Random pullback attractor of a non-autonomous local modified stochastic Swift-Hohenberg with multiplicative noise

(1)

Article

Random pullback attractor of a non-autonomous local

modified stochastic Swift-Hohenberg with

multiplicative noise

Yongjun Li 1*, Tinggang Zhao1 and Hongqing Wu1

1_{School of Mathematics, Lanzhou City University, Lanzhou, 730070, P.R. China; [email protected] (Y. Li);}

[email protected](T.Zhao); [email protected](H.Wu), * Correspondence: [email protected] (Y. Li)

Abstract: In this paper, we study the existence of the random -pullback attractor of a non-autonomous local modiﬁed stochastic Swift-Hohenberg equation with multiplicative noise in stratonovich sense. It is shown that a random -pullback attractor exists in H20(D) when its

external force has exponential growth. Due to the stochastic term, the estimate are delicate, we overcome this difficulty by using the Ornstein-Uhlenbeck(O-U) transformation and its properties.

Keywords: Swift-Hohenberg equation; Random -pullback attractor; Non-autonomous random dynamical system

1. Introduction

The Swift-Hohenberg(S-H) type equations arise in the study of convective hydrodynamical, plasma confinement in toroidal and viscous film flow, was introduced by authors in [1]. After that, Doelman and Standstede [2] proposed the following modified Swift-Hohenberg equation for a pattern formation system near the onset to instability

2 2 3

2

|

0,

t

u

+  +  +

u

au

+ 

b

u

+ =

u

(1.1)

where a and b are arbitrary constants.

We can see from the above equation that there exist two operatorsand2, these two operators have some symmetry, for anyuH D1₀( ), the inner product (u u, )= −  ( u u, )=

‖ ‖



u

2_{, for any}

2 0

( )

u



H

D

, (2u u, ) (=  u, u)=‖ ‖u2,is antisymmetry and 2is symmetry. We will use the symmetry principle study S-H equation.

The dynamical properties of the S-H equation are important for the studies pattern formation system have been extensively investigated by many authors; see [3-8]. Polat [8] establish the existence of global attractor for the system (1.1), and then Song et al. [7] improved the result inH .k

Recently for non-autonomous modiﬁed S-H equation:

2 2 3

(

2

|

( , ))

( )

du

+  +  +

u

u au b

+ 

u

+ −

u

g x t dt

=

lu dW t

, (1.2) it has also attracted the interest of many authors. If

l

=

0

, equation (1.2) becomes a non-autonomous modified S-H equaiton. Park [9] proved the existence of -pullback attractor when the external force has exponential growth, Xu et al.[10] established the existence of uniform attractor when the external force

g

(

x

,

t

)

satisﬁes translation bounded, these results need the spatial variable in two dimensions. When

l

=

0

, equation (1.2) becomes a non-autonomous stochastic S-H equation, if

|

b

|

<<

1

is a constant, Guo et al.[11] investigated the equation when

g

(

x

,

t

)

=

0

and proved the existence of random attractor which need the spatial variable in one dimension. For g x t( , )=0， to the best of our knowledge, the existence of random -pullback attractor for equation (1.2) has not yet considered.

(2)

In this paper, we consider the following one dimensional non-autonomous local modiﬁed stochastic S-H equation with multiplicative noise:

2 2 3

(

2

_xx _x

( , ))

( ),

[ , ),

du

+  +

u

+

au

+

bu

+

u

−

g x t dt

=

lu dW t

in D







(1.3)

0,

,

xx

u

=

u

=

x



D

(1.4)

( , )

,

u x



=

u x

_



D

(1.5)

Where

D

is a bounded open interval,

Δ

u

means

u

_xx, and

Δ

2

u

means

u

_xxxx,

|

b

|

<

4

,

a

and

l

are arbitrary constants, W(t)is a two-sided real-valued Wiener process on a probability space which will be speciﬁed later. For the external force g(x)∈ L2_loc(R,L2(D)), we assume that there exist M>0and

0 >

β such that

(1.6) The assumption is same as [8, 12], through simple calculation, for all t∈ R, we have

(1.7) An outline of this paper is as follows: In section 2, we recall some basic concepts about random -pullback attractors. In Section 3, we prove that the stochastic dynamical system generated by (1.3) exists a random -pullback attractor in H₀2(D).

2.2 Preliminaries

There are many research results on random attractors and related issues. The reader is referred to [13-19] for more details, we only list the deﬁnitions and abstract result

Let (X,‖‖_X) be a separable Banach space with Borel



-algebra ( )X and ( , , ) be a probability space. In this paper, the term -a.s.(the abbreviation for almost surely) denotes that an event happens with probability one. In other words, the set of possible exception may be non-empty, but it has probability zero.

Definition 2.1.([15,16,20,21]) ( , , , ( )_{t t} ) is called a metric dynamical systems if



:

 → 

is

( ( ) , )-measurable,and



0 is the identity on



,



s t+ = 

 

t s for all

t s

,



and



t

=

for all

t



.

Definition 2.2.([12,13,18]) A non-autonomous random dynamical system (NRDS)

( , )

 

on

X

over a metric dynamical system

( ,



, , ( )



_{t t}_

)

is a mapping

( , , ) :

t

X

,

( , , , )

t

x

( , , )

t

x

  

→

 

→

  

,

which represents the dynamics in the state space Xand satisﬁes the properties (i)

   

( , , )

is the identity on X;

(ii)

   

( , , )

t

=

( , ,

t s

    

_s₋_

) ( , , )

s

for all



 

s

t

; (iii)



→

  

( , , )

t

x

is -measurable for all

t





and

x



X

.

In the sequel, we use to denote a collection of some families of nonempty bounded subsets of X:

,

{ ( , )

( ) :

,

}.

D





D



=

D t





X

t







Definition 2.3.([12,13,18]) A set

B



is called a random -pullback bounded absorbing set for NRDS

( , )

 

if for any

t



and any

D



, there exists



0

( ,

t D



)

such that

( , ,

t

_ _t

) ( ,

D

_ _t

)

B t

( , )

   

₋

  

₋





for any

 



0.

Definition 2.4.([12,13,18]) A set

=

{ ( , ) :

A t



t



,





}

is called a random -pullback attractor for

{ , }

 

if the following hold:

(i)

A t

( , )



is a random compact set;

(3)

(iii) attracts all set in ; that is, for all B and -a.s.





,

lim

d

( ( , ,

t

_ _t

) ( ,

B

_ _t

), ( , ))

A t

0.

→−

   

−

  

−



=

Where

d

is the Hausdorff semimetric given by

( ,

)

sup inf

_X

a b B

dist B

b a

 

=

−

.

Definition 2.5.([14, 17]) A NRDS

( , )

 

on a Banach space

X

is said to be pullback flattening if for every random bounded set

B

 =

{ ( , ) :

B t



t



,



 

}

, for any





0

and





there exists a

( , , )

T B



 



t

and a finite dimensional subspace

X

such that (i)

(

( , ,

t

) ( ,

t

))

T

P

t

B



 



   

₋

  

₋



is bounded, and

(ii)

|| (

)(

( , ,

_t

) ( ,

_t

)) ||

_X

T

I

P

t

B



 



   

₋

  

₋





−



,

where

P X

:

→

X

_ is a bounded projector.

Theorem 2.1.([14, 17]) Suppose that

( , )

 

is a continuous NRDS on a uniformly convex Banach space

X

.

If

( , )

 

possesses a random −pullback bounded absorbing setsB ={ ( , ) :B t



t ,



} and

( , )

 

is pullback flattening, then there exists a random -pullback attractor ={ ( , ) :A t



t ,



}. 3. Random pullback attractor for modified Swift-Hohenberg

In this section, we will use abstract theory in section 2 to obtain the random -pullback attractor for equation (1.3)-(1.5). First we introduce an Ornstein-Uhlenbeck process,

0

( ( )) :

_t

(

_t

)( )

,

.

z

 

e



   

d

t

−

= −





We known from [6], it is the solution of Langevin equation

( ).

dz

+

zdt

=

dW t

( )

W t

is a two-sided real-valued Wiener process on a probability space

( ,



, )

, where

{



C

( , ) : (0)



0},

 =



=

is the Borel algebra induced by the compact open topology of



, and is the corresponding Wiener measure on

{ ,



}

. We identify



( )

t

with

W t

( )

, i.e.,

( )

( , )

( ),

.

W t

=

W t

 

=

t



Define the Wiener time shift by

( )

(

)

( ),

, ,

.

t

s

s t

t

t s

 

=



+ −







Then

( ,



, ,



t

)

is an ergodic metric dynamical system.

From [15,16,20], it is known that the random variable

z

( )



is tempered and there exists a

t



-invariant set of full measure

  

such that for all



 

:

0

| (

) |

1

lim

0,

lim

(

)

0,

| |

t t

s

t t

z

ds

t

 

→

=

→



=

(3.1)

and for any





0

, there exists

 

( )



0

, such that

0

| (

z

 

_t

) |



  

( )

+

| |, |

t



t

z

(

 

_s

)

ds

|



  

( )

+

| | .

t

(3.2) Let

v s

( , )



=

e

−lz( s t− )

u s

( )

_{, then}

dv

= −

le

−lz( s t− )

u s dz e

( )

+

−lz( s t− )

du

_{. Using Langevin}

equation, combined with the original equation (1.3), we get

( ) 2 ( ) ( )

2 2 3

2

(

)

lz s t

|

lz s t lz s t

( , ),

[ , ),

xx x

dv

v

a lz v be

v

e

v

e

g x s in D

ds

 ₋  ₋ −  ₋



+  +

+ −

+

=

 

(3.3)

0

[ , ),

v

=  =

v

on D

  



(3.4)

( )

( , )

lz s t

( )

.

(4)

Equation (1.3)-(1.5) are equivalent to equation (3.3)-(3.5), by a standard method, it can be proved that the problem (3.3)-(3.5) is well posed in

H

₀2

( )

D

, that is, for every





and

2 0( )

v_H D , there exits a unique solution 2 0

([ , ), ( ))

vC



 H D (see e.g.[2,8,22]). Furthermore, the solution is continuous with respect to the initial condition

v

_ in 2

0( )

H D . To construct a non-autonomous random dynamical system

{ ( , , )}

V t

 

for problem (3.3)-(3.5), we

define 2 2

0 0

( , , ) :

( )

V t

 

H

D

→

H

D

by

V t

( , , )

 

v

_ . Then the system

{ ( , , )}

V t

 

is a non-autonomous random dynamical system in 2

0( )

H D .

We now apply abstract theory in Section 2 to obtain the random -pullback attractors for non-autonomous modified Swift-Hohenberg equation, by the equivalent, we only consider the random -pullback attractor of equation (3.3)-(3.5).

For convenience, the

L D

p

( )

_{norm of}

u

_{will be denoted by}

p



‖‖

,

H

=

L D

2

( )

with a scalar product and the norm of Sobolev spaces k( )

p

W D by

‖‖



_{k p}_, , we regard the space 2 0( )

H D endowed with the norm

|| ||

u

2,2

= 

‖ ‖

u

,

c

or

c

( )



denote the arbitrary positive constants, which only

depend on



and may be different from line to line and even in the same line.

For our purpose that the following Gagliardo-Nirenberg inequality will be used.

Lemma 3.1. (Gagliardo-Nirenberg Inequality). Let

D

be an open, bounded domain of the Lipschitz class in

n

. Assume that

1

      

p

,1

q

,1

r

, 0

 



1

and let

(

) (1

) .

n

k

m

p



q



r

− 

−

+ −

Then the following inequality holds:

1

,

( )

,

k p r m q

u



c D

u

−

u



‖ ‖

‖ ‖ ‖ ‖

Lemma 3.2.For all

t





, the following inequality hold:

2 ( ) 2

( , ,

_t

)

( )(

t

1

t

( )),

v t

  

_₋



c



e

− −

v

_

+ +

e

−

H t

‖

‖ ‖

(3.6)

and

2 ( ) 2 ( ) ₂ ₍ ₎ ₂

( )(

1

( )).

t t t s

t s t l z dr _t _t

xx

e

  

v

ds

c

e

 

v

_

e



H t





−

− +



₋ ₋ ₋



+ +



‖ ‖

(3.7)

Proof. Let

v s

( )

or

v

denotes

v s

( , ,

  

_{s t}₋

)

be the solution of equation (3.3)-(3.5). Taking the inner product of equation (3.3) with

v

, we get

( )

2 2 2 2 4

4

2 ( ) 4 ( )

4

1

2(

, ) (

)

( , )

2

( ( , ), ).

s t

s t s t

lz

x

lz lz

d

v

v v

a lz

v

be

v v

v

ds

e

v

e

g x s v

 

   

−

− − −

+ 

+ −

+

=

‖‖ ‖ ‖

‖‖

Using Young inequality, we get

2 2

1

| (2

, ) |

4

.

4

v v

v





‖ ‖



+

‖ ‖

By integration by parts, we obtain

( ) 2 ( ) 2 ( ) 2 2

( , )

(

)

,

s t s t s t

lz lz lz

x _D x _D xx x

be

 −

v v

=

be

 −



v vdx

= −

be

 −



v v

+

vv dx

thus

( ) 2 ( ) 2

( , )

.

2

s t s t

lz lz

x _D xx

b

be

 −

v v

= −

e

 −



v v dx

Applying the Holder inequality and Young inequality, we get

2

( ) 2 ( ) 2 ( ) 2 2 2 ( ) 4

4 4

| ( , ) | |

| |

|

,

2

16

s t s t s t s t

lz lz lz lz

x _D xx xx xx

b

be

 

v v

e

 

v v dx

e

 

v



v

e

 

v



(5)

and

( ) 2

1

2 ( ) 2

| ( ( , ), ) |

( ( , ) .

4

s t s t

lz lz

e

−  −

g x s v



‖‖

v

+

e

−  −

‖

g x s

‖

For convenience, we take

1

4

 =

,

| | 2

b



(

| | 4

b



, the same conclusion hold), we obtain

2

2 ( ) 2 ( )

2 2 2 4 2

4

1

2(

5)

2(1

)

( ( , ) .

4

2

s t s t

lz lz

d

b

v

a lz

v

e

v

e

g x s

ds

 − −  −

+ 

+

− −

+

−



‖‖ ‖ ‖

‖‖

‖

Taking





0

such that

a

− − 



5

0

, we have

2

2 ( )

2 2 2 4

4

2 ( )

2 2

2(

)

2(1

)

4

1

2(

5)

( ( , ) .

2

s t s t lz lz

d

b

v

lz

v

e

v

ds

a

v

e

g x s

   



− − −

+ 

+

−

+

−

 −

− −

+

‖ ‖ ‖ ‖

‖ ‖

‖

By the Sobolev imbedding

L D

4

( )



L D

2

( )

and Young inequality, we get

2

2 ( ) 2 ( )

2 2 4

4 4

2(

5)

2(1

)

.

4

s t s t

lz lz

b

a



v

c v

e

 −

v

ce

−  −

−

− −

‖‖



‖‖



−

‖‖

+

Thus we obtain

2 ( )

2 2 2 2

2(

)

lz s t

(1 ( ( , ) ).

d

v

lz

v

ce

g x s

ds

 



− −

+ 

+

−



+

‖‖ ‖ ‖

‖‖

‖

Multiply this by 2 2 ( )

s r t

s l z dr

e

 −



  − _{and integrating from}



_to

t

_{, we have}

0 0

2 ( ) 2 ( )

2 2

2 ( ) 2 ( ) ₂ 2 ( ) 2 ( ) 2 ( ) ₂

2 ( ) 2 ( ) ₂ 2 ( ) 2 ( ) 2 ( )

( )

(1

( , ) )

(1

t

r t s

t t

r t r t s t

s

r r s t

t s t

t s t l z dr

t

t l z dr t s l z lz

t

t l z dr t s l z lz

v t

e

v ds

e

v

e

g x s

ds

e

v

e

g

                _           _ − − − − − − − − + − − + − − + − − − + − − + −



+







+





+



‖

‖ ‖

‖

‖ ‖

‖

2

( , ) )

x s

‖

ds

From (3.2), we get

0 0

2

l



_₋_t

z

(

 

r

)

dr



 

( )

+



(

t

−



), 2

l



_{s t}₋

z

(

 

r

) 2 (

−

lz

 

s t−

)



 

( )

+



(

t

−

s

).

Then we have

2 ( ) 2 ( )

2 2

( ) 2 ( ) 2

( ) 2 2

( )

( )(

(1

( , ) )

)

( )(

1

( , )

).

t r t s

t s t l z dr

t

t t s

t

t t s

v t

e

v ds

c

e

v

e

g x s

ds

c

e

v

e

g x s

ds

        _      _



− − + − − − − − − −



+





+



+ +



‖

‖ ‖

‖

‖ ‖

‖

Thus we get the desired results.

Lemma 3.3. For all

t





, the following inequality hold:

11 22

( )

2 ( ) 2 ₃ ₃

8 11

3 3

1

( , ,

)

( )[(1

)

(

( )

)]

t t t s t t

v t

c

e

v

e

v

t

e

H t

e

H

s ds

        

  





− − − − − − − −





+

−

+



‖

‖ ‖

(3.8)

Proof. Taking inner product of equation (3.3) with



2

v

, we have

2 2 2 2 2

( ) 2 2 2 ( ) 3 2 ( ) 2

1

2(

,

) (

)

2

( ,

)

( ,

)

( ( , ),

).

s t s t s t

lz lz lz

x

d

v

a lz

v

ds

be

 −

v

e

 −

v

e

−  −

g x s

v

 + 

+  

+ −

+



+



=



‖ ‖ ‖

‖

‖ ‖

(6)

By the H

older

inequality, Young inequality and Gagliardo-Nirenberg inequality, we get

2

1

2 2 2

2(

,

)

8

,

8

v

−  



‖



‖

+

‖ ‖



11 13

( ) 2 2 ( ) 2 2 ( ) 8 2 8

4

16 22

( )

2 2 3 3

| |

| ( ,

) | | |

1

,

8

s t s t s t

s t

lz lz lz

x x

lz

b e

v

b e

v

ce

v

ce

v

        − − − −















+

‖ ‖‖

‖

‖ ‖‖

‖

‖ ‖

11 1

2 ( ) 3 2 2 ( ) 3 2 2 ( ) 2 4 2 4

6

16 22

5 11

( )

2 ( ) 2 4 4 2 2 3 3

| ( ,

) |

1

8

s t s t s t

s t s t

lz lz lz

lz lz

e

v

e

v

ce

v

ce

v

ce

v

          − − − − −











=







+

‖ ‖‖

‖

‖‖ ‖‖

‖

‖‖ ‖

‖

‖ ‖

( ) 2

1

2 2 2 ( ) 2

( ( , ),

)

2

( , ) .

8

s t s t

lz lz

e

−  −

g x s



v



‖ ‖

v

+

e

−  −

‖

g x s

‖

Putting all these inequalities together, we deduce

16 22

( ) _{2 (} ₎

2 2 2 ₃ ₃ 2

2(

)

2(8

)

(

s t s t

( , ) ).

lz _lz

d

v

lz

v

a

v

c e

v

e

g x s

ds

  _{ }



− − −



+

−



+ −



+

‖ ‖

‖‖

‖

Multiplying this by

(

)

2 2 ( ) s

r t

s l z dr

s

−



e

 −



  − _{and integrating it over}

( , )



t

_{, we get}

2 2 ( ) ₂ 2 2 ( ) ₂

16 22

( )

2 2 ( ) ₂ _{2 (} ₎ ₂

3 3

(

)

( )

[

(

)

(

( , ) )

].

t s

r t r t

s

s t r t

s t

t

t l z dr s l z dr

lz

t s l z dr _lz

t

e

v t

c

e

v ds

s

e

v

e

v

e

g x s

ds

               _{ } 



− − − − ₋ − − − ₋



−









+

−



+



‖

‖ ‖

‖

Then we have

2 ( ) 2 ( )

2 2

16 22

2 ( ) 2 ( ) ( ) 2 ( ) 2 ( ) 2 ( ) ₂

3 3

1

( )

[(1

)

( , ) ].

t

r t s

t t

r t s t r t s t

s s

t t s l z dr

t s l z dr lz

t t t s l z dr lz

v t

c

e

v ds

t

e

v

ds

e

g x s

               



− − − − − − − + − − + + − − + −







+



−



+



‖

‖ ‖

‖

By (3.2), (3.6) and the inequality

(

a b

+

)

r



c a

(

r

+

b

r

) ( ,

a b



0,

r



1)

, we get

16 22 22

2 ( ) 2 ( ) ( )

( )

3 3 3

11

( ) 2 3

11 22 11 11

( )

3 3 3 3

11 22

( )

3 3

( )

(

1

( ))

( )

(

1

( ))

( )(

1

t

r t s t s

t s l z dr lz

t t

t s

t

t s s s

s s

t

t s

t

e

v

ds

c

e

v

ds

c

e

v

e

H s

ds

c

e

v

e

H

s ds

c

e

v

     _                   



− − − − + + ₋ ₋ − − − − − − − − − −





+ +



+ +



+ +



‖ ‖

8 11 3 s 3

( )

).

t t

e



e



H

s ds



− −



Thus we have

11 22

( )

2 ( ) 2 3 3

8 11

3 3

1

( , )

( )[(1

)

1

(

( )

( ))

)].

t t s s t t

v t

c

e

v

e

v

t

e

H t

e

H

s ds

      







− − − − − − −





+

−

+



‖

‖ ‖

We complete the proof of Lemma 3.3.

Let be the set of all function

r

:

→

(0,

+

)

such that

lim

t 2

( )

0

t

e r t



→−

=

and denote by

the class of all families

D

ˆ { ( ) :

=

D t

t



}

such that

D t

( )



B r t

( ( ))

for some

r t

( )



,

B r t

( ( ))

denote the closed ball in 2 0

( )

(7)

8 11

2 3 3

1

( )

2 ( )[1

(

( )

)]

s t t

r t

c



e

−

H t

e

− 

H

s ds

−

=

+



(3.10) By lemma 3.3 for any

D

ˆ



and

t



, there exists



₀

( , , )

D t

ˆ





t

such that

1 0

( , ,

_t

)

( ),

.

v t

  

_₋

r t

for any

 







‖

Since 0 3

11





  , simple calculation imply that

r t

₁

( )



, which say that the

B r t

( ( ))

₁ be a family of random -pullback bounded absorbing sets in

H

02

( )

D

and

{ ( ( ))}

B r t

1



.

Theorem 3.1. The non-autonomous random dynamical system to problem (1.1)-(1.3) possesses a unique random -pullback attractor in 2

0( )

H D .

Proof. We need only prove that the dynamical system (3.3)-(3.5) satisfies the pullback flattening condition. Since

A

−1_{is a continuous compact operator in} 2

0( )

H D , by the classical spectral theorem, there exists a sequence { }



_j _j₌₁ satisfing

1 2

0



 





j



,



j

→ +

,

as j

→ +

,

and a family of elements

{ }

e

j j 1



= of

H

02

( )

D

which are orthonormal in

H

such that

,

.

j j j

Ae

=



e

 

j

+

Let

H

_m

=

span e e

{ , ,

₁ ₂

,

e

_m

}

in

H

and

P

_m

:

H

→

H

_m be an orthogonal projector. For any

v



H

we write

1 2

(

)

.

m m

v

=

P v

+ −

I

P v

= +

v

Taking inner product of （3.3） with



2

v

2 in

H

, we get

2 2 2 2 2

2 2 2 2

( ) 2 2 2 ( ) 3 2 ( ) 2

2 2 2

1

2(

,

) (

)

2

(|

| ,

)

( ,

)

( ( , ),

).

s t s t s t

lz lz lz

d

v

a lz

v

ds

be

 −

v

e

 −

v

e

−  −

g x s

v



+ 

+  

+ −

+





+



=



‖

‖ ‖

‖

‖ ‖

By theHolderinequality, Young inequality and Gagliardo-Nirenberg inequality, we get

2 2 2 2

2 2

1

2(

,

)

8

,

8

v

−  



‖



‖

+

‖ ‖

( ) 2 2 2 2 2 2 ( ) 4

2 2 4

2 ( )

2 2 4(1 ) 4

2

2 ( )

2 2 3

2

4 ( )

2 2 2 6

2

1

( ,

)

2

8

1

8

1

(

)

8

4

1

,

8

2

s t s t

s t

lz lz

x x

lz

be

v

b e

v

ce

v

ce

v

ce

v

   

   

 



− −

−







+





+



=



+



=





+



+

‖

‖ ‖

‖

‖ ‖

‖

‖‖‖ ‖

(8)

2 ( ) 3 2 2 2 4 ( ) 3

2 2 6

4 ( )

2 2 3(1 ) 3

2

4 ( )

2 2 2

2

8 ( )

2 2 2 4

2

1

( ,

)

2

8

1

8

1

(

)

8

3

1

,

8

2

s t s t

s t s t s t lz lz lz lz lz

e

v

e

v

ce

v

ce

v

ce

v

           



− − − − − −

−







+





+



=



+



=





+



+

‖

‖‖

‖

‖‖

‖ ‖

‖

‖‖‖ ‖

‖

‖ ‖

‖‖

( ) 2 2 2 2 ( ) 2

2 2

1

( ( , ),

)

2

( , ) .

8

s t s t

lz lz

e

−  −

g x s



v



‖



v

‖

+

e

−  −

‖

g x s

‖

Putting all these inequalities together, we have

2 2 2 2

2 2 2

4 ( ) 8 ( ) 2 ( )

2 6 4 2

2(

)

18

(

lz s t lz s t lz s t

( , ) ).

d

v

a lz

v

ds

v

c e

 −

v

e

 −

v

e

−  −

g x s



+ 

+

−







+

‖

‖ ‖

‖

‖ ‖

‖‖

‖

2 2 2

2 2

n

v



‖



‖ ‖

 

‖

, which imply that

2 2

4 ( ) 8 ( ) 2 ( )

2 6 4 2

(

2 )

(

s t s t s t

( , ) ).

n

lz lz lz

d

v

lz

v

ds

c

v

e

 

v

e

 

v

e

 

g x s



− − − −



+

−







+

‖

‖ ‖

‖‖

‖

Multiply this by

(

)

2 ( ) s ns l z r t dr

s

−



e

 −



  − _{and integrating from}



_to

t

_{, we obtain}

2 ( ) ₂ 2 ( ) ₂

2

2 ( ) _{4 (} ₎ ₆ _{8 (} ₎ ₄ _{2 (} ₎ ₂

(

)

[ (1

)

(

)

(

( , ) )

]

t s

n r t n r t

s

n r t _{s t} _{s t} _{s t}

t

t l z dr s l z dr

t s l z dr _lz _lz _lz

t

e

v

c

s

e

v ds

s

e

v

e

v

e

g x s

ds

             _{ } _{ } _{ } 



− − − ₋ ₋ ₋ − − − ₋



−





+ −





+

−

+



‖

‖ ‖

‖

Thus we get

( ) 2 ( ) ( ) 2 ( ) 4 ( )

2 2 6

2

( ) 2 ( ) 8 ( ) ₄ ( ) 2 ( ) 2 ( ) ₂

1

[(1

)

( , )

( )[(1

t t

n _s r t n _s r t s t

t t

n r t s t n r t s t

s s

t s t l z dr t s t l z dr lz

t s t l z dr lz t s t l z dr lz

v

c

e

v ds

e

v ds

t

e

v ds

e

g x s

ds

c

                     





− − − − − − − − + − + + − + + − + −







+



+

−



+



+



‖

‖ ‖

‖

( )( ) 2 ( )( ) 6

( )( ) 4 ( )( ) 2

1 2 3 4

1

)

( , )

]

( )(

)).

n n n n t t

s t s t

t t

s t s t

e

v ds

e

v ds

t

e

v ds

e

g x s

ds

c

I

           





− − − − − − − −



+

−

+

=

+ + +



‖ ‖

‖

By simple calculation, we find that there exists

N



,

 

n

N

,

  

_n

− 

, and

( )

( ) 2 2

1

(1

)

,

n

( )

0

,

t _{s t} _{s t}

I

e

v ds

e

v s

as n

t

  



− −

 +



 



→

→ 

−



‖ ‖

‖

According to Lebesgue dominated convergent theorem, we obtain

1

0

.

I

→

as n

→ 

Using (3.6), we get

( )( ) ( ) 2 3

2

( )( ) 3 ( ) 6 3 3

3 ( ) 4

6 3

(

1

( ))

(

1

( ))

1

[

( )]

0

,

4

n

t _{s t} _s _s

t t

t

n n n

I

c e

e

v

e

H s

ds

c e

e

v

e

H s ds

e

c e

v

H t

as n

(9)

.

( )( ) ( ) 2 2

3

( )( ) 2 ( ) 4 2 2

2 ( ) 3

4 2

(

1

( ))

(

1

( ))

1

[

( )]

0

,

3

n

t

s t s s

t _{s t} _s _s

t t

t

n n n

I

c e

e

v

e

H s

ds

c e

e

v

e

H

s ds

e

c e

v

H t

as n

    

 

    

 

  









  



− − − − −

− − −

−



+ +



+ +



+

→

→ 

−



‖ ‖

，

and

( )( )

2 2

4

( , )

,

( , )

0

,

n

t

s t

t s

I

e



e



g x s

ds e

 

g x s

as n



− − −





‖

→

→ 

Again using Lebesgue dominated convergent theorem, we get

( )( ) 2

( , )

0

.

n s t

e

 − −

‖

g x s

‖

ds

→

as n

→ 

In summary, we obtain that the terms on the right hand of inequality (3.13) tend to 0 as

n

→ 

, which say that

‖

v t

₂

( , ,

  

_₋_t

)

‖

→

0

, i.e., the random dynamical system (3.3)-(3.5) satisfies pullback flattening.

5.

4.Conclusions

This paper extends the existence of pullback attractor of non-autonomous modified S-H equation to the case of non-autonomous stochastic modified S-H equation with multiplicative noise. In the concrete experiment, the random term in the equation is more consistent with the actual problem. For S-H equation with multiplicative noise, the external force has exponential growth, we have proved that the equation exists a random -pullback attractor in one dimension. In the future work, we will continue to investigate whether the same results can be obtained when the spatial dimension is two-dimensional or n-dimensional.

Author Contributions: All the authors have equal contribution to this study. All authors have read and agreed to the published version of the manuscript.

Funding: This research was supported by the National Natural Science Foundation of China(11761044, 11661048) and the key constructive discipline of Lanzhou City University (LZCU-ZDJSXK-201706).

Conflflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

References

1. J.B. Swift, P.C. Hohenberg, Hydrodynamics ﬂuctuations at the convective instability, Phys. Rev. A, 1977, 15,

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