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

), h(t)2ho >0, (H~) f E Co(O,T;H), f(r,O) =f(r,T). Khonglammftttinht6ngquatcuabaitoantalfty[;=1 Nghi<$mye'ucuabaitoan(4.1)- (4.4)du<;5Cthi€t l~ptubaitoan bie'nphansau: TImuEL2(0,T;V)nLOO(0,T;H)saDchoul EL2(0,T;H) va u(t) thoaphuclngtrinh bienphansau: (4.5) T T f\ul(t),vet)dt + f[(urCt),vrCt))+h(t)u(1,t)v(1,t)]dt° ° T T T + f(Fi(u(t)),v(t))dt= f(f(t),v(t))dt+uo fh(t)v(1,t)dt,° ° ° 'v'vEL2(0,T;V), 28 vadi~uki~nT - tudnhoan (4.6) u(O)=u(T). Khi do, taco dinh19sail Dinh Iy 4.1. Cho T>O va (Hl),(H~),(H~)dung.Khi do, bai loan (4.1)- (4.4)co duynhcitmQtnghi~mye'uT - tudnhoan U E L2(0,T;V) n LOO(O,T;H) saD r2/5u E L5/2(QT)' Chungminh.G6mnhi€u buGc. Btidc1.PhLtdngphapGalerkin. cho u/ E L2(0,T;H), Lffy mOtcd sa t11!cchufin {wi},j =1,2,...trongkhong gian Hilberttachdu<;1cV. Ta Hmurn(t)theod~ng rn (4.7) urn(t)=LCrnj(t)Wj' j=l trongdo crn/t) thoah~phudngtrlnhvi phanphituySn (4.8) (u~(t),Wj) +(urnr(t),Wjr ) +h(t)urn(1,t)Wj(1) +(Fi (Urn(t)),Wi) =(/(t),Wi)+uoh(t)w/l),1:::;j:::;m, vadi€u ki~nT - tu~nhoan (4.9) Urn(0) =urn (T). D~utieD,taxeth~phudngtrlnh(4.8)vadi€u ki~nd~u (4.9/) Urn(0) =uorn' trong do Uorn thuOc khong gian sinh boi cac ham {Wi},j=1,2,...,m.Khi do, ta duQcmOth~m phudngtrlnhvi phanthuongphituySnVGicac fin hamCrnj(t),j=1,2,...,m,va cacdi€u ki~nd~u(4.9/).D~thffydingt6nt~iurn(t)co d~ng 29 (4.7)thoa(4.8)va (4.9/)vdi hftukh~pndi tfen 0s,t s,Tm,vdi illQt TmE (O,T]. Cac danhgia tien nghi~illsail day chophepta 11yTm=T vdi illQi ill. Brioe 2.Ddnhgid liennghi~m. Nhanphudngtrlnhthlij cuah~(4.8)vdi cmj (t), saildo1a'y t6ng theoj , taduQc (4.10) ~llum(t)112+21Iumr(t)r+2h(t)u~(l,t)dt 1 +2frl Um(r,t)15/2dr° =2(f(t),um(t))+2uoh(t)um(t). Tu giathiet(H~)vaba'td£ngthlic(2.9),suyfa (4.11) 2\\umr(t)112+2h(t)u~(l,t)~C11Ium(t)II~ tfong do C1=mill {I,ho}. Dodo,tu(4.10),(4.11)suyfa 1 (4.12) ~llum(t)112+Clllum(t)II~ +2 frlum(r,t)15/2 dr m ° s,2(f(t),um(t))+2uoh(t)um(l,t) s,~llf(t)112+6'llum(t)112+~luoI21IhI1200 +6' ll um(t)ll v 2 6' 6' L (O,T) . S,~llf(t)112+~luoI21Ihll~oo(O'T)+26'llum(t)II~ vdi illQi 6'>0. ChQn6'>0 saGcho (4.13) CI-26'=C2 >0. Do do, tu (4.12),(4.13)tathuduQc (4.14) ~lIum(t)1I2+C21Ium(t)1I2dt 30 t::;~llum(t)112+C21Ium(t)II~+2 frlum(r,t)15!2dr & 0 ::;~llf(t)112+~luoI21IhI1200 =ht(t). (j (j L (O,T) Nhanbfttd~ngthuc(4.14)vdi eC2tvasaildotichphantaco t (4.15) lIum(t) 112::;lluoml12e-C2t +e-C2t fht(s)eC2Sds. 0 Cho T >0, taxethams6sail t - j(eC2t-lrt fht(s)eC2Sds, O<t::;T, (4.16) R(t)=] 0 ht(0)/C2, t=O. Khi do R ECo[o,T]vatad~tR =max~R(t). O<;,t<;,T N€u II uomll::; R tu(4.15),(4.16)chota (4.17) II um(t) II::;R nghlala Tm=T vdimQim. GQi Bm(O,R)la quacftudongtam0, bankfnhR trongkhong gianmchi~usinhbdi caeham wi'} =1,2,...,m,d6ivdi chuffn11.11 Xet anhX£;lFm:Bm(O,R)~ Bm(O,R)chobdicongthuc (4.18) Fm(uom)=um(T). Ta chungminhr~ngFmla anhX£;lco. Gia sa Uom'VomEBm(O,R)va d~tm(t)=Um(t)-Vm(t),trongdo Um(t)'Vm(t)la caenghi~mcuah~(4.8)tren [O,T]thoacaedi~u ki~ndftuum(O)=Uomva vm(O)=vomIftn ltim(t) thoah~phuongtrlnhvi phansailday (4.19) \~(t),wi)+(mr (t),wir)+h(t)m(1,t)wi (1) =- (I Um(t)1112Um(t)-I Vm(t)1112Vm(t),Wi),1::;}::;m vadi~uki~ndftu 31 (4.20) m(O)=uom-vom' Tinh loantuongtlfnhuchuang3,tadU<;1c (4.21) ~11m(t)112+ 11mr(t)r+ 2h(t)1m(1,t)I2dt =- 2(1um(t)11/2Um(t)-I Vm(t)11/2Vm(t),Um(t) - Vm(t))~0 Do (4.11),tu(4.21)suyra (4.22) ~11m(t)112+C111mr(t)II~~O Tich phanbfttd~ngthU'c(4.22),tadU<;1C 1 --TCI (4.23) II um(T)- vm(T)II ~e 2 II Uom- Vomll- - nghlala Fm:Bm(O,R)-) Bm(O,R)la anhX(;lco. Do d6 t6n t(;li duy nhftt UomE Bm(O,R) sao cho Uom=Fm(uom)=um(T). Do d6, voi m6i m t6n t(;limQt ham uomEBm(O,R) S110cho nghi<%mcuabai loangiatribandffu(4.8),(4.9/) la mQtnghi<%m T- tuffnhoancuah<%(4.8).Nghi<%mnayclingthoabfttd~ngthU'c (4.17)voihffuh€t tE[O,T]vanha(4.14)tasuyra t t 1 (4.24) II Um(t)112 +C2III Urnes)II ~ds+2Ids Irl um(r,s)15/2dr ~C3' 0 0 0 trongd6 C3la h~ngs6dQcl~pvoim. NhanphuongtrlnhthU'j cuah<%(4.8)voi c~/t), Iftyt6ngtheoj va saud6tichphantungphffntheobi€n ttu0d€n T, tac6 T 2 ITd 2 ~Iu~(t) II dt +- I-II Umr(t)II dt 0 2 0 dt T T +1. Ih(t)~[u;(1,t)]dt + I(I Um(t)11/2Um(t),u~(t)dt 2 0 dt 0 (4.25) 32 T T =f(f(t),u~(t))dt+uofh(t)u~(1,t)dt 0 0 Tu (4.9)tathfiydinghaibfitd~ngthucsaildaydung: T d 2 il f-II Umr(t)II dt=O, 0 dt T 1 R J .. 1/2 1 2 d 5/2 111 f(1um(t)I Um(t),um(t))dt=-frdr -IUm(r,t)1 dt 0 50 0 dt 1 =2fr(I um(r,T)15/2-I Um(r,O)15/2)dr =O. 50 Do do,d~ngthuc(4.25),nhotkh phantungphftntathuduQc (4.26) T . 2 T IT ]lu~(t)11dt= f(f(t),u~(t))dt+- fhl(t)u~(1,t)dt 0 0 20 T - Uofhl (t)um(I,t)dt. 0 Sail cling,nho(4.24),(4.26),suyfa bfitd~ngthucsail T 2 T T 2 (4.27) 2]1u~(t)II dt ~ fllf(t) 112dt + ]1u~(t)II dt 0 0 T T + ll hl ll fu~(1,t)dt+2Iuolll hl ll rlum(1,t)ldtLoo(O,T) Loo(O,T)JI0 0 T 2 T T ~ ]1U~(t)II dt + Slifer) 112dt +311hl!lrOO(O,T)IIi Um(t) II~dt 0 0 0 T +2~luolll hl ll rIIUm(t)llvdtLOO(O,T) JI0 T 2 T T ~ ]lu~(t)11dt+ fllf(t)112dt+31Ihl!lrOO(O,T)fllum(t)ll~dt 0 0 0 ( T J 1/2 +2~3T 1"0111hl"'(O,T) }Ium(t)II~dt 33 T 2 ~ ]IU~(t)11dt+C4, 0 trongdo C4la hangs6dQcl~pvoim. T 2 (4.28) ]1u~(t)II dt~C4voimQim. 0 M~tkhac,tli'(4.24),tacodanhgia t f In 3/5 1/2 1 5/3 (4.29) dsJI r 1 urn(r,s)I urn(r,s) dr 0 0 t 1 = Ids SriUrn(r,s)15/2dr ~!:.C3. 0 0 2 Bdoc 3. Qua gicii h(}n. Do (4.24),(4.28),(4.29)ta suyfa, t6nt:;timQtdayconcuaday {Urn},v~nky hit%uIa {urn}saocho (4.30) urn~ U trong Loo(O,T;H) ye'u *, (4.31) urn~U trong L2(O,T;V) ye'u, (4.32) u~~ u/ trong L2(O,T;H) ye'u, (4.33) r2/5urn~ r2/5u trongL5/2(QT)' Tru'dehe'ttanghit%mrang (4.34) u(O)=u(T). Voi mQivEH, tli'(4.9)taco (4.35) T f(u~(t),v)dt =(urn(t)- urn(O),v) =o. 0 Tli'(4.32),(4.35)suyra T T f(u~(t),v)dt~ f(u/ (t),v)dt =0 khim~ +00 0 0 Tinh toantu'dngtl!nhu'(4.35)taco 34 (4.36) (4.37) T (u(T) - u(O),v) =f\Ul (t),v)dt =0 vdi ffiQi v E H, 0 vadodo(4.34)dung. Dungb6 dS 2.11.vS tinhcoffipactcuaJ .L.Lions,ap dl;lngvao (4.31),(4.32)tacoth€ Iffyratuday{urn}ffiQt dayconvftnky hi~uIa {urn}saocho (4.38) urn ~ u ffi(;lnhtrong L2(0,T;H). Do dinhIy Riesz- Fischer,tu(4.38)tacoth€ Iffyratu {urn}ffiQt dayconvftnkyhi~uIa {urn}saocho (4.39) urn(r,t)~ u(r,t) a.e(r,t) trongQT=(0,1)x(O,T). D 11 1/2 I.". ".0 U H U U len tl;lcDen (4.40) 3/5 1 1 J/2 3/5 1 1 J/2 r urn(r,t) urn(r,t)~ r u(r,t) u(r,t) vdi a.e.(r,t) trongQT' Ap dl;lngb6 dS2.12,vdi / 3/5 1 1 1/2 3/5 1 . / J/2 N=2,q=53,Grn=r Urn urn,G=r u u. Tu (4.29),(4.41)suyra (4.41) r3/51urnlJ/2urn ~ r3/51 U 1J/2 u trong L5/\QT) y<5u. K " h.". ( ) 1 . ( iTCt J . 12 1",. ? h ~ Y lyU gi t =.J2sm T ,1= , ,... a ffiQtcosotn!cc uan trong khong gian Hilbert thlfc L2(O,T). Khi do t~p {giWj;i,j=I,2,...}I~pthanhffiQtco sdtrlfcchufintrongkhong ; 2 pan L (O,T;V). Nhanphuongtrlnhthili cua(4.8)vdi gi(t) saildoIffytichphan theot , 0~t ~T , taco 35 T T J( U~(t),wi)gi (t)dt + f(umr (t),wi r )gi (t)dt 0 0 T T + fh(t)um(1,t)w/l)gi (t)dt+ f(1Um(t)11/2Um(t),Wi)gi(t)dt 0 0 T T =f(/(t), Wi)gi(t)dt+fuoh(t)w/l)gi(t)dt 0 0 V} =1,2,...,m,Vi E N. D€ quagidi h~ncuas6 h~ngphi tuye'nI um(t)I ]/2 Um(t)trong (4.42) (4.42)tadungb6 dS sail B6d~4.1.Vi,}=1,2,...taco . T T }~~f(1Um(t)11/2Um(t),Wi)gi(t)dt = S(IU(t)11/2U(t),Wi)gi(t)dt. 0 0 Chungminh.Chily ding(4.41)tu'dngdu'dngvdi T ] T 1 (4.43) fdt fr3/51 uml1/2 umdr~ fdt fr3/51 U 11/2udr 0 0 0 0 1 VE (L5/\QT)) =L5/2(QT)' M~t khac, ta co T T] S(i Um(t)11/2Um(t),Wi )gi(t)dt = f SriUml1/2umWi(r)gi(t)dr dt 0 00 (4.44) T] =f f(r3/51Uml1/2um)(r2/5w/r)gi(t) }1rdt. 00 Do (4.44),b6 dS4.1sedu'Qchungminhne'utakh~ngdinhdu'Qc ding =r2/5wi(r)gi(t) EL5/2(QT)'Th~tv~y,do bfftd~ngthuc (2.7),taco Tl Tl 5/2 f fl15/2drdt=f Sriw/r)gi(t) I drdt 00 00 36 1 T f 114 1 ' 1 5/2 ~ 512 = r- 'irWj(r) dr jlgi(t)1 dt 0 0 5/2 1 T (4.45) ~(21IWjllv) fr-1I4drflgi(t)15/2dt0 0 T 15 '" II 11 5I 2 ~ 5I 2 =)'i2 Wj V jlgi(t)1 dt <+00.0 V~yb6d~4.1du<;5cchungminh. Cho m~ +00tfong(4.42),tli (4.30)- (4.32)vab6d~4.1,tasuy fa u thoaphuongtrlnhbie'nphan T T (4.46) f(u/(t),WjJgi(t)dt+ f(ur(t),Wjr)gi(t)dt 0 0 T T + fh(t)u(1,t)w/l) gi(t)dt + f(1u(t)11/2u(t),Wj)gi(t)dt 0 0 T T =f(f(t),Wj)gi(t)dt +itafh(t)wj(1)gi(t)dt, Vi,} E N. 0 0 Tli (4.46)suyfa phuongtrlnhsaildaydung. T T T (4.47) f(ul(t),v(t)Jdt+ f(ur(t),vr(t))dt+ fh(t)u(l,t)v(1,t)dt 0 0 0 T T T +f(1u(t)1112u(t),vet)dt=f(f(t), v(t))dt+itafh(t)v(1,t)dt 0 0 0 VvEL2(Q,T;V). V~ySt!t6nt'.linghi~mdu<;5cchungminh. Blioe 4. Tinhduynht{tnghifm. Gia sa u,v la hainghi~mye'ucua(4.1)- (4.4).Khi do W=u - v thoabai tmlnbie'nphansailday (4.48) T T T . f(wi(t),cp(t)Jdt +f(Wr(t),CPr(t))dt + fh(t)w(1,t)cp(1,t)dt 0 0 0 37 T+f(1U(t)11/2U(t)-I vet)11/2v(t),rp(t))dt =0,VrpEL2(0,T;V), 0 (4.49) w(O)=weT), vdi u,vEL2(0,T;V)nLOO(0,T;H),ul,/ EL2(0,T;H), 2/5 2/5 L5/2(Q )r u, r VET' T Ltty rp=w trong(4.48)va chliydug f\wi (t),wet)dt =O. 0 Khi dosadvng(4.11)va'(4.49)taduQc 1 T T (4.50) -Clllwll\ . ~]lwr(t)112dt+fh(t)w2(l,t)dt2 L (O,T,V) 0 0 T =- f(1 u(t) 11/2 U(t)-I V(t)11/2V(t),U(t)- vet))dt ~O. 0 Di~unayd~nd€n w=0 nghla1au=v. Dinh 1y4.1duQcchungminhhoanloan. 38 ._.