Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 10, pp. 1–23.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

DYNAMICS OF A NON-AUTONOMOUS STOCHASTIC WEAKLY

DAMPED PLATE MODEL WITH CRITICAL EXPONENT

LAN WEN, LU YANG

Abstract. In this article, we study the long-time behavior of the non-autonomous

stochastic weakly damped plate model with critical exponent. By decompos-
ing the solutions of the system and estimating the bounds of solutions in a

more regular space, we obtain random attractors, when the external term is

time-dependent and the nonlinearity has a critical growth.

1. Introduction

In this article, we consider the non-autonomous stochastic weakly damped plate
equation with critical nonlinearity and additive white noise,

utt + αut + ∆2u+ λu+ f(u, x) = g(x, t) + h(x) ˙W (t), x ∈ U, t > τ, τ ∈ R,

u(x, t)|∂U =
∂

∂ν
u|∂U = 0, t ≥ τ, τ ∈ R,

u(x, τ) = uτ (x), ut(x, τ) = u1τ (x), x ∈ U, τ ∈ R,

(1.1)

where U is an open bounded set of Rn with a smooth boundary ∂U , u(t) = u(x, t) is
a real-valued function on U × [τ,∞), τ ∈ R, λ > 0, α > 0, the external term g(·, t)
∈ Cb(R, H2

0 (U)), Cb(R, H2
0 (U)) denotes the set of continuous bounded functions

from R into H2
0 (U), h ∈ H2

0 (U). W is a two-sided real-valued Wiener process on
a probability space (Ω,F ,P), where Ω = {ω ∈ C(R,R) : ω(0) = 0}, the Borel σ-
algebra F on Ω is generated by the compact open topology, P is the corresponding
Wiener measure on F . For any t ∈ R, we can define a mapping θt on Ω by
θt(·) = ω(t+·)−ω(·) for ω ∈ Ω, then (Ω,F ,P, (θt)t∈R) is an ergodic metric dynamical
system.

The nonlinear term f satisfies the following assumptions:

(A1) f(u, x) = f1(u, x)+f2(u, x) and there exist positive constants c0, c1, c2, c3,
c4, c5, and functions βi ∈ L1(U), i = 1, 2, such that for x ∈ U , u ∈ R,

0 ≤ G1(u, x) ≤ c0uf1(u, x) ≤ c1G1(u, x),

G1(u, x) =

∫ u

0

f1(r, x)dr,

f1 ∈ C2(R,R), f ′1,u(0, x) = 0, |f ′′1,u(u, x)| ≤ c2|u|q−2,

(1.2)

2020 Mathematics Subject Classification. 37L05, 35B40, 35B41.
Key words and phrases. Weakly damped plate equation; pullback random attractor;

time-dependent external term; critical exponent.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted October 11, 2021. Published February 7, 2022.

1



2 L. WEN, L. YANG EJDE-2022/10

where 2 < q <∞ if n ≤ 4, and 2 < q ≤ n
n−4 if n ≥ 5;

f2(·, x) ∈ C1(R,R), |f ′2,u(u, x)| ≤ c3 (1 + |u|p) , 0 ≤ p < q − 1, (1.3)

c4|u|q+1 − β1(x) ≤ G(u, x) ≤ c5uf(u, x) + δ0u
2 + β2(x),

G(u, x) =

∫ u

0

f(r, x)dr,

for some 0 ≤ δ0 ≤ c5
8λ2

1
, where λ1 is the first eigenvalue of operator A = −4.

The plate equation arises in the nonlinear theory of oscillations, and our problem
(1.1) has strong background in mathematical physics, the main motivation of the
study comes not only from applications, but also from the mathematics, here the
time-dependent external forcing and critical nonlinearity require more complicated
techniques to some extent. In recent years the asymptotic behavior of the plate
equation has been considered extensively in many papers(see, e.g., [1, 4, 11, 12,
14, 25]). For the deterministic plate equation without noise (i.e., h ≡ 0), [11,
12] established the existence of global attractors for localized damping, in [1, 25],
the authors dealt with the plate equation with nonlinear damping; Von Karman
equation is also one of the most important plate models (see [5, 6, 13] for details).

As for the autonomous stochastic system (where the external term g is indepen-
dent of t), for the wave equation, if the nonlinearity f has a subcritical exponent,
the existence of random attractors have been investigated in [10, 28]. When f
has a critical exponent, the existence of random attractors have been considered in
[16, 26]. For the autonomous stochastic plate system, the authors in [17] proved the
existence of random attractors for plate model with strongly damping, [18] showed
the random attractors for plate equation with linear memory.

When the forcing term g is time-dependent, for the non-autonomous stochastic
wave equation, if f has a subcritical growth, the existence of random attractors were
studied in [15, 20, 22], and [29, 30] showed the upper bound of fractal dimension
of random attractors. When n = 3 and f has a critical exponent, the existence
and boundedness of fractal dimension of random attractors have been successfully
obtained for both additive noise and multiplicative noise, see [23, 24]. However,
to the best of our knowledge, the non-autonomous stochastic weakly damped plate
equation is less discussed, especially for the non-autonomous external term.

In this article, inspired by the ideas in [23, 24], we analyze the dynamical behav-
ior of the non-autonomous stochastic weakly damped plate equation. Under the
assumptions that the external term g is time-dependent, and the nonlinear team f
has a critical growth, by decomposing the solutions of system through two different
modes, we estimate the bounds of solutions in a higher regular space, and then es-
tablish the existence of random attractors. For the existence of random attractors,
some kind of compactness of the process is a key ingredient, when verifying the
pullback compactness, it is important to deal with the critical nonlinearity.

This article is organized as follows. In section 2, we give some preparations for
our consideration. In section 3, we estimate the bounds of solutions. In section
4, we decompose the solutions of the equation into two parts: one part decays
exponentially, another part is bounded in a higher regular space. In section 5, we
obtain the existence of random attractors of the system (1.1).



EJDE-2022/10 DYNAMICS OF A PLATE MODEL WITH CRITICAL EXPONENT 3

2. Preliminaries

In this section, we recall some basic concepts of pullback random attractors
[2, 7, 8].

We know that self-adjoint positive linear sectorial operator A = −4 has eigen-
values {λi}i∈N such that 0 < λ1 ≤ λ2 ≤ · · · ≤ λm ≤ . . . , λm → +∞ as m → +∞.
For r ∈ R, the r-th powers Ar of A can be defined. Denote V2r = D(Ar), and
it is a Hilbert space with inner product (u, v)2r = (Aru,Arv). The injection
Vr1 ↪→ Vr2 is compact for any r1 > r2 and V0 = L2(U), V2 = H2

0 (U). Write
Er = D(Ar+1)×D(Ar) for r ∈ R, and let E = H2

0 (U)× L2(U). Denote the inner
products and norms of L2(U), H2

0 (U) and E by

(u, v) =

∫
U

uv dx, ‖u‖2 = (u, u), ∀u, v ∈ L2(U),

(u, v)2 =

∫
U

∆u∆v dx, ‖u‖22 = (u, u)2, ∀u, v ∈ H2
0 (U),

(y1, y2)E = (u1, u2)2 + (v1, v2), ∀yi = (ui, vi)
T ∈ E, i = 1, 2,

‖y‖2E = ‖u‖22 + ‖v‖2, ∀y = (u, v)T ∈ E.
First of all, we transfer the stochastic differential equation (1.1) into a random

system without noise term. Write z(θtω) := −α
∫ 0

−∞ eαs(θtω)(s)ds (t ∈ R) as an
Ornstein-Uhlenbeck stationary process which can solve the equation dz + αzdt =
dW (t). From [2, 3, 9, 30], we know that t 7→ z(θtω) is continuous in t for almost
every ω ∈ Ω and

lim
t→+∞

e−γt|z(θ−tω)| = 0, ∀γ > 0; E[|z(θtω)|r] =
Γ( 1+r

2 )
√
παr

, (2.1)

for all r > 0 and t ∈ R, where Γ is Gamma function. Actually, as z(θtω) is Gauss
stationary process with expectation 0 and square variance 1

2α [2, 9, 20]. Then for
any r > 0 and t ∈ R, by [23],

E[|z(θtω)|r] =
1√
παr

∫ +∞

0

ξ
1+r
2 −1e−ξdξ =

Γ( 1+r
2 )

√
παr

.

For simplicity, in this paper, we write a.e. ω ∈ Ω as ω ∈ Ω; all the numbers ci
(i ∈ N) below are independent of (ω, τ, t). Let

v = ut + εu− h(x)z(θtω), ε =
λ2

1α

α2 + 3λ2
1

, t ≥ τ, τ ∈ R.

Then (1.1) can be changed into the stochastic system in the Hilbert space E,

ϕ̇+ Λϕ = F (ϕ, θtω, t), t ≥ τ,
ϕ(τ, ω) = ϕτ (ω) = (uτ , u1,τ + εuτ − h(x)z(θτω))T,

(2.2)

τ ∈ R, where

ϕ =

(
u
v

)
, Λ =

(
εI −I

A2 − ε(α− ε)I (α− ε)I

)
F (ϕ, θtω, t) =

(
h(x)z(θtω)

−f(u, x)− λu+ g(x, t) + εh(x)z(θtω)

)
.

Following the arguments in [16, 17, 18, 20, 26], it can be proved that for any
ϕτ = ϕ(τ, ω) ∈ E, problem (2.2) is well-posed in E; that is, the (weak) solution



4 L. WEN, L. YANG EJDE-2022/10

ϕ(·, τ, ω, ϕτ ) of (2.2) exists uniquely and globally for t ∈ [τ,∞), and ϕ(·, τ, ω, ϕτ ) ∈
C([τ,∞);E) can define a continuous cocycle on E, Φ : R+ × R × Ω × E → E,
(t, τ, ω, ϕτ ) 7→ Φ(t, τ, ω)ϕτ (ω) by

Φ(t, τ, ω)ϕτ (ω) = ϕ(t+ τ, τ, θ−τω, ϕτ (θ−τω))

=

(
u(t+ τ, τ, θ−τω, ϕτ (θ−τω))

ut(t+ τ, τ, θ−τω, ϕτ ) + εu(t+ τ, τ, θ−τω, ϕτ )− h(x)z(θtω)

)
over R and (Ω,F ,P, (θt)t∈R), where Φ(0, τ, ω)ϕτ (ω) = ϕτ (θ−τω) and

Φ(t, τ − t, θ−tω)ϕτ−t(θ−tω) = ϕ(τ, τ − t, θ−τω, ϕτ−t(θ−τω)).

We notice that the initial data uτ (x), u1τ (x) of (1.1) is independent of ω, con-
versely, u(t, τ, ω, x) and ut(t, τ, ω, x) depend on ω for t > τ .

Next, we recall the definition of random attractor and the existence criterion of
random attractor for cocycle Φ.

Definition 2.1. A family K = {K(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D(E) of nonempty
subsets of E is called a measurable D(E)-pullback attracting set for Φ if

(i) K is a measurable with respect to F in Ω;
(ii) for all τ ∈ R, ω ∈ Ω, and for every B ∈ D(E),

lim
t→+∞

dH (Φ(t, τ − t, θ−tω,B(τ − t, θ−tω)),K(τ, ω)) = 0,

where dH(·, ·) denotes the Hausdorff semi-distance between two subsets of
E.

Definition 2.2. A family A = {A(τ, ω) : τ ∈ R, ω ∈ Ω} ∈ D(E) of nonempty
subsets of E is called a measurable D(E)-pullback random attractor for Φ if

(i) A is a measurable in ω and compact in E for any τ ∈ R, ω ∈ Ω;
(ii) A is invariant, i.e., for any τ ∈ R, ω ∈ Ω, t ≥ 0, Φ(t, τ, ω,A(τ, ω)) =

A(τ + t, θtω);
(iii) A is an attracting set in D(E).

Lemma 2.3. [2, 3] If Φ has a compact measurable (with respect to F) D(E)-
pullback attracting set K in D(E), then Φ has a unique D(E)-pullback attractor A
in D(E) given by: for each τ ∈ R and ω ∈ Ω,

A(τ, ω) = ∩r≥0∪t≥rΦ(t, τ − t, θ−tω,K(τ − t, θ−tω)).

3. Boundedness of solutions

In this section, we obtain bounds for solutions. Define D(E) as the collection
of all tempered families of nonempty subsets of E with respect to (θt)t∈R [21],
which means, for every B = {B(τ, ω) ⊂ E : τ ∈ R, ω ∈ Ω} ∈ D(E), it holds that
for any γ > 0, ω ∈ Ω, limt→∞ e−γ|t|‖B(τ + t, θtω)‖E = 0, where ‖B(τ, ω)‖E =
supx∈B(τ,ω) ‖x‖.

Lemma 3.1. For any τ ∈ R, ω ∈ Ω, there exists a tempered variable M0(ω)(
independent of τ) such that for any set B ∈ D(E), there exists T (τ, ω,B) ≥ 0
such that the solution ϕ(τ, τ − t, θ−τω, ϕτ−t(θ−τω)) of (2.2) with ϕτ−t(θ−τω) ∈
B(τ − t, θ−tω) satisfies

‖ϕ(τ, τ − t, θ−τω, ϕτ−t(θ−τω))‖E ≤M0(ω), ∀t ≥ T (τ, ω,B),



EJDE-2022/10 DYNAMICS OF A PLATE MODEL WITH CRITICAL EXPONENT 5

that is, the closed tempered measurable ball B0(ω) = {ϕ ∈ E : ‖ϕ‖E ≤ M0(ω)} of
E satisfies

Φ(t, τ − t, θ−tω,B(τ − t, θ−tω)) ⊆ B0(ω), ∀t ≥ T (τ, ω,B). (3.1)

Proof. For any τ ∈ R, ω ∈ Ω, t ≥ 0, let ϕ(r) = ϕ(r, τ − t, θ−τω, ϕτ−t(θ−τω)) =
(u(r), v(r))T ∈ E (r ≥ τ − t) is a solution of (2.2) with ϕ(τ − t) = ϕτ−t(θ−τω) =
(uτ−t, u1,τ−t + εuτ−t − h(x)z(θ−tω))T ∈ E. From (A1) we obtain

(f(u, x), u) ≥ 1

c5
Ḡ(r)− 1

8
‖u‖22 −

β̄2

c5
,∫

U

|u|q+1dx ≤ 1

c4
[Ḡ(r) + β̄1], |f(u, x)| ≤ c6(1 + |u|)q,

(3.2)

here Ḡ(r) =
∫
U
G(u(r, x), x)dx, β̄i(x) =

∫
U
βi(x)dx, i = 1, 2. Taking the inner

product of (2.2) in E with ϕ(r), for r ≥ τ − t, we have

1

2

d

dt

[
‖ϕ(r)‖2E + 2Ḡ(r) + 2β̄1 + λ‖u‖2

]
+ (Λϕ,ϕ)E + ε(f(u, x) + λu, u)

= (f(u, x) + λu, h(x)z(θr−τω)) + (h(x)z(θr−τω), u)2

+ (g(x, r) + εh(x)z(θr−τω), v) .

(3.3)

From [9, 30] we obtain

(Λϕ,ϕ)E ≥
ε

2
(‖u‖22 + ‖v‖2) +

α

2
‖v‖2. (3.4)

So we can deduce that

(f(u, x) + λu, h(x)z(θr−τω))

≤ c7|z(θr−τω)|
(
‖h‖+ (

∫
U

|u|q+1dx)
q
q+1 ‖h‖Lq+1

+ λ(

∫
U

|u|q+1dx)
1
q+1 ‖h‖

L
q+1
q

)
≤ c7|z(θr−τω)|

(
‖h‖+

( 1

c4
[Ḡ(r) + β̄1]

) q
q+1 ‖h‖Lq+1

+ λ
( 1

c4
[Ḡ(r) + β̄1]

) 1
q+1 ‖h‖

L
q+1
q

)
≤ c7‖h‖|z(θr−τω)|+ c8

( 1

c4
[Ḡ(r) + β̄1]

) q
q+1 ‖h‖Lq+1 |z(θr−τω)|

+ c8λ
( 1

c4
[Ḡ(r) + β̄1]

) 1
q+1 ‖h‖

L
q+1
q
|z(θr−τω)|

≤ c7‖h‖|z(θr−τω)|+ ε

2c5
Ḡ(r) + c9β̄1 + c10‖h‖q+1

2 |z(θr−τω)|q+1

+ c11‖h‖
q+1
q

2 |z(θr−τω)|
q+1
q ,

(3.5)

where c11 depends on λ,

(h(x)z(θr−τω), u)2 ≤
2

ε
z2(θr−τω)‖h‖22 +

ε

8
‖u‖22, (3.6)

and

(g(x, r) + εh(x)z(θr−τω), v) ≤ 1

α

[
‖g‖2 + ε2z2(θr−τω)‖h‖2

]
+
α

2
‖v‖2, (3.7)



6 L. WEN, L. YANG EJDE-2022/10

where ‖g‖2 = supr∈R ‖g(·, r)‖2 <∞. From (3.3)-(3.7), we obtain that

d

dt
y(r) + ρy(r) ≤ q(θr−τω), ∀r ≥ τ − t, (3.8)

where ρ = min{ ε2 ,
ε

2c5
} and

y(r) = ‖ϕ(r)‖2E + 2Ḡ(u) + 2β̄1 + λ‖u‖2 ≥ ‖ϕ(r)‖2E ,

q(θr−τω) =
4

ε
z2(θr−τω)‖h‖22 +

2

α

[
‖g‖2 + ε2z2(θr−τω)‖h‖2

]
+ c7‖h‖|z(θr−τω)|+ 2c9β̄1 + c10‖h‖q+1

2 |z(θr−τω)|q+1

+ c11‖h‖
q+1
q

2 |z(θr−τω)|
q+1
q +

4ε

c5
β̄2 + 2ρβ̄1

≤ c12 + c13|z(θr−τω)|q+1,

where c13 depends on λ. By (3.2), we have

−β̄1 ≤ Ḡ(r) +
λ

2
‖u‖2 ≤ c14

(
1 +

∫
U

|u|q+1dx
)

+ λ‖u‖2 ≤ c15

(
1 + ‖u‖q+1

2

)
,

where c15 depends on λ. Using the Gronwall’s inequality to (3.8) on [τ − t, r] (r ≥
τ − t), we can deduce that for r ≥ τ − t,

y (r, τ − t, θ−τω, ϕτ−t(θ−τω))

≤ y (τ − t, τ − t, θ−τω, ϕτ−t(θ−τω)) e−ρ(r+t−τ)

+

∫ r

τ−t
q(θs−τω)e−ρ(r−s)ds,

(3.9)

where

y(τ − t, τ − t, θ−τω, ϕτ−t(θ−τω)) ≤ ‖ϕτ−t(θ−τω)‖2E + 2c15(1 + ‖uτ−t‖q+1
2 ) + 2β̄1,

and ∫ r

τ−t
q(θs−τω)e−ρ(r−s)ds ≤ c12

ρ
+ c13

∫ r

τ−t
|z(θs−τω)|q+1e−ρ(r−s)ds.

From (3.9) we obtain

‖ϕ(r, τ − t, θ−τω, ϕτ−t(θ−τω))‖2E

≤
(
‖ϕτ−t(θ−τω)‖2E + 2c15(1 + ‖uτ−t‖q+1

2 )
)
e−ρ(r+t−τ)

+ c16 + c13

∫ r

τ−t
|z(θs−τω)|q+1e−ρ(r−s)ds, ∀r ≥ τ − t.

So we have
‖ϕ(τ, τ − t, θ−τω, ϕτ−t(θ−τω))‖2E

≤
(
‖ϕτ−t(θ−τω)‖2E + 2c15(1 + ‖uτ−t‖q+1

2 )
)
e−ρt

+ c16 + c13

∫ 0

−∞
|z(θsω)|q+1eρsds.

(3.10)

For any set B(τ, ω) ∈ B ∈ D(E),

ϕτ−t(θ−τω) = (uτ−t, u1,τ−t + εuτ−t − h(x)z(θ−tω))T ∈ B(τ − t, θ−tω) ∈ D(E),



EJDE-2022/10 DYNAMICS OF A PLATE MODEL WITH CRITICAL EXPONENT 7

we obtain

lim sup
t→+∞

(
‖ϕτ−t(θ−τω)‖2E + 2c15(1 + ‖uτ−t‖q+1

2 )
)
e−ρt = 0. (3.11)

Writing

M2
0 (ω) = 2

(
c16 + c13

∫ 0

−∞
|z(θsω)|q+1eρsds

)
<∞, (3.12)

this is a tempered random variable, then from (3.10) and (3.11), there exists
T (τ, ω,B) ≥ 0 such that ϕ(τ, τ−t, θ−τω, ϕτ−t(θ−τω)) ∈ B0(ω) for all t ≥ T (τ, ω,B),
i.e., (3.1) holds. �

By (3.1), then there exists T (τ, ω,B0) ≥ 0 such that

ϕ(r, τ − t, θ−τω,B0(θ−tω)) ∈ B0(θr−τω), ∀t ≥ T (τ, ω,B0), τ − t ≤ r ≤ τ. (3.13)

From (2.1) and (3.12), for any τ ∈ R, we have E
(
M2

0 (θτω)
)

= 2
(
c16 + c13

1
ρ

Γ( q+2
2 )√

παq+1

)
and for k > 1,

E(M2k
0 (θτω))

≤ 22k
[
ck16 + ck13

(∫ 0

−∞
e

k
2(k−1)

ρsds
)k−1

E
(∫ 0

−∞
e
k
2 ρs|z(θs+τω)|(q+1)kds

)]
= 22k

[
ck16 + ck13

(2(k − 1)

kρ

)k−1 2

kρ

Γ( 1+(q+1)k
2 )

√
πα

(q+1)k
2

]
<∞.

(3.14)

4. Decomposition of solutions

In this section, we decompose the solution of (2.2) into two parts, one of them
decays exponentially and another one is ultimately pullback bounded in a more
regular space. For this goal, we make two methods of decomposing the solutions of
(2.2) with different initial data.

For any τ ∈ R and ω ∈ Ω, let

B1(τ, ω) = ∪t≥T (τ,ω,B0)ϕ(τ, τ − t, θ−τω,B0(θ−tω)) ⊆ B0(ω).

Let ϕ(r) = ϕ(r, τ − t, θ−τω, ϕτ−t(θ−τω))(r ≥ τ − t, t ≥ 0) be a solution of (2.2)
with ϕτ−t(θ−τω) ∈ B1(τ − t, θ−tω) ⊆ B0(θ−tω), then by (3.13) we know that
ϕ(r) ∈ B0(θr−τω) for all r ≥ τ − t,

‖ϕ(r, τ − t, θ−τω, ϕτ−t(θ−τω))‖E ≤M0(θr−τω). (4.1)

4.1. Decomposition of solution I. First, we decompose ϕ(r) = ϕ1(r) + ϕ2(r).
Here ϕ1(r) = (u1, v1)T and ϕ2(r) = (u2, v2)T satisfy

ϕ̇1 + Λϕ1 + F1(ϕ1, x) = 0, r > τ − t,
ϕ1(τ − t, τ − t, θ−τω, ϕτ−t(θ−τω)) = ϕτ−t(θ−τω),

(4.2)

and
ϕ̇2 + Λϕ2 + F2(ϕ,ϕ1, x) = F3(θr−τω, r), r > τ − t,

ϕ2(τ − t, τ − t, θ−τω, ϕτ−t(θ−τω)) = (0, 0)T,
(4.3)

where

v1 = u1,t + εu1, v2 = u2,t + εu2 − h(x)z(ω),

F1(ϕ1, x) =

(
0

f1(u1, x) + λu1

)
, F2(ϕ,ϕ1, x) =

(
0

f(u, x)− f1(u1, x) + λu2

)
,



8 L. WEN, L. YANG EJDE-2022/10

F3(ω, r) =

(
h(x)z(ω)

g(x, r) + εh(x)z(ω)

)
.

Next we appraise the part ϕ1.

Lemma 4.1. For any τ ∈ R, ω ∈ Ω, t ≥ 0, there exist a constant σ1 > 0 and
a tempered random variable M1(ω) > 0 (independent of t and τ) such that the
solution ϕ1(r) = ϕ1(r, τ − t, θ−τω, ϕτ−t(θ−τω)) of (4.2) satisfies

‖ϕ1(r, τ − t, θ−τω, ϕτ−t(θ−τω))‖E ≤M1(θ−tω)e−σ1(t+r−τ), ∀ r ≥ τ − t. (4.4)

Proof. As for (3.3), we take the inner product (·, ·)E of (4.2) with ϕ1 = (u1, v1)T.
Then from (1.2) and (3.4), the following inequality holds for r ≥ τ − t,

d

dt

[
‖ϕ1(r)‖2E + 2Ḡ1(r) + λ‖u1‖2

]
+ ε(‖u1‖22 + ‖v1‖2) +

2ε

c0
(Ḡ1(r) + λ‖u1‖2) ≤ 0,

here Ḡ1(r) =
∫
U
G1(u1(r, x), x)dx ≥ 0. Therefore,

d

dt
y1(r) + 2σ1y1(r) ≤ 0, ∀r ≥ τ − t, (4.5)

where

y1(r) = ‖ϕ1(r)‖2E + 2Ḡ1(u1) + λ‖u1‖2 ≥ ‖ϕ1(r)‖2E , σ1 = min{ε
2
,
ε

c0
}.

Using the Gronwall’s inequality in (4.5), we obtain that

‖ϕ1(r)‖2E ≤ y1(r) ≤ y1(τ − t)e−2σ1(t+r−τ)

≤
(
c17‖ϕτ−t(θ−τω)‖2E + 2c15(1 + ‖uτ−t‖q+1

2 )
)
e−2σ1(t+r−τ)

≤ c18

(
1 +Mq+1

0 (θ−tω)
)
e−2σ1(t+r−τ)

= M2
1 (θ−tω)e−2σ1(t+r−τ), ∀ r ≥ τ − t.

(4.6)

�

For the part ϕ2, we have the following estimate.

Lemma 4.2. For any τ ∈ R, ω ∈ Ω, t ≥ 0, there exists a positive-value random
variable M2(t, ω) > 0 such that for r ≥ τ − t, the solution ϕ2(r) = (u2, v2)T of
(4.3) satisfies

‖Aν+1u2(τ, τ − t, θ−τω, ϕτ−t)‖2 + ‖Aνv2(τ, τ − t, θ−τω, ϕτ−t)‖2 ≤M2(t, ω) (4.7)

with a positive constant

ν = min
{1

2
,
n− 4

4
,

4− (n− 4)p

4

}
> 0. (4.8)



EJDE-2022/10 DYNAMICS OF A PLATE MODEL WITH CRITICAL EXPONENT 9

Proof. Taking the inner product (·, ·)E with A2νϕ2 = (A2νu2, A
2νv2)T of (4.3), we

obtain

1

2

d

dt

(
‖Aν+1u2‖2 + ‖Aνv2‖2 + 2

∫
U

[f(u, x)− f1(u1, x) + λu2]A2νu2dx
)

+ (Λϕ2, A
2νϕ2) + ε

∫
U

[f(u, x)− f1(u1, x) + λu2]A2νu2dx

−
∫
U

[(
f ′1,u(u, x)− f ′1,u(u1, x)

)
u1,t + f ′2,u(u, x)u1,t

+f ′u(u, x)u2,t + λu2,t]A
2νu2dx

=
(
f(u, x)− f1(u1, x) + λu2, A

2νh(x)z(θr−τω)
)

+
(
h(x)z(θr−τω), A2νu2

)
2

+
(
g(x, r) + εh(x)z(θr−τω), A2νv2

)
, ∀r ≥ τ − t.

(4.9)

Similar to (3.4)-(3.7), we obtain that for r ≥ τ − t,

(Λϕ2, A
2νϕ2) ≥ ε

2
‖Aν+1u2‖2 +

ε

2
‖Aνv2‖2 +

α

2
‖Aνv2‖2,(

f(u, x)− f1(u1, x) + λu2, A
2νh(x)z(θr−τω)

)
≤ c19

(
1 +M2q

0 (θr−τω) +M2
0 (θr−τω) + z2(θr−τω)

)
,(

h(x)z(θr−τω), A2νu2

)
2
≤ 4

ε
z2(θr−τω)‖h‖22 +

ε

16
‖Aν+1u2‖2,(

g(x, r) + εh(x)z(θr−τω), A2νv2

)
≤ 2

α
[‖g‖22 + ε2z2(θr−τω)‖h‖22] +

α

4
‖Aνv2‖2,

where c19 depends on λ and ‖g‖22 = supr∈R ‖g(·, r)‖22 <∞. Using Hölder’s inequal-
ity, (1.2), (1.3), (4.1), and (4.4), we have the following estimates for r ≥ τ − t,∫
U

f ′u(u, x)u2,t ·A2νu2 dx

≤ c20

∫
U

(
1 + |u|q−1

)
|u2,t||A2νu2|dx

≤ c21

(∫
U

(1 + |u|q−1)
n
2 dx

)2/n(∫
U

|A2νu2|
2n

n−4+4ν dx
)n−4+4ν

2n
(∫

U

|u2,t|
2n

n−4ν dx
)n−4ν

2n

≤ c22

(
1 + ‖u‖q−1

2

)
‖Aν+1u2‖

[
‖Aνv2‖+ ‖εAνu2 + z(θr−τω)Aνh‖

]
≤ 1

2
c22

(
1 +Mq−1

0 (θr−τω)
)

(‖Aν+1u2‖2 + ‖Aνv2‖2)

+ c23

(
1 +M4q−4

0 (θr−τω) + z4(θr−τω)
)

+
ε

16
‖Aν+1u2‖2,∫

U

f ′2,u(u, x)u1,t ·A2νu2dx

≤ c3
∫
U

|u1,t|(1 + |u|p)|A2νu2|dx

≤ c3
(∫

U

|u1,t|2dx
)1/2(∫

U

(1 + |u|p)
2n

4−4ν dx
) 4−4ν

2n
(∫

U

|A2νu2|
2n

n+4ν−4 dx
)n+4ν−4

2n

≤ c24 (‖v1‖+ ε‖u1‖) (1 + ‖u‖p2) ‖Aν+1u2‖

≤ c25

[
1 +M4p

0 (θr−τω) +M4
1 (θ−tω)e−4σ1(t+r−τ)

]
+

ε

16
‖Aν+1u2‖2,



10 L. WEN, L. YANG EJDE-2022/10

and∫
U

[f ′1,u(u, x)− f ′1,u(u1, x)]u1,tA
2νu2dx

≤ c26

∫
U

|u1,t|
(
|u1|q−2 + |u|q−2

)
|u2||A2νu2|dx

≤ c26

(∫
U

|u1,t|
2n

4q−nq+2n dx
) 4q−nq+2n

2n
(∫

U

(|u1|q−2 + |u|q−2)
2n

(n−4)(q−2) dx
) (n−4)(q−2)

2n

×
(∫

U

|u2|
2n

n−4−4ν dx
)n−4−4ν

2n
(∫

U

|A2νu2|
2n

n−4+4ν dx
)n−4+4ν

2n

≤ c27(1 +
ε√
λ1

)M1(θ−tω)e−σ1(t+r−τ)

×
(
Mq−2

1 (θ−tω)e−(q−2)σ1(t+r−τ) +Mq−2
0 (θr−τω)

)
‖Aν+1u2‖2

≤ c28

[
Mq−1

0 (θr−τω) +Mq−1
1 (θ−tω)e−(q−1)σ1(t+r−τ)

]
‖Aν+1u2‖2.

Also we have∫
U

λu2tA
2νu2dx ≤ λc29

(
‖Aν+1u2‖2 + ‖Aνv2‖2 + z2(θr−τω)

)
.

By the above inequalities and (4.9), we obtain

d

dt
y2(r) ≤ m1(θr−τω)y2(r) + q1(θr−τω), ∀r ≥ τ − t, (4.10)

where

y2 = ‖Aν+1u2‖2 + ‖Aνv2‖2 + 2

∫
U

[f(u, x)− f1(u1, x) + λu2]A2νu2dx, (4.11)

m1(θr−τω) = c30[1 +Mq−1
0 (θr−τω) +Mq−1

1 (θ−tω)e−
4

n−4σ1(t+r−τ)]− ε

4
, (4.12)

q1(θr−τω) = c31[1 +M4q−4
0 (θr−τω) + z4(θr−τω) +M2q

1 (θ−tω)e−2qσ1(t+r−τ)],

y2 (τ − t, τ − t, θ−τω, ϕτ−t(θ−τω)) = (0, 0)T,

where c30, c31 depend on λ. By using the Gronwall’s inequality to (4.10) on [τ−t, r]
(r ≥ τ − t), we have

y2(r, τ − t, θ−τω, ϕτ−t(θ−τω)) ≤
∫ r

τ−t
q1(θξ−τω)e

∫ r
ξ
m1(r,ω)drdξ, (4.13)

for all r ≥ τ − t.
Note that∣∣ ∫

U

f2(u, x) ·A2νu2dx
∣∣ ≤ c32

∫
U

(
1 + |u|p+1

)
|A2νu2|dx

≤ c32

(∫
U

(
1 + |u|p+1

)2
dx
)1/2(∫

U

|A2νu2|2dx
)1/2

≤ c32

[
1 +M2p+2

0 (θr−τω) +M2
1 (θ−tω)e−2σ1(t+r−τ)

]
,

and ∫
U

∣∣[f1(u, x)− f1(u1, x) + λu2]A2νu2

∣∣ dx
≤ c33

∫
U

(1 + |u2|q + |u1|q + λ|u2|) |A2νu2|dx



EJDE-2022/10 DYNAMICS OF A PLATE MODEL WITH CRITICAL EXPONENT 11

≤ c33

(∫
U

(
1 + |u2|q + |u1|q + λ|u2|

)2
dx
)1/2(∫

U

|A2νu2|2dx
)1/2

≤ c34

[
1 +M2q

0 (θr−τω) +M2q
1 (θ−tω)e−2qσ1(t+r−τ)

]
.

Therefore,∫
U

[f(u, x)− f1(u1, x) + λu2]A2νu2dx

=

∫
U

[f2(u, x) + f1(u, x)− f1(u1, x) + λu2]A2νu2dx

≤ c35[1 +M2q
0 (θr−τω) +M2q

1 (θ−tω)e−2qσ1(t+r−τ)], ∀r ≥ τ − t,

(4.14)

where c34, c35 depend on λ. Thus, according to (4.11), (4.13) and (4.14), we obtain
that for r ≥ τ − t,
‖Aν+1u2(r, τ − t, θ−τω, ϕτ−t(θ−τω))‖2 + ‖Aνv2(r, τ − t, θ−τω, ϕτ−t(θ−τω))‖2

≤ 2y2 (r, τ − t, θ−τω, ϕτ−t(θ−τω))

+ 2c35[1 +M2q
0 (θr−τω) +M2q

1 (θ−tω)e−2qσ1(t+r−τ)]

≤ 2c35[1 +M2q
0 (θr−τω) +M2q

1 (θ−tω)e−2qσ1(t+r−τ)]

+ 2c31

∫ r

τ−t
[1 +M4q−4

0 (θξ−τω) + z4(θξ−τω)

+M2q
1 (θ−tω)e−2qσ1(t+ξ−τ)]e

∫ r
ξ
m1(θs−τω)drdξ.

(4.15)
Denoting

M2(t, ω) = 2c35[1 +M2q
0 (ω) +M2q

1 (θ−tω)e−2qσ1t]

+ 2c31

∫ 0

−t
[1 +M4q−4

0 (θξω) + z4(θξω)

+M2q
1 (θ−tω)e−2qσ1(t+ξ)]e

∫ 0
ξ
m1(θrω)drdξ,

(4.16)

then we can obtain (4.7) from (4.15) and (4.16). �

Motivated by [27, Proposition 1.4], Lemmas 4.1 and 4.2, we obtain the following
decomposition of solutions for (2.2).

Lemma 4.3. Let τ ∈ R, ω ∈ Ω and t ≥ 0. Then for any T > 0, the following
statements are valid.

(i) There exist two positive constants K̄, K1 and the solution ϕ(r) of (2.2) has
a decomposition: ϕ(r) = φ1(r) + φ2(r), where φ1(r), φ2(r) satisfy∫ τ

r

‖φ1 (ξ, τ − t, θ−τω, ϕτ−t(θ−τω))‖q−1
E dξ ≤ K0

σ1T
(τ − r) + K̄,

‖φ2(r, τ − t, θ−τω, ϕτ−t(θ−τω))‖q−1
E ≤ K1M2(T, ω), ∀r ≥ τ − t.

(4.17)

(ii) The first component u(r) of solution ϕ(r) of (2.2) has a decomposition that
u(r) = w1(r) + w2(r), where w1(r), w2(r) satisfy∫ τ

r

‖w1(ξ, τ − t, θ−τω, ϕτ−t(θ−τω))‖q−1
2 dξ ≤ K0

σ1T
(τ − r) + K̄,

‖Aν+1w2(r, τ − t, θ−τω, ϕτ−t(θ−τω))‖q−1 ≤ K1M2(T, ω), ∀r ≥ τ − t,
(4.18)

where ν is as in (4.8), K0 = E[Mq−1
1 (ω)].



12 L. WEN, L. YANG EJDE-2022/10

Proof. From (4.6), we know M2
1 (ω) = c18

(
1 +Mq+1

0 (ω)
)

, then we can find a pos-

itive constant c36 that

Mq−1
1 (ω) = c36

(
1 +M

q2−1
2

0 (ω)
)
.

As (θt)t∈R is measure-preserving and ergodic on (Ω,F ,P), from the Birkhoff ergodic
Theorem [19], we obtain

E[Mq−1
1 (θsω)] = E[Mq−1

1 (ω)]

= c36

(
1 + E

[
M

q2−1
2

0 (ω)
])

= c36

(
1 + 2

q2−1
2

[
c
q2−1

4
16 + c

q2−1
4

13

( 2q2 − 10

(q2 − 1)ρ

) q2−5
4

8

(q2 − 1)ρ

Γ( 4+(q+1)(q2−1)
8 )

√
πα

(q+1)(q2−1)
8

])
= K0 <∞, ∀s ∈ R,

and for any fixed T > 0, s ∈ R and ω ∈ Ω (in fact for a.e.ω ∈ Ω),

1

k

k∑
l=1

Mq−1
1 (θs+lTω)→ E[Mq−1

1 (θsω)] = K0, k →∞.

Therefore, for ω ∈ Ω, there exists a large integer k0(ω) <∞ satisfying

K0

2
≤ 1

k

k∑
l=1

Mq−1
1 (θs+lTω) ≤ 3K0

2
, ∀k ≥ k0(ω), ∀s ∈ R,

and

K0

2
k0(ω) ≤

k0(ω)∑
l=1

Mq−1
1 (θs+lTω) ≤ 3K0

2
k0(ω), ∀s ∈ R. (4.19)

Taking the expectation on (4.19), we obtain

1

2
E[k0] ≤ k0(ω) ≤ 3

2
E[k0], k0(ω) <∞,

so, we have E[k0] <∞.
(i) We construct functions φ1(r) and φ2(r). When T > 0 and k ∈ N, consider

equations (4.2) and (4.3) at the interval [τ − t+ (k − 1)T, τ − t+ kT ]. Note

φ1 = (w1, w̃1)T = ϕ1, φ1(τ − t+ (k − 1)T ) = ϕ(τ − t+ (k − 1)T ),

φ2 = (w2, w̃2)T = ϕ2, φ2(τ − t+ (k − 1)T ) = (0, 0)T.

By (4.4), as s ≥ 0, r ∈ [τ − t+ s+ (k − 1)T, τ − t+ s+ kT ], we obtain

‖φ1(r)‖q−1
E = ‖w1‖q−1

2 + ‖w̃1‖q−1

≤Mq−1
1 (θ−t+s+(k−1)Tω)e−(q−1)σ1(t−s−(k−1)T+r−τ),

(4.20)

and ∫ τ−t+s+kT

τ−t+s+(k−1)T

‖φ1(ξ)‖q−1
E dξ

≤
∫ τ−t+s+kT

τ−t+s+(k−1)T

Mq−1
1 (θ−t+s+(k−1)Tω)e−(q−1)σ1(t−s−(k−1)T+ξ−τ)dξ



EJDE-2022/10 DYNAMICS OF A PLATE MODEL WITH CRITICAL EXPONENT 13

≤
Mq−1

1 (θ−t+s+(k−1)Tω)

(q − 1)σ1
.

By (4.20), as k ≥ k0(ω) and s ≥ 0,∫ τ−t+s+kT

τ−t+s
‖φ1(ξ)‖q−1

E dξ

≤
(∫ τ−t+s+T

τ−t+s
+

∫ τ−t+s+2T

τ−t+s+T
+ · · ·+

∫ τ−t+s+kT

τ−t+s+(k−1)T

)
‖φ1(ξ)‖q−1

E dξ

≤ 1

(q − 1)σ1

[
Mq−1

1 (θ−t+sω) +Mq−1
1 (θ−t+s+Tω) + · · ·+Mq−1

1 (θ−t+s+(k−1)Tω)
]

≤ K0

σ1
k.

Thus, when τ − t ≤ r ≤ τ and τ − r = mT + r̄, m ∈ Z+, r̄ ∈ [0, T ), we obtain:
(a) If m ≥ k0(ω), then∫ τ

r

‖φ1(ξ)‖q−1
E dξ ≤ (

∫ r+T

r

+

∫ r+2T

r+T

+ · · ·+
∫ r+(m+1)T

r+mT

)‖φ1(ξ)‖q−1
E dξ

≤ K0

σ1T
(τ − r) +

K0

σ1
.

(b) If 0 < m < k0(ω), then∫ τ

r

‖φ1(ξ)‖q−1
E dξ ≤

(∫ τ−(k0−1)T

τ−k0T
+

∫ τ−(k0−2)T

τ−(k0−1)T

+ · · ·+
∫ τ

τ−t

)
‖φ1(ξ)‖q−1

E dξ

≤ K0

σ1
k0(ω).

Thus ∫ τ

r

‖φ1(ξ)‖q−1
E dξ ≤ K0

σ1T
(τ − r) + K̄,

where K̄ = 3K0

2σ1
E[k0] + K0

σ1
.

For t ≥ T , ϕ2(r) is the solution of (4.3) on the interval [r−T, r] with ϕ2(r−T ) =
(0, 0)T, by (4.7), we obtain

‖ϕ2(r, r − T, θ−τω, ϕr−T )‖q−1
Eν ≤ c37‖ϕ2(r, r − T, θ−τω, ϕr−T )‖2Eν
≤ K1M2(T, ω), ∀r ≥ τ − t.

(4.21)

Thus, combining (4.7) and (4.21), for any t ≥ 0 and r ≥ τ − t, we can choose φ2

like this, then (4.17) holds.
(ii) can be obtained directly from (i). �

4.2. Decomposition of solutions II. Let ϕ(r) be the solution of (2.2), and write
ϕ(r) = ϕL(r) + ϕN (r). Here ϕL(r) = (uL, vL)T and ϕN (r) = (uN , vN )T satisfy

ϕ̇L + ΛϕL + F1(ϕL, x) = 0, r > τ − t,
ϕL(τ − t, τ − t, θ−τω, ϕτ−t(θ−τω)) = ϕL,τ−t = (uτ−t, u1,τ−t + εuτ−t)

T,
(4.22)

and

ϕ̇N + ΛϕN + F2(ϕ,ϕL, x) = F̃2(θr−τω, r), r > τ − t,
ϕN (τ − t, τ − t, θ−τω, ϕτ−t(θ−τω)) = (0,−h(x)z(θ−tω))T, t ≥ 0,

(4.23)



14 L. WEN, L. YANG EJDE-2022/10

where uτ−t and u1,τ−t are independent of ω, and vL = uL,t + εuL, vN = uN,t +
εuN − h(x)z(ω). Next, we estimate the component ϕL.

Lemma 4.4. For any τ ∈ R and ω ∈ Ω, t ≥ 0, there exists a constant ML > 0
(independent of ω, t and τ) such that the solution ϕL(r) = ϕL(r, τ − t, ϕL,τ−t) of
(4.22) satisfies

‖ϕL(r, τ − t, ϕL,τ−t)‖E =
(
‖uL(r)‖22 + ‖vL(r)‖2

)1/2
≤MLe

−σ1(t+r−τ), ∀r ≥ τ − t.
(4.24)

Proof. Note that ϕL,τ−t = ϕτ−t(θ−τω)+(0, h(x)z(θ−tω))T ∈ B0(θ−tω) and ϕL,τ−t
is independent of ω, we replace θtω by ω, so we have

‖ϕL,τ−t‖2E ≤ 2M2
0 (ω) + 2‖h(x)‖2|z(ω)|2, ∀τ ∈ R, ω ∈ Ω, t ≥ 0.

Taking the expectation with respect to ω ∈ Ω, we obtain that

‖ϕL,τ−t‖2E = ‖uτ−t‖22 + ‖u1,τ−t + εuτ−t‖2

≤ 2 E[M2
0 (ω)] + 2‖h(x)‖2 · E[|z(ω)|2]

≤ 4
(
c16 + c13

1

ρ

Γ( q+2
2 )

√
παq+1

)
+ 2‖h(x)‖2

Γ( 1+2
2 )

√
πα

= 4
(
c16 + c13

1

ρ

Γ( q+2
2 )

√
παq+1

)
+

1

α
‖h(x)‖2.

As in (4.6) for r ≥ τ − t, we obtain

‖ϕL(r)‖2E ≤ [‖ϕL,τ−t‖2E + 2c15(1 + ‖uτ−t‖q+1
2 )]e−2σ1(t+r−τ) = M2

Le
−2σ1(t+r−τ),

where

M2
L = 4c16 + 4c13

1

ρ

Γ( q+2
2 )

√
παq+1

+
1

α
‖h(x)‖2 + c15

+ c15

(
4c16 + 4c13

1

ρ

Γ( q+2
2 )

√
παq+1

+
1

α
‖h(x)‖2

) q+1
2

.

�

The following estimate is same as in Lemma 4.2.

Lemma 4.5. For any τ ∈ R, ω ∈ Ω, there exist a random variable tν(ω) ≥ 0
and a tempered random variable Mν(ω) > 0 (independent of t and τ) such that the
solution ϕN (τ, τ − t, θ−τω, ϕτ−t(θ−τω)) of (4.23) satisfies: for t ≥ tν(ω),

‖Aν+1uN (τ, τ − t, θ−τω, ϕτ−t)‖2 +‖AνvN (τ, τ − t, θ−τω, ϕτ−t)‖2 ≤M2
ν (ω), (4.25)

where ν is as in (4.8).



EJDE-2022/10 DYNAMICS OF A PLATE MODEL WITH CRITICAL EXPONENT 15

Proof. Similar to (4.9), we take the inner product ((4.23), A2νϕN )E , then

1

2

d

dt

(
‖Aν+1uN‖2 + ‖AνvN‖2 + 2

∫
U

[f(u, x)− f1(uL, x) + λuN ]A2νuNdx
)

+ (ΛϕN , A
2νϕN ) + ε

∫
U

[f(u, x)− f1(uL, x) + λuN ]A2νuNdx

−
∫
U

[(f ′1(u, x)− f ′1(uL, x))uL,t + f ′2(u, x)uL,t + f ′(u, x)uN,t + λuN,t]A
2νuNdx

=
(
f(u, x)− f1(uL, x) + λuN , A

2νh(x)z(θr−τω)
)

+
(
h(x)z(θr−τω), A2νuN

)
2

+ (g(x, r) + εh(x)z(θr−τω), A2νvN ), ∀r ≥ τ − t.
(4.26)

From ϕN (r) = ϕ(r)− ϕL(r) and (4.1), (4.24),

‖ϕN (r)‖E ≤ML +M0(θr−τω), ∀r ≥ τ − t.

Using Hölder’s inequality, (A1) and Lemma 4.3, we obtain that: for r ≥ τ − t,∫
U

[f ′1(u, x)− f ′1(uL, x)]uL,tA
2νuNdx

≤ c2
∫
U

|uL,t|
(
|uL|q−2 + |u|q−2

)
|uN ||A2νuN |dx

≤ c2
(∫

U

|uL,t|
2n

4q−nq+2n dx
) 4q−nq+2n

2n
(∫

U

(
|uL|q−2

) 2n
(n−4)(p−2) dx

) (n−4)(p−2)
2n

×
(∫

U

|uN |
2n

n−4−4ν dx
)n−4−4ν

2n
(∫

U

|A2νuN |
2n

n−4+4ν dx
)n−4+4ν

2n

+ c2

(∫
U

|uL,t|
2n

4q−nq+2n dx
) 4q−nq+2n

2n
(∫

U

(
|w1|q−2

) 2n
(n−4)(p−2) dx

) (n−4)(p−2)
2n

×
(∫

U

|uN |
2n

n−4−4ν dx
)n−4−4ν

2n
(∫

U

|A2νuN |
2n

n−4+4ν dx
)n−4+4ν

2n

+ c2

(∫
U

|uL,t|
2n

4q−nq+2n dx
) 4q−nq+2n

2n
(∫

U

|uN |
2n

(n−4)(p−2) dx
) (n−4)(p−2)

2n

×
(∫

U

(
|w2|q−2

) 2n
n−4−4ν dx

)n−4−4ν
2n

(∫
U

|A2νuN |
2n

n−4+4ν dx
)n−4+4ν

2n

≤ c38

(
‖uL‖q−2

2 + ‖w1‖q−2
2

)
‖Aν+1uN‖2 + c39

[
1 +M4

0 (θr−τω)

+ ‖Aν+1w2‖4q−8
]

+
ε

16
‖Aν+1uN‖2,

and∫
U

[f ′(u, x)uN,t + λuN,t]A
2νuNdx

≤ c20

∫
U

[(
1 + |u|q−1

)
|uN,t|+ λuN,t

]
|A2νuN |dx

≤ c20

(∫
U

|uN,t|2dx
)1/2(∫

U

|A2νuN |2dx
)1/2

+ c20

(∫
U

|w1|
n(q−1)

2 dx
)2/n(∫

U

|A2νuN |
2n

n−4+4ν dx
)n−4+4ν

2n
(∫

U

|uN,t|
2n

n−4ν dx
)n−4ν

2n



16 L. WEN, L. YANG EJDE-2022/10

+ c20

(∫
U

|uN,t|2dx
)1/2(∫

U

(
|w2|q−1

) 2n
4−4ν dx

) 4−4ν
2n

×
(∫

U

|A2νuN |
2n

n+4ν−4 dx
)n+4ν−4

2n

+ c20

(∫
U

|λ|n2
)2/n(∫

U

|uN,t|
2n

n−4ν dx
)n−4ν

2n
(∫

U

|A2νuN |
2n

n+4ν−4 dx
)n+4ν−4

2n

≤ c40‖w1‖q−1
2

(
‖Aν+1uN‖2 + ‖AνvN‖2

)
+

ε

16
‖Aν+1uN‖2

+ c41

[
M4

0 (θr−τω) + z4(θr−τω) + ‖w1‖4q−4
2 + ‖Aν+1w2‖4q−4

]
,

where c40 is dependent of λ;∫
U

f ′2(u, x)uL,t ·A2νuNdx

≤ c42

∫
U

|uL,t| (1 + |u|p) |A2νuN |dx

≤ c43

(∫
U

|uL,t|2dx
)1/2(∫

U

(1 + |u|pdx)
2n

4−4ν dx
) 4−4ν

2n
(∫

U

|A2νuN |
2n

n+4ν−4 dx
)n+4ν−4

2n

≤ c44

[
1 +M2p

0 (θr−τω)
]

+
ε

16
‖Aν+1uN‖2.

From (4.26) and similar to (4.10), we have

d

dt
yN (r) +m2(r)yN (r) ≤ q2(θr−τω), ∀r ≥ τ − t, (4.27)

where

yN = ‖Aν+1uN‖2 + ‖AνvN‖2 + 2

∫
U

[f(u, x)− f1(uL, x) + λu2]A2νuNdx, (4.28)

m2(r) =
ε

2
− c45‖w1(r)‖q−1

2 − c45‖uL(r)‖q−2
2 ,

q2(θr−τω) = c46[1 +M2q
0 (θr−τω) + z4(θr−τω) + ‖w1‖4q−4

2 + ‖Aν+1w2‖4q−4],

yN (τ − t, τ − t, θ−τω, ϕτ−t(θ−τω)) ≤ ‖h‖22 · z2(θ−tω),

2

∫
U

[f(u, x)− f1(uL, x) + λuN ]A2νuNdx ≤ c47[1 +Mq+1
0 (θr−τω)],

where c45, c46, and c47 are dependent of λ. Using the Gronwall’s inequality in
(4.27), on [τ − t, r] (r ≥ τ − t), we have

yN (r) ≤ yN (τ − t)e−
∫ r
τ−tm2(s)ds +

∫ r

τ−t
q2(θξ−τω)e−

∫ r
ξ
m2(s)dsdξ. (4.29)

Let

T = T1 =
4c45K0

σ1ε

in (4.18), for τ − t ≤ r ≤ τ , we have∫ τ

r

‖w1(s)‖q−1
2 ds ≤ ε

4c45
(τ − r) + K̄,

‖Aν+1w2(r)‖q−1 ≤ K1M2(T1, ω).

(4.30)



EJDE-2022/10 DYNAMICS OF A PLATE MODEL WITH CRITICAL EXPONENT 17

For r ≥ ξ ≥ τ − t, from (4.30) and (4.32), we obtain that∫ τ

ξ

m2(s)ds =

∫ τ

ξ

ε

2
− c45‖w1(s)‖q−1

2 − c45‖uL(s)‖q−2
2 ds,

≥ ε

4
(τ − ξ)− c48,

(4.31)

where c48 = c45K̄ + c45
σ1(q−2)M

q−2
L . So we have

yN (τ − t)e−
∫ r
τ−tm2(s)ds ≤ ‖h‖22z2(θ−tω)e−

ε
4 t+c48

t→+∞−→ 0. (4.32)

From (4.20), we obtain∫ −(m−1)T1

−mT1

‖w1(r)‖4q−4
2 e

ε
4 rdr

≤
∫ −(m−1)T1

−mT1

M4q−4
1 (θ−mT1ω)e−(4q−4)σ1(r+mT1)e

ε
4 rdr

≤ 4

ε
e
ε
4T1M4q−4

1 (θ−mT1
ω)e−

ε
4mT1 , ∀m ≥ 1,

and ∫ 0

−∞
‖w1(r)‖4q−4

2 e
ε
4 rdr

=
(
· · ·+

∫ −(m−1)T1

−mT1

+ · · ·+
∫ −T1

−2T1

+

∫ 0

−T1

)
‖w1‖4q−4

2 e
ε
4 rdr

≤ 4

ε
e
ε
4T1

+∞∑
m=1

M4q−4
1 (θ−mT1

ω)e−εmT1/4.

From [2], we know that M4q−4
1 (ω) is tempered and M4q−4

1 (θtω) is continuous in t,
so we can find a tempered random variable ς(ω) (> 0) such that

M4q−4
1 (θ−mT1ω) ≤ ς(ω)e

ε
4q−4mT1 , ∀m ≥ 1,

then we obtain∫ 0

−∞
‖w1(r)‖4q−4

2 e
ε
4 rdr ≤ ς(ω)

4

ε
e
ε
4T1

+∞∑
m=1

e
ε(2−q)
4q−4 mT1 =

4

ε

e
ε

4q−4T1

1− e
ε(2−q)
4q−4 T1

ς(ω) <∞.

Hence∫ τ

τ−t
q2(θξ−τω)e−

∫ τ
ξ
m2(s)dsdξ

≤ c49

∫ τ

τ−t

[
1 +M2q

0 (θr−τω) + z4(θr−τω) + ‖w1‖4q−4
2 +M4

2 (T1, ω)
]
e−

ε
4 (τ−ξ)dξ

≤ c50

(
1 +M4

2 (T1, ω) +

∫ 0

−∞

(
M2q

0 (θrω) + z4(θrω) + ‖w1(r)‖4q−4
2

)
e
ε
4 rdr

)
<∞,

where c49 and c50 depend on λ.



18 L. WEN, L. YANG EJDE-2022/10

From (4.28), (4.29) and (4.32), for t ≥ 0, we can deduce that

‖Aν+1uN (τ, τ − t, θ−τω, ϕτ−t)‖2 + ‖AνvN (τ, τ − t, θ−τω, ϕτ−t)‖2

≤ 2yN (τ, τ − t, θ−τω, ϕτ−t(θ−τω)) + 2c47[1 +Mq+1
0 (ω)]

≤ 2‖h‖22 · z2(θ−tω)e−
ε
4 t+c48 + 2c51

[
1 +Mq+1

0 (ω)
]

+ c52(1 +M4
2 (T1, ω).

+

∫ 0

−∞

(
M2q

0 (θrω) + z4(θrω) + ‖w1(r)‖4q−4
2

)
e
ε
4 rdr).

(4.33)

By (4.32), there exists a random variable tν(ω) ≥ 0 such that

0 ≤ 2‖h‖22 · z2(θ−tω)e−
ε
4 t+c48 ≤ 1, ∀t ≥ tν(ω). (4.34)

We denote

M2
ν (ω)

= c52

(
1 +M4

2 (T1, ω) +

∫ 0

−∞

(
M2q

0 (θrω) + z4(θrω) + ‖w1(r)‖4q−4
2

)
e
ε
4 rdr

)
+ 1 + 2c51

[
1 +Mq+1

0 (ω)
]
,

(4.35)
which is tempered, then we have (4.25) from (4.33) and (4.34). �

Lemma 4.6. For any τ ∈ R, ω ∈ Ω, t ≥ 0, assume that Bν(τ, ω) ⊆ B1(τ, ω) ⊆
B0(ω) and Bν(τ, ω) ∈ D(Eν) where ν is same as in (4.8). Then there exist a

random variable t̃ν(ω) > 0 and a tempered random variable M̃ν(ω) > 0 (indepen-
dent of t and τ) such that the solution ϕ(τ, τ − t, θ−τω, ϕτ−t(θ−τω)) of (2.2) with
ϕτ−t(θ−τω) ∈ Bν(τ − t, θ−tω) ⊆ B0(θ−tω) ∩ D(Eν) satisfies

‖ϕ(τ, τ − t, θ−τω, ϕτ−t(θ−τω))‖2Eν
= ‖Aν+1u(τ, τ − t, θ−τω, ϕτ−t)‖2 + ‖Aνv(τ, τ − t, θ−τω, ϕτ−t)‖2

≤ M̃2
ν (ω), ∀t ≥ t̃ν(ω).

(4.36)

Proof. We take the inner product ((2.2), A2νϕ)E with A2νϕ = (A2νu,A2νv)T, we
obtain

1

2

d

dt

(
‖Aν+1u‖2 + ‖Aνv‖2 + 2

∫
U

[f(u, x) + λu]A2νudx
)

+ (Λϕ,A2νϕ) + ε

∫
U

[f(u, x) + λu]A2νudx−
∫
U

[f ′u(u, x)ut + λut]A
2νudx

=
(
f(u, x) + λu,A2νh(x)z(θr−τω)

)
+
(
h(x)z(θr−τω), A2νu

)
2

+
(
g(x, r) + εh(x)z(θr−τω), A2νv

)
, ∀r ≥ τ − t.

(4.37)

Then, for r ≥ τ − t,∫
U

[f ′(u, x)ut + λut]A
2νudx

≤ c43‖w1‖q−1
2

(
‖Aν+1u‖2 + ‖Aνv‖2

)
+

ε

16
‖Aν+1u‖2

+ c45

[
M4

0 (θr−τω) + z4(θr−τω) + ‖w1‖4q−4
2 + ‖Aν+1w2‖4q−4

]
,∫

U

[f(u, x) + λu]A2νudx ≤ c53

[
1 +Mq+1

0 (θr−τω)
]
,



EJDE-2022/10 DYNAMICS OF A PLATE MODEL WITH CRITICAL EXPONENT 19

where c53 depends on λ. Similar to (4.27) by using (4.37), we obtain

d

dt
ỹ1(r) + m̃1(r)ỹ1(r) ≤ q̃1(θr−τω), ∀r ≥ τ − t, (4.38)

where

ỹ1(r) = ‖Aν+1u‖2 + ‖Aνv‖2 + 2

∫
U

[f(u, x) + λu]A2νu dx,

q̃1(θr−τω) = c54

[
1 +M4

0 (θr−τω) + z4(θr−τω) + ‖w1‖4q−4
2 +M4

2 (T1, ω)
]
,

m̃1(r) =
ε

2
− c47‖w1(r)‖q−1

2 .

Using the Gronwall’s inequality in (4.38) on [τ − t, r] (r ≥ τ − t), we have

ỹ1(r) ≤ ỹ1(τ − t)e−
∫ r
τ−t m̃1(s)ds +

∫ r

τ−t
q̃1(θξ−τω)e−

∫ r
ξ
m̃1(s)dsdξ. (4.39)

Same as (4.31), for τ ≥ ξ ≥ τ − t, we obtain that∫ τ

ξ

m̃1(s)ds ≥ ε

4
(τ − ξ)− c50,

so ∫ τ

τ−t
q̃1(θξ−τω)e−

∫ τ
ξ
m̃1(s)dsdξ

≤ c55

(
1 +M4

2 (T1, ω) +

∫ 0

−∞

(
M4

0 (θrω) + z4(θrω) + ‖w1(r)‖4q−4
2

)
e
ε
4 rdr

)
.

From ϕτ−t(θ−τω) ∈ B0(θ−tω) ∩ D(Eν), we obtain

ỹ1(τ − t)e−
∫ r
τ−t m̃1(s)ds

≤
(
‖Aν+1uτ−t‖2 + ‖Aνvτ−t‖2 + 2c53

[
1 +Mq+1

0 (θr−τω)
])
e−

ε
4 t+c50

t→+∞−→ 0.

Hence, there exists a random variable t̃ν(ω) ≥ 0 satisfying

0 ≤ ỹ1(τ − t)e−
∫ τ
τ−t m̃1(s)ds ≤ 1, ∀t ≥ t̃ν(ω).

Based on (4.39), we obtain

‖Aν+1u(τ, τ − t, θ−τω, ϕτ−t)‖2 + ‖Aνv(τ, τ − t, θ−τω, ϕτ−t)‖2

≤ 2ỹ1(τ, τ − t, θ−τω, ϕτ−t(θ−τω)) + 2c53[1 +Mq+1
0 (ω)]

≤ c55

(
1 +M4

2 (T1, ω) +

∫ 0

−∞

(
M4

0 (θrω) + z4(θrω) + ‖w1(r)‖4q−4
2

)
e
ε
4 rdr

)
+ 1 + 2c53

[
1 +Mq+1

0 (ω)
]

= M̃2
ν (ω), ∀t ≥ t̃ν(ω).

(4.40)

�

Using Lemmas 4.1-4.6, by recursion we obtain the following result.

Lemma 4.7. For any τ ∈ R, ω ∈ Ω, t ≥ 0, let Bκ(τ, ω) ⊆ B1(τ, ω), Bκ(τ, ω) ∈
D(Eκ) and ϕτ−t(θ−τω) ∈ Bκ(τ−t, θ−tω). Then there exist Tκ(ω) ≥ 0 and tempered

random variables M̃κ(ω) > 0, Mκ(ω) > 0 (independent of t and τ) such that for
t ≥ Tκ(ω), the solution ϕ(τ, τ − t, θ−τω, ϕτ−t(θ−τω)) of (2.2) satisfies



20 L. WEN, L. YANG EJDE-2022/10

(i) for ν ≤ κ ≤ 1,

‖ϕ(τ, τ − t, θ−τω, ϕτ−t(θ−τω))‖2Eκ
= ‖Aκ+1u(τ, τ − t, θ−τω, ϕτ−t(θ−τω))‖2 + ‖Aκv(τ, τ − t, θ−τω, ϕτ−t(θ−τω))‖2

≤ M̃2
κ(ω);

(ii) for ν ≤ κ ≤ 1− ν,

‖ϕN (τ, τ − t, θ−τω, ϕτ−t(θ−τω))‖2Eκ+ν
= ‖Aκ+ν+1uN (τ, τ − t, θ−τω, ϕτ−t)‖2 + ‖Aκ+νvN (τ, τ − t, θ−τω, ϕτ−t)‖2

≤M2
κ(ω),

(4.41)

where ν is same as in (4.8).

5. Existence of a random attractor

In this section, by applying [29, Lemma 3.7] and Lemma 4.7, show the existence
of a random attractor for (1.1). Firstly, we consider the Lipschitz property of cocycle
Φ on B1(τ, ω). For any τ ∈ R, ω ∈ Ω and any ϕjτ (ω) = (ujτ (ω), vjτ (ω)) ∈ B1(τ, ω),
j = 1, 2, let

ϕj(r) = ϕj(r, τ, ω, ϕjτ (ω)) = (uj(r), vj(r)), r ≥ τ,

be solutions of (2.2) with initial data ϕjτ (ω), j = 1, 2, respectively, and let

ψ(r) = ϕ1(r)− ϕ2(r) = (u1(r)− u2(r), v1(r)− v2(r)) = (ξ(r), η(r)).

So that

ψ̇ + Λψ = F (ϕ1, θrω, r)− F (ϕ2, θrω, r), r ≥ τ,
ψτ (ω) = (ξτ , ητ ) = (ξτ , u1,1,τ − u2,1,τ + εξτ ) = (u1τ − u2τ , v1τ − v2τ ).

(5.1)

Lemma 5.1. There exists a tempered random variable C1(ω) > 0 such that for any
τ ∈ R, t ≥ 0, and ω ∈ Ω, it holds

‖ϕ(t+ τ, τ, θ−τω, ϕ1τ (θ−τω))− ϕ(t+ τ, τ, θ−τω, ϕ2τ (θ−τω))‖E
≤ e

∫ t
0
C1(θsω)ds‖ϕ1τ − ϕ2τ‖E .

(5.2)

Proof. From (4.1), for r ≥ τ , we obtain ‖ϕ1(r)‖E ≤M0(θrω), ‖ϕ2(r)‖E ≤M0(θrω).
Based on (5.1), we have the following inequality by taking the inner product (·, ·)E
with ψ(r),

d

dt
‖ψ(r)‖2E ≤

(c56

2α

(
1 +M2q−2

0 (θr−τω)
)
− ε+ λ

)
‖ψ(r)‖2E , ∀r ≥ τ. (5.3)

Using Gronwall’s inequality on (5.3), we obtain

‖ϕ1(r)− ϕ2(r)‖2E ≤ ‖ϕ1τ − ϕ2τ‖2Ee
∫ r−τ
0 ( c562α (1+M2q−2

0 (θτ+sω))−ε+λ)ds, ∀r ≥ τ.

As ω → θ−τω and r → t+ τ , (5.2) holds with

C1(ω) =
c56

2α

(
1 +M2q−2

0 (ω)
)

+ λ,

0 < E(C1(ω)) =
c56

2α

(
1 + E[M2q−2

0 (ω)]
)

+ λ ≤ c56

2αc18
K0 + λ <∞. �



EJDE-2022/10 DYNAMICS OF A PLATE MODEL WITH CRITICAL EXPONENT 21

Lemma 5.2. For any τ ∈ R, ω ∈ Ω, there exist a Tν(ω) ≥ 0, a random bounded

ball B̃1(ω) of E1 with radius b1(ω), a positive number σ̃ and a tempered random

variable Q̃(ω) > 0 such that the solution ϕ(τ, τ− t, θ−τω, ϕτ−t(θ−τω)) of (2.2) with
initial data ϕτ−t(θ−τω) ∈ B1(τ − t, θ−tω), it holds that

dE

(
ϕ(τ, τ − t, θ−τω,B1(τ − t, θ−tω)), B̃1(ω)

)
≤ Q̃(θ−tω)e−σ̃t, t ≥ Tν(ω).

Proof. Now we assume ϕτ−t(θ−τω) ∈ B1(τ − t, θ−tω), and K̃ν(ω) ⊂ Eν ⊂ E is the
random ball of Eν with radius Mν(ω) defined in (4.35), here ν is same as in (4.8).
By Lemma 4.4, we can deduce that for any t ≥ 0,

dE

(
ϕ(τ, τ − t, θ−τω,B1(τ − t, θ−tω)), K̃ν(ω)

)
≤MLe

−σ1t. (5.4)

For ϕτ−t(θ−τω) ∈ K̃ν(θ−tω), according to Lemmas 4.4–4.7, there exist t1ν(ω) ≥ 0

and a random ball K̃2ν(ω) of E2ν with the radius M2ν(ω) and satisfies

dE

(
ϕ(τ, τ − t, θ−τω, K̃ν(θ−tω)), K̃2ν(ω)

)
≤ P1ν(θ−tω)e−σ1t, t ≥ t1ν(ω), (5.5)

where P 2
1ν(θ−tω) = c̃1

[
1 +M4

ν (θ−tω)
]

+ 2‖h‖2 · |z(θ−tω)|2. From [29, Lemma 3.7],
Lemma 5.1 and (5.4) and (5.5), there exists T1ν(ω) ≥ t1ν(ω) ≥ 0 (independent of
τ), such that for t ≥ T1ν(ω),

dE

(
ϕ(τ, τ − t, θ−τω,B1(τ − t, θ−tω)), K̃2ν(ω)

)
≤ P2ν(θ−tω)e−

a1
2 σ1t,

where P2ν(θ−tω) = ML + P1ν(θ−(1−a1)tω) is tempered and

0 < a1 ≤
σ1

3 E[C1(ω)] + 4σ1
.

There exists an integer k̃ ≥ 1 such that 1 − ν ≤ (k̃ − 1)ν < 1 when ν > 0 is a

fixed positive constant. Repeating k̃ (≤ [ 1
ν ] + 2) steps as above recursion, we can

obtain that there exist Tk̃ν(ω) > 0 (independent of τ) and a random ball B̃1(ω) of
E1 with radius b1(ω) (defined in (4.41)) such that for t ≥ Tk̃ν(ω),

dE

(
ϕ(τ, τ − t, θ−τω,B1(τ − t, θ−tω)), B̃1(ω)

)
≤ Pk̃ν(θ−tω)e−

a
k̃−1
2 σ1t, (5.6)

where

0 < aj ≤
aj−1

2 σ1

3 E[C1(ω)] +
aj−1

2 σ1 + 3σ1
<∞, j = 2, · · · , k̃ − 1,

Pk̃ν(θ−tω) = ML + P(k̃−1)ν(θ−(1−ak̃−1)tω),

is tempered. �

Combining Lemmas 2.3 and 5.2, the existence of a random attractor for the RDS
Φ can be proved.

Theorem 5.3. The cocycle Φ associated with (2.2) possesses a D(E)-pullback ran-

dom attractor A ∈ D(E) such that for any τ ∈ R, ω ∈ Ω, A(τ, ω) ⊆ B̃1(ω)∩B0(ω)
and

‖A(τ, ω)‖E1 = sup
ϕ∈A(τ,ω)

‖ϕ‖E1 ≤ b1(ω),

where b1(ω) is the radius of the bounded ball B̃1(ω) ⊂ E1.



22 L. WEN, L. YANG EJDE-2022/10

Proof. Since E1 ↪→ E is a compact embedding, from (5.6) we know that B̃1(ω)
in Lemma 5.2 is a compact measurable D(E)-pullback attracting ball in E for
any τ ∈ R and ω ∈ Ω. According to Lemma 2.3, the RDS Φ possesses a D(E)-
pullback random attractor A ∈ D(E), that is for any τ ∈ R, ω ∈ Ω, and A(τ, ω) ⊆
B̃1(ω)∩B0(ω). Based on (4.35), (4.40) and (4.41), the radius b1(ω) of B̃1(ω) ⊂ E1

is given by

b21(ω) = c57

(
1 +M4k̃

2 (T1, ω) +M
(q+1)k̃
0 (ω)

)
+ c58

∫ 0

−∞

(
M4k̃

0 (θrω) + |z(θrω)|4k̃ + ‖w1(r)‖(4q−4)k̃
2

)
eεr/4dr.

(5.7)

�

Acknowledgments. The authors express their gratitude to the anonymous referee
for the valuable comments and suggestions.This research was partially supported
by the NSFC grant (12071192).

References

[1] M. Al-Gharabli, S. Messaoudi; Existence and a general decay result for a plate equation with
nonlinear damping and a logarithmic source term, J. Evol. Equ., 18 (2018), 105–125.

[2] L. Arnold; Random Dynamical Systems, Springer-Verlag, Berlin, 1998.

[3] T. Caraballo, P. Kloeden, B. Schmalfuss; Exponentially stable stationary solutions for sto-
chastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183–207.

[4] V. Carbone, M. Nascimento, K. Schiabel-Silva, R. Silva; Pullback attractors for a singularly

nonautonomous plate equation, Electron. J. Differential Equations, 2011 (2011), No. 77, 1–13.
[5] I. Chueshov, I. Lasiecka; Inertial manifolds for von Karman plate equations, Appl. Math.

Optim, (special issue dedicated to J. L. Lions). 46 (2002), 179–206

[6] I. Chueshov, I. Lasiecka; Global attractors for von Karman evolutions with a nonlinear bound-
ary dissipation, J. Differential Equations, 198 (2004), 196–231.

[7] H. Crauel, F. Flandoli; Hausdorff dimension of invariant sets for random dynamical systems,
J. Dynam. Differential Equations, 10 (1998), 449–474.

[8] H. Crauel, A. Debussche, F. Flandoli; Random attractors, J. Dynam. Differential Equations,

9 (1997), 307–341.
[9] X. Fan; Random attractors for damped stochastic wave equations with multiplicative noise,

Internat. J. Math., 19 (2008), 421–437.

[10] X. Fan; Random attractor for a damped sine-Gordon equation with white noise, Pacific J.
Math., 216 (2004), 63–76.

[11] A. Kh Khanmamedov; Global attractors for the plate equation with a localized damping and
a critical exponent in an unbounded domain, J. Differential Equations, 225 (2006), 528–548.

[12] A. Kh. Khanmamedov; Existence of a global attractor for the plate equation with a critical

exponent in an unbounded domain, Appl. Math. Lett., 18 (2005), 827–832.

[13] A. Kh Khanmamedov; Global attractors for von Karman equations with nonlinear interior
dissipation, J. Math. Anal. Appl., 318 (2006), 92-=101.

[14] T. Liu and Q. Ma; Time-dependent attractor for plate equations on Rn, J. Math. Anal. Appl.,
479 (2019), 315–332.

[15] H. Li, Y. You, J. Tu; Random attractors and averaging for non-autonomous stochastic wave

equations with nonlinear damping, J. Differential Equations, 258 (2015), 148–190.
[16] Y. Lv, W. Wang; Limiting dynamics for stochastic wave equations, J. Differential Equations,

244 (2008), 1–23.

[17] W. Ma, Q. Ma; Attractors for stochastic strongly damped plate equations with additive noise,
Electron. J. Differential Equations, 2013 (2013), No. 111, 1–12.

[18] X. Shen, Q. Ma; Existence of random attractors for weakly dissipative plate equations with

memory and additive white noise, Compu. Math. Appl., 73 (2017), 2258–2271.
[19] P. Walters; An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982.



EJDE-2022/10 DYNAMICS OF A PLATE MODEL WITH CRITICAL EXPONENT 23

[20] B. Wang; Random attractors for non-autonomous stochastic wave equations with multiplica-

tive noise, Discrete Contin. Dyn. Syst., 34 (2014), 269–300.

[21] B. Wang; Sufficient and necessary criteria for existence of pullback attractors for non-
compact random dynamical systems, J. Differential Equations, 253 (2012), 1544–1583.

[22] Z. Wang, L. Zhang; Finite fractal dimension of random attractor for stochastic non-

autonomous strongly damped wave equation, Compu. Math. Appl., 75 (2018), 3343–3357.
[23] Z. Wang, S. Zhou; Random attractor for stochastic non-autonomous damped wave equation

with critical exponent, Discrete Contin. Dyn. Syst., 37 (2017), 545–573.

[24] Z. Wang, S. Zhou; Random attractor and random exponential attractor for stochastic non-
autonomous damped cubic wave equation with linear multiplicative white noise, Discrete

Contin. Dyn. Syst., 38 (2018), 4767–4817.

[25] L. Yang, C. Zhong; Global attractor for plate eqution with nonlinear damping, Nonlinear
Anal., 69 (2008), 3802–3810.

[26] M. Yang, J. Duan, P. Kloeden; Asymptotic behavior of solutions for random wave equations
with nonlinear damping and white noise, Nonlinear Anal. RWA., 12 (2011), 464–478.

[27] S. Zelik; Asymptotic regularity of solutions of a nonautonomous damped wave equation with

a critical growth exponent, Commun. Pure Appl. Anal., 3 (2004), 921–934.
[28] S. Zhou, F. Yin, Z. Ouyang; Random attractor for damped nonlinear wave equations with

white noise, SIMA J. Appl. Dyn. Syst., 4 (2005), 883–903.

[29] S. Zhou, M. Zhao; Fractal dimension of random attractor for stochastic non-autonomous
damped wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 36

(2016), 2887–2914.

[30] S. Zhou, M. Zhao; Fractal dimension of random invariant sets for nonautonomous random
dynamical systems and random attractor for stochastic damped wave equation, Nonlinear

Anal., 133 (2016), 292–318.

Lan Wen
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Email address: wenl18@lzu.edu.cn

Lu Yang
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Email address: yanglu@lzu.edu.cn


	1. Introduction
	2. Preliminaries
	3. Boundedness of solutions
	4. Decomposition of solutions
	4.1. Decomposition of solution I
	4.2. Decomposition of solutions II

	5. Existence of a random attractor
	Acknowledgments

	References