Global stability of the Euler-Bernoulli beams excited by
multiplicative white noises
Zhenzhen Li
Division of Dynamics and Control, School of Mathematics and Statistics, Shandong University of Technology, ZiBo
255000, China
E-mail: zzlicncn@163.com
Kun Zhao
Beijing Electro-Mechanical Engineering Institute, Beijing 100074, P. R. China
E-mail: zhaokunhit@yeah.net
Hongkui Li
Division of Dynamics and Control, School of Mathematics and Statistics, Shandong University of Technology, ZiBo
255000, China
E-mail: Lhk8068@163.com
Juan L.G. Guirao
Department of Applied Mathematics and Statistics, Technical University of Cartagena, Hospital de Marina, Cartagena
30203, Spain
E-mail: juan.garcia@upct.es
Huatao Chen*
Division of Dynamics and Control, School of Mathematics and Statistics, Shandong University of Technology, ZiBo
255000, China
E-mail: htchencn@sdut.edu.cn
Reception: 05/01/2023 Acceptance: 10/03/2023 Publication: 30/03/2023
Suggested citation:
Zhenzhen Li, Kun Zhao, Hongkui Li, Juan L.G. Guirao and Huatao Chen. (2023). Global stability of the Euler-
Bernoulli beams excited by multiplicative white noises. 3C Tecnología. Glosas de innovación aplicada a la pyme,
12 (01), 386-412. https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
386
Global stability of the Euler-Bernoulli beams excited by
multiplicative white noises
Zhenzhen Li
Division of Dynamics and Control, School of Mathematics and Statistics, Shandong University of Technology, ZiBo
255000, China
E-mail: zzlicncn@163.com
Kun Zhao
Beijing Electro-Mechanical Engineering Institute, Beijing 100074, P. R. China
E-mail: zhaokunhit@yeah.net
Hongkui Li
Division of Dynamics and Control, School of Mathematics and Statistics, Shandong University of Technology, ZiBo
255000, China
E-mail: Lhk8068@163.com
Juan L.G. Guirao
Department of Applied Mathematics and Statistics, Technical University of Cartagena, Hospital de Marina, Cartagena
30203, Spain
E-mail: juan.garcia@upct.es
Huatao Chen*
Division of Dynamics and Control, School of Mathematics and Statistics, Shandong University of Technology, ZiBo
255000, China
E-mail: htchencn@sdut.edu.cn
Reception: 05/01/2023 Acceptance: 10/03/2023 Publication: 30/03/2023
Suggested citation:
Zhenzhen Li, Kun Zhao, Hongkui Li, Juan L.G. Guirao and Huatao Chen. (2023). Global stability of the Euler-
Bernoulli beams excited by multiplicative white noises. 3C Tecnología. Glosas de innovación aplicada a la pyme,
12 (01), 386-412. https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
ABSTRACT
This paper considers the global stability of the Euler-Bernoulli beam excited by multiplicative white
noise. Based on the theory of global random attractors, the Hausdorff dimensions of the global random
attractors for the system is obtained. According to the relationship between Hausdorff dimensions and
global Lyapunov exponents, the global stability of the stochastic beam is derived.
KEYWORDS
Global stochastic stability, Stochastic Euler-Bernoulli beam, Global Lyapunov exponents
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
387
1 INTRODUCTION
The dynamics of the beams are important problem in the elastokinetics [1]. Tajik [2] proposed the
stability analysis of motion equations of unbalanced spinning pre-twisted beam. Dai [3] investigated the
limit point bifurcations and jump of cantilevered microbeams according to Galerkin method and modal
truncation. The bifurcation and chaos for transverse motion of axially accelerating viscoelastic beams
was studied by Chen [4]. Pellicano [5] analyzed the linear subcritical behavior, bifurcation analysis
and stability of a simply supported beam subjected to an axial transport of mass. Based on the direct
method of multiple scales, Mao [6] dealt with problems of stability and saddle-node bifurcations for
supercritically moving beam. Using Melnikov method [7], Zhang [8, 9] investigated the multi-pulse
global bifurcations of a cantilever beam. Zhou [10] studied the chaos and subharmonic bifurcation of
a composite laminated buckled beam with a lumped mass. Applying the phase plane and positive
position feedback approach, Hamed [11] investigated the stability and bifurcation of the cantilever
beam system which carrying an intermediate lumped mass, to name but a few. For more details, one
can see refer to Refs [12–15] and the references therein.
Stochastic stability is one of the most important issue in the research areas of stochastic dynamics [16].
It is well known that, when the problem associated with stability are considered, there exists a very
useful tool named Lyapunov exponents which can be distinguished as local type and global type [17].
Generally speaking, the solution of system is stability when biggest Lyapunov exponent associated is less
than 0. With respect to the investigations on the stability and other dynamics behaviors for the beams
by using local Lyapunov exponents, one can refer to the Refs [18
–
20] and the references therein. Similarly,
the global Lyapunov exponents are also the powerful tools in studying the global dynamics for stochastic
beams. Unfortunately, calculating the global Lyapunov exponents is not an easy thing, but if we only
consider the global stability of the system, we can use the Hausdorff dimension of global attractors
associated with system to describe the signs of the biggest global Lyapunov exponent. The method
which can be used to get the Hausdorff dimension estimations associated with the global Lyapunov
exponents was due to Debussche [21]. Employing this method, together with the support relationship
between global random attractors and probability invariant measures proposed by Crauel [22,23]. Chen
et al [24] consider the global dynamics of the Euler-Bernoulli beams with additive white noises. With
respect to investigation on the global dynamics of the nonautonomous the Euler-Bernoulli beams by
global attractors theory, see Chen et al [25].
Let
D
= (0
,L
), this paper consider the Euler-Bernoulli beam equation excited multiplicative white
noise in the following form
utt +α(ut)−∆2u+β∥∇u∥2−p(−∆)u=σu ˙
W, (1)
with the hinged boundary condition
x=0:u=uxx = 0; x=L:u=uxx =0,(2)
and the initial value
t=τ:u=u0,u
t=u1,(3)
where
u
=
t
(
t, x
)
,x∈D
is the lateral displacement of the beam,
α
(
ut
)denotes the damping,
β>
0
,σ
are constants, the negative and positive of
p∈R
can show the stretch and compress of the beam.
∥∇u∥2
denotes the geometry of the beam bending for its elongation.
W
is the one dimensional two-sided
real-valued standard Wiener process, σu ˙
Wrepresents the multiplicative white noise.
Let
∥u∥≡∥u∥L2(D)
,
∥u∥s≡∥u∥Hs
0(D)
,(
u, v
)
≡
(
u, v
)
L2(D)
,(
u, v
)
s
=(
u, v
)
Hs
0(D)
, where
Hs
(
D
),
Hs
0
(
D
),
s∈R
are the usual Sobolev Spaces,for more detailed, see [26].
A
=∆
2
with boundary
condition
(2)
, then
D
(
A
)=
{u|u∈H4
(
D
)
H1
0
(
D
)
,
∆
u
=0
}
, and then
A
is self-adjoint, positive,
unbounded linear operators and
A−1∈L
(
L2
(
D
)) is compact. then, their eigenvalues
{λi}i∈N
satisfy
0
<λ
1≤λ2≤···→∞
and the corresponding eigenvalues
{ei}∞
i=1
form an orthonormal basis in
L2
(
D
).
Following the the mechanism in [27] p55, the power of (
−
∆)
s,s∈R
can also be defined,particularly,
D
(
A1
2
)=
H1
0
(
D
)
H2
(
D
). Moreover, for any
s1,s
2∈R,s
1>s
2
,
D
(
As1
)can be compact imbedding in
D(As2),and the following holds
∥u∥s1≥λ
s1−s2
2
1∥u∥s2,∀u∈D(As1).(4)
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
388
1 INTRODUCTION
The dynamics of the beams are important problem in the elastokinetics [1]. Tajik [2] proposed the
stability analysis of motion equations of unbalanced spinning pre-twisted beam. Dai [3] investigated the
limit point bifurcations and jump of cantilevered microbeams according to Galerkin method and modal
truncation. The bifurcation and chaos for transverse motion of axially accelerating viscoelastic beams
was studied by Chen [4]. Pellicano [5] analyzed the linear subcritical behavior, bifurcation analysis
and stability of a simply supported beam subjected to an axial transport of mass. Based on the direct
method of multiple scales, Mao [6] dealt with problems of stability and saddle-node bifurcations for
supercritically moving beam. Using Melnikov method [7], Zhang [8, 9] investigated the multi-pulse
global bifurcations of a cantilever beam. Zhou [10] studied the chaos and subharmonic bifurcation of
a composite laminated buckled beam with a lumped mass. Applying the phase plane and positive
position feedback approach, Hamed [11] investigated the stability and bifurcation of the cantilever
beam system which carrying an intermediate lumped mass, to name but a few. For more details, one
can see refer to Refs [12–15] and the references therein.
Stochastic stability is one of the most important issue in the research areas of stochastic dynamics [16].
It is well known that, when the problem associated with stability are considered, there exists a very
useful tool named Lyapunov exponents which can be distinguished as local type and global type [17].
Generally speaking, the solution of system is stability when biggest Lyapunov exponent associated is less
than 0. With respect to the investigations on the stability and other dynamics behaviors for the beams
by using local Lyapunov exponents, one can refer to the Refs [18
–
20] and the references therein. Similarly,
the global Lyapunov exponents are also the powerful tools in studying the global dynamics for stochastic
beams. Unfortunately, calculating the global Lyapunov exponents is not an easy thing, but if we only
consider the global stability of the system, we can use the Hausdorff dimension of global attractors
associated with system to describe the signs of the biggest global Lyapunov exponent. The method
which can be used to get the Hausdorff dimension estimations associated with the global Lyapunov
exponents was due to Debussche [21]. Employing this method, together with the support relationship
between global random attractors and probability invariant measures proposed by Crauel [22,23]. Chen
et al [24] consider the global dynamics of the Euler-Bernoulli beams with additive white noises. With
respect to investigation on the global dynamics of the nonautonomous the Euler-Bernoulli beams by
global attractors theory, see Chen et al [25].
Let
D
= (0
,L
), this paper consider the Euler-Bernoulli beam equation excited multiplicative white
noise in the following form
utt +α(ut)−∆2u+β∥∇u∥2−p(−∆)u=σu ˙
W, (1)
with the hinged boundary condition
x=0:u=uxx = 0; x=L:u=uxx =0,(2)
and the initial value
t=τ:u=u0,u
t=u1,(3)
where
u
=
t
(
t, x
)
,x∈D
is the lateral displacement of the beam,
α
(
ut
)denotes the damping,
β>
0
,σ
are constants, the negative and positive of
p∈R
can show the stretch and compress of the beam.
∥∇u∥2
denotes the geometry of the beam bending for its elongation.
W
is the one dimensional two-sided
real-valued standard Wiener process, σu ˙
Wrepresents the multiplicative white noise.
Let
∥u∥≡∥u∥L2(D)
,
∥u∥s≡∥u∥Hs
0(D)
,(
u, v
)
≡
(
u, v
)
L2(D)
,(
u, v
)
s
=(
u, v
)
Hs
0(D)
, where
Hs
(
D
),
Hs
0
(
D
),
s∈R
are the usual Sobolev Spaces,for more detailed, see [26].
A
=∆
2
with boundary
condition
(2)
, then
D
(
A
)=
{u|u∈H4
(
D
)
H1
0
(
D
)
,
∆
u
=0
}
, and then
A
is self-adjoint, positive,
unbounded linear operators and
A−1∈L
(
L2
(
D
)) is compact. then, their eigenvalues
{λi}i∈N
satisfy
0
<λ
1≤λ2≤···→∞
and the corresponding eigenvalues
{ei}∞
i=1
form an orthonormal basis in
L2
(
D
).
Following the the mechanism in [27] p55, the power of (
−
∆)
s,s∈R
can also be defined,particularly,
D
(
A1
2
)=
H1
0
(
D
)
H2
(
D
). Moreover, for any
s1,s
2∈R,s
1>s
2
,
D
(
As1
)can be compact imbedding in
D(As2),and the following holds
∥u∥s1≥λ
s1−s2
2
1∥u∥s2,∀u∈D(As1).(4)
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
Suppose
α
(
ut
)=
αA1
2ut
, here
α>
0is a constant,
A1
2ut
is called the damping with strongly form, then
following abstract form of Euler-Bernoulli beam equation excited multiplicative white noises
utt +αA1
2ut+Au +β∥∇u∥2−p(−∆)u=σu ˙
W,
x=0:u=uxx = 0; x=L:u=uxx =0,
u(x, τ)=u0(x),∂
tu(x, τ)=u1(x).
(5)
The rest of paper is organized as follows. Preliminaries and main lemmas are listed in Section 2. Section
3 is devoted to derive main results and proof associated. The proofs for the main lemmas are given in
Section 4. Finally, the conclusions are presented in section (4).
2 PRELIMINARIES AND MAIN LEMMAS
2.1 PRELIMINARIES
Let
E1
=
D
(
A1
2
)
×L2
(
D
)
,E
2
=
D
(
A3
4
)
×H1
0
(
D
)equipped with Graph norms and the induced inner
products, then they are all Hilbert spaces. Let
(X, ∥·∥
X)
be a complete separable metric space with Borel
σ
-algebra
B
(
X
)and (Ω
,F,P
)be a probability space. We consider Ω=
{ω|ω
(
·
)
∈C
(
R,R
)
,ω
(0) = 0
},F
is the
σ
-algebra and
P
is the Wiener measure. Set a family of measure preserving and ergodic
transformations θtω(·)=ω(·+t)−ω(·),∀t∈R, Consider the following system
dz +µzdt =dW
z(−∞)=0,(6)
and the solution of system (6) is given by
z(θtω) := −µ0
−∞
eµτ (θtω)(τ)dτ. (7)
z
(
θtω
)is Ornstein-Uhlenbeck process (in Short O-U process). The following results on O-U process
belong to Fan [28].
Lemma 1. The Ornstein-Uhlenbeck process z(θtω)defined in system (7) satisfies
E[|z(θtω)|]= 1
√πµ,E|z(θtω)|2=1
2µ,(8)
and there exists a constant t1(ω)>0satisfying
0
−t|z(θsω)|ds < 1
√πµt, 0
−t|z(θeω)|2ds < 1
2µt, ∀t≥t1,(9)
and the mapping t→ z(θtω)grows sublinearly, i.e.
lim
t→±∞
z(θtω)
t=0.
Moreover If µ≥2β, β > 0, then
Eeβs+t
s|z(θτω)|2dτ ≤e
βt
µ,∀s∈R,t≥0,(10)
when µ3≥r2,r ≥0, the following holds
Eers+t
s|z(θτω)|dτ ≤e
rt
√µ,∀s∈R,t≥0.(11)
The following is Random dynamical system which is due to Aronld [29].
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
389
Definition 1. The flow
{θt:Ω→Ω,t∈R}
is a family of measure preserving transformations in
probability space, such that (
t, ω
)
→θtω
is measurable,
θ0
=
id, θt+s
=
θt◦θs
for all
s, t ∈R
. Then
Ω,F, P, {θt}t∈Ris called a metric dynamical system.
Definition 2. A random dynamical system (RDS) on Polish space (
X, d
)with Borel
σ
algebra
B
(
X
)
on Ω,F, P, {θt}t∈Ris a measurable mapping
ϕ:R+×Ω×X→X
(t,ω,x)→ ϕ(t, ω)x
such that P−a.s.
1. ϕ(0,ω)=id on X.
2. ϕ(t+s, ω)=ϕ(t, θsω)◦ϕ(s, ω)for all s, t ∈R+.
The theory of global random attractors is as following, one can refer to Crauel and Flandoli [22, 30] and
Schmalfuss [31].
Definition 3. A random set
K
(
ω
)is said to absorb the set
B⊂X
for a RDS
ϕ
, if
P−a.s
. there
exists tB(ω)such that
ϕ(t, θ−tω)B⊂K(ω),∀t≥tB(ω).
Definition 4. Let
B⊂
2
X
is a collection of subsets of
X
, then a closed random set
A
(
ω
)is called
random attractor associated with the RDS ϕ, if P−a.s.
1. A(ω)is a random compact set.
2. A(ω)is invariant i.e. ϕ(t, ω),A(ω)=A(θtω)for all t≥0.
3. For every B∈B,
lim
t→∞ dist (ϕ(t, θ−tω)B,A(ω)) = 0,
where dist (·,·)denotes the Hausdorff semidistance defined by
dist(A, B) = sup
x∈A
inf
y∈Bd(x, y), A, B ⊂X.
The following random fixed point is important in investigating the global stability.
Definition 5. [29, 32] Let
φ
(
t, ω
)be a RDS,
a
(
ω
)is a random set and consists of one point (
P−a.s
).
a(ω)is called the random fixed point if the following holds
φ(t, ω)a(ω)=a(θtω),∀t∈R+.
The coming theorem is very useful to verify the existence of global random attractors in this paper.
Theorem 1. [24] Suppose
Sε
(
t, ω
)is a RDS on Polish space (
X, d
), and suppose that
ϕ
possesses an
absorbing set in
X
and for any nonrandom bounded set
B⊂X, lim
t→+∞Sε(t, θ−tω)B
is relative compact
P-a.s. Then ϕpossesses uniqueness random attractors defined by the following
A(ω)=
B∈X
ΛB(ω),
where union is taken over all bounded B⊂X, and ΛB(ω)given by
ΛB(ω)=
s≥0
t≥s
ϕ(t, θ−tω)B.
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
390
Definition 1. The flow
{θt:Ω→Ω,t∈R}
is a family of measure preserving transformations in
probability space, such that (
t, ω
)
→θtω
is measurable,
θ0
=
id, θt+s
=
θt◦θs
for all
s, t ∈R
. Then
Ω,F, P, {θt}t∈Ris called a metric dynamical system.
Definition 2. A random dynamical system (RDS) on Polish space (
X, d
)with Borel
σ
algebra
B
(
X
)
on Ω,F, P, {θt}t∈Ris a measurable mapping
ϕ:R+×Ω×X→X
(t,ω,x)→ ϕ(t, ω)x
such that P−a.s.
1. ϕ(0,ω)=id on X.
2. ϕ(t+s, ω)=ϕ(t, θsω)◦ϕ(s, ω)for all s, t ∈R+.
The theory of global random attractors is as following, one can refer to Crauel and Flandoli [22, 30] and
Schmalfuss [31].
Definition 3. A random set
K
(
ω
)is said to absorb the set
B⊂X
for a RDS
ϕ
, if
P−a.s
. there
exists tB(ω)such that
ϕ(t, θ−tω)B⊂K(ω),∀t≥tB(ω).
Definition 4. Let
B⊂
2
X
is a collection of subsets of
X
, then a closed random set
A
(
ω
)is called
random attractor associated with the RDS ϕ, if P−a.s.
1. A(ω)is a random compact set.
2. A(ω)is invariant i.e. ϕ(t, ω),A(ω)=A(θtω)for all t≥0.
3. For every B∈B,
lim
t→∞ dist (ϕ(t, θ−tω)B,A(ω)) = 0,
where dist (·,·)denotes the Hausdorff semidistance defined by
dist(A, B) = sup
x∈A
inf
y∈Bd(x, y), A, B ⊂X.
The following random fixed point is important in investigating the global stability.
Definition 5. [29, 32] Let
φ
(
t, ω
)be a RDS,
a
(
ω
)is a random set and consists of one point (
P−a.s
).
a(ω)is called the random fixed point if the following holds
φ(t, ω)a(ω)=a(θtω),∀t∈R+.
The coming theorem is very useful to verify the existence of global random attractors in this paper.
Theorem 1. [24] Suppose
Sε
(
t, ω
)is a RDS on Polish space (
X, d
), and suppose that
ϕ
possesses an
absorbing set in
X
and for any nonrandom bounded set
B⊂X, lim
t→+∞Sε(t, θ−tω)B
is relative compact
P-a.s. Then ϕpossesses uniqueness random attractors defined by the following
A(ω)=
B∈X
ΛB(ω),
where union is taken over all bounded B⊂X, and ΛB(ω)given by
ΛB(ω)=
s≥0
t≥s
ϕ(t, θ−tω)B.
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
Some powerful transformation on system (5) are derived as following. Set v1=ut, we get
ut=v1,
v1t=−αA1
2v1−Au −β∥∇u∥2−p(−∆)u+σu ˙
W,
x=0:u=ux= 0; x=L:u=uxx =0,
u(x, τ)=u0(x),v
1(x, τ)=u1(x),
(12)
let v2=v1+εu, here ε>0, we have
dU
dt =QU +X1(ω, U),Uτ=(u0,u
1+εu0)T,(13)
where
U=U1
U2=u
v2,Q=−εI, I
−A+εαA1
2−ε,−αA1
2−εI,
X1(ω, U)=X11(ω, U)
X12(ω, U)=0
−β∥∇u∥2−p(−∆)u+σu ˙
W.
System
(5)
and system
(13)
are equivalent, thus, the dynamical behavior of system
(5)
can be reflected
by system (13).
Set v=ut+εu −σuz (θtω), where z(θtω)O-U process formulated by (6), it gives that
dV
dt =QV +X2(θtω)+X3(V),Vτ=(u0,u
1+εu0−σu0z(θtω))T,(14)
where
V=V1
V2=u
v,X3(V)=X31(V)
X32(V)=0
−β∥∇u∥2−p(−∆)u,
X2(θtω)=X21 (θtω)
X22 (θtω)=σuz (θtω)
σµ−αA1
2+2ε−σz (θtω)uz (θtω)−σvz (θtω).
System (14) is a system with random coefficient, which can be studied ωby ω.
2.2 MAIN LEMMAS
In order to obtain the global stochastic stability of the system
(13)
based on the global random
attractors theory, the first step should be utilized to verify the system (13) can induce a RDS.
Lemma 2. For any
τ∈R
and initial value
Vτ∈E1
, system
(14)
possesses a unique local mild solution
V(t,τ,ω;Vτ)∈C([τ,τ +T],E
1),t∈[τ,τ +T],∀T>0.
Let
φ
(
t,τ,ω
)be the solution mapping determined by system
(14)
, which means
V(t,τ,ω;Vτ)
=
φ(t,τ,ω)Vτ, we have φ(t, 0,ω)=φ(0,−t, θtω),∀t≥0,∀ω∈Ω. Defining
S(t, ω) := φ(t, 0,ω),∀t≥0,∀ω∈Ω,
gives the RDS associated with system
(14)
, which together with the relationship between
ιε
:
(u, v1)T→
[
y, v1−σuz (θtω)
]
T
=
(u, v)T
implies that system
(13)
can also generate a RDS
Sε
(
t, ω
)with the
following from
Sε(t, ω)=ι−1(θtω),S(t, ω)ι(ω):E1→E1,
is the RDS induced by system
(13)
. The following are very important in proof for the existence of the
global random attractors for the system (14).
Let
p≤2
3λ
1
4
1,0<ε= min
1,
α8λ
1
4
1−12p
λ
1
4
110λ
1
4
1−15p+4α2λ
1
4
1
,(15)
then
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
391
Lemma 3. For any given U=[U1,U
2]T∈E1, the following holds
(QU ,U)E1≤−ε
2∥U∥2
E1−ε
4∥U2∥2−3pε
4∥A1
4U1∥2.
Based on the Lemma 3, we have
Lemma 4. For any given bounded set
B∈E1
, there exists a random variable
r1
(
ω
)
>
0and
TB
(
ω
)
≥
0,
for ∀t≥TB(ω), the following holds,
∥Sε(t, θ−tω)B∥E1≤r1(ω).
Furthermore,
E�r2
1(ω)<∞.
Lemma 4 shows that
Sε
(
t, ω
)has a global absorbing set. In order to obtain the existence of the global
random attractors for the system
Sε
(
t, ω
), besides the existence of the global absorbing set, we need
verify that the Sε(t, ω)is asymptotically compact.
To begin with, we decompose the solution
V
generated by system
(13)
with the initial value
Uτ
(
ω
)=
(u0,u
1+εu0)T
into two parts
U
=
Ua
+
Ub
=
(ua,u
a
t+εua)T
+
�ub,u
b
t+εubT, where Ua
solves
dUa
dt =QU a+0, σua˙
WT
Ua
τ(ω)=0,
(16)
and Ubsolves
dUb
dt =QU b+0,−β∥∇u∥2−p(−∆)u+σub˙
WT
Ub
τ=(u0,u
1+εu0)T.
(17)
Split the solution
V
of system
(14)
with the initial value
Vτ
(
ω
)=
(u0,u
1+εu0−σu0z(θ−τω))T
into
two parts
V
=
Va
+
Vb
=
(ua,v
a)T
+
�ub,v
bT
=
(ua,u
a
t+εua)T
+
�ub,u
b
t+εub−σuz (θtω)T,where Va
solves
dVa
dt =QV a+σuaz(θtω),σµ−αA 1
2+2ε−σz (θtω)uaz(θtω)−σvaz(θtω)T
Va
τ(ω)=0,
(18)
and Vbsolves
dVb
dt =QV b+σubz(θtω),σµ−αA 1
2+2ε−σz (θtω)ubz(θtω)−σvbz(θtω)T
+�0,β∥∇u∥2−p∆uT
Vb
τ=(u0,u
1+εu0−σu0z(θ−τω))T.
(19)
For the solutions of system (18) and (19), we have the following priori estimates respectively.
Lemma 5. The solution Uaof system (16) satisfies
lim
t→+∞
A1
4Ua
E1
=0.
Lemma 6. There exists r2(ω)>∞,T
2(ω)≥0, the solutions Ubsatisfies
Ub
E1≤r2(ω),∀t≥T2(ω).
Lemma 7. There exists r3(ω)>0,T
3(ω)≥0, the following holds
A1
4Ub
E1≤r3(ω),∀t≥T3(ω).
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
392
Lemma 3. For any given U=[U1,U
2]T∈E1, the following holds
(QU ,U)E1≤−ε
2∥U∥2
E1−ε
4∥U2∥2−3pε
4∥A1
4U1∥2.
Based on the Lemma 3, we have
Lemma 4. For any given bounded set
B∈E1
, there exists a random variable
r1
(
ω
)
>
0and
TB
(
ω
)
≥
0,
for ∀t≥TB(ω), the following holds,
∥Sε(t, θ−tω)B∥E1≤r1(ω).
Furthermore,
E�r2
1(ω)<∞.
Lemma 4 shows that
Sε
(
t, ω
)has a global absorbing set. In order to obtain the existence of the global
random attractors for the system
Sε
(
t, ω
), besides the existence of the global absorbing set, we need
verify that the Sε(t, ω)is asymptotically compact.
To begin with, we decompose the solution
V
generated by system
(13)
with the initial value
Uτ
(
ω
)=
(u0,u
1+εu0)T
into two parts
U
=
Ua
+
Ub
=
(ua,u
a
t+εua)T
+
�ub,u
b
t+εubT, where Ua
solves
dUa
dt =QU a+0, σua˙
WT
Ua
τ(ω)=0,
(16)
and Ubsolves
dUb
dt =QU b+0,−β∥∇u∥2−p(−∆)u+σub˙
WT
Ub
τ=(u0,u
1+εu0)T.
(17)
Split the solution
V
of system
(14)
with the initial value
Vτ
(
ω
)=
(u0,u
1+εu0−σu0z(θ−τω))T
into
two parts
V
=
Va
+
Vb
=
(ua,v
a)T
+
�ub,v
bT
=
(ua,u
a
t+εua)T
+
�ub,u
b
t+εub−σuz (θtω)T,where Va
solves
dVa
dt =QV a+σuaz(θtω),σµ−αA 1
2+2ε−σz (θtω)uaz(θtω)−σvaz(θtω)T
Va
τ(ω)=0,
(18)
and Vbsolves
dVb
dt =QV b+σubz(θtω),σµ−αA 1
2+2ε−σz (θtω)ubz(θtω)−σvbz(θtω)T
+�0,β∥∇u∥2−p∆uT
Vb
τ=(u0,u
1+εu0−σu0z(θ−τω))T.
(19)
For the solutions of system (18) and (19), we have the following priori estimates respectively.
Lemma 5. The solution Uaof system (16) satisfies
lim
t→+∞
A1
4Ua
E1
=0.
Lemma 6. There exists r2(ω)>∞,T
2(ω)≥0, the solutions Ubsatisfies
Ub
E1≤r2(ω),∀t≥T2(ω).
Lemma 7. There exists r3(ω)>0,T
3(ω)≥0, the following holds
A1
4Ub
E1≤r3(ω),∀t≥T3(ω).
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
The variation equations of system (13)
d
U
dt =Q
U+
X(U)
U,(20)
where
X(U)=
X1(U),
X3(U)
X2(U),
X4(U)
=0,0
β∥∇u∥2−p(−∆) ·+2β(∇u, ∇·)(−∆)u+σ˙
W,0.
On the other hand
d
V
dt =Q
V+
X2(V)
V+
X3(V)
V,(21)
here
V={
V1,
V2},
X2((θtω)) =
X21((θtω)),
X23((θtω))
X22((θtω)),
X24((θtω))
=σz (θtω),0
σ(µ−α+2ε−σz (θtω)) z(θtω),−σz (θtω),
X3(V)=
X31(V),
X33(V)
X32(V),
X34(V)
=0,0
β∥∇u∥2−p(−∆) ·+2β(∇u, ∇·)(−∆)u, 0.
Let
U
=
U1,
U2
be the solution of system
(20)
with initial
t
=0:
U0∈E1
and
V
=
V1,
V2
be the
solution of system (21) with initial value t=0:
V0=I∈E1.
Let U(1) =U(1)
1,U(1)
2,U(2) =U(2)
1,U(2)
2are two solutions of system (13), then
dU(1) −U(2)
dt =QU(1) −U(2)+X1U(1)−X1U(2),(22)
Analogously, let
V(1)
=
V(1)
1,V(1)
2
=
[u1,v
1],V(2)
=
V(2)
1,V(2)
2
=
[u2,v
2]
are two solutions of
system (14) with initial values V(1)
0,V(2)
0, where V(2)
0=V(1)
0+I,I=[I1,I
2]∈E1, then
dV(1) −V(2)
dt =QV(1) −V(2)+X2V(1)−X2V(2)
+X3V(1)−X3V(2).
(23)
In addition, set Γ= [Γ1,Γ2]T=U(1) −U(2) −
U, we have the following two Lemmas.
Lemma 8. For ∀u1,u
2∈D(A1
2), there exist constants c1(ω),c
2(ω)in (51), such that
X32 (u1)−
X32 (u2)
LD(A1
2),L2(D)≤c1(ω)
A1
2u1−A1
2u2
,
X32 (u1)
LD(A1
2),L2(D)≤c2(ω).
(24)
Furthemnore,
E(c1(ω)) <∞,E(c2(ω)) <∞.
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
393
Lemma 9. Let Sε(ω) := Sε(t, ω), then Sε(ω)is uniformly quasidifferentiable on A(ω).
The following relationship between Hausdorff dimension and global Lyapunov exponents which will
be used to assert the global stochastic stability.
Theorem 2. [21]Suppose that, for
k
=1
,...,d, sup
u∈A(ω)
ωk
(
DSε
(
ω, u
)) is integrable. Then we can
generalize the notion of global Lyapunov exponents introduced in Ref. [17] by setting
Λk=Eln sup
u∈A(ω)
ωk(DSε(ω, u))−Eln sup
u∈A(ω)
ωk−1(DSε(ω, u)),
for k≥2and
Λ1=Eln sup
u∈A(ω)
ω1(DSε(ω, u)).
And it is easy to see that there exists ωdsatisfying (26)(27) if and only if
Λ1+...+Λ
d<0.
3 MAIN RESULTS AND PROOFS
By Lemma 4, we get that
Sε
(
t, ω
)has a global absorbing set, which along with Lemma 5, Lemma 6 and
Lemma 7 gives that
Sε
(
t, ω
)is asymptotically compact. Thus, employing the Theorem 1, it is asserted
that there exist global random attractors A(ω),ω ∈Ωfor system 13.
Based on the results and the estimations on global random attractors, the Hausdorff dimension of
the global random attractors
A
(
ω
)can be got by the method proposed by Debussche [21]. The outline
of this method is as follows.
Firstly, it is verified that
Sε
(
ω
)is almost surely uniformly differentiable on
A
(
w
), i.e. for
∀u∈A
(
w
),
there exist a linear operator
DSε
(
ω, u
)in
L
(
H
), the space of continuous linear operator from
H
to
H
,
such that if uand u+hare in A(w):
|Sε(ω)(u+h)−Sε(ω)u−DSε(ω, u)h|≤K(ω)|h|1+α,(25)
where K(w)≥1,w ∈Ωis a random variable, and α>0is a constant.
Secondly, there exists an integrable random variable ωd, such that
ωd(DSε(ω, u)) ≤ωd(ω),∀u∈A(ω),P−a.s, (26)
and
E(ln (ωd)) <0.(27)
Thirdly, there exists a random variable α1such Palmost surely
α1≥1,α
1(DSε(ω, u)) ≤α1(ω),∀u∈A(ω),P−a.s, (28)
and
E(ln α1)<∞.(29)
Finally,
E(ln K)<∞.(30)
If the (26),(27),(28)(29) and (30) hold, the Hausdorff dimension of A(w)is less than d.
The main results in this paper is given in the following Theorem.
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
394
Lemma 9. Let Sε(ω) := Sε(t, ω), then Sε(ω)is uniformly quasidifferentiable on A(ω).
The following relationship between Hausdorff dimension and global Lyapunov exponents which will
be used to assert the global stochastic stability.
Theorem 2. [21]Suppose that, for
k
=1
,...,d, sup
u∈A(ω)
ωk
(
DSε
(
ω, u
)) is integrable. Then we can
generalize the notion of global Lyapunov exponents introduced in Ref. [17] by setting
Λk=Eln sup
u∈A(ω)
ωk(DSε(ω, u))−Eln sup
u∈A(ω)
ωk−1(DSε(ω, u)),
for k≥2and
Λ1=Eln sup
u∈A(ω)
ω1(DSε(ω, u)).
And it is easy to see that there exists ωdsatisfying (26)(27) if and only if
Λ1+...+Λ
d<0.
3 MAIN RESULTS AND PROOFS
By Lemma 4, we get that
Sε
(
t, ω
)has a global absorbing set, which along with Lemma 5, Lemma 6 and
Lemma 7 gives that
Sε
(
t, ω
)is asymptotically compact. Thus, employing the Theorem 1, it is asserted
that there exist global random attractors A(ω),ω ∈Ωfor system 13.
Based on the results and the estimations on global random attractors, the Hausdorff dimension of
the global random attractors
A
(
ω
)can be got by the method proposed by Debussche [21]. The outline
of this method is as follows.
Firstly, it is verified that
Sε
(
ω
)is almost surely uniformly differentiable on
A
(
w
), i.e. for
∀u∈A
(
w
),
there exist a linear operator
DSε
(
ω, u
)in
L
(
H
), the space of continuous linear operator from
H
to
H
,
such that if uand u+hare in A(w):
|Sε(ω)(u+h)−Sε(ω)u−DSε(ω, u)h|≤K(ω)|h|1+α,(25)
where K(w)≥1,w ∈Ωis a random variable, and α>0is a constant.
Secondly, there exists an integrable random variable ωd, such that
ωd(DSε(ω, u)) ≤ωd(ω),∀u∈A(ω),P−a.s, (26)
and
E(ln (ωd)) <0.(27)
Thirdly, there exists a random variable α1such Palmost surely
α1≥1,α
1(DSε(ω, u)) ≤α1(ω),∀u∈A(ω),P−a.s, (28)
and
E(ln α1)<∞.(29)
Finally,
E(ln K)<∞.(30)
If the (26),(27),(28)(29) and (30) hold, the Hausdorff dimension of A(w)is less than d.
The main results in this paper is given in the following Theorem.
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
Theorem 3. Let
d= min
n∈Z+|1
n
n
i=1
λ−1
2
i<
ε
2−8
ε�βr2+|p|2−2M7
√πµ −2M8
2µ
16β2r4
ελ
1
4
1
,
when
M7,M
8
are constants described by
(56)
stated in Section 4, then the Hausdorff dimension of
A
(
ω
)
dH(A(ω)) < d.
Obviously, employing and Theorem 2, we have that when
λ−1
2
1≤
ε
2−8
ε�βr2+|p|2−2M7
√πµ −2M8
2µ
16β2r4
ελ
1
4
1
,
the Hausdorff dimension of for system
(13)
is 0, which indicates that the global random attractors
consists of only one random fixed point which is global stability.
Proof. The solution mapping of system
(20)
denoted by
DSε
(
t, ω, U
), and
DS
(
t, ω, V
)signifies the
solution mapping of system
(21)
. Moreover, let
DSε
(
ω, U
) :=
DSε
(1
, ω, U
). By Lemma 9, we attain
Sε
(
ω
)is almost surely uniform differentiable, and the conditions
(25)
and
(30)
hold. In the light of
(90)
and
(91)
stated in the proof of Lemma 9 in Section 4, the
(28)
and
(29)
are satisfied. On the other
hand, we have
ωn(DS(ω, V)) = sup
U(i)
0
<1
i=1,...,n
exp 1
tt
0
Tr(Q+
X2(V)+
X3(V)) ◦Qn(s)ds,
where
Qn
(
s
)=
Qns, τ, Vτ;
V0
1,...,
V0
n
is the orthogonal projector from
E1
onto the space spanned
by
V1
(
t
)
,...,
Vn
(
t
), here
V
=
S
(
t, τ
)
Vτ
and
V1,...,
Vn
are the solution of system
(21)
with initial
values
V0
=
V0
1,...,
V0
n
respectively. For any given time
s
,
Vi
(
s
)=
{µi(s),ν
i(τ)},i
=1
,...,n
is an
orthonormal basis of Qn(s)E1, then
Tr(Q+
X2(V)+
X3(V)) ◦Qn(s)=
n
i=1 (Q+
X2(V)+
X3(V))Vi(s),Vi(s)E1
.
In term of Lemma 3, we get
(QV i,Vi)E1≤−ε
2∥Vi∥2
E1−ε
4∥νi∥2,
and
X3Vi,ViE1
=−�β∥∇u∥2−p(−∆)µi,ν
i−(2β(∇u, ∇µi)(−∆)u, νi)
≤
β∥∇u∥2−p(−∆)µi
∥νi∥+∥2β(∇u, ∇µi)(−∆)u∥∥νi∥
≤|β∥∇u∥2−p|∥∆µi∥∥νi∥+2β
λ
1
8
1∥∆u∥2∥∇µi∥∥νi∥
≤8
ε�βr2+|p|2∥∆µi∥2+16β2r4
ελ
1
4
1∥∇µi∥2+ε
4∥νi∥2,
(31)
moreover, we have
X2
V,
V≤M7|z(θtω)|+M8|z(θtω)|2
V1
2
H2+
V2
2
H2,(32)
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
395
where M7,M
8defined by (57) stated in Section 4. Hence, we can obtain
Tr(Q+
X2(V)+
X3(V)) ◦Qn(s)
≤−ε
2∥Vi∥2
E1+8
ε�βr2+|p|2∥∆µi∥2+16β2r4
ελ
1
4
1∥∇µi∥2
+M7|z(θtω)|+M8|z(θtω)|2
V1
2
H2+
V2
2
H2
≤−nε
2+8n
ε�βr2+|p|2+16β2r4
ελ
1
4
1
n
i=1
λ−1
2
i
+2nM7|z(θtω)|+M8|z(θtω)|2.
Thus,
ωn(DSε(ω, V)) = sup
U(i)
0
<1
i=1,...,n
exp 1
tt
0
Tr(Q+
X2(V)+
X3(V)) ◦Qn(s)ds
= exp
1
tt
0−nε
2+8n
ε�βr2+|p|2+16β2r4
ελ
1
4
1
n
i=1
λ−1
2
i+2nM7|z(θtω)|+M8|z(θtω)|2ds
,
which together with (9) gives that ∃T4(ω)>0,∀t≥T4,
ωn(DSε(ω, V))
≤exp
−nε
2+8n
ε�βr2+|p|2+2nM7
√πµ +2nM8
2µ+16β2r4
ελ
1
4
1
n
i=1
λ−1
2
i
.(33)
Let
T(θtω)=10
σz (θtω)1
.
Clearly, T(θtω)is a linear operator from E1to itself, then we have
DSε(t, ω)=T(θtω)DS(T,ω).
Let
On
be the space spanned by
e1,...,e
n
for any
n∈N
, then the quadratic form
χ∈On→
∥T
(
θtω
)
χ∥2
E1
is well defined, continuous, and nonnegative on
On
. Let
α1≥···≥αn
be the eigenvalues
associated with χ1,χ
2,...,χ
n∈Onsatisfying
(T(θtω)χi,T(θtω)χj)E1=α2
iδij .
Set χj=(ξj,η
j)T,j =1,2,...,n, it can be derived
α2
i=(T(θtω)χi,T(θtω)χi)E1
≤(ξj,ξ
j)H2+(σz (θtω)ξj+ηj, σz (θtω)ξj+ηj)
≤(ξj,ξ
j)H2+(ηj,η
j)+2σ|z(θtω)|∥ξj∥∥ηj∥+σ2|z(θtω)|2(ξj,ξ
j)
≤∥ξj∥2
H2+∥ηj∥2+1
λ
1
2
1�σ|z(θtω)|+σ2|z(θtω)|2∥ξj∥2
H2+σ|z(θtω)|∥ηj∥2
≤1+ σ
λ
1
2
1|z(θtω)|+σ2
λ
1
2
1|z(θtω)|2+λ−1
1+λ−1
1σ|z(θtω)|.
So
ln αi≤ln
σ
2λ
1
2
1|z(θtω)|+σ
2λ
1
2
1|z(θtω)|2+1
2λ1
+σ
2λ1|z(θtω)|
,
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
396
where M7,M
8defined by (57) stated in Section 4. Hence, we can obtain
Tr(Q+
X2(V)+
X3(V)) ◦Qn(s)
≤−ε
2∥Vi∥2
E1+8
ε�βr2+|p|2∥∆µi∥2+16β2r4
ελ
1
4
1∥∇µi∥2
+M7|z(θtω)|+M8|z(θtω)|2
V1
2
H2+
V2
2
H2
≤−nε
2+8n
ε�βr2+|p|2+16β2r4
ελ
1
4
1
n
i=1
λ−1
2
i
+2nM7|z(θtω)|+M8|z(θtω)|2.
Thus,
ωn(DSε(ω, V)) = sup
U(i)
0
<1
i=1,...,n
exp 1
tt
0
Tr(Q+
X2(V)+
X3(V)) ◦Qn(s)ds
= exp
1
tt
0−nε
2+8n
ε�βr2+|p|2+16β2r4
ελ
1
4
1
n
i=1
λ−1
2
i+2nM7|z(θtω)|+M8|z(θtω)|2ds
,
which together with (9) gives that ∃T4(ω)>0,∀t≥T4,
ωn(DSε(ω, V))
≤exp
−nε
2+8n
ε�βr2+|p|2+2nM7
√πµ +2nM8
2µ+16β2r4
ελ
1
4
1
n
i=1
λ−1
2
i
.(33)
Let
T(θtω)=10
σz (θtω)1
.
Clearly, T(θtω)is a linear operator from E1to itself, then we have
DSε(t, ω)=T(θtω)DS(T,ω).
Let
On
be the space spanned by
e1,...,e
n
for any
n∈N
, then the quadratic form
χ∈On→
∥T
(
θtω
)
χ∥2
E1
is well defined, continuous, and nonnegative on
On
. Let
α1≥···≥αn
be the eigenvalues
associated with χ1,χ
2,...,χ
n∈Onsatisfying
(T(θtω)χi,T(θtω)χj)E1=α2
iδij .
Set χj=(ξj,η
j)T,j =1,2,...,n, it can be derived
α2
i=(T(θtω)χi,T(θtω)χi)E1
≤(ξj,ξ
j)H2+(σz (θtω)ξj+ηj, σz (θtω)ξj+ηj)
≤(ξj,ξ
j)H2+(ηj,η
j)+2σ|z(θtω)|∥ξj∥∥ηj∥+σ2|z(θtω)|2(ξj,ξ
j)
≤∥ξj∥2
H2+∥ηj∥2+1
λ
1
2
1�σ|z(θtω)|+σ2|z(θtω)|2∥ξj∥2
H2+σ|z(θtω)|∥ηj∥2
≤1+ σ
λ
1
2
1|z(θtω)|+σ2
λ
1
2
1|z(θtω)|2+λ−1
1+λ−1
1σ|z(θtω)|.
So
ln αi≤ln
σ
2λ
1
2
1|z(θtω)|+σ
2λ
1
2
1|z(θtω)|2+1
2λ1
+σ
2λ1|z(θtω)|
,
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
for any 1≤i≤n. Since
ωn(T(θtω)) = α1α2...α
n,
we obtain
ln ωn(T(θtω)) ≤ln
nσ
2λ
1
2
1|z(θtω)|+nσ
2λ
1
2
1|z(θtω)|2+n
2λ1
+nσ
2λ1|z(θtω)|
,(34)
which combine with Lemma 1 gives that
E(ωn(T(θtω))) ≤E
nσ
2λ
1
2
1|z(θtω)|+nσ
2λ
1
2
1|z(θtω)|2+n
2λ1
+nσ
2λ1|z(θtω)|
≤M9,
(35)
where M9given by (58) stated in Section 4. Since (33) and (34), we get that
ln ωn(DSε(ω, U)) = ln ωn(T(θtω)) + ln ωn(DS(T,ω))
≤ln
nσ
2λ
1
2
1|z(θtω)|+nσ
2λ
1
2
1|z(θtω)|2+n
2λ1
+nσ
2λ1|z(θtω)|
−nε
2+8n
ε�βr2+|p|2+2nM7
√πµ +2nM8
2µ+16β2r4
ελ
1
4
1
n
i=1
λ−1
2
i,
then
ωn(DSε(ω, U)) ≤
nσ
2λ
1
2
1|z(θtω)|+nσ
2λ
1
2
1|z(θtω)|2+n
2λ1
+nσ
2λ1|z(θtω)|
×exp
−nε
2+8n
ε�βr2+|p|2+2nM7
√πµ +2nM8
2µ+16β2r4
ελ
1
4
1
n
i=1
λ−1
2
i
.
Employing (35), we obtain
E(ωn(DSε(ω, U)))
≤M9E
exp
−nε
2+8n
ε�βr2+|p|2+2nM7
√πµ +2nM8
2µ+16β2r4
ελ
1
4
1
n
i=1
λ−1
2
i
.
Let
ωn(ω)=M9exp
−nε
2+8n
ε�βr2+|p|2+2nM7
√πµ +2nM8
2µ+16β2r4
ελ
1
4
1
n
i=1
λ−1
2
i
,
then
E(ωn(DSε(ω, U))) ≤E(ωn(ω)) ,(36)
and
E(ln ωn(ω)) ≤ln M9
×E
−nε
2+8n
ε�βr2+|p|2+2nM7
√πµ +2nM8
2µ+16β2r4
ελ
1
4
1
n
i=1
λ−1
2
i
,(37)
together with Lemma 4 we find
ln M9×E
−nε
2+8n
ε�βr2+|p|2+2nM7
√πµ +2nM8
2µ+16β2r4
ελ
1
4
1
n
i=1
λ−1
2
i
≤+∞.
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
397
Therefore, we get dH(A(ω)) < d. Especially, if
λ−1
2
1≤
ε
2−8
ε�βr2+|p|2−2M7
√πµ −2M8
2µ
16β2r4
ελ
1
4
1
,
dH
(
A
(
ω
))=0, which merges with the Theorem 2 shows that the largest global Lyapunov exponent of
Sε(ω)is
λ1=E(ln ω1(ω))
≤E
−ε
2+8
ε�βr2+|p|2+2M7
√πµ +2M8
2µ+16β2r4
ελ
1
4
1
n
i=1
λ−1
2
i
<0,(38)
thus, we can conclude there exists a random fixed point of system
(14)
which is global stochastic
stability.
4 PROOFS FOR LEMMAS
This section is intended to complete the proofs of Lemmas listed in subsection 2.2.
Proof for Lemma 2: Firstly, we display
X2
(
U
)+
X3
(
U
):
E1→E1
satisfies local Lipschitz
condition. Since ∥X3(U)−X3(V)+X2(U, θtω)−X2(V,θtω)∥E1
≤∥X3(U)−X3(V)∥E1+∥X2(U, θtω)−X2(V, θtω)∥E1,
let
c>
0
,τ ∈R
are given constant, for
∀U,V∈E1,∥U∥E1≤c, ∥V∥E1≤c
, combining Lemmas 4 and
7, we get there exists a positive constant C1(T,τ,ω,c)such that
∥X3(U)−X3(V)∥E1
=
β∥∇U1∥2−p((−∆)U1−(−∆)V1)+β∥∇U1∥2−β∥∇V1∥2(−∆)V1
≤β∥∇U1∥2−p∥∆U1−∆V1∥+β(∥∇U1∥+β∥∇V1∥)∥∆V1∥∥∆U1−∆V1∥
≤C1(T,τ,ω,c)∥U−V∥E1.
On the other hand, by Lemma 1, there exists a positive constant C2(T,τ,ω,c)which satisfies
∥X2(U, θtω)−X2(V,θtω)∥E1
=|σz (θtω)|∥U1−V1∥H2+|σz (θtω)|∥V2−U2∥
+∥σ(µ+2ε−σz (θtω)) z(θtω)(U1−V1)∥+|σαz (θtω)|
A1
2V1−A1
2U1
≤|σz (θtω)|
A1
2V1−A1
2U1
+|σαz (θtω)|
A1
2V1−A1
2U1
+|σ(µ+2ε−σz (θtω)) z(θtω)|
λ
1
2
1
A1
2V1−A1
2U1
+|σz (θtω)|∥V2−U2∥
≤C2(T,τ,ω,c)∥U−V∥E1,
The indicated above conclude X2(U)+X3(U):E1→E1satisfies local Lipschitz condition.
The semigroup method (Theorem 2.5.4 in [33]) is employed to achieve the existence and uniqueness
of solution for system
(14)
. Based on Lemma 3.5 in Ref. [24] and Lemma 2.2.3 in Ref. [33], we have
Q
is
m
-accretive in
E1
, then it can induce a linear semigroup of contractions formulated by
eQt,t∈R+,
which
together with the assertion that
X2(U, θtω)
+
X3
(
U
):
E1→E1
satisfies local Lipschitz condition can
guarantee the system (14) possesses a unique local mild solution with the form
V(t,τ,ω;Vτ)=eQ(t−τ)Vτ+t
τ
eQ(t−s)(X2(θsω)+X3(V)(s)) ds,
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
398
Therefore, we get dH(A(ω)) < d. Especially, if
λ−1
2
1≤
ε
2−8
ε�βr2+|p|2−2M7
√πµ −2M8
2µ
16β2r4
ελ
1
4
1
,
dH
(
A
(
ω
))=0, which merges with the Theorem 2 shows that the largest global Lyapunov exponent of
Sε(ω)is
λ1=E(ln ω1(ω))
≤E
−ε
2+8
ε�βr2+|p|2+2M7
√πµ +2M8
2µ+16β2r4
ελ
1
4
1
n
i=1
λ−1
2
i
<0,(38)
thus, we can conclude there exists a random fixed point of system
(14)
which is global stochastic
stability.
4 PROOFS FOR LEMMAS
This section is intended to complete the proofs of Lemmas listed in subsection 2.2.
Proof for Lemma 2: Firstly, we display
X2
(
U
)+
X3
(
U
):
E1→E1
satisfies local Lipschitz
condition. Since ∥X3(U)−X3(V)+X2(U, θtω)−X2(V,θtω)∥E1
≤∥X3(U)−X3(V)∥E1+∥X2(U, θtω)−X2(V, θtω)∥E1,
let
c>
0
,τ ∈R
are given constant, for
∀U,V∈E1,∥U∥E1≤c, ∥V∥E1≤c
, combining Lemmas 4 and
7, we get there exists a positive constant C1(T,τ,ω,c)such that
∥X3(U)−X3(V)∥E1
=
β∥∇U1∥2−p((−∆)U1−(−∆)V1)+β∥∇U1∥2−β∥∇V1∥2(−∆)V1
≤β∥∇U1∥2−p∥∆U1−∆V1∥+β(∥∇U1∥+β∥∇V1∥)∥∆V1∥∥∆U1−∆V1∥
≤C1(T,τ,ω,c)∥U−V∥E1.
On the other hand, by Lemma 1, there exists a positive constant C2(T,τ,ω,c)which satisfies
∥X2(U, θtω)−X2(V,θtω)∥E1
=|σz (θtω)|∥U1−V1∥H2+|σz (θtω)|∥V2−U2∥
+∥σ(µ+2ε−σz (θtω)) z(θtω)(U1−V1)∥+|σαz (θtω)|
A1
2V1−A1
2U1
≤|σz (θtω)|
A1
2V1−A1
2U1
+|σαz (θtω)|
A1
2V1−A1
2U1
+|σ(µ+2ε−σz (θtω)) z(θtω)|
λ
1
2
1
A1
2V1−A1
2U1
+|σz (θtω)|∥V2−U2∥
≤C2(T,τ,ω,c)∥U−V∥E1,
The indicated above conclude X2(U)+X3(U):E1→E1satisfies local Lipschitz condition.
The semigroup method (Theorem 2.5.4 in [33]) is employed to achieve the existence and uniqueness
of solution for system
(14)
. Based on Lemma 3.5 in Ref. [24] and Lemma 2.2.3 in Ref. [33], we have
Q
is
m
-accretive in
E1
, then it can induce a linear semigroup of contractions formulated by
eQt,t∈R+,
which
together with the assertion that
X2(U, θtω)
+
X3
(
U
):
E1→E1
satisfies local Lipschitz condition can
guarantee the system (14) possesses a unique local mild solution with the form
V(t,τ,ω;Vτ)=eQ(t−τ)Vτ+t
τ
eQ(t−s)(X2(θsω)+X3(V)(s)) ds,
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
where, t≥τ, t, τ ∈R.
Proof for Lemma 3 : Since
(QU ,U)E1
≤−ε∥U1∥2
H2+ε∥U2∥2+εα
A1
2U1
∥U2∥−ε2∥U1∥∥U2∥−α
A1
4U2
2
≤−ε∥U1∥2
H2+ε∥U2∥2+ε
2
1−3p
2λ
1
4
1
A1
2U1
2+εα2λ
1
4
1
2λ
1
4
1−3p∥U2∥2−α
A1
4U2
2
≤−ε
2∥U1∥2
H2+ε∥U2∥2+εα2λ
1
4
1
2λ
1
4
1−3p∥U2∥2−α
A1
4U2
2−3pε
4∥A1
4U1∥2
≤−ε
2∥U1∥2
H2+
−α
λ
1
4
1
+ε+εα2λ
1
4
1
2λ
1
4
1−3p
∥U2∥2−3pε
4∥A1
4U1∥2
≤−ε
2∥U∥2
E1−ε
4∥U2∥2−3pε
4∥A1
4U1∥2,
where, pand εsatisfies (15). Thus complete the proof.
Before deriving the other proofs for Lemmas, the following estimations and quantities are introduced
2−β
A1
4u
2−pA1
2u, v≤−1
2β
d
dt β∥A1
4u∥2−p2
+εp2
4β+3pε
2∥A1
4u∥2−ε
4β[β∥A1
4u∥2−p]2
+4σ2|z(θtω)|2
7εβ β∥A1
4u∥2−p2,
2σA1
2uz (θtω),A1
2u≤ε
2∥A1
2u∥2+2σ2|z(θtω)|2
ε∥A1
2u∥2,
(39)
2σµ+2ε−αA1
2−σz (θtω)uz (θtω)−σvz (θtω),v
≤2
εσ2|z(θtω)|2∥v∥2+ε
2∥v∥2+σ2|z(θtω)|24
ε∥u∥2+ε
4∥v∥2
+ε
2∥v∥2+4σ2|z(θtω)|2(µ+2ε)2
ε∥u∥2+α2
ε∥A1
2u∥2
(40)
2σA1
2uaz(θtω),A1
2ua≤ε
2∥A1
2ua∥2+2σ2|z(θtω)|2
ε∥A1
2ua∥2,(41)
2σ(µ+2ε−αA1
2−σz (θtω))uaz(θtω)−σvaz(θtω),A1
2va
≤2
εσ2|z(θtω)|2∥va∥2+ε
2∥va∥2+σ2|z(θtω)|24
ε∥ua∥2+ε
4∥va∥2
+ε
4∥va∥2+8σ2|z(θtω)|2(µ+2ε)2
ε∥ua∥2+α2
ε∥A1
2ua∥2,
(42)
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
399
(−2β∥A1
4u∥2−pA1
2ua,A1
2vb)
≤3β2
λ
1
2
1∥A1
2u∥4
A1
2ua
2+3p2∥Va∥2
E1
+2d
dt
A1
2ub
2+2ε2
A1
2ub
2+2σ2|z(θtω)|2
A1
2ub
2,
−2β∥A1
4u∥2−p(A1
2ub,A1
2vb)
=−2β∥A1
4u∥2−p(A1
2ub,A1
2ut+εub−σuz (θtω))
≤−d
dt β∥A1
4u∥2−p∥A1
2ub∥2+β
2∥A1
2u∥3∥ut∥,
(43)
2σµ+2ε−αA1
2−σz (θtω)ubz(θtω)−σvz (θtω),A1
2vb
≤ε
2∥A3
4ub∥2+2σ2|z(θtω)|2
ε∥A1
4ub∥2+2
εσ2|z(θtω)|2∥A1
4vb∥2+ε
2∥A1
4vb∥2
+ε
2∥A1
4vb∥2+4σ2|z(θtω)|2(µ+2ε)2
ε∥A1
4ub∥2+α2
ε∥A3
4ub∥2
+σ2|z(θtω)|24
ε∥A1
4ub∥2+ε
4∥A1
4vb∥2,
(44)
and β
2∥A1
2u∥3∥ut∥≤β
2∥V∥3
E1(∥v∥+∥εu∥+∥σuz (θtω)∥)
≤β
2∥V∥3
E1(∥V∥E1+ε∥V∥E1+σ|z(θtω)|∥V∥E1)
≤β+βε
2+β
2|σz (θtω)|∥V∥4
E1,
By Lemma 4, Lemma 5 and Lemma7,
M(t,τ,ω)defined as follow is bounded,
M(t,τ,ω)=3β2
λ
1
2
1∥A1
2u∥4
A1
2ua
2+3p2∥Va∥2
E1+β
2∥A1
2u∥3∥ut∥
+2ε2+3pε
2+2−ε+2|p|∥A1
2ub∥2<+∞,
(45)
M0= max
8α2+2
ε+8(µ+2ε)2+4
ελ
1
2
1
,2
ε+ε
4
,(46)
M= max
8(µ+2ε)2+6
ελ
1
2
1
+8α2
ε,ε2+8
4ε,8
7ε
,(47)
M1=4βr2
1(ω)+3|p|λ
1
4
1
2λ
1
4
1
,(48)
M2=3βr4
1(ω)
λ
1
4
1
+2|p|r2
1(ω)+εp2
4β,(49)
M3= max
1,2
ε+ε
4,4(µ+2ε)2+2
ελ
1
2
1
+4α2
ε+2|p|+4
λ
1
4
1
+ε
4λ
1
2
1
,(50)
c1(ω) = max
1,6βr1(ω)
λ
1
4
1
,c
2(ω) = max
1,r2(ω)
2+2βr1(ω)
λ
1
8
1
,(51)
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
400
(−2β∥A1
4u∥2−pA1
2ua,A1
2vb)
≤3β2
λ
1
2
1∥A1
2u∥4
A1
2ua
2+3p2∥Va∥2
E1
+2d
dt
A1
2ub
2+2ε2
A1
2ub
2+2σ2|z(θtω)|2
A1
2ub
2,
−2β∥A1
4u∥2−p(A1
2ub,A1
2vb)
=−2β∥A1
4u∥2−p(A1
2ub,A1
2ut+εub−σuz (θtω))
≤−d
dt β∥A1
4u∥2−p∥A1
2ub∥2+β
2∥A1
2u∥3∥ut∥,
(43)
2σµ+2ε−αA1
2−σz (θtω)ubz(θtω)−σvz (θtω),A1
2vb
≤ε
2∥A3
4ub∥2+2σ2|z(θtω)|2
ε∥A1
4ub∥2+2
εσ2|z(θtω)|2∥A1
4vb∥2+ε
2∥A1
4vb∥2
+ε
2∥A1
4vb∥2+4σ2|z(θtω)|2(µ+2ε)2
ε∥A1
4ub∥2+α2
ε∥A3
4ub∥2
+σ2|z(θtω)|24
ε∥A1
4ub∥2+ε
4∥A1
4vb∥2,
(44)
and β
2∥A1
2u∥3∥ut∥≤β
2∥V∥3
E1(∥v∥+∥εu∥+∥σuz (θtω)∥)
≤β
2∥V∥3
E1(∥V∥E1+ε∥V∥E1+σ|z(θtω)|∥V∥E1)
≤β+βε
2+β
2|σz (θtω)|∥V∥4
E1,
By Lemma 4, Lemma 5 and Lemma7,
M(t,τ,ω)defined as follow is bounded,
M(t,τ,ω)=3β2
λ
1
2
1∥A1
2u∥4
A1
2ua
2+3p2∥Va∥2
E1+β
2∥A1
2u∥3∥ut∥
+2ε2+3pε
2+2−ε+2|p|∥A1
2ub∥2<+∞,
(45)
M0= max
8α2+2
ε+8(µ+2ε)2+4
ελ
1
2
1
,2
ε+ε
4
,(46)
M= max
8(µ+2ε)2+6
ελ
1
2
1
+8α2
ε,ε2+8
4ε,8
7ε
,(47)
M1=4βr2
1(ω)+3|p|λ
1
4
1
2λ
1
4
1
,(48)
M2=3βr4
1(ω)
λ
1
4
1
+2|p|r2
1(ω)+εp2
4β,(49)
M3= max
1,2
ε+ε
4,4(µ+2ε)2+2
ελ
1
2
1
+4α2
ε+2|p|+4
λ
1
4
1
+ε
4λ
1
2
1
,(50)
c1(ω) = max
1,6βr1(ω)
λ
1
4
1
,c
2(ω) = max
1,r2(ω)
2+2βr1(ω)
λ
1
8
1
,(51)
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
C(p)=
1,p⩽−2,
(1 −p+2
λ
1
4
1
),−2<p<λ
1
4
1−2
2,(52)
q=
σµ+2ε+2λ
1
2
1+αλ
1
2
1
2λ
1
2
1
,(53)
−M4=−ε+c2(ω),(54)
M5=
σµ+2ε+αλ
1
4
1
2λ
1
4
1
+|σ|,(55)
M6=ε−16
ε�βr2+|p|2−64β2r4
ελ
1
2
1
,(56)
M7=σ+σ
λ
1
2
1
+
σµ+2ε+αλ
1
2
1
2λ
1
2
1
,M
8=σ2
2λ
1
2
1
,(57)
M9=nσ
2√λ1πµ +nσ
2λ1√πµ +nσ
4µλ
1
2
1
+n
2λ1
.(58)
Proof for Lemma 4 : Taking the inner product of Vby V=[u, v]Tin E1, we get that
d
dt∥V∥2
E1≤2(QV ,V)E1+2σA1
2uz (θtω),A1
2u
+2−β
A1
2u
2−pA1
2u, v
+2σµ+2ε−αA 1
2−σz (θtω)uz (θtω)−σvz (θtω),v
.
(59)
Set
H(t,τ,ω)=H(u, v)= 1
2ββ
A1
4u
2−p2
+
A1
2u
2+∥v∥2.
By (40) and Lemma 3, we find that
d
dtH(u, v)≤−ε
2H(u, v)+εp2
4β+4σ2|z(θtω)|2
7εβ β∥A1
4u∥2−p2
+
α2+2
ε+(µ+2ε)2+4
ελ
1
2
1
σ2|z(θtω)|2∥A1
2u∥2
+2
ε+ε
4σ2|z(θtω)|2∥v∥2
≤−ε
2H(u, v)+Mσ2|z(θtω)|2H(u, v)+ εp2
4β,
(60)
where Mis formulated by (47). Then, for any t>0, the following holds
H(0,−t, ω)≤e−ε
2t+0
−tMσ2|z(θsω)|2dsH(−t, −t, ω)+εp2
4β0
−t
e−ε
2s+0
sMσ2|z(θkω)|2dk ds.
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
401
Since v=ut+εu −σuz (θtω), we get
∥U∥2
E1≤2
A1
2u
2+2
ut+εu −σuz (θtω)∥2+2∥σuz (θtω)
2
≤2
1+ σ2
λ
1
4
1|z(θtω)|2
∥V∥2
E1.
(61)
Thus, we have
∥U(0,−t, ω)∥2
E1≤2
1+ σ2
λ
1
4
1|z(θtω)|2
∥V(0,−t, ω)∥2
E1
≤2
1+ σ2
λ
1
4
1|z(θtω)|2
e−ε
2t+0
−tMσ2|z(θsω)|2dsH(−t, −t, ω)
+
1+ σ2
λ
1
4
1|z(θtω)|2
εp2
2β0
−t
e−ε
2s+0
sMσ2|z(θkω)|2dk ds.
(62)
Since the random variable z(θtω)is tempered, along with Lemma 1, we can infer that
2σ2
λ
1
4
1|z(θtω)|2e−ε
2t+0
−tMσ2|z(θsω)|2dsH(−t, −t, ω)→0,t→+∞,
and 2σ2
λ
1
4
1|z(θtω)|2εp2
4β0
−t
e−ε
2s+0
sMσ2|z(θkω)|2dk ds →0,t→+∞,
similarly
2e−ε
2t+0
−tMσ2|z(θsω)|2dsH(−t, −t, ω)→0,t→+∞,
and εp2
2β0
−t
e−ε
2s+0
sMσ2|z(θkω)|2dk ds < ∞.
Let
ρ1(ω)=εp2
2β0
−t
e−ε
2s+0
sMσ2|z(θkω)|2dk ds. (63)
According to (59) - (63), we obtain that there exists T1(ω)>0, such that
∥U(t, θ−tω)∥E1=∥U(0,−t, ω)∥E1≤r1(ω),∀t≥T1(ω),(64)
where r1(ω)=ρ1(ω).
On the other hand, utilizing (10), we obtain
E(ρ1(ω)) = εp2
2β0
−∞
e−ε
2sEeMσ20
s|z(θkω)|2dkds
≤εp2
2β0
−∞
e−ε
2s−sMσ2
µds
=εp2
2β0
T
e−ε
2s−sMσ2
µds −εp2
2βT
−∞
eε+Mσ2
µsds
=εp2
2β0
T
e−ε
2s−sMσ2
µds −εp2eε+Mσ2
µT
2βε+Mσ2
µ
<+∞.
(65)
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
402
Since v=ut+εu −σuz (θtω), we get
∥U∥2
E1≤2
A1
2u
2+2
ut+εu −σuz (θtω)∥2+2∥σuz (θtω)
2
≤2
1+ σ2
λ
1
4
1|z(θtω)|2
∥V∥2
E1.
(61)
Thus, we have
∥U(0,−t, ω)∥2
E1≤2
1+ σ2
λ
1
4
1|z(θtω)|2
∥V(0,−t, ω)∥2
E1
≤2
1+ σ2
λ
1
4
1|z(θtω)|2
e−ε
2t+0
−tMσ2|z(θsω)|2dsH(−t, −t, ω)
+
1+ σ2
λ
1
4
1|z(θtω)|2
εp2
2β0
−t
e−ε
2s+0
sMσ2|z(θkω)|2dk ds.
(62)
Since the random variable z(θtω)is tempered, along with Lemma 1, we can infer that
2σ2
λ
1
4
1|z(θtω)|2e−ε
2t+0
−tMσ2|z(θsω)|2dsH(−t, −t, ω)→0,t→+∞,
and 2σ2
λ
1
4
1|z(θtω)|2εp2
4β0
−t
e−ε
2s+0
sMσ2|z(θkω)|2dk ds →0,t→+∞,
similarly
2e−ε
2t+0
−tMσ2|z(θsω)|2dsH(−t, −t, ω)→0,t→+∞,
and εp2
2β0
−t
e−ε
2s+0
sMσ2|z(θkω)|2dk ds < ∞.
Let
ρ1(ω)=εp2
2β0
−t
e−ε
2s+0
sMσ2|z(θkω)|2dk ds. (63)
According to (59) - (63), we obtain that there exists T1(ω)>0, such that
∥U(t, θ−tω)∥E1=∥U(0,−t, ω)∥E1≤r1(ω),∀t≥T1(ω),(64)
where r1(ω)=ρ1(ω).
On the other hand, utilizing (10), we obtain
E(ρ1(ω)) = εp2
2β0
−∞
e−ε
2sEeMσ20
s|z(θkω)|2dkds
≤εp2
2β0
−∞
e−ε
2s−sMσ2
µds
=εp2
2β0
T
e−ε
2s−sMσ2
µds −εp2
2βT
−∞
eε+Mσ2
µsds
=εp2
2β0
T
e−ε
2s−sMσ2
µds −εp2eε+Mσ2
µT
2βε+Mσ2
µ
<+∞.
(65)
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
Proof for Lemma 5 : Taking the inner product (·,·)E1of (18) with A1
2Vagives
d
dt∥A1
4Va∥2
E1=2(QV a,A1
2Va)E1+ 2(σA1
2uaz(θtω), Aua)
+2 σ(µ+2ε−αA1
2−σz (θtω))uaz(θtω)−σvaz(θtω),A1
2va.
(66)
Substituting (41) and (42) into (66)„ we infer the following by combining with Lemma 3
d
dt∥A1
4Va∥2
E1≤−ε
2∥A1
4Va∥2
E1+4
ε+ε
4σ2|z(θtω)|2∥A1
4va∥2
+
8(µ+2ε)2
ελ
1
2
1
+8α2
ε+4
ελ
1
2
1
+2
ελ
1
2
1
σ2|z(θtω)|2∥A3
4ua∥2,
hence d
dt∥A1
4Va∥2
E1≤−ε
2+σ2|z(θtω)|2M0∥A1
4Va∥2
E1,
where M0is given by (46). Thus, we have
∥A1
4Va∥2
E1≤e0
−t−ε
2t+σ2|z(θsω)|2M0ds∥A1
4Va(−t, −t, ω)∥2
E1.
Since the random variable z(θtω)is tempered, along with Lemma 1 states
e0
−t−ε
2t+σ2|z(θsω)|2M0ds∥A1
4Va(−t, −t, ω)∥2
E1→0,t→+∞,
Thus, it can be obtained that
lim
t→+∞
A1
4Ua
(t, θ−tω)E1=0.
Proof for Lemma 6 : Taking the inner product (·,·)E1of (19) with Vbgives
d
dt∥Vb∥2
E1≤Υ(1) +Υ
(2),(67)
here
Υ(1) =2(QV b,Vb)+2σA1
2ubz(θtω),A1
2ub+2−β
A1
4ub
2−pA1
2ub,v
b
+2 σµ+2ε−αA1
2−σz (θtω)ubz(θtω)−σvz (θtω),v
b,
Υ(2) =2 β
A1
4ub
2−pA1
2ub,v
b+2−β
A1
4u
2−pA1
2u, vb,
(68)
in which Υ(1) is bounded. On the other hand
Υ(2) =2β
A1
4u−A1
4ua
2−pA1
2u, 1
2vb+2β
A1
4u−A1
4ua
2−p−A1
2ua,1
2vb
+2−β
A1
4u
2−pA1
2u, vb
≤β
λ
1
4
1
2∥V∥2+2∥Va∥2+|p|λ
1
4
1
β
(∥V∥2+∥Va∥2)+
β
λ
1
4
1∥V∥2+|p|
∥V∥2
+β
2λ
1
4
1
4∥V∥2+2∥Va∥2+3|p|λ
1
4
1
β
H(ub,v
b).
(69)
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
403
Set
H′(t,τ,ω)=H(ub,v
b)= 1
2ββ
A1
4ub
2−p2
+
A1
2ub
2+∥vb∥2,
taking account into (67), (68) and (69), we can get
d
dtH(ub,v
b)≤−ε
2H(ub,v
b)+
β
λ
1
4
1∥V∥2+|p|
∥V∥2+εp2
4β
+
Mσ2|z(θtω)|2+2β
λ
1
4
1∥V∥2+β
λ
1
4
1∥Va∥2+3|p|
H(ub,v
b)
+2β∥V∥2+2β∥Va∥2+|p|λ
1
4
1
λ
1
4
1
(∥V∥2+∥Va∥2).
(70)
Thus,
H′(0,−t, ω)≤e−ε
2t+0
−tMσ2|z(θsω)|2+β
2λ
1
4
1
4∥V∥2+2∥Va∥2+3|p|λ
1
4
1
β
ds
H′(−t, −t, ω)
+0
−t
e−ε
2s+0
sMσ2|z(θkω)|2+β
2λ
1
4
1
4∥V∥2+2∥Va∥2+3|p|λ
1
4
1
β
dk
×
2β∥V∥2+2β∥Va∥2+|p|λ
1
4
1
λ
1
4
1
(∥V∥2+∥Va∥2)+β∥V∥4
λ
1
4
1
+|p|∥V∥2+εp2
4β
ds.
Applying Lemma 4 and Lemma 5, we obtain
H′(0,−t, ω)≤e−ε
2t+0
−tMσ2|z(θsω)|2+M1dsH′(−t, −t, ω)
+M20
−t
e−ε
2s+0
sMσ2|z(θkω)|2+M1dk ds,
where
M1,M
2
are defined by
(48)
and
(49)
correspondingly. Since the random variable
z
(
θtω
)is
tempered, together with Lemma 1, we find
e−ε
2t+0
−tMσ2|z(θsω)|2+M1dsH′(−t, −t, ω)→0,t→+∞,
and
M20
−t
e−ε
2s+0
sMσ2|z(θkω)|2+M1dk ds < ∞.
Let
ρ2(ω)=M20
−t
e−ε
2s+0
sMσ2|z(θkω)|2+M1dk ds. (71)
As indicated above, we obtain that there exists T2(ω)>0such that
∥U∥E1(t, θ−tω)=∥U∥E1(0,−t, ω)≤r2(ω),∀t≥T2(ω),
where r2(ω)=ρ2(ω).
Proof for Lemma 7 : Taking the inner product (·,·)E1of (19) with A1
2Vbgives
d
dt∥A1
4Vb∥2
E1≤2(QV b,A1
2Vb)+2σA1
2ubz(θtω), Aub
+2 −β
A1
4u
2−pA1
2u, A1
2vb
+2 σµ+2ε−αA1
2−σz (θtω)ubz(θtω)−σvz (θtω),A1
2vb.
(72)
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
404
Set
H′(t,τ,ω)=H(ub,v
b)= 1
2ββ
A1
4ub
2−p2
+
A1
2ub
2+∥vb∥2,
taking account into (67), (68) and (69), we can get
d
dtH(ub,v
b)≤−ε
2H(ub,v
b)+
β
λ
1
4
1∥V∥2+|p|
∥V∥2+εp2
4β
+
Mσ2|z(θtω)|2+2β
λ
1
4
1∥V∥2+β
λ
1
4
1∥Va∥2+3|p|
H(ub,v
b)
+2β∥V∥2+2β∥Va∥2+|p|λ
1
4
1
λ
1
4
1
(∥V∥2+∥Va∥2).
(70)
Thus,
H′(0,−t, ω)≤e−ε
2t+0
−tMσ2|z(θsω)|2+β
2λ
1
4
1
4∥V∥2+2∥Va∥2+3|p|λ
1
4
1
β
ds
H′(−t, −t, ω)
+0
−t
e−ε
2s+0
sMσ2|z(θkω)|2+β
2λ
1
4
1
4∥V∥2+2∥Va∥2+3|p|λ
1
4
1
β
dk
×
2β∥V∥2+2β∥Va∥2+|p|λ
1
4
1
λ
1
4
1
(∥V∥2+∥Va∥2)+β∥V∥4
λ
1
4
1
+|p|∥V∥2+εp2
4β
ds.
Applying Lemma 4 and Lemma 5, we obtain
H′(0,−t, ω)≤e−ε
2t+0
−tMσ2|z(θsω)|2+M1dsH′(−t, −t, ω)
+M20
−t
e−ε
2s+0
sMσ2|z(θkω)|2+M1dk ds,
where
M1,M
2
are defined by
(48)
and
(49)
correspondingly. Since the random variable
z
(
θtω
)is
tempered, together with Lemma 1, we find
e−ε
2t+0
−tMσ2|z(θsω)|2+M1dsH′(−t, −t, ω)→0,t→+∞,
and
M20
−t
e−ε
2s+0
sMσ2|z(θkω)|2+M1dk ds < ∞.
Let
ρ2(ω)=M20
−t
e−ε
2s+0
sMσ2|z(θkω)|2+M1dk ds. (71)
As indicated above, we obtain that there exists T2(ω)>0such that
∥U∥E1(t, θ−tω)=∥U∥E1(0,−t, ω)≤r2(ω),∀t≥T2(ω),
where r2(ω)=ρ2(ω).
Proof for Lemma 7 : Taking the inner product (·,·)E1of (19) with A1
2Vbgives
d
dt∥A1
4Vb∥2
E1≤2(QV b,A1
2Vb)+2σA1
2ubz(θtω), Aub
+2 −β
A1
4u
2−pA1
2u, A1
2vb
+2 σµ+2ε−αA1
2−σz (θtω)ubz(θtω)−σvz (θtω),A1
2vb.
(72)
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
Set
H1(t,τ,ω)=H1(u, v)=β∥A1
4u∥2−p−2∥A1
2ub∥2+∥A1
4Vb∥2
E1,(73)
we find
H1(t,τ,ω)=∥A1
4Vb∥2
E1+β∥A1
2ub∥2∥A1
4u∥2−p∥A1
2ub∥2−2∥A1
2ub∥2
≥∥A1
4ub∥2+β∥A1
2ub∥2∥A1
4u∥2+ (1 −p+2
λ
1
4
1
)∥A3
4ub∥2
≥∥A1
4ub∥2+β∥A1
2ub∥2∥A1
4u∥2+C(p)∥A3
4ub∥2
≥0,
where C(p)is given by (52), which along with (43), (44) and (72) states that
d
dtH1(u, v)≤−ε
2H1(u, v)+H1(u, v)+
M(t,τ,ω)
+σ2|z(θtω)|2β∥A1
4u∥2−p−2∥A1
2ub∥2
+
4(µ+2ε)2+2
ελ
1
2
1
+4α2
ε+2|p|+4
λ
1
4
1
+ε
4λ
1
2
1
σ2|z(θtω)|2∥A3
4ub∥2
+2
ε+ε
4σ2|z(θtω)|2∥A1
4vb∥2
≤−ε
2H1(u, v)+�1+M3σ2|z(θtω)|2H1(u, v)+
M(t,τ,ω),
where
M
(
t,τ,ω
)and
M3
are defined by
(45)
and
(50)
respectively. Then by Lemma 4, Lemma 5 and
Lemma7, the following holds
H1(0,−t, ω)≤e−ε
2t+0
−t(1+M3σ2|z(θsω)|2)dsH1(−t, −t, ω)
+r0
−t
e−ε
2s+0
s(1+M3σ2|z(θkω)|2)dkds.
Since the random variable z(θtω)is tempered, applying Lemma 1, we find
e−ε
2t+0
−t(1+M3σ2|z(θsω)|2)dsH1(−t, −t, ω)→0,t→+∞,
r0
−t
e−ε
2s+0
s(1+M3σ2|z(θkω)|2)dkds < ∞.
then we get
H1(0,−t, ω)≤ρ30(ω),
where ρ30(ω)=r0
−te−ε
2s+0
s(1+M3σ2|z(θkω)|2)dkds.
Exploiting (61) and (73), we can obtain
∥A1
4Vb∥2
E1(0,−t)+β∥A1
2ub∥2(0,−t)∥A1
4u∥2(0,−t)
≤ρ30(ω)+ p+2
λ
1
4
1∥A3
4ub∥2(0 −t)
≤ρ30(ω)+ p+2
λ
1
4
1∥A1
4Vb∥2
E1(0,−t),
then
(1 −p+2
λ
1
4
1
)∥A1
4Vb∥2
E1(0,−t)≤ρ30(ω),
utilizing (52), we get
∥A1
4Vb∥2
E1(0,−t)≤λ
1
4
1
λ
1
4
1−p−2
ρ30(ω).(74)
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
405
By (72)-(74), we obtain that there exists T30(ω)>0, such that
∥A1
4Vb∥2
E1(0,−t)≤ρ3(ω),∀t≥T30(ω),(75)
where, ρ3(ω)= λ
1
4
1
λ
1
4
1−p+2
ρ30(ω). Since the relation between ∥A1
4Ub∥E1and ∥A1
4Vb∥E1,
∥A1
4Ub∥2
E1(0,−t, ω)
≤2
1+ σ2
λ
1
4
1|z(θtω)|2
∥A1
4Vb(0,−t, ω)∥2
E1
≤2rλ
1
4
1
λ
1
4
1−p+2
1+ σ2
λ
1
4
1|z(θtω)|2
0
−t
e−ε
2s+0
s(1+M3σ2|z(θkω)|2)dkds.
(76)
Since the random variable z(θtω)is tempered, we get the following holds by Lemma 1,
2rσ2
λ
1
4
1−p+2|z(θtω)|20
−t
e−ε
2s+0
s(1+M3σ2|z(θkω)|2)dkds →0,t →+∞,
and
2rλ
1
4
1
λ
1
4
1−p+20
−t
e−ε
2s+0
s(1+M3σ2|z(θkω)|2)dkds < ∞.
Thus, there exists a constant T3(ω)>0such that
A1
4Ub
E1
(t, θ−tω)=
A1
4Ub
E1
(0,−t, ω)≤r3(ω),∀t>T
3(ω),
Proof for Lemma 8 : For ∀κ∈D(A1
2), we find that
X32 (u1)−
X32 (u2)κ
≤
�β∥∇u1∥2−β∥∇u2∥2(−∆)κ
+∥2β(∇u1,∇κ)(−∆)u1−2β(∇u2,∇κ)(−∆)u2∥
≤β(∥∇u1∥ − ∥∇u2∥)(∥∇u1∥+∥∇u2∥)∥(−∆)κ∥
+2β∥(∇u1,∇κ)∥∥(−∆)u1−(−∆)u2∥+2β|(∇u1−∇u2,∇κ)∥∥(−∆)u2∥
≤β(∥∇u1∥ − ∥∇u2∥)(∥∇u1∥+∥∇u2∥)∥(−∆)κ∥
+2β∥∇u1∥∥∇κ∥∥(−∆)u1−(−∆)u2∥+2β∥∇u1−∇u2∥∥∇κ∥∥(−∆)u2∥,
Merging Lemma 4 with (51), we find
X32 (u1)−
X32 (u2)κ
≤β(∥∇u1∥ − ∥∇u2∥)(∥∇u1∥+∥∇u2∥)∥(−∆)κ∥
≤β
λ
1
4
1
(∥∆u1∥−∥∆u2∥)2r1(ω)∥(−∆)κ∥
+2β
λ
1
4
1
r1(ω)∥∆κ∥∥(−∆)u1+∆u2∥+2β
λ
1
4
1∥∆u1−∆u2∥∥∆κ∥r1(ω)
≤c1(ω)∥∆u1−∆u2∥∥∆κ∥,∀u1,u
2∈D(A1
2),
along with Lemma 4 demonstrates E(c1(ω)) ≤∞, hence
X32 (u1)−
X32 (u2)
LD(A1
2),L2(D)= sup
κ∈D(A1
2)
∥
X32 (u1)−
X32 (u2)κ∥
∥κ∥D(A1
2)
≤c1(ω)
A1
2u1−A1
2u2
.
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
406
By (72)-(74), we obtain that there exists T30(ω)>0, such that
∥A1
4Vb∥2
E1(0,−t)≤ρ3(ω),∀t≥T30(ω),(75)
where, ρ3(ω)= λ
1
4
1
λ
1
4
1−p+2
ρ30(ω). Since the relation between ∥A1
4Ub∥E1and ∥A1
4Vb∥E1,
∥A1
4Ub∥2
E1(0,−t, ω)
≤2
1+ σ2
λ
1
4
1|z(θtω)|2
∥A1
4Vb(0,−t, ω)∥2
E1
≤2rλ
1
4
1
λ
1
4
1−p+2
1+ σ2
λ
1
4
1|z(θtω)|2
0
−t
e−ε
2s+0
s(1+M3σ2|z(θkω)|2)dkds.
(76)
Since the random variable z(θtω)is tempered, we get the following holds by Lemma 1,
2rσ2
λ
1
4
1−p+2|z(θtω)|20
−t
e−ε
2s+0
s(1+M3σ2|z(θkω)|2)dkds →0,t →+∞,
and
2rλ
1
4
1
λ
1
4
1−p+20
−t
e−ε
2s+0
s(1+M3σ2|z(θkω)|2)dkds < ∞.
Thus, there exists a constant T3(ω)>0such that
A1
4Ub
E1
(t, θ−tω)=
A1
4Ub
E1
(0,−t, ω)≤r3(ω),∀t>T
3(ω),
Proof for Lemma 8 : For ∀κ∈D(A1
2), we find that
X32 (u1)−
X32 (u2)κ
≤
�β∥∇u1∥2−β∥∇u2∥2(−∆)κ
+∥2β(∇u1,∇κ)(−∆)u1−2β(∇u2,∇κ)(−∆)u2∥
≤β(∥∇u1∥ − ∥∇u2∥)(∥∇u1∥+∥∇u2∥)∥(−∆)κ∥
+2β∥(∇u1,∇κ)∥∥(−∆)u1−(−∆)u2∥+2β|(∇u1−∇u2,∇κ)∥∥(−∆)u2∥
≤β(∥∇u1∥ − ∥∇u2∥)(∥∇u1∥+∥∇u2∥)∥(−∆)κ∥
+2β∥∇u1∥∥∇κ∥∥(−∆)u1−(−∆)u2∥+2β∥∇u1−∇u2∥∥∇κ∥∥(−∆)u2∥,
Merging Lemma 4 with (51), we find
X32 (u1)−
X32 (u2)κ
≤β(∥∇u1∥ − ∥∇u2∥)(∥∇u1∥+∥∇u2∥)∥(−∆)κ∥
≤β
λ
1
4
1
(∥∆u1∥−∥∆u2∥)2r1(ω)∥(−∆)κ∥
+2β
λ
1
4
1
r1(ω)∥∆κ∥∥(−∆)u1+∆u2∥+2β
λ
1
4
1∥∆u1−∆u2∥∥∆κ∥r1(ω)
≤c1(ω)∥∆u1−∆u2∥∥∆κ∥,∀u1,u
2∈D(A1
2),
along with Lemma 4 demonstrates E(c1(ω)) ≤∞, hence
X32 (u1)−
X32 (u2)
LD(A1
2),L2(D)= sup
κ∈D(A1
2)
∥
X32 (u1)−
X32 (u2)κ∥
∥κ∥D(A1
2)
≤c1(ω)
A1
2u1−A1
2u2
.
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
On the other hand, since
X32 (u)
LD(A1
2),L2(D)= sup
κ∈D(A1
2)
∥
X32 (u)κ∥
∥κ∥D(A1
2)
=∥β∥∇u∥2−p(−∆)κ+2β(∇u, ∇κ)(−∆)u∥
∥∆κ∥
≤∥β∥∇u∥2−p(−∆)κ∥+∥2β(∇u, ∇κ)(−∆)u∥
∥∆κ∥
≤[β∥∇u∥2−p]2+2β
λ
1
8
1∥∆u∥2.
In term of Lemma 4 and (51), we find
X32(u)
LD(A1
2),L2(D)≤r2(ω)
2+2β
λ
1
8
1
r1(ω)=c2(ω),
and
E(c2(ω)) ≤∞.
Proof for Lemma 9 : Taking the inner product of (23) by V(1) −V(2) in E1illustrates
1
2
d
V(1) −V(2)
2
E1
dt =QV(1) −V(2),V(1) −V(2)E1
+X2V(1)−X2V(2),V(1) −V(2)E1
+(X32 (ω, u1)−X32 (ω, u2),v
1−v2),
(77)
Merging with (24), we have
∥X32 (ω, u1)−X32 (ω, u2)∥≤c2
A1
2(u1−u2)
.(78)
Furthermore, we get
(X32 (ω, u1)−X32 (ω, u2),v
1−v2)≤∥X32 (ω, u1)−X32 (ω, u2)∥∥v1−v2∥
≤c2(ω)
A1
2(u1−u2)
∥v1−v2∥
≤c2(ω)
2
V(1) −V(2)
2
E1
,
(79)
and X2V(1)−X2V(2),V(1) −V(2)E1
≤|σz (θtω)(µ+2ε−σz (θtω)) |+2λ
1
2
1|σz (θtω)|
2λ
1
2
1
(∥un−um∥2
H2+∥vn−vm∥2)
+|ασz (θtω)|
2
A1
2(un−um)
2+∥vn−vm∥2
≤|σz (θtω)(q−σz (θtω)) |+2λ
1
2
1|σz (θtω)|
2λ
1
2
1∥Vn−Vm∥2
E1
≤
q|z(θtω)|+σ2|z(θtω)|2
2λ
1
2
1
∥Vn−Vm∥2
E1,
(80)
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
407
here qis defined by (53), and
QV(1) −V(2),V(1) −V(2)≤−ε
2∥V(1) −V(2)∥2
E1−ε
4∥v1−v2∥2.(81)
Substituting (79), (80) and (81) into (77), we get
d
V(1) −V(2)
2
E1
dt ≤
−ε+c2(ω)+2q|z(θtω)|+σ2
λ
1
2
1|z(θtω)|2
V(1) −V(2)
2
E1
.
thus, we have
V(1) −V(2)
2
E1
(0,−t, ω)≤e0
−t−M4+2q|z(θsω)|+σ2
λ
1
2
1
|z(θsω)|2ds
∥I∥2
E1,(82)
taking into account the between
V
2
E1
and
U
2
E1
,
U(1) −U(2)
2
E1≤2
1+ σ2
λ
1
4
1|z(θtω)|2
e0
−t−M4+2q|z(θsω)|+σ2
λ
1
2
1
|z(θsω)|2ds
∥I∥2
E1,
here
M4>
0is formulated by
(54)
. Combine assertion that random variable
z
(
θtω
)is tempered with
Lemma 1, we find
2σ2
λ
1
4
1|z(θtω)|2e0
−t−M4+2q|z(θsω)|+σ2
λ
1
2
1
|z(θsω)|2ds
→0,t→+∞,
and
2e0
−t−M4+2q|z(θsω)|+σ2
λ
1
2
1
|z(θsω)|2ds
<∞,
then, ∃t′
0,∀t≥t′
0,
U(1) −U(2)
2
E1≤p′
1,set p1= max{1,p
′
1},we obtain
U(1) −U(2)
2
E1
(0,−t, ω)=
U(1) −U(2)
2
E1
(t, θ−tω)≤p1∥I∥E1.
and
p1(ω)≥1,E(p1(ω)) <∞,E(ln (p1(ω))) <∞.(83)
On the other hand, taking the inner product of
(21)
by
V
in
E1
, which together with Lemma 3 shows
(Q
V,
V)E1≤−ε
2∥
V∥2
E1−ε
4
V2
2,(84)
and
X3
V,
VE1
=−β∥∇u∥2−p(−∆)
V1,
V2−2β∇u, ∇
V1(−∆)u,
V2
≤|β∥∇u∥2−p|
∆
V1
V2
+2β
λ
1
4
1∥∆u∥2
∆
V1
V2
≤8
ε|β∥∇u∥2−p|
∆
V1
2+8
ε
2β
λ
1
4
1∥∆u∥2
∆
V1
2
+ε
4
V2
2
≤8
ε�βr2+|p|2
∆
V1
2+64β2r4
ελ
1
2
1
∆
V1
2+ε
4
V2
2,
(85)
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
408
here qis defined by (53), and
QV(1) −V(2),V(1) −V(2)≤−ε
2∥V(1) −V(2)∥2
E1−ε
4∥v1−v2∥2.(81)
Substituting (79), (80) and (81) into (77), we get
d
V(1) −V(2)
2
E1
dt ≤
−ε+c2(ω)+2q|z(θtω)|+σ2
λ
1
2
1|z(θtω)|2
V(1) −V(2)
2
E1
.
thus, we have
V(1) −V(2)
2
E1
(0,−t, ω)≤e0
−t−M4+2q|z(θsω)|+σ2
λ
1
2
1
|z(θsω)|2ds
∥I∥2
E1,(82)
taking into account the between
V
2
E1
and
U
2
E1
,
U(1) −U(2)
2
E1≤2
1+ σ2
λ
1
4
1|z(θtω)|2
e0
−t−M4+2q|z(θsω)|+σ2
λ
1
2
1
|z(θsω)|2ds
∥I∥2
E1,
here
M4>
0is formulated by
(54)
. Combine assertion that random variable
z
(
θtω
)is tempered with
Lemma 1, we find
2σ2
λ
1
4
1|z(θtω)|2e0
−t−M4+2q|z(θsω)|+σ2
λ
1
2
1
|z(θsω)|2ds
→0,t→+∞,
and
2e0
−t−M4+2q|z(θsω)|+σ2
λ
1
2
1
|z(θsω)|2ds
<∞,
then, ∃t′
0,∀t≥t′
0,
U(1) −U(2)
2
E1≤p′
1,set p1= max{1,p
′
1},we obtain
U(1) −U(2)
2
E1
(0,−t, ω)=
U(1) −U(2)
2
E1
(t, θ−tω)≤p1∥I∥E1.
and
p1(ω)≥1,E(p1(ω)) <∞,E(ln (p1(ω))) <∞.(83)
On the other hand, taking the inner product of
(21)
by
V
in
E1
, which together with Lemma 3 shows
(Q
V,
V)E1≤−ε
2∥
V∥2
E1−ε
4
V2
2,(84)
and
X3
V,
VE1
=−β∥∇u∥2−p(−∆)
V1,
V2−2β∇u, ∇
V1(−∆)u,
V2
≤|β∥∇u∥2−p|
∆
V1
V2
+2β
λ
1
4
1∥∆u∥2
∆
V1
V2
≤8
ε|β∥∇u∥2−p|
∆
V1
2+8
ε
2β
λ
1
4
1∥∆u∥2
∆
V1
2
+ε
4
V2
2
≤8
ε�βr2+|p|2
∆
V1
2+64β2r4
ελ
1
2
1
∆
V1
2+ε
4
V2
2,
(85)
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
with the similar calculation of (80), we have
X2
V,
V≤
M5|z(θtω)|+σ2
2λ
1
4
1|z(θtω)|2
V
2
E1
,(86)
where M5is given by (55). From (84), (85) and (86), we get
d∥
V∥2
E1
dt ≤−ε∥
V∥2
E1+2
M5|z(θtω)|+σ2
2λ
1
4
1|z(θtω)|2
V
2
E1
+2
8
ε�βr2+|p|2
∆
V1
2+64β2r4
ελ
1
2
1
∆
V1
2
≤
−M6+2M5|z(θtω)|+σ2
λ
1
4
1|z(θtω)|2
V
2
E1
,
(87)
where M6>0is defined by (56). Thus
∥
V∥E1≤e0
−t−M6+2M5|z(θsω)|+σ2
λ
1
4
1
|z(θsω)|2ds
∥I∥E1,
by the relation between
V
2
E1
and
U
2
E1
, we have
U
2
E1≤2
1+ σ2
λ
1
4
1|z(θtω)|2
∥
V∥2
E1
≤2
1+ σ2
λ
1
4
1|z(θtω)|2
e0
−t−M6+2M5|z(θsω)|+σ2
λ
1
4
1
|z(θsω)|2ds
∥I∥E1.
The random variable z(θtω)is tempered, which together with Lemma 1 gives that
2σ2
λ
1
4
1|z(θtω)|2e0
−t−M6+2M5|z(θsω)|+σ2
λ
1
4
1
|z(θsω)|2ds
→0,t→+∞,
and
2e0
−t−M6+2M5|z(θsω)|+σ2
λ
1
4
1
|z(θsω)|2ds
<∞,
then, ∃t′
2,∀t≥t′
2,
U
2
E1≤p′
2, set p2= max{1,p
′
2},we obtain
U
2
E1
(0,−t, ω)=
U
2
E1
(t, 0,θ
−tω)≤p2∥I∥E1,(88)
then we can get
E(p2(ω)) <∞.(89)
In the rest paper, the value of
U
at
t
=1is still denoted by
U
, and
U
(1) =
DSεω, U(1)I
, where
DSεω, U(1)is the linear solution mapping of system (20).
According to (84) and (89), we get that
DSεω, U(1)
L(E1,E1)= sup
I∈E1
∥
DSεω, U(1)I∥E1
∥I∥E1≤p2(ω),(90)
https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
409
and
p2(ω)≥1,E(ln (p2(ω))) <∞.(91)
On the other hand, Γsatisfies
∥Γ(ω)∥E1≤∥U(1) −U(2)∥E1+∥−
U∥E1
≤(p1+p2)∥I∥E1
≤c(ω)∥I∥E1,
where c(ω) = max{1,p
1+p2}. Obviously, when t=1
∥Γ(1,ω)∥E1≤c(ω)∥I∥E1.
Combining (91) with (89), we have
E(ln c(ω)) <∞,E(c(ω)) <∞.
Since
Sε
(
ω
) :=
Sε
(1
,ω
), merging with
(88)
and
(25)
, we can conclude that
Sε
(
ω
)is almost surly uniform
differentiable on A(ω).
CONCLUSIONS
This paper consider global stochastic stability of the Euler-Bernoulli beam equations excited by
multiplicative white noise. The system can induce a RDS which owns global random attractors,
moreover, Hausdorff dimension of the attractor is finite. Specially, when
λ−1
2
1≤
ε
2−8
ε(βr2+|p|)2−2M7
√πµ −2M8
2µ
16β2r4
ελ
1
4
1
,
the Hausdorff dimension is 0, which indicates that the stochastic Euler-Bernoulli beam possesses a
random fixed point which is global stochastic stability.
ACKNOWLEDGEMENT
This work is supported by Natural Science Foundation of Shandong Province (No. ZR2020MA054) and
National Natural Science Foundation of China (No. 12072178; No. ).
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https://doi.org/10.17993/3ctecno.2023.v12n1e43.386-412
3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254-4143
Ed.43 | Iss.12 | N.1 January - March 2023
410
and
p2(ω)≥1,E(ln (p2(ω))) <∞.(91)
On the other hand, Γsatisfies
∥Γ(ω)∥E1≤∥U(1) −U(2)∥E1+∥−
U∥E1
≤(p1+p2)∥I∥E1
≤c(ω)∥I∥E1,
where c(ω) = max{1,p
1+p2}. Obviously, when t=1
∥Γ(1,ω)∥E1≤c(ω)∥I∥E1.
Combining (91) with (89), we have
E(ln c(ω)) <∞,E(c(ω)) <∞.
Since
Sε
(
ω
) :=
Sε
(1
,ω
), merging with
(88)
and
(25)
, we can conclude that
Sε
(
ω
)is almost surly uniform
differentiable on A(ω).
CONCLUSIONS
This paper consider global stochastic stability of the Euler-Bernoulli beam equations excited by
multiplicative white noise. The system can induce a RDS which owns global random attractors,
moreover, Hausdorff dimension of the attractor is finite. Specially, when
λ−1
2
1≤
ε
2−8
ε(βr2+|p|)2−2M7
√πµ −2M8
2µ
16β2r4
ελ
1
4
1
,
the Hausdorff dimension is 0, which indicates that the stochastic Euler-Bernoulli beam possesses a
random fixed point which is global stochastic stability.
ACKNOWLEDGEMENT
This work is supported by Natural Science Foundation of Shandong Province (No. ZR2020MA054) and
National Natural Science Foundation of China (No. 12072178; No. ).
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