Global Attractor for a Class of Nonlinear Generalized Kirchhoff-Boussinesq Model

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1 Inernaional Journal of Modern Nonlinear Theory and Applicaion, 6, 5, 8-9 Publihed Online March 6 in SciRe hp://wwwcirporg/journal/ijmna hp://dxdoiorg/36/ijmna659 Global Aracor for a la of Nonlinear Generalized Kirchhoff-Bouineq Model Penghui Lv, Ruijin Lou, Guoguang Lin Mahemaical of Yunnan Univeriy, Kunming, hina Received 5 December 5; acceped March 6; publihed 7 March 6 opyrigh 6 by auhor and Scienific Reearch Publihing Inc Thi work i licened under he reaive ommon Aribuion Inernaional Licene ( BY hp://creaivecommonorg/licene/by// Abrac In hi paper, we udy he long ime behavior of oluion o he iniial boundary value problem for a cla of Kirchhoff-Bouineq model flow u u β u u div ( g ( u u h( u f ( x = We fir prove he wellne of he oluion Then we eablih he exience of global aracor Keyword Kirchhoff-Bouineq Model, Srongly Damped, Exience, Global Aracor Inroducion In hi paper, we are concerned wih he exience of global aracor for he following nonlinear plae equaion referred o a Kirchhoff-Bouineq model: where i a bounded domain in ( = in ( u u β u u div g u u h u f x u x, = u x ; u x, = u x, x ( (,, (,, u x = u x = x (3 N, and, How o cie hi paper: Lv, PH, Lou, RJ and Lin, GG (6 Global Aracor for a la of Nonlinear Generalized Kirchhoff-Bouineq Model Inernaional Journal of Modern Nonlinear Theory and Applicaion, 5, 8-9 hp://dxdoiorg/36/ijmna659 β are poiive conan, and he aumpion on g( u, h( u will be pecified laer Recenly, huehov and Laiecka [] udied he long ime behavior of oluion o he Kirchhoff-Bouineq plae equaion

2 P H Lv e al wih clamped boundary condiion u ku u = div f u f u f u ( (, u x u( x, =, = v wih where v i he uni ouward normal on Here k > i he damping parameer, he mapping f : and he mooh funcion f and f repreen (nonlinear feedback force acing upon he plae, in paricular, m When σ f u = u u, f u = u u f u u u u u f u =, alo conidering he ( wih a rong damping, hen ( become a cla of Krichhoff model ariing in elaoplaic flow, = = and { σ } (5 u div u u u u h u g u = f x (6 which Yang Zhijian and Jin Baoxia [] udied In hi model, Yang Zhijian and Jin Baoxia gained ha under raher mild condiion, he dynamical yem aociaed wih above-menioned IBVP poee in differen phae pace a global aracor aociaed wih problem (6, ( and (3 provided ha g and h aify he nonexploion condiion, wih ρ G lim inf (7 m ρg ( < <, G( = g( τ dτ, δ (,, θ,, β > uch ha g lim inf (8 m< N N m<, and h = h h and here exi conan ( h ( v v ( h ( v v θ ( h ( v v v v,,,, β Zhijian Yang, Na Feng and Ro Fu Ma [3] alo udied he global aracor for he generalized double diperion equaion ariing in elaic waveguide model In hi model, g aifie he nonexploion condiion, (9 u u u u u g u = f x ( g lim inf λ, g, ( p N where λ ( > i he fir eigenvalue of he, and < p < a N = ; p p a N 3 N T F Ma and M L Pelicer [] udied he exience of a finie-dimenional global aracor o he following yem wih a weak damping ( σ in (, u u u ku f u = h L ( xxxx x x wih imply uppored boundary condiion and iniial condiion u, = u L, = u, = u L, =, (3 xx xx 83

3 P H Lv e al (,, (,, (, u x = u x u x = u x x L ( ρ f ˆ = f τ d τ f, ρ >, p where σ z = z, p, k >, and f (, For more relaed reul we refer he reader o [5]-[8] Many cholar aume ( m div g u u = u u, o make hee equaion more normal; we ry o make a differen hypohei (pecified Secion, by combining he idea of Liang Guo, Zhaoqin Yuan, Guoguang Lin [9], and in hee aumpion, we ge he uniquene of oluion, hen we udy he global aracor of he equaion Preliminarie For breviy, we ue he follow abbreviaion: p p k k L = L, H = H, H = L, =, = p L p L k k wih p, and V = H H, where H are he L -baed Sobolev pace and H are he compleion of k ( in H for k > The noaion (, for he H-inner produc will alo be ued for he noaion of dualiy pairing beween dual pace In hi ecion, we preen ome maerial needed in he proof of our reul, ae a global exience reul, and prove our main reul For hi reaon, we aume ha (H g (, G( lim inf ( where m 3 g lim inf G = g τ dτ, < ρ <, and when N, where m < a N = ; 6 N m m N ρg ( 3 m g (,, ( a 3 N ; and m = a N 5 λ (H h and h ( u <, λ > i he fir eigenvalue of he Now, we can do priori eimae for Equaion ( Lemma Aume (H, (H hold, and ( u, u V H, f H uv, V H, and (-(3 aifie where v = u εu, ( Then he oluion (, V H k k (3 uv of he problem H uv, = u v e e ( λ λ < ε < min,,, and β βε ( H = v u u G u d, xv = u εu, hu here exi E and = >, uch ha ( uv u v E (, = > (5 V H Remark ( and ( imply ha here exi poiive conan and, uch ha ρ G, g G (6 8

4 P H Lv e al Proof of Lemma Proof Le v = u εu, hen v aifie ( ε ( ε ε β βε ( v v u v u u = div g u u h u f x (7 Taking H-inner produc by v in (7, we have d v v uv v uv v d ( ε ( ε ε (, β βε (, (, ( ( ( = div g u u, v h u, v f x, v (8 λ λ Since v = u εu and < ε < min,,, by uing Holder inequaliy, Young inequaliy and β Poincare inequaliy, we deal wih he erm in (8 one by one a follow, 3 v v (9 ( ε and uv u v v u ( ε ε (, ε ε ε ε λ λ ε u v ( βε d βε ( uv, = βε ( uu, εu = u βε u ( d (, (, (, ε d u v = u v = u u u = u ε u ( d ( div( g( u u, v = ( g( u u, u ε u = g( u u ud x ε ( g( u u, u d = G( u d x ε ( g( u u, u d By (9-(3, i follow from ha By (6, we can obain ( hu v ( f( x v ( d (, d v u βε u G u x d 3ε v u βε u ε g u u u β v,, ( ( = ( ( ( = ερ ( ( ερ ε ε g u u, u ε g u u dx ε ρg u dx Subiuing (5 ino (, we receive G u dx dx d x (3 ( (5 85

5 P H Lv e al d v u βε u ( G( u dx d 3ε v u βε u ερ ( G( u d x β v h u, v f x, v ε dx ερ d x ( ( By uing Holder inequaliy, Young inequaliy, and (H, we obain Then, we have (, (6 f x v f v f 8 v (7 ( hu, v = ( hu, v h ( u u vdx h ( u βε h u u v v u βε ( d 3ε v u βε u G u dx v u d h ( u 3βε u ερ ( G( u dx β v βε f ε dx ερ d x λ Becaue of < ε <, we ge β 3 Subiuing ( ino (9 ge Taking = ε ερ = { ε ερ} (8 (9 u 3 βε u u βε u ( ( d ( d v u βε u G u x d v ε u βε u ερ G u dx h ( u β v f ε dx d x ερ βε min,, min,, hen ( d d d : d H H f ε x x ερ = ( where = βε ( ( H v u u G u x, by uing Gronwall inequaliy,we obain d H H e ( e (3 From (H : m g, and m < a N = ; 6 N m m N a 3 N ; m = 86

6 a 5 N, we have P H Lv e al m 3 G u dx u, according o Embedding Theorem, hen H L m 3 βε βε k = min, = >, hen we have λ λ Then So, here exi E and ( V H k k, le H uv, = u v e e ( = >, uch ha ( uv lim, V H k ( uv u v E ( (5, V H = > (6 f H, h (, hen he oluion uv,, and Lemma In addiion o he aumpion of Lemma, if (H 3 : 3 ( uv, of he problem (-(3 aifie where λ λ v = u εu, < ε < min,, β = >, uch ha E and Proof Taking H-inner produc by d d ( H uv, = u v e e (7 9 3 k k, and ( βε H = v u u, hu here exi ( uv u v E 3 (, = > (8 v = u ε u in (7, we have (, (, (, v ε v ε ε u v βε u v v β v ( ( ( = div g u u, v h u, v f x, v (9 Uing Holder inequaliy, Young inequaliy and Poincare inequaliy, we deal wih he erm in (9 one by one a follow, and 3 v v (3 ( ε ε ε u v = u v u v λ ( ε ε (, ( ε ε (, ε ε ε v u u v λ 8 8 (3 βε d βε ( u, v = βε ( u, u ε u = u βε u (3 d ( ε d u, v = u, v = u, u u = u ε u (33 d Subiuing (3-(33 ino (9, we can obain ha 87

7 P H Lv e al ( d 7ε v u βε u v u βε u β v d 8 ( div g u u, v h u, v f x, v By uing Holder inequaliy, Young inequaliy, and (H, (H 3, we obain (, (, (3 f x v f x v f v f = 8 v (35 ( ( ( ( h ( u u, v ( h ( u u, v h u, v = h u u, v = h u u h u u, v h u u v h u u v β h u u h u u v β β By uing Gagliardo-Nirenberg inequaliy, and according he Lemma, we can ge n n n 3 u u u : = Then, we have By uing he ame inequaliy, we can obain ( ( β (36 hu, v β,,, 3, v h u h u u (37 ( div( g ( u u, v ( u, v ( u u, v = g u u g u u, v u u, v ( u v u u v β v u u u β β ( ( By uing Gagliardo-Nirenberg inequaliy, and according he Lemma, we can ge ( n m ( n ( 5 ( n : 6 u u u =, inequaliy, we have where n nε =, hen ( (, n 7 n n (38 u u u Then, by uing Young β div g u u v v u n n n n n n n n n n 6 7 u u n n β β ( ( β ε div g u u v v u 8( n ε β 6 7 u (39,,,,,,, ( 8 88

8 P H Lv e al Subiuing (35, (37, ( ino (3, we receive λ Becaue of < ε <, we ge β d 3ε v u βε u v u d βε u β v f 8 ( Taking = min, ε = ε, hen where 3 u βε u u βε u ( d d H H f 8 : = 9 H = v u βε u, by Gronwall inequaliy, we have βε βε Le k = min, = >, o we ge λ λ Then So, here exi E and (3 9 H H e ( e ( ( H uv, = u v e e (5 9 3 k k = >, uch ha ( uv 3 9 lim, k ( uv u v E 3 ( (6, = > (7 3 Global Aracor 3 The Exience and Uniquene of Soluion Theorem 3 Aume ha ( H g (, where G = g τ dτ, ρ G lim inf g lim inf < < and m 3 ρg ( 3 λ > i he fir eigenvalue of he, and when N, m g (,, 89

9 P H Lv e al where m < a N = ; 3 ( H ( u, u H H 6 N m m N λ, f H, h and h ( u < Then he problem (-(3 exi a unique mooh oluion a 3 N ; m = a N 5 3 ( uu, L [, ; H ( H ( Remark We denoe he oluion in Theorem 3 by S( u, u = ( u, u Then S compoe a 3 coninuou emigroup in Proof of Theorem 3 Proof By he Galerkin mehod and Lemma, we can eaily obain he exience of Soluion Nex, we prove he uniquene of Soluion in deail Aume uv, are wo oluion of (-(3, le w= u v, hen w x, = w x =, w x, = w x = and he wo equaion ubrac and obain Taking H-inner produc by By (H, (H w w w w= div g u u g v v h u h v β d d w in (3, we ge β w w w w ( ( = div g u u g v v, w h u h v, w ( ( h( u h( v, w = ( h( u h( v, w = ( h ( ξ w, w ( h ( ξ ( ξ h w w w w d ( ( ( = ( ( θ θ dθ = ( ( g ( Uθ Uθ g( Uθ d θ w, w ( Uθ d θ w, w div g u u g v v, w g U U d θ, w ( (, ( d, θ θ w w U w w w w U d θ θ w w ( m β w w U d w θ θ β where { } { } min uv, ξ max uv,, Uθ = θu θ v, < θ < By uing Gagliardo-Nirenberg inequaliy, and according he Lemma,we can ge ( n m θ ( n ( θ θ ( n U U U : = Then, we have ( 3 ( β Subiuing (33, (35 ino (3 div g u u g v v, w w,,, w (3 (3 (33 (3 (35 9

10 P H Lv e al Taking Then ( h ( ξ d ( w w β w 3 w w d ( h ( ξ 3 B = max, d d By uing Gronwall inequaliy, we obain So, we can ge Tha how ha w ( w ( w B w w (36 (37 ( B w w w w e (38 w becaue of w ( x =, w ( x = w =, w = Tha i Therefore w( x, = u = v We ge he uniquene of he oluion So he proof of he Theorem 3 ha been compleed 3 Global Aracor Theorem 3 [] Le X be a Banach pace, and { }( S : X X, S( = S S( (,, S( = I, here I i a uni operaor Se condiion S( i bounded, namely R >, u R, i exi a conan ( R, o ha X S u ( R ( [, ; I exi a bounded aborbing e B X X, namely, B X ( S B B S are he emigroup operaor on X S aify he follow, i exi a conan, o ha ; here B and B are bounded e 3 When >, S( i a compleely coninuou operaor Therefore, he emigroup operaor S( exi a compac global aracor A Theorem 33 Under he aume of Theorem 3, equaion have global aracor 3 where H 3 A= ω B = S B, {, :, 3 = 3 H H } B = uv uv u v E E, B i he bounded aborbing e of H and aifie S A= A> ; lim di S B, A =, here, 3 B and i i a bounded e, 9

11 P H Lv e al (, up inf 3 di S B A = S x y x B, y A Proof Under he condiion of Theorem 3, i exi he oluion emigroup 3 3 : S H H H H ( From Lemma -Lemma, we can ge ha { ( uv, 3 R}, S, here 3 X =, 3 B i a bounded e ha include in he ball (, 3 = 3 3, (, (, S u v u v u v R u v B H H H H 3 Thi how ha S( i uniformly bounded in 3 ( Furhermore, for any ( u, v, when max {, } 3 3, we have S u v u v E E, = H H So we ge B i he bounded aborbing e 3 (3 Since V H i compac embedded, which mean ha he bounded e in V3 H i he compac e in V H, o he emigroup operaor S( exi a compac global aracor A Theorem 33 i proved Acknowledgemen The auhor expre heir incere hank o he anonymou reviewer for hi/her careful reading of he paper, giving valuable commen and uggeion Thee conribuion grealy improved he paper Funding Thi work i uppored by he Naional Naural Science Foundaion of People Republic of hina under Gran 657 Reference [] huehov, I and Laiecka, I (6 Exience, Uniquene of Weak Soluion and Global Aracor for a la of Nonlinear D Kirchhoff-Bouineq Model AIM Journal, 5, [] Yang, Z and Jin, B (9 Global Aracor for a la of Kirchhoff Model Journal of Mahemaical Phyic, 5, Aricle ID: 37 [3] Yang, Z, Feng, N and Ma, TF (5 Global Aracor for he Generalized Double Diperion Equaion Nonlinear Analyi, 5, 3-6 hp://dxdoiorg/6/jna6 [] Ma, TF and Pelicer, ML (3 Aracor for Weakly Damped Beam Equaion wih p-laplacian Dicree and oninuou Dynamical Syem Supplemen, [5] Yang, Z and Liu, Z (5 Exponenial Aracor for he Kirchhoff Equaion wih Srong Nonlinear Damping and Supercriial Nonlineariy Applied Mahemaic Leer, 6, 7-3 hp://dxdoiorg/6/jaml59 [6] Kloeden, PE and Simen, J (5 Aracor of Aympoically Auonomou Quai-Linear Parabolic Equaion wih Spaially Variable Exponen Journal of Mahemaical Analyi and Applicaion, 5, 9-98 hp://dxdoiorg/6/jjmaa69 [7] Silva, MAJ and Ma, TF (3 Long-Time Dynamic for a la of Kirchhoff Model wih Memory Journal of Mahemaical Phyic, 5, Aricle ID: 55 [8] Lin, GG, Xia, FF and Xu, GG (3 The Global and Pullback Aracor for a Srongly Damped Wave Equaion wih Delay Inernaional Journal of Modern Nonlinear Theory and Applicaion,, 9-8 hp://dxdoiorg/36/ijmna39 [9] Guo, L, Yuan, ZQ and Lin, GG ( The Global Aracor for a Nonlinear Vicoelaic Wave Equaion wih Srong Damping and Linear Damping and Source Term Inernaional Journal of Modern Nonlinear Theory and Applicaion,, -5 hp://dxdoiorg/36/ijmna5 [] Lin, GG ( Nonlinear Evoluion Equaion Yunnan Univeriy Pre 9