Random Pullback Attractor of a Non-Autonomous Local Modified Stochastic Swift-Hohenberg with Multiplicative Noise

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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 2 stratonovich sense. It is shown that a random -pullback attractor exists in H0 (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

223 uuuaubuut + ++++=2||0, (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 1 2 have some symmetry, for any uHD 0 (), the inner product (,)(,)=uuuu −  =‖u‖ , for any 2 2 2 2 u H0 () D , (,)(,)==u uu u ‖u‖ ,  is antisymmetry and  is 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 in H.k Recently for non-autonomous modiﬁed S-H equation:

duuu+(2| 22 au + |( b 3  , ))( + 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.

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

223 (1.3) duuuaubuug+++++−=(2(, ))( ),[ xxxx tdtludW ,), tin D 

uuxD==xx 0,, (1.4) (1.5) u x( , u ) ,x D= 2 Where D is a bounded open interval, Δu means uxx , and Δ u means uxxxx, |b|< 4 , a and l are arbitrary constants, W(t) is a two-sided real-valued Wiener process on a probability space which will 2 2 be speciﬁed later. For the external force g(x)∈ Lloc(R,L (D)), we assume that there exist M > 0 and β > 0 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 2 (1.3) exists a random -pullback attractor in H0 (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]) (,,,()) tt is called a metric dynamical systems if  :  →  is

((),)  -measurable,and 0 is the identity on  , stts+ = for all ts,  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 (,,,()) tt is a mapping (tXXtxtx ,  ,):,(  , ,,→→  )( , ,) , which represents the dynamics in the state space X and satisﬁes the properties (i) (,,) is the identity on ;

(ii) (tt ,,)(  ss ,,)(= ,,)  s− for all  st; (iii) → (tx ,,) is -measurable for all t  and xX . 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 (,)tD such that

(tDB ,  ,)  t−− (tt  ,)(  , )   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) At(,) is a random compact set;

(ii) is invariant; that is, for -a.s.  , and   t , (,,)(,)(,)t   A  = A t t−  ;

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

lim((,,)(,),(,))0.dtBA t −−tt =  →−

Where d is the Hausdorff semimetric given by dist( B , )=− supinf b a X . bB 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 BBtt ={(,):,} , for any   0 and   there exists a T B( ,t , )   and a finite dimensional subspace X such that

(i) PtB((,,)(,))−−tt is bounded, and  T

(ii)|| ()((,,)(,))IPtB− || −−ttX ,  T

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 sets BBtt ={(,):,} and ( , ) is pullback flattening, then there exists a random -pullback attractor ={(,):,}Att .

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  zedt(()) :()( = ),. − tt− We known from [6], it is the solution of Langevin equation dz+= zdt dW( t ). Wt() is a two-sided real-valued Wiener process on a probability space (,,) , where  =={(,):(0)0},C is the Borel algebra induced by the compact open topology of  , and is the corresponding Wiener measure on { , } . We identify()t with Wt(), i.e., W( tW )(== ,)( ttt ),. Define the Wiener time shift by

t (ssttt )()(=+− 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  : t |z (t ) | 1 lim== 0, limz (s ) ds 0, (3.1) tt→||tt → 0 and for any   0 , there exists ( ) 0, such that t |ztzdst (  ) |++  (  )|  |, |  (  ) | ( )| |. (3.2) ts0 −lz() −−lz()()  lz   Let v( seu , )( s= ) st− , then dv= − les−− t u() s dz + e s t du . Using Langevin equation, combined with the original equation (1.3), we get

dv 223 lzlzlz(  )2  (  )( ) − + v +2 v + ( a − lz )| v ++= bev|( , ),[s−−− ts  ts,ev t ), eg x s in D  (3.3) ds xxx v=  v =0 on  D  [ ,  ), (3.4) −lz()st− (3.5) v(,)(). x == v e u x in D

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Equation (1.3)-(1.5) are equivalent to equation (3.3)-(3.5), by a standard method, it can be 2 proved that the problem (3.3)-(3.5) is well posed in HD0 (), that is, for every   and 2 2 v H D 0 (), there exits a unique solution vCHD([,),()) 0 (see e.g.[2,8,22]). Furthermore, the 2 solution is continuous with respect to the initial condition v in HD0 (). To construct a non-autonomous random dynamical system {Vt ( , , ) } for problem (3.3)-(3.5), we 22 define VtHDHD(,,) :()() 00→ by V(,,) t v . Then the system {Vt ( , , ) } is a 2 non-autonomous random dynamical system in HD0 (). 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). p 2 For convenience, the LD() norm of u will be denoted by‖‖p , H L= D () with a scalar k ‖‖ 2 product and the norm of Sobolev spaces WDp () by  kp, , we regard the space HD0 () 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 1,1,1,01pqr    and let nnn km−−+−()(1). pqr Then the following inequality holds:

1− ‖ucDuu‖k,, prm q ()‖ ‖ ‖ ‖

Lemma 3.2. For all t  , the following inequality hold: 2()2 −−−tt ‖v( tceveH , ,)( −t )(1( t‖ )),+ + ‖ ‖ (3.6) and

t t 2 (s−+  tl ) 2() zdr tt− 2( )2 −−−tt  evdsceveHs t ‖ ‖ +( + )(1( )). ‖ ‖ (3.7)  xx 

Proof. Let vs() or v denotes vs(,,) st−  be the solution of equation (3.3)-(3.5). Taking the inner product of equation (3.3) with v , we get

1 d 22224 lz()st− ‖vvv‖+‖ v ‖ a +2( lz  v , ) + bev ()( −+ , v‖ ) v‖ x ‖‖4 2 ds

2lzlz (s−− ts  t )(  )4 − +=ev eg‖‖ x4 s v ( ( , ), ). Using Young inequality, we get

1 22 | (2,+v ) |4. vvv ‖ ‖ ‖‖ 4 By integration by parts, we obtain lz()()() 2 lz   2 lz   2 2 bes− t(,)(), v v= be s − t v vdx = − be s − t v v + vv dx xDD x xx x thus

lz()() 22b lz   bes−− t(,). v v=− e s t v v dx x2 D xx Applying the H o lder inequality and Young inequality, we get b b b2 belz(s− t  )| ( vve2 , ) |= | lz (  s − t  ) vvdxe 2 |  | | lz (  s − t ‖ ) vv‖‖‖ 2 ‖ v‖ 2 + e 2 lz (  s − t ‖ ) v‖ 4 , x2D xx 2 xx44 xx 16

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and

−−lzlz()2() 221 egxsvvegxss−− ts t | ((,),) |((,). +‖‖ ‖ ‖ 4 1 For convenience, we take  = , | |b 2  (| |b 4  , the same conclusion hold), we obtain 4 db2 1 ‖vvalzveveg‖22242++−−+−‖ ‖ x2(5)2(1)(( s ‖ , ).‖ 2lzlz ()2s−− ts  () t‖ ‖ − ‖ ‖ ds 424 Taking   0 such that a − −  50, we have 2 db2224 2()lz st− ‖vvlzvev‖++−+−‖ ‖ 2()2(1) ‖‖ ‖‖4 ds 4 1  −−−+2(5)((,).avegx s ‖‖22−2()lz st− ‖ ‖ 2 42 By the Sobolev imbedding L D( )L ( D)  and Young inequality, we get 2 224 b 2()2()lzlz − −−−−+2(5)2(1).avcvevce ‖‖ ‖‖ s−− ts t‖‖ 444 Thus we obtain

d 2222 −2()lz  ‖vvlzvcegx‖++−+‖ ‖ s 2()(1((,)). ‖‖ st− ‖ ‖ ds s 22()slzdr−  rt− Multiply this by e  and integrating from  to t , we have t t 222 (s−+  t ) 2() l zdrrt− ‖v() tev‖ + ds s ‖ ‖  tt t −−2 ( +−−tl  ) zdrt +− 2()2  s ( r l−−− tr ) zlz ts 2() t 2 () 22 ++eveg x sds ‖ ‖ s (1(‖ , ) ) ‖   00 t −−2( +−−tl  )2( zdrt +−  s )2(rrs l t )2(zlz )2()2 − 2 ++eveg  −−ts t ‖ ‖ (1 ‖ (,))x sds‖   From (3.2), we get 00 2l z () dr  ()( + t − ),2 l z ()2(  − lz  )  ()( + t − s ).  −−tr s t r s− t Then we have t t 222 ()s−+ 2() tl zdr rt− ‖v() tev‖ + ds s ‖ ‖  t −−−−()2()2tt s ++ceveg( )((1( x sds ,‖ ))) ‖ ‖ ‖   t +ceveeg( + )(1(−−− x()22 ,tts sds )).‖ ‖ ‖ ‖  

Thus we get the desired results.

Lemma 3.3. For all t  , the following inequality hold:

1122 −−()t 2( )2 1 −−t 33 ‖++v( tcevev , ,)( −t )[(1)‖ ‖ ‖ ‖ ‖ t − (3.8) 8 11 t −  s ++e−t ( H ( te )( ) H )] 33 s ds − 2 Proof. Taking inner product of equation (3.3) with  v , we have 1 d ‖v‖2 +‖  2 v‖ 2 +2(  v ,  2 v ) + ( a − lz‖ ) v‖ 2 2 ds (3.9)

lz(s− t  )2 2 2 lz (  s − t  ) 3 2− lz (  s − t  ) 2 +be(,) vvex  + (,) vve  = ((,),). gxsv 

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By the H o l d e r inequality, Young inequality and Gagliardo-Nirenberg inequality, we get 1 −+2(,)8,vvvv2222 ‖ ‖ ‖ ‖ 8 1113 lzlzlz()()()s−−− ts ts t 22222 88 |||bevvbevvcevv (,) | || xx ‖ ‖4‖ ‖ ‖‖‖ ‖ 1622 1 lz()st− +‖ 22vcev‖ 33‖‖ , 8 111 2lzlzlz ()2s−−− ts ()2 ts t () 323222 44 evvevvcevvv| (,) |  ‖‖6‖ ‖ ‖ ‖‖‖‖ ‖ 511 1622 1 lz()st− =+cevvvcev2lz ()st− ‖ 222‖44‖‖ ‖ ‖ 33‖‖ 8

−−lzlz()2() 2222 1 egx svvegxs−− ts t ((,),)2(,). s + ‖ ‖ ‖ ‖ 8 Putting all these inequalities together, we deduce 1622 d lz()st− ‖+−+−++vlzvavc‖2222 2()2(8)(( eveg‖ x ,‖ s )). ‖ ‖ 33‖‖ −2lz ()st− ‖ ‖ ds s 22()slzdr−  rt− Multiplying this by ()se−  and integrating it over ( , ) t , we get ts t 2t−− 2()22() l zdrs r−− tr l t zdr 22 ()(tev− )[ tcev ds‖ ‖ ‖ ‖ 

s 1622 t lz() 22()s−  l zdrrt− 22st− −2lz () +−++()((seveveg , ) ) x sds ]. ‖ ‖ 33‖‖ st− ‖ ‖  Then we have t t 221 −−2 ( +t s ) 2 l zdr () rt− ‖+v( tcev‖ )[(1) ds s ‖ ‖ t − 

t 1622 t tt−−2 ( ++t s ) 2 l  zdr ()()r−− ts t lz −−2 ( +−t s ) 2 l  zdr () lz 2 () s  r−− ts t 2 ++ev dseg x s 33‖‖ s ‖ ( ,‖ ) ].  rrr By (3.2), (3.6) and the inequality ()()abc++ (,0,1) aba br , we get t 16 22 22 tt−2 (t − s ) + 2 l z (   ) dr + lz (   )  r−− t s t −− ()ts es 3‖ v‖ 3 ds c() e‖ v‖ 3 ds  11 t c( ) e−t e  s ( e − ( s − ‖ ) v‖ 2 + 1 + e −  s H ( s )) 3 ds  

11 22 11 11 t −()ss −  −  c( ) e−ts e ( e3‖ v‖ 3 + 1 + e 3 H 3 ( s )) ds   11 22 8 11 −−()t t −  s c( )( e33‖ v‖ + 1 + e−t e33 H( s ) ds ).   Thus we have 11 22 1 −−()t ‖v( t ,‖ )2  c ( )[(1 + ) e−− (ts‖ ) v‖ 2 + e33‖ v‖ + 1 t − 

8 11 t −  s ++e−t ( H ( t ) e33 H ( s )) ds )]. − We complete the proof of Lemma 3.3. Let be the set of all function r :→ (0, + ) such that limet r2 ( t )= 0 and denote by t→− the class of all families Dˆ ={():} D t t such that D( t ) B ( r ( t )) for some rt() , B( r ( t )) 2 denote the closed ball in HD0 () with radius rt(). Let

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811 t −  s 2 −t rtceHteHsds( )2()[1((=++  )())] 33 (3.10) 1 − ˆ ˆ By lemma 3.3 for any D and t  , there exists 0 ( ,D , ) t t  such that

‖vtrtforany(,,)(),. −t ‖ 10

3 Since 0  , simple calculation imply that rt1() , which say that the B r( t(1 )) be a 11 2 family of random -pullback bounded absorbing sets in HD0 () and { (B ( r)) t1 }  .

Theorem 3.1. The non-autonomous random dynamical system to problem (1.1)-(1.3) possesses a unique 2 random -pullback attractor in HD0 ().

Proof. We need only prove that the dynamical system (3.3)-(3.5) satisfies the pullback flattening −1 2 condition. Since A is a continuous compact operator in HD0 (), by the classical spectral theorem,  there exists a sequence {} jj=1 satisfing

0,,,→12 +→ +jj asj  2 and a family of elements {}e jj=1 of HD0 () which are orthonormal in H such that + Aeejjjj= ,.

Let Hspaneeemm= {,,,}12 in H and Pmm H: H → be an orthogonal projector. For any vH we write

vP=+−=+ vIPvvvmm(). 12 2 Taking inner product of （3.3） with  v2 in H , we get

1 d 22222 ‖+vvvva2222‖ +‖ lzv +−‖ 2(,) () ‖ ‖ 2 ds

lzlzlz()2s−−− ts  ()() ts t  22322 − ++=bevvevveg(|| ,)( x sv ,)(222 ( , ),). By the H o l d e r inequality, Young inequality and Gagliardo-Nirenberg inequality, we get

2222 1 −+2(,)8,vvvv ‖ ‖ ‖ ‖ 228 1 −belz(s−− t  )( v2 ,  2 v ) ‖  2 v‖ 2 + 2 b 2 e 2 lz (  s t ‖ ) v‖ 4 xx28 2 4 1 ‖ 2v‖ 2 + ce2lz (st− ‖ ) v‖ 4(1−‖ )  v‖ 4 8 2

11 =‖ 2v‖ 2 + ce2lz (st− ‖ ) v‖ 3‖  v‖ () = 842 11 ‖ 2v‖ 2 +‖  v‖ 2 + ce4lz (st− ‖ ) v‖ 6 , 822

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1 −+evvvev2lzlz ()4s−− ts () t (,)232223 ‖ ‖ ‖‖ 226 8 1 +‖ 223(1)3vcevv‖ 4lz ()st− ‖‖ −‖ ‖ 8 2

11 =+=‖ 222vcevv‖ 4lz ()st− ‖‖‖ ‖ () 832 11 ++‖ 2224vvcev‖ ‖ ‖ 8lz ()st− ‖‖, 822

−−lzlz()2() 2222 1 egxs svvegx−− ts t ((,),)2(,). s + ‖ ‖ ‖ ‖ 228 Putting all these inequalities together, we have

d 2 2 2 2 ‖v2‖ +‖  v 2‖ +2( a − lz‖ )  v 2‖ ds 18‖ v‖2 + c ( e4lz (s− t ‖ ) v‖ 6 + e 8 lz (  s − t ‖ ) v‖ 4 + e− 2 lz (  s − t ‖ ) g ( x , s‖ ) 2 ). 222 n‖vv22‖ ‖ ‖ , which imply that

d 22 ‖+−vlzv22‖ (2)n ‖ ‖ ds +++cvevevegx‖((,)).‖2642 s4()8()2()lzlzlzs−−− ts ts‖ t ‖ ‖‖ − ‖ ‖ s nrslzdr− t 2() − Multiply this by ()se−  and integrating from  to t , we obtain ts t nrt−− tnr2()2()  l t zdrs −− l zdr 22 ()[tevcsev−+ (1) −ds‖ ‖ ‖ ‖ 2  s t nrs− t 2()  l zdr − 4lzlzlz ()8  ()2 ()642  − +−++()((seeveveg , ) ) x ] sds s−−− ts ts‖ t ‖ ‖‖ ‖ ‖  Thus we get tt tt 226 1 nr(s tnr−  t ts) +− t 2 l  ++ zdrs ()( −−− t ) 2 l zdr () lz 4 () ‖++vcev‖ dsev[(1) ds ss‖ ‖ ‖‖ 2 t −  tt tt nr(s ts−  tnrt ) ++− ts 2  l t +− zdr () −−−− lzs 8  ()( t l ) zdr 2 lz()42 2 () ++ev dsegss x s ds ‖‖ ‖ (,)‖  tt 1 ()(−  )()(s −−  ts ) t − 26 +c( )[(1 ) evnn dsev‖ ds+‖ ‖‖ t −  tt ++ev()(nn− dseg  )()(s −−  ts‖ ) xt ‖ −s42 ds ‖ ( ,‖ )] 

=+cIIII( +)()).1234 +

By simple calculation, we find that there exists N  , nN, n −   , and t 1  (st− ) 2 ()st− 2 I(1 + ) e‖  vds‖   , en ‖  vs (‖ ) → 0 asn →  , 1 t −  According to Lebesgue dominated convergent theorem, we obtain

Ias1 →→ n0.  Using (3.6), we get t I c e(n −− )(st ) ( e−(ss − ‖ ) v‖ 2 + 1 + e −  H ( s )) 3 ds 2   t c e(n −− )(st ) ( e−3 (ss − ‖ ) v‖ 6 + 1 + e − 3  H 3 ( s )) ds   −3 (tt −  ) − 4  −t ee631 c[ e‖ v‖ + + H ( t )] → 0 as n →  , n−44   n −   n − 

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t IceeveHsds++ ()()n −−st(1(−−−()22 ))ss‖ ‖ 3   t ()()−−st −−−2 ()422ss . ++ceeveHsdsn (1( )) ‖ ‖   −−−2 ()3tt −t ee421 ++→→c[( evHtas )]0,  n ‖ ‖ ， nnn−−−33 and t −ts 22()()−−st Ieegx→→ sdsegx  ‖ sas(,),(,)0, n ‖ n ‖ ‖ 4  Again using Lebesgue dominated convergent theorem, we get (−− )(st ) 2 en ‖ g( x , s‖ ) ds→ 0 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 ‖vt2 (,,)0 −t ‖→ , 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.

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