Luận văn Khảo sát phương trình sóng phi tuyến trong không gian sobolev có trọng

CHUONG 3 KHAO SAT PHUONG TRINH Uti -(u" +~u,)+F(u,u,)=f(r,t\ VOl NHOM BlED KIEN THO NHA T~. Xetbaitmin(3.1)- (3.4)sail uu-( Urr+~Ur)+F(U,Uf)=f(r,t), O<r<l,O<t<T, I lill J;ur (r,t) 1 <+00,- u,(l,t) =hou(l,t)+h(t)Uf(1,t)+g(t), r~O+ (3.1) (3.2) u(r,O)=uo(r),uJr,O) =ul(r), F(u,uf)=f;(u)+F2(Uf)=lula-l u+lufIP-lUp (3.3) (3.4) 111.1.Du'av~bili tminbitn philo Xetbaitmin(3.1)- (3.4). Til (3.1)tasuyra r I [ ] T I f fr Ull-(urr+.lur)+F(u,ul) wdrdt=f frfivdrdt,VwED(O,T;V1).(3.5) 00 r 00 ChQn w(r,t)=~(t).v(r),trongd6 ~ED(O,T), VEVp k~th<jp(2.12)va (3.2),ta vi~tl~i(3.5)nhu'sail ref!{(ul(t),v)+a(u,v)+(F( u(t),ul(t»),v;] ~(t)dt JLdt T +f[h(t)ul(l,t)+g(t)Jv(1)~(t)dt 0 (3.6) r = f(!(t), v)~(t)dt,V; E D(O,T), Vv E VI. 0 HQcvienHuynhVan Tung Trang20 Tli (3.6)tathuduQc ~\ul(t),v)+a(u(t),v)+(F(u(t),ul(t)),v) +(h(t)UI(l,t) +g(t))v(l) = (J(t), v), 'v'vE Vi' a.e.t E (O,T). (3.7) Hamu thml(3.7),(3.3)duQcgQilanghi~mye'ucuabaitmin(3.1)- (3.4). 111.2.811t6n t~iva duynha'tnghi~mye'u Ta thanhl~pnh6mgicithie'thunha'tnhusail (HI) ho>0, 1::;a <3, 1::;fJ <3, (H2) f,frEL2(0,T;Vo)' (H3) gEH\O,T), g(O)=0, (H4) hEC2(~),h(t)"20,'v't>O,h(O)=O, / vat6nti;lih~ngsO'8 E (O,ho)saDcho hI(t)"2-8, 'v't"20, (H5) Uo E V2 ' UI E VI . Chtithich3.1. hoia hangs{fduangxudthi~ntrangb6d~2.5. Saildayla dinh19ve svt6nt~ivaduynha'tnghi~mcuabailoan(3.1)- (3.4) voi nh6mgicithie'thunha't. Binhly 1. ChotruacT>Ova(HI)- (Hs) thoG.Khi d6 bai loan (3.1)- (3.4) c6duynhdtm()tnghi~myeu uEL00(0,T; VI) n L2(0,T;V2)saocho ul E Loo(0,T; VI), Ull E LOO(O,T;Vo)' 1 1 ra+lu E Loo(O,T;La+l(Q»), rfJ+lul E LfJ+l(QT), u(l,.) E W1,00(0,T). Chung minh. Vi~cchungminhdinh191duQcchialamnhi€u buoc. Bu'ocl. Xa'pXlGalerkin w. ? Xet {Wj =A}lacdsdtrvcchuantrongkhonggianVIvai tichvo huangla a(.,.) nhutrongb6de2.6. TaHmnghi~mxa'pxi cuabailoanbie'nphan(3.7)duaid~ng m um(r,t)=L>mj(t)Wj(r), j=l (3.8) HQcvienHuynhVan Tung Trang21 trongdocachams6cmJt),j =1,m thmlh~phuongtrlnhviphanthuong (u~(t),WI)+a(um(t),WI)+(F( Um(t),u~(t)),Wi) + (h(t) U~(1,t) + g(t) ) WI(1) =(f(t),wJ,l'!;}'!;m, (3.9) clingvdi di€u ki~nd~u m um(O)=UOm=Lamiwi ~ Uo mqnhtrang V2,khi m ~ 00, J=I (3.10)m u~(O)=Ulm =LPmJwJ ~ UI mqnhtrang~, khi m ~ 00. J=I Vdi m6i T >0 cho trudc,ta se sttdl;lngdinh ly diem ba'td9ngSchauderde chungminhh~(3.9),(3.10)conghi~mcm=(cm!"'"cmm)tren[O,Tm]c [O,T]. Ta cob6d€ saildayv€ slft6nt(;linghi~mcuah~(3.9),(3.10). B6 d~3.1. ChotneucT >o va (HI) - (Hs)thoa.Khi do hf (3.9),(3.10)co nghifm cm=(cml"'" cmm)tren[O,Tm]c[O,T]. Chungminhb6d~3.1. H~(3.9),(3.10)duQcvie'tl(;linhusau c~/t) +AJCmJ(t) =II ~\2 [(F( um(t),u~(t)),wJ)+(h(t)u~(1,t)+g(t))WJ(1)-(f(t),Wi)J1 Cm/O) =amJ' C~/O)=PmJ' I'!; j '!;m, hay cm;(t)=amicostA t)+Jx; sin(At) (3.11) 1 II sinA (t-T) -II Wi W 0 jx; [(F(Um(T),U~(T)), WJ)+ h(T)U~(1,T)Wi(l) JdT - 1 II sinA (t-T) IlwJW 0 A [g(T)w/l)-(f(T),wJ)]dT,l'!;j'!;m. BoquachIs6m,khidoh~(3.11)duQcvie'tl(;linhusail c=Uc, (3.12) HQcvienHuynhVanTung Trang22 trongdo c=(cp...,cm),UC=(U c\,...,(Uc)m)' (3.13) t (U e)/t) =G/t)+ fN;U-r)(Ve)/r)dr, 0 (3.14) 1 t GU) =aNI (t)+ f3N(t)- 2 fNU-r) [g(r)w,(l)_1 fer), Wj)J dr, 1 11 II Ilw.11 1 . \1 0 (3.15) (Ve)/t) =- 1 2 [ /F [ i>iU)wpIe:U)Wi ] ,W; ) +h(t)Ie:(t)w;(1)Wj(1) } , (3.16) Ilwj II \ 1=1 1=1 1=1 sin(At) 1~j ~m. N;U) A' (3.17) Voi m6i 00, tad~t S={CEC1([O,Tm];IRm):IleI11~M}, Ilelll=llcllo+lle/llo' m Ilello=supleU)I\' leU)11=Ile;(t)I. O<;t<;Tm i=1 Thl S Ia t~pcon16idongvabi ch~ncua y =C1([o,Tm];IRm). Sa dl;mgdinhly diembcftdQngSchauder,chungtase chungminhr~nganhX(;l u: S --+Y duQcdinhnghiabdi (3.12)- (3.17)comQtdiembcftdQng.Biembcft dQngnaychinhIanghi~mcuah~(3.9),(3.10). a) ChungminhU(S)cS. Taco GIU)=-XaN. (t)+f3N/(t)1 III 11 (3.18) 1 t - 2 fN;U-r) [g(r)w/l)- ] dr, II Wj II 0 t (U e)~(t)=G~U)+fN;U-r)(Ve);(r)dr , 1~j ~m. 0 (3.19) Tli (3.14)- (3.19)tasuyra anhX(;lU: Y --+Y xacdinh. Cho eE S, tli (3.14) - (3.19) ta suy ra HQcvienHuynhVanTung Trang23 1 t T I(Ue)(tH~IG(t)11+ f1 fl(Ve)('Hd'~IIGllo+ ~IIVello v~ 0 v~ (3.20) m 1 m 1 T ~I . . 1 a; 1+f1 II /3;1+ f1 rT + f1 /3(M,T), FI v~ ;=1 VAl VAl t I(Ue)/(tH~ IG/(tH + fl(Ve)('Hd,~ IIG1 110+TmIIVello 0 (3.21) m m ~KII a;l+II /3;1+rT+Tm/3(M,T), 1=1 ;=1 trongdo m 1 T rT =~II wi W l(lg(t)w/l)l+I\i(t)'Wi)1)dt, (3.22) ~/3;CM,T) IIVello~/3(M,T)=L. 2' ;=1 II Wi II /3;(M,T)=SUP { /F (teiWi,td/Wi ) ,Wj ) :llelllRm~M'lldlllRm~M }\ I I I I (3.24) +K,'II Wi 11,11h1I,"(onsup{tlc, III w,II,:IIell,. s; M}. Ch1ithich3.2. Titb6di 2.20.itasuyrarling F( ~e,(t)w"~e:(t)W,)EVo, 'IeES. (3.23) Do d6 (3.24)luontbntc;zi. Tli (3.20)va(3.21)tasuyra II [Ie II, <; lOT + Tm(l +k)P(M,T) , (3.25) trongdo OJ, =(1+A Jtlaj 1+[1+~)(r, +tl PjI} (3.26) va /3(M,T)du'QCxacdinhbdi(3.23)va(3.24). ChonM va 0<1~<T saochoM?:2"" va To(I +J;., ]P(M,T)'; ~. Tli (3.25)tathudu'QcII UeIII ~M, voimQieE S. HQcvienHuynhVan Tung Trang24 b) Chungminh U lien tl,lCtrenS . Cho C E S, {Ck}C S va Ck~ C trong Y, ta co t (U Ck);(t) - (U c)/t) = fN;(t-T)[(VCk)/T)-(VC)/T)]dT, 0 t (U Ck)~(t)- (U c)~(t)= fN;(t- T)[(VCk)/T)-(V C)/T) ]dT, 0 Tm IIUCk - Ucllo::; JX: IIVCk - Vcllo, II(UCk)/ - (Uci 110::;TmIIVCk - Vcllo, IWe, - Uell,'; Tm(l+~Juve, -Veil,. m m Iw(1)1 I(VCk)(t)- (Vc)(tn::; h(t)II(c~i(t)-c:(t))w,(1)1 I J 2 . 1=1 J=I II Wi II +f~ /F (fck;(t)w;,fc~;(t)W; ) -F (fc/t)w;,fc:(t)w; )'Wj)j=1 II Wj II \ ;=1 ;=1 ;=1 ;=1 ( ~ )[ ~lw/l)I J / / ::;llhllr,w(o,T) f:1lw;(1)1 7:1llwjW IIck-c 110 ( ( m ) ( m ) m IwjI ) + F; LCk;(t)W; -F; LC/t)W;,L 2 1=1 1=1 J=1 II Wj II ( ( m / ) ( m / ) m IWjl ) + F; LCk;(t)W; -F; LC;(t)W;,L 2. ;=1 ;=1 j=1II Wj II m Di.itR=MIll w,III'Khi dotli(3.28)vab6d€ 2.20.iii,tadu'QC 1=1 I (V c, )(1) - (V c)(t) I,S K,' IIhilL"I'T) (tII w, "l~IIII :~II:; J II c~ - C'II, + (fIIWIIII )( f 1 2 J [ ~KR(a)llck-cllo+~KR(j3)llc~-c/lloJ.;=1 J=l IIwi II Tli (3.27)va (3.29)tasuyra U lien tl,lCtrenS. (3.27) (3.28) (3.29) HQcvienHuynhVan Tung Trang25 c) Chungminh US compactrongCI ([O,Tm];IRm). Do USc S, nenhQcacham us ={Ue:eE S}bi ch~nd€u theochu~n11.111trong CI([O,Tm];IRm).Ta chI c~nchungminh hQcac ham US={Ue:eES} Ia lien t1,lC d6ngb~cd6ivoichu~n11.111trongkhonggianCI([O,Tm];IRm). Cho eES, t,tlE[O,TmLtaco (Ue)j(t)-(Ue)/tl) , ( ( (3 30) =G/t)-Gj(t/)+ IN/t-r)(Vc)lr)dr- INj(tl-r)(Vc)/r)dr . a a ( =G/t)-G/tl)+ I[N/t-r)-N/tl -r)]cVC)/r)d~(, a - IN/tl -r)(Vc)j(r)dr, ( G(t)-G(t/) =a. [NI (t)-NI (tl) J +{3.[N.(t)-N.(tl) J1 1 1 j j 1 j j 1 ( - 2 I[N.(t-r)-N.(tl -r) J [g(r)w,.(1)_1f(r),w j. )J dr Ilwjllo 1 1 . \ (' +II W: 112fNj(t1 -r) [g(r)w/l)-\f(r), Wj)Jdr. (3.31) M~tkhactaco f INj(t)- N/tl) I:::;It- tl I, lIN~(t)-N~(t/)1:::;[X:lt-tll, Vt,tl E[O,Tm],l:::;j:::;m. (2.32) Tli (3.23),(3.30)- (3.32),tasuyra m I(Ue)(t)- (Ue)(tl) II=LI (Ue),(t) - (Ue)/tl) I j=1 (3.33) ,;[tlaj 1+F.tIP, I+r,}-tll+p(M,r{r. +.k )It-tl Danhgiatu'dngtt,I'nhu'(3.33),taco m I(Uei (t)-(Ue)1(t/)11=LI(Ue)~(t)-(Ue)~(t/)1 j=1 (3.34) ,; K[ Ktl aj 1+~Ilij I+r,] 1t-t' 1+1i(M,T)(KTm +1)1I-I' I. HQcvienHuynhVanTung Trang26 Tli (3.33)va(3.34)tasuyrahQcachamus Ia lient1;lCd6ngb~cd6ivoi chuffn11.111trongkhonggianC\[O,Tm];IRm).V~ytheodinhly Arzela- Ascolithl US compact.Tli cacketquaa),b),c)vadinhly di€m beltdQngSchauder,U co di€m beltdQngtrongS. Nhuv~ybaitoan(3.9),(3.10)t6nt(;linghi<$mUmtren[O,Tm].V~yb6d~ 3.1duQcchungminhxong. Cacdanhgiatiennghi<$msandaychopheptalelyTm=T,Vm. Chli thich 3.3. TrangchangminhSl! tbn tf;linghi~mxfipxl Galerkincua bili loan bifn phan (3.7) chung ta co thi thay cae gid thift yfu han so vrJi (HI)-(Hs) nhusau ~ { hO>0, 0<a<3, 0<{J<3, fEL\O,T;Vo), gEL\O,T), hECo(IR), UOEVI' ul EVo' V6'igid thift UoEVI, UIEVo thi (3.10) du(fcthaythf bili m um(O)=uom=IamjWj ~uo mf;lnhtrang~, khi m~oo, j=1 (3.35)m u~(0)=Ulm=I PmjWj ~ UI mf;lnhtrang ~p khi m ~ 00. j=1 Bu'oc2. Danh gia tit~nnghi~m Danh gia 1. Nhanphuongtrlnhthu j cuah<$(3.9)voi C~j(t)va lelyt6ngtheoj tli 1denm,taduQc Id [ ] 1 d J I -- lIu~(t)112+a(um(t),um(t))+-- rlum(t)la+ldr 2 dt a +1dt0 (3.36) 1 +frlu~(t)IJJ+l dr+h(t)lu~(1,t)12 +g(t)u~(1,t) 0 =(f(t),u~(t)). f)~t t Xm(t)=Ilu~(t)W+a(um(t),um(t))+2fh(r)lu~(1,r)12dr 0 2 1 t 1 +- fr Ium(t)Ia+ldr+2 f fr Iu~(r)IJJ+ldrdr. a+lo 00 (3.37) HQcvienHuynhVanTung Trang27 La'ytichphanhaivfS(3.36)theo t tu0 dfSnt, tadu(jc I I Xm(t)=Xm(0)+2f(J('l"),u~(r»)dr- 2fg(r)u~(l,r)dr 0 0 I I ~Xm(O)+fIIJ(r)112dr+fllu~(r)112dr 0 0 I +21g(t)um(l,t)I+2IiI (r)um(l,r) Idr 0 (vi g(O)=0) I ~Xm(O)+II J 11~2(o,T;Vo)+fXm(r)dr 0 2 h 1 I I +-l(t)+-.JLu~(l,t)+ - fll(r)12 dr+ hofu~(l,r)dr. ho 2 hOD 0 2 I Xm(O)=II Ulm W +a(uom'uom)+- SriuomIa+ldr a+1o 2 I I-a2 2 f 2 ( I ) a+1 ~ II u1m II +CI II Uom III +- r -Vr IUom I dr a+10 2Ka+1 I I-a ~ II Ulm W +CI II UOm II~ + 2 IIUOmII~+Ifr 2 dr a+1 0 4Ka+1 ~ II Ulm W + CI II UOmII~+ 2 II UOmII~+I. (a +1)(3-a) Tu (3.10),va(3.39)tasuyra Xm(O)~M?),'\1m, trongdo Mil) dQcl~pvdi ffi.Tu (3.38)- (3.40)tasuyra 1 I X (t)~gl(t)+-X (t)+2fX (r)dr,m 2 m m0 hay I Xm(t)~2g,(t)+4fXm(r)dr, 0 , (I) 2 2 21/2 trong do gj (t) =MT +II f IIL2(O,T;Vo)+-,;-1 g(t) I +-,;-11 g IIL2(o,T).0 0 Tu giii thifSt(H3)vado H'(O,T) c.Co([O,T])Dentasuyra gl (t)~Mi2) a.e.t E[0,T] . Ap dl;lngb6d€ Gronwallvao(3.41),tadu(jc Xm(t)S 2M?)e41~M?) , a.e.t E [O,T]. (3.38) (3.39) (3.40) (3.41) (3.42) (3.43) HQcvienHuynhVanTung Trang28 Ch6 thich3.4. Trangdanhgia 1, tachuasitdf:lnghtt nhomgid thitt thrinh{{t, thq.mchi co thi thaym(Jts{;'gid thitt bJi cacgid thittytu hantuangring,ch~ng hqn (Hi) dur;cthaybJi ho>0, 0<a <3, 0<fJ <3; (H3) dur;cthaybJi gEHi(O,T). Trang tru(jnghr;pnay, ta chi vifc c(Jng themVaGvt phdicua(3.38)dqi lur;ng 21g(O)Uom(1) I hichq.n;. . . \ Nhomgid thitt thrinh{[tnayth1;lcs1;lcandentrangdanhgia 2 sauday"vatrang cacburJcsaudb. Danhgia2. L1yd<;lohamhaiv€ cuaphuongtrlnhthuj cuah~(3.9)d6ivoi t, tadu<;lc (u:;(t),Wj)+a(u~(t),Wj)+(:tF(Um(t),U~(t)),Wj) +(hi (t)UI (1,t) +h(t)UII(l,t) +gl (t))w/l) (3.44) =(Ir(t),Wj)'l~j~m. Nhanphuongtrlnhthuj cuah~(3.44)vOi c~/t) va l1y t6ngtheoj tit 1d€n m, tadu<;lc 1 d 1 2dt[llu~(t)W+a(u~(t),U~(t))J+PJrlu~(t)IP-llu~(t)12dr0 (3.45) 1 d +-hl (t)-I u~(l,t) 12+h(t) Iu~(1,t)122 dt +l (t)U~(1,t)+(:tF;(um(t)),U~(t)) =(Ir(t),u~(t)), l~j~m. Bi;it Ym(t) = II u~(t) W +a(u~(t),u~(t))+(hi(t)+e) Iu~(1,t) 12 (3.46) t t 1 +2 fh(T) 1 u~(1,T) 12 dT +2fJ f fr IU~(T)113-11U~(T)12drdT. 0 0 0 L1y tichphan(3.45)theot tit0 d€n t, tadu<;lc HQcvieDHuynhVan Tung Trang29 tYm(t)=Ym(o)+elu~(1,T)121~:;)+ fhll(T)lu~(1,T)12dT 0 (3.47) t t ( d ) t +2f\ft(T),U~(T))dT-2f -F; (Um(T)),U~(T)dT-2 fl(T)U~(1,T)dT 0 0 dT 0 t ::;;ym(o)+eKj21Iulm II~+elu~(1,t)12 +llhll Ilc'(O,T)flu~(1,T)12dT 0 I t t' +2fllft(T)llllu~(T)lldT+2 fll~F;(Um&))IIIIU~(T)lldT 0 0 dT t +21l (O)Ulm(1)I+21l (t)u~(1,t) I+2fill (T)U~(1,T) IdT. 0 Ta co hoIu~(1,t)12::;;a( u~(t),u~(t))::;;Ym(t), (3.48) 2 h -£I ( )21l (t)u~(1,t)1::;;-I gl(t)12+~ hoIu~(1,t)12 ho-£I 2hO (3.49) 2 h -£I ::;; - lgI(t) 1 2 +~Y (t) h - £I 2h m '0 0 t t 2fll/(T)U~(1,T)ldT::;;~llgIIWL2 T +fYm(T)dT,h (0, )0 0 0 (3.50) II U~(t)II~::;;~a( u~(t),u~(t))::;;~ Ym(t). Co Co (3.51) Til (3.37),(3.43),(3.51),vab6d€ 2.20.ii,taduQc 1 II ~FJum(t))112=a2 frlum(t)12a-2Iu~(t)12dr 0 (3.52) 1 ::;;a2 K; II U~(t) II~ fl Um(t) 12a-2dr::;; a2 K; Ym(t) k\(a) II Um(t)ll~a-2 0 Co a2K2K (a) ( M(3) ) a-l a2 K2 K(a)(M(3) ) a-l < 2 I ~ Y (t) - 2 1 T- C C m - a Ym(t). 0 0 Co Til (3.47)- (3.50),(3.52)tathuduQc Ym(t) s Ym(0)+8KJz II uJm II~ +: Ym(t) +II it IIL2(0,T;Vo)0 t II hIt II. t + fYmCr)dT+ L"'(O,T)fYm(T)dT 0 ~ 0 - ( (3) )a-l t I aZK;K1(a) MT fYm(T)dT+ fYm(T)dT + a 0 0Co +Il (0)IZ +KJZII UJm II~ +~ Igl (t) IZ ho-8 h - 8 II gll 11~2 T f l +~Ym(t)+ (0, ) + Ym(T)dT, 2ho ho 0 hay t Ym(t)sgzm(t)+Mi4) fYm(T)dT, 0 (3.53) trangdo gZm(t)=h2~)8[Ym(0)+KIZ(8+1)IIUlmII~+llit 1I~2(O,T;Vo)J0 +~ [ ll(O)IZ +2Il(t)lz +llglllI~2(0'T) ] , ho-8 ho-8 ho (3.54) Mi4)=~ [ 3+ IIhIIIILoo(O,T)+aZK;KJ(a)(M?)r-l ] ho-8 h a.0 Co // (3.55) Nhan phuongtrlnhthlij cuah~(3.9)vOi c~(t) va la"yt6ngtheoj tli'1 de"nm,J taduQc Ilu~(t)IIZ+(Aum(t),u~(t»)+(F(um(t),U~(t»),u~(t») +(h(t)u~(1,t)+get)u~(1,t) = (J(t), u~(t»). Trang(3.56),chot=0,vachuyding g(O)=h(O)=0, taduQc Ilu~(O)llz+(Auom'u~(O»)+(F(uom,uJm)'u~(O»)= (J(O),u~(O»). (3.56) Suyra Ilu~(O)11s IIAum(O)II+11 F{uom,uJm)II+IIJ(O)II. Ap d\lngb6d€ 2.20.i,taduQc IIF(uOm'u1m)II s II I\(uom) II +II F;(uJm) II sK(a)II UOm II~ +K (,8)II U1mIii. (3.57) (3.58) HQcvienHuynhVan Tung Trang31 Tli (3.46),(3.57)va(3.58),tathudu'Qc Ym(O)=Ilu~(O)W+a(Ulm'ulm)+(h/(O)+B)u~m(1) ~ (II Aum(0) II+ IIF(uorn'Ulrn)II+11f(O) 11)2 (3.59) +[CI +KI2(e+llhIIICo(O,TJJllulrn II~ ~3(II UOrnII; +2k2(a) II UOrnII~a+2k2(fJ) II Ulrn 11~j3+11f(O) W) +[CI+KI2(e+II hi Ilco(o,TJJllulrnII~. Tli (3.10),(3.54),(3.59),gii thi€t (H3),vado HI (0,T) C.CO([0,Tn nentasuyra g2rn(t)~Mi5),a.e.tE[O,T]. (3.60) Tli (3.53),(3.60)vab6d~Gronwall,tadu'Qc (5) M(4)( (6) Yrn(t)~MT e T ~MT, a.e. tE[O,T]. (3.61) Tu'dngtlj (3.58),tacodu'Qc IIF(um(t),u~(t))II:S;K(a) IIum(t)lit +K(jJ)llu~(t)llf ( M(3) J ~ ( M(6) J ~ :s;K(a) ~ + K(jJ) ~ . (3.62) Tli (3.62),tasuyra IIF(urn,u~)lloo. ) ~Mr),a.e.tE[O,T],L (O,T,Vo (3.63) va I 1I"r;.F(u., u~)II"(Q,) ~ 01F (um(r),u~(r»)II' drr ,;JT M!') (3.64) Bu'oc3. Qua gioi h~n Tli (3.37),(3.43),(3.46),(3.61),va (3.64)ta co th€ trichtli day {urn}mQtday conv~nkyhi~ula {urn}'saocho Urn~ u trong D$J(O,T;VI)y€u *, u~~ul trong DJ(O,T;VI) y€u *, uti~Ull trong L"' (OT V ) Y€u *rn , , 0 , (3.65) (3.66) urn(l,t) ~ u(l,t) trong WI""(O,T)y€u *, (3.67) (3.68) HQcvienHuynhVan Tung Trang32 1 1 ra+IUm~ ra+lu trong LOO(O,T;La+l(o.)) ye'u *, (3.69) 1 1 rfJ+I U~ ~ rfJ+I Ul trong LfJ+I (QT) ye'u, (3.70) (3.71)FrF(um'u~)~X trongL2(QT)ye'u. Ap dl;lngb6 d€ v€ tinh compactcua Lions vao (3.65)- (3.68),va dophep nhungWI,OO(O,T)c.Co([O,T])lacompactnentacoth~trichtuday{um}mQtday conv§nkyhi~ula {um},saocho um~ u m(;lnhtrongL2(0,T;Vo), U~~ UI m(;lnhtrong L2(0,T;Vo), (3.72) (3.73) Um(1,t) ~ u(l, t) m(;lnhtrong CO([0,T]) . (3.74) Theodinhly Riesz- Fischertacoth~trichtuday{um}trong(3.72)va(3.73) mQtdayconv§nkyhi~uIa {um},saocho um(r,t)~ u(r,t), a.e.(r,t)EQT' ~. u~(r,t)~ ul(r,t), a.e.(r,t)EQT' (3.75) (3.76) Do F(u,ut)=IuIa-I U+1Ut113-1Ut lien tl;lc,nen tu (3.75)va (3.76) ta suy ra F(um(t),u~(t))~F(u(t),ul(t)),a.e.(r,t) E QT' (3.77) Ap dl;lngb6 d€ 2.15vOi q=n=2, Q=Qp Gm=FrF(um,u~),G=FrF(u,ul), tu (3.64)va (3.77),tadu'Qc FrF(um,u~)~FrF(u,ul) trongL2(QT)ye'u. (3.78) Tu (3.71)va(3.78)tasuyra F(um,u~)~ F(u,uJ trongL2(0,T;Vo)ye'u. (3.79) Nhan(3.9)voi rpED(O,T) tuyy, r6i lfiy tichphantheot tuOde'nT, tadu'Qc r T f(u~(t),w/)rp(t)dt + fa(um(t),wJrp(t)dt o o r T + f(F (um(t), u~(t) ), WI)rp(t)dt+ f(h(t) u~(1,t) +get) )wi (1)rp(t)dt o o T =f(f(t),wJrp(t)dt,VI o (3.80) HQcvienHuynhVanTung Trang33 Do(3.67)taco T T f(u~(t),wJqy(t)dt~f(UII(t),Wj)qy(t)dt,khim~+oo. 0 0 (3.81) Qua giOih(;lnkhi m~ +00trang(3.80)bdi (3.65),(3.68),(3.79)va (3.81),ta duQc [(Ull(t),Wi)+a(u(t),Wj)+(F( u(t),ul(t)), Wi)Jqy(t)dt T + f( h(t)ul (l,t)+ get))WJ(l)qy(t)dt 0 T = f(f(t)'Wj)qy(t)dt,Vi, VqyED(O,T). 0 (3.82) Tu (3.82)tathuduQc(3.7). D€ chungminh u la nghi<%myeu cua bai tmin(3.1)- (3.4)taconph,Uchung minh u(O)=u(pui(O)=u]' a) Chungminh u(O)=uo' Taco i f t I '\ urn(t)=uOrn+ urn(s)ds. 0 (3.83) Nhan(3.83)voi r Wi r6i l§y tichphantheor tu0 den1,taduQc t (urn(t),wJ=(uorn,wi)+f(u~(s),wJ)ds. 0 (3.84) L§y tichphantheot tu0 denT trang(3.84),taduQc T T f(urn(t),Wj)dt =T(uorn,wj)+f(u~(s),(T-s)wj)ds. 0 0 (3.85) Qua gioi h(;lnkhi m ~ 00 trang(3.85)bdi(3.10),(3.65)va(3.66)taduQc T T f(u(t),Wi)dt=T(uo,wj)+f(u\s),(T-s)wj)ds, Vi=1,2,... 0 0 (3.86) Trang(3.85)thayUrn bdi u, taduQc T T f(u(t),wi)dt=T(u(O),Wj)+f(UI(S),(T-s)wj)ds,Vj=1,2,... 0 0 (3.87) Sosanh(3.86)va(3.87)taduQc (u(O),v) =(uo,v), VvEVI' nghlala u(O)=UO' HQcvienHuynhVanTung Trang34 b) Chungminhul(O)=Ul' Tit (3.10),(3.66),(3.67)va ly lu~ntu'dngtv nhu'ph:1na) ta cling thu du'Qc Ul (0)=Ul ' M~itkhactU(3.67),(3.79)vagiathi€t (Hz), tasuyrarflng Au =f -uti -F(u,ut) E LZCO,T;Vo),nghla Ia u E Lz(O,T;Vz)' V~y sv t6n t<;linghi<%mdu'Qcchungminh . Bu'O'c4. Chung minh sl!duynha'tnghi~m Giasa Up Uz la hainghi<%my€u cuabaitoan(3.1)-(3.4)nhu'trongdinhly 1. Khi dow=Ul - Uz langhi<%my€u cuabaitoan (Wll(t),v)+a(wet),v)+h(t)Wi(l,t)v(1) +(F( Ul(t),u{(t))- F( uz(t),ui(I)),v)=0,\IvEVI' a.e.tE (O,T), WE LOO(O,T;V1)nLz(O,T;Vz)' wi E LOO(0,T;V1),WllE LOO(O,T;Vo),w(1,.)E W1,OO(O,T), (3.88) 1- ~ ra+lwE LOO(O,T;La+l(Q)),r,B+lwlE L,B+l(Qr), w(r,O)=wi(r,O)=0, VI WiELOO(O,T;V1),nenl1yv=wl, tit(3.88)tadu'Qc t 111wi (I) W +la( wet),w(t)) + fh(T) Iwi(1,T) IzdT 2 2 0 t =- f(F(Ul(T),U{(T))- F( uz(T),ui(T)), wi (T)}dT. 0 (3.89) f)~t t O"(t)=II wi (t) W +a(w(t),w(t))+2fh(T) IWi (1,T) Iz dT. 0 (3.90) Tit (3.89)tasuyra t O"(t)=-2 f\F(Ul(T),U{(T)) - F( uz(T),ui(T)), Wi(T)}dT 0 t =-2 f(~(Ul(T)) -~(Uz(T)),WI(T))dT 0 (3.91) t -2 f(lu{(T)I,B-l U{(T)-lui(T)I,B-l Ui(T))(U{(T)-ui(T))dT 0 HQcvienHuynhVan Tung Trang35 t~ -2 f\F; (Ul (T») - F; (uz (T) ), Wi (T ))dT 0 t t ~ fllF;(u,(T») -F;(Uz(T»)W dT +fiiwl(T)W dr. 0 0 Ap d\:mgb6d€ 2.20.iiitaduQc IIF; (UI(t»)- F; (uzU») III ~KR (a)11wU) II~~ KR(a) aU), Co (3.92) trongd6 R =~nII Ui II C (O,T;VI). Til (3.91)va(3.92),tathuduQc aU)~ ( 1+KR(a) ] fa(T)dT. Co 0 (3.93) Ap dt,mgb6d€ Gronwallvao(3.93),taduQc aU)=0, nghla1a UI=uz. V~ydinh1y1dffduQcchungminh. HQcvienHuynhVanTung Trang36