Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 263, pp. 1–25.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
RANDOM ATTRACTORS FOR STOCHASTIC LATTICE
REVERSIBLE GRAY-SCOTT SYSTEMS WITH ADDITIVE NOISE
HONGYAN LI, JUNYI TU
Abstract. In this article, we prove the existence of a random attractor of
the stochastic three-component reversible Gray-Scott system on infinite lattice with additive noise. We use a transformation of addition involved with
Ornstein-Uhlenbeck process, for proving the pullback absorbing property and
the pullback asymptotic compactness of the reaction diffusion system with
cubic nonlinearity.
1. Introduction
In the previous decades, chemical kinetics has produced a variety of phenomenon which had been translated into challenging mathematical problems. A classical
example is seen in the waves of the Belousov-Zhabotinskii reaction. Other examples have been produced and require fewer species interactions, but still yield very
interesting behavior [26]. The problem of dealing with chemical reactions of systems, together with a number of initial and final products whose concentrations
are assumed to be controlled throughout the reaction process is an important one
under quite realistic conditions [1] and [26]. It is necessary to consider at least a
cubic nonlinearity in the rate equations [28]. These models include the Brusselator
system [1, 13, 31], the Gray-Scott system [16, 18, 24, 34, 35, 39] and the Glycolysis
model [26, 29, 30].
At the same time, the dynamics of infinite lattice systems has drawn much attention from mathematicians and physicists (e.g., [3, 4, 5, 13, 32]). Lattice dynamical
systems (LDSs) are spatiotemporal systems with discretization in some variables
including coupled ODEs /PDEs and coupled map lattices and cellular automata
[7], which occur in a wide variety of applications, ranging from biology [23, 33] to
chemical reaction theory [14, 22], laser system [15], electrical engineering [6], material science [19], image processing and pattern recognition [8], [9]. In this paper,
we will deal with the existence of a random attractor for the stochastic threecomponent reversible Gray-Scott system with additive white noise on an infinite
2010 Mathematics Subject Classification. 37H10, 35B40, 35B41.
Key words and phrases. Random attractor; additive white noise;
stochastic reversible Gray-Scott system.
c
2015
Texas State University.
Submitted July 25, 2014. Published October 8, 2015.
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H. LI, J. TU
EJDE-2015/263
lattice as follows:
dui = [d1 (ui+1 − 2ui + ui−1 ) − (F + k)ui + u2i vi − u3i + N zi + f1i ]dt
+ αi dwi ,
(1.1)
dvi = [d2 (vi+1 − 2vi + vi−1 ) − F vi − u2i vi + u3i + f2i ]dt + αi dwi ,
dzi = [d3 (zi+1 − 2zi + zi−1 ) + kui − (F + N )zi + f3i ]dt + αi dwi ,
for i ∈ Z and t > 0, with initial conditions
ui (0) = ui,0 ,
vi (0) = vi,0 ,
zi (0) = zi,0 ,
i ∈ Z,
2
(1.2)
2
where Z denotes the set of integers, u = (ui )i∈Z ∈ ` , v = (vi )i∈Z ∈ ` and
z = (zi )i∈Z ∈ `2 , d1 , d2 , d3 , F, k, and N are positive constants, f1 = (f1i )i∈Z ∈ `2 ,
f2 = (f2i )i∈Z ∈ `2 , f3 = (f3i )i∈Z ∈ `2 , α = (αi )i∈Z ∈ `2 , {wi |i ∈ Z} is independent
Brownian motions.
A three-component reversible Gray-Scott model was introduced in [24], which is
derived according to the scheme of following two reversible chemical or biochemical
reactions:
2A + B kk1−1 3A, A kk2−2 P.
In the reversible Gray-Scott model (1.1), k is the effective production rate for the
first reaction, 1/F is the mean residence time in the dimensionless unit, N is the
normalized rate of the second reverse reaction, and the parameter α measures the
proportional strength of the white noise dw/dt to the respective components.
Note the general Gray-Scott equations have the cubic terms ±Gu3 , but here
we put G = 1, since the difficulties arise otherwise in the below estimations (3.9),
(4.4) and (4.15). This is one of the key observation in this paper. Equation (1.1)
can be regarded as a discrete analogue of the stochastic three-component reversible
Gray-Scott system on R:
ut = d1 4u − (F + k)u + u2 v − u3 + N z + f1 + αwt ,
vt = d2 4v − F v − u2 v + u3 + f2 + αwt ,
(1.3)
zt = d3 4z + ku − (F + N )z + f3 + αwt .
The asymptotic dynamics of solutions for the reversible Gray-Scott system (where
α = 0) have been studied by several authors, e.g. [16, 18, 21, 34, 35, 37, 39]. The
established results naturally focus on the existence of global attractors by showing
the absorbing property and the asymptotic compactness of the solution semigroups
for autonomous system [18, 21, 34, 35, 37, 39] or the skew-product flow for nonautonomous system [16].
For stochastic three-component reversible Gray-Scott system, the solution mapping defines a random dynamical system, which is a parametric dynamical system.
Random attractors are the appropriate objects for describing asymptotic dynamics
of such a parametric dynamical systems and have been studied by several authors
[17, 38]. Note that in these papers, only multiplicative white noise was considered.
In this paper we study stochastic reversible Gray-Scott system driven by additive
white noise as in (1.1). The impact of these two types of noise on the solutions of
stochastic reversible Gray-Scott system is quite different. When dealing with random attractors of stochastic equations, we usually transform the stochastic equation
into a pathwise one with random parameters. Additive noises are more challenge
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3
than multiplicative noises, there is still open to prove the existence of random attractor for stochastic Brusslator system with additive noise. And if the stochastic
three-component reversible Gray-Scott system is driven by additive white noise,
then there are several additional terms appearing after the equation is transformed
(see (4.1) in Section 4). These terms have great effect on the way to derive uniform
estimates of solutions. This is the topic we want to study as well as one of the
main contributions in this work. Beside this common difficulty, there exist new
challenges in analyzing the second pair of oppositely signed cubic terms ±u3 in
the u-equation and the v-equation, and the occurrence of the third component’s
z-equation, which do not occur in dealing with the case of Gray-Scott equations in
[17]. In addition, for the discrete time models governed by difference equations are
more appropriate than the continuous ones. However, very few investigations are
on this topic, especially for the stochastic three-component reversible Gray-Scott
system with additive white noise on infinite lattice, to the best of our knowledge.
This paper is organized as follows. In Section 2 we recall some basic concepts
and results related to random attractor for random dynamical systems. In Section
3 we formulate the problem and make assumptions to define a random dynamical
system generated by the stochastic three-component reversible Gray-Scott system
in an infinite lattice with additive white noise. In Section 4, we conduct uniform
estimate to prove the pullback absorbing property and the pullback asymptotic
compactness for the random dynamical system. Then the existence of random
attractors for (1.1) on infinite lattices is obtained.
2. Preliminaries
In this section, we introduce the definitions of the random dynamical system and
the random attractor, which are taken from [3, 10, 12, 20].
Let (H, d) be a complete separable metric space, (Ω, F, P) be a probability space,
R+ = [0, ∞).
Definition 2.1. (Ω, F, P, (θt )t∈R ) is called a metric dynamical system if θ : R×Ω →
Ω is (B(R) × F, F) measurable, θ0 = I, θs+t = θs ◦ θt for all s, t ∈ R, and θt P = P
for all t ∈ R.
Definition 2.2. A continuous random dynamical system (RDS) on H over a metric
dynamical system (Ω, F, P, (θt )t∈R ) is a mapping
ϕ : R+ × Ω × H → H,
(t, ω, x) 7→ ϕ(t, ω, x),
+
which is (B(R ) × F × B(H), B(H))-measurable and satisfies, for every ω ∈ Ω,
(i) ϕ(0, ω, ·) is the identity on H;
(ii) Cocycle property: ϕ(t + s, ω, ·) = ϕ(t, θs ω, ϕ(s, ω, ·)) for all t, s ∈ R+ ;
(iii) ϕ(·, ω, ·) : R+ × H → H is strongly continuous.
Definition 2.3. The set A ⊂ Ω is called invariant with respect to (θt )t∈R if for all
t ∈ R, θt−1 A = A.
Definition 2.4. A random bounded set B(ω) ⊂ X is called tempered with respect
to (θt )t∈R if for every ω ∈ Ω,
lim eβt d(B(θ−t ω)) = 0
t→∞
where d(B) = supx∈B kxkX .
for all β > 0,
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H. LI, J. TU
EJDE-2015/263
Definition 2.5. A random set K(ω) is called a pullback absorbing set in D, where
D is a collection of random sets of H, if for all B ∈ D and every ω ∈ Ω, there exists
a tB (ω) > 0 such that
ϕ(t, θ−t ω, B(θ−t ω)) ⊂ K(ω),
for all t ≥ tB (ω).
Definition 2.6. Suppose ϕ(t, ω) is a RDS, a random set A is called a random D
attractor if the following hold:
(i) A(ω) is a random compact set, i.e., ω → d(x, A(ω)) is measurable for every
x ∈ H and A(ω) is compact for every ω ∈ Ω;
(ii) A(ω) is strictly invariant, i.e., for every ω ∈ Ω and all t ≥ 0 one has
ϕ(t, ω, A(ω)) = A(θt ω);
(iii) A(ω) attracts all sets in D, i.e. for all B ∈ D and ω ∈ Ω we have
lim d(ϕ(t, θ−t ω, B(θ−t ω)), A(ω)) = 0,
t→∞
where d(X, Y ) = supx∈X inf y∈Y kx − ykH is the Hausdorff semi-metric
(here, X ⊆ H, Y ⊆ H).
Theorem 2.7 ([3, Proposition 4.1]). Let K(ω) ∈ D be an absorbing set for the
random dynamical system ϕ(t, θ−t ω)t≥0,ω∈Ω which is closed and which satisfies
for ω ∈ Ω the following asymptotic compactness condition: each sequence xn ∈
ϕ(tn , θ−tn ω, K(θ−tn ω)) with tn → ∞ has a convergent subsequence in H. Then the
random dynamical system ϕ has a unique global random attractor
A(ω) = ∩t≥tK (ω) ∪t≥τ ϕ(t, θ−t ω, K(θ−t ω)).
3. Existence and uniqueness of solutions
In this paper, we have the following standing assumption on the constants in
(1.1):
N + k ≤ 2/7F.
(3.1)
which will be used in (3.11), (4.6) and (4.20). We shall use Ci to denote generic or
specific positive constants which do not depend on the proportional strength α of
additive white noise in (1.1).
We first formulate the mathematical setting of our problem (1.1)-(1.2). Set
X
`2 = {u = (ui )i∈Z : ui ∈ R,
|ui |2 < +∞},
i∈Z
equipped it with the inner product and norm as follows:
X
hu, vi =
ui vi , kuk2 = hu, ui, u = (ui )i∈Z , v = (vi )i∈Z ∈ `2 .
i∈Z
Then `2 = (`2 , h·, ·i, k · k) is a Hilbert space. Set E = `2 × `2 × `2 be the product
Hilbert space. In view of the cubic term ±u2 v, ±u3 , we need u ∈ `6 , v ∈ `6 to make
(1.1) hold in `2 .
Define A, B and B ∗ to be linear operators from `2 to `2 as follows:
(Bu)i = ui+1 − ui ,
(B ∗ u)i = ui−1 − ui ,
(Au)i = −ui+1 + 2ui − ui−1 ,
for all i ∈ Z.
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RANDOM ATTRACTORS
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It is easy to show that A = BB ∗ = B ∗ B, and (B ∗ u, v) = (u, Bv) for all u, v ∈ `2 ,
which implies that (Au, u) ≥ 0 for all u ∈ `2 . Indeed, A, B and B ∗ are all bounded
operators from `2 to `2 , because
X
kBuk2 =
|ui+1 − ui |2 ≤ 4kuk2 , ∀u = (ui )i∈Z ∈ `2 ,
i∈Z
∗
2
kAuk2 ≤ kBB ∗ uk ≤ 16kuk2 ,
2
kB uk ≤ 4kuk ,
∀u = (ui )i∈Z ∈ `2 .
Let ei ∈ `2 denote the element having 1 at position i and all the other components
0. Define
X
W (t) ≡ W (t, ω) =
αi wi (t)ei , with (αi )i∈Z ∈ `2
i∈Z
as the white noise with values in `2 defined on the probabilities space (Ω, F, P),
where
Ω = {ω ∈ C(R, `2 ) : ω(0) = 0},
the σ-algebra F is generated by the compact open topology ([2], Appendices A.2
and A.3), and P is the corresponding Wiener measure on F. Let
θt ω(·) = ω(· + t) − ω(t),
t ∈ R,
then (Ω, F, P, (θt )t∈R ) is a metric dynamical system.
We rewrite system (1.1) with initial values Y0 = (u0 = (u0,i )i∈Z , v0 = (v0,i )i∈Z ,
z0 = (z0,i )i∈Z )T in a vector form as the integral equations in E = `2 × `2 × `2 :
Z th
u(t) = u0 +
− d1 Au(s) − (F + k)u(s) + u2 (s)v(s) − u3 (s)
0
i
+ N z(s) + f1 ds + W (t),
Z t
(3.2)
v(t) = v0 +
[−d2 Av(s) − F v(s) − u2 (s)v(s) + u3 (s) + f2 ]ds + W (t),
0
Z t
z(t) = z0 +
[−d3 Az(s) + ku(s) − (F + N )z(s) + f3 ]ds + W (t),
0
for t ≥ 0 and ω ∈ Ω.
Lemma 3.1. Assume that (3.1) holds and T > 0. Then the following properties
hold:
(1) For any initial data Y0 = (u0 , v0 , z0 )T ∈ E, there exists a unique solution
Y (t) = (u(t), v(t), z(t))T ∈ L2 (Ω, C([0, T ], E)) of equations (1.1).
(2) For all ω ∈ Ω, we have the following estimate
sup [ku(t)k2 + kv(t)k2 + kz(t)k2 ]
t∈[0,T ]
≤ 2(ku0 k2 + kv0 k2 + kz0 k2 ) + 6 sup kW (t)k2
t∈[0,T ]
Z
+ 2C0
T
(kW (s)k2 + kf1 k2 + kf2 k2 + kf3 k2 )ds.
0
Proof. Equations (3.2) can be written as an abstract first-order ODE in E as follows
Z t
Z t
Y (t) = Y0 +
ΘY (s)ds +
G(Y (s))ds + W (t), t > 0,
(3.3)
0
0
T
Y (0) = Y0 = (u0 , v0 , z0 ) ∈ E,
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H. LI, J. TU
EJDE-2015/263
where Y = (u, v, z)T , W (t) = (W (t), W (t), W (t))T and


−d1 A − (F + k)I
0
NI
,
0
−d2 A − F I
0
Θ=
kI
0
−d3 A − (F + N )I
 2

u v(s) − u3 (s) + f1
G(Y (s)) = −u2 v(s) + u3 (s) + f2  .
f3
It is easy to show that the linear bounded operator Θ maps E into E. Let B be a
bounded set in E, we have
kG(Y (1) ) − G(Y (2) )k2E
= k((u(1) )2 v (1) − (u(1) )3 − (u(2) )2 v (2)
+ (u(2) )3 − (u(1) )2 v (1) + (u(1) )3 + (u(2) )2 v (2) − (u(2) )3 , 0)T k2E
= 2k(u(1) )2 v (1) − (u(2) )2 v (2) k2 + 2k(u(1) )3 − (u(2) )3 k2
= 2k(u(1) )2 (v (1) − v (2) ) + v (2) (u(1) )2 − (u(2) )2 k2 + 2k(u(1) − u(2) )((u(1) )2
+ (u(2) )2 + u(1) u(2) )k2
≤ 2(2k(u(1) )2 (v (1) − v (2) )k2 + 2kv (2) (u(1) − u(2) )(u(1) + u(2) )k2 )
(3.4)
+ 3k(u(1) − u(2) )((u(1) )2 + (u(2) )2 )k2
≤ 4ku(1) k4 kv (1) − v (2) k2 + 4kv (2) k2 ku(1) − u(2) k2 ku(1) + u(2) k2
+ 6k(u(1) − u(2) )k2 (ku(1) k4 + ku(2) k4 )
≤ 4L2 (B)kY (1) − Y (2) k2E + 4L(B)4L(B)kY (1) − Y (2) k2E
+ 12L2 (B)kY (1) − Y (2) k2E
≤ 32L2 (B)kY (1) − Y (2) k2E ,
where L(B) is a positive constant depending only on B. (3.4) implies that G(Y )
is locally (with respect to Y ∈ E) Lipschitz from E to E. We obtain the property
(1) by the classical theory of ODEs.
To show the existence of a global solution of equations (3.2), we first transform
(3.2) into pathwise equations. Denote
û(t) = u(t) − W (t),
v̂(t) = v(t) − W (t),
ẑ(t) = z(t) − W (t).
Then equations (3.2) are transformed into the equations
Z t
û(t) = u0 +
[−d1 A(û(s) + W (s)) − (F + k)(û(s) + W (s))
0
+ (û(s) + W (s))2 (v̂(s) + W (s))]ds
Z t
+
[−(û(s) + W (s))3 + N (ẑ(s) + W (s)) + f1 ]ds,
(3.5)
0
Z
t
[−d2 A(v̂(s) + W (s)) − F (v̂(s) + W (s))
Z t
2
− (û(s) + W (s)) (v̂(s) + W (s))]ds +
[(û(s) + W (s))3 + f2 ]ds,
v̂(t) = v0 +
0
0
(3.6)
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Z
7
t
[−d3 A(ẑ(s) + W (s)) + k(û(s) + W (s))
ẑ(t) = z0 +
0
(3.7)
− (F + N )(ẑ(s) + W (s)) + f3 ]ds.
For each fixed ω ∈ Ω, equations (3.5)-(3.7) are pathwise equations. By the classical theory of ODEs, it follows that equations (3.5)-(3.7) has a local solution
(û(t), v̂(t), ẑ(t))T ∈ L2 (Ω, C([0, Tmax ), E)), where [0, Tmax ) is the maximal interval of existence of the solution of (3.5)-(3.7). Next, we show that the local solution
is a global solution.
For a fixed ω ∈ Ω, taking the inner product of equation (3.5)-(3.7) with (û, v̂, ẑ)T
in E, we obtain
kû(t)k2 + kv̂(t)k2 + kẑ(t)k2
2
2
2
Z
t
≤ ku0 k + kv0 k + kz0 k − 2d1
hA(û(s) + W (s)), û(s)ids
0
Z t
− 2(F + k)
hû(s) + W (s), û(s)ids
0
Z t
+2
h(û(s) + W (s))2 (v̂(s) + W (s)), û(s)ids
0
Z t
Z t
−2
h(û(s) + W (s))3 , û(s)ids + 2
hf1 , û(s)ids
0
0
Z t
Z t
+ 2N
hẑ(s) + W (s), û(s)ids − 2d2
hA(v̂(s) + W (s)), v̂(s)ids
(3.8)
0
0
Z t
Z t
hv̂(s) + W (s), v̂(s)ids − 2
h(û(s) + W (s))2 (v̂(s) + W (s)), v̂(s)ids
− 2F
0
0
Z t
Z t
3
+2
h(û(s) + W (s)) , v̂(s)ids + 2
hf2 , v̂(s)ids
0
0
Z t
Z t
− 2d3
hA(ẑ(s) + W (s)), ẑ(s)ids + 2k
hû(s) + W (s)), ẑ(s)ids
0
0
Z t
Z t
− 2(F + N )
hẑ(s) + W (s), ẑ(s)ids + 2
hf3 , ẑ(s)ids.
0
0
Grouping the following terms on the right-hand side of (3.8), we have
Z t
h(û(s) + W (s))2 (v̂(s) + W (s)), û(s)ids − 2
h(û(s) + W (s))3 , û(s)ids
0
0
Z t
Z t
2
−2
h(û(s) + W (s)) (v̂(s) + W (s)), v̂(s)ids + 2
h(û(s) + W (s))3 , v̂(s)ids
0
0
Z t
2
=2
h(û(s) + W (s)) (v̂(s) + W (s)), û(s) − v̂(s)ids
0
Z t
−2
h(û(s) + W (s))3 , û(s) − v̂(s)ids
0
Z t
=2
h(û(s) + W (s))2 (v̂(s) − û(s)), û(s) − v̂(s)ids
Z
t
2
0
8
H. LI, J. TU
= −2
EJDE-2015/263
Z tX
(ûi (s) + Wi (s))2 (ûi (s) − v̂i (s))2 ds ≤ 0.
(3.9)
0 i∈Z
From (3.8)-(3.9), we have
kû(t)k2 + kv̂(t)k2 + kẑ(t)k2
Z t
≤ ku0 k2 + kv0 k2 + kz0 k2 − 2d1
hAW (s), û(s)ids
0
Z t
Z t
− 2(F + k)
kû(s)k2 ds − 2(F + k − N )
hW (s), û(s)ids
0
0
Z t
Z t
hẑ(s), û(s)ids − 2d2
hAW (s), v̂(s)ids
+ 2(N + k)
0
0
Z t
Z t
Z t
2
− 2F
kv̂(s)k ds − 2F
hW (s), v̂(s)ids − 2d3
hAW (s), ẑ(s)ids
0
0
0
Z t
Z t
− 2(F + N )
kẑ(s)k2 ds − 2(F + N − k)
hW (s), ẑ(s)ids
0
0
Z t
Z t
Z t
+2
hf1 , û(s)ids + 2
hf2 , v̂(s)ids + 2
hf3 , ẑ(s)ids.
0
0
(3.10)
0
By Young’s inequality, we have the following estimates
4d21
F +k
kûk2 +
kAk2 kW k2 ,
4
F +k
F
3d2
−2d2 hAW, v̂i ≤ kv̂k2 + 2 kAk2 kW k2 ,
3
F
F +N
4d23
−2d3 hAW, ẑi ≤
kẑk2 +
kAk2 kW k2 ,
4
F +N
F +k
4(F + k − N )2
−2(F + k − N )hW, ûi ≤
kûk2 +
kW k2 ,
4
F +k
2(N + k)hû, ẑi ≤ (N + k)kûk2 + (N + k)kẑk2 ,
F
−2F hW, v̂i ≤ kv̂k2 + 3F kW k2 ,
3
F +N
4(F + N − k)2
−2(F + N − k)hW, ẑi ≤
kẑk2 +
kW k2
4
F +N
F +k
4
2hf1 , û(s)i ≤
kûk2 +
kf1 k2 ,
4
F +k
F
3
2hf2 , v̂(s)ids ≤ kv̂k2 + kf2 k2 ,
3
F
F +N
4
2hf3 , ẑ(s)ids ≤
kẑk2 +
kf3 k2 .
4
F +N
−2d1 hAW, ûi ≤
By (3.1), we have
N +k ≤
Thus, we obtain
kũ(t)k2 + kv̂(t)k2 + kẑ(t)k2
F +k
,
4
N +k ≤
F +N
.
4
(3.11)
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RANDOM ATTRACTORS
Z
4d21
F +k t
kû(s)k2 ds +
≤ ku0 k2 + kv0 k2 + kz0 k2 +
kAk2 kW (s)k2 ds
4
F
+
k
0
Z t
Z
F +k t
2
kû(s)k ds +
kũ(s)k2 ds
− 2(F + k)
4
0
0
Z
Z t
4(F + k − N )2 t
2
+
kW (s)k ds + (N + k)
kû(s)k2 ds
F +k
0
0
Z t
Z
Z
F t
3d22 t
2
2
+ (N + k)
kẑ(s)k ds +
kṽ(s)k ds +
kAk2 kW (s)k2 ds
3 0
F 0
0
Z
Z t
Z t
F t
2
2
kṽ(s)k ds + 3F
kW (s)k2 ds
− 2F
kv̂(s)k ds +
3 0
0
0
Z
Z t
F +N t
4d23
+
kẑ(s)k2 ds +
kAk2 kW (s)k2 ds
4
F +N 0
0
Z t
Z
F +N t
− 2(F + N )
kẑ(s)k2 ds +
kẑ(s)k2 ds
4
0
0
Z
Z
Z t
4(F + N − k)2 t
F +k t
4
+
kW (s)k2 ds +
kû(s)k2 ds +
kf1 k2 ds
F +N
4
F +k 0
0
0
Z
Z
Z
3 t
F +N t
F t
kv̂(s)k2 ds +
kf2 k2 ds +
kẑ(s)k2 ds
+
3 0
F 0
4
0
Z t
4
2
+
kf3 k ds
F +N 0
Z t
Z t
≤ ku0 k2 + kv0 k2 + kz0 k2 − (F + k)
kû(s)k2 ds − F
kv̂(s)k2 ds
0
0
Z t
2
4d
1
kAk2 kW (s)k2 ds
− (F + N )
kẑ(s)k2 ds +
F +k
0
Z
Z
3d22 t
4(F + k − N )2 t
2
kW (s)k ds +
kAk2 kW (s)k2 ds
+
F +k
F 0
0
Z t
Z t
4d23
2
+ 3F
kW (s)k ds +
kAk2 kW (s)k2 ds
F +N 0
0
Z
Z t
Z
4(F + N − k)2 t
4
3 t
+
kW (s)k2 ds +
kf1 k2 ds +
kf2 k2 ds
F +N
F +k 0
F 0
0
Z t
4
+
kf3 k2 ds .
F +N 0
Then
kũ(t)k2 + kv̂(t)k2 + kẑ(t)k2
Z t
Z t
≤ ku0 k2 + kv0 k2 + kz0 k2 − (F + k)
kû(s)k2 ds − F
kv̂(s)k2 ds
0
0
Z t
4d2
2
2 Z t
3d
4d
1
3
− (F + N )
kẑ(s)k2 ds +
+ 2+
kAk2 kW (s)k2 ds
F +k
F
F +N
0
0
Z
4(F + k − N )2
4(F + N − k)2 t
+
+ 3F +
kW (s)k2 ds
F +k
F +N
0
9
10
H. LI, J. TU
+
4
F +k
Z
t
kf1 k2 ds +
0
3
F
EJDE-2015/263
t
Z
kf2 k2 ds +
0
Z
4
F +N
Z
t
kf3 k2 ds
0
t
(kû(s)k + kv̂(s)k + kẑ(s)k2 )ds
≤ ku0 k + kv0 k + kz0 k − F
0
Z t
Z t
2
2
+ C1
kAk kW (s)k ds + C2
kW (s)k2 ds
0
0
Z t
+ C3
(kf1 k2 + kf2 k2 + kf3 k2 )ds
0
Z t
2
2
2
≤ ku0 k + kv0 k + kz0 k + C0
(kW (s)k2 + kf1 k2 + kf2 k2 + kf3 k2 )ds, (3.12)
2
2
2
2
2
0
where
3d2
4d23
4d21
+ 2+
,
F +k
F
F +N
4(F + k − N )2
4(F + N − k)2
(3.13)
C2 =
+ 3F +
,
F +k
F +N
3
4
4
, ,
}, C0 = max{C1 kAk2 + C2 , C3 }.
C3 = max{
F +k F F +N
Hence, from (3.12), we obtain that kû(t)k2 + kv̂(t)k2 + kẑ(t)k2 is bounded by a
continuous function, which implies the global existence of a solution on interval
[0, T ]. Therefore, for all ω ∈ Ω, it follows that
C1 =
sup [ku(t)k2 + kv(t)k2 + kz(t)k2 ]
t∈[0,T ]
= sup [kû(t) + W (t)k2 + kv̂(t) + W (t)k2 + kẑ(t) + W (t)k2 ]
t∈[0,T ]
(3.14)
≤ 2(ku0 k2 + kv0 k2 + kz0 k2 ) + 6 sup kW (t)k2
t∈[0,T ]
Z
+ 2C0
T
(kW (s)k2 + kf1 k2 + kf2 k2 + kf3 k2 )ds.
0
Thus, the proof is complete.
Theorem 3.2. Equation (3.2) generates a continuous RDS {ϕ(t, ω)}t≥0,ω∈Ω over
(Ω, F, P, (θt )t∈R ), where
ϕ(t, ω, (u0 , v0 , z0 )) = (u(t, ω, u0 ), v(t, ω, v0 ), z(t, ω, z0 )),
for all t ≥ 0, ω ∈ Ω.
The proof of the above theorem is similar to that of [3, Theorem 3.2]. We omit
it.
4. Existence of random attractors
In this section, we prove the existence of a random attractor for the random
dynamical system generated by (1.1). To convert the stochastic wave equation
to a pathwise one with random parameters, we introduce an Ornstein-Uhlenbeck
process in `2 on the metric dynamical systems (Ω, F, P, (θt )t∈R ) given by the Wiener
process:
Z 0
y(θt ω) = −(F + N + k)
e(F +N +k)s (θt ω)(s)ds, t ∈ R, ω ∈ Ω.
−∞
EJDE-2015/263
RANDOM ATTRACTORS
11
The above integral exists for any path ω with a sub-exponential growth, and y solve
the following Itô equations respectively:
dy + (F + N + k)ydt = dw(t),
t > 0.
Furthermore, there exists a θt -invariant set Ω0 ⊂ Ω of full P measure such that
(1) the mappings s → y(θs ω), is continuous for each ω ∈ Ω;
(2) the random variables ky(θt ω)k is tempered.
Let
ũ(t) = u(t) − y(θt ω),
ṽ(t) = v(t) − y(θt ω),
z̃(t) = z(t) − y(θt ω).
Then we obtain
ũt = −d1 A(ũ + y(θt ω)) − (F + k)ũ + (ũ + y(θt ω))2 (ṽ + y(θt ω))
− (ũ + y(θt ω))3 + N z̃ + f1 + 2N y(θt ω),
ṽt = −d2 A(ṽ + y(θt ω)) − F ṽ − (ũ + y(θt ω))2 (ṽ + y(θt ω))
(4.1)
3
+ (ũ + y(θt ω)) + f2 + (N + k)y(θt ω),
z̃t = −d3 A(z̃ + y(θt ω)) + kũ − (F + N )z̃ + f3 + 2ky(θt ω),
with the initial value conditions
ũ(0, ω, ũ0 ) = ũ0 (ω) = u0 − y(ω),
ṽ(0, ω, ṽ0 ) = ṽ0 (ω) = v0 − y(ω),
z̃(0, ω, z̃0 ) = z̃0 (ω) = v0 − y(ω).
Lemma 4.1. Let (3.1) hold. There exists a θt -invariant set Ω0 ⊂ Ω of full P
measure and an absorbing random set K(ω), ω ∈ Ω0 , for the random dynamical
system ϕ(t, ω), i.e. for all B ∈ D and all ω ∈ Ω0 , there exists TB (ω) > 0 such that
ϕ(t, θ−t ω, B(θ−t ω)) ⊂ K(ω)
for all t ≥ TB (ω).
Moreover, K ∈ D.
Proof. Taking the inner product of (4.1) with (ũ, ṽ, z̃)T in E, we obtain
1 d
kũk2
2 dt
= −d1 hAũ, ũi − d1 hAy(θt ω), ũi − (F + k)kũk2 + h(ũ + y(θt ω))2 (ṽ + y(θt ω)), ũi
− h(ũ + y(θt ω))3 , ũi + hN z̃, ũi + hf1 , ũi + 2N hy(θt ω), ũi,
1 d
kṽk2
2 dt
= −d2 hAṽ, ṽi − d2 hAy(θt ω), ṽi − F kṽk2 − h(ũ + y(θt ω))2 (ṽ + y(θt ω)), ṽi
+ h(ũ + y(θt ω))3 , ṽi + hf2 , ṽi + (N + k)hy(θt ω), ṽi,
1 d
kz̃k2 = −d3 hAz̃, z̃i − d3 hAy(θt ω)), z̃i + khũ, z̃i − (F + N )kz̃k2
2 dt
+ hf3 , z̃i + 2khy(θt ω), z̃i.
(4.2)
12
H. LI, J. TU
EJDE-2015/263
Summing the three equations, we obtain
d
[kũk2 + kṽk2 + kz̃k2 ] + 2d1 hAũ, ũi + 2d2 hAṽ, ṽi + 2d3 hAz̃, z̃i
dt
+ 2(F + k)kũk2 + 2F kṽk2 + 2(F + N )kz̃k2
= −2d1 hAy(θt ω), ũi − 2d2 hAy(θt ω), ṽi − 2d3 hAy(θt ω)), z̃i + 2(N + k)hz̃, ũi
+ 2hf1 , ũi + 4N hy(θt ω), ũi − 2h(ũ + y(θt ω))2 (ṽ + y(θt ω)), ṽi
(4.3)
+ 2h(ũ + y(θt ω))2 (ṽ + y(θt ω)), ũi − 2h(ũ + y(θt ω))3 , ũi
+ 2h(ũ + y(θt ω))3 , ṽi + 2hf2 , ṽi + 2(N + k)hy(θt ω), ṽi + 2hf3 , z̃i
+ 4khy(θt ω), z̃i.
Similar to (3.9), we have
− 2h(ũ + y(θt ω))2 (ṽ + y(θt ω)), ṽi + 2h(ũ + y(θt ω))2 (ṽ + y(θt ω)), ũi
− 2h(ũ + y(θt ω))3 , ũi + 2h(ũ + y(θt ω))3 , ṽi
= 2h(ũ + y(θt ω))2 (ṽ + y(θt ω)), ũ − ṽi − 2h(ũ + y(θt ω))3 , ũ − ṽi
= −2h(ũ + y(θt ω))2 (ũ − ṽ), ũ − ṽi
X
= −2
(ũi + yi (θt ω))2 (ũi − ṽi )2 ≤ 0.
(4.4)
i∈Z
By Young’s inequality, we have the estimates
4d21
F +k
kũk2 +
kAy(θt ω)k2 ,
4
F +k
F
3d2
−2d2 hAy(θt ω), ṽi ≤ kṽk2 + 2 kAy(θt ω)k2 ,
3
F
4d23
F +N
kz̃k2 +
kAy(θt ω)k2 ,
−2d3 hAy(θt ω), z̃i ≤
4
F +N
16N 2
F +k
kũk2 +
ky(θt ω)k2 ,
4N hy(θt ω), ũi ≤
4
F +k
2(N + k)hũ, z̃i ≤ (N + k)kũk2 + (N + k)kz̃k2 ,
−2d1 hAy(θt ω), ũi ≤
F
3(N + k)2
kṽk2 +
ky(θt ω)k2 ,
3
F
F +N
16k 2
4khy(θt ω), z̃i ≤
kz̃k2 +
ky(θt ω)k2 ,
4
F +N
F +k
4
2hf1 , ũi ≤
kũk2 +
kf1 k2 ,
4
F +k
F
3
2hf2 , ṽids ≤ kṽk2 + kf2 k2 ,
3
F
F +N
4
2hf3 , z̃ids ≤
kz̃k2 +
kf3 k2 .
4
F +N
By (4.3)-(4.5), we obtain
(4.5)
2(N + k)hy(θt ω), ṽi ≤
d
[kũk2 + kṽk2 + kz̃k2 ] + 2(F + k)kũk2 + 2F kṽk2 + 2(F + N )kz̃k2
dt
F +k
4d21
F
3d2
F +N
≤
kũk2 +
kAy(θt ω)k2 + kṽk2 + 2 kAy(θt ω)k2 +
kz̃k2
4
F +k
3
F
4
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RANDOM ATTRACTORS
13
4d23
F +k
kAy(θt ω)k2 + (N + k)kũk2 + (N + k)kz̃k2 +
kũk2
F +N
4
4
F +k
16N 2
F
+
kf1 k2 +
kũk2 +
ky(θt ω)k2 + kṽk2
F +k
4
F +k
3
3
F
3(N + k)2
F +N
2
2
2
+ kf2 k + kṽk +
ky(θt ω)k +
kz̃k2
F
3
F
4
4
F +N
16k 2
+
kf3 k2 +
kz̃k2 +
ky(θt ω)k2 .
F +N
4
F +N
From (3.1) and (3.11), we have
+
d
[kũk2 + kṽk2 + kz̃k2 ] + (F + k)kũk2 + F kṽk2 + (F + N )kz̃k2
dt
3d2
4d23
4d21
kAy(θt ω)k2 + 2 kAy(θt ω)k2 +
kAy(θt ω)k2
≤
F +k
F
F +N
16N 2
3(N + k)2
+
ky(θt ω)k2 +
ky(θt ω)k2
F +k
F
4
3
4
16k 2
ky(θt ω)k2 +
kf1 k2 + kf2 k2 +
kf3 k2
+
F +N
F +k
F
F +N
≤ C1 kAy(θt ω)k2 + C4 ky(θt ω)k2 + C3 (kf1 k2 + kf2 k2 + kf3 k2 )
(4.6)
≤ C5 (ky(θt ω)k2 + kAy(θt ω)k2 + kf1 k2 + kf2 k2 + kf3 k2 ),
where C1 and C3 are defined in (3.13), C4 = 16N 2 /(F + k) + 3(N + k)2 /F +
16k 2 /(F + N ), and C5 = max{C1 , C3 , C4 }. By Gronwall’s inequality, it follows
that
kũ(t, ω, ũ0 (ω))k2 + kṽ(t, ω, ṽ0 (ω))k2 + kz̃(t, ω, z̃0 (ω))k2
C5
≤ e−F t [kũ0 (ω)k2 + kṽ0 (ω)k2 + kz̃0 (ω)k2 ] +
(kf1 k2 + kf2 k2 + kf3 k2 )
F
Z t
+ C5
e−F (t−s) (ky(θs ω)k2 + kAy(θs ω)k2 )ds.
(4.7)
0
Note that the random variable y(θt ω) is tempered and y(θt ω) is continuous in t.
Therefore, it follows from Proposition 4.3.3 in [2] that there exists a tempered
function l(ω) > 0 such that
ky(θt ω)k2 + kAy(θt ω)k2 ≤ l(θt ω) ≤ l(ω)eF |t|/2 .
(4.8)
Replacing ω by θ−t ω in (4.7) and using (4.8), we obtain
kũ(t, θ−t ω, ũ0 (θ−t ω))k2 + kṽ(t, θ−t ω, ṽ0 (θ−t ω))k2 + kz̃(t, θ−t ω, z̃0 (θ−t ω))k2
C5
(kf1 k2 + kf2 k2 + kf3 k2 )
≤ e−F t [kũ0 (θ−t ω)k2 + kṽ0 (θ−t ω)k2 + kz̃0 (θ−t ω)k2 ] +
F
Z t
+ C5
e−F (t−s) (ky(θs−t ω)k2 + kAy(θs−t ω)k2 )ds
0
C5
≤ e−F t [kũ0 (θ−t ω)k2 + kṽ0 (θ−t ω)k2 + kz̃0 (θ−t ω)k2 ] +
(kf1 k2 + kf2 k2 + kf3 k2 )
F
Z 0
+ C5
eF τ (ky(θτ ω)k2 + kAy(θτ ω)k2 )dτ
−t
≤ e−F t [kũ0 (θ−t ω)k2 + kṽ0 (θ−t ω)k2 + kz̃0 (θ−t ω)k2 ]
14
H. LI, J. TU
+
EJDE-2015/263
C5
2C5 l(ω)
(kf1 k2 + kf2 k2 + kf3 k2 ) +
.
F
F
(4.9)
Define R2 (ω) = 2[C5 (kf1 k2 + kf2 k2 + kf3 k2 ) + 2C5 l(ω)]/F ; since l(ω) is a tempered
function, then R(ω) is also tempered. Define
K̃(ω) = {(ũ, ṽ, z̃) ∈ `2 × `2 × `2 , kũk2 + kṽk2 + kz̃k2 ≤ R2 (ω)}.
Then K̃(ω) is an absorbing set for the random dynamical system
(ũ(t, ω, ũ0 ), ṽ(t, ω, ṽ0 ), z̃(t, ω, z̃0 ));
that is, for every B ∈ D and every ω ∈ Ω0 , there exists TB (ω) such that
Φ(t, θ−t ω, B(θ−t ω)) ⊂ K̃(ω)
for t ≥ TB (ω).
Let
K(ω) = {(u, v, z) ∈ `2 × `2 × `2 , kuk2 + kvk2 + kzk2 ≤ R12 (ω)},
where
R12 (ω) = 2R2 (ω) + 6ky(θt ω)k2 .
Then, K(ω) is an absorbing random set for the random dynamical system ϕ(t, ω)
since
ϕ(t, ω, (u0 , v0 , z0 ))
= Φ(t, ω, (u0 − y(ω), v0 − y(ω), z0 − y(ω))) + (y(θt ω), y(θt ω), y(θt ω))
= (ũ(t, ω, u0 − y(ω)) + y(θt ω), ṽ(t, ω, v0 − y(ω)) + y(θt ω), z̃(t, ω, z0 − y(ω))
+ y(θt ω))
and K ∈ D. This completes the proof.
To prove the pullback asymptotic compactness for the dynamical system ϕ, we
first prove the following lemma.
Lemma 4.2. Let (3.1) hold. Assume the initial functions (u0 (ω), v0 (ω), z0 (ω)) ∈
K(ω), where K(ω) is the absorbing set in Lemma 4.1. Then for every ε > 0, there
exist T (ε, ω) > 0 and N (ε, ω) > 0 such that the solution
(u(t, ω, u0 (ω)), v(t, ω, v0 (ω)), z(t, ω, z0 (ω)))
of (1.1) satisfies
X
h
ku(t, θ−t ω, u0 (θ−t ω))k2 + kv(t, θ−t ω, v0 (θ−t ω))k2
|i|≥N (ε,ω)
i
+ kz(t, θ−t ω, z0 (θ−t ω))k2 < ε,
for all t ≥ T (ε, ω) > 0.
Proof. We choose a smooth function ρ such that 0 ≤ ρ ≤ 1 for all s ∈ R and
(
0, if |s| < 1,
ρ(s) =
(4.10)
1, if |s| > 2,
and there exists a positive constant C6 , such that |ρ0 (s)| ≤ C6 for s ∈ R.
EJDE-2015/263
RANDOM ATTRACTORS
15
We first consider the random equation (4.1). Let r be a fixed positive integer
|i|
which will be specified later. Taking the inner product of (4.1) with ρ( |i|
r )ũ, ρ( r )ṽ
and ρ( |i|
r )z̃ in E, respectively, we obtain
1 d X |i| 2
ρ
|ũi |
2 dt
r
i∈Z
= −d1 hAũ, ρ
X |i| |i| |i| ũi − d1 hAy(θt ω), ρ
ũi − (F + k)
ρ
|ũi |2
r
r
r
i∈Z
|i| |i| + h(ũ + y(θt ω))2 (ṽ + y(θt ω)), ρ
ũi − h(ũ + y(θt ω))3 , ρ
ũi
r
r
|i| |i| |i| + N hz̃, ρ
ũi + hf1 , ρ
ũi + 2N hy(θt ω), ρ
ũi,
r
r
r
1 d X |i| 2
ρ
|ṽi |
2 dt
r
i∈Z
= −d2 hAṽ, ρ
X |i| |i| |i| ṽi − d2 hAy(θt ω), ρ
ṽi − F
ρ
|ṽi |2
r
r
r
i∈Z
(4.11)
|i| |i| − h(ũ + y(θt ω))2 (ṽ + y(θt ω)), ρ
ṽi + h(ũ + y(θt ω))3 , ρ
ṽi
r
r
|i| |i| + hf2 , ρ
ṽi + (N + k)hy(θt ω), ρ
ṽi,
r
r
1 d X |i| 2
ρ
|z̃i |
2 dt
r
i∈Z
= −d3 hAz̃, ρ
X |i| |i| |i| z̃i − d3 hAy(θt ω), ρ
z̃i − (F + N )
ρ
|z̃i |2
r
r
r
i∈Z
|i| |i| |i| + khũ, ρ
z̃i + hf3 , ρ
z̃i + 2khy(θt ω), ρ
z̃i.
r
r
r
Recalling the property |ρ0 (s)| ≤ C6 , we have
|i| ũi
r
X
|i| =
(B ũ)i Bρ
ũ
r
i
i∈Z
h
X
|i + 1| |i| i
=
(B ũ)i ρ
ũi+1 − ρ
ũi
r
r
i∈Z
h |i| |i + 1| i
X
|i| =
(ũi+1 − ũi ) ρ
(ũi+1 − ũi ) + ρ
−ρ
ũi+1
r
r
r
hAũ, ρ
i∈Z
X ρ0 (ξ)
|i| |ũi+1 − ũi |2 −
(ũi+1 − ũi )ũi+1
r
r
i∈Z
i∈Z
C6 X
2C6
≥−
(|ũi+1 |2 + |ũi ||ũi+1 |) ≥ −
kũk2 .
r
r
≥
X
ρ
i∈Z
(4.12)
16
H. LI, J. TU
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Similarly, we have
X
|i| ṽi =
(Bṽ)i Bρ
r
i∈Z
X
|i| z̃i =
(B z̃)i Bρ
hAz̃, ρ
r
hAṽ, ρ
i∈Z
|i| 2C6
ṽ ≥ −
kṽk2 ,
r
r
i
2C6
|i| z̃ ≥ −
kz̃k2 .
r
r
i
(4.13)
Summing the three equations in (4.11), we find that
X |i| 1 d X |i| 2
ρ
|ũi | + |ṽi |2 + |z̃i |2 + (F + k)
ρ
|ũi |2
2 dt
r
r
i∈Z
+F
X
i∈Z
i∈Z
X |i| |i| 2
ρ
|ṽi | + (F + N )
ρ
|z̃i |2
r
r
i∈Z
2C6 d1
2C6 d2
2C6 d3
|i| ≤
kũk2 +
kṽk2 +
kz̃k2 + (N + k)hz̃, ρ
ũi
r
r
r
r
|i| |i| + h(ũ + y(θt ω))2 (ṽ + y(θt ω)), ρ
ũi − h(ũ + y(θt ω))3 , ρ
ũi
(4.14)
r
r
|i|
|i|
− h(ũ + y(θt ω))2 (ṽ + y(θt ω)), ρ
ṽi + h(ũ + y(θt ω))3 , ρ
ṽi
r
r
|i| |i| |i| + hf1 , ρ
ũi + hf2 , ρ
ṽi + hf3 , ρ
z̃i
r
r
r
|i| |i| |i| ũi − d2 hAy(θt ω), ρ
ṽi − d3 hAy(θt ω), ρ
z̃i
− d1 hAy(θt ω), ρ
r
r
r
|i| |i| |i| + 2N hy(θt ω), ρ
ũi + (N + k)hy(θt ω), ρ
ṽi + 2khy(θt ω), ρ
z̃i.
r
r
r
Grouping from the fifth term to eighth term of the right-hand side of (4.14), we
have
|i| |i| ũi − h(ũ + y(θt ω))3 , ρ
ũi
h(ũ + y(θt ω))2 (ṽ + y(θt ω)), ρ
r
r
|i| |i| − h(ũ + y(θt ω))2 (ṽ + y(θt ω)), ρ
ṽi + h(ũ + y(θt ω))3 , ρ
ṽi
r
r
|i| = h(ũ + y(θt ω))2 (ṽ + y(θt ω)), ρ
(ũ − ṽ)i
r
(4.15)
|i| (ũ − ṽ)i
− h(ũ + y(θt ω))3 , ρ
r
|i| 2
= h(ũ + y(θt ω)) (ṽ − ũ), ρ
(ũ − ṽ)i
r
X |i| =−
ρ
(ũi + yi (θt ω))2 (ũi − ṽi )2 ≤ 0.
r
i∈Z
Then (4.14) can be reduced to
X |i| d X |i| 2
ρ
|ũi | + |ṽi |2 + |z̃i |2 + 2(F + k)
ρ
|ũi |2
dt
r
r
i∈Z
+ 2F
i∈Z
X
i∈Z
X |i| |i| 2
ρ
|ṽi | + 2(F + N )
ρ
|z̃i |2
r
r
i∈Z
EJDE-2015/263
RANDOM ATTRACTORS
17
4C6 d1
4C6 d2
4C6 d3
|i| kũk2 +
kṽk2 +
kz̃k2 + 2(N + k)hz̃, ρ
ũi
(4.16)
r
r
r
r
|i| |i| |i| ũi + 2hf2 , ρ
ṽi + 2hf3 , ρ
z̃i
+ 2hf1 , ρ
r
r
r
|i| |i| |i| − 2d1 hAy(θt ω), ρ
ũi − 2d2 hAy(θt ω), ρ
ṽi − 2d3 hAy(θt ω), ρ
z̃i
r
r
r
|i| |i| |i| + 4N hy(θt ω), ρ
ũi + 2(N + k)hy(θt ω), ρ
ṽi + 4khy(θt ω), ρ
z̃i.
r
r
r
For the second term to forth term in the right-hand side of (4.16), we have
X |i| |i| ũi =
ρ
f1i ũi
hf1 , ρ
r
r
≤
i∈Z
=
X
|i|≥r
≤
ρ
|i| f1i ũi
r
4 X
F + k X |i| 2
|ũi | +
ρ
|f1i |2 ,
4
r
F +k
|i|≥r
hf2 , ρ
hf3 , ρ
|i| ṽi =
r
X
|i|≥r
|i|≥r
|i| F X |i| 2
3 X
ρ
f2i ṽi ≤
ρ
|ṽi | +
|f2i |2 ,
r
3
r
F
|i|≥r
(4.17)
|i|≥r
X |i| X
F + N X |i| 2
4
|i| z̃i =
ρ
f3i z̃i ≤
ρ
|z̃i | +
|f3i |2 .
r
r
4
r
F +N
|i|≥r
|i|≥r
|i|≥r
For the fifth term to seventh term in the right-hand side of (4.16), we have
|i| − 2d1 hAy, ρ
ũi
r
|i| = −2d1 hBy, B ρ
ũ i
r
|i + 1| X
|i| = −2d1
(yi+1 − yi ) ρ
ũi+1 − ρ
ũi
r
r
i∈Z
= −2d1
X
ρ
|i|≥r−1
X |i| |i + 1| (yi+1 − yi )ũi+1 + 2d1
ρ
(yi+1 − yi )ũi
r
r
|i|≥r
X
F + k X |i| 2
≤
ρ
|ũi | + C7
|yi |2 ,
4
r
|i|≥r
|i|≥r−1
X
|i| F X |i| 2
−2d2 hAy, ρ
ṽi ≤
ρ
|ṽi | + C8
|yi |2 ,
r
3
r
|i|≥r
−2d3 hAy, ρ
(4.18)
|i|≥r−1
X
|i| F + N X |i| 2
z̃i ≤
ρ
|z̃i | + C9
|yi |2 ,
r
4
r
|i|≥r
|i|≥r−1
where Cj , j = 7, 8, 9 are positive constants depending only on d1 , d2 , d3 , F, N, k.
Direct computation shows that
X |i| X |i| |i| 2(N + k)hz̃, ρ
ũi ≤ (N + k)
ρ
|ũi |2 + (N + k)
ρ
|z̃i |2 ,
r
r
r
i∈Z
i∈Z
18
H. LI, J. TU
4N hy(θt ω), ρ
EJDE-2015/263
|i| F + k X |i| 2 16N 2 X |i| ũi ≤
ρ
|ũi | +
ρ
|yi (θt ω)|2 , (4.19)
r
4
r
F +k
r
i∈Z
i∈Z
F X |i| 2 3(N + k)2 X |i| |i| ṽi ≤
ρ
|ṽi | +
ρ
|yi (θt ω)|2 ,
2(N + k)hy(θt ω), ρ
r
3
r
F
r
i∈Z
i∈Z
|i| F + N X |i| 2
16k 2 X |i| 4khy(θt ω), ρ
z̃i ≤
ρ
|z̃i | +
ρ
|yi (θt ω)|2 .
r
4
r
F +N
r
i∈Z
i∈Z
From (4.16)-(4.19), (3.1) and (3.11), we have
X |i| d X |i| 2
ρ
|ũi | + |ṽi |2 + |z̃i |2 + (F + k)
ρ
|ũi |2
dt
r
r
i∈Z
i∈Z
X
+F
i∈Z
X |i| |i| 2
ρ
|ṽi | + (F + N )
ρ
|z̃i |2
r
r
i∈Z
4C6 d2
4C6 d3
4 X
3 X
4C6 d1
kũk2 +
kṽk2 +
kz̃k2 +
|f1i |2 +
|f2i |2
≤
r
r
r
F +k
F
|i|≥r
X
X
4
+
|f3i |2 + (C7 + C8 + C9 )
|yi (θt ω)|2
F +N
|i|≥r
+
16N 2
+
F +k
|i|≥r
(4.20)
|i|≥r−1
3(N + k)2
16k 2 X
+
|yi (θt ω)|2 .
F
F +N
|i|≥r
By Gronwall’s inequality, we obtain that for t ≥ Tk = Tk (ω) ≥ 0,
X
ρ
i∈Z
|i| |ũi (t, ω, ũ0 (ω))|2 + |ṽi (t, ω, ṽ0 (ω))|2 + |z̃i (t, ω, z̃0 (ω))|2
r
≤ e−F (t−Tk )
X
i∈Z
Z
ρ
|i| |ũi (Tk , ω, ũ0 (ω))|2 + |ṽi (Tk , ω, ṽ0 (ω))|2 + |z̃i (Tk , ω, z̃0 (ω)|2
r
t
4C d
4C6 d2
4C6 d3
6 1
+
kũk2 +
kṽk2 +
kz̃k2 eF (τ −t) dτ x
r
r
r
Tk
X
X
1 4 X
3
4
+
|f1i |2 +
|f2i |2 +
|f3i |2
F F +k
F
F +N
|i|≥r
|i|≥r
|i|≥r
Z t
X
+ (C7 + C8 + C9 )
eF (τ −t)
|yi (θτ ω)|2 dτ
Tk
+
16N 2
F +k
|i|≥r−1
2
+
(4.21)
3(N + k)
16k 2 +
F
F +N
Z
t
Tk
eF (τ −t)
X
|yi (θτ ω)|2 dτ.
|i|≥r
Replace ω by θ−t ω. We then estimate each term on the right-hand side of (4.21).
From (4.7)-(4.8) with t replaced by Tk and ω by θ−t ω, it follows that
EJDE-2015/263
RANDOM ATTRACTORS
19
|i| h
|ũi (Tk , θ−t ω, ũ0 (θ−t ω))|2 + |ṽi (Tk , θ−t ω, ṽ0 (θ−t ω))|2
r
i∈Z
i
+ |z̃i (Tk , θ−t ω), z̃0 (θ−t ω)|2
≤ e−F (t−Tk ) e−F Tk [kũ0 (θ−t ω)k2 + kṽ0 (θ−t ω)k2 + kz̃0 (θ−t ω)k2 ]
e−F (t−Tk )
X
ρ
C5
(kf1 k2 + kf2 k2 + kf3 k2 )
F
Z Tk
+ C5
e−F (Tk −s) (ky(θs−t ω)k2 + kAy(θs−t ω)k2 )ds
+
(4.22)
0
−F t
[kũ0 (θ−t ω)k2 + kṽ0 (θ−t ω)k2 + kz̃0 (θ−t ω)k2 ]
F
2
C5
+ e−F (t−Tk ) (kf1 k2 + kf2 k2 + kf3 k2 ) + C5 l(ω)e− 2 (t−Tk ) .
F
F
≤e
Thus, there exists a T1 (ε, ω) > Tk (ω) such that if t > T1 (ε, ω), then
X |i| h
|ũi (Tk , θ−t ω, ũ0 (θ−t ω))|2 + |ṽi (Tk , θ−t ω, ṽ0 (θ−t ω))|2
e−F (t−Tk )
ρ
r
i∈Z
(4.23)
i 1
2
+ |z̃i (Tk , θ−t ω), z̃0 (θ−t ω)| < ε.
4
Next we estimate second term on the right-hand side of (4.21),
Z t
4C6 d2
4C6 d1
kũ(τ, θ−t ω, ũ0 (θ−t ω))k2 +
kṽ(τ, θ−t ω, ũ0 (θ−t ω))k2
r
r
Tk
4C6 d3
kz̃(τ, θ−t ω, ũ0 (θ−t ω))k2 eF (τ −t) dτ
+
r
Z
4C6 d t F (τ −t) −F τ
≤
e
e
[kũ0 (θ−t ω)k2 + kṽ0 (θ−t ω)k2 + kz̃0 (θ−t ω)k2 ]
r
Tk
C5
(kf1 k2 + kf2 k2 + kf3 k2 )
+
FZ
τ
+ C5
eF (s−τ ) (ky(θs−t ω)k2 + kAy(θs−t ω)k2 )ds dτ
(4.24)
0
4C6 d ≤
[kũ0 (θ−t ω)k2 + kṽ0 (θ−t ω)k2 + kz̃0 (θ−t ω)k2 ](t − Tk )e−F t
r
C5
4
+ 2 (kf1 k2 + kf2 k2 + kf3 k2 ) + C5 l(ω) ,
F
F
where d = max{d1 , d2 , d3 }. Recall the fact that (ũ0 (θ−t ω), ṽ0 (θ−t ω), z̃0 (θ−t ω)) ∈
K(θ−t ω), which implies that kũ0 (θ−t ω)k2 +kṽ0 (θ−t ω)k2 +kz̃0 (θ−t ω)k2 ≤ R2 (θ−t ω),
and R(ω) is tempered. Thus there exists T2 (ε, ω) > Tk (ω) and N1 (ε, ω) > 0 such
that for t > T2 (ε, ω) and r > N1 (ε, ω), we have
Z t
2C6 d1
2C6 d2
kũ(τ, θ−t ω, ũ0 (θ−t ω))k2 +
kṽ(τ, θ−t ω, ũ0 (θ−t ω))k2
r
r
Tk
(4.25)
2C6 d3
1
2
F (τ −t)
+
kz̃(τ, θ−t ω, ũ0 (θ−t ω))k e
dτ < ε.
r
4
20
H. LI, J. TU
EJDE-2015/263
Since f1 , f2 , f3 ∈ `2 , there exists N2 (ε, ω) > 0 such that for r > N2 (ε, ω),
1
X
1 4 X
3 X
4
|f1i |2 +
|f2i |2 +
|f3i |2 < ε.
F F +k
F
F +N
4
|i|≥r
|i|≥r
(4.26)
|i|≥r
Finally, we estimate the last term on the right-hand side of (4.21). Let T ∗ > 0 to
be determined later. We have for t > T ∗ + Tk
t
Z
eF (τ −t)
(C7 + C8 + C9 )
Tk
+
Z
3(N + k)
16k 2
16N
+
+
F +k
F
F +N
0
X
eF s C10
Tk −t
0
|yi (θτ −t ω)|2 dτ
|i|≥r−1
2
2
=
Z
X
Z
t
eF (τ −t)
Tk
|yi (θτ −t ω)|2 dτ
|i|≥r
|yi (θs ω)|2 ds + C5
|i|≥r−1
X
X
|yi (θs ω)|2 ds
(4.27)
|i|≥r
X
X
eF s C10
|yi (θs ω)|2 ds + C5
|yi (θs ω)|2 ds
≤
−T ∗
Z
|i|≥r−1
|i|≥r
−T ∗
eF s C10 ky(θs ω)k2 ds + C5 ky(θs ω)k2 ds .
+
Tk −t
Using (4.8), we have
−T ∗
Z
Tk −t
∗
F
2C11
l(ω)e− 2 T .
eF s C10 ky(θs ω)k2 + C5 ky(θs ω)k2 ds ≤
F
(4.28)
Thus, by choosing
T∗ >
2
16C11 l(ω)
ln
,
F
Fε
for t > T ∗ + Tk , we have
Z
−T ∗
Tk −t
ε
eF s C10 ky(θs ω)k2 + C5 ky(θs ω)k2 ds < .
8
(4.29)
For the fixed T ∗ , from Lebesgue’s theorem there is an N3 (ε, ω) such that for r >
N3 (ε, ω),
Z
0
ε
X
X
eF s C10
|yi (θs ω)|2 ds + C5
|yi (θs ω)|2 ds < .
8
−T ∗
|i|≥r−1
(4.30)
|i|≥r
Therefore, by letting
T (ε, ω) = max{T1 (ε, ω), T2 (ε, ω), T ∗ (ε, ω) + Tk (ω)},
Ñ (ε, ω) = max{N1 (ε, ω), N2 (ε, ω), N3 (ε, ω)},
(4.31)
EJDE-2015/263
RANDOM ATTRACTORS
21
for t > T (ε, ω) and N > Ñ (ε, ω), we have
X |ũi (t, θ−t ω, ũ0 (θ−t ω))|2 + |ṽi (t, θ−t ω, ṽ0 (θ−t ω))|2
|i|>2r
+ |z̃i (t, θ−t ω, z̃0 (θ−t ω))|2
X |i| ≤
ρ
|ũi (t, θ−t ω, ũ0 (θ−t ω))|2
r
(4.32)
i∈Z
+ |ṽi (t, θ−t ω, ṽ0 (θ−t ω))|2 + |z̃i (t, θ−t ω, z̃0 (θ−t ω))|2 < ε,
which implies
X |ui (t, θ−t ω, u0 (θ−t ω))|2 + |vi (t, θ−t ω, v0 (θ−t ω))|2
|i|≥N (ε,ω)
+ |zi (t, θ−t ω, z0 (θ−t ω))|2
X
=
(|ũi (t, θ−t ω, ũ0 (θ−t ω)) + y(θ−t ω)|2 + |ṽi (t, θ−t ω, ṽ0 (θ−t ω))
|i|≥N (ε,ω)
+ y(θ−t ω)|2 + |z̃i (t, θ−t ω, z0 (θ−t ω)) + y(θ−t ω)|2 )
X
≤2
(|ũi (t, θ−t ω, ũ0 (θ−t ω))|2 + |ṽi (t, θ−t ω, ṽ0 (θ−t ω))|2
(4.33)
|i|≥N (ε,ω)
+ |z̃i (t, θ−t ω, z0 (θ−t ω))|2 ) + 6
X
|y(θ−t ω)|2 < 8ε,
|i|≥N (ε,ω)
provided N (ε, ω) is large enough. This completes the proof of the lemma.
We are now ready to show the pullback asymptotic compactness of the random
set K(ω).
Lemma 4.3. For ω ∈ Ω, the set K(ω) is pullback asymptotically compact in the
sense of each sequence (un , vn , zn ) ∈ ϕ(tn , θ−tn ω, K(θ−tn ω)) with tn → ∞ having
a convergent subsequence in `2 × `2 × `2 .
Proof. We follow the method in [3]. Let ω ∈ Ω0 for each sequence {tn }∞
n=1 :
t1 , t2 , · · · , tn → ∞ as n → ∞, and
(un (tn , θ−tn ω, xn ), vn (tn , θ−tn ω, yn ), zn (tn , θ−tn ω, %n )) ∈ ϕ(tn , θ−tn ω, K(θ−tn ω));
this implies that there exists (xn , yn , %n ) ∈ K(θ−tn ω) such that
(un (tn , θ−tn ω, xn ), vn (tn , θ−tn ω, yn ), zn (tn , θ−tn ω, %n )) = ϕ(tn , θ−tn ω, (xn , yn , %n )).
Since K(ω) is a bounded absorbing set, for large n, ϕ(tn , θ−tn ω, (xn , yn , %n )) ∈
K(ω); thus there exists (u, v, z) ∈ `2 × `2 × `2 , and a subsequence (u0n , vn0 , zn0 ) =
ϕ(tn , θ−tn ω, (xn , yn , %n )) such that
(u0n (tn , θ−tn ω, xn ), vn0 (tn , θ−tn ω, yn ), zn0 (tn , θ−tn ω, %n )) → (u, v, z)
2
2
2
(4.34)
(u0n , vn0 , zn0 )
weak in ` × ` × ` . Next, we show that
is also strongly convergent in
the norm k · k in `2 × `2 × `2 , i.e., for each > 0 there is N ∗ (, ω) > 0 such that for
n ≥ N ∗ (, ω),
k(u0n (tn , θ−tn ω, xn ), vn0 (tn , θ−tn ω, yn ), zn0 (tn , θ−tn ω, %n )) − (u, v, z)k ≤ .
22
H. LI, J. TU
EJDE-2015/263
In fact, from Lemma 4.2, for any > 0, there exists an N ∗ (, ω) and a K1 (, ω)
such that for n ≥ N ∗ (, ω),
X
0
|u0ni (tn , θ−tn ω, xn )|2 + |vni
(tn , θ−tn ω, yn )|2
|i|≥K1 (,ω)
+
(4.35)
0
|zni
(tn , θ−tn ω, %n )|2
1
≤ 2 .
8
Since (u, v, z) ∈ `2 × `2 × `2 , there exists K2 () > 0 such that
X
1
|ui |2 + |vi |2 + |zi |2 ≤ 2 .
8
(4.36)
|i|≥K2 ()
Let K(, ω) = max{K1 (, ω), K2 ()}. From the weak convergence (4.34), we have
for each |i| ≤ K(, ω) as n → ∞,
0
0
(u0ni (tn , θ−tn ω, xn ), vni
(tn , θ−tn ω, yn ), zni
(tn , θ−tn ω, %n )) → (ui , vi , zi ),
which implies that there exists N2∗ (, ω) > 0 such that for n ≥ N2∗ (, ω),
X 0
|u0ni (tn , θ−tn ω, xn ) − ui |2 + |vni
(tn , θ−tn ω, yn ) − vi |2
|i|≤K(,ω)
+
(4.37)
0
|zni
(tn , θ−tn ω, %n )
− zi |
2
1
≤ 2 .
2
Combining (4.35)-(4.37), we obtain that for n ≥ N ∗ (, ω),
ku0n (tn , θ−tn ω, xn ) − uk2 + kvn0 (tn , θ−tn ω, yn ) − vk2 + kzn0 (tn , θ−tn ω, %n ) − zk2
X 0
≤
|u0ni (tn , θ−tn ω, xn ) − ui |2 + |vni
(tn , θ−tn ω, yn ) − vi |2
|i|≤K(,ω)
0
+ |zni
(tn , θ−tn ω, %n ) − zi |2 +
X
|u0ni (tn , θ−tn ω, xn ) − ui |2
|i|≥K(,ω)
+
≤
0
− vi | + |zni
(tn , θ−tn ω, %n ) − zi |2
X |ui |2 + |vi |2 + |zi |2 + 2
|u0ni (tn , θ−tn ω, xn )|2
0
|vni
(tn , θ−tn ω, yn )
1 2
+2
2
X
|i|≥K(,ω)
2
|i|≥K(,ω)
0
0
+ |vni
(tn , θ−tn ω, yn )|2 + |zni
(tn , θ−tn ω, %n )|2
≤
1 2 1 2 1 2
+ + = 2 .
2
4
4
The proof is complete.
Combining Lemmas 4.1 and 4.2 with Lemma 4.3, we obtain our main result,
which is a direct consequence of Theorem 2.7.
Theorem 4.4. The random dynamical systems {ϕ(t, ω)}t≥0,ω∈Ω possess a random
attractor in `2 × `2 × `2 .
Remark 4.5. Equation (1.1) can be reduced to two ordinary differential equations
describing kinetics of cubic chemical or biochemical reactions with additive noise
EJDE-2015/263
RANDOM ATTRACTORS
23
on an infinite lattice, such as two following models: stochastic reversible Selkov
equations [27, 36]:
dui = (−d1 (Au)i − aui + u2i vi − u3i )dt + αi dwi ,
dvi = (−d2 (Av)i − bvi − u2i vi + u3i )dt + αi dwi ,
(4.38)
stochastic reversible Glocolysis equations [25]:
dui = (−d1 (Au)i − aui + bvi + u2i vi − u3i )dt + αi dwi ,
dvi = (−d2 (Av)i − bvi − u2i vi + u3i )dt + αi dwi ,
(4.39)
with the conditions (1.2). Equations (4.38)-(4.39) are useful models for study of
cooperative processes in chemical kinetics. By introducing an Ornstein-Uhlenbeck
process, we transform the stochastic equation into a pathwise one with tempered
random variables:
Z 0
y(θt ω) = −b
ebs (θt ω)(s)ds, t ∈ R, ω ∈ Ω.
−∞
Note that y solves the Itô equation
dy + bydt = dw(t),
t > 0.
Let
ũ = u − y(θt ω), ṽ = v − y(θt ω).
Then, similar to Lemmas 4.1–4.3 and Theorem 4.4, we can prove that
(1) The random dynamical system governed by stochastic reversible Selkov
equations (4.38) has a random attractor in E.
(2) If 2b ≤ a holds, the random dynamical system governed by stochastic reversible Glocolysis equations (4.39) has a random attractor in E.
Acknowledgements. We would like to thank the anonymous referee for the carefully reading the manuscript and for making very useful suggestions. The first author is supported by the National Natural Science Foundation of China (11101265)
and by the Shanghai Education Research and Innovation Key Project (14ZZ157).
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Hongyan Li
College of Management, Shanghai University of Engineering Science, Shanghai 201620,
China
E-mail address: hongyanlishu@163.com
Junyi Tu
Department of Mathematics and Statistics, University of South Florida, Tampa, FL
33620, USA
E-mail address: junyi@mail.usf.edu