Khảo sát phương trình Parabolic phi tuyến trong miền hình cầu

hoacacdiSuki~nsau: Nghi~mySucuabaitoangiatrtbienvaband~u(3.1)-(3.4)duQ'c thanhl~pnhusau: Tim uEL2(0,T;V)nLoo(0,T;H)saochou(t)thoabaitoanbiSnphan sau d -(u(t), v)+a(t)(ur(t),vr)+a(t)h(t)u(l,t)v(l)+(F(u(t)),v)dt =(f(t),v)+lIoa(t)h(t)v(l),VvEV,a.e.,t E(O,T), vadiSuki~nd~u (3.5) (3.6) u(O)=UO' H9CvienNguyen Vii Dziing (HI) UoEH, (HJ lIo E JR, (HJ a, hE W1,oo(0,T),aCt) ao>0, (H4) f E L2(0,T;H). Khao satphuangtrinhparabolicphi tuyin trongmi~nhinhcdu trang18 Khi dotacodinhly sau Dinhly3.1.ChoT>O va(H)-(H4)i/ung.Khii/o,battoim(3.1)-(3.4)co duynhdtmr}tnghi?myiu UELZ(O,T;V)nL"'(O,T;H),saocho (3.7) tu E Loo(o,r; v), tu'E L2(0,r; H), r2/P E LP(Qr). Chungminh.ChUngminhg6mnhiSubu6c. Bmyc1.PhuO'ngphapGalerkin.Ky hi~ubai {wJ j =1,2,...la mQtca So' tr\Icchu~ntrongkh6nggianHilberttachduQ'CV. Ta tim Um(t) theod~ng (3.8) m Um (t)=2:>m/t)Wj' j=l trongdoCmj(t),1~j ~m thoah~phuangtrinhvi phanphituySn (u~(t),Wj)+a(t)(umr,Wjr)+a(t)h(t)um(1,t)w/1) +(F(um(t)),Wj) =(J(t), Wj) +uoa(t)h(t)wj(1),1~j ~m, (3.9) (3.10) um(O)=uOm' trongdo (3.11) UOm~ Uom~nhtrong H. DSthfiyr~ngv6i mQim t6nt~imQtnghi~mum(t)theod~ng(3.8)thoa(3.9) va(3.10)hAukh~pnaitren0~t ~Tmv6i mQtTm'0<Tm~T. Cac danhgia tiennghi~msaudaychopheptalfiyTm=T v6i mQim. BU'O'c2.Danhgia tieDnghi~m. TaseIAnluQ'thiStl~phaidanhgiatiennghi~mdu6iday.Khokhanchinh a dayla s6h~ngphi tuySnF(um(t))=Ium(tf-zUm(t) thalligiavaophuang trinh,dodovi~cdanhgiatinhbi ch~nvasaudoquagi6ih~ncuas6h~ng phi tuySnnaylamQtkhokhan. a)DanhgiGthunhdt.Nhanphuangtrinhthu j cuah~(3.9)bai Cmj(t)va t6ngtheoj, taco H9CvienNguyln VflDzflng Khaosatphuongtrinhparabolicphi tuyin trongmiJn hinhcdu trang19 1 ~llum(t)112+2a(t)llumr(t)112+2u;(1,t)+2 fr2Ium(t)IPdr (3.12) dt 0 =2(1-a(t)h(t)u;(l,t)+2(f(t),um(t))+2uoa(t)g(t)um(l,t). Tirb~tdingthuc(2.9),tasurradng 2a(t)lIumr(t)112+2u; (l,t) ~2aollumr(t)1I2+2u; (l,t) (3.13) ~2mill{I,ao}~Iumr(t)112+u; (1,t))~allium(t)II~, ,. 4. { }val al =-mm 1,aa .3 Ta sur tir(2.6)-(2.8),(3.12),(3.13)r~ng :tIIUm(t)112+aIIIUm(t)II~+2fr2Ium(r,t)IP dra ~ 211- a(t)h(t)1 ~llumr(t)112+ (3+ 1/ fJ ~Ium(t)II2] + 211f(t)llllum(t)11+ 4Iuaa(t)h(t)1 IIum(t)llv ~ 2(1+IlahllJ~llumr(t)II~+(3+1/ fJ~lum(t)112] +11/(t)112+Ilum(t)II2 +~IuDnahll: +2fJllum(t)II~ ~ ~/uanah!l: +Ilf(t)112+2fJ(2 +liGht )llum(t)II~ + [1+ 2(3 + 1/ fJ Xl + Ilahll",)]llum(t)112,V fJ > 0, trong doky hi~uIHI",= II-llroo(a,T) dSchichu~ntrongL"'(O,r). (3.14) ChQnfJ >0 saocho (3.15) 2fJ(2+lIahIIJ~ al' Do do,tir(3.14),(3.15)tathuduQ'c I ~llum(t)1I2+~alllum(t)II~+2fr2Ium(r,t)IPdr (3.16) dt 2 a ~~IUanah!l: +IIf(t)112+[1+2(3+1/fJX1+IIahIIJ]llum(t)112. Tir (3.16),l~ytichphantheot,vasud\lng(3.10),(3.11)taduQ'c H9CvienNguyin VuDzung Khiw satphuongtrinhparabolicphi tuyin trongmi~nhinhcdu trang20 (3.17) 1 t t I Ilum (t)112 +-al filum(s)ll~ds+2Ids fr21um(r,s)IPdr 2 0 0 0 ~lluomll'+~luol'llahll:+~lf(S)II'ds t + [1 + 2(3 +1/ fJ Xl + lIaht)] filum(s )112ds 0 t ~M}2)+M~) filum(s)112ds, 0 trong doM j1) ,M i2) la cach~ngs6chiph\!thuQcvaoT vaduQ'chQnnhu sail Mil) =1+2(3+lIfJX1 +Ilaht), M;""llao.II'+(~lu,I'llahll:)T+pli(s)II'ds, ';1m. Ap d\!ngb6dSGronwall,tathuduQ'ctir(3.17)r~ng 1 t t I Ilum(t)112+-al fllum(s)ll~ds+2Ids fr2Ium(r,sf dr(3.18) 2 0 0 0 ~Mi2) exp(tMil»)~ M T' \;fm,\;ft,O~t~Tm~T, i.e.,Tm=T. b)Danhgia thuhaloNhan(3.9)b6i t2C~j(t)vat6ngtheoj, taco 21[tu~(t)112+~ [ a(t)lltumr(t)112+a(t)h(t)t2u;(1,t)+~t2Jr2IUm(r,t)IPdr ]dt p 0 =Ilu r(t)112~~2a(t)]+u;(1,t)~~2a(t)h(t)]m dt dt I +~tfr21um(r,t)IPdr+2(tf(t),tu~(t)) p 0 +2uo~~2a(t)h(t)um(1,t)]- 2uoum(1,t)~ ~2a(t)h(t)Jdt ~ Tich phan (3.19)d6i v6'ibiSn thai giantir 0 dSnt, sail do s~pxSp l~icac s6 h~ng,ta thu duQ'c (3.19) H9CvienNguyln VflDzflng Khao satphuongtrinhparabolicphi tuyin trangmi~nhinhcdu trang21 t 1 2 fllsu~(s)112ds +a(t)lltumr(0112+t2u~(l,t) +3..t2fr21um(r,tf dr 0 p 0 =[1- a(t)h(t) ]t2u~(l,t) + f[S2a(s) J IJumr(S)112ds 0 (3.20) t J t 1 +f[s2a(s)h(s)u~(l,s)ds+i fsdsfr2Jum(r,sfdr 0 Po 0 t +2f(sf(s),su~(s))ds+2uot2a(t)h(t)um(l,t) 0 t -2uo f[s2a(s)h(s)Jum(l,s)ds. 0 Dungb~td~ngthuc(2.9),taco (3.21) a(t)/itumr(t)112+t2u~(l,t)~~/ltum(t)II~,VtE[o,rl Vm.2 Dungcaeb~td~ngthuc(2.6),(2.8),(2.9)vav6i 13>0 nhutrong(3.15),ta danhgiakhongkhokhancaes6h~ngavSphaicua(3.20)nhusau [1- a(t)h(t)] t2u~(1,t) ::;(1+ Ilahll ) (.alltumr(0112+ (3 + 1/ 13~Itum(0112) (3.22) 00 ::;(1 +lIGht )~lltum(Oll~+(3+1/ fJ)t2M T ) f[S2a(s)]'/lumrCs)1I2ds+f[S2a(s)h(s)J u~(l,s)ds 0 0 (3.23) ~[(t2a)ro +4&2ah)J}IUm(S)II~£i, 2 [ ' , ]~-MT (t2a) +4 (t2ah) ,a] 00 00 (3.24) 21uofiB' a(s)h(s) J u. (l,s)ds ,; ~uolll{t'ahfII. ~Iu. (S)II,ds ,; 4juolllV'ahfll.~Olu.(s)ll~dSr ';~UoIIIV'ahfll..JT~, HQcvienNguyin ViiDziing KhilOsatphuongtrinhparabolicphi tuyin trongmi~nhinhcdu trang22 (3.26) 4 t 1 4 t 1 - fsdsfr2Ium(r,s)IPdr5,-t Idsfr2Ium(r,s)IPdr (3.25) PooP 0 0 4 Mr 2TMr5,-T-= , P 2 P 2Iuot2a(t)h(t)um(1,t)/5,2IuoltllahIIJtum(1,t)/ 5,41uoItllaht Iitum(t)llv 5,plltum(t)II~+;(uotjjaht)2, t t t (3.27) 12f(sf(s),su~(s»)ds5,fllsf(s)112ds+ fllsu~(s)112ds. . 0 0 0 (3.28) Dodo,tasuytu(3.20)-(3.27)f~ng t ~Isu~(s)112ds+~al11tum(t)II~ 0 4 5, (1+liGhtX3+1/p)T2M?) +~M,[11&2aill.+411&'ahill.J+2T;, +Fls!(s)1I"b +41u,IIIV'ahfII. § ~: MT 4 ( ) 2 ~ +pT2 uollhlL 5,Mp trongdo if r la cach~ngsachiph\!thuQCvaoT. M~itkhactu (3.18),tacodanhgia t 1 ,t 1 , Ids]r2/P'F(um(r,s)fdr=Idsfr21Ium(r,s)IP-rdr (3.29) 0 0 0 0 t 1 1 =Ids fr2Ium(r,sf dr 5,-Mr 5,Mr, 0 0 2 BU'o-c3.Qua gio-ih~n. Do (3.18),(3.28),(3.29)tasuyfa f~ng,tant~iillQtdayconcuaday {uJ, v~nky hi~ula {um}saocho H9CvienNguyln ViiDzfmg KhilOsatphu:angtrinhparabolicphi tuyin trongmiJn hinhc6u trang23 (3.30) (3.31) (3.32) Urn ~ U trongLOO(O,T;H) ySu*, Urn~ U trong L2(O,T;V) ySu, turn~ tu trong Loo(O,T;V)ySu *, " , (3.33) (tuJ ~ (tu)trongL2(O,T;H)yell, (3.34) r2/PUrn~ r2/Pu trong Y (Qr) ySu. Dungb6dS2.11vStinhcompactcuaLions[3],apd\mgvao(3.32),(3.33) tacothStrichfa tuday {urn}mQtdayconvankyhi~ula {urn} saocho (3.35) turn~ tu m(;lnhtrong L2(O,T;H). Theodinhly Riesz-Fischer,tu (3.35)tacothStrichramQtdayconcuaday {urn}vankyhi~ula {urn}saocho (3.36) Urn(r,t) ~ u(r,t) a.e (r,t) trong Qr =(O,l)x(O,T). Do F(u) =lujP-2u lien t1,1C,ta co (3.37) F(urn(r,t))~ F(u(r,t)) a.e (r,t) trong Qr' Ap d1,1ngb6 dS2.12vS S\l'hQit1,1ySutrong Lq(Qr) v6i , 2/ ' 2/ ' I I P-2 2/ ' 2/ ' 1I P-2 N =2, q=p, Grn=r P F(urn)=r P urn urn' G =r P F(u) =r P u U. Tasuytu(3.29),(3.37)r~ng Grn~ G trongLP'(Qr) ySu, hay (3.38) r2/P'lurnIP-2urn~r2/P'luIP-2utrongLP'(Qr)ySu. Gia sir rpEC1([O,TJ),rp(T)=o. Nhanphuongtrinh(3.9)v6i rp,r6i tichphan haivStheobiSnt, taduqc KhilOsatphuangtrinhparabolicphi tuyin trongmi~nhinhcdu trang24 (3.39) T T - (Uorn,wi )rp(O)- f(Urn(t), Wi)rp'(t)dt+ fa(t)( Urnr (t),Wir)rp(t)dt 0 0 T T +fa(t)h(t)urn(l,t)w/l)rp(t)dt + f(F(urn(t)),Wi)rp(t)dt 0 0 T T = f(/(t), Wi)rp(t)dt+ fa(t)h(t)w/l)rp(t)dt, 1:5,j :5,m. 0 0 DS quagi6i h~ncuasf>h~ngphi tuySnF(urn(t))=Iurn(t)IP-2Urn(t) trong (3.39), tasud\lngb6dSsail nBd~3.1 T T l~~oof(F(urn(t)),wi )rp(t)dt= f(F(u), Wi)rp(t)dt. 0 0 ChUngminhb6d~3.1. Chuyr~ng(3.38)tUOTIgdUOTIgv6i TIT 1 f fr2/P'IUrn(t)IP-2Urn(t)(r,t)dtdr~ f fr2/P'lu(t)IP-2u(t)(r,t)dtdr, (3.40) 00 00 V E (U' (QT))' =U(QT)' M~tkhac,taco T T 1 f(F(urn(t)),Wi)rp(t)dt= ffr21urn(t)IP-2Urn(t)Wi (r)rp(t)drdt (3.41) 0 00 = ff~2/ p'IUrn(tW-2 Urn(t)~r2/Pwi(r)rp(t) }irdt. 0 0 Do (3.40),b6 dS3.1seduQ'chungminhnSutanghi~ml~iduQ'cr~ng (r,t)=r2/Pwi(r)rp(t)ELP(QT)'Th~tv~y,dob~td~ngthuc(2.7),taco (3.42) TIT 1 ffl(r,t)IPdrdt =ffr2Iwi(r)rp(tfdrdt 0 0 0 0 1 T =fr2-plrwi(rf dr ]rp(t)IPdt 0 0 I T :5,(FsIIWillv! fr2-Pdr flrp(tWdt 0 0 H9CvienNguyln ViiDziing KhilOsatphuO'ngtrinhparabolicphi tuyin trongmiJn hinhcdu trang25 T ~ 3 ~ p (~IIWjllvr flqJ(t)la+ldt < +00. V~ybE>d@3.1 duQ'cchungminh ho~mt~t. Cho m~ +00trong (3.39), ta sur ra tir (3.11), (3.30), (3.31) va bE>d@3.1, r&ngu thoaphuangtrinhbiSnphan (3.43) T T - (uo,Wj}qJ(O)- f(u(t), Wj}qJ'(t)dt+ fa(t)(ur(t), Wjr}qJ(t)dt 0 0 T T + fa(t)h(t)u(l,t)wj(1)qJ(t)dt+ f(F(u(t)), Wj}qJ(t)dt 0 0 T T = f(f(t), Wj}qJ(t)dt+liofa(t)h(t)wj(l)qJ(t)dt, 0 0 1~j ~m, \tqJE c1([0,T]), qJ(T)=O. Dodotaco (3.44) T T - (uo,v)qJ(O)- f(u(t), v)qJ'(t)dt+fa(t)(ur (t),Vr)qJ(t)dt 0 0 T T +fa(t)h(t)u(l,t)v(l)qJ(t)dt+f(F(u(t)),v)qJ(t)dt 0 0 T T = f(f(t), v)qJ(t)dt.+liofa(t)h(t)v(l)qJ(t)dt, 0 0 \t qJE C1([0,T]), qJ(T) = 0, \tv E V. L~yqJE D(O,T), tir(3.44)tasurra T d T f- [(u(t),v)}P(t)dt+ fa(t)(ur(t),Vr)qJ(t)dt 0 dt 0 T T +fa(t)h(t)u(l,t)v(l)qJ(t)dt+f(F(u(t)),v)qJ(t)dt 0 0 T T = f(f(t), v)qJ(t)dt+liofa(t)h(t)v(l)qJ(t)dt, 0 0 \tqJED(O,T),\tVEV. Do do,taco (3.45) KhilOsatphuongtrinhparabolicphi tuyin trangmiJn hinhcdu trang26 (3.46) ~[(u(t),v)]+a(t)(ur(t),vr)+a(t)h(t)u(l,t)v(l)+(F(u(t)),v) =(J(t),v)+uaa(t)h(t)v(l),Vv E V, dungtrongD(O,T)vadodohAllhSttrong(O,T). Cho q>E C1([0,TJ),q>(T)=o. Nhan phuangtrinh (3.46)cho q>,saudo tich phanhaivStheobiSnthaigiantathuduQ'c (3.47) T T - (u(O),v)q>(O)- f(u(t), v)q>'(t)dt+ fa(t)(ur (t),Vr)q>(t)dt a a T T +fa(t)h(t)u(l,t)v(l)q>(t)dt+f(F(u(t)),v)q>(t)dt a a T T = f(J(t), v)q>(t)dt+uafa(t)h(t)v(l)q>(t)dt, a a v q>Eel ([O,TJ), q>(T)=0, Vv E V. SOsanh(3.44)v6i (3.47),tathuduQ'c (3.48) - (u(O),v)q>(O)= -(ua, v)q>(O),V q>Eel ([O,T]), q>(T)= 0, Vv E V, ma(3.48)tuangduangv6idiSuki~ndAti (3.49) u(O)=Ua' Ta chuyr~ng,tiT(3.30)-(3.34)taco u E L2(0,T;V)nD"(0,T;H), tuEL"'(O,T;V), tu' E L2(0,T; H), r21puE LP(QT} V~ySlJt6nt~inghi~mduQ'chungminh. BtrO'c4.Tinh duy nh§t nghi~m. Tru6chSttacAnb6dSsauday BB d~3.2.GiGsir w Ia nghi?myiu Gilabai toimsau (3.50) Wt - a(t)( Wrr +~Wr)=f(r,t), 0<r <1,0<t <T, (3.51) I;~~rwr(r,t)!<+00,-Wr(l,t)=h(t)w(l,t), Khao satphuongtrinhparabolicphi tuyin trongmi~nhinhcdu trang27 (3.52) w(r,O)=0, (3.53) WE L2(0,T;V)nL"'(0,T;H), twE L"'(O,T;V), tw'E L2(0,T;H). Khi d6 (3.54) ~llw(t)112+ fa(s)~IWr(s)112+h(S)W2(1,s)]ds 2 0 t - f(J(s), w(s))ds=0, a.et E (O,T). 0 Chti thich 2.B6 dS3.2Ii mQtS\ft6ngquathoacuab6dStrongcu6nsach cuaLions [3]chotruemghQ'PkhonggianSobolevcotrQng.ChUngminhb6 dS3.2co thStimthfiytrong[2].Bay gia, ta sechUngminhtinh duynhfit nghi~m. Gicisiru vi v Ii hainghi~mySucuabii toan(3.1)-(3.4).Khi do w=u- v Ii nghi~mySu cua bii toan (3.50)-(3.52)v6i vS phcii cua (3.50) Ii J(r,t) =-lu(t)IP-2u(t)+Iv(t)IP-2V(t).Dung b6 dS3.2,ta co ding thuc sau (3.55) ~llw(t)112+ fa(s)~lwr(s)112+h(S)W2(1,S)]ds 2 0 t =- f(lu(sf-i u(s) -lv(s)la-iv(s),w(s))ds~O. 0 Do tinhchfitdandi~utangcuahim s6th\fcu~ lulP-2u.TIT(3.55)tasuyra r&ngw=o. Tinh duynhfitduQ'chungminh. V~ydinhIy 3.1duQ'chUngminhxong. H9CvienNguyln ViiDziing ._.