THE FINITE HAUSDORFF AND FRACTAL DIMENSIONS OF THE GLOBAL ATTRACTOR FOR A CLASS KIRCHHOFF-TYPE EQUATIONS

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1 European Journal of Maheaics and Copuer Science Vol 4 No 7 ISSN HE FINIE HAUSDORFF AND FRACAL DIMENSIONS OF HE GLOBAL ARACOR FOR A CLASS KIRCHHOFF-YPE EQUAIONS Guoguang Lin & Xiangshuang Xia Deparen of aheaics Yunnan Universiy Kuning 659 CHINA gglin@ynueducn @qqco ABSRAC In his paper we sudy he Hausdorff and Fracal diensions of he global aracor for a class Kirchhoff-ype equaions wih srongly daped ers and source ers We esify he dynaical syse associaed wih above-enioned odels is Freche differenial under suiable condiions Furherore we obain a precise esiae of Hausdorff and Fracal diensions of he global aracor Keywords: Kirchhoff-ype equaions; he global aracor; Freche differenial; Hausdorff and Fracal diensions Maheaics Classificaion:35B437L3 INRODUCION Guoguang Lin and Xiangshuang Xia [] sudied he exisence of a global aracor for a Kirchhoff ype equaions wih srongly daped ers and source ers Furherore In his paper we consider he finie Hausdorff and Fracal diensions of he global aracor for he above enioned odels as following: u M ( u v u u g ( u v f ( x ( v M ( u v ( v ( v g ( u v f ( x ( u( x u( x u ( x u( x x (3 v( x v( x v ( x v( x x (4 u( x v( x x (5 where is a bounded doain in R n wih a sooh boundary is a consan and fi ( x( i is a given source er Moreover M ( u v is a scalar funcion hen he assupions on M and gi ( u v will be specified laer Generally speaking he Hausdorff diensions has been sudied by soe auhors In 99s S Zhou[] obained a ore precise esiae of he upper bound of he Hausdorff diension of aracor for srongly daped nonlinear wave equaions u u u h( u f ( u g x u( x x u( x u ( x u ( x u ( x x (6 Meanwhile He also obained Hausdorff diension decreases as he srong daping grew for large daping Laer Xiaoing Fan and S Zhou [3] go he Hausdorff diension of Kernel secions which Progressive Acadeic Publishing UK Page 94 wwwidpublicaionsorg

2 European Journal of Maheaics and Copuer Science Vol 4 No 7 ISSN decreases as he srong daping grew for large srong daping under soe condiions paricularly in he auonoous case In recenly years he sudies abou he esiae of diensions is developing In 4 Meixia Wang Cuicui ian and Guoguang Lin [4] go he upper bound esiaion of he Hausdorff and Fracal diensions of aracor u uxxxx uxx ux (7 In 7 Yunlong Gao[5] considered he Hausdorff diension of a global aracor for a class of srongly daped Higher-order Kirchhoff ype equaions: (8 q u ( u ( u ( u g( u f ( x ( x [ u( x u ( x u ( x u ( x x (9 i u u( x i L x ( ( i v hanks o he previous resuls of he Hausdorff we invesigae he Hausdorff diension of he high and low order coupled Kirchhoff-ype equaions Firsly we obain he soluion s seigroup is Freche differenial hen we esablish a ore precise esiae of he Hausdorff and Fracal diensions he Esiaes of he Hausdorff Diensions for he Global Aracor In his paper soe inner produc nors abbreviaions and soe assupions( H ( H6 and noaions needed in he proof of our resuls is refer o[] Differeniabiliy of he seigroup In order o esiae diensions we assue: ( H7 : here exiss such ha d ( d M ( u v d ( d ( and M ( u v C ( ( H 8 :For every M here exis k k( M such ha : g u u v g u v k u u v v ( iu ( ( iu ( ( g u v v g u v k u u v v (3 iv( ( iv( ( for anyu u H ( ; v v H ( ; u u M v v M and ( he inner produc and he nor in E space are defined as follows: i ( ui vi pi qi E( i we have ( E ( u u ( v v ( p p ( q q (4 u v p E q (5 Progressive Acadeic Publishing UK Page 95 wwwidpublicaionsorg

3 European Journal of Maheaics and Copuer Science Vol 4 No 7 ISSN (3 (3 Seing ( u v p q E p u u q v v in{ 5 (5 6 5 (5 6 } he Eq (-(5 4 4 is equivalen o H( F( (6 where u p v q H ( p ( p u ( ( u (7 ( q q v ( ( v F( [ M ( u v ]( u g( u v f( x [ M ( u v ]( v g( u v f( x (8 Lea For ( u v p q E we ge ( H ( E ( p q (9 E 4 Proof By (4(7 we ge ( H ( ( u p u ( p ( p u ( ( u p E ( v q v ( q ( q v ( ( v q u p p ( u p ( u p q q ( v q ( v q v ( By eploying holder s inequaliy Young s inequaliy and Poincare inequaliy we process he ers in ( we have ( u p u p u p ( ( v q v q v q ( ( u p u p (3 ( v q v q (4 By he value of and subsiuing (-(4 we have Progressive Acadeic Publishing UK Page 96 wwwidpublicaionsorg

4 European Journal of Maheaics and Copuer Science Vol 4 No 7 ISSN ( H ( E ( u ( p ( p p ( ( ( v q q q ( u v p q ( p q 4 ( p q E 4 (5 he proof is copleed he linearized equaions of (- (5 we have U M ( u v ( U M '( u v [( u U ( v V ]( u ( U g u ( u v U g v( u v V (6 V M ( u v ( V M '( u v [( u U ( v V ]( v ( V gu ( u v U gv( u v V (7 U( x V ( x (8 x x U( x U( x V ( x V( x (9 where ( E( u v p q S( ( u v p q is he soluions of (-(5 wih ( u v p q Given ( u v p q and S( : E E he soluion S( ( u v p q E by sand ehods we can show ha for ( E he linear iniial boundary value proble(6-(9 possess a unique soluion ( U( V( P( Q( L (( ; E heore For R he apping S( : E Eis Freche differeniable on E I s differenial a ( u v u v is he linear operaor on E : ( ( U( V( P( Q( where U ( V ( is he soluion of(6-(9 Proof le ± ( u v p q E ( u v p q E wih E R R E we denoe ( u v p q S( ( u %% v p q % S( We can ge he Lipchiz propery of S ( on he bounded se of E ha is ± c 9 E E S( S( e ( ( Le u % u U % v v V is he soluion of proble: M u v h ( ( ( ( M u v h ( ( ( ( ( ( (3 Progressive Acadeic Publishing UK Page 97 wwwidpublicaionsorg

5 European Journal of Maheaics and Copuer Science Vol 4 No 7 ISSN ( ( (4 Seing s u v % s u% % v hen h ( M ( s M (% s( u% M '( s[( u U ( v V ]( u g( u v g( u%% v g u( u v U g v( u v V (5 h ( M ( s M (% s( % v M '( s[( u U ( v V ]( v g( u v g( u%% v gu( u v U gv( u v V (6 aking he scalar produc of each side of ( wih (wih and adding he ogeher d d ( M ( s ( ( ( h ( h d d (7 Seing u u u% v v %by v ( H 7 he ean value heore and Lea5we have (( M ( s M (% s( u% % % % % ( M '[ s ( s]( u u v v ( u ( M '[ s ( % s][( u u u% ( v v % v]( u% (8 ({ M '[ s ( % s] M '( s}[( u u u% ( v v % v]( u% M ''( s [ s ( % s s][( u u u% ( v v % v]( u% M ''( s ( ( % s s[( u u u% ( v v % v]( u% c ( u v ( u v (9 ( c u v ( M '( s[( u u u% ( v v % v][( u% ( u] ( M '( s[( u u u% ( v v % v] u c ( u v u 3 (3 4( c u v ( M '( s[( u u u% ( v v % v]( u M '( s[( u u ( v v]( u ( M '( s[( u u u% ( v v % v ( u u ( v v]( u 5( c u v (3 herefore inegrae (8-(3 we have Progressive Acadeic Publishing UK Page 98 wwwidpublicaionsorg

6 European Journal of Maheaics and Copuer Science Vol 4 No 7 ISSN (( M ( s M (% s( u% M '( s[( u U ( v V ]( u (3 c u v 6( And siilarly (( M ( s M (% s( % v M '( s[( u U ( v V ]( v c ( u v 7 (33 By( H 4 ( H8 Lea5hanks o ( g ( u v g ( u v g ( u v u u g ( u v g ( u v g ( u v u u [ g ( u ( u v g ( u v] u u u 8( c u v u ( g ( u v g ( u v g ( u v v v g ( u v g ( u v g ( u v v v [ g ( u v ( v g ( u v] v v v 9( c u v v (34 (35 Hence by Young s inequaliy we obain ( g( u v g( u%% v g u ( u v U g v( u v V ( g ( u v g ( u% v g ( u% v g ( u%% v g ( u v U g ( u v V u v ( g ( u v g ( u% v g ( u v u ( g ( u% v g ( u%% v g ( u v v u v ( g ( u v ( g ( u v u v ( c u c v g ( u v g ( u v u v L L In he siilar way ( g ( u v g ( u%% v g ( u v U g ( u v V u v (36 ( c u c v g ( u v g ( u v 3 u v L L (37 By ( H 4 ( H 7 (3-(33(36-(37and Young s inequaliy we have d [ ( ] c4 [ ( ] d 4 4 c ( u v u v (38 By Gronwall s inequaliy and ( we obain 5 Progressive Acadeic Publishing UK Page 99 wwwidpublicaionsorg

7 European Journal of Maheaics and Copuer Science Vol 4 No 7 ISSN (39 Since 4 4 c4 ( 5 ( c e u v u v d c e ( ( ( ² ( ( U ( ( E c7 4 6 E E E c7 c6e E E ( ( ( (4 as ( in E he proof is copleed he Hausdorff Diensions and Fracal Diensions for he Global Aracor Consider he firs varision of (6 wih iniial condiion: ' P( ( ( ( ( E (4 Where ( U V P Q E P U U Q V V and ( u v p q E is a soluion of (6 I I I I P( I ( ( I ( I ( ( I ( (4 (43 ( g u ( u v gv( u v gu( u v gv( u v ( [ M( u v ]( U M '( u v [( u U ( v V]( u [ M( u v ]( V M '( u v [( u U ( v V]( v (44 I s easy o prove fro Lea ha (4is a well-posed proble in E he apping S ( : u v p u u q v v u( v( p( q( ( u( v( p( q( is Freche differeniable on E for E is differenial a u v p q is he linear operaor on ( U( V( P( Q( where ( U( V ( P( Q( is he soluion of (4 Lea[6] For any orhonoral faily of eleens of E L n we have ( E Progressive Acadeic Publishing UK Page wwwidpublicaionsorg

8 European Journal of Maheaics and Copuer Science Vol 4 No 7 ISSN n n v v v v ( ( v[ (45 where is he eigenvalue of ( Proof his is a direc consequence of Lea VI 63 of [6] heore If ( H ( H4( H6 hold saisfying ( Rk 4 6 ( Rk hen here exiss such ha he Hausdorff and 4 6 Fracalof diensions of he global aracor in E saisfies n v v dh ( in n n N ( n 6 c 35 (46 n v v df ( in n n N ( n 6 c 35 (47 where R is as in Lea5and ( n ( p n n p n n n (48 n n or p n n Proof Le n Nbe fixed Respec o soluions n of (4A a given ie s le Q ( n s denoe he orhogonal proecion in E ono span( s ( s n ( s ( E = L n be an orhonoral basis of Le y s Q ( s E span ( s ( s ( s (49 n n Wih consider he inner produc ( E and nor E Assue ( s ( u( s v( s p( s q( s (5 hen ( s M s E By y and Leawe have E ( P( ( s y ( s y ( s E 4 (5 By( H 4 ( ( ( s y ( s y ( s E ( g ( u v ( g ( u v ( g ( u v ( g ( u v u v u v c c c c (5 hen by he Soblev ebedding heore: Progressive Acadeic Publishing UK Page wwwidpublicaionsorg

9 European Journal of Maheaics and Copuer Science Vol 4 No 7 ISSN v v v v v H ( H ( H ( H ( H ( v [] (53 herefore ( ( ( s y ( s y ( s E c c c c c 4 4 (54 By( H 6 ( H 7 and Young s inequaliy we have ( ( ( s y ( s y ( s 37( E ( R k R k ( R k R k ( ( R k R k Choose saisfying ( ( R k R k ( (56 We obain n p ( s (( P( ( s ( ( s ( ( s y ( s y ( s n E (57 If 6c ( Rk n ( ( ( n 8 n R k c35 n c35( n ( 35 n hen q liinf sup sup sup p ( s ds n c ( ( n n n 35 R E ( 6c35 n (58 which iply ha we obain (46(47 by Lea 4 of [] he proof of heore is copleed ACKNOWLEDGEMENS he auhors would like o hanks for he anonyous referees for heir valuable coens and suggesions sincerely hese conribuions increase he value of he paper Progressive Acadeic Publishing UK Page wwwidpublicaionsorg

10 European Journal of Maheaics and Copuer Science Vol 4 No 7 ISSN REFERENCES []GuoguangL&XiangshuangX(7 he Global Aracor for he Kirchhoff-ype equaions wih srongly daped ers and source erseuropean Journal of Maheaics and Copuer ScienceVol4No []Zhou s(999 Global aracor for srongly daped nonlinear wave equaions Funcional differenial equaions [3]XiaoingF&SZhou(4 Kernel secions for non-auonoous srongly daped wave equaions of non-degenerae Kirchhoff-ype Applied Maheaics and Copuaion [4] MeixiaWangCuicuiian &GuoguangLin(4Global Aracor and Diension Esiaion for a D Generalized Anisoropy Kuraoo-Sivashinsky Equaion Inernaional Journal of Modern Nonlinear heory and Applicaion363-7 [5]GuoguangL&YunlongG(7he Global and Exponenial Aracors for he Higher- Order Kirchhoff-ype Equaion wih Srong Linear DapingJournal of Maheaics Research:Vol [6]ea R(998Infinie Diensional Dynaics Syses in Mechanics and physics Springer New York Progressive Acadeic Publishing UK Page 3 wwwidpublicaionsorg