Luận văn Phương trình sóng tuyến tính với điều kiện biên chứa phương trình tích phân tuyến tính

13 CHu'ONG2. sTjTON Ti}I vA DUY NHAT NGHItM Trangchuangnay,chungtoi trlnhbaydinhly t6nt<;tiva duynha'tcua nghi<$my€u loancl;lcchobailoan:TimmQtc~pham(u(x,t),pet))thoa Utt- Uxx +Ku +AUt=F(x,t), 0<x<1,0<t<T, (2.1) Ux(0,t) =P(t), (2.2) ux(1,t)=-KJu(1,t), (2.3) u(x,O)=uo(x),Ut(x,O)=uJ(x), (2.4) pet)=get)+hu(O,t)- £k(t- s)u(O,s)ds, (2.5) trangd6 uo'uJ'F, g, k la cachams6chotruocthoacacdi~uki<$naod6mata sechirasau. ChungminhduQcdlja vao phuongphapGalerkin lien h<$voi cac danhgia lien nghi<$m,tu d6 rutra cacdayconhQitl;ly€u v~nghi<$mtrongcackhonggian hamthichhQpnhovaomQts6phepnhungcompact. Truoclientathanhl~pcacgiathi€t sau (HI) UoEH2, uJEH\ (H2) K ? 0,A ? 0, h? 0,KJ >0, (H3)F, F; EL\QT); QT=Qx(O,T), (H4) k E H2(0,T), (Hs) g E H2(0,T). 14 Dfnhly 2.1.Gidsa (HI) - (Hs) dung.Khi dovelimJi T >0, t6nt(Iiduynh[{tmQt nghi?myfu (u,P) cuahili toan(2.1)- (2.5)saDcho UELOO(0,T;H2), utELOO(O,T;HI), utIELOO(0,T;L2), u(O,.), u(I,.), P E wi,OO(0,T). ChUngminh.G6m nhi~ubuck B1ioc1.Xap xl Galerkin. Xet mQtday{Wj}la mQtcosdde'mduQctrongH2. Ta tlmnghi~mxffpXl cuabaitmin(2.1)- (2.5)duoid~ng m um(t)=I Cmj(t)Wj' j=i (2.6) trongdo Cmj(t) thoamanh~phuongtrlnhvi phansau (u~(t),Wj) +(umx(t),Wjx)+Kium(1,t)w/l) +Pm(t)w/O) +(Kum(t)+AU~(t),Wj) =(F(t), Wj), 1~j ~m, (2.7) Pm(t)=g(t)+hum(0,t)- 1k(t- s)Um(0, S)ds, (2.8) m Um(O)=UOm=IamjWj ~ Uom~nhtrongH2, j=i (2.9) m u~(O)=Uim=IfJmjWj ~ UI m~nhtrongHi. j=I H~(2.7)- (2.9)duQcvie'tl~iduoid~ng { m m m ~aij(c~j(t)+AC~j(t))+~bijcmj(t)=/;(t)+~dij 1k(t-s)cm/s)ds, Cmi(O)=ami'C~i(O)=fJmi' 1~i ~m, (2.10) 15 trongdo { f'(~ =(F(t),Wi )- g(t)W,(0), Gij=(Wi' W) ), dij=Wi(O)W)(0),bH_IW ,w )+Kl w.(l)w.(l) +Ka.. +hdH'l} \ IX JX I J l} l} (2.11) Khi dotacob6d~sail B6 d~2.1. Gidsit (HI)- (Hs) ill dung.V6'imQiT >0 effdink,khidoh~(2.10)- (2.11)coduynhatmQtnghi~mtren[0,T]. . ChUngminh. Bo quachi s6 m, tavie'tepap Pi tu'dngling thaycho emPamPPmi vakyhi~ul'.li e(t)=(e](t), e2(t),...,em(t)f ' el (t) =(e{(t), ei (t), ..., e~(t)t' ell (t) =(e{1(t), e~(t),..., e~(t)t' (2.12) A =(aij)'B =(bij)'D =(dij)' Ta vie'tl'.lih~(2.10)- (2.21)du'did'.lngvecto fA (ell(t)+Ael(t))+Be(t)=fer)+1k(t- s)De(s)ds, 1ceO)=a, el(0) =p. (2.13) . Chuyding A khadaova khongma'tinht6ngquattagiaSITdng A>O.Tich phan(2.13)tathudu'Qc e(t)=J(t)-1 e-W-r)drJ: A-IBe(T)dT +1e-A(t-r)drIdT Ik(T-s)De(s)ds, (2.14) vdi ](t)=a+ ~(l-e-At)p+ le-w-r)drIA-If(T)dT. (2.15) m Vdi m6i T> 0, ta d~tY =CO([O,T];IRm),Ilell=suple(t)II' le(t)l]=Llei(t)I. Ky O"'t",r i=1 m hi~uchufin cuamatr~nA =(aij) la !IAII=~~xLlaijl. LJ_m i=1 Chuyr~ngtaconco IAc(t)11~IIAlllc(t)II' Vc E Y, Vt E [O,T]. Bay giO,vdi m6i c =(cpc2,...,cm)E Y, tadinhnghlatmintii'H xacdinhbCJi Hc(t) =](t) -£ e-W-r)drrA-IBc(T)dT +£e-A(t-r)drrdT [k(T-S)A-IDc(s)ds, VtE[O,T]. Tu (2.17)tasuyradng H bi€n Y vaochinhno. Ta sechungminhr~ng,t6nt~inEN, saocho Hn ==H[Hn-I]:Y ~ Y, lamt)tanhx~co. Baygio,vdimQic,dEY, vdimQit E[O,T],taco Hc(t)-Hd(t) =£e-W-r)dr rdT[k(T-S)A-ID[c(s)-d(s)]ds -£ e-W-r)drrA-IB[c(T)-d(T)]dT. Ta sechungminh IHnc(t)- Hnd(t)11~ (2~)!D;t2nIIc-dll, Vn E N, trongdo DT = IIA-IDllllkIILI(O,T)+IIA-IBII. Th~tv~y,taco IHc(t)- Hd(t)11~ £e-W-r)dr dT[lk(T- s)IIIA-IDIIIc(s)-d(s)11ds +£e-W-r)drrIIA-IBlllc(T)-d(T)11dr. Dodotasuytu(2.20)r~ng IHc(t)- Hd(t)11~IIA-IDII£ e-W-r)drrdT[lk(B)ldBllc - dll +IIA-IBII£e-w-r)rdrllc- dll 16 (2.16) (2.17) (2.18) (2.19) (2.20) 17 ~~(1IA-1DllllkIILl(0,T)+IIA-IBII)t211e- dll 1 =- DTt211e- dll.2 (2.21) V~y(2.19)dungvoi n =1.Giil sit (2.19)dungvoi n ~1.Ta c6 IHn+le(t)- Hn+ld(t)11=IH[Hne](t) - H[Hnd](t)11 ~ £e-w-r)dr f: dr Ilk(r-s)IIIA-1DIIIHne(s)-Hnd(s)11 ds +1e-Jc(t-r)drEIIA-IBIIIHne(r) - Hnd(r)11 dr ~ le-w-r)dr Edr Ilk(r-s)IIIA-IDII(2~)!D;s2nlle-dllds +re-Jc(t-r)drf' II A-IB II~Dnr2nlle-dlldr1 Jo (2n)! T ~IIA-1DII(~:)!lie- dllllkilLI(O,T)1e-W-r) dr Er2ndr +IIA-1BII(~:)!lie- dll£e-W-r) dr rr2ndr Dn 2n+2 < L- II A-1Dllllkll t Ile-d il - (2n)! LI(O,TJ(2n+1)(2n+2) Dn t2n+2 +(2;)!IIA-IBII(2n+1)(2n+2)lle-dll = D;+I t2n+21Ie-dll. (2n +2)! (2.22) Do d6(2.19)dungvoi mQinEN. Til (2.19),tathudu<jc II Hne- Hnd ll ~~ D;T2nlie- dll~~ (.jD;T)2n lie- dll.(2n)! (2n)! (2.23) Do ~(JDTT)2n ~ 0 khi n ~ +00nen c6 no~1 sao cho 1 (JDTT )2110 <1.(2n)! (2no)! 18 V~y Brio:Y ~ Y hi anhx~co.Ta suyradingB comQtdi~mba'tdQngduynha't trong Y, nghla la h~(2.10)- (2.11)co mQtnghi~mduy nha'tum(t)tren do~n [0,T]. Bd8c2.DanhgialiennghifmI Thay(2.8)vao(2.7).Khi donhanphuongtrinhthuj cua(2.7)bdi C~j(t) va la'yt6ngtheoj, saudotichphantheobie"nthaigiantu0 de"nt, taduQc 8m(t)=8m(0)-2 £g(s)u~(O,s)ds +2£ u~(O,s)ds1k(S-T)Um(O,T)dT +2£(F(s),u~(s»)ds, (2.24) trongdo 8m(t) =llu~(t)112+llumx(t)112+Kllum(t)112 +K1u~(l,t)+hu~(O,t)+2-1£llu~(s)112ds. (2.25) sa d\lngcacghlthie"t(H4),(Hs)' la'ytichphantUngph~ntheobie"nthaigiantu 0 de"nt , taduQc 8m(t) =8m(0)+2g(0)uOm(0) -2g(t)um(0,t) +2£ g' (s)um(0,s)ds +2um(0,t)£k(t-T)Um(O,T)dT-2k(0)£u~(O,s)ds -2 £um(O,s)dslk'(s-T)um(O,T)dT +2£(F(s),u~(s»)ds. (2.26) Khi do,sad\lng(2.9),(2.25)taco Sm(0)+2Ig(O)uom(O)I~C1voimQim. (2.27) 19 sa dt,mgbfftd~ngthuc 2abO, (2.28) tathudu'<Jctu (2.26)ding Sm(t) <;;;,C1 +~g2(t)+/3u~(O,t)+~ Lli (s)12ds+/3£ u~(O,s)ds+/3u~(O,t)/3 /3 +~I £k(t-r)um(O,r)drl' +2Ik(O)1£u;(0,s)ds+~£IIF(s)II'ds +fJ rllu~(s)112ds+r[fJU;(O,S)+~Irk'(s- T)Um(O,T)dTI}s =C1 +~ [g2(t)+ Llg!(s)r ds+ L!IF(s)112ds] +2/3u~(0,t) +2(/3+Ik(O)!)£ u~(O,s)ds+/3£llu~(s)112ds +~11: k(t- T)Um(O,T)dTI' +~£dslrk'(s- T)Um(O,T)df (2.29) Chli Y ding, IIvxl12+K1v2(1)~Collvll:1,voimQi vEH1, (2.30) trongdo Co=.!.mill{I,K1}.3 M~tkhactu(2.25)vab6d~1.1,taco lum(0, 01 <;;;,Ilurn(Ollc(o) <;;;,/ilium (OIIHl <;;;,(/i / JCo ).JSrn (t) ==Co.JSm (t). (2.31) Tu(2.30)- (2.31)tacodanhgiasan 20 Sm(t)~c1+~[g2(t)+£lgl(S)12ds+£IIF(s)112ds]+2fJC~Sm(t) +2C~(fJ+lk(O)1)£Sm(s)ds+fJC~£Sm(s)ds 1 1 1 2 1 I s 1 2 +fJ £k(t-r)um(O,r)dr +fJ £ds lkl(s-r)um(O,r)dr . (2.32) M~tkhac,apdl.mgbfttd£ngthilcCauchy- Schwarz,tacodanhgiachohairich phancu6itrongv€ phaicua(2.32),nhusau Tichphanthilnhftt: 1 I 1 2 1 fJ £k(t-r)um(O,r)dr ~ fJ £e(B)dB£u~(O,r)dr ~(C~I fJ) £e(B)dB £Sm(r)dr ~(C~I fJ)llkll~2(O,T)£Sm(r)dr. (2.33) Tichphanthilhai: ~idsI L' k' (s - r)um(0,r)drl' ,; ~ilk'(17)1'dBi u~(0,r)dr -2 ~C;t £lkl(B)12dB£Sm(r)dr -2 tCot li e 11 2 ['Sm(r)dr. ~ fJ [2(O,T).10 (2.34) ChQnfJ saocho 0<2fJC~~1/2. Sa dl.mgcaedanhgiatrenvatu(2.30)- (2.34), taco Sm(t)~gl(t)+M}l)£Sm(r)dr, (2.35) 21 trongdo gl(t)=2CI+; (g2(t)+Ilg/II:2(0,T)+IIFII~2(QT))' -2 - 2 - 2 2C° II 11 2 M}I) =2CofJ+4CO(fJ+lk(O)I)+-(I+T) k HI' fJ (2.36) Tu HI(O,T)4CO([O,TJ) va giathie't(Hs), tasuyradingt6nt~ih~ngsf)M?) ChIph\lthuQcvaoT saocho IgI(t)I::;;M?),a.e,tE[O,T]. (2.37) Dodo s (t) <M(2) + M(1) rs (-r)d-rm - T T Jo m ' O::;;t::;;T. (2.38) Ap d\lngb6d~Gronwall,tathuduQc Sm(t)::;;M?) exp(tM}l))::;;MT VtE[O,T]. (2.39) Bu'oc3.Dcinhgiciliennghi~mII: £)~oham(2.7)theobie'nt taduQc (u:(t), wi) +(u~x(t),Wix)+Klu~(t)w/l) +P~(t)W/O) +(Ku~(t)+AU~(t),Wi) =(F;, Wi)' 1::;;j::;;m. (2.40) Nhanphuongtrlnhthli j cua(2.40)bdi c~(t), la'yt6ngtheoj, saudotich phantheobie'nthaigiantu0 de'nt , saukhis~pxe'pl~itaco Xm(t)=Xm(O)-2 Lg/(s)u~(O,s)ds +2£[k(O)um(O,S)+rk/(S-T)Um(O,T)dT}~(O,s)ds +2£(FI (s),u~(s))ds, (2.41) 22 trongdo Xm(t) =Ilu~(t)llz+llu~xIIz +Kllu~(t)llz +K1Iu~(l,t)lz+hlu~(O,t)lz+2A£llu~(T)lrdT. (2.42) Tichphantungphantheobie'nthaigiantu 0 de'nt cua(2.41),taco Xm(t)=Xm(O)+2g1(O)Ulm(0) -2gl(t)u~(0,t)+2£glf(s)u~(O,s)ds +2[ k(O)um(0,t) + £ kl (t - s)um(0,s)ds] u~(0, t) -2k(0)uOm(0)Ulm(0)-2 £[k(O)U~(O,T)+kl(O)um(O,T) t +re(T-s)um(O,s)ds]u~(O,T)dT+2f(F1(s),u~(s))ds0 =X m(0) +2g1(0)u1m(0) -2k(0)uom(O)Ulm(0) +kl(O)u;m(0) -kl (O)u~(0,t) -2g1(t)u~(0,t) +2k(0)um(0,t)u~(0,t) +2£ gll(s)u~(0,s)ds -2k(0) £IU~(O,T)lzdT +2u~(0,t)£kl(t-s)um(O,s)ds -2 £ u~(O,T)dTrkif (T- s)um(0,s)ds+2£(FI (s),u~(s))ds. (2.43) Bautien,tasuyratU(2.9),(2.42),(H4),(Hs) r~ng IXm(0) +2g1(0)u1m(0) - 2k(0)uom(O)Ulm(0)+kl (O)u;m(0)1 ~Cz+llu~(O)llz, (2.44) trongdo Cz chIphl;lthuQcV~lOUo'Upg, k, K, Kp h. Mi;Ukhactu(2.7)taco 23 "u~(O)IIZ-(uomxx,u~(O»)+(Kuom+AU1m'U~(0»)=(F(O),u~(O»). Dodo Ilu~(O)" ~ Iluomxxll+ Klluom 11+A11Ulm11+ IIF(O)/I. (2.45) Ta suyratIT(2.9)va(2.45)r~ng IIU~(O)II~ C3, (2.46) trongdo C3lah~ngs6chIphl;lthuQcvaouo'Ul'F, K, A. TIT (2.42),vab6d€ 1.1taco IIU~(t)IICO(Q)~J2llu~(t)tl ~CoJXm(I). (2.47) TIT(2.42),(2.44)va(2.46),tathuduQc X m(I) ~Cz +C; +Ikl(0)1u~(0,I) +21gl(I)u~(0,1)1+2Ik(0)um(0,I)u~(0,1)1 +2£lgll(S)U~(O,s)lds+2Ik(0)1Llu~(O,T)lzdT +2Iu~(0,t)1£lkl(I-S)Um(O,s)lds +2£IU~(O,T)ldTrlkll(T-S)Um(O,s)lds +2£I(F1(s),u~(s»)lds. (2.48) Sando,sadl;lngbfitd~ngthlic(2.31),(2.39),(2.47),tathuduQctIT(2.48),r~ng Xm(t) ~Cz +C; +Ikl(0)1C~MT +21gl(1)1CoJXm (I) +2Ik(0)IC~~MT.~Xm(l)+2CoLlgll(S)IJxm(s)ds +2Ik(0)lc~£Xm(T)dT+2C~~MT£lkl(B)ldB.JXm(T) 24 +2C~~MT tie(B)1 dB.!) Xm (-r)dT + £IIF(T)112dT+ ! Xm(T)dT. (2.49) Ta sad\lngbfttd~ngthuc(2.28)vdi (3=1/4,tu(2.49)taco Xm(t)~C2+ci +Ikl(0)1C~MT+4(lg!(t)1Cor +(1/4)Xm(t) +4e(0)C~MT+(1/4)Xm(t)+4C~£Ig!!(s)12ds +£ Xm(s)ds +2Ik(0)1C~£ Xm(T)dT+4C~MT(£Ik! (B)ldBr +(1/4)Xm(t) +C~MTt(!le(B)ldBr +£Xm(T)dT +£IIF(T)112dT+ £Xm(T)dT 3 -2 I I -4 ~4Xm(t) +C2+ci +COMTk/(O) +4CoMTe(0) +4C~ Il g// 1/22 +4C~MT "kll~I(OT) +C~MTT llk// 1121 +IIFII~2 (Q )L (O,T) , L (O,T) T +4C~Ig/ (tJr +(3+24C~/k(OJ/J[ Xm(rJdr. (2.50) Chli Y dingH1(0,T)4 CO ([O,T]),tu(2.50)vacacghlthi€t (H3)-(Hs), tasuyra dng Xm(t)~M?)+Mf4)£Xm(T)dT, VtE[O,T], (2.51) -2 trongdo Mf4)=12+8Co Ik(o)1 va M?) la h~ngs6chiph\lthuQcvaoT, k, g va F. Sad\lngb6d6Gronwall,tu(2.51)taco Xm(t)~Mf3)exp(tMf4))~M;,Vt E [O,T]. (2.52) 25 M~itkhac,tasuyratu(2.8),(2.25)va(2.39),r~ng IlprnIIWl,OO(o,T)~M?), 'v'te[O,T]. (2.53) Blioc 4: Quagiai ht;m Tu (2.25),(2.39),(2.42)va (2.53),tasuyra r~ngt6nt~imQtdayconcuaday {(urn,Prn)}v~nkyhi~ula {(urn,Prn)}'saocho Urn~U trongLoo(O,T;HI)ye'u*, (2.54) U~~ ul trong Loo(0,T; HI) ye'u*, (2.55) U: ~dl trongLoo(0,T;L2)ye'u*, (2.56) Urn(0,t)~ u(O,t)trongWI,oo(O,T)ye'u*, (2.57) Urn(1,t)~ u(1,t) trongwl,oo(0,T) ye'u* , (2.58) Prn(t)~ pet) trongWl,oo(0,T) ye'u*. (2.59) Do b6d6compactcuaLions( xemb6d6 1.8), tasuyratu(2.54),(2.54),(2.57) va(2.58)r~ng,t6nt~imQtdayconv~nkyhi~ula {urn}'saocho Urn~ U m~nhtrong L2(Qr ), (2.60) U~~ ul m~nhtrong L2(Qr ), (2.61) Urn(0,t)~ u(O,t) m~nhtrongCO([0,TD, (2.62) Urn(1,t) ~ u(1,t) m~nhtrongCO([0,TD. (2.63) Tu (2.8),(2.62),taco Prn(t)~ g(t)+hu(O,t)- 1k(t- s)u(O,s)ds==pet) m~nhtrongCO([0,TD . (2.64) Tu (2.59)va(2.64)tathuduQc 26 P(t)=P(t). (2.65) QuagiOih~ntrong(2.7),(2.9),tac6u thoamanphuongtrlnh (u//,v)+(ux,vx)+Klu(l)v(l) +P(t)v(O)+(Ku +Au/,v) =(F, v), '\IvEHI, U(O)=uo' d (0)=UI' (2.66) M~tkhac,tac6tu (H3)va(2.66)ding Uxx=ul/+Ku+Ad -FELOO(0,T;L2(o.)). Dod6 U E Loo(0, T; H2). Slf t6nt~inghi~mduQcchungminhho~mt§t. Btioc 5.Sl!duynhdtnghi~m Giii sa (up~),(U2'~)1ahainghi~my€u cuabaitoan(2.1)- (2.5)saocho Ui ELOO(0,T;H2), u; ELOO(O,T;HI), u;/ELOO(0,T;L2), Ui(0,.), Ui(1,.), 1>;E Wl,oo(0,T), i =1,2. (2.67) Khi d6 U =UI - U2' P =~- ~thoamanbaitoanbi€n phansau (UI/,v) +(ux,vx)+KIU(l)v(I)+P(t)v(O)+(Ku+Ad,v)=0, '\IvEH\ (2.68) U(O)=ul (0) =0, pet)=hu(O,t)- £k(t- s)u(O,s)ds. Ta l§y v=u/ trong(2.68),saud6tichphantheot , tac6 8(t)=21d(0,r)drIk(r-s)u(O,s)ds, (2.69) trongd6 27 5(t)=IIU'(t)II2 +Ilux(t)II2+hu2(0,t) +KIU2(I,t) +Kllu(t)112+2.-t£llu'(s)W ds. (2.70) Tichphantungphgntheot (jv~phaicua(2.69)ta du'<;Jc 5(t) =2u(0,t) £ k(t - s)u(O,s)ds -2k(0)£U2 (0,r)dr -2 £u(O,r)drrk'(r-s)u(O,s)ds. (2.71) M~tkhactu(2.30),(2.70)vab6d~1.1,tathudu'<;Jc lum(0,t)1~Ilum(t)llqo)~hilUm (t)IIHI~(h / .JCo)~5(t) ==Co~5(t). (2.72) Tu (2.71),(2.72)va(2.28)tasuyrading 15(t)1~2C~~5(t)£lk(t-s)I~5(s)ds+2Ik(0)IC~£5(r)dr +2C~£~5(r)drrlkl(r-s)I~5(s)ds. ~(1/2)5(t) +2C~£k2(B)dB£5(s)ds+2Ik(0)IC~£ 5(s)ds 2 ( r 2 ) 1/2 r +2CoJi 11kl(B)IdB 15(s)ds I I £~-5(t) +-mT 5(s)ds,2 2 (2.73) trongdo mT=4Ik(0)IC~+4C~Ilkll~2(OT ) +4c~JT lleI1 2 ., L (O,T) (2.74) Do do,tu(2.73),tasuyrading 5(t)~mT£5(s)ds VtE[O,T]. (2.75) Theob6d~Gronwalltaco 5 ==0 vadinhIf 2.1du'<;JcchUngminhhoantfft .