ROBUST EXPONENTIAL ATTRACTORS FOR MEMORY RELAXATION OF PATTERN FORMATION EQUATIONS

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1 IJRRAS 8 () Augus ROBUST EXONENTIAL ATTRACTORS FOR EORY RELAXATION OF ATTERN FORATION EQUATIONS WANG Yuwe, LIU Yongfeng & A Qaozhen* College of ahemacs and Infomaon Scence, Nohwes Nomal Unvesy, Lanzhou Gansu 77, Chna. *Emal: maqzh@nwnu.edu.cn. ABSTRACT In hs pape, we pove he exsence of he obus exponenal aacos fo memoy elaxaon of paen fomaon, equaons n he phase-space and we mpove esuls n []. Key wods: aen fomaon equaons; Robus exponenal aacos.. INTRODUCTION Le [ ll, ] be a bounded doman wh smooh bounday,we consde he paen fomaon equaons u u u u g( u) (.) subjec o he Dchle bounday condon u ( ),. g C, g, We assume wh such ha g( ) c( )., (.) g () lm nf, (.) whee s he fs egenvalue of wh Dchle bounday condon. Accodng o [6] he long-em dynamcs beween he absac fs ode evoluon equaon u Au B u and he memoy elaxaon of (.4) (.4) u s Au s B[ u s ] ds (.5) ae close when s suffcenly small. ee, A: D A u u :[ ], and be a lbe space s a scly posve self-adjon lnea opeao wh compac nvese B : D( A ) s a nonlnea opeao. As dealed n [] hee s a complee equvalence beween (.5) and he followng sysem u ssds, (.6) T Au B[ u]. In hs pape, we se, A wh he doman 4 D A and 7

2 IJRRAS 8 () Augus Yuwe & al. Robus Exponenal Aacos fo emoy Relaxaon g u, B u hen he memoy elaxaon of equaon (.) s equvalen o he followng evoluon sysem u s sds, (.7) T Au A u u g u. The paen fomaon equaon poays he chemcal eacon and he flame combuson phenomenon (see [.] fo deal). Exsence of he global aacos fo he equaon (.) has been suded by A. ION n []. The global aaco s compac, fully nvaan and aacve fo he semgoup, bu may aaco ajecoes a a slow ae. Convesely, an exponenal aaco s a compac se, posvely nvaan unde he acon of he semgoup, whch aacs exponenally fas ajecoes depang fom he bounded ses. Thus exponenal aacos ae expeced o be moe obus han global aacos unde peubaons. In hs pape, we sudy he exsence of he obus exponenal aaco fo (.7) n he phase-space. I s woh nong ha he exponenal aacos aac bounded subses of he whole phase-space. The man esuls of hs pape s Theoem.8.. RELIINARIES Le, and denoe he nne poduc and he nom n L, especvely. Fo, he ( ) / heachy of compacly nesed lbe spaces D( A ) wh he sandad nne poducs Noce ha ( ) ( ) u, u A u, A u.. Idenfyng wh s dual space, hee holds., s s, fo ae.. s Denong (, ), we assume ha W s nonnegave,and ha he exponenal decay condon holds fo some. Fuhemoe,fo,, we se s Then, we noduce he weghed lbe spaces (.) s, hen we have s ds, s sds. (.) L ;, endowed wh he nne poducs, s s, s ds. We make use of he nfnesmal geneao of he gh-anslaon semgoup on, he lnea Opeao Ts s ( s beng he dsbuonal devave wh espec o s ) wh doman On accoun of (.), hee holds (see [6,]) Fo D T :,. s,,we defne he poduc Banach spaces T,, DT. (.), f, (.4) f. When =, we nepe he pa ( u, ) as u, and he nom educes o he fs componen only. We shall need he lfng map L : and he pojecon map Q on, gven by 8

3 IJRRAS 8 () Augus Yuwe & al. Robus Exponenal Aacos fo emoy Relaxaon Lu = ( u,), Q ( u, ) =. We ecall he compac and dense njecons n [] s DA ( ) DA ( ), s, (.5) and he connuous embeddng hen fom If, Fo, and R we se s ( ) DA.7, we have he followng nequaly 6 s L, u { : }. B R z z R s [, ). (.6). (.7) Defnon. [6] A famly { }, of compac subses of s sad o be a obus famly of exponenal aacos f he followng condons holds s posvely nvaan fo S. () Each () Thee exs and a posve nceasng funcon such ha, fo evey R hee holds ds S B R, R e. () The facal dmenson of n s unfomly bounded wh espec o. (v) Thee exss a connuous nceasng funcon Θ : [; ] [; ) wh Θ() = such ha sym ds (, L ) ( ). sym whee ds and ds denoe he nonsymmec and symmec ausdoff dsance beween ses espec- vely. Fo a fxed, () () ae nohng bu he usual condons defnng an exponenal aaco (bu obseve ha, conay o he ognal defnon see [9] fo deal) we eque he aacon popey on he whole phase-space), whle condon (v) chaacezes he obusness popey. Gven z ( u, ) ( z u f ), we denoe he weak soluon a me o (.7) wh nal daa z as ( u( ), ), f, S z (.8) u( ), f. Condons on B. We assume ha B :. Fo evey R,hee exss C = C(R) such ha sup B u B u C u u, (.9) Assumpon (S). Fo evey [,] phase-space Wheneve z R. u R u R sup B u C. sysem (.7) geneaes a songly connuous semgoup S oeove, fo evey gven R,hee exss K = K(R) such ha K, (.) on he S z S z Ke z z (.) 9

4 IJRRAS 8 () Augus Yuwe & al. Robus Exponenal Aacos fo emoy Relaxaon Theoem. [6] Le he followng assumpons hold. ( ) fo,, hee exss R such ha B R s an absobng se fo S on, unfomly wh espec o.namely, gven any R, hee exss a me,dependng only on R,such ha S R R,. oeove, fo evey R,hee exssc C R, ( ) Thee exs fo some and. ( such ha sup S z C, z R R such ha R sasfes. ds S R, R e. )Gven any R > hee exs, and, :, (possbly dependng on R) wh z, z, he map S adms he decomposon S z S z L z, z N z, z, L and N sasfy L z, z z z, fo lage enough such ha fo evey Whee Besdes N z, z N z, z z z. fulfls he Cauchy poblem fo some w sasfyng, fo all T >, T,,, w z z (, T)., Then,hee exss a famly of obus exponenal aacos. Remak. Condon () wh s acually mpled by he songe condon (4) Fo evey R, hee exss C = C(R),such ha sup B u B u C u u. u R Lemma.[4] Le X be a Banach space,and le C, X sup E z m and Le E : X be a funcon such ha E z, fo some m, and evey z. In addon, assume ha fo evey z he funcon connuously dffeenable and sasfes he dffeenal nequaly d E z z k X d Fo some and k boh ndependen of z hen, E z sup E : k m,. X X k a E z s

5 IJRRAS 8 () Augus Yuwe & al. Robus Exponenal Aacos fo emoy Relaxaon Lemma.4 [5] Le be an absoluely connuous posve funcon on dffeenal nequaly d g( ) ( ) h, d fo almos evey, whee g and h ae funcons on, such ha fo some m and fo some m. Then fo some g y dy m,,,, and sup h y dy m, e,, m, and m e / e. Lemma.5 [7] Le,, be subses of X such ha X v,, X ds S L e whch sasfes fo some v ds S, L e, fo some v, v and L, L. Assume also ha fo all z, z S( ) K j ( j,,) hee holds v S( ) z S( ) z L e z z fo some v and some L. Then follows ha X X v vv ds X S, Le, whee v v v v. ROBUST EXONENTIAL ATTRACTORS IN and L L L L. he We need o vefy condons on B, assumpons (S), and (), (), () n heoem.. In fac, B[u] = g(u) s easly seen o fulfl condons (.9),(.) and (4). By means of he Galekn scheme adaped o sysems wh memoy (see [8] fo deal), one can show ha, fo evey,, sysem (.7) geneaes a songly connuous semgoup S on he phase-space Lemma. Fo evey gven R,hee exs K = K(R) such ha S ( ) z S ( ) z K Ke z z, (.) Wheneve z R.. oof : Se u% u u, ( s) ( s) ( s), hen we ge u ( s) ( s)ds =, = / T Au A u u g( u) g( u). ulplyng (.) by Au%n and (.) by n% n, we oban d / ( u ), =,, ( ) ( ), T A u u g u g u d. (.) (.)

6 IJRRAS 8 () Augus Yuwe & al. Robus Exponenal Aacos fo emoy Relaxaon Because of (.),(.) and (.5), we have he followng esmaes A / u, u, ( s) A c( u / u ds ), ( s) ds c( u ), g( u) g( u), cg( u) g( u) ( ( ) ( ) c g u g u ) ( c u g ( u ) g( u ), ), c( u ence ( ) ( u c u d By Gonwall lemma,we complee he poof. d ), ). (.4) Lemma. Condon () holds fo =. oof : Fo any gven z u v (, ), we consde he funconal E( z) z u u G( u), u,, g whee G() = ( ) d, fo some whch wll be chosen small enough so ha he followng esmaes hold. Fom (.) and (.), hee exss (,) such ha g ( u ), u ( ) u c, (.5) G( u), ( ) u c, (.6), u. (.7) 4 The las wo nequales, ogehe wh (.), yeld c E( z) c( z ). (.8) We now fx z wh z R, and we denoe ( u( ), ) S ( ) z. ulplyng (.7) by Au A u u g( u) n ence,we ge and (.7) by n, we oban u, Au u, A u u, u u, g( u), u T,. d u u u G u u s s ds d ( (( ), ) ( ) ( ). (.9)

7 IJRRAS 8 () Augus Yuwe & al. Robus Exponenal Aacos fo emoy Relaxaon Besdes, ecallng (.), and mulplyng (.7) by u n, s saghfowad o oban d u u T u u u u g u u d Theefoe, E E( S ( ) z) sasfes,,, ( ),. (.) d E ( s ) ds u u u d (.) g ( u ), u ( s ) u, ( s ) ds ( s ) ( s ) Usng (.5) (.7), we oban d ( E ( S ( ) z )) u ( ) ( ) s s ds d u u ( s) u, ( s) ds ( s) ( s) c. (.) Because of (.7) and (.7), he ems on he gh-hand sde can be conolled as u u u, ( s ) u, ( s ) ds ( ) ( ), u s s ds c ( ) ( ) s s c. 4 By he foce of (.), and choose small enough we ge d E ( S ( ) z ) S ( ) z. c (.) d 8 In vew of (.8), fom lemma., hee exss ( R) such ha E ( S ( ) z ) sup{ E ( ) : c },. Usng (.8), and subsequenly negang (.) on (, ), we mee he clam. Remak. In vew of (.7), and negang (.) wh on, we fnd he followng Inegal esmae fo some ( R). Lemma.4 Unde he above assumpons,hee exs such ha u A u C,. u ( y) dy, (.4)

8 IJRRAS 8 () Augus Yuwe & al. Robus Exponenal Aacos fo emoy Relaxaon oof : ulply (.) wh Au n, we have d A u Au A u g ( u ), Au A u, Au d. (.5) we have he followng conols ( ), ( ) ( ( ) ) Au g u Au Au g u dx Au g u dx g( u) dx, 4 ence we have because of (.), we ge Au A u, Au A u Au dx Au A u A u. 4 6 d d A u g u dx 6 g( u) dx C ( u ) dx C C 6 u. L We se A = subjec o he Dchle bounday condon hen we have D( A) = ( ) ( ) = D( A ) L ( ) = D( ), A Lemma.5 Thee exs and an nceasng funcon such ha, f S ( ) z e ( z ) And u ( y) dy ( ) Fo some ( R) and ( R). oof :Fo small enough we noduce he enegy funconal z ( ), (.6) Rfo some., hen E( ) S ( ) z u u g( u), u u, c. Choose he consan c appeang n E () lage enough s appaen ha S ( ) z E( ) S ( ) z c. (.7) ulplyng (.7) by, hs yelds Au A u u g( u) n, n and (.7) by n, n d E ( ) ( s ) ( s ) ds u d u u g( u), u g ( u ) u, u ( s ) ( s ) ( s) u, ( s) ds. (.8) 4

9 IJRRAS 8 () Augus Yuwe & al. Robus Exponenal Aacos fo emoy Relaxaon Agung as n he pevous poof and akng no accoun lemma., we have ence,we oban oeove, usng he embeddng ( s) ( s) ( s) u, ( s) ds ( ) ( ). s s c (.9) d E ( ) E ( ) ( s ) ( s ) ds d g ( u) u, u u, c. (.) D( A ) L ( ) L L. we have he conols g( u) u, u c( u ) u A u ( ) c ( u ) u A u L Explong(.) and c( u ) u u (.) u, c, povded ha s small enough we end up wh d E E c c u c u d 8 E. (.) Because of (.4), we apply lemma.4 o ge ( ) E( ) ce() e c. Fnally, negang he dffeenal nequaly (.) wh on (, ), and usng (.7), we complee he poof. Remak.6 In pacula, fo, sasfed fo = as well. seng R ( R) and C ( C), we see ha condon () s Remak.7 Fom he esmaes of (.9) we know ha he sobolev embeddng s he maxmal bu how o conol a hghe gowh condon han (.) s open. In ode o pove () analogously o wha obseved n [] a funcon g C ( R) such ha (.),(.) adms a decomposon g g g sasfyng Nex, fo any fxed z B R g ( ) c ( ), g (), g ( ), g( ) c. ( ), we make he decomposon S ( ) z D ( ) z K ( ) z whee D ( ) z ( v( ), ) and K ( ) z ( w( ), ) ae he soluons o he poblems (.) 5

10 IJRRAS 8 () Augus Yuwe & al. Robus Exponenal Aacos fo emoy Relaxaon and v ( s) ( s) ds, T Av A v v g( v), D () z z w ( s) ( s) ds, T Aw A w w g( u) g( v), K () z (.4) (.5) By a slgh modfcaon of lemma., can be shown ha hee exs and such ha D z ( ) e, Lemma.8 Gven [, ], f z R, hee exss ( R) such ha 4 4 K ( ),. 4 (.6) oof : Fom he pevous esuls we know K ( ) z c. Fo small enough We noduce he enegy funconal E ( ) K ( ) z w w g ( u ) g ( v ), w w, c. Choose he consan c appeang n E () lage enough s appaen ha K ( ) z E( ) K ( ) z c. (.7) Agung as n he pevous poof we oban d E ( ) E ( ) ( g ( u ) g ( v )) u, w d 4 4 Due o (.),(.5) and ( ) g ( u ) u, w g ( v ) w, w c. (.8) D( A ) L ( ), we ge L L L4 ( g( u) g( v)) u, w c w u A w 4 8 c u A w c A w u A w 6

11 IJRRAS 8 () Augus Yuwe & al. Robus Exponenal Aacos fo emoy Relaxaon c u w c u E, on accoun of (.),(.5), we oban g ( v) u, w c u A w L 4 8 c u A w c u c u E, because of (.5),(.7),(.), hs yelds 4 4 L L4 g( v) w, w c v w A w c v w w cv L4 L4 ence d E ( ) E ( ) he h c, d whee h c v ( ) c u. Due o (.4) lemma.4 and he nequaly, we ge Fo some c. Accodng o lemma.4,we mee he clam. E. h ( y ) dy c [ ( ) 4 ], Lemma.9 Condon () holds. oof : Successve applcaons of lemma.6 fo,,,, ogehe wh he exponenal decay (.4), 4 4 we consuc a sequence of fve balls sang fom B ( R ), each exponenally aaced by he nex one. Thus on accoun of he connuous dependence esmae (.), he ansvy of he exponenal aacon lemma.5 enals he desed popey (). Ou man heoem eads as follows. Theoem. Assume ha g C ( ), and sasfes (.),(.) wh g(). Then hee exs obus exponenal aacos { } fo he semgoup of opeaos S () geneaed by sysem (.7). As a saghfowad consequence we have he followng coollay. Coollay. The semgoup S () acng on he phase-space possesses a conneced global aaco A. In pacula he facal dmenson of A n s unfomly bounded wh espec o. 7

12 IJRRAS 8 () Augus Yuwe & al. Robus Exponenal Aacos fo emoy Relaxaon REFERENCES [].. aken Advanced Syneges Insably heaches of self-oganzng sysems and devces Spnge- elag. Beln (98); []. A. C. Newell T. asso J. Lega Ode aamee Equaons fo aens Annual Revew of Flud echancs. 5 (99); []. A. Ion A. Geogescu On he exsence and on he facal and hausdoff dmensons of some global aaco Nonlnea Analyss TA 8 (997): [4].. Belle. aa Aacos fo semlnea songly dampedwave equaon on R Dscee Conn. Dynam. Sysems 7 (): [5].. Gassell. aa Asympoc behavo of a paabolc-hypebolc sysem Commun. ue Appl. Anal. 8 (4):49-8. [6]. S. Ga. Gassell A. anvlle and. aa emoy elaxaon of fs ode evoluon equaons Nonlneay.8 (5): [7]. Fabe. Galusnsk C. anvlle A. Zelk S. Unfom exponenal aacos fo a sngulaly peubed damped wave equaon. Dscee Conn. Dynam. Sysems (4):-8. [8]. Gassell and aa. Exsence of a unvesal aaco fo a fully hypebolc phase-feld sysem. J. Evol. Eqns 4 (4):7-5. [9]. A.EdenC.FoasB.NcolaenkoR.Temam Exponenal Aacos fo Dsspave Evoluon Equaons New Yok:asson.as:Wely(994). []. R.Temam Infne-dmensonal dynamcal sysems n mechancs and physc Spnge New Yok(997). []. Aea J. Cavalho A.N. ale J.K. A damped hypebolc equaon wh ccal exponen Comm. aal Dffe. Eqs.7 (99): []. Gassell. aa. Unfom aacos of nonauonomous sysems wh memoyevoluon Equaons Semgoups and Funconal Analyss (ogess n Nonlnea Dffeenal Equaons and he Applcaons vol 5) ed A Loenz and B Ruf (Boson A:Bkhause)():

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( )

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( ) 5-1. We apply Newon s second law (specfcally, Eq. 5-). (a) We fnd he componen of he foce s ( ) ( ) F = ma = ma cos 0.0 = 1.00kg.00m/s cos 0.0 = 1.88N. (b) The y componen of he foce s ( ) ( ) F = ma = ma