectories of
the process {U(𝑡, 𝜏, 𝜔)}:
N (𝜔) = {𝑢 (⋅, 𝜔) | 𝑢 (⋅, 𝜔) satisfies Definition 6
and ‖𝑢 (𝑠, 𝜔)‖𝐸 ≤ 𝑀𝑢 (𝜔) for 𝑠 ∈ R} .
(15)
Definition 8. The random set
N (𝑠, 𝜔) = {𝑢 (𝑠, 𝜔) | 𝑢 (⋅, 𝜔) ∈ N (𝜔)} ⊆ 𝐸
(16)
Definition 9. A random set W𝜏,Σ (𝐵)(𝜔) = ⋂𝑡≥𝜏 𝐵𝑡 (𝜔) is called
the uniformly (with respect to 𝜎 ∈ Σ) random (pullback)
omega-limit set of 𝐵(𝜔) which can be characterized as
follows, analogously to that for semigroups,
𝑦 ∈ W𝜏,Σ (𝐵) (𝜔) ⇐⇒ there are sequences {𝑥𝑛 } ⊂ 𝐵 (𝜃𝜏−𝑡𝑛 𝜔) ,
{𝜎𝑛 } ⊂ Σ, {𝑡𝑛 } ⊂ R𝜏 ,
such that 𝑡𝑛 󳨀→ +∞ and
U𝜎𝑛 (𝑡𝑛 , 𝜏, 𝜃𝜏−𝑡𝑛 𝜔) 𝑥𝑛 󳨀→ 𝑦
where 𝑑(𝐵) = sup𝑥∈𝐸 ‖𝑥‖𝐸 .
U (𝑡, 𝜏, 𝜔) : 𝐸 󳨀→ 𝐸,
Definition 6. A random curve 𝑢(𝑠, 𝜔), 𝑠 ∈ R, is said to be
a complete trajectory of the process {U(𝑡, 𝜏, 𝜔)} if for P a.e.
𝜔∈Ω
(𝑛 󳨀→ ∞) .
(17)
Let 𝐵(𝜔) ∈ B(𝐸), and its Kuratowski measure of
noncompactness 𝜅(𝐵) is defined by
𝜅 (𝐵) = inf {𝛿 > 0 | 𝐵 admits a finit
covering by sets of diameter ≤ 𝛿} .
(18)
Definition 10. A family of processes {U𝜎 (𝑡, 𝜏, 𝜔)}, 𝜎 ∈ Σ,
is said to be uniformly (with respect to 𝜎 ∈ Σ) random
(pullback) omega-limit compact if, for P a.e. 𝜔 ∈ Ω and
any 𝜏 ∈ R, the set 𝐵𝑡 (𝜔) is bounded for every 𝑡 and
lim𝑡 → ∞ 𝜅(𝐵𝑡 (𝜔)) = 0.
We now present a method to verify the uniformly (with
respect to 𝜎 ∈ Σ) random (pullback) omega-limit compactness.
Definition 11. A family of processes {U𝜎 (𝑡, 𝜏, 𝜔)}, 𝜎 ∈ Σ, is
said to satisfy uniformly (with respect to 𝜎 ∈ Σ) condition
4
Abstract and Applied Analysis
(C) if, for P a.e. 𝜔 ∈ Ω and any fixed 𝜏 ∈ R, tempered set
𝐵(𝜔) ∈ B(𝐸), 𝜀 > 0, there exist 𝑡0 = 𝑡(𝜏, 𝐵, 𝜀, 𝜔) ≥ 𝜏 and a
finite dimensional subspace 𝐸1 of 𝐸 such that
Definition 13 (cf. [41]). A function 𝜑 ∈ 𝐿2loc (R; 𝐸) is said to be
normal if, for any 𝜀 > 0, there exists 𝛿 > 0 such that
sup ∫
(i) Π(⋃𝜎∈Σ ⋃𝑡≥𝑡0 U𝜎 (𝑡, 𝜏, 𝜃𝜏−𝑡 𝜔)𝐵(𝜃𝜏−𝑡 𝜔)) is bounded,
and
(ii) ‖(𝐼 − Π)(⋃𝜎∈Σ ⋃𝑡≥𝑡0 U𝜎 (𝑡, 𝜏, 𝜃𝜏−𝑡 𝜔)𝑥)‖ ≤ 𝜀, ∀𝑥 ∈
𝐵(𝜃𝜏−𝑡 𝜔),
where Π : 𝐸 → 𝐸1 is a bounded projector.
Therefore, we have the following results.
Theorem 12. Let Σ be a metric space, and let 𝑇(𝑡) be a
continuous invariant semigroup 𝑇(𝑡)Σ = Σ on Σ. A family
of processes {U(𝑡, 𝜏, 𝜔)}, 𝜎 ∈ Σ, acting in 𝐸 is (𝐸 × Σ, 𝐸)continuous (weakly) and possesses the compact uniformly (with
respect to 𝜎 ∈ Σ) random attractor AΣ (𝜔) satisfying
AΣ (𝜔) = W0,Σ (𝐵0 (𝜔)) = W𝜏,Σ (𝐵0 (𝜔))
= ⋃ N𝜎 (0, 𝜔) ,
(19)
∀𝜏 ∈ R,
𝑡∈R
𝑡+𝛿
𝑡
󵄩2
󵄩󵄩
󵄩󵄩𝜑(𝑠)󵄩󵄩󵄩𝐸 𝑑𝑠 ≤ 𝜀.
(24)
We denote by 𝐿2𝑛 (R; 𝐸) the set of all normal functions in
Obviously, 𝐿2𝑛 (R; 𝐸) ⊂ 𝐿2𝑏 (R; 𝐸), and it is proved
in [41] that 𝐿2𝑛 (R; 𝐸) is a closed subset of 𝐿2𝑏 (R; 𝐸). Let a
fixed symbol 𝜎0 (𝑠) = 𝑓0 (𝑠) = 𝑓0 (⋅, 𝑠) be normal functions in
𝐿2loc (R; 𝐸). That is, the family of translation {𝑓0 (𝑠 + 𝜂), 𝜂 ∈
R} forms a normal function set in 𝐿2loc ([𝑇1 , 𝑇2 ]; 𝐸), where
[𝑇1 , 𝑇2 ] is an arbitrary interval of the time axis R. Therefore,
𝐿2loc (R; 𝐸).
Σ (𝜎0 ) = Σ (𝑓0 ) = {𝑓0 (𝑥, 𝑠 + 𝜂) | 𝜂 ∈ R}𝐿2
loc
(R;𝐸) .
(25)
After integrating (8), one can easily see that
𝑑
(𝛼2 𝑢 − 𝛼12 Δ𝑢) 𝑑𝑥 = ∫ 3 𝑓𝑑𝑥 + ∫ 3 𝑄𝑊𝑡 𝑑𝑥.
∫
𝑑𝑡 [0,𝑙]3 0
[0,𝑙]
[0,𝑙]
(26)
𝜎∈Σ
if it
(i) has a bounded uniformly (with respect to 𝜎 ∈ Σ)
random absorbing set 𝐵0 (𝜔), and
(ii) satisfies uniformly (with respect to 𝜎 ∈ Σ) condition
(C).
Moreover, if 𝐸 is a uniformly convex Banach space, then
the converse is true.
3. Functional Setting
We consider the probability space (Ω, F, P) where
𝑇
Ω = {𝜔 = (𝜔1 , 𝜔2 , . . . , 𝜔𝑛 ) ∈ 𝐶 (R, R𝑛 ) : 𝜔 (0) = 0} . (20)
F is the Borel 𝜎-algebra induced by the compact-open
topology of Ω and P the corresponding Wiener measure on
(Ω, F). Then we will identify 𝑊(𝑡) with
𝑇
𝑊 (𝑡) = (𝑤1 (𝑡) , 𝑤2 (𝑡) , . . . , 𝑤𝑛 (𝑡)) = 𝜔 (𝑡)
for 𝑡 ∈ R.
(21)
Define the time shift by
𝜃𝑡 𝜔 (⋅) = 𝜔 (⋅ + 𝑡) − 𝜔 (𝑡) ,
𝜔 ∈ Ω, 𝑡 ∈ R.
(22)
Then {𝜃𝑡 }𝑡∈R is a family of measure preserving transformations on probability space (Ω, F, P) in Definition 1.
Next we define a symbol space Σ(𝜎0 ) for (8). We assume
that the function 𝑓(⋅, 𝑡) =: 𝑓(𝑡) ∈ 𝐿2loc (R; 𝐸) is translation
bounded. That is, for 𝑓(𝑠) ∈ 𝐿2𝑏 (R; 𝐸), we have
𝑡+1
󵄩󵄩 󵄩󵄩2
󵄩 󵄩2
󵄩󵄩𝑓󵄩󵄩𝐿2 = 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿2 (R;𝐸) = sup ∫
𝑏
𝑏
𝑡
𝑡∈R
󵄩2
󵄩󵄩
󵄩󵄩𝑓(𝑠)󵄩󵄩󵄩𝐸 𝑑𝑠 < ∞.
(23)
On the other hand, because of the spatial periodicity of
solution, we have ∫[0,𝑙]3 Δ𝑢𝑑𝑥 = 0. Then we have
𝑑
𝛼2 𝑢𝑑𝑥 = ∫ 3 𝑓𝑑𝑥 + ∫ 3 𝑄𝑊𝑡 𝑑𝑥.
∫
𝑑𝑡 [0,𝑙]3 0
[0,𝑙]
[0,𝑙]
(27)
That is, the mean of solution is invariant provided that the
means of the forcing term and the perturbing term are zero. In
this paper, we will consider the forcing term, perturbing term
and initial values with spatial means that are zero. That is, we
assume ∫[0,𝑙]3 𝑓𝑑𝑥 = 0, ∫[0,𝑙]3 𝑄𝑊𝑡 𝑑𝑥 = 0, and ∫[0,𝑙]3 𝑢𝜏 𝑑𝑥 = 0.
Next we introduce some essential functional spaces.
(i) We denote by V = {𝜑 : 𝜑 a vector-valued
trigonometric polynomial defined on [0, 𝑙]3 , such that
∇ ⋅ 𝜑 = 0 and ∫[0,𝑙]3 𝜑(𝑥)𝑑𝑥 = 0}, and we let
𝐻 and 𝑉 be the closure of V in 𝐿2 ([0, 𝑙]3 )3 and in
𝐻1 ([0, 𝑙]3 )3 , respectively. We can observe that 𝐻⊥ , the
orthogonal complement of 𝐻 in 𝐿2 ([0, 𝑙]3 )3 , is {∇𝑝 :
𝑝 ∈ 𝐻1 ([0, 𝑙]3 )} (cf. [38, 41]).
(ii) We denote by 𝑃 : 𝐿2 ([0, 𝑙]3 )3 → 𝐻 the 𝐿2 orthogonal
projection, usually referred to as Helmholtz-Leray
projector, and by 𝐴 = −𝑃Δ the Stokes operator with
domain 𝐷(𝐴) = 𝐻2 ([0, 𝑙]3 )3 ⋂ 𝑉. Note that, in the
case of periodic boundary condition, 𝐴 = −Δ|𝐷(𝐴) is
a self-adjoint positive operator with compact inverse.
Hence, the space 𝐻 has an orthogonal basis {𝑒𝑗 }∞
𝑗 of
eigenfunctions of 𝐴, that is, 𝐴𝑒𝑗 = 𝜆 𝑗 𝑒𝑗 , with
0 < 𝜆1 ≤ 𝜆2 ≤ ⋅ ⋅ ⋅ ≤ 𝜆𝑗 ≤ ⋅ ⋅ ⋅ ,
𝜆 𝑗 → +∞, as 𝑗 → ∞.
(28)
In fact, these eigenvalues have the form |𝑘|2 4𝜋/𝐿2
with 𝑘 ∈ Z3 /{0}.
Abstract and Applied Analysis
5
(iii) We denote by (⋅, ⋅) the 𝐿2 ([0, 𝑙]3 )3 inner product
and by ‖ ⋅ ‖2 the corresponding 𝐿2 ([0, 𝑙]3 )3 norm. By
virtue of Poincaré inequality one can show that there
is a constant 𝑐 > 0, such that
𝑐‖𝐴𝑢‖2 ≤ ‖𝑢‖𝐻2 ≤ 𝑐−1 ‖𝐴𝑢‖2
for every 𝑢 ∈ 𝐷 (𝐴) ,
󵄩
󵄩
󵄩
󵄩
𝑐󵄩󵄩󵄩󵄩𝐴1/2 𝑢󵄩󵄩󵄩󵄩2 ≤ ‖𝑢‖𝐻1 ≤ 𝑐−1 󵄩󵄩󵄩󵄩𝐴1/2 𝑢󵄩󵄩󵄩󵄩2
(29)
for every 𝑢 ∈ 𝑉.
Moreover, one can show that 𝑉 = 𝐷(𝐴1/2 ) (cf. [38,
42]). We denote by 𝑉󸀠 the dual of 𝑉. Hereafter 𝑐
will denote a generic scale invariant positive constant
which is independent of the physical parameters in
the equation.
(iv) Following the notation for the Navier-Stokes equations we denote yhat 𝐵(𝑢, V) = 𝑃[(𝑢 ⋅ ∇)V], and we
set 𝐵(V)𝑢 = 𝐵(𝑢, V) for every 𝑢, V ∈ 𝑉. That is, for
every fixed V ∈ 𝑉, 𝐵(V) is a linear operator acting on
𝑢. Note that
(𝐵 (𝑢, V) , 𝑤) = − (𝐵 (𝑢, 𝑤) , V)
for every 𝑢, V, 𝑤 ∈ 𝑉.
(30)
̃ V) = −𝑃(𝑢 × (∇ × V)) for every 𝑢, V ∈
We also denote that 𝐵(𝑢,
𝑉. Using the identity
and in particular
⟨𝐵̃ (𝑢, V) , 𝑢⟩𝑉󸀠 ≡ 0 for every 𝑢, V ∈ 𝑉.
Furthermore, one has
󵄨󵄨
󵄨󵄨
󵄨 ≤ 𝑐‖𝑢‖2 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝐴1/2 𝑤󵄩󵄩󵄩󵄩1/2 ‖𝐴𝑤‖1/2
󵄨󵄨⟨𝐵̃ (𝑢, V) , 𝑤⟩
󸀠󵄨
2
󵄨󵄨
󵄩
󵄩2 󵄩
󵄩2
𝐷(𝐴) 󵄨󵄨
(37)
for every 𝑢 ∈ 𝐻, V ∈ 𝑉, and 𝑤 ∈ 𝐷(𝐴) and by symmetry one
has
󵄨󵄨
󵄨󵄨
󵄨󵄨⟨𝐵̃ (𝑢, V) , 𝑤⟩
󵄨󵄨 ≤ 𝑐󵄩󵄩󵄩󵄩𝐴1/2 𝑢󵄩󵄩󵄩󵄩1/2 ‖𝐴𝑢‖1/2 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩 ‖𝑤‖2
2 󵄩
󵄨󵄨
󵄩
󵄩2
󵄩2
𝐷(𝐴)󸀠 󵄨󵄨
(38)
for every 𝑢 ∈ 𝐷(𝐴), V ∈ 𝑉, and 𝑤 ∈ 𝐻. Also
󵄨󵄨
󵄨󵄨
󵄨󵄨⟨𝐵̃ (𝑢, V) , 𝑤⟩
󵄨󵄨
󵄨󵄨
𝐷(𝐴)󸀠 󵄨󵄨
󵄩󵄩 1/2 󵄩󵄩1/2
󵄩𝐴 𝑢󵄩󵄩 ‖V‖2 ‖𝐴𝑤‖2
≤ 𝑐 (‖𝑢‖1/2
2 󵄩
󵄩
󵄩2
for every 𝑢 ∈ 𝑉, V ∈ 𝐻, and 𝑤 ∈ 𝐷(𝐴). In addition,
󵄨󵄨
󵄨󵄨
󵄨󵄨⟨𝐵̃ (𝑢, V) , 𝑤⟩
󵄨󵄨
󵄨󵄨
𝐷(𝐴)󸀠 󵄨󵄨
󵄩
󵄩󵄩 1/2 󵄩󵄩
󵄩1/2
󵄩𝐴 𝑤󵄩󵄩
≤ 𝑐 (󵄩󵄩󵄩󵄩𝐴1/2 𝑢󵄩󵄩󵄩󵄩2 ‖𝐴𝑢‖1/2
2 ‖V‖2 󵄩
󵄩
󵄩2
for every 𝑢 ∈ 𝐷(𝐴), V ∈ 𝐻, and 𝑤 ∈ 𝑉.
one can easily show that
(𝐵̃ (𝑢, V) , 𝑤) = (𝐵 (𝑢, V) , 𝑤) − (𝐵 (𝑤, V) , 𝑢)
= (𝐵 (V) 𝑢 − 𝐵∗ (V) 𝑢, 𝑤)
(32)
for every 𝑢, V, 𝑤 ∈ 𝑉, where 𝐵∗ (V) denotes the adjoint
operator of the linear operator 𝐵(V) defined above. As a result
we have
𝐵̃ (𝑢, V) = (𝐵 (V) − 𝐵∗ (V)) 𝑢
(40)
󵄩󵄩 1/2 󵄩󵄩1/2
󵄩𝐴 𝑤󵄩󵄩 )
+‖V‖2 ‖𝐴𝑢‖2 ‖𝑤‖1/2
2 󵄩
󵄩
󵄩2
(31)
𝑗=1
(39)
󵄩
󵄩󵄩
󵄩1/2
+‖V‖2 󵄩󵄩󵄩󵄩𝐴1/2 𝑢󵄩󵄩󵄩󵄩2 󵄩󵄩󵄩󵄩𝐴1/2 𝑤󵄩󵄩󵄩󵄩2 ‖𝐴𝑤‖1/2
2 )
3
(𝑏 ⋅ ∇) 𝑎 + ∑𝑎𝑗 ∇𝑏𝑗 = −𝑏 × (∇ × 𝑎) + ∇ (𝑎 ⋅ 𝑏) ,
(36)
for every 𝑢, V ∈ 𝑉.
(33)
The next lemma will present some properties of the
̃
bilinear operator 𝐵.
Lemma 14 (cf. [19]). The operator 𝐵̃ can be extended continuously from 𝑉 × 𝑉 with values in 𝑉󸀠 , and it satisfies
󵄨
󵄩󵄩 1/2 󵄩󵄩1/2 󵄩󵄩 1/2 󵄩󵄩 󵄩󵄩 1/2 󵄩󵄩
󵄨󵄨 ̃
󵄩𝐴 𝑢󵄩󵄩 󵄩󵄩𝐴 V󵄩󵄩 󵄩󵄩𝐴 𝑤󵄩󵄩 ,
󵄨󵄨⟨𝐵 (𝑢, V) , 𝑤⟩ 󸀠 󵄨󵄨󵄨 ≤ 𝑐‖𝑢‖1/2
2 󵄩
󵄩
󵄩2 󵄩
󵄩2 󵄩
󵄩2
󵄨
𝑉󵄨
󵄩
󵄨󵄨 ̃
󵄨
󵄩󵄩 1/2 󵄩󵄩1/2
󵄩󵄩
󵄩
󵄨󵄨⟨𝐵 (𝑢, V) , 𝑤⟩ 󸀠 󵄨󵄨󵄨 ≤ 𝑐󵄩󵄩󵄩𝐴1/2 𝑢󵄩󵄩󵄩 󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩 ‖𝑤‖1/2
󵄩𝐴 𝑤󵄩󵄩
2 󵄩
󵄩
󵄨
󵄩
󵄩2 󵄩
󵄩2
󵄩2
𝑉󵄨
(34)
We now apply the result in Section 2 to the stochastic
nonautonomous Camassa-Holm equations. To associate a
family of processes U𝜎 (𝑡, 𝑠, 𝜔) with the stochastic equations
over (Σ, 𝑇(𝑡)) and (Ω, F, P, (𝜃𝑡 )𝑡∈R ), we need to convert
the stochastic equations with a random additive term into
deterministic equations with a random parameter.
First we define the bounded linear operator 𝑄 in (8) as
follows:
𝑛
𝑄𝑥 = ∑𝑥𝑗 ℎ𝑗 ,
𝑗=1
𝑇
𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ R𝑛 ,
(41)
where ℎ𝑗 ∈ 𝐷(𝐴), 𝑗 = 1, 2, . . . , 𝑛.
Given 𝑗 = 1, 2, . . . , 𝑛, consider the Ornstein-Uhlenbeck
equation:
𝑑𝑦𝑗 + 𝜇𝑦𝑗 𝑑𝑡 = 𝑑𝑤𝑗 (𝑡) .
(42)
One can easily check that a solution to (42) is given by
for every 𝑢, V, 𝑤 ∈ 𝑉. Moreover,
⟨𝐵̃ (𝑢, V) , 𝑤⟩𝑉󸀠 = −⟨𝐵̃ (𝑤, V) , 𝑢⟩𝑉󸀠 ,
4. Stochastic Nonautonomous
Camassa-Holm Equations
for every 𝑢, V, 𝑤 ∈ 𝑉
(35)
𝑦𝑗 (𝑡) = 𝑦𝑗 (𝜃𝑡 𝜔𝑗 ) = −𝜇 ∫
0
−∞
𝑒𝜇𝜏 (𝜃𝑡 𝜔𝑗 ) (𝜏) 𝑑𝜏,
𝑡 ∈ R.
(43)
6
Abstract and Applied Analysis
It is known that the random variable |𝑦𝑗 (𝜔𝑗 )| is tempered and
that 𝑦𝑗 (𝜃𝑡 𝜔𝑗 ) is P a.e is continuous. Now we put 𝑧(𝜃𝑡 𝜔) = (𝛼02 +
𝛼12 𝐴)−1 ∑𝑚
𝑗=1 ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 ). By (42) we have
𝑑 (𝛼02 𝑧 (𝜃𝑡 𝜔) + 𝛼12 𝐴𝑧 (𝜃𝑡 𝜔))
+ 𝜇 (𝛼02 𝑧 (𝜃𝑡 𝜔) 𝑑𝑡 + 𝛼12 𝐴𝑧 (𝜃𝑡 𝜔)) 𝑑𝑡
By a Galerkin method as in [19], it can be proved that if
𝑓(𝑥, 𝑡) ∈ 𝐿2loc ((0, 𝑇); 𝐻) and V𝜏 ∈ 𝑉, for P a.e. 𝜔 ∈ Ω, (48)
has a unique solution satisfying for any 𝑇 > 𝜏
V (𝑡, 𝜔, V𝜏 ) ∈ 𝐶 ([𝜏, 𝑇) ; 𝑉) ⋂ 𝐿2 ([𝜏, 𝑇) ; 𝐷 (𝐴)) ,
𝑑V
∈ 𝐿2 ([𝜏, 𝑇) ; 𝐻)
𝑑𝑡
(44)
𝑛
= ∑ℎ𝑗 𝑑𝑤𝑗 (𝑡) .
and such that, for almost all 𝑡 ∈ [𝜏, 𝑇) and for any 𝑤 ∈ 𝐷(𝐴),
𝑗=1
Employing Cauchy-Schwarz’s inequality, we get
⟨
󵄩2 󵄩
󵄩2 󵄩 1/2
󵄩2
󵄩󵄩
󵄩󵄩𝑧 (𝜃𝑡 𝜔)󵄩󵄩󵄩2 + 󵄩󵄩󵄩󵄩𝐴 𝑧 (𝜃𝑡 𝜔)󵄩󵄩󵄩󵄩2 + 󵄩󵄩󵄩𝐴𝑧 (𝜃𝑡 𝜔)󵄩󵄩󵄩2
𝑛
󵄨
󵄨2
≤ 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 ,
(45)
𝑗=1
To show that (8) corresponds to a process {U𝜎 (𝑡, 𝜏, 𝜔)},
we let V = 𝑢 − 𝑧(𝜃𝑡 𝜔), where 𝑢 is a solution of (8). Then for
V(𝑡, 𝜔), we have
𝜕 2
(𝛼 V + 𝛼12 𝐴V)
𝜕𝑡 0
− (V + 𝑧 (𝜃𝑡 𝜔))
× (∇ × (𝛼02 V + 𝛼12 𝐴V + 𝛼02 𝑧 (𝜃𝑡 𝜔) + 𝛼12 𝐴𝑧 (𝜃𝑡 𝜔)))
1
∇𝑝
𝜌0
= 𝑓 (𝑥, 𝑡) + 𝜇𝛼02 𝑧 (𝜃𝑡 𝜔) + 𝜇𝛼12 𝐴𝑧 (𝜃𝑡 𝜔)
(46)
(47)
We apply 𝑃 to (46) and use the notation in Section 3 to obtain
the equivalent system of equations
𝜕 2
(𝛼 V + 𝛼12 𝐴V)
𝜕𝑡 0
+ ]𝐴 (𝛼02 V + 𝛼12 𝐴V + 𝛼02 𝑧 (𝜃𝑡 𝜔) + 𝛼12 𝐴𝑧 (𝜃𝑡 𝜔))
+ 𝐵̃ (V + 𝑧 (𝜃𝑡 𝜔) , 𝛼02 V + 𝛼12 𝐴V
(48)
+𝛼02 𝑧 (𝜃𝑡 𝜔) + 𝛼12 𝐴𝑧 (𝜃𝑡 𝜔))
= 𝑃𝑓 (𝑥, 𝑡) + 𝜇𝛼02 𝑧 (𝜃𝑡 𝜔) + 𝜇𝛼12 𝐴𝑧 (𝜃𝑡 𝜔)
satisfying the initial condition
V (𝑥, 𝜏, 𝜔) = V𝜏 (𝑥, 𝜔) .
+𝛼02 𝑧 (𝜃𝑡 𝜔) + 𝛼12 𝐴𝑧 (𝜃𝑡 𝜔)) , 𝑤⟩𝐷(𝐴)󸀠
+ ⟨𝐵̃ (V +
𝑧 (𝜃𝑡 𝜔) , 𝛼02 V
+
(51)
𝛼12 𝐴V
+𝛼02 𝑧(𝜃𝑡 𝜔) + 𝛼12 𝐴𝑧(𝜃𝑡 𝜔)) , 𝑤⟩𝐷(𝐴)󸀠
Now for any 𝑓(𝑥, 𝑡) ∈ Σ(𝑓0 ), (48) with 𝑓 instead of 𝑓0
possesses a corresponding process {U𝑓 (𝑡, 𝜏, 𝜔)} acting on 𝑉.
It is analogous to the proof in [27] to prove that, for P a.e
𝜔 ∈ Ω, the family {U𝑓 (𝑡, 𝜏, 𝜔) | 𝑓 ∈ Σ(𝑓0 )} of processes is
(𝑉 × Σ(𝑓0 ); 𝑉)-continuous. Let
is solution of (48) satisfying
(52)
󵄩󵄩
󵄩
󵄩󵄩V𝑓 (𝑥, 𝑡, 𝜔)󵄩󵄩󵄩 ≤ 𝑀𝑓 (𝜔) ∀𝑡 ∈ R}
󵄩
󵄩𝐻
defined in the periodic box [0, 𝑙] , satisfying
V (𝑥, 𝜏, 𝜔) = V𝜏 (𝑥, 𝜔) = 𝑢𝜏 (𝑥) − 𝑧 (𝜃𝜏 𝜔) ∈ 𝐻.
+ ] ⟨𝐴 (𝛼02 V + 𝛼12 𝐴V
N𝑓 (𝜔) = {V𝑓 (𝑥, 𝑡, 𝜔) for 𝑡 ∈ R | V𝑓 (𝑥, 𝑡, 𝜔)
3
∇ ⋅ V = 0,
𝑑 2
(𝛼 V + 𝛼12 𝐴V) , 𝑤⟩
𝑑𝑡 0
𝐷(𝐴)󸀠
= (𝑃𝑓 (𝑥, 𝑡) + 𝜇𝛼02 𝑧 (𝜃𝑡 𝜔) + 𝜇𝛼12 𝐴𝑧 (𝜃𝑡 𝜔) , 𝑤) .
+ ]𝐴 (𝛼02 V + 𝛼12 𝐴V + 𝛼02 𝑧 (𝜃𝑡 𝜔) + 𝛼12 𝐴𝑧 (𝜃𝑡 𝜔))
+
(50)
(49)
be the so-called kernel of the process {U𝑓 (𝑡, 𝜏, 𝜔)}.
5. Uniformly Random Attractor for Stochastic
Nonautonomous Camassa-Holm Equation
In [19], the authors have shown that the semigroup corresponding to the autonomous system possesses a global attractor. In [21–24], the authors have proved that the deterministic
version of nonautonomous system has a uniform attractor.
The main objective of this section is to obtain the existence of
uniformly (with respect to 𝑓 ∈ Σ(𝑓0 )) random attractor for
the stochastic nonautonomous Camassa-Holm equations in
𝑉 and 𝐷(𝐴).
Lemma 15. Let {𝐵(𝜔)}𝜔∈Ω be tempered and V𝜏 (𝜔) ∈ 𝐵(𝜔).
Then the process {U𝑓 (𝑡, 𝜏, 𝜔)} corresponding to (48) possesses
a uniformly (with respect to 𝑓 ∈ Σ(𝑓0 )) random absorbing set
K0 (𝜔) in 𝑉.
Abstract and Applied Analysis
7
𝑛
󵄨
󵄨2
+ 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨
Proof. Letting 𝑤 = V in (51), we have
𝑗=1
1 𝑑
󵄩
󵄩
󵄩2
󵄩2
(𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 ) + ] (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V‖22 )
2 𝑑𝑡
≤
𝑛
= −] ( ∑𝐴ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 ) , V)
𝑛
]𝛼12
󵄨
󵄩
󵄨
󵄩2
‖𝐴V‖22 + 𝑐 ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 (𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
2
𝑗=1
𝑛
󵄨
󵄩
󵄨2
󵄩2
+ 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 (𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
𝑗=1
𝑛
− (𝐵̃ (V + 𝑧 (𝜃𝑡 𝜔) , 𝛼02 V + 𝛼12 𝐴V + ∑ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 )) , V)
𝑗=1
𝑗=1
𝑛
󵄨
󵄨2
+ 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 .
𝑗=1
𝑛
+ (𝑃𝑓 (𝑥, 𝑡) , V) + ( ∑𝜇ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 ) , V) .
(54)
𝑗=1
(53)
Now we estimate the second term of (53) on the right-hand
side. Applying Lemma 14 we have
󵄨󵄨
󵄨󵄨
𝑛
󵄨󵄨
󵄨
󵄨󵄨(𝐵̃ (V + 𝑧 (𝜃 𝜔) , 𝛼2 V + 𝛼2 𝐴V + ∑ ℎ 𝑦 (𝜃 𝜔 )) , V)󵄨󵄨󵄨
󵄨󵄨
𝑡
𝑗 𝑗
𝑡 𝑗
0
1
󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
𝑗=1
󵄨
󵄨󵄨
󵄨󵄨
𝑛
󵄨󵄨
󵄨󵄨
= 󵄨󵄨󵄨󵄨(𝐵̃ (𝑧 (𝜃𝑡 𝜔) , 𝛼02 V + 𝛼12 𝐴V + ∑ ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 )) , V)󵄨󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
𝑗=1
󵄨
󵄨
󵄩
󵄩1/2 󵄩
󵄩1/2
≤ 𝑐󵄩󵄩󵄩󵄩𝐴1/2 𝑧(𝜃𝑡 𝜔)󵄩󵄩󵄩󵄩2 󵄩󵄩󵄩𝐴𝑧(𝜃𝑡 𝜔)󵄩󵄩󵄩2
󵄩󵄩
󵄩󵄩
𝑛
󵄩󵄩 2
󵄩󵄩 󵄩
󵄩
2
󵄩
× 󵄩󵄩󵄩𝛼0 V + 𝛼1 𝐴V + ∑ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2
󵄩󵄩
󵄩󵄩
𝑗=1
󵄩
󵄩2
󵄩󵄩
󵄩󵄩
𝑛
󵄩󵄩
󵄩󵄩
󵄩󵄩 󵄩󵄩󵄩󵄩 2
2
+ 𝑐󵄩󵄩𝐴𝑧 (𝜃𝑡 𝜔)󵄩󵄩2 󵄩󵄩𝛼0 V + 𝛼1 𝐴V + ∑ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄩󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
𝑗=1
󵄩
󵄩2
󵄩󵄩 1/2 󵄩󵄩1/2
󵄩𝐴 V󵄩󵄩
× ‖V‖1/2
2 󵄩
󵄩
󵄩2
󵄩
󵄩
󵄩2
󵄩
󵄩󵄩
≤ 𝑐󵄩󵄩󵄩󵄩𝐴1/2 𝑧(𝜃𝑡 𝜔)󵄩󵄩󵄩󵄩2 ‖V‖22 + 𝑐󵄩󵄩󵄩𝐴𝑧(𝜃𝑡 𝜔)󵄩󵄩󵄩2 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2
By Poincaré’s inequality, we have
󵄨󵄨 𝑛
󵄨󵄨
𝑛
󵄨󵄨
󵄨󵄨 ]𝛼2 󵄩
󵄨
󵄩2
󵄨2
󵄨
󵄨
] 󵄨󵄨( ∑𝐴ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 ) , V)󵄨󵄨󵄨󵄨 ≤ 0 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 + 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 ,
6
󵄨󵄨 𝑗=1
󵄨󵄨
𝑗=1
󵄨
󵄨
]𝛼2 󵄩 1/2 󵄩2 󵄩 󵄩2
󵄨 󵄩 󵄩
󵄨󵄨
󵄨󵄨(𝑃𝑓 (𝑥, 𝑡) , V)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉󸀠 ‖V‖2 ≤ 0 󵄩󵄩󵄩󵄩𝐴 V󵄩󵄩󵄩󵄩2 + 𝑐󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉󸀠 ,
6
󵄨󵄨 𝑛
󵄨󵄨
2
𝑛
󵄨󵄨
󵄨
󵄨󵄨( ∑ 𝜇ℎ 𝑦 (𝜃 𝜔 ) , V)󵄨󵄨󵄨 ≤ ]𝛼0 󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩2 + 𝑐 ∑ 󵄨󵄨󵄨𝑦 (𝜃 𝜔 )󵄨󵄨󵄨2 .
󵄨󵄨
󵄨󵄨
󵄩󵄩
󵄨󵄨 𝑗 𝑡 𝑗 󵄨󵄨
󵄩󵄩2
𝑗 𝑗
𝑡 𝑗
6
󵄨󵄨 𝑗=1
󵄨󵄨
𝑗=1
󵄨
󵄨
(55)
Associating with the above inequalities and employing
Poincaré’s inequality, we have
𝑑
󵄩
󵄩2
(𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
𝑑𝑡
󵄩
󵄩2
+ ]𝜆 1 (𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
𝑛
󵄨
󵄩
󵄨
󵄩2
≤ 𝑐 ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 (𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
𝑗=1
𝑛
󵄨
󵄩
󵄨2
󵄩2
+ 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 (𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
]𝛼2
+ 1 ‖𝐴V‖22
4
󵄩
󵄩󵄩
󵄩2
󵄩󵄩
+ 𝑐󵄩󵄩󵄩󵄩𝐴1/2 𝑧(𝜃𝑡 𝜔)󵄩󵄩󵄩󵄩2 󵄩󵄩󵄩𝐴𝑧(𝜃𝑡 𝜔)󵄩󵄩󵄩2 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2
󵄩
󵄩󵄩
󵄩
+ 𝑐󵄩󵄩󵄩󵄩𝐴1/2 𝑧(𝜃𝑡 𝜔)󵄩󵄩󵄩󵄩2 󵄩󵄩󵄩𝐴𝑧(𝜃𝑡 𝜔)󵄩󵄩󵄩2
𝑗=1
𝑛
󵄨
󵄨2 󵄩 󵄩2
+ 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 + 𝑐󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉󸀠 .
𝑗=1
(56)
𝑛
󵄨
󵄨2 󵄩
󵄩2
+ 𝑐 ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2
𝑗=1
]𝛼2
󵄩
󵄩
+ 𝑐󵄩󵄩󵄩𝐴𝑧 (𝜃𝑡 𝜔)󵄩󵄩󵄩2 ‖V‖22 + 1 ‖𝐴V‖22
4
󵄩2
󵄩
󵄩2
󵄩
󵄩2 󵄩
+ 𝑐󵄩󵄩󵄩𝐴𝑧(𝜃𝑡 𝜔)󵄩󵄩󵄩2 ‖V‖22 + 𝑐󵄩󵄩󵄩𝐴𝑧(𝜃𝑡 𝜔)󵄩󵄩󵄩2 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2
𝑛
󵄨
󵄨2 󵄩
󵄩2
󵄩
󵄩2
+ 𝑐󵄩󵄩󵄩𝐴𝑧(𝜃𝑡 𝜔)󵄩󵄩󵄩2 + 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2
𝑗=1
Applying Gronwall’s lemma, we have
󵄩
󵄩2
𝛼02 ‖V (𝑡, 𝜔)‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑡, 𝜔)󵄩󵄩󵄩󵄩2
󵄩2
󵄩
󵄩2
󵄩
≤ 𝑒𝐽1 (𝜏,𝑡,𝜔) (𝛼02 󵄩󵄩󵄩V𝜏 (𝜔)󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V𝜏 (𝜔)󵄩󵄩󵄩󵄩2 )
𝑛
𝑡
󵄨
󵄨2 󵄩
󵄩2
+ ∫ (𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨󵄨 + 𝑐󵄩󵄩󵄩𝑓 (𝑠)󵄩󵄩󵄩𝑉󸀠 ) 𝑒𝐽1 (𝑠,𝑡,𝜔) 𝑑𝑠,
𝜏
𝑗=1
(57)
8
Abstract and Applied Analysis
is convergent. For any 𝑡 > 𝜏, there exists 𝐾 ∈ N such that
−𝐾 < 𝜏 − 𝑡, and we have
where
𝑛
𝑡
󵄨
󵄨
𝐽1 (𝜏, 𝑡, 𝜔) = − ]𝜆 1 𝑡 + ]𝜆 1 𝜏 + 𝑐 ∑ ∫ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝑠
𝜏
𝑗=1
𝑛
𝑡
󵄨
󵄨2
+ 𝑐 ∑ ∫ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝑠.
𝜏
∫
(58)
0
𝜏−𝑡
󵄩2 𝐽 (𝑠,0,𝜔)
󵄩󵄩
𝑑𝑠
󵄩󵄩𝑓 (𝑠 + 𝑡)󵄩󵄩󵄩𝑉󸀠 𝑒 1
=∫
𝑗=1
𝜏−𝑡
Replacing 𝜔 by 𝜃−𝑡 𝜔 in (57) and (58), we have
󵄩
󵄩2
𝛼02 󵄩󵄩󵄩V (𝑡, 𝜃−𝑡 𝜔)󵄩󵄩󵄩2
+
+∫
Then by piecewise integration, we have
0
󵄩
󵄩2
∫ 󵄩󵄩󵄩𝑓0 (𝑠 + 𝑡 + 𝜂)󵄩󵄩󵄩𝑉󸀠 𝑒𝐽1 (𝑠,0,𝜔) 𝑑𝑠
−𝐾
󵄨
󵄨2 󵄩
󵄩2
(𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨󵄨 + 𝑐󵄩󵄩󵄩𝑓 (𝑠 + 𝑡)󵄩󵄩󵄩𝑉󸀠 ) 𝑒𝐽1 (𝑠,0,𝜔) 𝑑𝑠.
𝐾
𝑗=1
Note that
are stationary and ergodic (cf. [24]),
then it follows from ergodic theorem that
1 0 󵄨󵄨
󵄨
󵄨
󵄨
∫ 󵄨󵄨𝑦 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏 = 𝐸 (󵄨󵄨󵄨󵄨𝑦𝑗 (𝜔𝑗 )󵄨󵄨󵄨󵄨) .
𝑡 → ∞ 𝑡 −𝑡 󵄨 𝑗
By differential mean value theorem, for each 1 ≤ 𝑗 ≤ 𝐾, there
exists 𝑠𝑗 ∈ [−𝑛 + 𝑗 − 1, −𝑛 + 𝑗], such that
(60)
𝐾
−𝑛+𝑗
∑∫
𝑗=1 −𝑛+𝑗−1
lim
1 0 󵄨󵄨
𝑐
󵄨2
󵄨
󵄨2
∫ 󵄨󵄨𝑦 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏 = 𝐸 (󵄨󵄨󵄨󵄨𝑦𝑗 (𝜔𝑗 )󵄨󵄨󵄨󵄨 ) = .
𝑡 → ∞ 𝑡 −𝑡 󵄨 𝑗
𝜇
𝑚
𝐽 (𝑠, 0, 𝜔)
1 0󵄨
󵄨
lim 1
= lim (]𝜆 1 + 𝑐 ∑ ∫ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏
𝑠 → −∞
𝑠 → −∞
𝑠
𝑠
𝑠
𝑗=1
𝑚
1 0󵄨
󵄨2
+𝑐 ∑ ∫ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏)
𝑗=1 𝑠 𝑠
𝑐
𝑐
− ).
√𝜇 𝜇
Since
{|𝑦𝑗 (𝜔𝑗 )|}𝑛𝑗=1
0
(63)
−∞ 𝑗=1
𝑡+1
𝑡
󵄩󵄩
󵄩2
󵄩󵄩𝑓0 (𝑠)󵄩󵄩󵄩𝑉󸀠 𝑑𝑠 < 𝑀𝑓0 .
(68)
𝐽1 (𝑠𝑗 ,0,𝜔)
is convergent.
It is easy to check that the series ∑∞
𝑗=1 𝑒
Associating with (65)–(68), for any 𝑡 > 𝜏, we have
0
󵄩2 𝐽 (𝑠,0,𝜔)
󵄩󵄩
𝑑𝑠 ≤ 𝑟2 (𝜔) ,
󵄩󵄩𝑓(𝑠 + 𝑡)󵄩󵄩󵄩𝑉󸀠 𝑒 1
𝜏−𝑡
∫
(69)
∞
𝑟2 (𝜔) = 𝑀𝑓0 ∑𝑒𝐽1 (𝑠𝑗 ,0,𝜔) .
𝑗=1
(70)
∫
0
𝜏−𝑡
𝑛
󵄨
󵄨2 󵄩
󵄩2
(𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨󵄨 + 𝑐󵄩󵄩󵄩𝑓 (𝑠 + 𝑡)󵄩󵄩󵄩𝑉󸀠 ) 𝑒𝐽1 (𝑠,0,𝜔) 𝑑𝑠 ≤ 𝑟3 (𝜔) ,
𝑗=1
(71)
as 𝑠 󳨀→ −∞.
𝑛
󵄨
󵄨2
∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑒𝐽1 (𝑠,0,𝜔) 𝑑𝑠
󵄩󵄩
󵄩󵄩𝑓0 (𝑠 + 𝑡 +
Therefore (64) and (69) imply that, for 𝑡 > 𝜏, the second term
on the right-hand side of (59) can be bounded by
where 𝑟3 (𝜔) = 𝑐𝑟1 (𝜔) + 𝑐𝑟2 (𝜔). For V𝜏 (𝜔) ∈ 𝐵(𝜔) being
tempered, there exists 𝑇1 (𝜔) > 0 such that, when 𝑡 ≥ 𝑇2 (𝜔),
are tempered, the integral
𝑟1 (𝜔) = ∫
−𝑛+𝑗−1
󵄩2
𝜂)󵄩󵄩󵄩𝑉󸀠 𝑑𝑠.
where
𝑚
𝐽1 (𝑠, 0, 𝜔)
1 0󵄨
󵄨
= ]𝜆 1 + 𝑐 ∑ ∫ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏
𝑠
𝑠
𝑠
𝑗=1
𝐽1 (𝑠, 0, 𝜔) 󳨀→ −∞,
𝑗=1
𝑡∈R
So for P a.e. 𝜔 ∈ Ω, there are 𝑇0 (𝜔) > 0 and 𝜇0 > 0 such that,
for 𝑠 ≥ 𝑇1 (𝜔) and 𝜇 ≥ 𝜇0 ,
1 0󵄨
󵄨2
+𝑐 ∑ ∫ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏 < 0,
𝑗=1 𝑠 𝑠
−𝑛+𝑗
sup ∫
(62)
𝑚
(67)
𝐾
By the property of normal function, there exists 𝑀𝑓0 , which
just depends on 𝑓0 , such that
Employing (61), we have
= (]𝜆 1 −
󵄩2 𝐽 (𝑠,0,𝜔)
󵄩󵄩
𝑑𝑠
󵄩󵄩𝑓0 (𝑠 + 𝑡 + 𝜂)󵄩󵄩󵄩𝑉󸀠 𝑒 1
= ∑ 𝑒𝐽1 (𝑠𝑗 ,0,𝜔) ∫
(61)
lim
(66)
𝑗=1
On the other hand, we have
𝑐
1 0 󵄨󵄨
󵄨
󵄨
󵄨
,
∫ 󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏 = 𝐸 (󵄨󵄨󵄨󵄨𝑦𝑗 (𝜔𝑗 )󵄨󵄨󵄨󵄨) =
𝑡 → ∞ 𝑡 −𝑡
√𝜇
−𝑛+𝑗
󵄩󵄩
󵄩2 𝐽 (𝑠,0,𝜔)
= ∑∫
𝑑𝑠.
󵄩󵄩𝑓0 (𝑠 + 𝑡 + 𝜂)󵄩󵄩󵄩𝑉󸀠 𝑒 1
−𝑛+𝑗−1
(59)
{|𝑦𝑗 (𝜃𝑡 𝜔𝑗 )|}𝑛𝑗=1
lim
(65)
0
𝑛
𝜏−𝑡
󵄩󵄩
󵄩2 𝐽 (𝑠,0,𝜔)
𝑑𝑠
󵄩󵄩𝑓0 (𝑠 + 𝑡 + 𝜂)󵄩󵄩󵄩𝑉󸀠 𝑒 1
󵄩
󵄩2
≤ ∫ 󵄩󵄩󵄩𝑓0 (𝑠 + 𝑡 + 𝜂)󵄩󵄩󵄩𝑉󸀠 𝑒𝐽1 (𝑠,0,𝜔) 𝑑𝑠.
−𝐾
󵄩
󵄩2
𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V(𝑡, 𝜃−𝑡 𝜔)󵄩󵄩󵄩󵄩2
󵄩
󵄩2
󵄩
󵄩2
≤ 𝑒𝐽1 (𝜏−𝑡,0,𝜔) (𝛼02 󵄩󵄩󵄩V𝜏 (𝜃−𝑡 𝜔)󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V𝜏 (𝜃−𝑡 𝜔)󵄩󵄩󵄩󵄩2 )
0
0
(64)
󵄩
󵄩2
󵄩
󵄩2
𝑒𝐽1 (𝜏−𝑡,0,𝜔) (𝛼02 󵄩󵄩󵄩V𝜏 (𝜃−𝑡 𝜔)󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V𝜏 (𝜃−𝑡 𝜔)󵄩󵄩󵄩󵄩2 ) < 𝑟3 (𝜔) .
(72)
Abstract and Applied Analysis
9
Letting 𝑇3 (𝜔) = max{𝑇1 (𝜔), 𝑇2 (𝜔)} and 𝑟4 (𝜔) = 2𝑟3 (𝜔), when
𝑡 ≥ 𝑇3 (𝜔), we have
󵄩
󵄩2
󵄩
󵄩2
𝛼02 󵄩󵄩󵄩V(𝑡, 𝜃−𝑡 𝜔)󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V(𝑡, 𝜃−𝑡 𝜔)󵄩󵄩󵄩󵄩2 ≤ 𝑟4 (𝜔) ,
𝑛
+ 𝑐∑ ∫
𝑗=1 𝑡
𝑛
+ 𝑐∑ ∫
𝑡+1
𝑗=1 𝑡
We define
(76)
(74)
In conclusion, K1 (𝜔) is a uniformly random absorbing set
for {U(𝑡, 𝜏, 𝜔)} in 𝑉, which complete the proof.
Lemma 16. Let {𝐵(𝜔)}𝜔∈Ω be tempered and V𝜏 (𝜔) ∈ 𝐵(𝜔).
Then the process {U𝑓 (𝑡, 𝜏, 𝜔)} corresponding to (48) possesses
a uniformly (with respect to 𝑓 ∈ Σ(𝑓0 )) random absorbing set
K2 (𝜔) in 𝐷(𝐴).
As the previous consideration in Lemma 15, for 𝑠 ∈ [𝑡, 𝑡 + 1)
where 𝑡 ≥ 𝑇2 (𝜔), we have
󵄩
󵄩2
󵄩
󵄩2
𝛼02 󵄩󵄩󵄩V(𝑠, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑠, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 ≤ 𝑟5 (𝜔) ,
󵄩
󵄩2
𝛼02 ‖V (𝑡 + 1, 𝜔)‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑡 + 1, 𝜔)󵄩󵄩󵄩󵄩2
+ ]∫
∫
𝑡+1
𝑡
󵄩
󵄩2
󵄩
󵄩2
(𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑠, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩𝐴V (𝑠, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2 ) 𝑑𝑠
≤ 𝑟6 (𝜔) ,
(78)
where
𝑛
0
󵄨
󵄨
𝑟6 (𝜔) = 𝑐𝑟5 (𝜔) ∑ ∫ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝑠
−1
󵄩2
󵄩
(𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑠, 𝜔)󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V (𝑠, 𝜔)‖22 ) 𝑑𝑠
𝑡
𝑗=1
𝑛
0
󵄨
󵄨2
+ 𝑐𝑟5 (𝜔) ∑ ∫ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝑠
−1
󵄩
󵄩2
≤ (𝛼02 ‖V (𝑡, 𝜔)‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑡, 𝜔)󵄩󵄩󵄩󵄩2 )
𝑛
+ 𝑐∑ ∫
𝑡+1
𝑗=1 𝑡
×
𝑛
+ 𝑐∑ ∫
𝑡+1
𝑗=1 𝑡
+
󵄩
󵄩2
𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑠, 𝜔)󵄩󵄩󵄩󵄩2 ) 𝑑𝑠
𝑛
𝑡+1
𝑗=1 𝑡
󵄨󵄨
󵄨2
󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨 𝑑𝑠.
󵄨
󵄨
Replacing 𝜔 by 𝜃−𝑡−1 𝜔 in (75), we have
]∫
𝑡+1
𝑡
𝑛
󵄩
󵄩2
󵄩
󵄩2
(𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑠, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩𝐴V (𝑠, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2 ) 𝑑𝑠
≤ 𝑐∑ ∫
𝑡+1
𝑗=1 𝑡
(79)
𝑛
0
󵄨
󵄨2
+ 𝑐 ∑ ∫ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝑠 + 𝑟5 (𝜔) .
−1
𝑗=1
(75)
Now we let 𝑤 = 𝐴V in (51), and we have
1 𝑑
󵄩
󵄩2
(𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V‖22 )
2 𝑑𝑡
󵄨󵄨
󵄨2
󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨
󵄨
󵄨
󵄩2
󵄩
× (𝛼02 ‖V (𝑠, 𝜔)‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑠, 𝜔)󵄩󵄩󵄩󵄩2 ) 𝑑𝑠
+ 𝑐∑ ∫
𝑗=1
󵄨󵄨
󵄨
󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨
󵄨
󵄨
(𝛼02 ‖V (𝑠, 𝜔)‖22
(77)
where 𝑟5 (𝜔) = 𝑒]𝜆 1 𝑟4 (𝜔).
Associating (76) with (77), we have
Proof. Integrating (56) into [𝑡, 𝑡+1] where 𝑡 ≥ 𝑇3 (𝜔), we have
𝑡+1
󵄨󵄨
󵄨2
󵄨󵄨𝑦𝑗 (𝜃𝑠−𝑡−1 𝜔𝑗 )󵄨󵄨󵄨 𝑑𝑠
󵄨
󵄨
󵄩
󵄩2
󵄩
󵄩2
+ (𝛼02 󵄩󵄩󵄩V (𝑡, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑡, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 ) .
K1 (𝜔) = {V (𝜔) ∈ 𝑉 | 𝛼02 ‖V(𝜔)‖22
󵄩
󵄩2
+𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V(𝜔)󵄩󵄩󵄩󵄩2 ≤ 𝑟4 (𝜔)} .
󵄨󵄨
󵄨2
󵄨󵄨𝑦𝑗 (𝜃𝑠−𝑡−1 𝜔𝑗 )󵄨󵄨󵄨 (𝛼02 󵄩󵄩󵄩󵄩V (𝑠, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩22
󵄨
󵄨
󵄩
󵄩2
+𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑠, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 ) 𝑑𝑠
(73)
∀𝑡 ≥ 𝑇3 (𝜔) , 𝑓 ∈ Σ (𝑓0 ) .
𝑡+1
󵄨󵄨
󵄨
󵄨󵄨𝑦𝑗 (𝜃𝑠−𝑡−1 𝜔𝑗 )󵄨󵄨󵄨 (𝛼02 󵄩󵄩󵄩󵄩V (𝑠, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩22
󵄨
󵄨
󵄩
󵄩2
+𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑠, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 ) 𝑑𝑠
󵄩
󵄩2
+ ] (𝛼02 ‖𝐴V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴3/2 V󵄩󵄩󵄩󵄩2 )
𝑛
= −] ( ∑ 𝐴ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 ) , 𝐴V)
𝑗=1
− (𝐵̃ (V + 𝑧 (𝜃𝑡 𝜔) , 𝛼02 V + 𝛼12 𝐴V
𝑛
+ ∑ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 )) , 𝐴V)
𝑗=1
𝑛
+ (𝑃𝑓 (𝑥, 𝑡) , 𝐴V) + (∑𝜇ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 ) , 𝐴V) .
𝑗=1
(80)
10
Abstract and Applied Analysis
Employing Lemma 14, the bilinear term in (80) can be
bounded by
Associating with all the above inequalities, we have
𝑑
󵄩
󵄩2
(𝛼2 󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V‖22 )
𝑑𝑡 0 󵄩
𝑛
(𝐵̃ (V + 𝑧 (𝜃𝑡 𝜔) , 𝛼02 V + 𝛼12 𝐴V + ∑ ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 )) , 𝐴V)
󵄩
󵄩2 2
≤ 𝑐(𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
𝑗=1
2
󵄩
󵄩2
+ 𝑐(𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V‖22 )
󵄩1/2 󵄩
󵄩
󵄩1/2
≤ 𝑐 (󵄩󵄩󵄩󵄩𝐴1/2 V + 𝐴1/2 𝑧 (𝜃𝑡 𝜔)󵄩󵄩󵄩󵄩2 󵄩󵄩󵄩𝐴V + 𝐴𝑧(𝜃𝑡 𝜔)󵄩󵄩󵄩2
𝑛
󵄨
󵄨2
+ 𝑐 ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨
󵄩󵄩
󵄩󵄩
𝑛
󵄩󵄩 2
󵄩󵄩 󵄩
󵄩
2
󵄩
×󵄩󵄩󵄩𝛼0 V + 𝛼1 𝐴V + ∑ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝐴3/2 V󵄩󵄩󵄩󵄩2 )
󵄩󵄩
󵄩󵄩
𝑗=1
󵄩
󵄩2
𝑗=1
𝑛
󵄨
󵄨4
+ 𝑐 ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨
󵄩󵄩
󵄩󵄩
𝑛
󵄩󵄩
󵄩󵄩
󵄩󵄩 󵄩󵄩󵄩󵄩 2
2
+ 𝑐 (󵄩󵄩𝐴V + 𝐴𝑧 (𝜃𝑡 𝜔)󵄩󵄩2 󵄩󵄩𝛼0 V + 𝛼1 𝐴V + ∑ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄩󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
𝑗=1
󵄩
󵄩2
𝑗=1
󵄩
󵄩2
+ 𝑐 (𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
󵄩
󵄩2
× (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V‖22 )
󵄩󵄩 3/2 󵄩󵄩1/2
󵄩𝐴 V󵄩󵄩 )
×‖𝐴V‖1/2
2 󵄩
󵄩
󵄩2
𝑛
󵄨
󵄩
󵄨2
󵄩2
+ 𝑐 ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 (𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
2
󵄩
󵄩2 ]𝛼
≤ ]𝛼12 󵄩󵄩󵄩󵄩𝐴3/2 V󵄩󵄩󵄩󵄩2 + 0 ‖𝐴V‖22
4
+
𝑐(𝛼02 ‖V‖22
+
𝑗=1
󵄩 󵄩2
+ 𝑐󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉󸀠
󵄩
󵄩2 2
𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
𝑛
󵄨
󵄩
󵄨2
󵄩2
+ 𝑐 ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V‖22 ) .
2
󵄩
󵄩2
+ 𝑐(𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 + 𝛼02 ‖𝐴V‖22 )
+ 𝑐 (𝛼02 ‖V‖22 +
󵄩
󵄩
󵄩2
󵄩2
𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 ) (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2
𝑗=1
(83)
+ 𝛼02 ‖𝐴V‖22 )
Applying Gronwall’s lemma on [𝑠, 𝑡 + 1] where 𝑠 ∈ (𝑡, 𝑡 + 1)
and 𝑡 ≥ 𝑇3 (𝜔), we have
𝑛
󵄨
󵄩
󵄨2
󵄩2
+ 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 (𝛼02 ‖V‖22 + 𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
𝑗=1
󵄩
󵄩2
𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V(𝑡 + 1, 𝜔)󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V (𝑡 + 1, 𝜔)‖22
𝑛
󵄨
󵄩
󵄨2
󵄩2
+ 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 + 𝛼02 ‖𝐴V‖22 )
󵄩
󵄩2
≤ (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑠, 𝜔)󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V (𝑠, 𝜔)‖22 ) 𝑒𝐽2 (𝑠,𝑡+1,𝜔)
𝑗=1
𝑛
󵄨
󵄨4
+ 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 .
𝑗=1
+ 𝑐∫
(81)
𝑠
+ 𝑐∫
The other terms are bounded by
󵄨󵄨 𝑛
󵄨󵄨
𝑛
󵄨󵄨
󵄨󵄨 ]𝛼2
󵄨
󵄨2
󵄨
󵄨
] 󵄨󵄨( ∑𝐴ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 ) , 𝐴V)󵄨󵄨󵄨󵄨 ≤ 0 ‖𝐴V‖22 + 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 .
4
󵄨󵄨 𝑗=1
󵄨󵄨
𝑗=1
󵄨
󵄨
]𝛼2
󵄨 󵄩 󵄩
󵄩 󵄩2
󵄨󵄨
2
󵄨󵄨(𝑃𝑓 (𝑥, 𝑡) , 𝐴V)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉󸀠 ‖𝐴V‖2 ≤ 0 ‖𝐴V‖2 + 𝑐󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉󸀠 .
4
󵄨󵄨󵄨 ]𝛼2
󵄨󵄨󵄨 𝑛
𝑛
󵄨
󵄨
󵄨
󵄨2
𝜇 󵄨󵄨󵄨󵄨( ∑ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 ) , 𝐴V)󵄨󵄨󵄨󵄨 ≤ 0 ‖𝐴V‖22 + 𝑐 ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 .
4
󵄨󵄨
󵄨󵄨 𝑗=1
𝑗=1
󵄨
󵄨
(82)
𝑡+1 𝑛
󵄨
󵄨2
∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑒𝐽2 (𝜏,𝑡+1,𝜔) 𝑑𝜏
𝑗=1
𝑡+1 𝑛
󵄨
󵄨4
∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑒𝐽2 (𝜏,𝑡+1,𝜔) 𝑑𝜏
𝑠
𝑗=1
𝑡+1
󵄩2 𝐽 (𝜏,𝑡+1,𝜔)
󵄩󵄩
𝑑𝜏
󵄩󵄩𝑓(𝜏)󵄩󵄩󵄩𝑉󸀠 𝑒 2
+ 𝑐∫
𝑠
+ 𝑐∫
𝑡+1 𝑛
𝑠
󵄨
󵄨2
∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨
𝑗=1
󵄩
󵄩2
× (𝛼02 ‖V (𝜏, 𝜔)‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V (𝜏, 𝜔)󵄩󵄩󵄩󵄩2 ) 𝑒𝐽2 (𝜏,𝑡+1,𝜔) 𝑑𝜏
+ 𝑐∫
𝑡+1
𝑠
󵄩
󵄩2 2
(𝛼02 ‖V (𝜏, 𝜔)‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V (𝜏, 𝜔)󵄩󵄩󵄩󵄩2 ) 𝑒𝐽2 (𝜏,𝑡+1,𝜔) 𝑑𝜏,
(84)
Abstract and Applied Analysis
11
Integrating (87) with respect to 𝑠 over (𝑡, 𝑡+1) where 𝑡 ≥ 𝑇3 (𝜔)
and employing the property of normal function in (68), we
have
where
𝐽2 (𝑠, 𝑡, 𝜔)
= 𝑐∫
𝑡
(𝛼02 ‖V (𝜏, 𝜔)‖22
𝑠
+
󵄩
󵄩2
󵄩
󵄩2
𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑡 + 1, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩𝐴V (𝑡 + 1, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2
󵄩
󵄩2
𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V (𝜏, 𝜔)󵄩󵄩󵄩󵄩2 ) 𝑑𝜏
𝑡
󵄩
󵄩2
+ 𝑐 ∫ (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V (𝜏, 𝜔)󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V (𝜏, 𝜔)‖22 ) 𝑑𝜏
𝑠
≤ 𝑟8 (𝜔) ,
(85)
where
𝑟8 (𝜔) = 𝑟6 (𝜔) 𝑒𝑟7 (𝜔) + 𝑐𝑒𝑟7 (𝜔) 𝑟5 (𝜔)2
𝑡 𝑛
󵄨
󵄨2
+ 𝑐 ∫ ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏.
𝑠
0 𝑛
󵄨
󵄨2
+ 𝑐𝑒𝑟7 (𝜔) ∫ ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏
−1
𝑗=1
𝑗=1
Replacing 𝜔 by 𝜃−𝑡−1 𝜔 in (84) and (85), we have
0 𝑛
󵄨
󵄨4
+ 𝑐𝑒𝑟7 (𝜔) ∫ ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏
−1
𝑗=1
𝐽2 (𝜏, 𝑡 + 1, 𝜃−𝑡−1 𝜔)
≤ 𝑐∫
𝑡+1
𝑡
𝑡+1
0 𝑛
󵄨
󵄨2
+ 𝑐𝑒𝑟7 (𝜔) 𝑟5 (𝜔) ∫ ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏.
−1
󵄩
󵄩2
+ 𝛼12 󵄩󵄩󵄩𝐴V(𝜏, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2 ) 𝑑𝜏
𝑛
+ 𝑐∑ ∫
𝑗=1
󵄩
󵄩2
(𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V (𝜏, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2
𝑡
𝑡+1
𝑗=1 𝑡
We define
(86)
󵄨󵄨
󵄨2
󵄨󵄨𝑦𝑗 (𝜃𝜏−𝑡−1 𝜔𝑗 )󵄨󵄨󵄨 𝑑𝜏
󵄨
󵄨
So much for that we have proved the existence of bounded
uniformly (with respect to 𝑓 ∈ Σ(𝑓0 )) random absorbing sets
in 𝑉 and 𝐷(𝐴). Next we derive the existence of uniformly
(with respect to 𝑓 ∈ Σ(𝑓0 )) random attractors in 𝑉 and 𝐷(𝐴).
󳵻
= 𝑟7 (𝜔) .
And then
󵄩
󵄩2
󵄩2
󵄩
𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑡 + 1, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩𝐴V(𝑡 + 1, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2
󵄩
󵄩2
󵄩2
󵄩
≤ (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V(𝑠, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩𝐴V (𝑠, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2 ) 𝑒𝑟7 (𝜔)
+ 𝑐𝑒
𝑟7 (𝜔)
𝑟5 (𝜔) + 𝑐𝑒
0
𝑛
󵄨
󵄨2
∫ ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏
−1
𝑡
A1 (𝜔) = AΣ(𝑓0 ) (𝜔) = W0,Σ(𝑓0 ) (K1 ) = ⋃ N𝑓 (0, 𝜔) ,
(91)
𝑗=1
where K1 (𝜔) is the uniformly (with respect to 𝑓 ∈ Σ(𝑓0 ))
random absorbing set in 𝑉 and N𝑓 (𝜔) is the kernel of
the process {U𝑓 (𝑡, 𝜏, 𝜔)}. Furthermore, the kernel N𝑓 (𝜔) is
nonempty for all 𝑓 ∈ Σ(𝑓0 ).
𝑗=1
𝑡+1
Theorem 17. If 𝑓0 (𝑥, 𝑠) is a normal function in 𝐿2loc (R, 𝑉󸀠 ),
then the process {U𝑓0 (𝑡, 𝜏, 𝜔)} corresponding to (48) possess a
compact uniformly (with respect to 𝜏 ∈ R) random attractor
A1 (𝜔) in 𝑉 which coincides with the uniformly (with respect to
𝑓 ∈ Σ(𝑓0 )) random attractor AΣ(𝑓0 ) of the family of processes
{U𝑓 (𝑡, 𝜏, 𝜔)}, 𝑓 ∈ Σ(𝑓0 ) :
𝑓∈Σ(𝑓0 )
0 𝑛
󵄨
󵄨4
+ 𝑐𝑒𝑟7 (𝜔) ∫ ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏
−1
+ 𝑐𝑒𝑟7 (𝜔) ∫
(90)
In conclusion, K2 (𝜔) is a uniformly random absorbing set
for {U(𝑡, 𝜏, 𝜔)} in 𝐷(𝐴), which complete the proof.
𝑗=1
2
󵄩
󵄩2
K2 (𝜔) = {V (𝜔) ∈ 𝐷 (𝐴) | 𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V(𝜔)󵄩󵄩󵄩󵄩2
+𝛼12 ‖𝐴V(𝜔)‖22 ≤ 𝑟8 (𝜔)} .
𝑛
0
󵄨
󵄨2
≤ 𝑐𝑟5 (𝜔) + 𝑐𝑟6 (𝜔) + 𝑐 ∑ ∫ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏
−1
𝑟7 (𝜔)
(89)
+ 𝑐𝑒𝑟7 (𝜔) 𝑀𝑓0
󵄩
󵄩2
(𝛼02 󵄩󵄩󵄩V (𝜏, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2
󵄩
󵄩2
+ 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V (𝜏, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 ) 𝑑𝜏
+ 𝑐∫
(88)
∀𝑡 ≥ 𝑇3 (𝜔) , 𝑓 ∈ Σ (𝑓0 ) ,
󵄩2
󵄩󵄩
󵄩󵄩𝑓(𝜏)󵄩󵄩󵄩𝑉󸀠 𝑑𝜏
Proof. We only have to verify condition(C). As the previous
section, for fixed 𝑚, let 𝐻1 be the subspace spanned by {𝑒𝑗 }𝑚
𝑗=1
and 𝐻2 the orthogonal complement of 𝐻1 in 𝐻. We write
𝑛
0
󵄨
󵄨2
+ 𝑐𝑒𝑟7 (𝜔) 𝑟5 (𝜔) ∫ ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏.
−1
𝑗=1
(87)
V = V1 + V 2 ,
V1 ∈ 𝐻1 , V2 ∈ 𝐻2 for any V ∈ 𝐻.
(92)
12
Abstract and Applied Analysis
𝑛
𝑛
󵄨
󵄨
󵄨2
󵄨4
+ 𝑐 ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑗 𝜔𝑗 )󵄨󵄨󵄨󵄨 + 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑗 𝜔𝑗 )󵄨󵄨󵄨󵄨
The proof of boundary of V1 is similar to the proof in
Lemma 15. We need to estimate V2 , where V = V1 + V2 is a
solution of (48). Letting 𝑤 = V2 in (51), we have
𝑗=1
𝑛
󵄨
󵄩
󵄨2
󵄩2
+ 𝑐 ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑗 𝜔𝑗 )󵄨󵄨󵄨󵄨 (𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
1 𝑑
󵄩
󵄩2
󵄩
󵄩2
󵄩 󵄩2
󵄩
󵄩2
(𝛼2 󵄩󵄩V 󵄩󵄩 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V2 󵄩󵄩󵄩󵄩2 ) + ] (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V2 󵄩󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩𝐴V2 󵄩󵄩󵄩2 )
2 𝑑𝑡 0 󵄩 2 󵄩2
𝑗=1
𝑛
󵄨
󵄩
󵄨2
󵄩2
+ 𝑐 ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑗 𝜔𝑗 )󵄨󵄨󵄨󵄨 (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V‖22 ) .
𝑛
= −] ( ∑ 𝐴ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 ) , V2 )
𝑗=1
𝑗=1
(94)
− (𝐵̃ (V + 𝑧 (𝜃𝑡 𝜔) , 𝛼02 V + 𝛼12 𝐴V
By Poincaré’s inequality, note that
󵄨󵄨
󵄨󵄨 𝑛
󵄨󵄨
󵄨󵄨
󵄨
] 󵄨󵄨󵄨( ∑𝐴ℎ𝑗 𝑦𝑗 (𝜃𝑗 𝜔𝑗 ) , V2 )󵄨󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑗=1
󵄨
󵄨
𝑛
+ ∑ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 )) , V2 )
𝑗=1
≤
𝑛
+ (𝑃𝑓 (𝑥, 𝑡) , V2 ) + 𝜇 ( ∑ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 ) , V2 ) .
𝑗=1
(93)
Employing Lemma 14, the second term on the right-hand
side of (93) can be bounded by
󵄨󵄨
󵄨󵄨
󵄨󵄨 ̃
2
2
󵄨󵄨󵄨(𝐵 (V + 𝑧 (𝜃𝑡 𝜔) , 𝛼0 V + 𝛼1 𝐴V
󵄨󵄨
󵄨
󵄨󵄨
󵄨󵄨
+ ∑ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 )) , V2 )󵄨󵄨󵄨󵄨
󵄨󵄨
𝑗=1
󵄨
󵄩1/2 󵄩
󵄩
󵄩1/2
× 󵄩󵄩󵄩󵄩𝐴1/2 V2 󵄩󵄩󵄩󵄩2 󵄩󵄩󵄩𝐴V2 󵄩󵄩󵄩2
󵄩2
V2 󵄩󵄩󵄩󵄩2 +
×
󵄩
󵄩2
(𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2
+
𝑐(𝛼02 ‖V‖22
+
󵄩
󵄩2
𝑐(𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2
+
+
𝑑
󵄩
󵄩2
󵄩 󵄩2
(𝛼02 󵄩󵄩󵄩V2 󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V2 󵄩󵄩󵄩󵄩2 )
𝑑𝑡
󵄩
󵄩2
󵄩 󵄩2
+ ]𝜆 𝑚+1 (𝛼02 󵄩󵄩󵄩V2 󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V2 󵄩󵄩󵄩󵄩2 )
󵄩 󵄩2
≤ 𝑐󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉󸀠 + 𝐺 (𝑡, 𝜔) ,
where
𝐺 (𝑡, 𝜔)
󵄩
󵄩2 2
= 𝑐(𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
𝑛
󵄨
󵄨2
+ 𝑐 ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨
𝑗=1
󵄩
󵄩2
+ 𝑐 (𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
𝛼12 ‖𝐴V‖22 )
󵄩
󵄩2
× (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V‖22 )
󵄩
󵄩2 2
𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
+ 𝛼12 ‖𝐴V‖22 )
𝑛
]𝛼02 󵄩󵄩 1/2 󵄩󵄩2
󵄨
󵄨2
󵄩󵄩𝐴 V2 󵄩󵄩 + 𝑐 ∑ 󵄨󵄨󵄨𝑦𝑗 (𝜃𝑗 𝜔𝑗 )󵄨󵄨󵄨 .
󵄩
󵄩
󵄨
󵄨
2
8
𝑗=1
2
󵄩
󵄩2
+ 𝑐(𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V‖22 )
]𝛼12 󵄩󵄩
󵄩2
󵄩𝐴V 󵄩󵄩
2 󵄩 2 󵄩2
󵄩
󵄩2
+ 𝑐 (𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
1/2
󵄩𝐴
8 󵄩
(95)
Associating with (93)–(95) and applying Poincaré inequality,
we have
󵄩1/2
󵄩
󵄩1/2 󵄩
≤ 𝑐󵄩󵄩󵄩V + 𝑧 (𝜃𝑡 𝜔)󵄩󵄩󵄩2 󵄩󵄩󵄩󵄩𝐴1/2 V + 𝐴1/2 𝑧 (𝜃𝑡 𝜔)󵄩󵄩󵄩󵄩2
󵄩󵄩
󵄩󵄩󵄩
𝑛
󵄩󵄩 󵄩
󵄩
󵄩
× 󵄩󵄩󵄩󵄩𝛼02 V + 𝛼12 𝐴V + ∑ ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄩󵄩󵄩󵄩 󵄩󵄩󵄩𝐴V2 󵄩󵄩󵄩2
󵄩󵄩
󵄩
󵄩󵄩2
𝑗=1
󵄩
󵄩
󵄩
+ 𝑐󵄩󵄩󵄩󵄩𝐴1/2 V + 𝐴1/2 𝑧 (𝜃𝑡 𝜔)󵄩󵄩󵄩󵄩2
󵄩󵄩
󵄩󵄩
𝑛
󵄩󵄩
󵄩󵄩
× 󵄩󵄩󵄩󵄩𝛼02 V + 𝛼12 𝐴V + ∑ ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄩󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩
𝑗=1
󵄩
󵄩2
]𝛼02 󵄩󵄩
󵄩
𝑛
]𝛼02 󵄩󵄩 1/2 󵄩󵄩2
󵄨
󵄨2
󵄩󵄩𝐴 V2 󵄩󵄩 + 𝑐 ∑󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨 ,
󵄩2
󵄨
󵄨
8 󵄩
𝑗=1
2
󵄨 󵄩 󵄩 󵄩 󵄩 ]𝛼 󵄩 1/2 󵄩2 󵄩 󵄩2
󵄨󵄨
󵄨󵄨(𝑃𝑓, V2 )󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉󸀠 󵄩󵄩󵄩V2 󵄩󵄩󵄩 ≤ 0 󵄩󵄩󵄩󵄩𝐴 V2 󵄩󵄩󵄩󵄩2 + 𝑐󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉󸀠 ,
8
󵄨󵄨
󵄨󵄨 𝑛
󵄨󵄨
󵄨󵄨
󵄨
󵄨
𝜇 󵄨󵄨( ∑ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 ) , V2 )󵄨󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑗=1
󵄨
󵄨
≤
𝑛
≤
𝑗=1
2
𝑛
󵄨
󵄨4
+ 𝑐 ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨
𝑗=1
(96)
Abstract and Applied Analysis
13
𝑛
󵄨
󵄩
󵄨2
󵄩2
+ 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 (𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
Replacing 𝜔 by 𝜃−𝑡−2 𝜔 in (84), we have
𝑗=1
󵄩
󵄩2
󵄩
󵄩2
𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V(𝑡 + 1, 𝜃−𝑡−2 𝜔)󵄩󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩𝐴V(𝑡 + 1, 𝜃−𝑡−2 𝜔)󵄩󵄩󵄩2
𝑛
󵄨
󵄩
󵄨2
󵄩2
+ 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V‖22 ) .
𝑗=1
(97)
Applying Gronwall Lemma on [𝑡, 𝑡 + 1] where 𝑡 ≥ 𝑇3 (𝜔), we
have
󵄩
󵄩2
󵄩
󵄩2
𝛼02 󵄩󵄩󵄩V2 (𝑡 + 1, 𝜔)󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V2 (𝑡 + 1, 𝜔)󵄩󵄩󵄩󵄩2
󵄩2
󵄩
󵄩2
󵄩
≤ 𝑒−]𝜆 𝑚+1 (𝛼02 󵄩󵄩󵄩V2 (𝑡, 𝜔)󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V2 (𝑡, 𝜔)󵄩󵄩󵄩󵄩2 )
(98)
𝑡+1
+ ∫ 𝐺 (𝑠, 𝜔) 𝑒−]𝜆 𝑚+1 𝑑𝑠
𝑡
𝑡+1
+ 𝑐∫
𝑡
𝑡
+ 𝑐∫
−]𝜆 𝑚+1
𝐺 (𝑠, 𝜃−𝑡−1 𝜔) 𝑒
𝑡+1
𝑡
𝑡+1 𝑛
󵄨
󵄨2
∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏−𝑡−2 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑒𝐽2 (𝜏,𝑡+1,𝜃−𝑡−2 𝜔) 𝑑𝜏
𝑡
+ 𝑐∫
𝑗=1
𝑡+1 𝑛
󵄨
󵄨4
∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏−𝑡−2 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑒𝐽2 (𝜏,𝑡+1,𝜃−𝑡−2 𝜔) 𝑑𝜏
𝑡
𝑗=1
𝑡+1 𝑛
󵄨
󵄨2
∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏−𝑡−2 𝜔𝑗 )󵄨󵄨󵄨󵄨
𝑗=1
󵄩
󵄩2
× (𝛼02 󵄩󵄩󵄩V(𝜏, 𝜃−𝑡−2 𝜔)󵄩󵄩󵄩2
󵄩
󵄩2
+ 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V(𝜏, 𝜃−𝑡−2 𝜔)󵄩󵄩󵄩󵄩2 ) 𝑒𝐽2 (𝜏,𝑡+1,𝜃−𝑡−2 𝜔) 𝑑𝜏
+ 𝑐∫
󵄩
󵄩2
(𝛼02 󵄩󵄩󵄩V2 (𝑡, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2
𝑡+1
𝑡
󵄩
󵄩2
+𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V2 (𝑡, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 )
+∫
+ 𝑐∫
𝑡
Replacing 𝜔 by 𝜃−𝑡−1 𝜔 in (98), we have
󵄩
󵄩2
𝛼02 󵄩󵄩󵄩V2 (𝑡 + 1, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2
󵄩
󵄩2
+ 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V2 (𝑡 + 1, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2
𝑡+1
󵄩
󵄩2
+𝛼12 󵄩󵄩󵄩𝐴V(𝑠, 𝜃−𝑡−2 𝜔)󵄩󵄩󵄩2 ) 𝑒𝐽2 (𝑠,𝑡+1,𝜃−𝑡−2 𝜔)
+ 𝑐∫
󵄩2 −]𝜆
󵄩󵄩
󵄩󵄩𝑓(𝑠)󵄩󵄩󵄩𝑉󸀠 𝑒 𝑚+1 𝑑𝑠.
≤ 𝑒−]𝜆 𝑚+1
󵄩
󵄩2
≤ (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V(𝑠, 𝜃−𝑡−2 𝜔)󵄩󵄩󵄩󵄩2
󵄩
󵄩2
(𝛼02 󵄩󵄩󵄩V(𝜏, 𝜃−𝑡−2 𝜔)󵄩󵄩󵄩2
󵄩
󵄩2 2
+𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V(𝜏, 𝜃−𝑡−2 𝜔)󵄩󵄩󵄩󵄩2 ) 𝑒𝐽2 (𝜏,𝑡+1,𝜃−𝑡−2 𝜔) 𝑑𝜏
(99)
+ 𝑐∫
𝑑𝑠
𝑡+1
𝑡
󵄩2 −]𝜆
󵄩󵄩
󵄩󵄩𝑓(𝑠)󵄩󵄩󵄩𝑉󸀠 𝑒 𝑚+1 𝑑𝑠.
󵄩2 𝐽 (𝜏,𝑡+1,𝜃−𝑡−2 𝜔)
󵄩󵄩
𝑑𝜏.
󵄩󵄩𝑓(𝜏)󵄩󵄩󵄩𝑉󸀠 𝑒 2
(103)
Analogously to the previous estimates, we have
Now we need to estimate the second term on the right-hand
side of (99). As the previous consideration in Lemma 15, for
𝑠 ∈ [𝑡, 𝑡 + 1) where 𝑡 ≥ 𝑇3 (𝜔), we have
󵄩
󵄩2
󵄩
󵄩2
(100)
𝛼02 󵄩󵄩󵄩V(𝑠, 𝜃−𝑡−2 )󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑠, 𝜃−𝑡−2 )󵄩󵄩󵄩󵄩2 ≤ 𝑟9 (𝜔) ,
𝐽2 (𝜏, 𝑡 + 1, 𝜃−𝑡−2 𝜔)
𝑡+1
≤ 𝑐∫
𝑡
󵄩
󵄩2
(𝛼02 󵄩󵄩󵄩V (𝜏, 𝜃−𝑡−2 𝜔)󵄩󵄩󵄩2
󵄩
󵄩2
+𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V(𝜏, 𝜃−𝑡−2 𝜔)󵄩󵄩󵄩󵄩2 ) 𝑑𝜏
where 𝑟9 (𝜔) = 𝑒2]𝜆 1 𝑟4 (𝜔). As the previous consideration in
Lemma 16, we have
∫
𝑡+1
𝑡
󵄩
󵄩2
(𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V(𝑠, 𝜃−𝑡−2 𝜔)󵄩󵄩󵄩󵄩2
󵄩
󵄩2
+𝛼12 󵄩󵄩󵄩𝐴V(𝑠, 𝜃−𝑡−2 𝜔)󵄩󵄩󵄩2 ) 𝑑𝑠 ≤ 𝑟10 (𝜔) ,
+ 𝑐∫
𝑡+1
𝑡
(101)
𝑛
0
𝑟10 (𝜔) = 𝑐𝑟9 (𝜔) ∑ ∫
𝑗=1 −2
𝑛
+ 𝑐∑ ∫
󵄨󵄨
󵄨
󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨 𝑑𝑠
󵄨
󵄨
𝑡+1
𝑗=1 𝑡
󵄨󵄨
󵄨2
󵄨󵄨𝑦𝑗 (𝜃𝜏−𝑡−2 𝜔𝑗 )󵄨󵄨󵄨 𝑑𝜏
󵄨
󵄨
≤ 𝑟11 (𝜔) ,
0
󵄨
󵄨2
+ 𝑐𝑟9 (𝜔) ∑ ∫ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝑠
−2
𝑗=1
𝑛
0
󵄨
󵄨2
+ 𝑐 ∑ ∫ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝑠 + 𝑟9 (𝜔) .
−2
𝑗=1
(104)
󵄩
󵄩2
+𝛼12 󵄩󵄩󵄩𝐴V (𝜏, 𝜃−𝑡−2 𝜔)󵄩󵄩󵄩2 ) 𝑑𝜏
where
𝑛
󵄩
󵄩2
(𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V (𝜏, 𝜃−𝑡−2 𝜔)󵄩󵄩󵄩󵄩2
(102)
where
𝑛
0
󵄨
󵄨2
𝑟11 (𝜔) = 𝑐𝑟9 (𝜔) + 𝑐𝑟10 (𝜔) + 𝑐 ∑ ∫ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏. (105)
−2
𝑗=1
14
Abstract and Applied Analysis
Integrating (103) with respect to 𝑠 ∈ (𝑡, 𝑡+1) associating (100),
(101), and (104) with (68), we have
󵄩
󵄩2
󵄩
󵄩2
𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V(𝑡 + 1, 𝜃−𝑡−2 𝜔)󵄩󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩𝐴V(𝑡 + 1, 𝜃−𝑡−2 𝜔)󵄩󵄩󵄩2
≤ 𝑟12 (𝜔) 𝑒𝑟7 (𝜔)
0 𝑛
󵄨
󵄨2
+ 𝑐𝑒𝑟7 (𝜔) ∫ ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏
−1
𝑗=1
0 𝑛
󵄨
󵄨4
+ 𝑐𝑒𝑟7 (𝜔) ∫ ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏 + 𝑐𝑒𝑟7 (𝜔) 𝑀𝑓0
−1
(106)
≤ 𝑟12 (𝜔) ,
𝑗=1
+ 𝑐𝑟5 (𝜔)2 𝑒𝑟7 (𝜔)
where
0 𝑛
󵄨
󵄨2
+ 𝑐𝑟5 (𝜔) 𝑒𝑟7 (𝜔) ∫ ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏
−1
𝑟11 (𝜔)
𝑟12 (𝜔) = 𝑐𝑟10 (𝜔) 𝑒
𝑗=1
𝑛
0
󵄨
󵄨2
+ 𝑐𝑒𝑟11 (𝜔) ∫ ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏
−1
󳵻
= 𝑟13 (𝜔) .
𝑗=1
(108)
𝑛
0
󵄨
󵄨4
+ 𝑐𝑒𝑟11 (𝜔) ∫ ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏
−1
𝑗=1
(107)
Associating with the above inequalities, we have
0 𝑛
󵄨
󵄨2
+ 𝑐𝑟9 (𝜔) 𝑒𝑟11 (𝜔) ∫ ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝜏
−1
𝑡+1
∫
𝑡
𝑗=1
+ 𝑐𝑟9 (𝜔)2 𝑒𝑟11 (𝜔) + 𝑐𝑀𝑓0 𝑒𝑟11 (𝜔) .
𝑟14 (𝜔) = 𝑐𝑟5 (𝜔)2 + 𝑐𝑟13 (𝜔)2
0 𝑛
󵄨
󵄨2
+ 𝑐 ∫ ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝑠
−1
𝑗=1
󵄩
󵄩2
󵄩
󵄩2
𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V (𝑠, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩𝐴V(𝑠, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2
+ 𝑐𝑟5 (𝜔) 𝑟13 (𝜔)
󵄩
󵄩2
(𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V(𝑡, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2
0 𝑛
󵄨
󵄨4
+ 𝑐 ∫ ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝑠
−1
󵄩
󵄩2
+𝛼12 󵄩󵄩󵄩𝐴V(𝑡, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2 ) 𝑒𝐽2 (𝑡,𝑡+1,𝜃−𝑡−1 𝜔)
0 𝑛
󵄨
󵄨2
+ 𝑐𝑟5 (𝜔) ∫ ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝑠
−1
󵄨
󵄨2
∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏−𝑡−1 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑒𝐽2 (𝜏,𝑡+1,𝜃−𝑡−1 𝜔) 𝑑𝜏
𝑡
𝑗=1
𝑗=1
𝑡+1 𝑛
𝑡
𝑗=1
𝑡+1 𝑛
󵄨
󵄨2
∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏−𝑡−1 𝜔𝑗 )󵄨󵄨󵄨󵄨
+ 𝑐∫
𝑡
𝑗=1
󵄩
󵄩2
× (𝛼02 󵄩󵄩󵄩V (𝜏, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2
󵄩
󵄩2
+𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V(𝜏, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 ) 𝑒𝐽2 (𝜏,𝑡+1,𝜃−𝑡−1 𝜔) 𝑑𝜏
𝑡+1
+ 𝑐∫
𝑡
𝑡+1
𝑡
𝑗=1
According to the property of {𝜆 𝑗 }∞
𝑗=1 mentioned in Section 3,
for P a.e 𝜔 ∈ Ω and ∀𝜀 > 0, there exists 𝑀 such that for
𝑚≥𝑀
󵄩
󵄩2
󵄩
󵄩2
𝑒−]𝜆 𝑚+1 (𝛼02 󵄩󵄩󵄩V2 (𝑡, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V2 (𝑡, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 )
≤
󵄩
󵄩2
(𝛼02 󵄩󵄩󵄩V(𝜏, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2
󵄩
󵄩2 2
+𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V(𝜏, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 ) 𝑒𝐽2 (𝜏,𝑡+1,𝜃−𝑡−1 𝜔) 𝑑𝜏
+ 𝑐∫
0 𝑛
󵄨
󵄨2
+ 𝑐𝑟13 (𝜔) ∫ ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑠 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑑𝑠.
−1
󵄨
󵄨4
∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝜏−𝑡−1 𝜔𝑗 )󵄨󵄨󵄨󵄨 𝑒𝐽2 (𝜏,𝑡+1,𝜃−𝑡−1 𝜔) 𝑑𝜏
+ 𝑐∫
󵄩2 𝐽 (𝜏,𝑡+1,𝜃−𝑡−1 𝜔)
󵄩󵄩
𝑑𝜏
󵄩󵄩𝑓(𝜏)󵄩󵄩󵄩𝑉󸀠 𝑒 2
(110)
𝑗=1
𝑡+1 𝑛
+ 𝑐∫
(109)
where
Replacing 𝑠, 𝑡 + 1, and 𝜔 by 𝑡, 𝑠, and 𝜃−𝑡−1 𝜔, respectively, in
(84) where 𝑠 ∈ (𝑡, 𝑡 + 1) and 𝑡 ≥ 𝑇3 (𝜔), we have
≤
𝐺 (𝑠, 𝜃−𝑡−1 𝜔) 𝑑𝑠 ≤ 𝑟14 (𝜔) ,
∫
𝑡+1
𝑡
𝑟5 (𝜔)
≤ 𝜀𝑟5 (𝜔) ,
𝑒]𝜆 𝑚+1
−]𝜆 𝑚+1
𝐺 (𝑠, 𝜃−𝑡−1 𝜔) 𝑒
𝑐∫
𝑡+1
𝑡
𝑟 (𝜔)
𝑑𝑠 ≤ 14]𝜆
≤ 𝜀𝑟14 (𝜔) ,
𝑒 𝑚+1
𝑐𝑀𝑓
󵄩2 −]𝜆
󵄩󵄩
󵄩󵄩𝑓(𝑠)󵄩󵄩󵄩𝑉󸀠 𝑒 𝑚+1 𝑑𝑠 ≤ ]𝜆 0 ≤ 𝜀.
𝑒 𝑚+1
(111)
Abstract and Applied Analysis
15
Therefore, we deduce from (99) that, for P a.e. 𝜔 ∈ Ω,
+ (𝑃𝑓 (𝑥, 𝑡) , 𝐴V2 )
𝑛
+ 𝜇 ( ∑ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 ) , 𝐴V2 ) .
󵄩
󵄩2
󵄩
󵄩2
𝛼02 󵄩󵄩󵄩V2 (𝑡, 𝜃−𝑡 𝜔)󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V2 (𝑡, 𝜃−𝑡 𝜔)󵄩󵄩󵄩󵄩2
≤ 𝜀 + 𝜀𝑟5 (𝜔) + 𝜀𝑟14 (𝜔) ,
𝑗=1
(114)
∀𝑡 ≥ 𝑇3 (𝜔) , 𝑓 ∈ Σ (𝑓0 ) ,
(112)
which indicates {U𝑓 (𝑡, 𝜏, 𝜔)}, 𝑓 ∈ Σ(𝑓0 ) satisfying uniform
(with respect to 𝑓 ∈ Σ(𝑓0 )) condition(C) in 𝑉. According to
Theorem 12, the proof is completed.
Applying Lemma 14, the second term on the right-hand side
of (114) can be bounded by
󵄨󵄨
󵄨󵄨
󵄨󵄨 ̃
2
2
󵄨󵄨(𝐵 (V + 𝑧 (𝜃𝑡 𝜔) , 𝛼0 V + 𝛼1 𝐴V
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
+ ∑ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 )) , 𝐴V2 )󵄨󵄨󵄨󵄨
󵄨󵄨
𝑗=1
󵄨
In the following we prove the existence of uniformly
random attractor for the families of processes {U𝑓 (𝑡, 𝜏, 𝜔)},
𝑓 ∈ Σ(𝑓0 ) corresponding to (48) in 𝐷(𝐴).
Theorem 18. If 𝑓0 (𝑥, 𝑠) is a normal function in 𝐿2loc (R, 𝑉󸀠 ),
then the process {U𝑓0 (𝑡, 𝜏, 𝜔)} corresponding to (48) possess a
compact uniformly (with respect to 𝜏 ∈ R) random attractor
A2 (𝜔) in 𝐷(𝐴) which coincides with the uniformly (with
respect to 𝑓 ∈ Σ(𝑓0 )) random attractor AΣ(𝑓0 ) of the family
of processes {U𝑓 (𝑡, 𝜏, 𝜔)}, 𝑓 ∈ Σ(𝑓0 ) :
𝑛
≤
󵄩
󵄩2
× (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V‖22 )
A2 (𝜔) = AΣ(𝑓0 ) (𝜔) = W0,Σ(𝑓0 ) (K2 ) = ⋃ N𝑓 (0, 𝜔) ,
𝑓∈Σ(𝑓0 )
󵄩
󵄩2 2
+ 𝑐(𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
(113)
where K2 (𝜔) is the uniformly (with respect to 𝑓 ∈ Σ(𝑓0 ))
random absorbing set in 𝐷(𝐴) and N𝑓 (𝜔) is the kernel of
the process {U𝑓 (𝑡, 𝜏, 𝜔)}. Furthermore, the kernel N𝑓 (𝜔) is
nonempty for all 𝑓 ∈ Σ(𝑓0 ).
2
󵄩
󵄩2
+ 𝑐(𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V‖22 )
𝑛
𝑛
󵄨
󵄨
󵄨2
󵄨4
+ 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 + 𝑐 ∑ 󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨
Proof. In Lemma 16, we have proved that the semigroup of
processes {U𝑓 (𝑡, 𝜏, 𝜔)}, 𝑓 ∈ Σ(𝑓0 ), has a uniformly random
absorbing set in 𝐷(𝐴). Now we testify that the semigroup
of processes corresponding to (48) satisfies uniform (with
respect to 𝑓 ∈ Σ(𝑓0 )) condition (C). Analogously to the proof
in Lemma 16, we easily check that V1 is bounded in 𝐷(𝐴).
Letting 𝑤 = 𝐴V2 in (51), we have
𝑗=1
𝑛
= −] ( ∑𝐴ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 ) , 𝐴V2 )
𝑗=1
− (𝐵̃ (V + 𝑧 (𝜃𝑡 𝜔) , 𝛼02 V + 𝛼12 𝐴V
𝑛
+ ∑ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 )) , 𝐴V2 )
𝑗=1
𝑗=1
𝑛
󵄨
󵄩
󵄨2
󵄩2 2
+ 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 (𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
𝑗=1
𝑛
2
󵄨
󵄩
󵄨2
󵄩2
+ 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 + 𝛼12 ‖𝐴V‖22 ) .
𝑗=1
1 𝑑
󵄩
󵄩2
󵄩
󵄩2
(𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V2 󵄩󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩𝐴V2 󵄩󵄩󵄩2 )
2 𝑑𝑡
󵄩
󵄩2
󵄩
󵄩2
+ ] (𝛼02 󵄩󵄩󵄩𝐴V2 󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩󵄩𝐴3/2 V2 󵄩󵄩󵄩󵄩2 )
2
]𝛼02 󵄩󵄩
󵄩2 ]𝛼 󵄩 3/2 󵄩2
󵄩󵄩𝐴V2 󵄩󵄩󵄩2 + 1 󵄩󵄩󵄩󵄩𝐴 V2 󵄩󵄩󵄩󵄩2
8
2
󵄩
󵄩2
+ 𝑐 (𝛼02 ‖V‖22 + 𝛼12 󵄩󵄩󵄩󵄩𝐴1/2 V󵄩󵄩󵄩󵄩2 )
(115)
Note that
󵄨󵄨 𝑛
󵄨󵄨
𝑛
󵄨󵄨
󵄨󵄨 ]𝛼2 󵄩
󵄨
󵄨2
󵄩2
󵄨
󵄨
] 󵄨󵄨( ∑ ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔j ) , 𝐴V2 )󵄨󵄨󵄨󵄨 ≤ 0 󵄩󵄩󵄩𝐴V2 󵄩󵄩󵄩2 + 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 ,
8
󵄨󵄨 𝑗=1
󵄨󵄨
𝑗=1
󵄨
󵄨
]𝛼2 󵄩
󵄨󵄨
󵄩
󵄩2 󵄩 󵄩 2
󵄨 󵄩 󵄩 󵄩
󵄨󵄨(𝑃𝑓 (𝑥, 𝑡) , 𝐴V2 )󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉󸀠 󵄩󵄩󵄩𝐴V2 󵄩󵄩󵄩2 ≤ 0 󵄩󵄩󵄩𝐴V2 󵄩󵄩󵄩2 + 𝑐󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉󸀠 ,
8
󵄨󵄨 𝑛
󵄨󵄨
𝑛
󵄨󵄨
󵄨󵄨 ]𝛼2 󵄩
󵄨
󵄨2
󵄩2
𝜇 󵄨󵄨󵄨󵄨( ∑ℎ𝑗 𝑦𝑗 (𝜃𝑡 𝜔𝑗 ) , 𝐴V2 )󵄨󵄨󵄨󵄨 ≤ 0 󵄩󵄩󵄩𝐴V2 󵄩󵄩󵄩2 + 𝑐 ∑󵄨󵄨󵄨󵄨𝑦𝑗 (𝜃𝑡 𝜔𝑗 )󵄨󵄨󵄨󵄨 .
8
󵄨󵄨 𝑗=1
󵄨󵄨
𝑗=1
󵄨
󵄨
(116)
16
Abstract and Applied Analysis
Associating with (114)–(116) and applying Poincaré’s inequality, we have
𝑑
󵄩
󵄩2
󵄩
󵄩2
(𝛼2 󵄩󵄩󵄩𝐴1/2 V2 󵄩󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩𝐴V2 󵄩󵄩󵄩2 )
𝑑𝑡 0 󵄩
󵄩
󵄩2
󵄩
󵄩2
+ ]𝜆 𝑚+1 (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V2 󵄩󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩𝐴V2 󵄩󵄩󵄩2 )
(117)
Applying Gronwall’s lemma over 𝑠 ∈ [𝑡, 𝑡+1] where 𝑡 ≥ 𝑇2 (𝜔),
we have
󵄩
󵄩2
󵄩
󵄩2
𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V2 (𝑡 + 1, 𝜔)󵄩󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩𝐴V2 (𝑡 + 1, 𝜔)󵄩󵄩󵄩2
+∫
𝑡+1
𝑡
𝐺 (𝑠, 𝜔) 𝑒−]𝜆 𝑚+1 𝑑𝑠 + 𝑐 ∫
𝑡+1
𝑡
(118)
󵄩
󵄩2
𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V2 (𝑡 + 1, 𝜃−t−1 𝜔)󵄩󵄩󵄩󵄩2
References
󵄩
󵄩2
+ 𝛼12 󵄩󵄩󵄩𝐴V2 (𝑡 + 1, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2
󵄩
󵄩2
≤ (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V2 (𝑡, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2
𝑡+1
𝑡
+ 𝑐∫
𝑡
(119)
𝐺 (𝑠, 𝜃−𝑡−1 𝜔) 𝑒−]𝜆 𝑚+1 𝑑𝑠
𝑡+1
󵄩󵄩 󵄩󵄩2 −]𝜆 𝑚+1
𝑑𝑠.
󵄩󵄩𝑓󵄩󵄩𝑉󸀠 𝑒
Analogously to the consideration in Theorem 17, for P a.e.
𝜔 ∈ Ω and ∀𝜀 > 0, there exists 𝑀 such that, for 𝑚 > 𝑀,
󵄩
󵄩2
󵄩2
󵄩
𝑒−]𝜆 𝑚+1 (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V2 (𝑡, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩𝐴V2 (𝑡, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2 )
≤
∫
𝑡+1
𝑡
𝑟12 (𝜔)
≤ 𝜀𝑟12 (𝜔) ,
𝑒]𝜆 𝑚+1
𝐺 (𝑠, 𝜃−𝑡−1 𝜔) 𝑒−]𝜆 𝑚+1 𝑑𝑠 ≤
𝑐∫
𝑡+1
𝑡
𝑟14 (𝜔)
≤ 𝜀𝑟14 (𝜔) ,
𝑒]𝜆 𝑚+1
𝑐𝑀𝑓
󵄩2 −]𝜆
󵄩󵄩
󵄩󵄩𝑓(𝑠)󵄩󵄩󵄩𝑉󸀠 𝑒 𝑚+1 𝑑𝑠 ≤ ]𝜆 0 ≤ 𝜀.
𝑒 𝑚+1
(120)
Therefore, we deduce from (119) that, for P a.e 𝜔 ∈ Ω,
󵄩
󵄩2
󵄩
󵄩2
𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V2 (𝑡 + 1, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩𝐴V2 (𝑡 + 1, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2
≤ 𝜀 + 𝜀𝑟12 (𝜔) + 𝜀𝑟14 (𝜔) ,
generates a process corresponding to (8). Note that the two
processes are equivalent by (122). It is easy to check that
V𝜎 (𝑡, 𝜏, 𝜔) has a uniformly (with respect to 𝜎 ∈ Σ) random
attractor provided U𝜎 (𝑡, 𝜏, 𝜔) possesses a uniformly (with
respect to 𝜎 ∈ Σ) random attractor. As a result Theorem 17
and Theorem 18 imply that (8) has a uniformly (with respect
to 𝑓 ∈ Σ(𝑓0 )) random attractor in 𝑉 and 𝐷(𝐴).
The authors would like to thank the reviewers for the valuable
suggestions and comments. The work is supported by the
National Natural Science Foundation (no. 11171115).
Replacing 𝜔 by 𝜃−𝑡−1 𝜔 in (118), we have
+∫
(122)
Acknowledgments
󵄩󵄩 󵄩󵄩2 −]𝜆 𝑚+1
𝑑𝑠.
󵄩󵄩𝑓󵄩󵄩𝑉󸀠 𝑒
󵄩
󵄩2
+𝛼12 󵄩󵄩󵄩𝐴V2 (𝑡, 𝜃−𝑡−1 𝜔)󵄩󵄩󵄩2 ) 𝑒−]𝜆 𝑚+1
Now we introduce a homeomorphism Φ(𝜃𝑡 𝜔)𝑢 = 𝑢 +
𝑧(𝜃𝑡 𝜔), 𝑢 ∈ 𝐸, whose inverse homeomorphism Φ−1 (𝜃𝑡 𝜔)𝑢 =
𝑢 − 𝑧(𝜃𝑡 𝜔). Then the transformation
V𝜎 (𝑡, 𝜏, 𝜔) = Φ (𝜃𝑡 𝜔) ∘ U𝜎 (𝑡, 𝜏, 𝜔) ∘ Φ−1 (𝜃𝜏 𝜔)
󵄩 󵄩2
≤ 𝐺 (𝑡, 𝜔) + 𝑐󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝑉󸀠 .
󵄩2
󵄩
󵄩2
󵄩
≤ (𝛼02 󵄩󵄩󵄩󵄩𝐴1/2 V2 (𝑡, 𝜔)󵄩󵄩󵄩󵄩2 + 𝛼12 󵄩󵄩󵄩𝐴V2 (𝑡, 𝜔)󵄩󵄩󵄩2 ) 𝑒−]𝜆 𝑚+1
which indicates {U𝑓 (𝑡, 𝜏, 𝜔)}, 𝑓 ∈ Σ(𝑓0 ) satisfying uniform
(with respect to 𝑓 ∈ Σ(𝑓0 )) condition (C) in 𝐷(𝐴). According
to Theorem 12, the proof is completed.
∀𝑡 ≥ 𝑇3 (𝜔) , 𝑓 ∈ Σ (𝑓0 ) .
(121)
[1] R. Camassa and D. D. Holm, “An integrable shallow water
equation with peaked solitons,” Physical Review Letters, vol. 71,
no. 11, pp. 1661–1664, 1993.
[2] R. Camassa, D. D. Holm, and J. M. Hyman, “An new integrable
shallow water equation,” Advances in Applied Mechanics, vol. 31,
pp. 1–33, 1994.
[3] H. R. Dullin, G. A. Gottwald, and D. D. Holm, “On asymptotically equivalent shallow water wave equations,” Physica D, vol.
190, no. 1-2, pp. 1–14, 2004.
[4] H. R. Dullin, G. A. Gottwald, and D. D. Holm, “Camassa-Holm,
Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves,” Fluid Dynamics Research, vol. 33,
no. 1-2, pp. 73–95, 2003.
[5] A. Constantin and W. A. Strauss, “Stability of the CamassaHolm solitons,” Journal of Nonlinear Science, vol. 12, no. 4, pp.
415–422, 2002.
[6] A. Bressan and A. Constantin, “Global conservative solutions
of the Camassa-Holm equation,” Archive for Rational Mechanics
and Analysis, vol. 183, no. 2, pp. 215–239, 2007.
[7] A. Constantin and W. A. Strauss, “Stability of peakons,” Communications on Pure and Applied Mathematics, vol. 53, no. 5, pp.
603–610, 2000.
[8] A. Constantin and J. Escher, “Global existence and blow-up for a
shallow water equation,” Annali della Scuola Normale Superiore
di Pisa, vol. 26, no. 2, pp. 303–328, 1998.
[9] G. Rodrı́guez-Blanco, “On the Cauchy problem for the
Camassa-Holm equation,” Nonlinear Analysis, vol. 46, no. 3, pp.
309–327, 2001.
[10] A. Constantin and J. Escher, “Wave breaking for nonlinear
nonlocal shallow water equations,” Acta Mathematica, vol. 181,
no. 2, pp. 229–243, 1998.
[11] Z. Xin and P. Zhang, “On the weak solutions to a shallow water
equation,” Communications on Pure and Applied Mathematics,
vol. 53, no. 11, pp. 1411–1433, 2000.
Abstract and Applied Analysis
[12] A. Bressan and A. Constantin, “Global dissipative solutions of
the Camassa-Holm equation,” Analysis and Applications, vol. 5,
no. 1, pp. 1–27, 2007.
[13] Y. Long, Z. Li, and W. Rui, “New travelling wave solutions
for a nonlinearly dispersive wave equation of Camassa-Holm
equation type,” Applied Mathematics and Computation, vol. 217,
no. 4, pp. 1315–1320, 2010.
[14] E. A. Olson, “Well posedness for a higher order modified
Camassa-Holm equation,” Journal of Differential Equations, vol.
246, no. 10, pp. 4154–4172, 2009.
[15] L. Tian, C. Shen, and D. Ding, “Optimal control of the viscous
Camassa-Holm equation,” Nonlinear Analysis, vol. 10, no. 1, pp.
519–530, 2009.
[16] D. Henry, “Infinite propagation speed for a two component
Camassa-Holm equation,” Discrete and Continuous Dynamical
Systems B, vol. 12, no. 3, pp. 597–606, 2009.
[17] J. Escher, O. Lechtenfeld, and Z. Yin, “Well-posedness and blowup phenomena for the 2-component Camassa-Holm equation,”
Discrete and Continuous Dynamical Systems A, vol. 19, no. 3, pp.
493–513, 2007.
[18] J. B. Li and Y. S. Li, “Bifurcations of travelling wave solutions for
a two-component Camassa-Holm equation,” Acta Mathematica
Sinica, vol. 24, no. 8, pp. 1319–1330, 2008.
[19] C. Foias, D. D. Holm, and E. S. Titi, “The three dimensional
viscous Camassa-Holm equations, and their relation to the
Navier-Stokes equations and turbulence theory,” Journal of
Dynamics and Differential Equations, vol. 14, no. 1, pp. 1–35,
2002.
[20] T. Caraballo, A. M. Márquez-Durán, and J. Real, “The asymptotic behaviour of a stochastic 3D LANS-𝛼 model,” Applied
Mathematics and Optimization, vol. 53, no. 2, pp. 141–161, 2006.
[21] T. Caraballo, A. M. Márquez-Durán, and J. Real, “Pullback and
forward attractors for a 3D LANS-𝛼 model with delay,” Discrete
and Continuous Dynamical Systems A, vol. 15, no. 2, pp. 559–578,
2006.
[22] D. Wu, “Finite dimensional uniform attractors for the nonautonomous Camassa-Holm equations,” Abstract and Applied
Analysis, vol. 2009, Article ID 952657, 15 pages, 2009.
[23] T. Tachim Medjo, “Averaging of a 3D Lagrangian averaged
Navier-Stokes-𝛼 model with oscillating external forces,” Communications on Pure and Applied Analysis, vol. 10, no. 4, pp.
1281–1305, 2011.
[24] T. Tachim Medjo, “A non-autonomous 3D Lagrangian averaged
Navier-Stokes-𝛼 model with oscillating external force and its
global attractor,” Communications on Pure and Applied Analysis,
vol. 10, no. 2, pp. 415–433, 2011.
[25] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,
North-Holland, Amsterdam, The Netherlands, 1992.
[26] A. Haraux, Systèmes Dynamiques Dissipatifs et Applications,
Masson, Paris, France, 1991.
[27] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of
Mathematical Physics, American Mathematical Society, Providence, RI, USA, 2002.
[28] L. Arnold, Random Dynamical Systems, Springer, Berlin, Germany, 1998.
[29] H. Crauel and F. Flandoli, “Attractors for random dynamical
systems,” Probability Theory and Related Fields, vol. 100, no. 3,
pp. 365–393, 1994.
[30] I. Chueshov, Monotone Random Systems Theory and Applications, Springer, Berlin, Germany, 2002.
17
[31] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite
Dimensions, Cambridge University Press, Cambridge, UK, 1992.
[32] B. Øksendal, Stochastic Differential Equations, Springer, Berlin,
Germany, 2000.
[33] H. Crauel, A. Debussche, and F. Flandoli, “Random attractors,”
Journal of Dynamics and Differential Equations, vol. 9, no. 2, pp.
307–341, 1997.
[34] F. Flandoli and B. Schmalfuss, “Random attractors for the
3D stochastic Navier-Stokes equation with multiplicative white
noise,” Stochastics and Stochastics Reports, vol. 59, no. 1-2, pp.
21–45, 1996.
[35] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American
Mathematical Society, Providence, RI, USA, 1988.
[36] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, UK, 2001.
[37] G. R. Sell and Y. You, Dynamics of Evolutionary Equations,
Springer, New York, NY, USA, 2002.
[38] R. Teman, Infinite-Dimensional Dynamical Systems in Machanics and Physics, Springer, New York, NY, USA, 1998.
[39] A. O. Çelebi, V. K. Kalantarov, and M. Polat, “Attractors for
the generalized Benjamin-Bona-Mahony equation,” Journal of
Differential Equations, vol. 157, no. 2, pp. 439–451, 1999.
[40] F. Morillas and J. Valero, “Attractors for reaction-diffusion
equations in R𝑛 with continuous nonlinearity,” Asymptotic
Analysis, vol. 44, no. 1-2, pp. 111–130, 2005.
[41] S. Lu, H. Wu, and C. Zhong, “Attractors for nonautonomous 2D
Navier-Stokes equations with normal external forces,” Discrete
and Continuous Dynamical Systems A, vol. 13, no. 3, pp. 701–719,
2005.
[42] O. Ladyzhenskaya, Attractors for Semigroups and Evolution
Equations, Cambridge University Press, Cambridge, UK, 1991.
[43] P. W. Bates, K. Lu, and B. Wang, “Attractors for lattice dynamical
systems,” International Journal of Bifurcation and Chaos in
Applied Sciences and Engineering, vol. 11, no. 1, pp. 143–153, 2001.
[44] P. W. Bates, H. Lisei, and K. Lu, “Attractors for stochastic lattice
dynamical systems,” Stochastics and Dynamics, vol. 6, no. 1, pp.
1–21, 2006.
[45] P. W. Bates, K. Lu, and B. Wang, “Random attractors for stochastic reaction-diffusion equations on unbounded domains,” Journal of Differential Equations, vol. 246, no. 2, pp. 845–869, 2009.
[46] B. Wang, “Random attractors for the stochastic BenjaminBona-Mahony equation on unbounded domains,” Journal of
Differential Equations, vol. 246, no. 6, pp. 2506–2537, 2009.
[47] X. Han, W. Shen, and S. Zhou, “Random attractors for stochastic
lattice dynamical systems in weighted spaces,” Journal of Differential Equations, vol. 250, no. 3, pp. 1235–1266, 2011.
[48] Z. Wang, S. Zhou, and A. Gu, “Random attractor of the
stochastic strongly damped wave equation,” Communications in
Nonlinear Science and Numerical Simulation, vol. 17, no. 4, pp.
1649–1658, 2012.
[49] Z. Wang, S. Zhou, and A. Gu, “Random attractor for a stochastic
damped wave equation with multiplicative noise on unbounded
domains,” Nonlinear Analysis, vol. 12, no. 6, pp. 3468–3482, 2011.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Applied Mathematics
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Mathematical Problems
in Engineering
Journal of
Mathematics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Discrete Dynamics in
Nature and Society
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Abstract and
Applied Analysis
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Journal of
Stochastic Analysis
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014