9 Inernionl Symoium on Comuing, Communicion, nd Conrol (ISCCC 9) Proc of CSIT vol () () IACSIT Pre, Singore Eonenil Decy for Nonliner Dmed Equion of Suended Sring Jiong Kemuwn Dermen of Mhemic, Fculy of Science, King Mongku Iniue of Technology dkrbng Abrc Thi er i concerned wih he energy decy of he globl oluion for IBVP o nonliner dmed equion of uended ring wih uniform deniy o which nonliner ouer force work For hi uroe, we emloyed he energy mehod [Wo-Y] nd derive he decy eime by he nonliner dming erm long he refined mehod of [M-Ik] Keyword: nonliner, decy eime, uended ring, dmed equion Inroducion In hi work, we will udy he energy decy of he globl oluion of hevy nd fleible ring wih uniform deniy uended from ceiling under grviy Suoe h nonliner ouer borbing force nd nonliner dming work o he ring in horizonl direcion in vericl lne For he derivion of he uended ring equion, ee [K-G-S], [Ko] nd [Y] e Ω be cylindricl domin (, ) (, T) nd be he finie lengh of he ring Then he bove nonliner roblem i formuled he following IBVP q u (,) u (,) u u = β u u, (,) Ω, (P) u (, ) =, (, T), u (,) = φ( ), u (,) = ψ( ), (, ), where i econd order differenil oeror of he form nd, β = (, ) = ( ), q > > re conn nd We noe h i he ecil ce of he differenil oeror μ (ee ) nd degenere he origin [Y] hve hown he eience of lmo eriodicc -oluion o IBVP for he liner equion of uended ring equion wih he qui-eriodic forcing erm, nd everl eriodic roblem of he nonliner uended ring equion hve been udied in [Y]-[Y4] nd [Y-N-M] [Wo-Y] udied he eience nd uniquene of ime-globl clicl oluion wih monoonou cubic nonliner erm u 3, when he iniil d re lrge The uroe of hi er i o how decy eime of globl oluion o he roblem (P) Noe h u q β u i he borbing erm when β < nd liner ce when β = In [Wo- Y], we roved he eience of ime-globl wek oluion o nonliner equion of uended ring wihou he dming erm by energy mehod bed on he oenil well Thi reerch, i o how he globl oluion of he roblem (P), Becue we hve he imilr energy ideniy o he roblem in [Wo-Y], Correonding uhor Tel: (83)8-94; f: 66-36-434-3 (e 685) E-mil ddre: jiui@homilcom 39
herefore we cn conruc he oenil well for he globl oluion by he me wy [Wo-Y] To derive he decy eime of he globl oluion, we ly n roch of [M-Ik], which dicued he globl eience nd energy decy o he wve equion of Kirchhoff Tye wih Nonliner Dming Term Conrry o [M-Ik], we need o ly he roerie of he oeror nd ome Poincre-Sobolev ye inequliie on ome rorie funcion ce uible for our roblem (ee [Y]) Funcion Sce, Oeror, he Bic Inequliie Definiion of Funcion Sce In hi ecion, we briefly review noion nd reul, which will be emloyed ler e R nd Z be he e of nonnegive number nd he e of nonnegive ineger, reecively n e nd Z For ny oen e O in R ( O ) nd H ( O) re he uul ebegue nd Sobolev ce, reecively In he following we ume h ll funcion re rel-vlued e μ > We denoe (, ; μ ) by Bnch ce whoe elemen f ( ) re meurble in (, ) nd ify μ / f( ) (, ) I norm i / f ( ) (, ; ) μ f d μ = In riculr, (, ; μ ) i Hilber ce wih inner roduc ( f, g ) = μ f ( ) g ( ) d μ (, ; ) Denoe H (, ; μ ) by Hilber ce, whoe elemen f ( ) nd heir weighed derivive j/ ( j f ) ( ), j =,,, belong o, norm (, ; μ ), where ( j f ) ( ) men he j -h derivive of f ( ) I / μ j ( j) μ = ( ) H (, ; ) j= f f d Denoe W (, ; μ ) by Bnch ce, whoe elemen f ( ) nd heir weighed derivive j/ ( j f ) ( ), j =,,, belong o (, ; μ ) I norm i / μ j ( j), μ = ( ) W (, ; ) j= f f d e T > nd Ω= (, ) (, T) We denoe ( Ω ; μ ) by Bnch ce, whoe elemen f (, ) re meurble in Ω nd ify j/ j k μ / f(, ) ( Ω ) I norm i / f (, ) ( ; ) μ f dd μ = Ω Ω We denoe H ( Ω ; ) by Hilber ce, whoe elemen f (, ) nd heir weighed derivive μ f(, ), j k, belong o ( Ω ; μ ) I norm i / μ j j k (, ) Ω j k f μ = f dd H ( Ω; ) H (, μ ; ) i ubce of H (, ; μ ) whoe elemen f ( ) ify f( ) = Similrly, H ( Ω ; μ ) i ubce of H ( Ω ; μ ) whoe elemen f (, ) ify f(, ) = for lmo ll (, T) K (, ; μ ) i ubce of H (, ; ) whoe elemen f = f( ) ify j =,,[( ) / ] Noe h K μ μ (, ; ) = (, ; ), μ K(, ; ) = H μ (, ; ), μ μ μ K = H H (, ; ) (, ; ) (, ; ) j H μ μ (, ; ) for 3
The Bic Proerie of μ We recll he more generl uended ring oeror μ inroduced by [K-G-S], [Ko] for μ > : μ = ( ) μ Clerly, μ coincide wih our differenil oeror for μ = Prooiion ([Y]) e μ be in () for μ > Then we hve he following erion: For f K (, ; μ ) nd g K (, ; μ ), we hve Thi lemm i he Poincre ye inequliy in μ ( μ f, g) μ = ( ) ( ) (, ; ) f g d μ H (, μ ; ) emm [Y] e μ > Then, for u H (, ; μ ), we hve u u μ (, ; ) μ (, ; ) emm 3 [Y] e nd μ > Then, for (), u W (, ; μ ) we hve μ μ μ u ( ) d c( u ( ) d u( ) d), where he conn c > deend on μ,, emm 4 [N] e Φ ( ) be non-increing nd nonnegive funcion on [, T ], T < uch h r Φ() k( Φ() Φ ( )) on [, T ], where k i oiive conn, nd r i nonnegive conn Then, we hve: (i) if we ume r >, hen where = { } [ ] m, r / r Φ() ( Φ () k r[ ] ) on [, T ], k (ii) if r =, hen Φ() Φ() e [ ] on [, T ], where k = log( k/( k )) 3 Energy eime nd Poenil well of (P) The ol energy of (P) coni of he oenil energy nd kineic energy defined follow E( u ; ) = Eu ( (, ), u(, )) = Ku ( (, )) Ju ( (, )) e E ( uu, ) be denoed by E( uu, ) = u( ) d Ju ( ) β q The oenil energy i defined J ( w) = w w( ) d q (, ; ), where = ( ( )) for w H w w d which i equivlen o H nd he kineic energy i defined K( w) = ( w (, )) d for w H ( Ω ; ) The nonliner erm in (P) ifie he following condiion (A) [Wo-Y] (A) f ( u, ) i of C -cl in (, u) [, ] Ru nd monoone decreing in u R, nd ifie r (, ) r C u uf u C u for ny [, ] nd u R Here C, C > re conn nd r > From (A), here ei conn λ uch h J ( λ u) i monoone increing in λ (, λ ) for ny fied u By he me wy [Wo-Y], he oenil well W for (P) round he origin i defined by 3
W = { u H (, ; ); J( λu) < d, λ } e λ = λ ( u) > be he fir vlue of λ which J ( λ u ) r o decree ricly The deh d of he oenil well W i defined by d = inf J( λ ( u) u) u H (, ; )\{} We ee [Wo-Y] h < d < nd W re oen nd bounded in H (, ; ) We ume he following condiion on he iniil d (B) e φ W nd ψ (, ; ) φ nd ψ ify he following condiion ( ψ φ ) β q ( ) ( ) q w( ) d< d Min Theorem Aume (A) nd (B) Then roblem (P) h wek oluion u H ( Ω ; ) ifying u(, ) W for ll (, T) Furhermore, we hve he decy eime: if =, hen k E( u ( ), u ( )) Ce on [, ), nd if >, hen /( ) Eu ( ( ), u ( )) C( ) on [, ), where kc, nd C re ny oiive conn deending on iniil d Proof Min Theorem Mulilying eq (P) by u nd inegring over [, ] (, ), we hve (3) = β u u dd u udd u u dd u u dd e u conider he econd erm of eq(33) (3) u ud = u ( u) d = ( u) d = ( u) We ue Prooiion o obin (33) uudd = ( u) d u ( ) u ( ) = where u = ( ) u d From (3) - (33), we hve β q u dd = ( ( )) ( ) ( ) ( ) (34) u d u u u d q = E () E ( ) = D (), when we e E( ) = E( u, u ) Mulilying eq(p) by u nd inegring over [, ] (, ), we hve (35) ( ) = J () d u() dd (( u), u( ) u ( ), u( )) u() u(), u () d Noe h [ ] nd [ ] (36), /4 3/4, By lying Holder' inequliy o he econd erm of eq(35), we hve ( )/( ) /( ) ( )/( ) {( )}/ u() dd u() d d /( ) ( ) /( ) ( ) ( ) /( ) = me u d d = me D (, ) ( ) (, ) ( ) The hird erm of eq(35) i eimed by he men vlue heorem /{( )} (37) u( i) me(, ) D( ) From emm 3, he l erm of eq (35) i eimed by /( ) /( ) () () () () u u dd u d u d d () () = u u d q 3
, W C u u ( ) d, nd le u conider he oenil energy nd emm, E () Ju ( ()) = ( ) q ( ) ( ) u d β u d q hen we hve (38) (39) (3) u, W u u dd ce D From (35) - (38), we obin () () () () ( ) ( ) ( ) ( ) J () d me(, ) D() 4 me(, ) D() u u() c E() D() Hence, i follow from (36) nd (39) h E () d () (()) u d Ju d ( ( ) ( ) ( ) ( ) c D D E D E ( ) ) = 4 Acknowledgemen I would like o ere my dee griude o Profeor MYmguchi for hi vluble uggeion I would like o lo hnk Profeor TMuym for giving me he inigh of hi reerch work 5 Reference [] R A Adm, Sobolev Sce, Acdemic Pre, 975 [] B G Korenev, Beel Funcion nd heir Alicion, Tylor nd Frnci Inc, [3] N S Kohlykov, E V Gliner nd M M Smirnov, Differenil Equion of Mhemicl Phyic, Mocow, 96 (in Ruin) Englih Trnlion: Norh-Hollnd Publ Co, 964 [4] T Muym nd RIkeh, On Globl oluion nd energy decy for he wve equion of Kirchhoff Tye wih Nonliner Dming Term, J Mh Anl Al 4, (996), 79-753 [5] M Nko, Aymoic biliy of he bounded or lmo eriodic oluion of he wve equion wih nonliner diiive erm, J Mh Anl Al 58 (977), 336-343 [6] J Sher, The eience of globl clicl oluion of he iniil-boundry vlue roblem for u u 3 = f, Arch Rionl Mech Anl, (966), 9-37 [7] C J Trner, Beel Funcion wih Some Phyicl Alicion, Hr Publihing Co, New York, 969 [8] G N Won, Theory of Beel Funcion, Cmbridge Univeriy Pre, 96 [9] J Wongwdi nd M Ymguchi, Globl oluion of IBVP o nonliner equion of uended ring, Tokyo J Mh 3, No (7), 543-556 [] J Wongwdi nd M Ymguchi, Globl clicl oluion of IBVP o nonliner equion of uended ring, Tokyo J Mh 3, No (8), 35-373 [] M Ymguchi, Almo eriodic ocillion of uended ring under Quieriodic liner force, J Mh Anl Al 33, No (5), 643-66 [] M Ymguchi, Free vibrion of nonliner equion of uended ring, rerin [3] M Ymguchi, Globl mooh oluion of IBVP o nonliner equion of uended ring, J Mh Anl Al 34, No (8), 798-85 [4] M Ymguchi, Infiniely mny eriodic oluion of nonliner equion of uended ring, FUNKCIAAJ EKVACIOJ-SERIO INTERNACIA 5, No (8), 45-67 [5] M Ymguchi, T Ngi nd K Mukne, Forced ocillion of nonliner dmed equion of uended ring, J Mh Anl Al 34, No (8), 89-7 33