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Ieraioal Joural of Moder Noliear Theory ad Applicaio 6 5 85- hp://wwwcirporg/oural/ia ISSN Olie: 67-9487 ISSN Pri: 67-9479 The Global Aracor ad Their Haudorff ad Fracal Dieio Eiaio for he

1 Ieraioal Joural of Moder Noliear Theory ad Applicaio hp://wwwcirporg/oural/ia ISSN Olie: ISSN Pri: The Global Aracor ad Their Haudorff ad Fracal Dieio Eiaio for he Higher-Order Noliear Kirchhoff-Type Euaio wih Srog Liear Dapig Yulog Gao Yuig Su Guoguag Li Depare of Maheaic Yua Uiveriy Kuig Chia How o cie hi paper: Gao YL Su YT ad Li GG (6 The Global Aracor ad Their Haudorff ad Fracal Dieio Eiaio for he Higher-Order Noliear Kirchhoff-Type Euaio wih Srog Liear Dapig Ieraioal Joural of Moder Noliear Theory ad Applicaio hp://dxdoiorg/436/ia6548 Received: Ocober 6 Acceped: Noveber 6 Publihed: Noveber 3 6 Copyrigh 6 by auhor ad Scieific Reearch Publihig Ic Thi work i liceed uder he Creaive Coo Aribuio Ieraioal Licee (CC BY 4 hp://creaivecooorg/licee/by/4/ Ope Acce Abrac I hi paper we udy he logie behavior of oluio o he iiial boudary value proble for a cla of rogly daped Higher-order Kirchhoff ype euaio: u + u + D u u+ g u = f x A fir we prove he exiece ad uiuee of he oluio by priori eiaio ad he Galerki ehod The we obai o he exiece of he global aracor A la we coider ha he eiaio of he upper boud of Haudorff ad fracal dieio for he global aracor are obaied Keyword Noliear Higher-Order Kirchhoff Type Euaio The Exiece ad Uiuee The Global Aracor Haudorff Dieio Fracal Dieio Iroducio I hi paper we are cocered wih he exiece of global aracor ad Haudorff ad Fracal dieio eiaio for he followig oliear Higher-order Kirchhoff-ype euaio: [ u( x u ( x u ( x u ( x x u + u + D u u+ g u = f x x Ω + ( = = Ω ( i u u( x = = i= x Ω ( + (3 i v DOI: 436/ia6548 Noveber 3 6

2 Y L Gao e al where > i a ieger coa ad > i a poiive coa Moreover Ω i a bouded doai i R wih he ooh boudary Ω ad v i he ui ouward oral o Ω g( u i a oliear fucio pecified laer Recely Maria Ghii ad Maio Gobbio [] udied pecral gap global oluio for degeerae Kirchhoff euaio Give a coiuou fucio :[ + [ + hey coider he Cauchy proble: where [ Ω u x + ux d x ux = x Ω T (4 u = u u = u (5 Ω R i a ope e ad u ad u deoe he gradie ad he Laplacia of u wih repec o he pace variable They prove ha for uch iiial daa ( u u here exi wo pair of iiial daa ( ( ˆ ˆ u u u u for which he oluio i global ad uch ha u = u + uˆ ˆ u = u+ u Yag Zhiia Dig Pegya ad Lei Li [] udied Logie dyaic of he Kirchhoff euaio wih fracioal dapig ad upercriical olieariy: where α u M u u+ u + f u = g x x Ω > (6 u = u x = u x u x = Ω u x (7 α Ω i a bouded doai R N wih he ooh boudary Ω ad he olieariy f ( u ad exeral force er g will be pecified The ai reul are focued o he relaiohip aog he growh expoe p of he olieariy f ( u ad well-poede They how ha (i eve if p i up o he upercriical rage ha i p < N + 4α ( N 4 α + he well-poede ad he logie behavior of he o- luio of he euaio are of he characer of he parabolic euaio; (ii whe N + 4α N + 4 p < he correpodig ubcla G of he lii oluio exi N 4α + N 4 + ad poee a weak global aracor Yag Zhiia Dig Pegya ad Liu Zhiig [3] udied he Global aracor for he Kirchhoff ype euaio wih rog oliear dapig ad upercriical olieariy: φ + u σ u u u u+ f u = h x i Ω (8 u x = u x = u x u x = u x x Ω (9 Ω where Ω i a bouded doai i ad f ( are oliear fucio ad N R wih he ooh boudary Ω ( σ φ ( h x i a exeral force er They prove ha i ricly poiive iffe facor ad upercriical olieariy cae here exi a global fiie-dieioal aracor i he aural eergy pace edowed wih rog opology Li Fucai [4] udied he global exiece ad blow-up of oluio for a higher-order 86

3 Y L Gao e al oliear Kirchhoff-ype hyperbolic euaio: Ω r p u + Du d x u+ u u = u ux Ω > ( i u u( x = = i= x Ω > ( i v u x = u x u x = u x ( where pr Ω i a bouded doai R wih a ooh boudary Ω ad a ui ouer oral v Seig + p+ E = u + D u u p+ ( + p+ Aue ha p aifie he codiio: p for N > ; p > for N N (3 Their ai reul are he wo heore: Theore Suppoe ha p r ad codiio (3 hold The for ay iiial daa ( u u H ( Ω H ( Ω H ( Ω he oluio of ( - ( exi globally Theore Suppoe ha p > ax { r } ad codiio ( hold The for ay iiial daa ( u u H ( Ω H ( Ω H ( Ω he oluio of ( - ( blow up a fiie ie i L p + or provided ha E < Li Ya [5] udied The Aypoic Behavior of Soluio for a Noliear Higher Order Kirchhoff Type Euaio: β Ω u + D u d x u+ u + g u = i Q = Ω + (4 i u u( x = = i= o Σ = Γ ( + (5 i v where Ω i a ope bouded e of ui oral vecor The fucio u x = u x u x = u x i x Ω (6 R wih ooh boudary Γ ad he g C aifie he followig codiio: G li if G ( g( r d r; = (7 ( g li if = (8 where γ ( γ ( 3 γ ( 4 C > uch ha γ < + = < = = Furherore here exi C G ( g li if (9 A la Li Ya udied he aypoic behavior of oluio for proble (4 - (6 For he o of he cholar repreeed by Yag Zhiia have udied all kid of low order Kirchhoff euaio ad oly a all uber of cholar have udied he 87

4 Y L Gao e al blow-up ad aypoic behavior of oluio for higher-order Kirchhoff euaio So i hi coex we udy he high-order Kirchhoff euaio i very eaigful I order o udy he high-order oliear Kirchhoff euaio wih he dapig er we borrow oe of Li Ya [5] parial aupio ( - (3 for he oliear er g i he euaio I order o prove ha he lea we have iproved he reul fro aupio ( - (3 uch ha < C The uder all aupio we prove uu L + ; H Ω H Ω ha he euaio ha a uiue ooh oluio ( ad obai he oluio eigroup S : H H H H Ω Ω Ω Ω ha global aracor Fially we prove he euaio ha fiie Haudorff dieio ad Fracal dieio by referece o he lieraure [7] For ore relaed reul we refer he reader o [6] [7] [8] [9] [] I order o ake hee euaio ore oral i ecio ad i ecio 3 oe aupio oaio ad he ai reul are aed Uder hee aupio we prove he exiece ad uiuee of oluio he we obai he global aracor for he proble ( - (3 Accordig o [6] [7] [8] [9] [] i ecio 4 we coider ha he global aracor of he above eioed proble ( - (3 ha fiie Haudorff dieio ad fracal dieio Preliiarie For coveiece we deoe he or ad calar produc i L ( Ω by ad f = f ( x p p k k k k L = L ( Ω H = H ( Ω H = H ( Ω = = p Accordig o [5] we pree oe aupio ad oaio eeded i he proof of g u C Ω aifie ha G = g r d r he our reul For hi reao we aue oliear er (H Seig G( (H If L p L ; li if ; ( ( where r < + ( = r < ( = r = ( g li up = ( r 3 4 (H 3 There exi coa C > uch ha C G ( (H 4 There exi coa C > uch ha where p ; g li if (3 ( p p g C + (4 g C + (5 88

5 Y L Gao e al For every γ > by (H -(H 3 ad apply Poicaré ieualiy here exi coa C ( γ > uch ha J( u + γ D u + C( γ u H ( Ω (6 ( g( u u CJ ( u γ Du C( γ u H + + Ω (7 i idepede of γ Ω Ω Ω where J( u = G( u d x< C Ω Lea Aue (H -(H 3 hold ad ( u u H L f x L The he oluio ( uv of he proble ( - (3 aifie ( uv L ( + ; H ( Ω L( Ω ad ( C C + + D u + v y e + + C + (8 4 C 6λ C λ + λ + + where v = u + u < < i λ + λ 4 i he fir eigevalue of i H ( Ω ad y u u + = + D u + D u + + J( u + C + + γ w C = f + C ( γ CC ( γ + γ = > γ λ = > { λ ( } w= i + Thu here exi ha ( uv D u v E H Proof We ake he calar produc i E ad = Ω > uch = + > (9 Ω L Ω L of euaio ( wih ( u u D u u g( u v ( f ( x v v = u + u The = ( Afer a copuaio i ( we have d = + ( d ( u v v v ( uv ( ( ( + d u v = D u + D v D u ( d d + + D u u v = D u + D u d ( g( u v J( u ( g( u u Collecig wih ( - (4 we obai fro ( ha + (3 d = + (4 d d + v D u + D u + J u v + u v d + ( ( + D v D u + D u + g u u = f x v (5 89

6 Y L Gao e al Sice v = u + u ad 4 C 6λ C + λ + + < < by uig Hölder i- 4 λ i + λ eualiy Youg ieualiy ad Poicaré ieualiy we deal wih he er i (5 oe by oe a follow: u v u v D u v (6 λ D v λ v (7 By (7 we ca obai ( g( u u CJ( u D u C( γ (8 λ where γ = > λ Becaue of f ( x L Ω we ca obai ( By (6 - (9 i follow fro ha f f x v f v + v (9 d v D u + D u + J( u + ( λ v d + + D u + D u + CJ( u f + C ( γ λ By Youg ieualiy ad < < < we have + λ D u + D u + ( D u ( ( + D u D u + ( By ( we ge + ( λ v D u + D u + C J u + + = ( λ v + ( + D u + + ( D u D u CJ( u w v + D u + C J u + + w v D u + D u + CJ( u + { λ } where w= ( + i (3 9

7 Y L Gao e al By ( ad ubiuig (3 io ( we receive d + v D u + D u + + J( u d w v D u + D u + + CJ u + + w f + C γ (4 Sice ( C + + 6λ C < < ad < C < we ge 4 { } w= i λ + C (5 By (6 ad ( we have + D u + D u + + J u + C + + ( D u J( u C( γ + + where γ = > Cobiig wih (5 ad (6 forula (4 io ( γ d + v D u + D u + + J( u + C( γ d C v D u + D u + + J( u + C( γ + + w f + C ( γ CC ( γ + + y = v D u + D u + + J u + C γ + + i iplified a We e d d y C y C where C = f + C( γ CC( γ (6 (7 The (7 + (8 w + Fro cocluio (6 we kow y So by Growall ieualiy we obai y y e + (9 C C C γ + + D u D u + + where y = u + u D u + D u + + J( u + C By geeralized Youg ieualiy we have + The we ge ( + ( D u D u + + (3 9

8 Y L Gao e al By (6 ad (3 we have y v D u D u J u C = ( γ + ( ( v + D u + D u J u C( γ ( D u ( v { }( v D u i ( v D u = Cobiig wih (9 ad (3we obai C C + + D u + v y( e + + ( C + The C + + ( uv = D u + v + H ( Ω L ( Ω ( li C + So here exi E ad = Ω > uch ha ( uv D u v E H (3 (3 (33 = + > (34 Ω L Ω Lea I addiio o he aupio of Lea (H - (H 4 hold If (H 5 : f x H u u H H uv of he pro- ( Ω ad ( ( Ω ( Ω The he oluio ble ( - (3 aifie ( uv L ( + ; H ( Ω H ( Ω ad D u D v where ( z ( D f + C 3 e α + + (35 T α T v u u = + ad T i{ if D u } z D u D u D u D u H Ω λ i he fir eigevalue of i = + + α { i λ M} = Thu here exi E ad = = Ω > uch ha ( uv D u D v E H = + > (36 Ω H Ω Proof Takig L -ier produc by ( v = u + u i ( we have u + u + D u u+ g u v = f x v (37 Afer a copuaio i (37 oe by oe a follow d ( u ( v = D v D v + ( D u D v d d D v D v D u D v d λ (38 9

9 Y L Gao e al ( ( D u ( u ( v d u v = D v D u D u (39 d D u d d = D u D u D u + D u D u d d By Youg ieualiy we ge g ( u D v ( (4 g u v g u D v (4 Nex o eiae + ad Youg ieualiy we have By p K > uch ha u L p io g u i (4 By (H 4 : ( p g C p ( + d Ω p p ( + + Ω p ( C + C u dx g u C u x Ω C C u C u dx p p L ( Ω C Ω+ C u ad Ebedig Theore he p H Ω L K D u (4 Ω So here exi D u bouded by lea The (4 ur Ω 3( Collecig wih (43 fro (4 we have g u C pc K Ω (43 ( g( u v By f ( x H ( C D v 3 (44 Ω ad Youg ieualiy we obai f x v = ( D f x D v D f + D v (45 Iegraig (38 - (4 (44 - (45 fro (37 eail d D v + ( D u D u + D v ( + D v d d + D u + D u D u D f C + d λ By Poicaré ieualiy uch ha λ D v D v So (46 ur io d D v + ( D u D u + ( λ D v d d + D u + D u D u D f C + 3 d λ 3 (46 (47 93

10 Y L Gao e al Fir we ake proper uch ha λ > ad D u a The we aue ha here exi M > uch ha M > ad M D u D u D u d λ o > by La- d < + The forula i iplified d M D u + D u M (48 By Growall ieualiy we ge d λ M ( M λ D u < D u e + M O accou of Lea we kow Naely we prove ha here are M > ake (49 D u i bouded So he hypohei i rue d < + (5 M D u D u D u d λ Subiuig (5 io (47 we receive + ( + ( λ ( 3 d D v D u D u D v d + M D u D u D f + C Takig α { i λ M} = he where ( (5 d z + αz D f + C 3 (5 d z = D v + D u D u By Growall ieualiy we have D f + C 3 e + (53 α α z z where = + + ( Le T = i{ if D u } o we ge The z D u D u D u D u So here exi D v D u z ( D f + C 3 e α + + (54 T α T D f + C 3 li ( uv = D u + D v (55 H ( Ω H ( Ω α T E ad = Ω > uch ha ( uv D u D v E H = + > (56 Ω H Ω 94

11 Y L Gao e al 3 Global Aracor 3 The Exiece ad Uiuee of Soluio Theore 3 Aue (H - (H 4 hold ad ( u u H H f ( x H ( Ω v u u ( u( x v( x L ( ; H H Ω Ω = + So Euaio ( exi a uiue ooh oluio + Ω Ω (3 Proof By he Galerki ehod Lea ad Lea we ca eaily obai he exiece of Soluio Nex we prove he uiuee of Soluio i deail Aue uv are wo oluio of he proble ( - (3 le w= u v he w x = w x = w x = w x = ad he wo euaio ubrac ad obai w w D u u D v v g( u g( v = (3 By uliplyig (3 by w we ge ( w w D u u D v v g u g v w = (33 d w w = w (34 d ( w w ( D u ( u D v ( vw = ( ( + ( ( = D w (35 ( D u w w D u D v v w d = D u D w D u D u D w d + D u D v vw Exploiig (34 - (36 we receive d ( w + D u D w + D w d ( g( u g( v w = D u D u D w D u D v v w (36 (37 I (37 accordig o Lea ad Lea uch ha ( θ ( θ ( θ ( D u D w D u D v v w D u D w + 4 D u + θ D v D u D w v w + C4 D w C5 D w w C C C4 ( + D w + w 5 5 where < θ < C ( > ad 4 C5 θ > are coa (38 95

12 Y L Gao e al By (H 4 we obai ( g( u g( v w ( θ ( θ ( g u v ww g ( θu ( θ v w w = + + p ( θ ( θ Ω p ( θ ( θ C ( + u + v dx w w (39 C + u+ v w w ( θ λ CC p D w w 6 where C C ( θ p λ ( θ λ ( θ λ CC p w + CC p D w = > i coa Fro he above we have ( ( θ ( d C w D u D w C CC w D w d For (3 becaue D u So we have d d where D u i bouded The here exi > ( w + D u D w 7 ( θ C5 C4( + + CC 6 w ( θ C4 C5 CC ( C w + D u D w D u D w ( θ ( θ C5 C4 C5 CC 6 C7 = i C4 + + CC wall ieualiy for (3 we obai C 7 ( uch ha (3 (3 By uig Gro- w + D u D w w + D u D w e = (3 Hece we ca ge w + D u D w = Tha how ha w = D u D w = (33 Tha i Therefore w( x = (34 u = v (35 So we ge he uiuee of he oluio 96

13 Y L Gao e al S are he eigroup op- + = = where I i a ui 3 Global Aracor Theore 3 [] Le E be a Baach pace ad { } eraor o E S : E E S( τ S S( τ( τ S I operaorse S( aify he follow codiio: R > u R i exi a coa S i uiforly bouded aely E o ha ( [ I exi a bouded aborbig e B o ha C R S u C R + ; (36 E E aely B E i exi a coa S B B (37 ; where B ad B are bouded e 3 Whe > S( i a copleely coiuou operaor Therefore he eigroup operaor S( exi a copac global aracor Theore 33 Uder he aue of Lea Lea ad Theore 3 euaio have global aracor ( B = ω = S B (38 τ τ { : H H H H } where B = uv H Ω H Ω uv = u + v R + R B i he bouded aborbig e of H S = > ; li di S B = here H ad aifie ( ( B H H ad i i a bouded e ( y H H di S B = up if S x y (39 x B Proof Uder he codiio of Theore 3 i exi he oluio eigroup S( S : H H H H here E = H ( Ω H ( Ω ( Fro Lea o Lea we ca ge ha B H ( Ω H ( Ω i a uv R bouded e ha iclude i he ball { H } H H H H H H H R + C ( u v B S u v = u + v u + v + C Thi how ha S( i uiforly bouded i H ( Ω H ( Ω ( Furherore for ay ( u v H ( Ω H ( Ω whe ax { } have H H H H (3 we S u v = u + v R + R (3 So we ge B i he bouded aborbig e (3 Sice E : = H Ω H Ω E : = H Ω L ( Ω i copac ebedded which ea ha he bouded e i E i he copac e i E o he eigroup operaor S( exi a copac global aracor 97

14 Y L Gao e al 4 The Eiae of he Upper Boud of Haudorff ad Fracal Dieio for he Global Aracor We rewrie he proble ( - (3: Le Au u + Au+ Au Au+ gu = f x i Ω R + (4 u x = u x ; u x = u x x Ω (4 i u + u( x Ω = = ( i = i Ω R (43 i v = u where Ω i a bouded doai i N wih ooh boudary Ω i poiive coa ad i poiive ieger The liearized euaio of he above euaio a follow: U + AU = FU (44 U = ξ U = ζ (45 Le U H ( Ω proble (44 - (45 have a uiue oluio U i he oluio of proble (44 - (45 We ca prove ha he ( ( U L T H Ω U L T L Ω The euaio (44 i he liearized euaio by he Euaio (47 Defie he L : L U u u ϕ = u u appig ζ = here = le u u { } { u u } ϕ = ϕ + ξζ = + ξ + ζ le ϕ R E ϕ = ϕ = { } S ϕ ϕ ϕ S u u { } = E ϕ R E = H ( Ω ( Ω Ł Lea 4 [6] Aue H i a Hilber pace E i a copac e of H S : E H i a coiuou appig aify he follow codiio S ( E = E > ; If S( i Fréche differeiable i exi i a bouded liear differeial operaor + L ( ϕ CR; LE ( E > ha i ϕ ϕ ( ϕ ( S S L u v { ξζ } E E { ξζ} The proof of lea 4 ee ref [6] i oied here Accordig o Lea 4 we ca ge he followig heore : Theore 4 [6] [7] Le i he global aracor ha we obai i ecio 3I ha cae ha fiie Haudorff dieio ad Fracal dieio i 6 H ( Ω H ( Ω ha i dh ( df ( 5 5 Proof Firly we rewrie he euaio (4 (4 io he fir order abrac evoluio euaio i E Rϕ uu u + i a ioorphic ap- Le Ψ= = { + } le R :{ uu } { uu u} pig So le i he global aracor of { S( } of { S } he R i alo he global aracor ad hey have he ae dieio The Ψ aifie a follow: 98

15 Y L Gao e al { u } T u u T T where Ψ= { + } ( Ψ = = Ψ +ΛΨ+ g Ψ = f (46 Ψ = + (47 { } { } uu u g g u f f x Λ = I I + A u A A I A I T (48 Ψ : = F Ψ = f ΛΨ g Ψ (49 P = F ( Ψ (4 P P g P where P { UU U} T g P g( uu +Λ + Ψ = (4 { } T = + Ψ = The iiial codiio (45 ca be wrie i he followig for: ωω { ξζ} E P = = (4 We ake N he coider he correpodig oluio: ( P = P P P; P E of he iiial value: ( ω = ω ω ω; ω E i he Euaio (4 - (4 So here i ( d e TrF S τ Ψ Q τ τ = ω ω ω P P P fro E ψ ( τ = S ( τ Ψ we ge S ( τ :{ u v = u+ u} { u( τ v( τ = u ( τ + u( τ } ψ ( τ = { u( τ v ( τ = u ( τ + u( τ } here u i he oluio of proble (4-(43; repree he ouer produc T repre he race Q ( τ = Q ( τ Ψ ; ω ω ω E i a orhogoal proecio fro he pace E = o he ubpace paed by { P( τ P( τ P ( τ } For a give ie τ le φ ( τ = ξ ( τ ζ ( τ = φ ( τ i he { } { } = adard orhogoal bai of he pace Q ( τ pa P ( τ P ( τ P ( τ Fro he above we have E = ( Ψ( τ ( τ = ( Ψ( τ ( τ φ ( τ φ ( τ TrF Q F Q = ( F ( ( τ φ ( τ φ ( τ = Ψ = where ( i he ier produc i E F φ φ φ φ g u ξ ξ E The ({ ξζ }{ ξζ } ( ξξ ( ζζ E ( Ψ = ( Λ ( ( ( ( A u ( A ( A ( ξ ζ E E = + ; E E Λ φ φ = ξ + ξ ζ + ξ ζ + ζ ζ ζ = ξ + ( ( ξ ζ + λ A u ( ξ ζ + λ ζ ζ a + (43 (44 99

16 Y L Gao e al where + + A u λ ( λ + + A u λ a : = i Now uppoe ha { } E Ψ = { u u + u } E u D( A ; D( A = { u Au } The here i a [ ] o ake he appig g : D( A ρ ( u u accordig o heore 33 i a bouded aborbig e i ie here are he followig reul: R A up u D( A Au < R = up Aξ < ; A { ξζ } where g ( u ξ ζ ee: obaied: ρ ( g u r < v g u ξ ζ r ξ ζ ( E A he ae (45 Copreheive above ca be F Ψ φ φ a ξ + ζ + r ξ ζ (46 a r ( ξ + ζ + ξ a ξ + ζ = φ = due o φ ( τ i a adard orhogoal bai i E E Q τ So { } ( F ( E = a Ψ τ φ τ φ τ + ξ (47 = a r Alo o all akig ξ λ (48 = = So TrF a r Ψ + (49 ( ( τ Q( τ λ a = Le u aue ha { u u } i euivale o { } ( Ψ Ψ = u u+ u R The TrF S τ Q τ dτ = = (4 up up Ψ R ω E ω E = li up Accordig o (49 (4 o a r + a a r + a = = λ λ (4

17 Y L Gao e al Therefore he Lyapuov expoe of (or R i uiforly bouded a r µ + µ + + µ + λ (4 a Fro wha ha bee dicued above i exi a ad r are coa he ( λ = = a (43 6r a r 5a λ (44 a = r a = (45 + a λ i i= ( + ax (46 5 Accordig o he referece [6] [7] we iediaely o he Haudorff dieio ad 6 dh df 5 5 fracal dieio are repecively 5 Cocluio I hi paper we prove ha he higher-order oliear Kirchhoff euaio wih liear dapig i L (( + ; H ( Ω H ( Ω ha a uiue ooh oluio ( uu Fur- her we obai he oluio eigroup S : H ( Ω H ( Ω H ( Ω H ( Ω ha global aracor Fially we prove he euaio ha fiie Haudorff dieio L + ; H Ω H Ω ( ad Fracal dieio i Ackowledgee The auhor expre heir icere hak o he aoyou reviewer for hi/her careful readig of he paper givig valuable coe ad uggeio Thee coribuio grealy iproved he paper Fud Thi work i uppored by he Naioal Naural Sciece Foudaio of People Republic of Chia uder Gra 5676 Referece [] Ghii M ad Gobbio M (9 Specral Gap Global Soluio for Degeerae Kirchhoff Euaio Noliear Aalyi hp://doiorg/6/a99 [] Yag ZJ Dig PY ad Li L (6 Logie Dyaic of he Kirchhoff Euaio wih Fracioal Dapig ad Supercriical Nolieariy Joural of Maheaical Aalyi Applicaio hp://doiorg/6/aa6479 [3] Yag ZJ Dig PY ad Liu ZM (4 Global Aracor for he Kirchhoff Type Euaio wih Srog Noliear Dapig ad Supercriical Nolieariy Applied Maheaic Leer 33-7 hp://doiorg/6/al44

18 Y L Gao e al [4] Li FC (4 Global Exiece ad Blow-Up of Soluio for a Higher-Order Kirchhoff- Type Euaio wih Noliear Diipaio Applied Maheaic Leer hp://doiorg/6/a374 [5] Li Y ( The Aypoic Behavior of Soluio for a Noliear Higher Order Kirchhoff Type Euaio Joural of Souhwe Chia Noral Uiveriy [6] Tea R (998 Ifiie Dieioal Dyaic Sye i Mechaic ad Phyic Spriger New York [7] Wu JZ ad Li GG (9 The Global Aracor of he Boie Euaio wih Dapig Ter ad I Dieio Eiaio Joural of Yua Uiveriy [8] Yag ZJ (7 Logie Behavior of he Kirchhoff Type Euaio wih Srog Dapig o R Joural of Differeial Euaio N hp://doiorg/6/de784 [9] Yag ZJ ad Liu ZM (5 Expoeial Aracor for he Kirchhoff Euaio wih Srog Noliear Dapig ad Supercriical Nolieariy Applied Maheaic Leer hp://doiorg/6/al59 [] Li GG ( Noliear Evoluio Euaio Yua Uiveriy Pre Kuig Subi or recoed ex aucrip o SCIRP ad we will provide be ervice for you: Accepig pre-ubiio iuirie hrough Eail Facebook LikedI Twier ec A wide elecio of oural (icluive of 9 ubec ore ha oural Providig 4-hour high-ualiy ervice Uer-friedly olie ubiio ye Fair ad wif peer-review ye Efficie ypeeig ad proofreadig procedure Diplay of he reul of dowload ad vii a well a he uber of cied aricle Maxiu dieiaio of your reearch work Subi your aucrip a: hp://paperubiiocirporg/ Or coac ia@cirporg

EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D. S. Palimkar

Ieraioal Joural of Scieific ad Research Publicaios, Volue 2, Issue 7, July 22 ISSN 225-353 EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D S Palikar Depare of Maheaics, Vasarao Naik College, Naded