Luận văn Phương trình sóng phi tuyến liên hệ với một bài toán cauchy cho phương trình vi phân thường

Phuang trinh song phi tuye'nlien h~vai mQthili loan Cauchy cho phuang trinh vi phdn thuang trang14 Chlidng2 811t6nt~ivaduynha'tnghi~m Trangchu'dngnay,chungtoitdnhbaydinhly t5nt~ivaduynhat cuanghi«$mye'utoaDc1,1cchobai toaD:TIm mQtc~pham (u,P) thoa (2.1) (2.2) (2.3) (2.4) (2.5) Uti -Uxx +f(u,ut)=F(x,t), xeQ =(0,1),0<t<T, ux(O,t)=P(t), u(1,t)=0, u(x,O)=uo(x), ut(x,O)=u1(x), t P(t) =g(t)+H(u(O,t»- Jk(t-s)u(O,s)ds, 0 trongdo uo,ul,J,F,g,H,k la cachamchotrudcthoacacdi~uki«$nma chungtased~trasau. Chungminhdu'<jcd1,1'av o phu'dngphapGalerkin lien h«$vdi cac danh gia tieDnghi«$m,rut ra cac day con hQi t1,1ye'utrangcac khong gian hamthichh<jpnhamQts6cacphepnhungcompact.Trongphin nay, dinhly Schauderdu'<jcsa d1,1ngtrongvi«$chungminht5nt~inghi«$m xapxiGalerkin.Clingchuydingphu'dngphaptuyentlnhhoatrangcac baibaa[6,11,14]khongsad1,1ngdu'<jctronglu~nvannayvatrangcac b~bao~,4,5,7,9,1~. (AI) (A2) (A3) Trudche'tathanhl~pcacgiathie'tsau: UoE HI, u1E L2; g e H1(O,T) VT >0; keH1(O,T) VT>O vak(O)=O; H(Jc vien TrJn Thi Hu~ Chi PhllangIrinhsongphi luytnlien hf ViJi millhili loanCauchychophuangtrinhviphilnthuang trang 15 (A4) FEL2(Ox(0,T)) VT>O; (As) Hams6 HEel (R)thoaH(O)=0 vat6nt':limOth~ngs6 ho>0saoeho ~ H(1])= fH(s)ds~- ho. 0 ydimQi11E R; (F) Hams6 I: R2 ~ R thoa1(0,0)=0vacaedi€u ki~n: (FJ) I Ia dondi~ukhonggiamd6ivdibie'nthuhai,i.e., (f(u,v)-/(u,v))(v-v)~O Vu,v,vER; T6n t(;lihaih~ngs6a, ~E (0,1]va hai hams6Iien t\le BI,B2:R+~R+ saoeho: (F2) I/(u,v)-I(u,v)!~BI([ul)lv-vla Vu,v,v E R; (F3) I/(u,v)- l(u,v)1~B2(lvl)lu-uIPvu,v,vER; Khi dotacodinhIy sau DjnhIy2.1.Gillsa(AI)- (As)va (F;)- (F3)dung.Khi dov{JimJi T>0, t6ntr;limQtnghi~mye'u(u,P) cilahaitoan (2.1)- (2.5)saDcho (2.6) u E L"'(O,T;V), UtE L"'(O,T;L2), ut(O,t)E L2(0,T), (2.7) P E H1(0,T). H(}fl mla, ne'u ~=1 trong(F3)vacaehamsf;'H, B2 thoathemcae dduki~n, (A6) (F4) HE c2(R), H' (s)>- 1 VsER; B2(Iv J) E L2(Qr ) Vv E L2 (Qr), VT>O. Khi donghi~mcila hai toanfa duynha't. H(Jc vien Tr{mThi Huf Chi PhUiJng trinh song phi tuytn lien h~w1i rnQtbdi roan Cauchy cho phuiJng trinh vi phtin thuang trang16 Chti thich2.1.Ke'tquanay mf:lnhhonke'tquatrong[9].Th~tv~y, tuongungvdicungbaito0, cacgiathie'tsaudaydiidungtrong[9]khongdn thie'tsti'dl,mga day: (2.8)00, (2.9) B),B2la cachamkhonggiam. ChU'ngminh.Chungminhg6mnhi€u budc. Bu(Jc1.Xapxl Galerkin.XetmQtcosatn!cchuffnd~cbi~ttrongV. w/x) =~,Xt+A~)COS(AjX),Aj =(2j -I);, j =1,2,... duQcthanhl~ptucac hamriengcuatoantti'Laplace-82/&2 .D~t (2.10) m um(t)=L>m/t)Wj' j=1 trongdocm/t)thoamQth~phuongtrlnhphituye'nsauday (u~(t),w) +a(um(t),Wj)+Pm(t)W/O)+(f(Um(t),U~(t»,w) =(F(t),w), 1~j ~m, (2.11) (2.12) I Pm(t)=g(t)+H(um(O,t»- fk(t - s)Um(O,s)ds, 0 (2.13) ! ".(0) ="Om=~amjWj ->"0 m,nh(mngH', U~(0)=Ulm=L/3mjWj ~ U) m<;lnhtrong L2. j=) H~phuongtrlnh(2.11)- (2.13)duQcvie'tIf:lidudidf:lng (2.14) c~/t)+A~Cmj(t)=~(Pm(t)w/O) +(f(um(t),u~(t»,Wj)- (F(t),Wj)} IIWjl1 (2.15) I Pm(t)=g(t)+H(um(O,t»- fk(t - s)um(O,s)ds, 0 H(Jc vien Tr&n Thi Hu~ Chi PhUl1ngtrinh song phi tuytn lien h~vai milt hili loan Cauchy cho phul1ngtrinh vi phdn thuilng trang17 (2.16) Cm/O)=amj'C~j(O)=fJmj'1::;j::; m. B6d@2.1:Nghifmcuabili loanCauchysauday { CII (t)+A 2C(t)=q(t),t>0, c(O) = a, CI (0) = fJ, (2.17) cho beJiding thuc (2.18) sin(A t) I fsin(A(t - .)]yet)=acos(A.t)+fJ + q(.)d..A 0 A Chungminh:Chungminheongthue(2.18)khongkh6khan,taboqua. Ap dl,mgb6d~2.1ehohc$(2.14)-(2.16)vdi C(t)=C~i(t),A=Aj' a=amj' fJ=fJm;' q(t)=-=4(Pm(t)w/O) +(J(Um(t),u~(t)),Wj) - (F(t),Wj))' Ihl! Ta du'<;1ehc$(2.14)-(2.16)180tu'dngdu'dngvdi hc$phu'dngtrlnh vi rich phansau sin(Aj t) I sin(A/t - .)] Cm/t)=amjCOS(Al)+fJmj +f q(.)d. Aj 0 A} =am;cos(Al) +fJm;sin(Al) . A-} -~ rsinlAj(t- .)J(p (.)w(0)+(f(um('),u~(.)),Wj)- (F(.), w;)~. IlwJ 1 Aj m} ) sin(AA=amjcos(Al +fJmj ;: } - ~ !sinlAj(t-. )J[(g(.)+ H(um(0,.))- rk(. - s)um(O,.)d. )w; (0) Ilwj II Aj + (J(um(.),u~(,)),Wj)j1. H{Jc vien Trdn Thi Hu~ Chi Phllcmgtrinhsongphi tuye'nlienhevai mQtbiii roanCauchychophllcmgtrinhvipMn thllilng trang18 =amiCOS(A/)+ /3 . sin(A/)mJ Aj - ~ !sinlAj(t- r:)j(g(r:)wj(0)-(F(r:1w})~r: IIW}II A} - ~ !sinlA}(t- r:)j(H(Um(0,r:))w)(0)+(f(um(r:1u~(r:)1Wj)~r: IIWjl1 A} +w}(oj (inlA) (t- r:)j(rk(r:- s)UJO, r:)dr:)dr:. Ilw)II A} Ta vie't l(;li (2.19)nhu'sau cm/t) =Gmj(t) (2.20) - ~ fN/t - r:)[H(um(O,r:))w/O)+(f(Um(r:),u~(r:)),Wj )~r: IIWjl1 0 W (0)t r +~ fN/t - r:)dr:fk(r:- s)um(O,s)ds, 1:s;j:S;m, IIWjl1 0 0 trongdo (2.21) sin(Al) N}(t)= , Aj Gm/t)=amjN;(t)+/3mjNj(t) 1 t +~ fN/t- r:)[(F(r:),w) - w/O)g(r:)~r:. IIWjl1 0 Khi dotacob6d€ sau. B6d@2.2.Gidsa(AI)-(As) va(F;)-(FJ) fadung.VeJiT >° elfdfnh, khidoh~(2.20)-(2.21)conghi~mCm =(Cml,cm2'."'Cmm)trenmQtkhodng [O,TJc [O,T). Chungminh.Ta boquachIsaill, h~(2.20)-(2.21)vie'tl(;lidu'oid(;lng: (2.22) c=Uc, trongdoc=(CpC2"'.'Cm) , Uc=«UC)p(UC)W.,(UC)m), H9CvienTrimTh;Huf Chi - Phuangtrinhsongphi tuyln lienh? vai miltbdi loanCauchychophuangtrinhvipMn thuilng trang19 (2.23) / (Uc)}(t)=G}(t)+SNit - r)(Vc)/r)dr,° (2.24) / (Vc)/t) =~/c(t),c'(t))+fk(t- s)/2/e(s))ds,° (2.25) G/I) =amjN;CI)+/imjN/1)+11:11'INjU- T)[(F(T),Wi)- Wj(O)gCT)¥T. /,} : R2m~ R, 12}:Rm~ R , (2.26) /,/e,d)=-=-4 [ H (fe,w/(o))w/O)+!I [ fe,w, ,fd,w, J 'Wi )] ' Ihll 1=1 \ 1=1 I=! (2.27) - W/O) m < .J;./e)- 2 Le,w,(O),l_J~m. Ihll 1=1 Vdi m6iTm>O,M>0, tad~t s= ~ECI~O,TJRm):llelll~M}, m Ilelll=Ilello + Ile'ilo' lie11o=SUp le(t)I!, le(t)l!=Lle,(OI.O~/~Tm 1=1 D€ tha'yding S la t?P can15idongva bi ch?ncua Y=CI~O,TJRm). Dungdinhly di~mba'tdQngSchaudertase chungminhrangtmintll' U: S~ Y xac dinhbdi (2.23)-(2.27) co diSmba'tdQng.Di~mba'tdQng nayla nghi~mcua(2.20). DftutientachungminhdingU bie'nt?P s vaochlnhno. il ChliY ding (Ve)}EcoQo,TmlR)vdi mQiCEC!~O,TJRm),do do tasuytu (2.23),vad£ngthuc (2.28) I (Ue)~(t)=G~(O+ fN~(t- r)(Ve)/r)dr,° HQc vien Tran Th; Hu? Chi PhU{fngtrinhsongphi tuytnlien h~va; mQthili loanCauchychophu{fngtrinhviphanthlli1ng trang20 r~ng U: Y ~ Y . Cho eES, tasur tit (2.23), (2.28)r~ng I(Ue)(t~1=fl(Ue)j(t~ j=1 =~IG)t)+fNj(t-rXve))r)drl m / m ~IIGj(t~ + JIINj(t-r~I(Ve)j(r~dr j=1 0 j=1 / m 1 1 ~ ~IG(tl +If; A-j(Ve)j(r1dr 1 / ~ IG(t~1 +~fl(veXrtdr 1/ ~IG(t~1+~jllvellodr 1 ~ IG(tl + ~Tmllvello' Tucmgtvtaco (2.29) 1 I(Ue)(t)11~ IG(t)11+~TmllVcllo . Dodo (2.30) I (Ue) (t)ll ~ IG'(t)11+TmllVello' M~tklulc,tasur tit (A3),(A4),(F2),(F3),va(2.24)r~ng !IVello = sup I(VeXt~1 O$/$Tm = sup fl(Ve)j(t~ O$/$Tmj=1 m m / ~ supIIJ;j(e,e'~+ supI ~k(t-s)f2j(e~ds O$/$Tmj=1 O$/$Tmj=10 Hl}c vien Tr&n Thi Hu~ Chi Phuang trlnh song phi tuytn lien h~vai milthili roan Cauchy cho phuang trlnh vi phan thullng trang21 m m / :s; I sup1J;)(c,c'~+ I flk(t-s)supIfz)(c~ds )=1O:>/:>T. )=1 0 09:>T. m m / :s;IN1(r.),M) + I flk(t-s~Nz(rz),M}is )=1 )=10 m m / :s;IN1 (r.),M) + INz (rzj'M)Jlk(t - s~ds )=1 )=1 0 m m / :s;IN1 (r.),M) + INz ([z),M) Jlk(s~ds )=1 )=1 0 m m :s;IN1(r.),M) + IllkIILJ(O,nNz(rz),M). )=1 j=1 Suyfa (2.31) IIVcllo::;i[NI(fIj,M)+llkIILI(o,T)Nz(/z),M)] ==P(M,T), )=1 vdi ffiQi cE S, tfong d6 (2.32) N1(fIj,M)=sup~J;/y,z)I:llyIIRm::;M,llzIIRm::;M}, (2.33) Nz{/zJ'M) =sup~/z/y)1: IlyllRm::;M}. Do do,tIT(2.29)-(2.31) tathudu'<jc 1 supIUc(t~1:s;supIG(t).+ -Tmllvcllo O:>/:>Tm O:>l:>Tm ~ 1 :s; sup IG(t). + - TmIIVcllo O:>/:>T ~ 1 :s;IIGllo'+ ~Tmllvcllo' TuO'ngtg taco sup IUc(t~1:s; jiG/jiG.+ TmIIVcllo.O:>/:>T Suyfa IIUclio +IIUCllo ::;IIGllo' +IIG'llo'+(~1 + 1)TmIIVcllo' Ht;Jcvien Trdn Thj Huf Chi Phllang trinh songphi tuye'nlien h~vai mtlthili loan Cauchy cho phuang trinh vi phdn thui1ng trang22 (2.34) IIUcll1 ::;IIGIII.+(1+~)rmP(M,T), trongd6 IIGIII. = IIGllo' + IIG'llo'=sup IG(t)11+ supIG'(t)l,. O,s/,sT Os/sT ChQnM va Tm> 0 saocho (2.35) M >211GIII.va (1+~)rmP(M,T) ::;Mh' Do d6 IIUcll,::;M voi mQi cES, nghiala, toantit'U biEn t~pS vaocrnnh n6. ii/ TiEp theota chungminh loan tit'nay Ia lien t\,1ctren S. Cho c,dES, tac6 (2.36) / (Uc)/t)-(Ud)/t)=SNit- .)[(vc)/.)-(Vd)/.)}i., 0 m I m suplJuc)At)-(Ud)j(t~ = ]Nj~-'~supLI(vc)j(.)-(Vd)j(.~d.. ~~~ 0 ~~~ Dod6 (2.37) 1 IIUc-Udllo::;-TmIIVc-Vdllo'~ Tu'dngtvtadingthudu'Qctuba'td~ngthuc (2.38) , , / (Uc)At) - (Ud)At) =fN~(t- .)[(vc)/.) - (Vd)/.)}i., 0 o~~fm ~I(uc)'j(t)- (Ud)'j~~=IIN~(t- '~o~~fmI(vc)j(')- (Vd)j('~ ::;IIIVc-vdtd.::;TmIIVc-Vdllo' ding (2.39) II(uc) - (Ud)110 ::;Tmllvc-vdllo' HQc vien Trdn Thi Hu~ Chi Phl1angtrinh songphi tuye'nlien h~VlJi m(jthili loan Cauchy cho phuang trinh vi phan thuang trang23 Baygio,tadn mQtdanhgias6h~ngliVe- Vdllo'Taco (2.40) (Vet(t)-(Vd)it)=hie(t),e'(t))-hid(t),d'(t)) , +Sk(t-s)[f2ie(s))-fzid(s))}is. 0 Tli cacgia thie't(A3),(A4),(F2),(FJ, va(2.40),tasuyfadingt6nt~imQt hftngs6 KM>0 saocho I(Ve)j(t)- (Vd)j(t~ = Ihj(e(t1e'(t))+!k(t - S)f2j(e(s))ds- hid(t 1d'(t))- !k(t - S)f2; (d(s))dsl s !flj(e(t),e'(t))-~j(e(t1d'(t)~+ l~j(e(t1d'(t))-~j(d(t),d'(t)~ +!Jk(t- s)lf2j (e(s))- f2j (d(s)~ds s Bl~e(t))le'(t)-d'(tr + B2~d'(t))le(t)-d(tt (2.41) +!Ik(t - s)lf2j (e(s)) - f2j (d(s )~ds, liVe- Vdllos KM ~Ie- dll: +lie'- d'lI: +(1 +IlkIIL'(o.rJie- diU, vdimQie,dES. V~ycacdanhgia (2.37),(2.39)va (2.41)chungtodng u: S ~ Y la lien tl,1c. iii/ Bay gio ta se chungminhfftngt~pus la mQtt~pcon compactcua Y. ChoeES, t,tlE[O,TmJ.Tli(2.23),tavie'tl~i (Ue)/!)-(Ue)it') =Git)-Git') (2.42) , +flNit-r)-N/t'-r)}:Ve)/r)dr 0 ,° -SNit'- r)(Ve)/r)dr. t Hl}cvienTrdnThiHu~Chi Phuangtrinhsongphi tuye'nlienh~vai m(jthili loanCauchychophuangtrinhvipMn thuang trang24 Chuydng tuba'tdiingthuc (2.43) IN/t)-N/s)l:;; It-sl "it,sE[O,TJ, tathudu'<jctIT(2.31)ding (2.44)I(Ue)(t)-(Ue)(t')ll=tl(Ue)j(t)-(Ue)i~'~ j=l =~!Gj (t)- Gj (t')+f[Nj(t - r)- Nj(t' - r)XVe)j(r)dr - jNj (t'- rXVe)j(r)dr! m m , ::;IIGj(t)-G~(t'~+I ~[Nj(t-r)-Nit'-r)XVet(r~dr j=i j=1 0 m " +I ~Nj(t'-r~I(Ve)j(r~dr j=1, I t' ::;IG~)-G(t'~1+ It-t'l fllvellodr+ ~ fllvellodr 1 ::;IG(t)- G(t'~1+ It- t'IIIVelloTm+ ;; It- t'IIiVello :;;IG(t)- G(t')I)+ (Tm+l) It- t'IIIVello :;;IG(t)-G(t')li+,8(M,T)( m+l) It-t'!. Tu'dngUf,tu(2.28),(2.31)va(2.35)taclingthudu'<jc " I (Ue)(t)-(Ue)(t')=G~(t)-Gj (t') + l[Nj'(t-r)-Nj'~'-r)JveJr)dr , , - SNj(t'-r)(Ve)j(r)dr, HQcvienTrdnThi Hu~Chi PhlNngtrinhsongphituytnlienhf va; rnQthili roan Cauchy cho phu(}ngtrinh vi philn thuang trang25 dodo (2.45)I(Uc)'(t)-(Uc)'(t't ~~IGj'(;)-Gj' (t'~ + ~ !I[Nj' (t-r)-Nj' (t'-r)Jvc)j(r~dr +~flNj' (t'- r~1(Vc)j(r~dr ~IG'(;)-G'(tt + AmTmIIVcllolt-t'l, ~!G'(t)- G'(t')ll+[J(M,T) (AmTm+1) It- t'l. Do usc S va tucaedaubgia(2.44),(2.45)tasuyradinghQcaeham us={uc,cEs},la bi ch~nva lien Wcd6ngb~cdo'ivoi chugnII-II. cua khonggianY.Ap d1,mgdinh19Arzela-Ascolivaokh6nggianY, tasuy rar~ngus la compacttrongY. Do dinh19di~mbatdQngSchauder,ta co cES saDcho c=Uc,ma di~mbatdQngnayla nghi~mcuah~(2.20). Bdd~2.2duQcchungminhhoantat.. Dungbd d~ 2.2 voi T>0, co' dinh, h~ (2.14)-(2.16)co nghi~m (Um(t),Pm(t))trenmQtkhoang [O,TmJ.Cac daubgia tieDnghi~mtrenday chopheptalayTm=T voimQim. Buae2.Caedanhgiatiennghi?m.Thay(2.15)vao(2.14),taduQc (u; (t),Wj)+a(um(t),Wj) +[g(t)+H(um(O,t))- !k(t- S)um(O,s)ds]W/O) +(f(um(t),u~(t)),Wj)=(F(t), w). Saudonhanphuongtrlnhnftybdi c~/t)valaytdngrheaj, taco H(Jc vien Tr6n Thi Huf Chi PhUdngtrinhsongphi tuytnlienh~vai mQthili loanCauchychophulJngtrinhviphewthuang trang26 (u~(t),u~(t) +a(um(t),U~(t)) +[g(t) +H(um(O,t))- fk(t - S)um(O,s)ds}~(O,t) +<J(um(t),U~(t)),U~(t» =(F(t), U~(t». Tichphantungph~nrheabie'nthaigiantu 0 de'nt,nhacacgiathie't (Az),(F;), taco (2.46) Sm(t)::; -2H(um(O,t))+2H(uom(O))+Sm(O)+ 2g(O)uoJo) t -2g(t)um(O,t) + 2 Jg'(s)uJO,s)ds 0 t t -2 J<J(um(s),O),u~(s)ds + 2J(F(s),u~(s))ds 0 0 t s + Ju~(O,s)dsJk(s - r )um(O,r)dr , 0 0 trongdo (2.47) Sm(t)=Ilu~(t)llz+llum(t)II~. Khi do,sadl;1ng(2.16),(2.47),(A4) vab6d61,taco (2.48) -2H(um(O,t))+ 2H(uom(O))+Sm(O)+ 2Ig(O)uom(O)1 1 , . . , ::;3C], VOl illQI m va t, trongdo C]la illQthAngsochiphl;1thuQcvaouo"upH,hovag. Sadl;1ngb6d6 1.1illQtl~nnuavaba'tdingthuc (2.49) 1 2ab::;-aZ+3bz,Va,bER, 3 ta thu du'<,1c (2.50) 1- 2g(t)um(O,t) + 2 fgl(S)Um (0,S)dSI H9C vien Trdn Thi Hu~ Chi Phllangtrinhsongphi tuy[nlien h~vai mrthili loanCauchychophllangtrinhviphfinthlliJng trang27 I ::;2lg(t~IUm(O,t~+ 2 flg'(s~lum(O,s~ds 0 ::;3g2(t)+ ~lum(O,tt+2[flgl(st dS)[f'Um(O,stdS) 1 I 1 I ::;3g2(t)+- SJt) +3flgl(stds+ -;;-fluJo,st ds3 0 .) 0 2 In, 1 2 1 Il f::;3g (t)+3jlg(s) ds+-Sm(t)+- Sm(s)ds. 0 3 3 0 T x d' b,:! dl< 1 1 b'" d2 h' aP bq ,. 1 1 kh. d 'a van ling 0 e . , at ang t tic ab::;-+- VOl -+-=1 I 0 P q P q tu (p;)ta suyfa dng (2.51) 1-2f(J(Um(S),O)'U~(S))dSI::; 2If(J(Um(S),O)- f(O,O)'U~(S))dSI I (l+fJ) ::;2B2(O)fSm(s)~ ds 0 (2.52) ~ Sm(s) 1 ] ::;2B2(O) y-+ 2/ ds 0 71+/3 /1-/3 =B2(OXl+/3)!SJs}ds+(I-/3)B2(O~. 1-2f(F(S)'U~(S))dSI::;2~IF(S)llllu~(s)lldS I I ::; fIIF(s)112ds + fllu~(s)112ds 0 0 I I ::; fIIF(s)112ds + fSm(s)ds. 0 0 Chli Y ding tichphansaucungtrang(2.46)vie'tl<;lisaukhi sli'd\lllg tich phan Hrc vienTrc1nThjHu~Chi Phuc/ng trinhsongphi tuytnlienhf vai milt hili toan Cauchy cho phuc/ngtrinh vi phan thuang trang28 (2.53) I .. 1=2fu~(O,s)dsfk(s - r)um(O,r)dr 0 0 I = 2um(O,t)fk(t - r)um(O,r)dr 0 I .. - 2fum(O,s)dsfk'(s- r)um(O,r)dr. 0 0 Do d6 (2.54) III ~ 2IUm(O,t~l!k(t-r)UJo,r)irl + 2!(lum(O,s~rlkl(s-r~luJo,r~dr)ds t ~2 ~Sm(t)~k(t- r)I~Sm(r)dr 0 I .. +2NSm(s)dsflk'(s- r)I~Sm(r)dr 0 0 ==I] + 12, Sf)h~ngthunh:1tI, trongvfi phai cua(2.54)du<;1cdanhgia nhusailnho vaob:1td~ngthuc(2.49).Ta c6 (2.55) I I, =2~Sm(t)flk(t-r)I~Sm(r)dr 0 s ~Sm(t}+{Fk~-<J,jSJf}l<)' [ 11 1, ] 2 1 I /2 I 12 ~ :3Sm(t)+3 (flk(t- rf dr) (fsJr)ir) 1 I t =- Sm(t)+3 fk2(s)dsfSm(r)dr. 3 0 0 H(}cvien TrIm Thj Hur Chi Phwng trinhsongphi tuytnlienh~viii m(jthili loanCauchychophlldngtrinhviphanthlllJng trang29 TudngW's6h~ngthuhai 12trongv€ phai cua(2.54)duQcdanhgia nhu saunhov~lOba'tdiingthucCauchy-Schwartz.Ta co (2.56) 12 I s =2 NSm(s)dsflk'(s- T)I~Sm(T)dT 0 0 [( I ) ~ ( l s ) 2 ) ~ 1 ::;2 fSm{s)ds llflk'{S-T~~T ds ( ) 2 1 lis ::;- fSm(s)ds+ 3Ids flk'{s- T~~T3 0 0 0 ::; ! fSm(S)ds+ 3fdS(flk'{S-T~~T) 2 3 0 0 0 1 I 1 I ::; - fSm(s)ds+3t flk'(sfds fSm(T)dr. 3 0 0 0 Tit (2.44)-(2.56)tathuduQc 1 ( 1 1 I ) 1 (2.57) III::; -Sm(t) + -+3fe(s)ds+3tflk'(s)12ds fSm(T)dT. 3 3 0 0 0 Ta suytit(2.46),(2.48),(2.50)- (2.53)va(2.57)ding 1 I 1 1 I Sm(t)::;-C1 +3g2{t)+3flg'{sfds+- Sm(t)+ - fSm(s)ds3 0 3 30 1 I + (I+p)B2(O)fSm(s)ds+{I- p)B2(O)t+ fIIF{s~12ds 0 0 1 1 ( 1 I 1 ) I + fSm(s)ds+- Sm(t)+ -+3 fe(s)ds+3t]k'(sfds fSm(T)dT 0 3 3 0 0 0 1 I 2 I ::; -C1 +3g2{t)+3flg'{sfds +-Sm(t) + fIIF{s~12ds 3 0 3 0 ( 5 1 1 ) I + -+3fk2(s)ds+3t]k'(s)12ds fSm(T)dT 3 0 0 0 H9CvienTrJn Thj Hu~Chi PhUdngtrinhsongphi tuytnlien hf vai mQthili roanCauchychophudngtrinhviphiinthuang trang30 hay (2.58) tfongdo (2.59) (2.60) I + (1+p)B2(O)fSm(s)ds+(I-p)B2(O)t, 0 I Sm(t)::;;D1(t)+ D2(t)fSm(T)dT,° I I D)(t)=C1 +3(l-p)B2(O)t+9g2(t)+9 flg'(s)12ds+3fIIF(sfds,° ° I I D2(t) =5 + 3(1+p)B2(O)+ 9 fk2(s)ds+9tflk'(sf ds.° ° Vi H1(O,T)<-+Co([O,T]),tucacgiathiet(AJ, (AJ taguyfading (2.61) ID/t)I::;;c7),a.e., 'v'tE[O,T],(;=1,2), tfongdo c7) la ffiQth~ngs6chiph\!thuQcvao T. Do b6d~Gronwall, tathudtt<;1ctu (2.58)-(2.61)ding (2.62) Sm(t)::;;Cj.l)exp(tCj.2»)::;;Cn 'v'tE [O,T], 'v'T>O. I Baygio,tadn danhgiatichphan~u~(O,sfds.° f)~t (2.63) (2.64) Km(t)=:tsin(Al) )=1 X 'J rm(t) = ~w/o{a., costAi) + Pm)SiO~~i)] J,. m ' fsin[A/t- .)] ( I Wi )--.;2I . !(Um(.),Um(.))-F(')'- 11 .. 11 d.. J=1 o AJ WJ Khi do tu (2.13),(2.20),(2.21),taviet l<;tium(O,t)nhttsau H(Jc vienTrdnThi Huf Chi Phuangtrinhsongphi tuyln lienh? va; m(jthili loanCauchychophuangtrinhviphanthlli1ng trang31 (2.65) I um(O,t)=Ym(t)- 2 fKm(t-T)Pm(T)dL 0 Ta dn b6d€ sauday. B6 d~2.3.Tant{lim(Jthlingso'C2>0 vam(Jthamdu(Jnglient~cD(t) d(Jc19pvdi m saDcho (2.66) I I ]Y~(T)12dT ~ C2+ D(t) ]IJ(Um(T),U~(T))- F(T)112dT, 0 0 V'tE [O,T], V'T>O. Chungminhb6d€ 2.3cothStlmtha'ytrong[2]. B6 d~2.4.Tant{lihaihlingsO'du(JngC?)va C}4)chiph~thu(JcvaoT saDcho (2.67) Ids I JK~(S - T)Pm(T)dT I 2 ~ C}3)+ C}4)Ids ]u~(O,T)12dT , 0 0 0 0 V'tE [O,T], V'T>O. Chungminh.Tichphantungph~n,taco (2.68) s s fK~(s - T)Pm(T)dT=Km(s)Pm(O)+ fKm(s- T)P~(T)dL 0 0 Khido If K~{s- T)PJT)dTI~ IKm{s)pJo~+ flKm{s- T~lp~{T~dT ~ IKJs)Pm{O~+(flKm{s- Tt dT)Yz( fIP~{TtdT)Yz ~ IKm{s)Pm{O~+(fIKm{rtdr)Yz(fIP~{TtdT)Yz. Blnhphu'dnghaiv€ saudonchphantheostathudu'<jc (2.69) Ids IlK:(s- r)P.(rJd{ H(}cvien TrJn Thi Hu? Chi Phu(/fIgtrinhsongphi tuytnlienhevai mQthili roanCauchychophu(/fIgtrinhviphanthuang trang32 t I ,\' S ~ 2 P,;(o)fK;(s)ds + Ids fK;(r)dr fIP;(r)12dr 0 0 0 0 ~ 2 fK;(s)ds [ p,;(O)+Ids]p;(r)ldr ] . 0 0 0 Chuydingtu(2.15)taco (2.70) Pm(O)=g(O) + H(uom(O», (2.71) r P;(r) =g'(r) + H'(um(O,r»u~(O,r)- fk'(r-s)um(O,s)ds. 0 Dung ba'td~ngthuc (a+b+cy ~3(a2+b2+C2~\ia,b,cE R, ta suy tu (2.62), (2.70),(2.71) vaA4 rang (2.72) IP;(rt ~3Ig'(rt+3IH'(um(O,r)tlu~(O,rt +3[flk'(r-s~luJo,s)dslr ~ 3Ig'(rt+3maxIH'(stlu~(O,rtISI$JCr +3(flk'(r-st ds)(flum(O,stds). Suy fa ,\ S ,\ (2.73) ]P;(r)12dr ~ 3 flg'(r)12dr+ 3 maxIH'(sf flu~(O,r)12dr 0 0 Isl$JCr 0 r r +3 rdr flk'(r)12dr fu;(O,r)dr 0 0 S ,\ ~ 3flg'(r)12dr +3 maxIH'(s)12flu~(o,rf dr 0 ISI$JCr 0 +3 rdr Ilk'(rf dr Iu;'(O,r)dr 0 0 ,\ S ~ 3 flg'(r)12dr+ 3 max IH'(s)12flu~(O,r)12dr 0 Isl$JCr 0 ' H{Jc vien Tr&n Thi Hue Chi PhlJang trinhsongphi tuytnlienh~vai mQthili loan Cauchy cho phlJang trinh vi phan th11i1ng trang33 s s +3s flk'(r)12dr fu;'(O,r)dr. 0 0 Do do,tasuytu(2.71)- (2.73)ding Ids IIK~('- fJP.(f)d{ (2.74) ::; 2 fK;'(S)ds [ p,;(O)+ Ids]p~(r)ldr ]0 0 0 ::; 2 IK;'(S)dS[{g(O)+H(Uom(O))Y+![3 [Ig'(rt dr +3maxIH'(st [ Iu~(o,rt dr +3s[Ik'(rt dr[u;,(O,r)dr]dsIsl,;;.JCr ::; 2 IK;'(S)dS [{g(O)+H(uom(O))Y+!dS[3!Ig'(rt dr +3maxIH'(st[Iu~(o,rt dr +3s'Ik'(rt dr' u;,(o,r)dr]Isl,;;.JCr -'>-'> I I ::;2 fK;,(s)ds [(g(O)+H(uom(O))Y+3t]g'(r)12dr 0 0 I s +3maxIH'(s)12Ids flu~(0,r)12dr Isl';;.JCr 0 0 3 I I +-t2 flk'(rt dr fu;,(0,r)dr ].2 0 0 Chuyding vdi m6i T >0,Km~ K m<:lnhtrong L2(0,T) khi m~ +00va dungcaegiathie't(AI)- (A4)vacaeke'tqua(2.16),(2.62)va (2.74)ta thudu'<;Jc(2.67).B6 d~2.4du'<;Jcchungminhhoanta't.. B6 d~2.5.TImt{Iihaihdngso'dLtangcis)va c?) chiph1;lthuQcVaG TsaDcho (2.75) I ~u~(O,r)12dr::; cis), '\it E [O,T], '\iT> O. 0 H{Jc vien Tr&n Thi Hu~ Chi PhUC1ngtrinh song phi tuytn Win h~vui mQthili loan Cauchy cho phUC1ngtrinh vi phan thuang trang34 (2.76) I ~P~(rfdr~C?), \tte[O,T],\tT>O. 0 Chungminh.VI (2.76)Ia hc$quacua(2.62)va (2.75),tachidn chung minhding(2.75). I u~(O,t)=r~(t)- 2 JK~(t- r)Pm(r)dr, 0 lu~(O,tt~ 2Ir~~t+8[!IK~(t-r~IPm(r~drJ. Tu (2.67),sudt,}ngb5 d€ 2.3va 2.4,tathudu'<;lc Ilu~(0,s)12ds ~2 ]r~(s)12ds +8 Ids I }K~(S- r)Pm(r)dr I 2 0 0 0 0 (2.77) I ~2C2+2 D(t) ]lj(um(r),u~(r»-F(r)112dr 0 I s +8C~3)+ 8 C~4)Jds ]u~(0,r)12dr. 0 0 M~tkhactu(2.62)vacaeghithie't(r;), (f;) tathudu'<;lc Ilj(um(t),u~(t))- F(t)112~21lj(um(t1u~(t)f +211F(t~12 ~4 max BI2(s~Iu~(t)II~~"+4B; (0~IumII~P+21IF(t)112Isl~v'C,- ~4 max BI2(s~lu~(t~12a+4B;(0)lumll~P+21IF(t)112,I* v'C,- VI 0<a ~1ta co 11-IIL2"~II-II. V~y (2.78)Ilj(um(t),u~(t))-F(t)II2~ 1Ij(um(t),u~(t))112+2 IF(t)II2 ~ 4 maxBI2(S)IIu~(tta+4B;(0) Ilum(t)II~P+211F(tt. j..!sv'C,- Dodo,sudt,}ng(2.47),(2.62)va(2.78)taco H(Jc vienTrdnThi Huf Chi Phllangtrinhsongphi tuye'nlienh~va; miltbiJi roanCauchychophllangtrinhviphanthttiJng trang35 (2.79) Ilf(x,t,um(t),u~(t))- F(x,t)11~ C?). Cu6icling,tu(2.77)va(2.79)tathuduQCbatd~ngthuc (2.80) I I s flu~(o,s)12ds~C}8)+8 C}4)Ids ~u~(o,1ldT,° 0 0 madi€u naydin de'n(2.75),dob6d€ Gronwall. B6 d€ 2.5duQcchungminhhoantat.. Bl1fJc3. Qua gifJi hc;ln. Tu (2.15),(2.47),(2.62),(2.75),(2.76)va (2.79),tasuyrading,t6nt<;li mQtdayconcuaday{(um,Pm)},vin kyhi~ula {(um,Pm)},saocho (2.81) (2.82) (2.83) (2.84) (2.85) (2.86) Um~ U trong L"'(O,T;V) ye'u *, U~~ u' trongL"'(O,T;L2)ye'u*, Um(O,t)~ u(O,t)trong L"'(O,T)ye'u *, u~(O,t)~ u'(O,t)trongL2(O,T)ye'u, f(um'u~)~ X trong L"'(O,T;L2)ye'u *, Pm~ P trongH1(O,T)ye'u. Do b6 d€ compactcuaLions (xem[8]),ta suyra tu (2.62),(2.75), (2.81)va (2.82)ding,t6nt<;limQtdayconvin kyhi~ula {uJ saocho (2.87) (2.88) um(O,t)~ u(O,t) m<;lnhtrong coao,TD, um~u m<;lnhtrongL2(Qr)va a.e. (x,t)trongQr' VI H lient\Ic,tasuyratu(2.15),(2.87)ding (2.89) I Pm(t) ~ get)+H(u(O,t» - fk(t- s)u(O,s)ds=pet) 0 m<;lnhtrongcoao,TD. Tu (2.86)va(2.89)tathuduQc H9c vienTrdnThi Huf Chi Phuongtrinhsongphi tuytnlienh~vui m(jthili loanCauchychophuongtrinhviphanthudng trang36 (2.90) p ==P a.e. trong QT' QuagiOih~ntrong(2.14)nhC1v~lO(2.81),(2.82),(2.85),(2.89)va (2.90) tathudu'Qc d -(u'~1v) + a(u(t1v)+ P(t)v(O)+ (x(t),v)=(F(t),v),'v'VEV.dt Ta coth€ chungminhIDeomQtcachtu'dngu,tnhu'trong[10]ding (2.91) (2.92) u(O)=Uo' u'(O) =u1. Khi do,d€ chungminhs1,1't6nt~icuanghi~mbai loan(2.1)- (2.5),ta chidn chungminhdingX=j(u,u').BaygiC1chungtasedn b6d~sau day. B6d~2.6.Gillsa u Langhi~mcuabai (Dansau (2.93) (2.94) Uti - Uxx +X=F, x E n =(0,1),0<t <T, uAO,t)=P(t), u(1,t)=0, (2.95) (2.96) u(x,O)=uo(x), u,(x,O)=u)(x), U E L"'(O,T;V), u/ E L"'(0,T;L2), u,(O,t) E L2(0,T). Khidotaco (2.97) / / !11u'~)12 + !llu(t)I~+ Jp(s)u'(O,s)ds+ J(X(s)- F(s1u'(s))ds2 2 0 0 ~~llu)112+~lluoll~a.e.t E [O,T]. Hdn mla, ne'uUo=u1 =0 thi (2.97)xdyradl1ngthuc. Chungminhb6d~2.6coth€ tlmthffytrong[2]. BaygiC1,tu(2.14)- (2.16)taco (2.98) f(J(um(s1u~(s))-F(s1u~(s))ds= !IIUlm112+!lluomll~ 0 2 2 H(Jc vien Tr&n Th; Huf Chi PhUdng trinh song phi tuye'nlien h~vOi milt hili toan Cauchy cho phudng trinh vi phan thudng trang3? - !llu~(t~12- !llu)t~l~- fp)s)u~(O,s}ds.2 2 0 Do b6 d€ 2.6,ta suytu (2.16),(2.81),(2.82),(2.84),(2.89)va (2.98) r~ng I (2.99) limsupf(J(u)s),u~(s»-F(s1u~(s»)ds m--H'" 0 ~ !llull12+ !lluoll~- !llu'(t)122 2 2 I - !llu(t)I~-fp(s)u'(O,s}ds2 0 I ~ f(X(s)-F(s),u'(s»)ds,a.e.tE[O,T]. 0 B~ngeachdungl~plu~ngi5ngnhutrang[10]tachungminhduQcr~ng X=f(u,u') a.e.trangQT'S\!'t6ntf;linghi~mduQcchungminh. ElidC4. Sf!duynhatnghi~m. Bay giOtagia sli'r~ngfJ =1trong(f;) va H thoa (.4,). Gia sli' (up~),(u2,~)la hainghi~mcuabai tmin(2.1)- (2.5).Khi d6 u=U1-u2, P =~-~ thoabaitminsauday Ull - u""+X =0, 0<x <1,0<t <T, ux(O,t)=pet), u(1,t)=0, u(x,O) = Ul (x,O) = 0, x=f(upU~)- f(u2,u;), (2.100) I P(t) =P. (t)- P2(t)=H(u1(O,t))- H(u2(O,t))- fk(t - s)u(O,s)ds, 0 u; E L"'(O,T;V), u: E L"'(0,T;L2), u: (O,t)E L2(0,T), P'EH1(0,T), i=I,2. H9C vien Tr&n Thi Huf Chi PhUdngtrinhsongphi tuytnlienhevbi m(jthili loanCauchychophudngtrinhviphanthullng trang38 Sad\mgb6d€ 2.6voi Uo=Ul=0, F= 0, tathuduQc (2.101) I ~"u'(t~12+ ~"u(t~l~+ fp(s)u'(O,s~s2 2 0 I + f(i(s1u'(s))ds=0 a.e. t E [O,T]. 0 E>~t (2.102) o-~)=Ilu'~~12+Ilu(t~I~,HJ(t) =H(uJ(O,t)) - H(U2(O,t)). Thay P (t), i vao(2.101)taco ~llu'(t~12+ ~"u(t~l~2 2 +I(H(u((O,s))-H(U2(O,S))-rk(s-r)u(O,r~r)u'(O,s~s + I(J(UI'U;)- f(U2,U;),U'(s))ds =0, vachuyr~nghamf la khonggiamd6ivoibienthuhai,taco (2.103) I o-(t) +2 fH((s)u'(o,s~s 0 I ~ 2 ]If(u((s1u;(s))- f(U2(S),U;(s)~lllu'(s~lds 0 I s +2 fu'(O,s~sfk(s-r)u(O,r~r. 0 0 Dunggiathiet(F;)taguyrar~ng (2.104) Ilf(UI (s1U;(s)) - f(U2 (s1U;(s)~I ~ IIB2 ~u; (s ~~lllu(s ~Iv . Dungnchphantungphftntrongnchphancu6icungtrang(2.103),ta duQc (2.105) I s J =2 fu'(O,s~sfk(s-r)u(O,r~r 0 0 H(Jc vienTrtinThi Hue Chi PhUlJng trinh song phi tuytn lien hf vcR mQthili loan Cauchy cho phulJng trinh vi phtin thuang trang39 1 I s =2u(0,t)fk(t-r)u(O,r)dr- 2 fu(O,s)dsfk'(s-r)u(O,r)dr. 0 0 0 Ta Suyfa tu(2.102)va(2105)ding (2.106) ( I ) ~ ( I ) ~ IJI :s;2 ~ fe(r)dr fu(r)dr ( I ) ~I +2Ji }k'(rtdr fu(r)dr 1 1 1 :s;13,u(t)+ - fe{r)dr fu{r)dr 1310 0 ( I ) ~1 +2Ji }k'(rtdr fu(r)dr, V13,>0. f)~t (2.107) M = max IluiIIL~ ( r.v)' ml =mill H'(s), m2=maxIH"(s~.1=',2 0, . !slsM IslsM Tu giathie'tA6taco ml>-1. M~tkhae,donehphantungphffnva (2.107),tasuyfading 1 1 [ I d ] 2 fH,(s)u'(O,s)ds=2 f f-H(uJO,s)+Ou(o,s))deu'(O,s)ds 0 0 ode I =U2(0,t)fH'(U2(0,S)+0u(0,s))de 0 (2.108) 1 I f u2(0,s)dsfH"(U2(0,S)+Ou(O,s)Xu;(O,s)+Ou'(o,s))de 0 0 1 ~ mJ U2(0,t)- m2 f u2(0,s)~u;(0,s)+lu;(0,s~}is 0 1 ~ m) U2(0,t) - m2 f u(s) ~u;(o,s)+lu;(O,s)}is. 0 Tu(2.103)- (2.105)va (2.108),tathudu'<Je H(Jc vien Tr&n Thi Huf Chi Phlldngtrinhsongphi tuytnlienh~vdi mi;thili loanCauchychophlldngtrinhviphiinthllilng trang40 o-(t)+2!Ht (s)u'(O,s)ds ~2LIIB2~u;(s~)lllu(s)Ivllu'(s)Ids+ IJI. Ma m)u2 (O,t)- m2 !o-(s~u; (O,s~ + lu; (O,s~}is+ o-(t) ~ o-(t)+2!Ht(s)u'(O,s)ds ~ 2 ! IIB2~u; (s~)llIu(s)Iv Ilu'(s )Ids + IJI ~2!IIB2~u;(s))p-(s)ds+ IJI, (2.109) I o-(t)+ mtU2(0,t)~ m2J o-(s)~u;(O,s~+lu;(O,s)l}is 0 I +]IB2~u;(s))10-(s)ds+IJI ==17(t). 0 ChuY tit(2.107),taco (2.110) (1+mt)u2(0,t)~o-(t)+ mtu2(0,t)~ 17(t). Ta suytit(2.106),(2.109)va(2.110)ding (2.111) o-(t)+ [ml + P2(1 + mJ] U2(0,t) ~ (1 + P2)17(t) I ~(1 + P2) ~m2~u;(0,s~+lu~(0,s))+IIB2~u;(s))Ip.(s)ds 0 +(1 + P2)PIo-(t) ( 1 I ( I ) Yz J I +(1+ P2) - Jk2(r)dr+2~ JIk'(rtdr Jo-(s)ds, PIO 0 0 VPt >0,VP2 >0. ChQn PI>0, P2>0saDcho ml+ P2(1 + m)) :2:~,(1 + P2)PI~ ~va d~t H9CvienTrim ThiHu~Chi Phwng trinhsongphi tuytnlienh~VIii mIlthili loanCauchychophudngtrinhviphdnthui'lng trang41 (2.112) R](t) =2(1 + fJ2)[m2~u;(0,s~+lu;(0,s~)+IIB2~u;(s~~1 + ~IIlkll~2(O.T)+2#WIIL2(O'T)]' Khi d6tu(2.111)va (2.112)tac6 (2.113) I IT(t) + U2(0,t) ::; fR1(sXlT(S)+U2(0,S))1s. 0 i.e. IT(t)+ U2(0,t)==0 nhob6 d~Gronwall. Dinh ly 2.1du'Qchungminhhoanta-t.. Chuthich2.2.Di~ukic$nk(O)=0chilakYthu~t,tac6th6boqua. Trongtru'onghQpriengcuaH vdi H(s)=hs,h>0,dinhly saudayla hc$ quacuadinhly 2.1. Dinh If 2.2.Gill sa (AI) - (AJ va (FI) - (F3)dung.Khi do VlJimJi T>0,hai loan (2.1)- (2.5)coit nhdtmQtnghi?mye'u(u,P)thoa(2.6), (2.7). Hdn mla, ne'u ~=1 trong(p;)vahamB2thoa (F4),khi do nghi?m nayladuynhdt. Dinhly 2.2chocungke'tquatrong[10]nhu'nggiathie't:"BJ kh6ng giam"dildungtrong[10]thlkh6ngdn thie'td day. Trongtru'onghQpriengvdi k(t)==0,ke'tquasaudayla hc$quacua dinhly 2.1. DinhIf 2.3.Gill sa (AJ, (Az),(A4)va (f;) - (p;)dung.Khi do,WlimJi T >0, hai loan (2.1)-(2.4) tlJdngungwYiP =g co it nhdtmQtnghi?m ye'uuthoa(1.4). H(Jc vienTrdnTh;Huf Chi Phuangtrinhsongph; tuytnlien heva; miltbili loanCauchychophuangtrinhviphiinthul1ng trang42 HCInmla nlu ~=1 trang(F;)vahamH vaB2 zJn iuf/tthoacaegid thilt (FJ va (A6).khidonghi~mnayia duynhai Chti thich2.3.Giangnhttchuthfch2.2,dinh192.3clingchocungket quanhtttrang[7]nhttnggiatiller"B( kh6nggiam"dii dungtrong[7] thikh6ngdn thiet(jday.