Luận văn Xấp xỉ tuyến tính cho một vài phương trình sóng phi tuyến

[ Chuang21 8 Chltdng2 KHAO SAT PHUONGTRINH SONGPHI TDYEN LIEN KET VOl DIED KI]tN BIEN HONH(jP 1.Mdd§u Trangchuang2,chungtoixetb~litoangiatribienvagiatriband~usanday UI/-uxx=f(X,t,u,ux,U,),XEO, O<t<T, (2.1) (2.2)uAO,t)-hou(O,t)=go(t),uAl,t)+h,u(l,t)=g,(t),O<t<T, u(x,O)=uo(x),u,(x,O)=Ut(x),XEO, (2.3) vdiho,hi Ia cach~ngs6khongamthotnidc,s6 h:~mgphituye'nf clingla hamtho tnidcthuQclop cl([O,l]x[0,00)XR3). Trangchliongnay,tasethie'tl~pmQtdinhly t<Snt~iva duynha'tWi ghHye'u cuabai toan(2.1)-(2.3)b~ngphliongphapxa'pXl tuye'ntinhke'thQpvdi phuong phapGalerkinvaphuongphapcompactye'u.Sand6chungWi kh{wsatva'nd~khai tri8nti~mc~nCllaWi giiUbai toan(2.1)-(2.3)theothams6be s khi s6h~ngphi tuye'nftrong(2.1)dliQcthaybdi f(x,t,u,ux'u,)+sg(x,t,u,ux'u,). Ta thanhl~pcacgiathie'tsan (HI) ho>0, hI :2:0, (H2) UoEH2 , U, EH' , (H3) f ECI([0,l]x[0,oo)xR3), (H4) gO,glEC3([0,oo)). .X6t hams6phl,1 q>(x,t)= h 1h [gl(t)eho(X-I)go(t)e-hIX].0 + I (2.4) D~t { BOV=vx(O,t)-hov(O,t) , O<t<T. BIV=vA1,t)+h1v(1,t) Khi d6,vdiphepd6ibie'n (2.5) w(x,t)=u(x,t)-q>(x,t),x EO, 0<t <T, (2.6) thlwthoamanphlidngtdnh Chll(Jng 21 9 W/t-W",,=](X,t,w,wx,W,),xEQ, O<t<T, (2.7) voidi8uki~nbienh6nh<;fpthu~nha't { Bow=0 . .. , O<t<T, Blw=O vadi8uki~nd~u (2.8) { w(x,O)=uo(x)-<p(x,O)=wo(x), xEQ . w,(x,O)=UI(X)-<P,(x,O)=WI(x), (2.9) trongd6 { ](x, .. t,w,Wx,w,)=f(x . , . t,w +<p,wx+<Px'w,+<PJ-<PII(X,t)+<Pxx(x,t), ... ... .. (2.10) wo(x)=uo(x)- <p(x,O),WI(x) = UI(x) -<PI(x,O) , thoa ] ECI(QX[0,oo)xR3), WoEH2, WI EHI. (2.11) Nhu'v~y tUbairoanbienh6nh<;fpkh6ngthu~nha't(2.1)-(2.3)vOiph6pbie'n d6i (2.6)se tu'angdu'angvoi bai roanbienh6nh<;fpthu~nnha't(2.7)-(2.9).Do d6, kh6nglamma'tinht6ngquattac6th~giaSLYding gi =0, i =0,1. 2.Su't6utaivaduynha'tI<iighHcuabaitoanbienh6nho'pthuflunha't TrenHI rasad~mgmQtchugntltangdltangsan: (2.12) i ( I ) ~ IlvllHI= V2(0)+flv'(xtdx . 0 (2.13) Trongchu'angnay,tadjnhnghiad~ngsongtuye'ntinhtrenHI nhu'sau: I a(u,v)= fu'(x)v'(x)dx+hou(O)v(O)+hlu(l)v(I),\lu, v E HI 0 (2.14) Khi d6tac6cacb6d8sau B6d€ 2.1 Ph6pnhungHI .CO(Q)lacompactva Ilvllco(o)~.J2l1vIIHI,\ v EHI. B6 d82.11amQtke'tquaquellthuQcmachungminhcllan6c6th~rimtha'y trongnhi8utaili~ulienquailde'n19thuye'tv8kh6nggianSobolev,ch~ngh~n[20]. B6 d€ 2.i Voi giathie't(HI), d~ngsongtuye'ntinhd6ixungdjnhnghiabdi (2.14) lientl;1C,cu'ongbuctrenHI xHI , nghiala : Chuang2)10 (i) la(u,v)I~CllluIIHlllvIIHI'Vu,v EH1, (ii)a(u,u)~Collull~l'Vu EHI. VOl Co =rnin{l,ho},C1=rnax{1,ho,2hd. Ch(fngminh:Sad\mgbttd£ngthucSchwartzvab6d€ 2.1taco(i)dung. Chungminh(ii) thld~danghentaboqua. BiJdi 2.3 T6n t~imQtcosdHilberttn1cchu£n{Wj}ci'taL2 g6mcacvectorriengWj (fngVOltririengAj saocho 0 <A, ~A, 2 ~ ... ~A,.~ ... , UrnA,.=co , I J j-too J (2.15) a(Wj,v)=A,j(Wf'V) ,voimQi vEHI,j=1,2,.... (2.16) Honnfi'aday {Wj/~} clingla cosdtr\fcchua':nHilbertcua HI tu'ongling VOltichvohtfonga(.,.). M~tkhac,chungtaclingcohamWjthoamanbattoangiatribiensan: -~Wj=A,jWj , trong0, (2.17) (2.18)w;(O)-hoWj(O)=w;(l)+hIWj(l)=O, Wj ECOO(O) . Chung minhb6 d€ 2.3 co th~tlm trong[20] (dinh196.2.1,p.l37, VOl V =HI, H =L2va a(.,.)dinhnghlanhu'(2.14». ,, Voi M >0, T >0 tad~t Ko=Ko(M,T,f) =supl/(x,t,u,v,w)1 Kl =KI (M,T,f) =sup(I/:1+1//1+11:1+11:1+1/,~D(x,t,u,v,w), (2.19) (2.20) suptrong(2.19),(2.20)dtfQc1tytrenmi€n 0 ~x ~1,0 ~t ~T, lul,lvl,lwl~JiM. W(M,T) ={vELoo(0,T;H2):V ELoo(O,T;HI), v ELoo(0,T;L2); IlvIILOO(O.T;H2)~ M, IlvIlLOO(O.T;Hl)~ M, IlvIILOO(O.T;L2) ~ M }. (2.21) Tie'pthea,taxaydlfngday{urn}trongW(M,T)b~ngquin~p.Day {urn}se dtfQCchungminhhQit\l v€ Wi gi.Hcuabattoan(2.1)-(2.3)trongW(M,T)(voi st1 chQnllfaM vaT thichhQp). ChQns6h~ngbandffuUoEW(M,T). Giasar~ng urn-]E W(M,T). (2.22) 11 Ta lien ke'tb~litoan(2.1)-(2.3) vOibai taanbignphantuye'ntinhsan: TIm UrnE W(M,T) thoa +a(Um,v)=,vdimQi v E HI, Um(O)= UO,um(O)= ul' (2.23) (2.24) trangd6 Fm(x,t)=f(x,t,um~l(x,t),V'um-l(x,t),Um-1(x,t)). Slf t6ntq.ieliaUrnehabdi dinh19duoiday. (2.25) Dillh IV2.1([15]) Giii sa (Hl)-(H3) dung.Khi d6t6ntq.icaehAngs6M >Ova T> 0 saGeha: vOimQiUoE W(M,T) ehatrude,t6ntq.imQtdayquinq.ptuye'ntinh{urn}cW(M,T) xaedinhbdi(2.23)-(2.25). Chungminh:Chungminhbaag6mbabltde. Eltde1 : DungphuongphapxffpXl Galerkind~xay dlfngWi giiii xffpXl U~k)(t ella(2.23)-(2.25). GQi {wj} la cosatrlfeehuffneuaHI nhutrangb6d~2.3(wj =W j /F;). f)~t k . U~k\t)=LC~)(t)Wj , j=l . (2.26) trangd6 c~J(t) thoah~phuongtrlnhviphantuye'ntinhsan: (u;:)(t),wj)+a(u~k)(t),Wj)=(Fm(t),wj)'1::;j::; k, (2.27) (k) (0) - .(k) (0) - um =UOk'um =Ulk' (2.28) vOi k- "(k) - H2UOk=L...,aj Wj ~ Uo trang, j=l (2.29) k- "A(k) - tr HIUlk = L...,tJj Wj ~ Ul ong'. j=l (2.30) TItgiii thie't(2.22),t6ntq.iTlk)>0 saGehabaitaan(2.27),(2.28)e6duynhfft Wigiai u~)(t) tren[o,Tlk)]. Ca'edanhgiasaudaytrangbude2ehopheptaIffyT~k)=T, vdiffiQikvavdi ffiQim. . DM.Jl.H.TlfN!!IEN. .... THlf \/U:fv 12 BlfOc2 :Danhgiatiennghi~m. * Trang(2.27)thayWjbdi it~)(t) taco 1 d ll o(k) ()11 2 1 d ( (k)() " (k)()) _ ( () o(k)())2"dt urn t +2 dta urn t ,urn t - Frnt ,urn t , saudo tichphantheot tadu'<;ic I p~k)(t)=p~)(0)+2f(Frn(t),it~)(~))d];, . 0 (2.31) trongdo p~:)(t)=IIU~)(t)1I2+a(u~k)(t),u~)(t)). * Trang(2.27)thayWjbdi- :. ~wi'khido.1 (U~nk)(t),~Wj)+a(u~)(t),~Wj)=(Frn(t),~wJ ' hay a(u~)(t),wJ +(~u~)(t),~Wj)=a(Frn(t),wJ. ThayWjbdi it~k)(t) trongd~ngthuctren,ke'th<;ipvdi (2.18)saudo m'ytich phantheot, tadli<;ic 2 I q~)(t)=a(it~)(t),it~:)(t))+II~u~)(t)11=q~k)(O)+fa(Frn(~),it~)(~))d~. 0 (2.32) * Dgoham(2.27)theot,saudothayWjbdi u~)(t) taco " i :tllu~)(tf+ ~ :ta(it~k)(t),it~)(t))=(F~(t),U~)(t)). Tichphanhaive'theot 2 I rlk) (t)=Ilu~k) (t)11 +a(it~)(t),it~)(t))=rlk) (0)+2f(F~(~),u~)(~))d~. 0 (2.33) Tit (2.31)-(2.33)d~nde'n s~)(t)==p~) (t)+q~)(t)+rlk)(t)=s~)(0) I I I +2f(Frn(~),it~:)(~))d~+2J a(Frn(~),it~)(~))d~+2 f(F~ (~),u~)(~))d~ 0 0 0 (2.34) Cactichphand ve'ph.;H(2.34)Hinltt<;itdl.t<;icdanhgiadlididay. + TichphanthTinha't Tit (2.19)va(2.22)taco 13 2If(Fm('t)'U~,~)('t))d't l ~2fIIFmllllu~)lld't~2KofJP~nk)('t)d't. 0 0 0 (2.35) +Tfchphiinthahai Dob6de2.2taco 2Ifa(Fm('t)'U~)('t))d't l ~ 2C, fllFmIIHlllu~k)IIHId't. 0 0 (2.36) Tli (2.19),(2.20)va (2.22)tatlmdl1<;1C IlFmll~1=IIVFml12+F;(O,t), F;(O,t) ~K;, I IIVFml12 = fU.:+f:Vum-t+f~u~um-l +f:Vum-t)2dx 0 I ~ fOf:12+If:12+lf~J +If:12)(1+IVum-112+1~Um-112+lvum-tI2)dx 0 ~ 4K~(1+lIum-lll~2+IIUm-III~I) ~ 4KN1+2M2). V~y IIVFml12~4K~(1+2M2). , Vadodo IlFmll~14K~(1+2M2)+K;. ,, (2.37) Tli (2.32),(2.36),(2.37)taco I 2C ( I I 2IJa(Fm,u~))d't~ ~ 2K1.J1+2M2+Ko)J"q~:)('t)d't. 0 ~Co 0 (2.38) +Tfchphiinthaba Ta co 2IJ(F~'U~k))d't l ~2JIIF~lIllu~)lld't. 0 . 0 (2.39) Tli (2.20)va(2.22)tathudl1<;1C 1 .IIF~112=fUr'+f:Um-l+f~uVUm-l+f:urn-J2dx 0 ~4K~(1+Ilurn-11l2+IIVUrn-tIl2+Ilum-,1I2) ~4K~(1+3M2) . Dodotli (2.39)tasuyfa [ Chuang2114 t I / 2If(F,~,u~nk))dr~4KJ.J1+3M2 f)r,~k)(r)d-r:. 0 0 Tli (2.34),(2.35),(2.38),(2.40)tathudu'<;fc / s~)(t) ~ S~k)(0) +K f~s~)(-r:)d-r:, 0 trangd6 -. ..2CJ ( .J .. 2 ) . I 2 - ( )K-2Ko+-JC; 2KJ 1+2M +Ko +4Ktv1+4M -K M,T,f . Tie'ptheotadanhgias6h:;mgS~nk)(0).Taco S~)(0)=lIu~)(0)112+2a(Ulk'Ulk)+/IUtkI12+1Ii1uokI12+a(uOk,uOk)' Trang (2.27),thayWjbdi u~nk)(t), sandoH(yt =0 ta du'<;fC Ilu~)(O)f -(i1UOk'U~)(O))=(f(x,O,uo,VuO,Ut),u~I~)(O)). Tli daySHYfa Ilu~:)(o)/f ~ IIi1uOkll+Ilf(x,O,uo,Vuo,Ut)ll. (2.40) (2.41) (2.42) (2.43) (2.44) Ta SHYtli (2.29),(2.30),(2.43),(2.44)dng t6nt~imQts6M >0 dQcl~pvdik vamsaocho S~)(O)~M2/4, vOimQikvarn. Ta lu'uy, vdigi:ithie't(H3)'SHYfa tli (2.19),(2.20)f~ng lirnTK;(M,T,f)=0, i =O,I. T~O+ Ke'th0saocho TK(M,T,f):$M. va k,.=2(1+~1+-k-J TK,(M,T,f) <1. Cu6i clingtaSHYfa tli (2.41),(2.45)f~ng M2 / s~)(t)~4+ K f~s~)(-r:)d-r:,0~t ~Tlk).0 M~Hkhac,ham set)=(M12+K t12)2 (2.45) (2.46) (2.47) (2.48) (2.49) (2.50) [ Cha(Jng2115 la Wi gi:Hcvcd'.lic1laphuongtrlnhtichphanVolterraphituye'nsandaytren[O,T] vOinhankh6nggiam~ (xem[12]). M2 t set)=-+ KJ.Js(t)d-r;,o~t ~T,4 0 (2.51) vadodotu (2.49)-(2.51)tanh~nduQC s~nk)(t)~s(t)~M2,'\ItE[O,Tlk)]. Tu daytaco Tn~k)=T, vdimQimvakvataSHYratudayding (2.52) U~)EW(M,T). (2.53) Budc 3 : QuagiOih'.ln Tu (2.53),t6nt'.limQtdaycon {U~f)}cua {U~k)}va t6nt'.liUrnsaGcho u~j)~ UrntrongL"'(O,T;H2) ye'u*, U~JJ~ Urntrong Loo(0,T; HI} ye'u* , U~:f) ~ Urn trongLoo(O,T;L2}ye'u* , (2,54) thoa UrnEW(M,T) . (2.55) Tu (255) quagioi h'.lntrong(2.27),(2.28)taco thSkiSmtrad6 dangding Urn thoaman(2.23),(2.24)trongLOO(O,T)ye'u*. Binh 192.1chungminhhoanta'tD Dink IV2.2([15]) Gia sl1'(H1)-(H3)dung.Khi dot6nt'.liM >0,T>o saGchobaitmln(2.1)~(2.3) coduynha'tmQtWigiaiye'uUEW(M,T). M?t khac,dayquin'.lptuye"ntinh{Urn}xac dinhbdi (2.22)-(2.24)hQitl}m'.lnh v€ Wi giai ye"u U trong kh6ng gian WI(T) ={uELoo(O,T;HI):U ELoo(O,T;L2)}. (2.56) Honnuataclingcodanhgiasais6 lIurn- uIIL"'(O,T;HI)+Ilurn- UIlL"'(O,T;L2)~ Ck;, vdimQim, (2.57) trongdo0<kT<1xacdinhbdi(2.48)vaC lah~ngs6chIphl}thuQcT,UO,Ul,vakr. Chungminh: atSut6ntaiWigiaiU: [ . . . ChU'dng5Zl16 Tntoe he'tta 1U'LI9r~ngWI(T) 1akh6nggianBanachd6i voi chuffn(xem[11]) Ilullw,(T)= IluIlL<O(O.T;H')+lIuIIL<O(O,T;L2)' Ta sechungmintdng {urn}1adayCauchytrongWI(T) . f)~t Vrn=Urn+\- Urn .Khid6Vmthoamanbai tmlnbie'nphansan: . { , 'Iv eH', vrn(O)- vrn(O)- 0. (2.58) U(y v::::;urn trong(2.58)vasad\mggiathie't(H3),taSHYtli dint 192.1,san khitichphantheet tac6 I Ilv,n!l2+a(Vrn,Vrn)=2f(Frn+1-Frn,vrn)d't 0 (2.59) t . c:; 2(1+.fi)KI fOlv rn~11I+Ilvrn-lllH')llvrnIld't° sa d\mgb6d~2.2(ii)va(2.59)tathudttQc Ilv,nII2+collvrnll~l c:;2(1+.fi)KITllvrn-lllw,(T)llv,nllw,{T)' '\It E[O,T]. Tli (2.60)dfinde'n Ilvrnllw\(T)c:;kTllvrn-lllw,(T)'voi mQi m . (2;60) (2.61) Vi v~y IIUrn+p- urnllw,(T)c:;lIuI - uollw,(T)1~~' voimQim,p .T (2.62) Ke'thQp(2.48)va (2.62)tac6{urn}la dayCauchytrong'WI(T),dod6t6nt~i UEWI(T) saocho Urn-+U trongWI(T) m~nh. (2.63) B~ngeachapd\mgmQtl91u~ntu'Cfngt\tmachungtadasad\mgtrongdint19 (2.1),tac6th~1a'yramQtdaycan{urnAcua{urn}saccho Urnj-+U trongL"'(0,T;H2) ye'u", Urnj-+u trongL"'(O,T;HI) ye'u", (2.64) (2.65) ~ it trongL"'(O,T;L2) ye'u", uEW(M,T). (2.66) (2.67) Ap dl;mgdint19Riesz-Fischer,tli (2.63),t6nt~idayconclla {Urnj-I}vfink9 hi~u1a{Urnrl}saceho: [ Chlt{jng2117 Um-I ~ U} h.h (x,t) EQT ' (2.68) (2.69)VUm-1~ Vu h.h (x,t) EQT '} itm-I ~it} h.h (x,t)EQT . (2.70) Dof lient1,lC,apd~mgdinh19hQi19bich~nLebesgue,tIT(2.68)-(2.70)taco Fm~ f(x,t,u,ux,Ut}trongL2(QT)m~nh. (2.71) } Mi;itkhacVI II F II{ , ) 5:Ko,voimQij,m} L'" O,T;L- (2.72) nentacoth~trichduQctIT{Fmj}mQtdayconvlingQiEl {Fmj}saocho Fmj~ F trongL""'(O,T;L2) ye'lh. (2.73) Sosanh(2.71)va(2.73)suyra F(x,t)=f(x,t,u,ux'ut),h.h(x,t)EQT . (2.74) V~y Fmj~ f(x,t,u,Ux'u,) trongL""'(0,T;L2)ye'u*. (2.75) Qua giOih~n(2.23),(2.24),b~ngs~l'ke'thQpvoi (2.64),(2.66),(2.75),ta thu du'Qcu thoabai toanbie'nphansail : (u,v)+a(u,v)=(J(x,t,u,ux'ut),v) , voi mQiv EH1, u(O)= 110,it(O)= 111, trbngL""'(O,T)ye'u* . bl Su'daynha'tWigi:H Gia saulvau21ahaiWi giaiye'ucllabaitoan(2.1)-(2.3),thoaUiEW(M,T), i=I,2. Di;itu = Ul - U2,khidoula loi giaicuabaitoanbie'nphansail: { (ii.v)+a(u.v)~(F, - F" v).'Iv EH' . u(O)=it(O)=0, (2.76) trongdo F;(x,t)=f(x,t,u;>VUi'itJ, i =1,2. La'yv=ittrong(2.76),sailkhitichphantheot taco Chu;;ng2118 r lIum(t)1I2+a(u(t),u(t))=2f(FI -F2,u)d1:° (2.77)r ~ 2Kl J(llu( 1:)11+ IIVu(1:)11+lIu{1:)11)lIu(1:)lId1:. o D?t z(t)=lIu(t)1I2+a(u(t),u(t)). (2.78) Khi d6,tasuytu(2.77)ding Z(t)QK,(2+~go) .fzW<. (2.79) Sad~mgb6dc3Gronwalltasuyra z(t)=0, hayul =U2 . Danhgiasais6(2.57)duQcSHYtu(2.62),(2.63)b~ngcachchop ~ 00. V~ytadachungminhxongdinh1y2.20 3.Khai triin tiemdin cllalotgiai Trongphfinnay,tagiasar~ng(ho,hi)va (uo,Ell) IfinIttQthoamancacgia thie't(HI) ,(H2).Taduavaogiathie'tsau: (Hs)f, g EC1(nX[0,00)xR3). Chungtaxetbairoannhi6usailday,trongd6 &Iathallis6be: u/t -uxx =Fe(x,t,u,ux,ur)'x En, O<t\<T, u.JO,t) - hou(O,t)=Ux(I,t) +hIu(I,t)= 0, U(X,O)=Uo(X),Ur(X,O)=Ul(x), Fe(x,t,u,Ux,UJ=I(x,t,u,Ux,Ur)+sg(X,t,u,Ux,Ur)' Trudche'ttachuyr~ngne'uj,g thoa(Hs), khid6cacdanhgialiennghit$m cua day xa"p xi Galerkin {u~)}trongchungminhdinh Iy 2.1 tttongungvOif =F li , vdi lei<1,thoaman (Pe) u~)EW(M,T), (2.80) trongd6 cac h~ngs6 M, T dQcI~pvdi &. Th~tv~y,trongqua trlnh chungminh, chungtachQncach~ngs6duongM vaT trong(2.43)-(2.45),(2.47),(2.48)matrong d6cacd:;tiluQngII/(x,O,uo'Vuo,ul)11va Ki(M,T,f), i =0,1sedttQcthaybdi 11/(x,O,uo'Vuo,uJ +IIg(x,O,uo'Vuo,uJI va Ki(M,T,f)+ K;(M,T,g) rheathutv . Chuang2119 VI v~y, giaih,~mu{;trongcackhonggianhamthichhQpcuaday {U~k)}khi k~ +00,saild6 m~ +00,laWigiiiiyehcuabattoan(PE)thoaman UE E W(M,T) . (2.81) Khi d6,theocachtu'angt\1'yai chungminhdinh192.2,tac6 th~chungminh dttQcdinggiGih:;mcliahQ{UE}trongcackhonggianhamthichhQpkhi 8~ a la Wi giiiiye'uduynhfftUocliabattoan(Po)(ungVOl8=a) thoaman UoEW(M,T) . (2.82) Hannuatac6dinh19sail: DinkIv 2.3[15] Giii StY(HI), (Hz)va (Hs)dung.Khi d6t5ntqicachangs6M >a vaT> a saorho,VOlmQi8,181<1,battoan(PE)c6duynhfftmOtWi giiii ye'uU{;EW(M,T)thoa manmOtu'aclu'Qngti~mc~n IIUE-UOIIWI(T) ::;Cl81, (2.83) trongd6CIa mOthangs6chIph\!thuQcvaoho'hI' T, Ko(M,T,g), K, (M,T,J) , va UoEW(M,T) la Wigiiiiye'uduynhfftcuabatto<ln(Po)ungVOl8=a. Chungminh: B~tU=uE- uo'Khi d6Uthoamanbattoanbie'nphansail: (u,v)+a(u,v)=(l,v)+8(g,v),vvEHI, u(a)=u(a) =a, l(x,t) =f(X,t,UE'V'UE'UE)-f(x,t,uo'V'uo,uo)' g =g(X,t,UE,V'UE,UJ. Trong(2.84),Iffyv=U,sailkhitichphantheot tadttQc , , (2.84) Ilu(t)112+a(U(t).U(t))S ZKJ (M.T,f)(Z+ ~;0)I(llull'+a(u.u))dH I +fllullzd't+8zK;(M,T,g)T. 0 (2.85) Dod6 1114(1)112+a(U(I),U(I))«; (2K, (M, T,f{ 2+~~O ) +1) J(llu112+a(u,u))dt+ +8zK;(M,T,g)T. (2.86) Bangcachapd\!ngb6d~Gronwall,tti'(2.86)tathudu'Qc Ilu(t)1I2+a(u(t),u(t)) :::;E2K~(M,T,g)TeXP((2K[(M,T,f{2+~~O)+I}). Vt E[O,T]. Ke'thQpVOlb6 d€ 2.2taco lIue-uollwi(T)=llullwi(T):::;cjEI,l I<1.. Chuang2120 (2.87) (2.88) V~ydinh192.3du'Qchungminh.0 MQtke'tquamachungtai tlmdu'Qctie'ptheosailla v€ khattri~nti~mc~n cilalot giaiye'uuede'nctp2theoE,VOllEIdlinho. Chungtaidu'athemgiathie'tsail (H6)f EC2(n x[0,00)xR3),g ECI(n x[0,00)x R3). GQi UoEW(M,T) la Wi giai ye'ucuabatroan (Po)nhu'trongdinh192.3.GQi ul EW(M,T) (voiM, Tthich hQp)la Wi giai ye'uduynhtt cuabai roan {. LU' =_F,(X.~uPVup",), x en, 0 <t <T, (p[) Bjul- 0, l - 0,1, UI(X,O)=u[(x,o)=o, trongdo 82 .82L=--- 8t2 8x2 FI (X,t;UI ,'lu[ ,UI) =uJu (x,t,uo''luo,uo)+'luJu, (x,t,uo''\luo'uo)+ +uJ,,(x,t,uo'luo,uo)+g(x,t,uo''luo,uo), vaBjdinhnghIanhu'(2.5). Gia sti'ueEW(M,T)la Wigiaiye'uduynhttcilabairoan(Pe).Bi;lt v=ue- Uo- SUI =ue - h, khidov thoamanbatroansail Lv=f[v +h]- f[h]+E(g[V+h]- g[hD+a(s,x,t),x En, 0 <t <T, Bjv=0, i =0,1, v(x,O)=v(x,O)=O. trongdo.. a(s,x,t)=f[uo+EUI]-f[uol- E(uJuluo]+'luJuJuo] +uJ" ruG])+ +E(g[Uo + SUI] - g[UoD C1 day,d~lamgQncachvie't,tasti'd\mgk9hi~u (2.89) (2.90) (2.91) 21 J[U] =J(x,t,u,Ux,Ut) (2.92) Chung ta stl'dl,mgkhai triSnTaylor de'ncffp2 rho J[uo +SUI]va de'ncffp1 rho g[Uo+SUI] t<:ti(x,t,uo'VUo,ito). Sail do do tinh bi ch~n cua cac ham upVupup i =0,1,trongkhonggianham £,"o(O,T;HI), tathudu'c;Ic la(s,x,t)1 :::;Ks2, h.h (x,t) E QT' (2.93) voi K =9M2K2(M,T,J)+3.J2MKI(M,T,g), K2(M,T,f) = sup(lJ:~[u]l+IJ:;ux[u]1+IJ;~[u]1+IJ:~x[u]1 + IJ:~[u]1+IJ:;u [u]1), (2.94) (2.95) trongdoSupIffy tren 0:::;x:::;1,0::;t:::;T, lul,IVul,lul:::;M.J2. Bay giOtadinhnghladayham{vm}nhu'sail Vo=0, Lvm=J[Vm-l+h]- J[h]+ s(g[Vm-I+h]- g[h])+a(s,x,t), x EQ, o <t <T, (2.96) Bjvm=O,i=O,I, vm(x,O)=vm(x,O)=0,m~1. Voi m=1,tacobaitoan . { LvI; ,*:x,t), x E . Q, 0< t <T, BjVl =0, l=O,1, VI(x,O)=VI(x,O)=0 , \, (2.97) Tichvahu'onghaive'(2.97)voi VI' sail doIffytichphantheot,ke'thc;Ip(2.93) taco IIvll12+ a(vI' VI) :::;2Ks2TllvIIILOO(O,T,L2) Vi v~y Ilv,llw,(T),; 2(1 +.J~J\'Te2 . (2,98) Ta sechungminht6nt<:timQth~ngs6cr dQcI~pmva S saorho Ilv.llw\(T):5(I+'/~JCT.2, 1'11m. (2.99) Tichvahu'onghaive'(2.96)voi Vm ' sailkhi Iffytichphantheot taco: Chuang2122 IIvml12+a(Vm,Vm) I ~2f(llj[vm-l +h]-j[h]II+llg[Vm-l +h]-g[h]I/)llvmIIL"'(O,T;L2)d1:+2KTf:21IvmIIL"'(O,T;L2).0 (2.100) Bi;H Ym=llvmllw,(T)' (2.101) Tli (2.100),(2.1O1)suyra Ym~ crym-(+8 , (2.102) trongdo " ~ 2(1+.J~)I +F2)T(Kt(M.T,f) +Kt(M.T.g)), O~2(1+j-JRTE" (2.103) Vdi giatrithichhQpciiaT,giil sadng crthoa cr<1. (2.104) BaygiOtadn b6d~dltdiday,manocoth~duQcchungminhra-td~dang. Bdd~2.4 GEl saday{ym}thoa 0 ~Ym~crym-l+8, m=1,2,..., Yo =0, trongdo 0 <cr<1 va 8 ~0 la cach~ngs6chotrudc.Khi do (2.105), , < 8 ,.. _ 12Ym -_1 ' Vdl mQl m - , ,....-cr (2.106) Ta suyradng, tli (2.101)-(2.104)va(2.106)r~ng Ilv.llwdTi"(1+~~JcT,', (2.107) trongdo - 1 CT =2KT-1-cr (2.108) M~tkhac,dayqui n~ptuye'ntinh {vm}dinhnghlabdi (2.96)hQit\1m~nh trongW1(1)v~Wi giili v cuabai toan(2.90).VI v~yquagidi h~n(2.107)khi m~ootato 23 Ilu, - u,- sulllw",,:S;(1+~~JCTS'. V?y, tacodinhIy du'oiday. (2.109) Dinh IV2.4 [15] Gia sa (HI), (H2)va (H6)dung. Khi d6 t6n t~icae h~ngs6 M >0, T >0 saGcho,voi mQi E,181<l,bai toan (Pe) c6 duy nha'tmQt loi giai ye'u duy nha't ueEW(M,T)thoa u'oclttQngti~mc?n de'nca'phai nhlt (2.109),cac ham Uo' UI la c3.cloig aiye'ucuacaebaitoan(po)va(Pt)I~n1u'<;Jt. . 4.Choyv~b?litoanvoidi~ukienbienh6nhopkhongthuftnnha't(2.1)-(2.3) Trangph~nnay,chungWi rUtra ke'tquacuabai toangia tri bienkh6ng thll§nha't(2.1)-(2.3)lingvOitntongh<;Jpgl :;t:0,g2:;t:0va thoagia thie't(H4)' Tli phepbie'nd6i(2.4),(2.6)vahtuy(2.11),tad~nbai roan(2.1)-(2.3)t6ngquatve, vi~cgiaibaitoanbienh6nh<;Jpthu~nha't(2.7)-(2.9). Gia sLY(Ht)- (H4) thoaman.Taxayd~tngdayham{wm}bdi: ChQns6h~ngd~utieRWoEW(M,T). Giasa wm,..tEW(M,T), talienke'tbairoan(2.7)-(2.9)voibaitoan: TIm wmEW(M,T) thoamanbai ta<lnbie'nphantuye'nHnh: { (Wm,v): ~(Wm.'v)=?:,v;, \:IvEH1, Wm(0) - wo' W m(0)- w" (2.110) trongd6 Wo(X)=u'o(x)-<p(x,O), Wt(x)=u((x)-<p,(x,O), Fm(x,t) =l(x,t,wm-1'V'wm-"Wm-t)' l(x,t,w, V'w,w) = f(x,t,w +<p,V'w+V'<p,W+<j))-<pl/+<Pxx' Theodinh1y(2.1)vOi wo'wpl thaychoUo'Upf I~n1u'<;Jt,thl t6nt~ihai hAngs6M>O, T>O saoehoday{wm}xacdinhbdi(2.110),voiWoEW(M,T) cho tru'oe.M~tkhae,theodinhIy 2.2thlday {wm}hQiW m~nhtrongWI(T)ve,Wi giai duynha'tWEW(M,T) euabaitoan(2.7)-(2.9). (2.111) Khi d6tae6dinhIy sail: Dinh IV2.5 Gi~sa(HJ - (H4) dung.Khi d6t6nt~icaeh~ngs6 M >0, T >0 saGeho: Chlt(Jng2124 (i) Voi mQi UoEW(M,T) cho tru'dc,t6n t:;timQt day qui n:;tptuye'nHnh {urn}c W(M,T) xacdinhbdi : U =w +<p m?:1rn rn' , Wmxacdinhtu(2.110),(2.111)ungvdigiatriband~uwo=Uo + <p. (ii) B~litoan(2.1)-(2.3)t6ntqi duynha'tmQtWi giai ~e'uUE W(M,T). (iii) urn~ U trongWI(T) mqnh. Honnaataclingcodaubgia Ilurn- ullwl(T)::;;Ck; , vdi mQi m (2.112) trongdok lah~ngs6du'ongthoamanT kT ~ 2(1+h)( 1+.J~}K.( M, T,J) <1 , (vdiM >0, T> 0 thichhQp),C la h~ngs6chituythuQcvaoT,Wo,WIva kT . B6i voibaitoankhaitri€nti<$mc~ncuaWigiaitheothams6be 8,taxetbai toannhi~usanday: u -u =F (x,t,u,u ,U),xEQ, O<t<T,II xx s x, (pJ ux(O,t)-hou(O,t)=go(t),ux(1,t)+h1u(1,t)=gl(t), u(x,O)=Uo(x),u,(x,O)=UI(x), . \ F (x,t,u,u,u .)=f (x,t,u,u,u )+8g(X,t,u,~,u) .s x, x , x , (2.113) Giathie't(HJ, (HJ, (HJ va(HJ ladung. Theodinh1)12.5,baitoan(po)ungvoi8=0 vabaitoan(ps)l~nlu'Qtcoduy nha'tWi giaiuova USEW(M,T), trongdoM, rta cach~ngs6du'ongthichhQpdQc l~pvoithams6be 8. Khi d6 u=Us - UolaWigiaiye'ucuabaitoan t o"~ - u~ =I[ u,]- I[ u,]+Bg[u,], XEQ, 0<t<T , Bou=B\u =0, u(O)=u,(O)=0 . (2.114) Chungminhtu'ongtl1trongdinh1)12.3tacodaubgia Ilullw,(T)=Ilus - uollw,(T)::;;ej81, 181< 1 , (2.115) trongdoCIah~ngs6chituythuQcvaoho,hi' T, Ko(M,T,g), K1(M,T,f). Ch£lang2il25 Khi dotacodinhly Dillh LV2.6 Ghi sa (HI), (H2),(H4)va (Hs) dung.Khi do tan t(;1icaeh~ngs6 va M >0, T >0 saocho, vOimQi&,lEIdube , bai loan(Pg)co duynha'tmQtWi gliB ye'u UcEW(M,1) thoa 11'ocl11'<;1ngtic%mc~nsau IIUE -UOIIWI(T) :::;CIEI, . (2.116) vOiC la ri1Qth~ngs6nh11'danoid tIeD. . . B6i VOlehaitri~ntic%mc~nde'nca'phal,tacodinhly saudaymachungminh cophgndi8uchinhchutIt sovOichunglninhcuadinh1y2.4,vi v~ytaboqua. Dillh Ii 2.7 Gia sa (HI), (H2), (H4)va(H6)dung.Khi do tan t(;1icae h~ngs6 M >0, T >0 saocho,voimQi&, lEI<1, bai loan (PE) tan t(;1iduynha'tmQtWi giai ye'uduynha'tuE E W(M,T) thoa11'oc111'<;1ngtic%mc~nde'nca'phainh11'sau Ilu,-u, -8u,llwdT}';(1+ .J~JCT82, (2.117) trongdo UO,UIEW(M,T) Ign111'<;1tchinhIa cac10igiai ye'ucuacacbai loan (po) (ung VOl E =0) vabai loan (PI),ung VOl E= 1, go(t) =gl(t)=0,uo(x)=UI(X)=0, FE= uJu [uo]+VuJu [uo]+itJul uo]+g[uo],CTIa h~ngs6chiph\!thuQcvaoT,M,. . . K((M,T,f),K2(M,T,f),K,(M,T,g).. \ 4.Xet mottrtt(fm~hdpenth~coa'. g chobAitoaDbienkhong'thnfifinha't Trangphgncu6icuach11'dng2naychungWixemxetmQtvi d\!v8khaitri~n tic%mc~nVOltr11'ongh<;1p . f =0, g = g(it)=it3. (2.118) BgutieD,chungt6ixettr1iongh<;1pt6ngquatValgthoamangiathie'tsau. gECN(R). (2.119) GQi uE. Ia101giaicuabailoan .. j LUE= Eg(itE) . ' 0 . " . < . x < " 1,0.( t <T, (p ) B,u =g, (t), i=O,1,. E IE. I :. u&(O)=uo' u&(O)=UI' . VOl M:>0, T >0 thichh<;1pta tim UO,UIEW(M,T) 19nl11'<;1tla Wigiaicua cacbailoansau 26 [ £UO . = . 0, ° <:X . <I, 0 <t <T (po) B,uo~g,(t), i =0,1, uo(O)=Uo' uo(O)=Ut' [ LUt =-g . ( . u . o! . .~ O<x<l, O<t<T (PI) Biut - 0, l.,- 0,1, Ut (0) =ut(0) =O. Voi 2 S',,'ps'"N , gQi up E W(M,T) Ia Wi giai bai loan Lu =F =F (x,t,u,u,...,u ),O<x<I, . O<t<T,p p pOI p-I (pp) i Bjup=0 , i =0,1, . u (0)=u (0)=0,p p trongdo Fz =g'(uo)ul' p-I .<XI.<xz .r:t.p-z . L ( ) ) L U Uz ...u zF =g'(u)u + g k (U. . I. p-, P C:. 3. p 0 p-t 0 a !a !...a ! k=Z <XI+<XZ+...+<Xp_Z=kt Z p-z p-Z Lir:t.j=p-l j=1 (2.120) f)ijt N v=u&-h=u&-uo-LCPUp' p=! (2.121) Khi dovIa Wigiaibailoan { ..LV=-c[g(~: Ii) . - . g . (Ii)] +a(c,x,t), B.v- 0 , l - 1,2,I v(x,o)=v(x,O)=0, 0 <x <I, 0 <t <\T, (2.122) trongdo N a(e,x,t)=c[g(li)- geLiD)]- LcP Fp' p=z (2.123) f)?t Kk(T,g)=suplg(k)(UO(x,t»)I,k =1,2,...,N, O$x$l O$t$T (2.123) KN(M,T,g)= sup Ig(N)(v)1 'vj$J2(N+t)M Khi do, taco b6 d€ sailday Bddi 2.5 Giasa(HI), (Hz),(H4)va(2.119)dung,khidot6nt~ih~ngs6K sao cho (2.124) IIa(c,x,t)IIL"'(O.T;L2)s'"KlclN+1 (2.125) 27 6.day K chiph\l thuQcvaoN, M, T va cachangs6 Kk(T,g), k=1,2"",N-1, KN(M,T,g) . Chungminh Tntongh<;1pN =1chungminhd~dang,taboquachungminhchitie't.CJ day tachixet trl1ongh<;1pN ~2, N B~HU =LgPUP ,Bang eachkhai tri~nTaylor cho g(uo+U) quanhdi~m p=l Uo de'nca'pN , Taco ( ' ) ( , ) .. N-l (k)( , ) (N)( ,.' g h - g Uo =g(uo+U)- g(uo)=~g Uo Uk + g Uo+8U) .Nft k! N! U (2.126) 0<8<1 ( N J k 'k . i, k!. al 2. a2 . N.aN U = Lg Ui = L. a!a La !(guJ (guJ ...(g UN) i=1 al+a2+...+aN=k I 2 N ai nguyen;;,0 (2.127) Tli'(2.126),(2,127)taco N-[ N(N-l) g(Ji) - g(uo)=LC[p,g]gP+ LC[N -l,p,g]gP + RN(g,g) p=l p=N (2,128) trongdo P . C[p,g] =Lg(k)(uo)L k=l al+a2+...+ap=k P Liai=p i=l N-l 7a- [ ,. ]-~ (k)( ; ).~~CN -l,p,g - L..Jg UoL..J ,a, k=l' lal=k I](a)=p . al. a2 . ap u1 U2 ,.,up a1!a21,..ap! (2.129) ", (2.130) 7a RN(g,g) =g(N)(UO+8U) L ~gl](a)a. lal=N trong(2.130),tadffsitd\mgcackyhi~udachis6san: a =(al,a2,.."aN) EZ:, la!=a1+a2+..,+aN' a!=a[!a2!...aN!, 11(a)=a1+2a2+...+NaN, (2.131) - - ( ) RN -a - al a2 aNV-Vi'V2"",VN E ,V -VI VI ",VN' V~ytli'(2.120),(2,123),(2.128)-(2.131)a(g,x,t) dl1<;1cvie'tl~inhl1san ( ~N~ Ja(g,x,t)=g ~C[N -l,p,g]gP +RN(g,g), (2,132) 28 Do cacham ui'i =1,2,...,Nbi ch~ntrongkhonggianhamL"'(O,T;HI), ta suyfa tli (2.123),(2.124),(2.130)ding Ila(s,.,.)IIL"'(OoT;L2)::;KlsIN+' trongdo - 2 . N-I (MN)k . (MNt - K =(N -2N)L k! Kk(T,g)+ N! KN(M,T,g) k=1 B6 de2.5daduQCchungminhxong.. Titp thea,taxetdayham{vm}djnhnghlabdi v =0,0 (2.133) Lvm=s[g(vm~I+Ji)-g(Ji)]+a(s,x,t),xEQ, O<t<T, By =0, i=O,l,, m vm(x,O)=Vm(x,O)=o ,m ~1. Vdi m=1,taco LvI =a(s,x,t), xED, O<t<T, B;v,=O,i=O,l, VI(x,O)= v(x,O)=0. Nhanhaivtcua(2.135)vdi VI'tasuyfatU (2.125)ding IIvI1I2+a(vi'VI)::;2KlsIN+ITllvIIIL"'(ooT;L2). Tli (2.136)taco Ilv,II,. (OT'"')+11'\11,"(oT"')S 2(1+-J~ }\:lgIN"T 0 \ Ta sechungminhdingt6nt'.limoth~ngs6CT, dQcl~prrt\vas saorho Il v I"'( . 1)+I.lv ll.",( .2 ) ::;CTlsIN+IT, Isl::;l, \1m (2.138)m LOoT, H m L O,T,L Nhanhaivt cua(2.134)vdivm' saukhirichphanrheat tathudu'Qc (2.134) (2.135) (2.136) (2.137) I IlvmJ +a(vm,v,J::;sfllg(vm+Ji)-g(Ji)/ldrllv,nIIL"'(OoT;L2) 0 (2.139) +2KlsIN+ITllv mIIL"'(ooT;L2). D~t1m=IIvJWI(T) =IlvJL"'(OoT;HI) +IlvJL"'(OoT;L2). Tli (2.139)suyfa 1m ::;cr1m-1+-0 , (2.140) trongdo 29 a~ 2(1+ ;--)1 +J2)TKI (M,T,g) , 1;~ 2(1+.j~JKTIEIN", (2.141) Voi giatricilaT >0thichhcjp,giasli'r~ngcrthoa cr<1. (2.142) Ap dl,lngb6d~2.4,tac6 - < 8 ".' .- 12Y - ~ , Val mOl m - , ,....m I-a . . (2.143) hay Ilvm!lwI(r)~crlslN+I, (2.144) trongd6 - ( 1 J - 1 Cr =2 1+ JC: KT~. C I-a0 D:lY ql1in~p tuye'ntinh{vm}dinhnghiabai(2.134)hQitl,lm~nhv~Wi giaiv Gilabai toan(2.122)trongWI(T). Vi V?yquagiOih~n(2.144)khi m~ 00 tadu'cjc (2.145) N uE -L: sPup l1 = IIvllw,(r) ::;;CrlslN+I p=o WI(T) (2.146) Khi d6tac6 Dinh IV2.8 . Gia sli'(HJ, (HJ, (HJ dungva gEcN(R). Khi d6t~nt~icach~ngs6 M >O,T >O.saochoVOlmQis,!sl<1,battoan(PE)c6duynha'tmQtWi giaiye'u uEEW(M,T) thoamQtu'oclu'cjngti~mc?nde'nca'pN +1nhu'(2.146),trongd6cac ham up' p =O,1,2,...,N19nlu'cjtla cacWigiaicilacacbattoan(pp),p =O,1,2,...,N. Trongdo~nsanclingtaxetg Cl,lthenhu(2:118),khid6 g ECoo(R). Ta cling hill yding g(k)=0, Vk~4, dod6phgndutrongkhattrienTaylorcilag xua'thi~n trong(2.132)la RN(s,g)=O,N~4. Dod6 a(s,x,t) trong(2.132)vie'tl~i N(N-I) ( 3 - }a(E,x,t) =E~i?l(uo) ~~~E't](a)=p Dod6ta c6danhgia lIa(s,.,.)too(o,r:L2)::;;(N3 - 2N2)(N2+3N+3)M3IsIN+I. (2.147) (2.148) (2.149) "Chuang2130 Cacbi€u thucFp trongb~liroan(pp) Fo=0, FI =g(uo)' F2=g'(uo)ul' F ,( ', ),1',, ( , ) ,2 3 =g Uo U2+2g Uo UI' F =g'(u)u +g"(u")uu +1-g"'(u)U3.4 03 0123! 01' 3 Fp =g'(uo)up-I+Lg(k)(Uo)L k=2 p-2 l>;=k ;=1 p-2 I>Xj=p-1 ;=1 Ual Ua2 ,ap-2 I 2 ...U p-2 a)a2!...a ! ' 4 ~p ~N.p-2 Cu6icungtacoke'tqua " , Dinh"JV 2.9 , Gia SlYcacgiathie'tcuadinhly2.8du'qcthoamanva(2.118)la dung,Khi do t6rit~icach~ngsO'duongM va T saochobili roan(p&)co duynha'tWi giaiye'u U&EW(M,T) thoau'ocluqngti~mc~n N u& - LSPUp ll ~ <$'(N,T)lsIN+1, p=o wItT) trongdo " <$'(N,T)=2 ( 1+ F;1 \N3 -2N2)(N2 +3N+3)T~, \c) I-a0 cr=24{1+J2)(1 +~~}M"