Luận văn Khảo sát phương trình parabolic phi tuyến trong miền hình cầu

KhilOsatphuongtrinhparabolicphi tuyin trangmi~nhinhcdu trang28 CHUONG4 sV TONT~I,DUYNHA.TvA ONDJNHNGHl~MT-TUANHoAN CUAPHUONGTRINH NHIET PHI TUYEN. Trongchuangnay,chungt6inghienClmnghi~mT-tu~nhoancua baitoangiatribienphituySnsauday (4.1) (4.2) (4.3) (4.4) Ut-(Urr+~Ur)+F(U)=f(r,t), O<r<l, O<t<T, I lim rur(r,t) ! <+00,ur(l,t) +h(t){u(l,t)- uo)=0,r~O+ u(r,O)=u(r,T), II P-z F(u) =u u, trongd6 2~p <3, Uo lelcach&ngs6chotruac,h(t),f(r,t) lelcachams6 chotruacT-tu~nhoantheot,thoacacgiii thiStsau (Hz) (H;) (H~) UoE JR, hE wi""(O,T),h(O)=her),h(t)~ho>0, f E CO([0,T}H), f(r,O) =f(r,T). Nghi~mySucuabaitoan(4.1)-(4.4)duQ'cthanhl~pnhusautu baitoan biSnphansau: Tim u ELZ(O,T;V)nL'"(O,T;H) saocho u' E LZ(o,T;H)vau(t)thoaphuang trinhbiSnphansau: (4.5) T T f(u'(t),vet)dt+J[(ur(t),vr(t))+h(t)u(l,t)v(l,t)}it 0 0 T + f(F(u(t)),v(t))dt° T T = f(f(t), v(t))dt+uofh(t)v(l,t)dt,'\IvE LZ(O,T;V), 0 0 vadi@uki~nT-tu~nho~m H9CvienNguyln ViiDzilng Khaosatphuongtrinhparabolicphi tuyin trongmi~nhinhcdu trang29 (4.6) U(O)=u(T). Trong ph~nmlY,chungtoi sechungminhbai tmln(4.1)-(4.4)co duynhatmQtnghi~my~uT-tu~nhoanvanghi~mnayclingemdinhd6ivai I, h,uo. 4.1.S1}'tAnt~ivaduynhAtcuanghi~my~uT-tuftnhoan Lienquaild~ns\l't6nt~ivaduynhatnghi~my~uT-tu~nhoancua baitmln(4.1)-(4.4)tacodinhly sau. Dinh If 4.1.ChoT>0 va(H), (H~),(H~)dung.Khi do,hai loan(4.1)- (4.4)co duynhdtmr}tnghi?myiu T-tudnhoanU E L2(0,T;V)nL"'(0,T;H), saDcho u'EL2(0,T;H),rI/PUELP(Qr). Chung minh.Chungminhg6mnhiSubuac. Bmyc1. PhU'O'ngphap Galerkin. Ky hi~ubai {Wj},j=1,2,...la mQtcO'sa tr\l'cchu~ntrongkhonggianHilberttachduQ'CV. Ta tim Urn (t) theod~ng (4.7) rn urn(t)=LCmj(t)Wj' j=l trongdo Crnj(t), 1::;j ::;m thoah~phuO'ngtrinhvi phanphituy~n (u~(t),Wj) +(urnr,Wjr) +h(t)urn(1,t)w/1) +(F(urn(t)), Wj) =(/(t), Wj)+uoh(t)wj(l),1::;j::; m, (4.8) vadiSuki~nT-tu~nhoan (4.9) Urn (0)= Urn(T). f)~utien,taxeth~phuO'ngtrinh(4.8)vadiSuki~nd~u ( 4.9') Urn(0)=UOrn' trongdo UOrn la hamtrongkhonggian m chiSusinh bai cac ham Wj' j =1,2, Khi do, ta thuduQ'cmQth~m phuO'ngtrinhvi phanthuemg H9CvienNguyln ViiDziing Khaosatphuongtrinhparabolicphi tuyin trongmiJn hinhcdu trang30 phi tuySnv6i cac~nhamCmj(t),1<5,j <5,m,vacacdiSuki~nd~u(4.9').DS th~yr~ngt6nt~ium(t) co d~ng(4.7)thoa(4.8)va (4.9')v6i h~ukhApnO'i tren0<5,t <5,Tm'v6imQtTm'0<Tm<5,T.Cacdanhgiatiennghi~msaudaycho pheptal~yTm=T v6i mQim. Bmyc2.Danh gia tH~nghi~m. NhanphuO'ngtrinhthu j cuah~(4.8)b6i Cmj(t),vasaudol~ytfmgtheoj, taduQ'c 1 ~llum(t)112+21IUmr(t)112+2h(t)u;(l,t)+2Jr2Ium(r,t)IPdr (4.10) dt 0 =2(f(t),um(t))+2uoh(t)um(l,t). Tir giii thiSt(H~) vab~td~ngthuc(2.9),tasuyrar~ng (4.11) 211umr(t)112+2h(t)u~(l,t) ~CiliUm(t)II~, v6i CI =mill{I,ho}. Do do,tasuytir (4.10),(4.11)r~ng :tllum(t)112+CIIIUm(t)II~+2fr2Ium(r,t)IP dr0 <5,2(f(t),um(t))+2uoh(t)um(l,t) <5, ~llf(t)112+51IUm(t)112+;Iuonh[ +51IUm(t)II~ =~llf(t)112+;Iuonhll: +251IUm(t)II~,'15>0. (4.12) ChQn5>0 saocho (4.13) CI-25=C2>0. Do do,tir(4.12),(4.13)tathuduQ'c H9CvienNguyln ViiDziing Khao satphuongtrinhparabolicphi tuyin trangmiJn hinhcdu trang31 ~llum (t)112+ C211um(t)112dt I ~ ~llum(t)112+C21Ium(t)II~+2fr2Jum(r,tWdr 0 ~;luol21lhll:+~Ilf(t)112=~(t). Nhanb~td~ngthuc(4.14)b6i eCz! vasaudol~ytichphantheot tathu (4.14) duQ'c ! (4.15) Ilum(t)112~lIuomI12e-Cz!+e-Cz! f~(s)eczsds. 0 Cho T >0, taxethams6sau ~ { (eel!-It f~(s)eczSds,O<t~T,(4.16) R(t)= 0 hi (0) / C2, t = O. KhidoRE Co[O,T].Ta d~tR =max~R(t).Ta thuduQ'ctir(4.15),(4.16)r~ngo,;,!,;,r nSu lIuomll~R, khido (4.17) Ilum(t)11~R, i.e.,Tm=T v6i mQi m. GQi Bm(O,R)la quac~udongtam0, bankinh R trongkhonggianm chiSu sinhb6i cachamWj' j =1,2,...,d6iv6i chu~n11.11. Xet anhXI;!Fm :BJO,R) ~ BJO,R)chob6icongthuc (4.18) Fm(Uom)= Um(T). TasechungminhdngFm lamQtanhXI;!co. Giasu UOm'VOmEBJO,R) vad~tm(t)=um(t)-vm(t),trongdo um(t)va vm(t) la cacnghi~mcuah~(4.8)tren [O,T]thoacacdiSuki~nd~uum(O)=UOmva Vm(0)=VOm'l~nluQ't.Khi do, m(t) thoah~phuangtrinh vi phansauday H9CvienNguyln ViiDziing Khaosatphuongtrinhparabolicphi tuyin trongmiJn hinhcdu trang32 «D~ (t), Wj ) + «D mr(t), Wjr) + h(t)<Dm(1,t)Wj (1) (4.19) / p-z p-z \. =-\lum(t)1 um(t)-lvm(t)1vm(t),Wj/'l~J~m, vadiSuki~nd~u (4.20) <Dm(O)=Uam-v am' B~ngcachtinhtoangi6ngnhu6 chuang3,tathuduQ'c ~II<Dm (t)llz + 211<Dmr (t)llz+2h(t)l<Dm(l,t)lz (4.21) dt = -2(lum(t)IP-zUm(t) -Iv m(t)IP-zVm(t),um(t) - vm(t)) ~ 0 Nhavao(4.11),tasurtir(4.21)r~ng (4.22) ~11<Dm(t)llz+C111<Dmr(t)II~ O.dt Tich phanb~td~ngthuc (4.22),ta thu duQ'c -.!.TC[ (4.23) IIUm(T)-vm(T)II~eZ IIUam-vamll, i.e.,Fm:Bm(O,R)~ BJO,R) laanhx'ilco.Dodotant'iliduynh~tUamE Bm(O,R) saocho Uam= Fm(uam)= Um(T). Do do, v6i mQi m, tan t'ili m9t ham UamE Bm(O,R)saochonghi~mcuabai toangiatrj band~u(4.8),(4.9')la m9tnghi~mT-tu~nhoancuah~(4.8). Nghi~mnayclingthoab~td~ngthuc(4.17)v6i h~uhStt E [o,T]vanha (4.14),tasurra t t 1 (4.24) IIUm(t)llz+Cz filum(s)lI~ds+2fds frzlum(r,s)IPdr ~C3, a a a trongdo C3lam9th~ngs6d9Cl~pv6i m. M~tkhac,b~ngcachnhanphuangtrinhthu j cuah~(4.8)b6i c~,l~ytlmg theoj vasaudol~ytichphand6iv6i biSnthaigiantir0 dSnT, tathu duQ'c H(JcvienNguyin VflDzflng KhilOsatphlfO'ngtrinhparabolicphi tuyin trongmiJn hinhcdu trang33 T ITd IT d fllu~(t)112dt+-f-llumr(t)112dt+- fh(t)-u~(l,t)dt 0 2 0 dt 2 0 dt T (4.25) +f(lum(t)IP-2Um(t),u~(t)}dt 0 T T = f(J(t),u~ (t))dt+Uofh(t)u~(l,t)dt. 0 0 Tir (4.9') tath~yr~ng Td f-Ilumr (t)112dt =0, 0dt T 1 f( )f(lum(t)IP-2um(t),u~(t)}dt=~fr2dr ~Ium(r,tr t0 P 0 0 dt = ~fr2 ~um(r,T)IP -Ium (r,O)IP ~r = O. Po Do do,d~ngthuc(4.25),nhaitchphantUngph~n,tathuduQ'c T T IT d T (4.26) fllu~(t)112dt=f(J(t),u~(t))dt+- fh'(t)-u~(l,t)dt-uofh'(t)um(I,t)dt. 0 0 2 0 dt 0 Sail cling, nha VelO(4.24),(4.26),tasuyrab~td~ngthucsail T T T T 2 fllu~(t)112dt ~ fIIJ(t)112dt + fllu~ (t)112dt +IIh'll",fu~(I,t)dt 0 0 0 0 T (4.27) +21uolllh't]um(l,t)~t 0 T T T ~ fllu~(t)1I2dt +]IJ(t)1I2dt+411h'tfilum(t)II~dt 0 0 0 T +41UolIlh'IL filum(t)lIvdt 0 T T T ~ fllu~(t)1I2dt +fIlJ(t)112dt +411h'tfilum(t)II~dt 0 0 0 +4v'Tlil, Illh'll.(~~m (')II~dtr T ~ fllu~(t)112dt +C4 , 0 KhilOsatphuongtrinhparabolicphi tuyin trongmiJn hinhcdu trang34 trongdo C4la ill9th~ngs6d9Cl~pv6'im. V~y T (4.28) ~Iu~(t)112dt~C4, v6'i illQi m. 0 M~tkhac,tir(4.24)tacodanhgia (4.29) Ids flr2!p'IUrn(r,s)IP-2Urn(r,sf' dr = Ids fr21urn(r,s)IPdr ~!C3' 0 0 0 0 2 BU'o-c3. Qua gio-ih~n. Do (4.24),(4.28),(4.29)tasur ra r~ng,tfmt<;liill9t dayconcuaday {uJ, v~nky hi~ula {urn}saocho (4.30) Urn~ U trong LOfO(O,T;H)ySu*, (4.31) Urn~u trong L2(0,T;V)ySu, (4.32)u~~u' trongL2(0,T;H)ySu, (4.33) r2!Purn~r2!putrongY(QT) ySu. Tru6'chSt,tanghi~illl<;lir~ng (4.34) u(O)=u(T). V6'iillQi vEH, tacotir(4.9)r~ng T (4.35) f(u~(t),v)dt=(urn(T) - Urn(0),v)=o. 0 Tasurtir(4.32)va(4.35)r~ng T T (4.36) f(u~(t),v)dt~ f(u'(t),v)dt=0,khi m~ +00, 0 0 Tinh toantuongt\1'nhu(4.35),taclingcod~ngthuc T (4.37) (u(T)-u(O),v)=f(u'(t),v)dt=O,VvEH, 0 vadodo(4.34)dung. H9CvienNguyln ViiDziing KhilOsatphuongtrinhparabolicphi tuyin trangmi~nhinhc6u trang35 Dungb6dS2.11vStinhcompactcuaLions [3],apd\1ngvao(4.31),(4.32) tacothStrichratirday{urn}mQtdayconv§,nkyhi~ula {urn}saocho (4.38) Urn -) U m~nhtrong L2(O,T;H). Theodinhly Riesz-Fischer,tir(4.38)tacothStrichramQtdayconcuaday {urn}v§,nkyhi~ula{uJ saocho (4.39) Urn(r,t) -) u(r,t) a.e.(r,f) trong QT=(0,1)x(O,T). Do Uf-7IUIP-2u lien t\1C,taco (440) 21' I I P-2 21 ' I I P-2 . r P Urn(r,f) Urn(r,f) -) r P u(r,f) u(r,f) a.e. (r,f) trong QT' Ap d\1ngb6dS2.12vSS\lhQit\1ySutrongLq (QT) v6i , 2/' 2/' II P-2 2/' 2/' II P-2N=2,q=p,Grn=r PF(urn)=r Purn urn,G=r PF(u)=r Pu u. Tir (4.29),(4.40)r~ng (4.41) r2IP'lurnIP-2urn-)r2IP'luIP-2utrongy'(QT) ySu. Ky hi~ug;(f)=~sinC:).i =1,2,...lamQtccysa tf\lCchu~ntrongkhong gianHilbertth\lc L2(0,T).Khi dot~p{g;Wj:i, i=1,2,...}cfingthanhl~pmQt ccysatr\lcchu~ntrongkhonggianL2(0,T;V). Nhanphucyngtrinhthil i cua(4.8)cho g;(f),vasaildol~ytichphand6iv6i biSn thai gian f, 0~f ~T, tathu duQ'c T T f( u~(f), Wj ;g; (f)<if+f( urnr(f), Wjr;g; (f)df 0 0 (4.42) T T + fh(t)urn(1,f)w/1)g; (f)df +f(lurn(f)IP-2Urn(f), Wj)g; (f)df 0 0 T T =f(f(f), Wj ;g;(f)df+fuoh(f)W/1)g;(f)df,Vi =1,2,...,m,Vi E N. 0 0 DSnghiencUuvSvi~cquagi6ih~ncuas6h~ngphituySnlurn(f)IP-2 Urn(f) trong(4.42),tasud\1ngb6dSsail H9CvienNguyin VfiDzfing Khao satphuongtrinhparabolicphi tuyin trongmiJn hinhcdu trang36 BBd~4.1. T T J~~oof([urn(t)IP-2Urn(t), wi )gi(t)dt = f\lu(t)IP-2U(t),Wi)gi(t)dt, Vi, j =1,2,... 0 0 ChungminhbBd~4.1. Chuyr~ng(4.41)tuO'ngduO'llgv6i TIT 1 fdt fr2/P'[urn(t)IP-2urn(t)(r,t)dr~ ffr2/P'lu(t)la-lu(t)(r,t)dt (4.43) 0 0 00 VE (U' (QT))=LP(QT). M?t khac,taco T T 1 f(lurn(t)IP-2Urn(t),Wi)gi(t)dt = f fr21urn(t)IP-2Urn(t)w/r)gi(t)drdt (4.44) 0 00T 1 (= ffr2/p'IUrn (t)IP-2Urn (t)Xr2/PWi(r)g;Ct)}irdt. 0 0 Do (4.44),b6dS4.1seduQ'chUngminhnSutakhAngdinhduQ'cr~ng (r,t)= r21 PWj (r)q:>(t) E U (QT)'Th~tv~y,dobfitdAngthuc(2.7),taco TJ T 1 f f[(r,t)IPdrdt = f ~r2w/r)q:>(tfdrdt 0 0 0 0 1 T = fr2-PIrw/rf dr flq:>(t)IPdt 0 0 (4.45) 1 T ~(FsIIWillvr fr2-Pdr]q:>(tWdt 0 0 T ~~(Fsllwill r nq:>(t)la+Jdt<+00.3- P v l' V~yb6dS4.1duQ'chUngminhhmlntfit. Cho m~ +00 trong(4.42),tasuyratll (4.30),(4.31),(4.32)vab6dS4.1, r~ngu th6aphuO'ngtrinhbiSnphan Khao satphuongtrinhparabolicphi tuyin trongmi~nhinhcdu trang37 (4.46) T T f(U'(t),wi )g;Ct)dt+ f(u,(t), Wi')g;Ct)dt a a T T + fh(t)u(l,t)Wi(l)gj(t)dt + f(lu(t)IP-2u(t),Wi)gj(t)dta a T T =f(/(t),Wi )gj(t)dt +Uafh(t)w/l)g;Ct)dt, Vi, j E N. a a V~y,tasuytu(4.46)r&ngphuangtrinhsaildaydung T T T f(u'(t),v)dt+f(u,(t),v,)dt+fh(t)u(l,t)v(l)dt a a a T + f\lu(t)IP-2u(t),v)dta (4.47) T T =f(/(t),v(t))dt+uafh(t)v(l,t)dt, Vv E L2(O,T;V} a a V~ys\l't6nt~inghi~mduQ'chUngminhxong. Bmyc4.Tinh duynh~tnghi~m. Giasuu vav lahainghi~mySucuabaitmin(4.1)-(4.4).Khi do w=u-v thoabaitoanbiSnphansailday T T f(w'(t), lp(t))dt + f[(W,(t),lp,(t)) +h(t)w(l, t)lp(1,t)}it a a T (4.48) +f(lu(t)IP-2u(t)-lv(t)IP-2vet),lp(t)dt=0, a VlpE L2(0,T;V), (4.49) w(O)=weT), v6i u, VE L2(0,T; V)nD'(O,T;H), u', V'EL2(0,T;H), r2lpu,r21pvEH(QT} T L~y lp =w trong(4.48)va chuy r&ngf(w'(t),w(t))dt=o. Khi do su d\lng a (4.11)va(4.49),tathuduQ'c Khao satphLfangtrinhparabolicphi tuyin trongmi~nhinhcdu trang38 (4.50) 1 r r 2C11Iw(t)II~2(o,r;v)~ fllwr(t)112dt +fh(t)w2(1,t)dt° 0 r = - f(lu(t)1p-2 u(t) -lv(t)IP-2 vet),u(t) - vet))dt ~O. 0 DiSunaydfindSnw=0,i.e.,u=v.Dinhly 4.1dugcchUngminhhoant&t. 4.2.S1}'Andinb cuangbi~my~uT-tuftnboan Trong phanll<lYchungt6i sekh~lOsattinh 6n dinh d6i v6i f, h,Uocua nghi~mySuT-tuanhoancuabaitmin(4.1)-(4.4). Tuang ung v6i f, h,uo, Ian lugt thoacacgia thiSt(H2), (H~),(H~),bai loan (4.1)-(4.4) co duy nh&t mQt nghi~m ySu T-tuan hoan u E L2(0,T;V)nL"'(0,T;H), saochou' E L2(0,T;H), r21puE LP(Qr)' Nghi~mnay ph\!thuQcvaou=u(f,h,uo)Tasechungminhnghi~mnay6ndinhd6iv6i f, h,UotheoillQtnghlamatasequidinhsau. Tru6chSttad~t H ={hE wi"" (O,T), h(O)=her),h(t)? ho>a}, y ={JE CO([O,T];H),f(r,O) =f(r,T)}. Khi do,tacodinhly saudaylienquaildSntinh6ndinhcuanghi~mySu Binb Iy 4.2. Nghi<?mu=u(f, h,uo)6ndtnhdr5ivai f, h,uo,theongh'ia Niu (fk,hk,uOk)'(f,h,uo)EYxHxJR, saocho fk ~ f trong CO([O,T];H), (4.51) hk~h trong w1""(0,T), UOk~ u trong JR, thi (4.52) Uk~u trongL2(0,T;V)va r21puk~r2lputrongLP(Qr), trongdo Uk=U(fk,hk,uOk)'u =u(f,h,uo). Khaosatphuongtrinhparabolicphi tuyin trongmiJn hinhcdu trang39 Chung minh. TruachSttad~themcackyhi~u Vk=Uk-u, Jk =fk - f, hk=hk-h, UOk=UOk-uo. ChovE L2(O,T;v) tllY y,truhaid~ngthucsau: (4.53) T T T f(u~(t),vet))dt + f[(Ukr(t),Vr(t)) +hk(t)Uk(1,t)v(1,t)]dt+ f(F(Uk (t)),vet))dt 0 0 0 T T = f(fk (t),v(t))dt+UOkfhk(t)v(l,t)dt, 0 0 Uk(0) =Uk(T), (4.54) T T T f(U'(t),vet)dt+ f[(ur(t),Vr(t))+h(t)u(l,t)v(l,t)]dt+ f(F(u(t)), vet)dt 0 0 0 T T =f(f(t),v(t))dt+UOk fh(t)v(l,t)dt, 0 0 U(O)=u(T), tathuduQ'c (4.55) T T f(v~(t),vet))dt + f[(Vkr(t),Vr(t)) + (hk(t)Uk(l,t) - h(t)u(l, t))v(1,t)]dt 0 0 T +f(F(Uk(t))-F(u(t)),v(t))dt 0 T T =f((Jk(t)}v(t))dt+f(UOkhk(t)-uoh(t))v(1,t)dt. 0 0 Chnv=Vk'trong(4.55)vasaukhi chuy r~ng T 1Td 1 1 (4.56) f((v~(t)),Vk(t))dt=- f-lIvk (t)lldt=-llvk(T)II--llvk(0)11=0, 0 20 dt 2 2 tathuduQ'c Khaosatphuangtrinhparabolicphi tuyin trongmi~nhinhcdu trang40 (4.57) T T fllvkr(t)112dt + f(hk(t)Uk(l,t) - h(t)U(l,t))vk(l,t)dt 0 0 T + f(F(Uk(t))- F(u(t)),Vk(t))dt 0 T T = f(Jk (t),Vk(t)}dt+ f(UOkhk(t) - uOh(t))vk(l,t)dt, 0 0 hay T T T flhr (t)112dt+ fhk(t)vi (l,t)dt +fhk(t)u(l,t)vk(l,t)dt 0 0 0 T +f(F(Uk(t))-F(u(t)),Vk(t))dt 0 (4.58) T T = f(Vk (t)) Vk(t)}dt+UOkfhk (t)Vk(l,t)dt 0 0 T +uo fhk(t)vk(l,t))dt. 0 Chu y r&ng T T T (4.59) ]IVkr(t)112dt+ fhk(t)vi (l,t)dt ~ C) ]h(t)II~dt=c)llvkll~2(O,T;V)' 0 0 0 trongd6 C)=mill{1,ho}.Dung bfitd~ngthuc (4.60) '\Ip~2,3Cp>O:~XIP-2X-lxIP-2XXX-y)~cplx-yIP'\Ix,yeIR, tasuyra T T ) (4.61) f(F(Uk(t))- F(u(t)),Vk(t))dt~Cpfdt fr21uk(t) - u(t)IPdr =Cp IIr2/PVkII;p(Qr)' 0 0 0 B&ngeachsird\lngbfitd~ngthuc(4.60),tasuyratll (4.58)-(4.61)r&ng c)llvkll~2(O,T;V)+Cpllr2/Pvkll;p(Qr) (4.62) T T ~ - fhk(t)u(l, t)vk(1,t)dt + f\Jk (t),Vk(t)}dt 0 0 T T +UOkfhk (t)Vk(l,t)dt +Uofhk(t)Vk(l,t)dt 0 0 H9CvienNguyln ViiDzilng Khaosatphuongtrinhparabolicphi tuyin trongmiJn hinhcdu trang41 T T S 411hkt fllu(t)llvIlvk(t)llvdt + Illkllco([O,T];H) fllvk (t)llv dt T T + 21uOkIllhkt ~h (t)llv dt + 211hkt Iuol fllvk (t)IIv dt° ° S 411hkIIJuIIL2(O,T;V) IlvkIIL2(O,T;V)+ Illk Ilco([o,T];H)Frllvk t2(O,T;V) + 21uOkIllhkt Frllvk t2(O,T;V) + 211hkt IUDIFrllvk IIL2(O,T;V) =[2(21Iut2(O'T;V)+Frluolllhkt +Frlllkllco([O,T];H)+2FrllhkIUuOkl]lvkt2(O,T;V) ==8k IIVkt2(O,T;V)' tfongdo (4.63) 8k =2(21IuIIL2(O'T;V)+Frluol)llhkt +Frlllkllco([O,T];H)+2FrllhktluOkl. Ta Suyfa tu (4.62)dng 1 (4.64) Ilvk(t)112( . ) S-8k'L O,T,V C ) 2 ~ II 21 li P 1 82 (4.65) c)llvkIIL2(O,T;V)+CprPVkLP(QT)SC) k' 1 821P 11 21p II SD~k' (4.66) r Vk LP(QT) ~C)Cp 1 1 821P II II 21pV II <-8k + D~ k . (4.67) Ih L2(O,T;V)+ r k LP(QT)- C) ~C)Cp Tu giathiSt(4.51),taco Il hk II ~ 0, Il lk II ~ 0, IUOk I ~ 0, 00 CO([O,T];H) vadays6~Ihkt } bi ch~n,nen8k ~ O. V~y 1 1 821P ~ o. I II II 21pV II <-8k + D~ k IVk L2(O,T;V)+ r k LP(QT) - c) ~C)Cp Dinh ly 4.2duQ'chungminhhoant~t.