The Estimates of the Upper Bounds of Hausdorff Dimensions for the Global Attractor for a Class of Nonlinear

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Advaces i Pure Mahemaics 8 8 - hp://wwwscirporg/oural/apm ISSN Olie: 6-384 ISSN Pri: 6-368 The Esimaes of he Upper Bouds of Hausdorff Dimesios for he Global Aracor for a Class of Noliear Coupled

1 Advaces i Pure Mahemaics hp://wwwscirporg/oural/apm ISSN Olie: ISSN Pri: The Esimaes of he Upper Bouds of Hausdorff Dimesios for he Global Aracor for a Class of Noliear Coupled Kirchhoff-Type Equaios Guoguag Li Mig Zhag Deparme of Mahemaics Yua Uiversiy Kumig Chia How o cie his paper: Li GG ad Zhag M 8 The Esimaes of he Upper Bouds of Hausdorff Dimesios for he Global Aracor for a Class of Noliear Coupled Kirchhoff-Type Equaios Advaces i Pure Mahemaics 8 - hps://doiorg/436/apm88 Received: December 7 7 Acceped: Jauary 6 8 Published: Jauary 9 8 Absrac This paper deals wih he Hausdorff dimesios of he global aracor for a class of Kirchhoff-ype coupled equaios wih srog dampig ad source erms We obai a precise esimae of upper boud of Hausdorff dimesio of he global aracor Keywords Kirchhoff-Type Equaios The Global Aracor Hausdorff Dimesio Copyrigh 8 by auhors ad Scieific Research Publishig Ic This work is licesed uder he Creaive Commos Aribuio Ieraioal Licese CC BY 4 hp://creaivecommosorg/liceses/by/4/ Ope Access Iroducio Guohuag Li Mig Zhag [] sudied he iiial boudary value problem for a class of Kirchhoff-ype coupled equaios ad obaied he exisece of he global aracor Nex i his paper we cosider he Hausdorff dimesios for he global aracor for he followig Kirchhoff-ype equaios: β β u M u + v u u+ g uv = f x v M u + v v v+ g uv = f x u x = u x u x = u x x Ω 3 v x = v x v x = v x x Ω 4 u Ω = v = 5 Ω where Ω is a bouded domai i R wih he smooh boudary Ω β > DOI: 436/apm88 Ja 9 8 Advaces i Pure Mahemaics

2 is a cosa M s is a oegaive ly dampig erms g uv ad ad f Jigzhu Wu Guoguag Li [] cosider a class of damped Bossiesq equaio: C fucio x are give forcig fucio u ad v are srog- g uv are oliear source erms f x where k+ u αu uxx u f x x + + = Ω > 6 u x = u x 7 u x = u x+ 8 Ω R α > he hey obai he exisece of he global aracor ad he limied of Hausdorff dimesio ad he limied of Fracal dimesio Xiaomig Fa Shegfa Zhou [3] cosider he followig o-auoomous srogly damped wave equaio of o-degeerae Kirchhoff-ype: p u α u β + γ u d x u+ h u + f u = g x x Ω > τ 9 Ω u x = τ x Ω τ τ u x = u x u x = u x x Ω τ τ where u= u x is a real-valued fucio o [ τ ope bouded se of Ω + τ R Ω is a R = 3 wih a smooh boudary Ω α > is C RL Ω is he se of coi- called he srog dampig β > ρ > γ h C RR ; f C R RR ; g Cb RL Ω b uous bouded fucios from R io L Ω Ad he hey obaied a precise esimae of upper boud of Hausdorff dimesio of kerel secios which decreases as he srog dampig grows for large srog dampig uder some codiios paricularly i he auoomous case Guoguag Li Yulog Gao [4] cocered he followig oliear Higher-order Kirchhoff-ype equaios: q m m m α β [ u + u + + u u+ g u = f x x Ω + u x = u x u x = u x x Ω 3 i u u x = = i= m x Ω [ + i 4 v where m > is a ieger cosa α > β > are cosas ad q is a real umber Ω is a bouded domai of R wih a smooh boudary Ω ad v is he ui ouward ormal o Ω g u is a oliear fucio specified laer Ad hey obaied he exisece of he global aracor I his case hey cosidered ha he esimaio of he upper bouds of Hausdorff for he global aracors is obaied Hausdorff Dimesios of he Global Aracor I his paper some ier produc orms abbreviaios ad some assumpios DOI: 436/apm88 Advaces i Pure Mahemaics

3 H - H 4 ad oaios eeds i he proof of our resuls i refer o [] Differeiabiliy of he Semigroup I order o esimae dimesios we suppose: H 5 For every k = k L such ha: iu L > here exis iu γ γ L Ω δ δ g u v g u+ u v k u u + k v v iv iv γ γ L Ω δ δ g u v g u v+ v k u u + k v v where uuvv H Ω ; u u v v L ; i = H 6 There exiss cosa µ µ µ such ha d µ θ + ω > d < µ M s µ µ = d µ θ + ω < d We defie A = E H H L L λ ; δ > ; 3 = Ω Ω Ω Ω The ier produc ad orm i E space are defied as follows: ϕ = u v p q E i = we have i i i i i ϕ ϕ = Au Au Av Av p p q q E 4 ϕ ϕ Au E E Av p q ϕ = = Seig ϕ = uvpq T E p= u + εu q = v + εv he Equaios -5 is equivale o where ϕ + H ϕ = F ϕ 6 H ϕ εu p εv q = ε p + β Ap + ε u + εβ Au εq + β Aq + ε v + εβ Av F ϕ = M u + v Au g u v + f x M u + v Av g u v + f x 7 8 Lemma For ay uvpq T E ϕ = we have DOI: 436/apm88 3 Advaces i Pure Mahemaics

4 Proof By 3 we ge ϕ ϕ ϕ E E H ε β 4 4 Ap β Aq 9 H A u p Au A v q A v E ϕ ϕ = ε + ε ε p β Ap ε u εβ Au p εq β Aq ε v εβ Av q = ε Au ε p + β A p + ε up εβ AuA p + ε Av ε q + β Aq + ε vq εβ AvAq By usig holder iequaliy ad Youg s iequaliy ad Poicare iequaliy we deal wih he erms i 9 by as follows: ε ε ε up Au p λ ε β β εβ AuA p Au A p ε ε ε vq Av q 3 λ ε β β εβ AvAq Av Aq 4 3 λ λβ By < ε < mi 4 + βλ 4 io we obai ad subsiuig -4 H ϕ ϕ ε Au + Av E ε ε β λ ε β + ε p + q + Ap + Aq ε ε β ε Au Av ε + + ε p + q λ β β + Ap + Aq + Ap + Aq 4 4 DOI: 436/apm88 4 Advaces i Pure Mahemaics

5 + ε ε β ε Au Av λ λβ ε β + ε p + q + Ap + Aq 4 4 ε β Au + Av + p + q + Ap + Aq 4 4 Proof fiished The liearized equaios of -5 he above equaios as follows: β u v β u v U + M u + v u U + v V Au + M u + v AU + AU + g u v U + g u v V = V + M u + v u U + v V Av + M u + v AV + AV + g u v U + g u v V = U x = V x = > 8 x Ω x Ω ξ ζ U x = U x = ξ 9 V x = V x = ζ ξ ζ ξ ζ he liear iiial boudary value problem 6- pos- where ξ ζ ξ ζ E u v u v S u v u v wih u v u v Α Give u v u v Α ad S : E E he soluio S u v u v E by sad mehods we ca show ha for ay E sess a uique soluio U V U V L + ; E Theorem For ay > R > he mappig S : E E differeiable o Is differeial a T = is he soluio of where U Proof is Freche ϕ = u v u v is he liear operaor o ξ T T ζ ξ ζ E: U V P Q V is he soluio of 6- Le ϕ = u v u v T E ϕ ξ ζ ξ ζ T ϕ R E ϕ we deoe R E = u + v + u + v + E wih T T uu = S ϕ uu Sϕ = We ca ge he Lipchiz propery of S o he bouded ses of E ha is E E C ϕ S ϕ e ξ ζ ξ ζ S Le θ = u u U ω = v v V is he soluio of problem θ + M u + v Aθ + βaθ = h ω + M u + v Aω+ βaω = h 3 DOI: 436/apm88 5 Advaces i Pure Mahemaics

6 where Le θ = θ = 4 ω = ω = 5 h = M u + v M u + v Au + M u + v u U + v V Au g uv + g uv + g uvu + g uvv 6 u v h = M u + v M u + v Av s = u + v + M u + v u U + v V Av g uv + g uv + g uvu + g uvv u v s = u + v so we ca ge M s M s Au M u v u U v V Au ξ γ θ ω = M u + u u u + v + v v v Au Ad M s u + u u u + v + v v v A u u M s u u u u + v v v v Au M s u + v Au The we have + u + v g uv g uv g u uv u u g u uv + g uv g uv + g uv v v g uv g uv g uv g uvu g uvv = + ω v v M s γ u + u u u + v + v v v Au θ C u u + v v A θ θ M u + v u u u + u + v v v + v A u u θ C u u + v v A θ M u + v u u u u + v v v v Au θ 3 C u u + v v A θ M u + v u θ + v ω Au θ 4 C θ + ω A θ DOI: 436/apm88 6 Advaces i Pure Mahemaics

7 By usig 3-33 we have M s M s Au + M s u U + v V Au θ 5 4 C u u + v v A θ + C θ + ω A θ Similarly M s M s Av + M s u U + v V Av ω 6 7 C u u + v v A ω + C θ + ω A ω Ad by usig H 5 g uv g uv + g u uv u u g u uv θθ + g uv g uv + g v uv v v g v uv ωθ δ+ δ θ θ θ ω θ δ δ+ 9 θ δ+ δ+ θ θ ω θ θ C u u + v v u u + C C u u v v + v v C + + C u u + v v Similar gu uv gu xv u u gu uv θω + g u v g u y v v g u v v v v ωω δ+ δ+ ω θ ω ω ω C u u + v v + C + 3 So we ca ge δ+ δ+ 4 4 θ ω µ θ ω d θ ω µ θ ω d C u u + v v + u u + v v + C ϕ The by usig Growall s iequaliy ad we obai θ ω β θ ω C4 δ+ δ e d 39 C u u + v v + u u + v v 4 δ+ ξ ζ ξ ζ ξ ζ ξ ζ E E C e + C6 7 The we ge ϕ U ξ ζ ξ ζ E δ ξ ζ ξ ζ ξ ζ ξ ζ E E E C6 C7 e + 4 as ξ T ζ ξ ζ i E The proof is compeed DOI: 436/apm88 7 Advaces i Pure Mahemaics

8 The Upper Bouds of Hausdorff Dimesios for he Global Aracor Cosider he firs variaio of 6 wih iiial codiio: T ψ + ϕψ=γ ϕψ+γ ϕψψ = ξ ζ ξ ζ > 4 P E where UV PQ T E uvpq T E ψ = P= U + εu Q= V + εv ad ϕ = is a soluio of 4 P ϕ ϕ ε I I ε I I = ε I εβ ε I β ε I εβ ε I β ϕ Γ = g u uv g uv v gu uv gv uv 4 43 Γ = 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 is easy o show form Theorem ha 4 is a well-posed problem i E u v p = u + εu q = v + εv he mappig S ε τ : { } { u τ v τ p τ u τ εu τ q τ v τ εv τ } E for ay is differeial a u v p q T ϕ = is he liear opera- o or o = + = + is freche differeiable ξ ζ ξ ζ T E : U V P Q where T U V P Q is he soluio of 4 Lemma [5] For ay orhoormal family of elemes of E T ξ ζ ξ ζ = we have ν ν ν ν ξ ζ λ + + λ ν [ = = E 45 Proof This is a direc cosequece of Lemma VI 63 of [5] Theorem If H -H 6 hold saisfyig he here exiss β > such ha he Hausdorff dimesio of global aracor Α i E saisfies ε dh Α mi N λ λ + < 46 = 6 C 4 where R is as i Lemma 6 i [] Proof Le N be fixed Cosider m soluios ψ ψ ψ of 4 DOI: 436/apm88 8 Advaces i Pure Mahemaics

9 A a give ime τ le B τ deoe he orhogoal proecio i E oo spa ψ s ψ s ψ s { } Le y T s = ξ ζ ξ ζ E B s E spa{ ψ s ψ s ψ s } wih respec o he ier produc ad orm E Suppose he ϕτ M s > τ By E = be a orhoormal basis of = 47 u v p q T E ϕτ = τ τ τ τ Α 48 y = ad Lemma we have E ε β β P ϕ s y s y s ξ E ζ Γ ϕ s y s y s E C ξ ξ + C ζ ξ C ξ ζ + C ζ ζ The by he Sobolev embeddig heorem: 5 H Ω H Ω H Ω 5 Therefore Γ ϕ s y s y s E C ξ ξ + C ζ ξ 3 + C ξ ζ + C ζ ζ 4 5 β β C ξ + ζ + ξ + ζ By Youg s iequaliy we have Γ ϕ s y s y s E µ ξ ξ + Rk ξ ξ + Rk ζ ξ + µ ζ ζ + Rk ξ ζ + Rk ζ ζ λ Rk λ Rk λ µ λ µ So exis β saisfyig We obai λ µ λ Rk 53 β ε ξ λ µ λ Rk β ε ζ DOI: 436/apm88 9 Advaces i Pure Mahemaics

10 = ϕ +Γ ϕ +Γ ϕ p s P s s s y s y s = ε β β β β ξ ζ + ξ + ζ = = ξ ζ E λ µ λ µ = λ Rk λ Rk + C + 6 ε + C λ + λ 8 If λ λ 7 = ε C + 6 = 7 he τ + q s = lim if sup sup sup p s ds R E τ τ φ ϕτ Α ε C 7 λ + λ < 6C7 = Proof fiish Refereces [] Li GG ad Zhag M 7 The Global Aracors for a Class of Noliear Coupled Kirchhoff-Type Equaios Europea Joural of Mahemaics ad Compuer Sciece 4 [] Wu JZ ad Li GG 9 The Global Aracor ad Is Dimesios Esimaio of Bossiesq Equaio wih Dampig Term Joural of Yua Uiversiy [3] Fa XM ad Zhou SF 4 Kerel Secios for No-Auoomous Srogly Damped Wave Equaios of No-Degeerae Krichhoff-Type Applied Mahemaics ad Compuaio hps://doiorg/6/amc3847 [4] Li GG ad Gao YL 7 The Global ad Expoeial Aracors for he Higher-Order Kirchhoff-Type Equaio wih Srog Liear Dampig Joural of Mahemaics Research 9 [5] Tema R 988 Ifiie-Dimesioal Dyamical Sysems i Mechaics ad Physics Applied Mahemaical Scieces 68 Spriger-Verlag New York hps://doiorg/7/ DOI: 436/apm88 Advaces i Pure Mahemaics

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