Random Attractors for Stochastic Reaction-Diffusion Equations with Distribution Derivatives on Unbounded Domains

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1 Alied Maheaics Published Olie Seeber 5 i SciRes h://wwwscirorg/oural/a h://dxdoiorg/436/a5659 Rado Aracors for Sochasic Reacio-Diffusio Equaios wih Disribuio Derivaives o Ubouded Doais Eshag Mohaed Ahed Ali Dafallah Abdelaid Lig Xu Qiaozhe Ma * College of Maheaics ad Saisics Norhwes Noral Uiversiy Lazhou Chia Eail: ahedesag@gailco aid_dafallah@yahooco @63co * aqzh@wueduc Received Augus 5; acceed Seeber 5; ublished 5 Seeber 5 Coyrigh 5 by auhors ad Scieific Research Publishig Ic his wor is licesed uder he Creaive Coos Aribuio Ieraioal Licese (CC Y h://creaivecoosorg/liceses/by/4/ Absrac I his aer we rove he exisece of rado aracors for a sochasic reacio-diffusio equaio wih disribuio derivaives o ubouded doais he olieariy is dissiaive for large values of he sae ad he sochasic aure of he equaio aears saially disribued eoral whie oise he sochasic reacio-diffusio equaio is recas as a coiuous rado dyaical syse ad asyoic coacess for his deosraed by usig uifor esiaes far-field values of soluios he resuls are ew ad aear o be oial Keywords Sochasic Reacio-Diffusio Equaio Rado Aracors Disribuio Derivaives Asyoic Coacess Ubouded Doai Iroducio he udersadig of he asyoic behavior of dyaical syse is oe of he os iora robles of oder aheaical hysics; oe way o aac he roble for dissiaive deeriisic dyaical syses is o cosider is global aracors his is a ivaria se ha aracs all he raecories of he syse Is geoery ca be very colicaed ad reflecs he colexiy of he log-ie dyaical of he syses I his aer we ivesigae he asyoic behavior of soluios o he followig sochasic reacio-diffusio equaios wih disribuio derivaives ad addiive oise defied i he sace : * Corresodig auhor How o cie his aer: Ahed EM Abdelaid AD Xu L ad Ma QZ (5 Rado Aracors for Sochasic Reacio-Diffusio Equaios wih Disribuio Derivaives o Ubouded Doais Alied Maheaics h://dxdoiorg/436/a5659

2 E M Ahed e al wih iiial daa where λ is a osiive cosa; ( ( λ ( ( du u u d= f u g x Dg d hd w ( D ( ( = u x = u x i ( x = is disribuio derivaives; g g L ( ( = ; f is a ; ad { } oliear fucio saisfyig cerai dissiaive codiios; h is give fucios defied o w = is ideede wo sided real-valued wieer rocesses o robabiliy sace which will be secified laer Sochasic differeial equaios of his ye arise fro ay hysical syses whe rado saio-eoral forcig is ae io accou I order o caure he esseial dyaics of rado syses wih wide flucuaios he coce of ullbac rado aracors was iroduced i [] beig a exesio o sochasic syses of he heory of aracors for deeriisic equaios foud i []-[5] for isace he exisece of such rado aracors has bee sudied for sochasic PDE o bouded doais; see eg [6] [7] ad for sochasic PDE o ubouded doais see eg [8] [9] ad he refereces herei I he rese aer we rove he exisece of such a rado aracor for sochasic reacio-diffusio Equaio ( defied i which is o fouded Noice ha he uboudedess of doai iroduces a aor difficuly for rovig he exisece of a aracor because Sobolev ebeddig heore is o loger coac ad so he asyoic coacess of soluios cao be obaied by he sadard ehod I he case of deeriisic equaios his difficuly ca be overcoe by he eergy equaio aroach iroduced by all i [] ad he eloyed by several auhors o rove he asyoic coacess of deeriisic equaios i ubouded doais his idea was develoed i [5] o rove asyoic coacess for he deeriisic versio of ( o I his aer we rovide uifor esiaes o he far-field values of soluios o circuve he difficuly caused by he uboudedess of he doais he ai coribuio of his aer is o exed he ehod of usig ail esiaes of he case sochasic dissiaive PDEs ad rove he exisece of rado aracor for he sochasic reacio-diffusio equaio wih disribuio derivaives o he ubouded doai he aer is orgaized as follows I Secio we recall soe reliiaries ad absrac resuls o he exisece of a ullbac rado aracor for rado dyaical syses I Secio 3 we rasfor ( io a coiuous rado dyaical syse Secio 4 is devoed o obai he uifor esiaes of soluio as hese esiaes are ecessary for rovig he exisece of bouded absorbig ses ad he asyoic coacess of he equaio I Secio 5 we firs esablish he asyoic coacess of he soluio oeraor by givig uifor esiaes o he ails of soluios ad he rove he esiaes of a rado aracor We deoe by ad ( he or ad he ier roduc i L ( ad use o deoe he or i L ( Oherwise he or of a geeral aach sace X is wrie as he leers C ad X Ci ( i = are geeric osiive cosas which ay chage heir values for lie o lie or eve i he sae lie Preliiaries ad Absrac Resuls As eioed i he iroducio our ai urose is o rove he exisece of a rado aracor For ha aer firs we will recaiulae basic coces relaed o rado aracors for sochasic dyaical syses he reader is referred o [6] []-[3] for ore deails Le ( X X be a searable Hilber sace wih orel σ-algebra (X ad le ( Ω FP be a robabiliy sace Defiiio ( Ω FP ( θ is called a eric dyaical syse if θ : Ω Ω is ( ( FF - easurable θ is he ideiy o Ω θs = θ θs for all s ad θ P P = for all Defiiio A coiuous rado dyaical syse (RDS o X over a eric dyaical syse ( Ω FP ( θ φ: Ω X X x φ x is a aig ( ( which is ( ( F ( X ( X easurable ad saisfies for P-ae ( φ( iy o X; ( φ( s = φ( θ s φ( s for all s (3 ( : X X all Hereafer we always assue ha φ is a coiuous RDS o X over Ω FP ( θ is he ideφ is coiuous for ( 79

3 E M Ahed e al Defiiio 3 A rado bouded se { ( } Ω of X is called eered wih resec o ( θ P-ae ( ( θ β li e d = for all β > where d( su x x X Defiiio 4 Le D be a collecio of rado subses of X ad { K( } Ω D he { K ( } = if for is called a rado absorbig se for φ i D if for every D ad P-ae here exiss ( > such ha ( ( K( ( φ θ θ for all Defiiio 5 Le D be a collecio of rado subses of X he φ is said o be D-ullbac asyoical- { } = ly coac i X if for P-ae φ( θ x ad x ( θ wih { ( } Ω D has a coverge subsequece i X wheever Defiiio 6 Le D be a collecio of rado suses of X he a rado se { A( } of X is called a D-rado aracor (or D-ullbac aracor for φ if he followig codiios are saisfied for P-ae ( A( is coac ad d( xa ( is easurable for every x X ; ( { A( } is ivaria ha is ( ( ( φ A = A θ for all ; (3 { A( } Ω aracs every se i D ha is for every { ( } = D ( φ( θ ( θ ( li d A = where d is he Hausdorff sei-eric give by ( d Y Z = su y Yif z Z y z for ay Y X ad Z X X he followig exisece resul for a rado aracor for a coiuous RDS ca be foud i [8] [3] Firs recall ha a collecio D of rado subses is called iclusio closed if wheever E ( is a arbirary rado se ad F ( is i D wih E( F( for all he E ( us belog o D Defiiio 7 Le D be a iclusio-closed collecio of rado subses of X ad φ a coiuous RDS o X over ( Ω FP ( θ Suose ha { K ( } is a closed rado absorbig se for φ i D ad φ is D-ullbac asyoically coac i X he φ has a uique D-rado aracor { A( } which is give by ( = φ( θ ( θ A K s I his aer we will ae D as he collecio of all eered rado subses of ( sochasic reacio-diffusio equaio i has a D-rado aracor L ad rove he 3 he Reacio-Diffusio Equaio o wih Disribuio Derivaives ad Addiive Noise wih iiial codiio ( ( λ ( ( du u u d = f u g x Dg d h d w x > (3 ( ( u x u x x = where λ is a osiive cosa g g L ( h ( ( H W are disribuio derivaive { } sace which will be secified below ad f C ( = (3 for soe D = w = are ideede wo-side real-valued wieer rocesses o a robabiliy wih he followig assuios: x 79

4 E M Ahed e al ( s s f s s α s s α (33 ( 4 s 4 s f s α3 s 3 s α (34 f ( s L (35 for s ad where L α α4 4 are osiive cosas ad λ > 4 Ω FP where I he sequel we cosider he robabiliy sace ( { ( ( : ( C } Ω= = = F is he orel σ-algebra iduced by he coac-oe oology of Ω ad P he corresodig wieer Ω F he we ideify wih easure o ( Defie he ie shif by ( he Ω FP ( θ ( ( ( ( ( ( W = for ( = ( ( Ω θ is a eric dyaical syse We ow associae a coiuous rado dyaical syse wih he sochasic reacio-diffusio equaio over ( Ω FP ( θ o his ed we eed o cover he sochasic equaio wih a rado addiive er i o a deeriisic equaio wih a rado araeer Give = cosider he Oe-diesioal Orseiuhlebec equaio he soluio of (36 is give by ( dz λz d = d w (36 λτ ( θ ( ( Noe ha he rado variable z ( is eered ad ( z = z θ λ e τ d τ for roosiio 433 i [] ha here exiss a eered fucio ( where r ( saisfies for P-ae = ( ( ( z z r( z θ is P-ae coiuous herefore i follows r > such ha (37 ( θ ( he i follows for (37 (38 ha for P-ae ( ( ( λ θ θ ( = Puig z( θ = hz = ( θ by (36 we have dz λzd = hd w r e r (38 λ z z e r (39 he exisece ad uiqueess of soluios o he sochasic arial differeial Equaio (3 wih iiial codiio (3 which ca be obaied by sadard Faou-Galeri ehods o show ha roble (3 (3 geeraes a rado syse we le υ( = u( z( θ where u is a soluio of roble (3 (3 he υ saisfies = υ λυ υ = f υ z θ g Dg z θ ( ( ( (3 y a Galeri ehod oe ca show ha if f saisfies (33-(35 he i he case of a bouded doai wih Dirichle boudary codiios for P-ae ad for all υ (3 has a uique soluio L 793

5 E M Ahed e al wih ( ( υ [ ( (( υ C L L ; H υ υ = υ for every > oe ay ae he doai o be a sequece of alls wih radius aroachig o deduce he exisece of a wea soluio o (3 o furher oe ay show ha υ( υ is uique ad coiuous wih resec o L for all Le υ i ( υ( ( θ ( ( u u = u z z he he rocess u is he soluio of roble (3 (3 we ow defie a aig φ : Ω L ( L ( by ( ( ( ( ( ( ( φ u = u u = υ u z z θ for all u Ω L (3 he φ is saisfies codiios (-(3 i Defiiio herefore φ is a coiuous rado dyaical syse associaed wih he sochasic reacio-diffusio equaio o I he ex wo secios we esablish uifor esiaes for he soluios of roble (3 (3 ad rove he exisece of a rado aracor for φ 4 Uifor Esiaes of Soluios I his secio we drive uifor esiaes o he soluios of (3 (3 defied o whe wih he urose of rovig he exisece of a bouded rado absorbig se ad he asyoic coacess of he rado dyaical syse associaed wih he equaio I aricular we will show ha he ails of he soluios ie soluios evaluaed a large values of x are uiforly sall whe he ie is sufficiely large We always assue ha D is he collecio of all eered subses of L ( wih resec o ( Ω FP ( θ he ex lea shows ha φ has a rado absorbig se i D Lea 4 Assue ha g g L ( ad (33-(35 hold he here exiss { K( } D such ha K = D ad P-ae { ( } Ω is a rado absorbig se for φ i D ha is for ay { ( } here is ( > such ha ( ( K( ( φ θ θ for all Proof We firs derive uifor esiaes o υ( u( z( θ ( we have = fro which he uifor esiaes o u Mulilig (3 by υ ad he iegraig over d υ λ υ υ = f( υ z( θ υd x ( g υ ( z( θ υ ( Dg υ d (4 For he oliear er by (33-(35 we obai ( υ ( θ υd f ( υ z( θ ( υ z( θ dx f ( υ z( θ z( θ dx f ( u udx f ( u z( θ dx 4 ( 4 ( f z x = = α u dx u dx α u z θ dx u z θ dx ( ( ( d ( α u u α u z θ x u z θ α u u α u C z( θ ( 4 u 4 z θ α u 4 ( ( ( 4 ( v z θ α u C z θ z θ α u 4 v C( z( θ z( θ o he oher had he ex wo ers o he righ-had side of (4 are bouded by (4 794

6 E M Ahed e al ( ( g υ z θ υ λ υ g z θ υ (43 λ he las er o he righ-had side of (4 is bouded by ad g = g = he i follows fro (4-(44 ha where ( g = g g Dg υ g υ g υ (44 d υ λυ α u ( 4 v d g g z( θ C ( z( θ z( θ λ Noe ha z( θ = hz ( θ ad h ( ( H W bouded as followig = y (39 we fid ha for P-ae i follows fro (45 (46 ha all which ilies ha for all Le λ λ 4 ( ( θ ( θ ( θ = (45 herefore he righ-had side of (45 is C z z C = P C (46 λτ ( ( τ 3 P θ C e r for all τ (47 d ( u 4 v P( C 4 d υ λ υ α θ (48 d ( 4 P( C4 d υ λ υ θ (49 = Alyig Growall s lea we fid ha for all y relacig by λ C υ( υ ( υ ( ( θ τ (4 λ ( τ 4 e e P τ d λ θ we ge fro (4 ad (47 ha for all C C υ θ υ θ υ θ (4 ( ( λ ( r( 3 4 e λ λ Noe ha φ( u( υ( u( z( z( θ = So fro (4 we ge ha for all ( u ( φ θ θ ( u ( z( z( = υ θ θ θ ( u ( z( z( u( θ z( θ Cr 3 ( C4 z( ( u( θ z( θ Cr 3 ( C4 z( λ e λ 4e υ θ θ θ y assuio { ( } D is eered O he oher had by defiiio ( herefore if u ( θ ( θ he here is ( > such ha for all ( (4 z is also eered 795

7 E M Ahed e al which alog wih (4 shows ha for all ( Give ( ( θ ( θ ( λ 4e u z Cr 3 C4 ( ( u( Cr 3 ( C4 z( φ θ θ (43 { ( } 3 4 ( = ( ( ( K u L : u Cr C z he { K( } Ω D furher (43 idicaes ha { K ( } Which colees he Proof We ex drive uifor esiaes for u i H ( ad for u i L ( Lea 4 Assue ha g L ( ad (33-(35 hold le { ( } ad P-ae he soluios u( u ( he for every of (3 wih υ( = u( z( saisfy for all is a rado absorbig se for φ i D = D ad u ( ( of roble (3 (3 ad υ( υ ( ( ( ( ( ( λ ( s λ e u s θ u θ ds e υ θ C r (44 ( θ ( θ υ ( θ ( ( λ ( s λ e u s u ds e C r (45 where C is a osiive deeriisic cosa ideede of r is he eered fucio i (37 Proof Firs relacig by ad he relacig by θ i (4 we fid ha Mulily he above by λ ( ( ( ad ( ( λ( s s ( e λ ad he silify o ge υ θ υ θ e υ θ e P θ d s C λ ( ( ( υ θ υ θ e ( ( λ( s λ( P s s C υ θ θ λ e e d e y (47 he secod er o he righ-had side of (46 saisfies λ ( s e P Fro (46 (47 i follows ha ( θ s ds λτ λ( λτ P( θ Cr 3 ( Cr 3 ( τ τ τ λ = e d e d e λ ( y (48 we fid ha for ( ( e υ θ υ θ ( Cr( λ( λ( λ e υ θ 3 e Ce λ λ ( s ( ( ( ( υ υ α e u s u ds ( ( ( ( ( λ( s λ s 4 e u s u ds e u s u ds ( ( ( ( λ( λ s λ( s υ υ θ s e e P ds C e d s (46 (47 (48 (49 Droig he firs er o he lef-had side of (49 ad relacig by θ we obai ha for all 796

8 E M Ahed e al ( ( ( ( ( λ( s λ s α e u s θ u θ ds e u s θ u θ ds ( ( θ λ ( s 4 θ e u s u ds λ ( C λτ ( ( P( τ e υ θ υ θ e θ d τ λ y (47 he secod er o he righ-had side of (4 saisfies for all (4 λτ λτ e P( θ τ dτ Cr 3 ( e d τ Cr 3 ( (4 he usig (4 ad (4 i follows fro (4 ha his colees he roof ( ( ( λ( s λ s λ λ ( s λ 4 ( ( α e u s θ u θ ds e u s θ u θ ds ( θ ( θ υ ( θ ( ( e u s u ds e C r ad (33-(35 hold Le = { ( } D ad u ( ( he for P-ae here exiss ( > such ha he soluios u( u ( ad υ( υ ( of (3 wih υ( = u( z( saisfy for all u( s θ u( θ ds C( r( Lea 43 Assue ha g g L ( where C is a osiive deeriisic cosa ad ( ( θ ( θ ( ( u s u ds C r r is he eered fucio i (37 Proof Firs relacig by ad he relacig by i (44 we fid ha Noe ha e ( ( ( λ( s λ of roble (3 (3 ( ( ( e u s θ u θ ds e υ θ C r (4 λ ( s λ e for s [ ] hece for (4 we have λ ( λ ( ( θ ( θ λ e d u s u s ( C( r( ( θ ( θ ( ( υ θ e u z C r e Sice u ( ad z ( are eered here is ( > such ha for all ( λ ( e u( θ z( θ C( r which alog wih (43 shows ha for all ( ( ( λ ( θ ( θ ( ( (43 u s u ds e C r (44 he fro (4 usig he sae ses of las rocess alyig o (45 we ge ha λ ( θ ( θ ( ( u s u ds e C r (45 he above uifor esiaes is a secial case lea 4 he he lea follows fro (44-(45 g L D u Lea 44 Assue ha g ( he for P-ae here exiss ( saisfies for all ( ad (33-(35 hold le = { ( } ad ( ( > such ha he soluio u( u ( of roble (3 (3 797

9 E M Ahed e al ( θ ( θ ( ( u s u ds C r where C is a osiive deeriisic cosa ad r ( is he eered fucio i (39 Proof Le ( be he osiive cosa i lea 43 ae ( ad s ( fid ha y (39 we obai u s ( θ u( θ ( s u( z( s = υ θ θ θ ( s u( z( s υ θ θ θ λ ( s r( ( s ( s = ( by (3 we (46 z θ C z θ Ce r Ce (47 Now iegraio (46 wih resec o s over ( by lea 43 ad iequaliy (47 we have ( θ ( θ 5 6 ( u s u d s C Cr (48 he he lea follows fro (48 Lea 45 Assue ha g g L ( ad (33-(35 hold le { ( } D > such ha for all he for P-ae here exiss ( ( θ ( θ ( ( u u C r where C is a osiive deeriisic cosa ad ( r is he eered fucio i (39 Proof aig he ier roduc of (3 wih υ L we ge ha d d i ( υ λ υ υ = ad u ( ( (49 f( u υd x ( g z( θ υ ( Dg υ = We esiaes he firs er i he righ-had side of (49 by (33 (34 we have ( υd = ( ( ( θ f u x f u u f u z dx f = ( u u dx f ( u z( θ dx u ( ( ( ( d ( 4 4 L u α u z θ x u z θ dx α4 α4 L u u dx z x 4 u 4 z ( θ d ( θ α α L u u z u z ( θ ( θ ( ( θ z( θ C u u u C z O he oher had he secod er o he righ-had side of (49 is bouded by ( g υ ( z( θ ( υ υ g z θ (43 (43 he las er o he righ-had side of (49 is bouded by Dg υ g υ g υ (43 ( 798

10 E M Ahed e al y (49-(43 we ge ha Le d d υ λ υ ( ( ( θ ( θ C u u u C z z g g z ( θ ( ( ( θ ( θ C u u u C z z ( θ ( θ ( θ ( (433 P = C z z (434 Sice z( θ = hz ( θ ad h ( ( H W such ha = which alog wih (39 shows ha y (433 (434 we have here are osiive cosas C ad C ( ( θ ( θ ( θ P C z z C = d d ( θ ( P Ce r C for all (435 ( ( θ υ C u u u P (436 Le ( be he osiive cosa i lea 43 ae ( ad s ( over (s o ge ha ( ( ( s ( ( d s τ ( ( ( ( ( ( υ υ υ υ θ τ ( C u τ u u τ u u τ u dτ s ( s ( ( d τ ( ( ( ( ( ( υ υ θ τ ( C u τ u u τ u u τ u d τ Now iegraig he above equaio wih resec o s over ( we fid ha ( ( ( s ( s ( τ ( ( ( ( ( ( υ υ υ υ d θ dτ ( C u τ u u τ u u τ u d τ Relacig by θ we obai ha ( ( υ θ υ θ υ( s θ υ( θ ds ( θ d τ τ ( ( τ θ ( ( ( θ τ θ θ ( τ θ u( θ d τ C u u u u u y lea 43 ad 44 i follows fro (437 ad (435 ha for all ( he iegrae 436 (

11 E M Ahed e al ( ( 3 4 ( ( s υ θ υ θ C Cr P θ ds λ s C 3 Cr 4 ( Ce r( c d s C5 Cr 6 ( he by 438 ad 39 we have for all ( which colees he roof ( θ ( θ υ( θ υ( θ ( ( ( z( C5 Cr 6 ( u u = z υ θ υ θ ad (33-(35 hold le = { ( } D ad u ( ( he for every > ad P-ae here exiss = ( > ad = ( > soluio υ( υ ( of (3 wih υ( = u( z( saisfies for all υ( θ ( ( υ θ x d x Lea 46 Assue ha g g L ( x Proof Choose a sooh fucio θ defied o such ha θ ( s for all s ad (438 such ha he θ ( s he here exiss a cosa C such ha ( s L ( ad iegraig over we fid ha for s = for s θ C for ay s ulilyig (3 by d x x θ υ dx λ d θ υ x ( υ θ υdx d x = θ f ( u υdx ( g z( θ θ υdx d Dg θ υ x We ow esiae he ers i (439 as follows firs we have x x x x ( υ θ υdx = d d θ υ x υθ υ x x x x = υ θ dx υθ υd x x Noe ha he secod er o he righ-had side of (44 is bouded by y (44 (44 we fid ha x x x υθ υdx υ θ υ dx C C υ υ d x ( υ υ x x C ( υθ υdx d θ υ x ( υ υ x θ υ i (439 (44 (44 (44 8

12 E M Ahed e al Fro (439 he firs er o he righ-had side we have x x x f u dx f u udx f u z d x θ ( υ = θ ( ( θ ( θ y (33 he firs er o he righ-had side of (443 is bouded by x θ f uux d α u θ dx u θ d x ( y (34 he secod er o he righ-had side of (443 is bouded by f ( u θ z ( θ dx x 3 3 ( d ( α u θ z θ x u θ z θ dx x α d ( d u θ x C z θ θ x x 3 z( θ θ dx 3 u θ d x he i follows fro (443-(445 we have ha 3 x f u θ υdx α u θ dx u θ dx ( ( ( ( x 3 θ θ θ θ u dx C z z d x For he secod er o he righ-had side of (439 we have g z θ θ υdx ( ( ( ( x λ θ υ dx d g z θ θ x λ For he las er o he righ-had side of (439 we have ha x θ υ = θ υ υθ Dg dx g dx g dx C g dx g dx υ θ υ C ( g υ λ θ d d g x θ υ x λ Fially by (439 (44 ad (447 (448 we have ha (443 (444 (445 (446 (447 (448 8

13 E M Ahed e al d θ υ dx λ d θ υ x d λ 3 x θ υ dx α u θ dx λ C 3 ( g υ υ u g g λ θ d x λ Noe ha (449 ilies ha ( ( ( ( C z θ z θ z θ θ d x λ d x θ υ dx λ d θ υ x d ( ( ( ( C z θ z θ z θ θ d x λ (449 C ( g υ υ g λ g θ d x (45 λ y lea 4 ad 45 here is ( ( ( H ( Now iegraig (45 over ( = > such ha for all ( ( υ θ υ θ C r we ge ha for all ( x λ θ υ( υ ( dx e θ υ ( υ ( dx C λ( s e ( g υ( s υ ( υ( s υ ( ds λ( s e g λ g θ dd xs λ ( ( ( ( λ( s e C z θ z θ z θ θ d xs d λ Relacig by θ we obai fro (45 ha for all ( x λ θ υ( θ υ( θ dx e θ υ ( θ υ( θ dx C λ ( s e ( g υ( s θ υ( θ υ( s θ υ( θ ds λ( s e g λ g θ dd xs λ ( ( θs ( θs z( s λ( s e C z z θ θ dd xs λ (45 (45 (453 8

14 E M Ahed e al I wha follows we esiae he ers i (453 Firs relacig by ad he relacig by (4 we have he followig bouds for he firs er o he righ-had side of (453 λ( x e θ υ ( θ υ( θ d ( υ ( θ P ( θs s C λ( λ λ( s e e e d ( λ( λτ C P λ( λτ ( C 3 ( υ θ θ τ λ e e e τ d ( λ e υ θ e e Cr dτ λ( λ λ( e υ( θ Ce Cr 3 ( e λ where we have used (47 y (454 we fid ha give > y lea 4 here is ( x here is ( λ( x e θ υ ( θ υ( θ d x 3 3 θ i (454 = > such ha for all (455 = > such ha he fourh er o he righ-had side of (453 saisfies C Ad hece here is R R ( λ( s C e υ( s θ υ ( θ ds ( r( = > such ha for all 3 ad R Firs relacig by s ad he relacig by side of (453 saisfies C λ( s e ( s ( d s υ θ υ θ (456 θ i (4 we fid ha he hird er o he righ-had C λ( s e ( s ( ds υ θ υ θ C λ C λ( s s λτ ( s C λ( s e ( ds e e P( d d e d s s υ θ θτ τ C λ C C s λτ ( e ( υ( θ e P( d ds θ τ τ C λ C C s λτ e ( υ( θ e P( θ d ds τ τ C e λ C C s ( υ( θ Cr 3 ( e d ds τ C λ C 8C e ( υ( θ Cr 3 ( λ his ilies ha here exis = ( > ad R R ( 4 4 λτ = such ha for all 4 ad R C λ( s e ( s ( d s υ θ υ θ (457 he he secod er o he righ-had side of (453 here exis = ( > ad R = R ( such ha for all 5 ad R3 we have ha C λ( s Cλ λ( e g e g (458 83

15 E M Ahed e al Noe ha g g L ( herefore here is R R ( λ = such ha for all R4 4 4 d g λ g x x For he five er o he righ-had side of (453 we have λ( s e dd g λ g θ xs λ λ( s e g λ g dd xs x λ λ( s e d s (459 Noe ha z( θ = hz ( θ ad h H ( W ( = ha for all R4 ad = x Hece here is R = R ( such ( ( ( ( λ h x h x h x x 4Cr ( r( d i 4 4 (46 where r ( is he eered fucio i (37 ad C is he osiive cosa i he las er o he righ-had side of (46 y (46 ad (37 (38 we have he followig bouds for he las er o he righ-had side of (453: x ( ( s ( s ( s ( s x C λ e zθ zθ zθ θ dd xs λ( s C e z z z xs ( ( θs ( θs ( θs dd ( ( θs ( θs ( θs = ( z ( θs z ( θs ds λ( s C e h z h z h z dd x s x λ λ( s e r ( = λ λ( s λ λτ e r( θs ds e r( d r θ r τ τ ( λ r ( λτ ( e r d τ Le = ( = ax { } ad R = R ( = { R R } ( (455-(46 ha for all 5 ad R5 we have which shows ha for all 5 ad R θ υ ( θ υ ( θ dx 5 (46 ax he i follows fro (453 υ( θ υ( θ x θ υ ( θ υ( θ x x his colees he roof Lea 47 Assue ha g g L ( he for every > d d 5 ad (33-(35 hold Le = { ( } D ad u ( ( ad P-ae here exiss = ( > ad R = R ( > such ha for 84

16 E M Ahed e al all u ( θ u ( θ ( x d x x R Proof Le ad R be he cosa i lea 46 y (46 ad (37 we have for all ad R ( = ( ( x R x R x R = ( = z dx h z dx h z dx z ( r he by (46 ad lea 46 we ge ha for all ad R which colees he roof 5 Rado Aracors ( θ ( θ = υ( θ υ( θ ( u u d x z dx x R x R ( ( ( υ θ υ θ x z x x R x R = d d 3 (46 I his secio we rove he exisece of a D-rado aracor for he rado dyaical syse φ associaed wih he sochasic reacio-diffusio Equaios (3 (3 o I follows fro lea 4 ha φ has a closed rado absorbig se i D which alog wih he D-ullbac asyoic coacess will ily he exisece of a uique D-rado aracor he D-ullbac asyoic coacess of φ is give below ad will be roved by usig he uifor esiaes o he ails of soluios g L ad (33-(35 hold he he rado dyaical syse ϕ is D- Lea 5 Assue ha g ( ullbac asyoically coac i L ( ; ha is for P-ae he sequece φ( θ u ( θ has a coverge subsequece i L ( rovided { ( } = D ad ( ( u θ θ Proof Le { ( } = D ad u ( ( θ θ we have ha Hece here is L ( { φ( θ u ( } ( θ L is bouded i = η such ha u o a subsequece ( u ( ( L { } he by lea 4 for P-ae φ θ θ η wealy i (5 Nex we rove he wea covergece of (5 is acually srog covergece Give > by lea 47 = ad R = R ( such ha for all here is ( Sice ha for all N x R here is N N ( ( ( φ θ u θ d x (5 = such ha for every N Hece i follows fro (5 x R ( ( O he oher had by lea 4 ad 45 here ( Le N N ( φ θ u θ d x (53 = such ha for all ( u ( C ( r( φ θ θ H = be large eough such ha for N he by (54 we have ha for all N (54 85

17 E M Ahed e al Deoe by R x : x R lows fro (55 ha u o a subsequece ( ( H ( ( ( φ θ u θ C r Q he se { } y he coacess of ebeddig H ( QR L ( QR which shows ha for he give > Noe ha L ( ( u ( ( L Q R φ θ θ η srogly i here exiss N N ( 3 = 3 such ha for all N3 ( u ( L ( QR η herefore here exiss R R ( (55 i fol- φ θ θ η (56 x R le R = ax { R R } ad N { N N N } 3 which shows ha 4 3 = such ha ( x η d x (57 = ax y (53 (56 ad (67 we fid ha for all N4 ( u ( L ( QR ( ( ( φ θ θ η x R3 x R3 ( φ θ u θ η d x φ θ u θ η dx 5 ( u ( ( L φ θ θ η srog i as waed Now we are i a osiio o rese our ai resul: he exisece of a D-rado aracor for φ i L ( heore 5 Assue ha g g L ( ad (33-(35 hold he he rado dyaical syse φ has a uique D-rado aracor i K i D by lea 4 ad is D-ullbac Proof Noice ha φ has a closed rado absorbig se { ( } asyoically coac i by lea 5 Hece he exisece of a uique D-rado aracor for φ follows fro roosiio 7 iediaely Foudaio er his wor was suored by he NSFC (334 Refereces [] Fladoli F ad Schalfuß (996 Rado Aracors for he 3D Sochasic Navier-Soes Equaio wih Mulilicaive Noise Sochasics ad Sochasic Reors h://dxdoiorg/8/ [] Aoci F ad Prizzi M ( Reacio-Diffusio Equaios o Ubouded hi Doais oological Mehods i Noliear Aalysis [3] Hale JK (988 Asyoic ehavior of Dissiaive Syses Aerica Maheaical Sociey Providece [4] Robiso JC ( Ifiie-Diesioal Dyaical Syses Cabridge Uiversiy Press Cabridge h://dxdoiorg/7/ [5] Wag (999 Aracors for Reacio-Diffusio Equaios i Ubouded Doais Physica D h://dxdoiorg/6/s67-789(9834- [6] Crauel H Debussche A ad Fladoli F (997 Rado Aracors Joural of Dyaics ad Differeial Equaios h://dxdoiorg/7/f95 [7] Caraballo Laga JA ad Robiso JC ( A Sochasic Pichfor ifurcaio i a Reacio-Diffusio Equaio Proceedigs of he Royal Sociey A h://dxdoiorg/98/rsa89 86

18 E M Ahed e al [8] aes PW Lu KN ad Wag X (9 Rado Aracors for Sochasic Reacio-Diffusio Equaios o Ubouded Doais Joural of Differeial Equaios h://dxdoiorg/6/de857 [9] Rosa R (998 he Global Aracor for he D Navier-Soes Flow o Soe Ubouded Doais Noliear Aalysis h://dxdoiorg/6/s36-546x( [] all JM (4 Global Aracors for Daed Seiliear Wave Equaios Discree ad Coiuous Dyaical Syses 3-5 h://dxdoiorg/3934/dcds43 [] Arold L (998 Rado Dyaical Syses Sriger-Verlag erli h://dxdoiorg/7/ [] Crauel H Debussche A ad Fladoli F (997 Rado Aracors Joural of Dyaics ad Differeial Equaios h://dxdoiorg/7/f95 [3] Crauel H ad Fladoli F (994 Aracors for Rado Dyaical Syses Probabiliy heory ad Relaed Fields h://dxdoiorg/7/f