Nghiên cứu một số phương trình nhiệt phi tuyến trong không gian sobolev có trọng

lamm~ttinht6ngquattal~y&=1. Nghi~mye'ucua bat toangia tri bien va ban d~u(3.1)- (3.4) duQcthanhl~pnhusau: TImuEI3(O,T;V)nLOO(O,T;H)saDchou(t) thoabili loanbitn phansau (3.5) d - (u(t),v)+(ur(t),vr) +h(t)u(l,t)v(l)+(Fi (u(t)),v)dt 16 =(/(t), v)+uoh(t)v(I),Vv EV, a.e.,t E (O,T), va di~uki~nddu (3.6) u(O)=uo' Khi d6tac6djnhIy sail Dinh If 3.1. ChoT >0 va (HI) - (H4) dung.Khi do,bai loan (3.1) (3.4) co duy nhat mQt nghi~m ytu u EL2(O,T;V)nLOC)(O,T;H) saocho (3.7) tuELOC)(O,T;V),tu/ EL2(O,T;H), r2/5uEL5/2(QT)' Chungminh. G6mnhiSubu'oc. BuGe1. PhudngphapGalerkin. La'y {wi},j =1,2,...la m<)tco sa tr1!cchu~ntrongkh6ng gian Hilbert tachdu'<JcV. Ta tlm um(t)theod~ng (3.8) m . um(t)=LCmj(t)Wj' j=l trongd6 cmj (t), 1~j ~m thoah~phu'ongtrlnhvi phanphi tuye'n (3.9) (U~(t),Wj) +(umr(t),Wjr) +h(t)um(1,t)Wj(1) +(Fi (um(t)),Wj) =(/(t), Wj) +uoh(t)wj(1),1~j ~m, (3.10) um(O)=uom' trongd6 (3.11) Uom~ Uom~nhtrongH. D~tha'yding voi m6im,t6nt~imQtnghi~mum(t)c6 d~ng(3.8) thoa(3.9)va (3.10)hftukhiipnoi tren O~t~Tm,voi mQtTmnao d6, O<Tm~T. 17 Cac danhgia tien nghil$msauday cho phepta 1a"yTm=T voi mQlm. Btioc2. Ddnhgidtiennghi~m. Ta se1~n1u'Qtthi€t 1~phaidanhgiatiennghil$mdu'oiday.Kh6 khanchinhd ph~nnay1asf)h~ngphi tuye'n Fl(um(t))=I Um(t)11/2Um(t) thalli gia vao phu'dng trlnh do d6 vil$cdanhgiatinhbich~nvaquagioih~ncuasf)h~ngnaycling 1am(>tkh6khan.Tuynhien,voi sf)h~ngphituye'nCl;lth~tfong tfu'onghQpnaykh6nggayfa nhi~utfdng~isovoi sf)h~ngphi K ~ " tuyentongquat. a) Danbgia1.Nhanphu'dngtrlnhthlij cuahl$(3.9)voi cm/t) vat6ngtheoj, tac6 (3.12) ~llum(t)112+21Iumr(t)f +2u~(1,t)dt 1 + 2 Sri Um(r,t) 15/2dr° =2(1-h(t))u~(1,t)+2(f(t),um(t))+2iioh(t)um(1,t) Tli ba"td£ngthlic(2.9),tasur fa rang (3.13) 211Umr(t) 112+ 2u~(1,t) 211Um(t) II~. Ta surtli (3.12),(3.13)ding (3.14) 1 ~IIUm(t)112 +II Um(t)II ~+2SriUm(r,t)15/2dr m ° ~211- h(t)1[,811Um,(t) !I'+(2+1/,8)11Um(t) 112] +211fer) 1I11um(t) II+ 21iioh(t) I~II Um(t)11v ~ 2( 1+IIh IILoo(o,T)) [fill Um(t) II~ +(2+1/P)IIUm(t)112] 18 +11/(1)112+IIUm(t)112+2~luillhll~(O,T) +2Pllum(t)II~ =~liioI21IhI1200+IIJ(t) 112+2/3(2+llhIILOO(OT») llum(t)II~2/3 L (O,T) , +[1+2(2+11/3)(1+llhIILoo(O,T»)]llum(t)112,'\ /3>0 ChQn/3>0 saocho (3.15) 2/3(2+II h IILOO(O,T»)~ 112. Tir (3.14),(3.15)taduQc 1 (3.16) ~llum(t)112+.!.llum(t)II~+2frlum(r,t)15/2dr dt 2 0 ,;;2~lulllhll~(O'T)+11/(1)112 + [1 + 2(2 + 11/3)(1+ IIh lIroo(O,T»)]II um(t) 112 . Lffytichphan(3.16)theot,vasad1;lng(3.10),(3.11)taco t t 1 (3.17) lIum(t)112+.!.~lum(s)ll~ds+2fdsfrlum(r,s)15/2dr 20 0 0 t ~lluoml12 +~liioI2 11h11200 + ~IJ(s)112 ds 2/3 L (O,T) ;1 t +[1+2(2+11/3)(1+II hIILoo(O,T»)]~Iurnes) 112ds 0 t ~M ?)+ M}l) ~IUrnes)112ds, 0 trongdo M}l),M}l)1acach[lngs6 chi ph1;lthuQcvao T va duQc chQnnhusail: M}l) = 1+ 2(2 + 11/3)(1+ II h II LOO(O,T»)' Mf2) 211uoml12+C~I ill II hII~oo(O'T»)T+JI/(S) 112dy, '1m. Nhob6d~Gronwall2.13,tir(3.17)taduQc 19 (3.18) t t 1 II Um(t)112 +~JII UrneS)II ~ds+2IdsSriUm(r,s)15/2dr 20 0 (2) ( (1) )~MT exptMT ~MT, '\1m,'\It, O~t~Tm~T, Tm=T.nghla 1a b)Danhgia2. Nhan(3.9)voi t2C~j(t)vat6ngtheoj, taco 21Itu~(t)112+ ~ [ II tumr(t)r+h(t)t2u;(1,t)+4t2frl Um(r,t)15/2dr ]m 5 0 =2tllUmr(t)112+u;(1,t)~[t2h(t)]dt 1 +~tfrlum(r,t)15/2dr+2(tf(t),tu~(t)) 5 0 +2uo~[t2h(t)um(1,t)]- 2uoum(1,t)~[t2h(t)] dt dt TJ:chphan(3.19)theobie'nthaigiantu0 de'nt saildo s~pxe'p1(;li cacs6h(;lngtaduQc (3.19) t (3.20) 2~lsu~(s)112ds+lltumr(t)r +t2u;(1,t) 0 1 4 2f I 1 5/2+-t r Um(r,t). dr 5 0 t t =[1- h(t)]t2u;(1,t)+2fsllUmr(s)112ds+ f[s2h(s)]/u;(1,s)ds 0 0 tit +8 fsds frlum(r,s)15/2dr+2 f(sf(s),su~(s))ds 5 0 0 0 t +2Ui2h(t)um(1,t)- 2uof[s2h(s)ium(1,s)ds 0 Dungbit dAngthuc(2.9),taco (3.21) Iitumr(t)r +t2u;(1,t)~~lltum(t)II~,'\ItE[O,T],'\1m. 20 Dungcaeba'td~ngthuc(2.6),(2.8),(2.9)va voi 13>0 nhu'tfong (3.15),tadanhgiakhongkh6khancaes6h(;tng(j v€ phili cua (3.20)nhu'sau (3.22) [1- h(t)]t2u;(I,t) ,:; (I +II hIILw(o,T))[piltum,(t)112 +(2+1/ P)II tum(t) 112J ::;(1+II h IILoo(O,T))[fJlltum(t) II~ +(2+1/13)t2MT 1 t t (3.23) 2IsilUmr(S)112ds+ I[s2h(s)]/U~(1,s)ds 0 0 ,;[ 2T +311(t2h)tw(o,n]]1Um(S)II~ds0 ::;2MT [ 2T+3 11 (t2hi ll ] , Loo(O,T) (3.24) t 21ito I[s2h(s)] / um(1,s)ds 0 t ::;2F3litol ll (t2hi ll rllum(s)llv ds Loo(O,T)JI0 ::;21itolll (t2hi ll ~6tMT 'LOO(O,T) (3.25) 21uot2h(t)um(t) I,,; PII tum(t) II ~+;(uillhIIL~(O.T)r. t t t 2 (3.26) 21 J(sf(s),su~(s))ds ::; III sf(s) 112ds + ~lsu~(s)1Ids 0 0 0 Do d6,tu (3.20)- (3.26)suyfa t (3.27) ~Isu~(s) 112ds +~II tum(t) II~ 0 4 ::;(1+II h !lroo(O,T))(2+1/fJ}t2MT +2MT(2T +311(t2hi!lroo(O,T)) 16 t t +- III SUm(S) IIv ds +III s f(s) 112ds 5 0 0 +~(uJIIh Ilno,T) r +21 uolll(t2h) / 11t"(O.T).J6t MT 21 t t ~M?) + 16]1surn(s)IIv ds ~M~4)+.!. ]1surn(s)II ~ds, 3 0 40 trongdo M?) ,M~4)=M?) +2~6la cach~ngs6chiphl;lthuQcT Dob6d€ Gronwall,tu(3.27)suyra t 2 1 (3.28) ~Isu~(s)II ds +-II turn(t)II~~M~4)et~Mf4)eT =M?). 0 4 M~tkhac,tu(3.18)tacodanhgia t 1 f n 3/5 1 5/3 (3.29) dsJlr F}(urn(r,s)) dr 0 0 t 1 = Ids frlurn(r,s)15/2dr ~.!.MT~MT 0 0 2 Bu'dc3. Quagidi h(ln Do (3.18),(3.28),(3.29)ta suyfa, t6n t~imQtday con cua day {urn}v~nkyhi~ula {urn}saocho (3.30) (3.31) (3.32) (3.33) (3.34) urn~ u trong LOO(O,T;H)ye"u*, urn~ u trongL2(O,T;V) ye"u, turn~ tu trong LOO(O,T;V)ye"u*, (turn)/~ (tu)/ trongL2(O,T;H) ye"u, r2/5urn~ r2/5u trong L5/2(QT) ye"u. Dung b6 d€ 2.11v€ tinh compactcua J.L.Lions, ap dl;lngvao (3.32),(3.33)taco th€ trichra tu day {urn}mQtday conv~nky hi~ula {urn}saocho (3.35) turn~ tu m~nhtrongL2(O,T;H). Theodinhly Riesz- Fischer,tu (3.35)taco th€ la'yra tu {urn} mQtdayconv~nky hi~ula {urn}saocho (3.36) urn(r,t)~u(r,t) a.e. (r,t)trong QT=(O,I)x(O,T). 22 Do Fi (u)= I U III 2 u lien Wc, lien (3.37) Fi(um(r,t))~ Fi (u(r,t)) a.e. (r,t) trongQT' Ap dlJng b6 d~ 2.12, vdi N = 2, q = 5/3, Gm=r3!5Fi(um)=r3!5IumI1l2um,G=r3!5Fi(u)=r3!5IuI1l2u. Tu (3.29),(3.37)suyra (3.38)r3!5luml1l2um~r3!5IuI1l2utrongL5!3(QT)ye'u Gia sa cpE c1([O,T]),cp(T)=O.Nhanphu'dngtrlnh(3.9)vdi cp, saild6 tichphantungphftntheobie'nt, tadu'<jc T T - \uom'wi )cp(O)- f\ um(t)'Wi )cp!(t)dt + f\ umr (t), wi r)cp(t)dt 0 0 (3.39) T T + fh(t)um(1,t)w;Cl)cp(t)dt+ f\Fi (um(t)),wi)cp(t)dt 0 0 T T =f\1(t),wi )cp(t)dt+Uofh(t)w;Cl)cp(t)dt,1~j ~m 0 0 D~ qua gidi hC;lncua so'hC;lngphi tuye'nFi(um(t))trong(3.39)ta sadlJngb6d~sail Bfl d~3.1.Taco T T lim f\Fi (um(t)),wi )cp(t)dt = f\Fi (u(t)),wi )cp(t)dtm-++00 0 0 Chung minh. Chti yding (3.38)tu'dngdu'dngvdi TIT 1 (3.40) fdt fr3!51uml1l2umdr ~ fdt fr3!51u 1112udr 0 0 0 0 ! \7'E (L5!\QT)) =L5!2(QT)' Mi;Hkhac,tac6 T Tl (3.41) f\Fi (um(t)),wi)cp(t)dt=f Sri umIII 2umWi(r)cp(t)dr dt 0 00 23 T 1 f f( 3/5 1 1 1/2 )( 2/5 )= J r Urn Urn r w;Cr)lp(t drdt. 00 Do (3.40), ta chungminh =r2/5w;Cr)lp(t)E L5/2(QT). Th~tv~y,do bfftd~ngthuc (2.7), ta co TIT 1 f~ 5/2 ff . i 1 5/2 (3.42) JI1 drdt= r w;Cr)lp(t) drdt 00 00 1 T f 1/4 1 r 1 5/2 ~ 1 5/2 = r- -vrw;Cr) drJllp(t) dt 0 0 5121 T ~(21Iwjllv) fr-1/4drfllp(t)15/2dt0 0 T 15 /,.. 11 11 512 ~ 5/2 =3-v2 Wj v Jllp(t)1 dt<+oo.0 Dodo,b6d~3.1.ducJcchungminh. Cho m~ +00tfong(3.39),tu (3.11),(3.30),(3.31)vab6d~3.1 tasuyfa u thoaphuongtrlnhbie'nphan T T - (uo,Wj)lp(O)- f(u(t),Wj)lpl(t)dt+f(ur(t),Wjr)lp(t)dt 0 0 T T (3.43) +fh(t)u(1,t)w;Cl)lp(t)dt+f(Fi(u(t)),Wj)lp(t)dt, 0 0 T T =f(/(t),Wj )lp(t)dt+uofh(t)w;Cl)lp(t)dt , 0 0 V rpE C1([O,T]) ,lp(T) =0, Vi =1,2,...,m. Dodotaco T T - (uo,v)lp(O)- f(u(t), v)lpl(t)dt + f(ur(t), Vr)lp(t)dt . 0 0 T T (3.44) + fh(t)u(1,t)v(1)lp(t)dt+ f(Fi (u(t)),v)lp(t)dt 0 0 T T =f(/(t), v)lp(t)dt+Uofh(t)v(1)lp(t)dt, 0 0 24 '\IcpE C1([0, T]), lp(T) =0,'\IvE V La'y cpED(O,T), tu (3.44)suyra T d T (3.45) f[-(u(t),v)]cp(t)dt+ f(ur(t),vr)cp(t)dt 0 m 0 T T + fh(t)u(l,t)v(l)cp(t)dt+ f(Fi (u(t)),v)cp(t)dt 0 0 T T = f(/(t), v)cp(t)dt+Uofh(t)v(l)cp(t)dt,'\IcpED(O,T),'\IvEV . 0 0 Dodotaco d (3.46 -( u(t),v)+(ur(t),vr)+h(t)u(l,t)v(l)+(Fi(u(t)),v)dt =(/(t), v)+uoh(t)v(l), '\IvEV dungtrongD(O,T)vadodoh~uhe'trong(O,T). Cho lpE c1([O,T]),cp(T)=o.Nhanphuongtrlnh(3.46)vdi cp,sail dotichphantungph~ntheobie'nthaigiantadu<jc T T - (u(O),v)cp(O)- f(u(t), v)cp/(t)dt+ f(ur(t),vr)cp(t)dt 0 0 T T (3.47) + fh(t)u(l,t)v(l)cp(t)dt+ f(Fi (u(t)),v)cp(t)dt 0 0 T T =f(/(t), v)cp(t)dt+Uofh(t)v(l)cp(t)dt, 0 0 '\IcpE C1([0,T]),cp(T)=0,'\IvEV. So sanh(3.44),(3.47)tadu<jc (3.48) - (u(O),v)cp(O)=-(uo,v)cp(O) '\IcpE C1([0,T]), cp(T)=0,'\IvE V, ma(3.48)tuongduongvdidi~uki~nd~u (3.49) u(O)=uo' Ta chu yding,tu(3.30)- (3.34),taco uEL2(0,T;V)nLOO(0,T;H),tuELOO(O,T;V)va 25 / 2 ( . ) 2/ S Ls/2(Q )tu E L O,T,H, rUE T . V~yst!t6nt<;tinghit%mdU<;1cchungminh. Bu'oc4. Tinh duynha'tnghit%m Trudche't,tacfinb6dSsailday. B6 d~3.2.Gidsaw Ianghifmytucuahailoansau 1 - (3.50)Wt-(wrr +-wr)=f(r,t), O<r<l, O<t<T,r (3.51) Ilim vlrwr(r,t) I <+00,wr(1,t)+h(t)w(1,t)=O, r~O+ (3.52) w(r,O)=O, { wEL2(0,T;V)n LOO(O,T;H), (3.53) . twELOO(O,T;V),tw/EL2(O,T;H). Khido t (3.54) ~llw(t)112+ Kllwr(s)112+h(s)w(1,s)]ds 2 0 t - f(J(s), w(s)}ds=0, a.e.t E(O,T). 0 Chti thich.B6dS3.2la t6ngquathoacuab6dStrongcu6nsach cuaJ.L.Lions [2]chotruongh<;1pkh6nggianSobolevco trQng. Chungminhcuab6dS3.2cothS!lmtha'ytrong[8]. GiasauvavIa hainghit%mye'ucuabaitoan(3.1)- (3.4).Khi do w =u - v la nghit%mye'"ucuabaitoan(3.50)- (3.52)vdive' phai J(r,t)=-lu(t)I1/2u(t)+lv(t)I1/2v(t).Dungb6dS3.2,ta cod~ngthucsail (3.55) t ~llw(t)112+ f[llwr(s)112+h(s)w2(1,s)]ds 2 0 t =- f(1 u(s) 11/2U(S)-I V(S)11/2V(S),W(S))ds:::;0, 0 26 do tinhchiltdondi~utangcua I u11/2u. Tir (3.55)ta suyfa ding w =O.Tlnh duy nhilt du'<;1cchungminh. V~ydinhly (3.1)du'<;1cchungminhKong. 27 ._.