Luận án Phương pháp giải tích hàm trong phương trình phi tuyến

PHƯƠNG PHÁP GIẢI TÍCH HÀM TRONG PHƯƠNG TRÌNH PHI TUYẾN ĐẶNG ĐÌNH HẢI Trang nhan đề Mục lục Mở đầu Chương_1: Phương trình sóng tựa tuyến tính với số hạng tiêu hao phi tuyến. Chương_2: Phương trình vi phân mô tả chuyển động tuần hoàn của một vệ tinh quanh quỹ đạo elliptic của nó. Chương_3: Đặc điểm hình thành và xu hướng diễn thế. Chương_4: Hình thái và sinh trưởng. Chương_5: Cấu trúc quần thể. Tài liệu tham khảo

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3), . N \L 1 S~Cunx.t)unx.(qdxl:::; C2Ii\7untCt)l\lIvunCt)1IJl. 1. 1. $:" (1/4)iiVunCt)!l2 + C22 II \7unt ( t ) II2 (1.'37 ) Do (A.U va CUI f\f C un, un t )un Ct ) I dx'::; k ii un C t ) II2 51... + k Ii unCt)lil/untCt)!1 ~ C3kY/2 )\lVunCt) U2 + CkY/2 )\Jvun t Ct )112 (1.38) ~\ -' Ke den, -/ r\gCth/'Ct)lclx~yJigCt)!lIIYUn(t)II{(1/4)I\\]un(t)1I2 + "t(/lg(t)1\2J J2.. ( 1. 39 ) rl1 (1.36 )-(J.39); ta. BUY ra ~t(un(t),Unt(t» + (1/2 - 3kY/2)I\\1un(t)!\2:[. ¥ IIg(t)1\2 + + (Y + C22 + kY/2)HVunt(t)112 (1.40) Nhan(1.35) vii c, c= max(2?,1+2(o+ C22)/clha c?ng vao (1.40), ta dJoe <p~(t) + [(1/2)c(Cl-3kY) - (Y+ C22 + kY/2)}ilvunt(t) 2 + + (1/2)(1 - 3kY -eko)llvun(t)1\2 ~ (c¥/2C1 + Y )\(g(t)1!2 (1.41) trong d~ cf (t) = (c/2)(l1un..(t)n2+lIvun(t)U2)+ (un(t),Un t (t».n " .Lei.co (1/2)cC1 - ('(+ C22) - (k4/2)(1+3e) ~ (1/2)(C1-kY(1+3c»~ C1/4 , (V 2)/ ' /2 ( ) (1.42)ao e ~ 1 + 2 0+ C2 Cl va k {C1 ¥ 1+3c, va (1/2)(1 - 3KO - eke) ~ do k ~. 1/(2o(c+3». 1/4 (1.43) I ra co \(un(t),un, (t))I~RUvun(t)\I\\un (t)I!':::' ([f/2)(II1Jun(t)1!2+ 1\Unt (t)1!2)~ ~ , I tv do BUY ra '::C1(\\untCt)i!2+fiV,/'Ct)I\2)::::; cf~Ct) ~. azCilunt(t)!l2+ n YunCt)112)(l.Lf4) , trong do a1 = O/2)(e -I?), a2 = (1/2)(e +!y-) I Ket hop (1.41)-(1.44), ta dude ;:>, cp'(t) + bcp (t).:;:' Ce--?~n n trong d~ b =a;lmin(C//4,1/4) I voi t ~ r ( 1. 45) I'a can bo~de' Bau ~, ~, ,?, ,. ~ ' , " , Bo de 1.2. Gia BU ~1a mot ham kha vi tren IR thoa cb' ( t) + I b cf( t) ~ b e-~t0 1 I voi t ~ T <per)~ 02 trang db b. (i = 0,1,2) 113. cach5ng so' duang..'/1'> O.l ' I, 'rhi 4'(t) ~. '0 e -'1]0 t3 I voi t ~r I - " vdi !:lot ~o > 0, trong do b3 la m~~~~ngso chi tuy thu9c via T va b. (i= 0, 1t 2) .- l I " Chun~ ~inh '00 de 1.2. I-"? t Xet <j..:(t) = b~e 0, trong do 0< '}) < ~ /0 'I1T sac cha b3 ) max(b2e '0 t bl/(b 0 - 1) 0»' min(bat~) va b3 du?c ch?n , 'rhi -~t -'ht ~!:I (t) + b ,t,(t) = b~(b -'J )e 10 ) b e)o y 0' ) 0 0 1 "V(:r) = b~e-'~oT > <I'Cr) ~ Bat k = ~ - '/i .rhi V tEfR k I (t) + 0 k( t) ~ 0 vdi t ~ r 0 kef) ~ ° , I tli do BUY ra k(t) ~ ebo(f-t)k(T) ~ 0 t ~ T 3;) de' 1.2 chloc cht1~gIn\nh. Ap d~llg05'de'1.2 vB-a(1. 45)t ta dlloc 0 .. '1J t '-}o(t) ~ Ce 0 t t oj,- I'n I trong do C > 0, ')0) O. Vayt do (1. 44) , 1\ui\(t)\i2 +l!vull(t)i\2 ~- Ce-'?ot - I ' tu do suy ra, do (1.20) va (1.21), t t ~ r . () '12 1'~ ( o )\j 2/ 1 " " f(j .n ()j ,2, n () .,2 ) ./ -'i1t I\Uttli +\YUl; .::::- lffiln \Uttl+\1'lu tit ~Ce/o , n voi t ~ r. IDinh 1./ 1 'I2 ,t>:'c m:i.::.h , , .",., ~ Trong cae d~nh ly 1.3-1.5 dlioi day, ta gia su f chi tuy thuoc VaG lit' I I ~/ "", ) ' rruoc net, ta gia su (A.2 va (A.4) f: IR~IR 1i~n tuc th~a 2 (f(x)- f(y»(x-y) ~ - clx-YI , voi m?i x,YE-iR, trong db c > 0 , va ! f ( x ) I ::; 1'; Cl + i x 10( + 1 ) I I ~ I vdi [:lei x€iR, trong do H> 0,« 1a hang so trong (A.2). , 'ra co Dinh ly 1.3. Gia s~ (A.2),(A.4) du9c thoa . Gia s~ l' >0, UoE Hol, U , E-L2 va gf-L2(Q.n)' fhi bai toaD bien vdi dieu ki~n ban dau (1.1)-- - .I. ( I ..' ~ ~I ~ r. J ", ' ~I 1.3) co duy nhat mot nghiem yeti u ~ 0,1' voi cae tint chat sau da.r UELOO(O,TjH 1)0 UtE L 00 C 0 ,r jL 2) n LP ( 0 , l' j Viol, P ) , ., ., n ~ I "' Chung minh. Gia su (u )la cae xap xi Ga1erkin cJa ~1.1)-(1.3) n.l. nn( ) - '\ c , (t hv,u ~ - L- nK K 1 , f ".. \ trong do cae w. thoa he 'k ' 0- 1', (un tt (t),VL) + (vunCt),vw~) + 2::. CR(un t ),w k ) + (f(un t Ct),w,)K .\: 1 r x. ,x. K- ]. 1 = (g(t)'Wk) l~k~n n( ) 1 u 0 = u ---j u trong Hon 0 0 . n r,, ) -' ' "'yo ,,~.2 U t'V - l'ln---i>ul ~~Ol1o 1, ( 1. 46 ) ~ ~, I ~ I Nhan moi phu~ng trinh trong (1.46) vdi c~k(t), t5ng thee k va tieh phan, ta duge N t (1/2)(l\un t (t)i!2 +\I\1unCt)1(2) + I. S f~(un t )un dxds, 1 o x. x.tSL 1 1 = t t 1,/')\1'", ,2,' !,-r" 112, J« 'f n \ "n )fc' )d ( (~ ) n ( r »ct-, '-I'-'\'li" l II,.!. L.. I - 1.,U t /)L< t ,'o) S + j g,J ,U t V Sn on 0 0 " I (1. 47) Do (A.4) va bat dang thUG Young, t oS(f(unt),Unt)(S)dS 4 -CicUUnt(s)Ui,p + 1) Do (A..2), t 5 f{)(un t )un t axis ~ c2 IIUnt (s)j\ Pio x, x. ,pJ2.,. l l - (1.48) - c3 (1. 49) Ki~ h~p (1.47)-(1.49), ta d~~e t 11ur\(t)U2 +iJvun(t)1j2 + 1i!unt(s)lli ds < C (1.50)0 ,p -', I 2 ., I Nhan moi phuong trinh trong (1.46) vdi ten, (t), ton" theo k va tieh- ~ nK W phan, ta du~c t r 2. n ( ) ,,2 J sllu, t Sllds 0 't; 2 N un tCt) 2 UntCt) + t Iff Xi fiC-C)d<:.dx+t [f r(-c)dt:dx 1JlO .:.'1..0 t t = _t2(vunCt),'Vunt(t» + fs2/ivur\(S)U2dS + 2 Ss('Vun(s),V Unt(s»ds0 0 N t un t(s) t Unt(S) , + 2 I fsf J Xi f3 ('r.) dT..dxds + 2 f S r f r(i: )d1:.dxds + 1 0 .)'2..O' 0 .J'L 0 t + Js2(untt(s),g(S»dS0 (1.51) Do (1.50) va (A,4); -C4 UntCt) ~ f f rCddt:dx'::; e_(l +\I Unt(t)Hi ) .,'1..0 ) ,p (1. 52) Do (A. 2 ) N un (t) c". II unt (t)li 1 P - C 7~L r f xit p(1:)dL:dx.~c 8(1+ilunt Ct)Ii;. (1.53) 0 ,p 152.0 -11) rJ (1.51)-(1.53), ta SHY ra t 2 n 2 Js IJu tt(s)1\ ds < C0 (1.54) , va tunt bi ch~n trong H1(~T) Do (A.4) va (1.50), (1.55) f(unt) bi ch~n trong Lq(~T) Binh nghia A: W 1,p--} w-l,q nhu trong0 , n .. ," t Lq(A . r ",-1,:{ ) h.U t b~ cnan rang ,; v¥ (1. 56 ) (1.16), ta co (1.57) rJ cac dap~ gia (1.50),(1.55)-(1.57), ta tien nann nhu trong chu~g mint cua d~nh ly 1.1 de' nh~n du.oc su ton t~i cua mot nghi~m duy nha't cua (1.1)_(1.3), va a~nh 1y 1.3 du?c chu~g mint. ,I~' J. .,- ~I ~ ~ -, -, Ke tiep, ta xet s~ ton t~i, duy nhat va tiem c~n cua nghiem cua " , i_", -' (1.1)-(1.3) voi ~ co the BUY bien. n ." '> .La gla su (A.5) ~ thu~e 1d~ C1 trin 8, pea) =0 v~ , 0 ~ ~ (x)~ C(l +Le.(x)!) " I c(+l . . ( )\ (1 ' .C<+l )°1 x \ ~ I ~ x .:; C2 + Jx I , , , , vdi moi x f if{I trong do 0<~0, C, Cl va C2 13. cae hang so duong. (A.6) f thuoc lop Cl tren G, f(O) =a va f') O. vat p = c<.+2, q = p/(p-l). 'l'a co Binh ly 1.4. Gi;i s~ (A.5), (A.6) chloc thoa. Gi1. s~ N = 1, u E H ln H2, , \ 0 a U1 E H 1, gE-LQoa+jW 1,P)nL2 1 (1:'+;1..2). Th:L bai to~n (1.1)-(1.3) Cb0 0 oc .1 pc ""~:::'.f- Y'r~~ ,",--'r,.;;,." ,,~~ 1 tr;; 1"+ T O'.~;. ..' h r/" t d- T .,1.:.\ '.L~'-'.v ' ". "~._"" ~ !l ~ 1""0 " J. , C ~ln C.13 sail ay U to- L 00 OR+ j H 1£1 H2)0 UtE LOO(Lct+;Ho1)(1LPClR+j'No1,P) - , Eon nue'l rceu II g( t)1/ i =O(h( t)) khi t ~ 00,q J -(1+2/c<) I " -1Jt ,,' tron£!: do net) =t ~ neue>()O~ net) = e ,')/0, netiC<= a, thi 1\u t ( t ) ii2 + II u ( t) i\2 =0 (h (t)) khi t ~ 00x xx 0 i ,.)!,,~ ,I , . ( ..01 f. 1 tronf; do h (t) :.=l, ", neu c<) ° va h t) = e /0c,'T) ") 0, neu 0<= 0. . 0 - - 0 10 , ~", J ~ ~ Chung minh. Khong giam tong quat, ta gia su JL =(0,1). .' ~ ( ) , J,' .~., ~ ..1 A \Gla su w, 1a cac Ham rleng cua -A tren ti n.l. -t...>.W.=J\kwk 'K 0 k vat n tinCt) = L c , (t)vf, 1 nK K ! J , ~ trong do cac c k tnoan. CULtt (t),w~) + (un (t),w, ) + (~Cun t (t»,w k ) + (fCun t Ct»'W k).. x A, X X ,x n ( ~ ) t H 1 ~2 u iJ =u '- u rong- n.t1 on --? 0 ," 0 = (g(t),wk) 1 ~ k ~ n (1.58) n ( ~ ) T 1 u .I- U =ul -~ 111 trong H~ _D 0 '- ., ., Nh~n m6i phddng trinh trong (1.53) bdi c~k(t) v~tang theo k cho Cl/2)~t(I\Ul\(t)li2 + Hunx(t)1!2) + Jj)(unxtCt»unxtCt)dX + SL + ffCUn Ct»un (t)dx =fgCt)un t(t)dx ft t t ~ Do (A.5), C1. 59) f0( un .. ( t ) ) un t ( t ) dx ~, C11!un t ( t ) 111P J2. XV x ,p 1 , . ' \ Do bat dang thuG Young, ; ( 1.60) ,\I?;(t)U\.Ct)\dX [; (Cl/2)J!Unt(t)lIi. p ~L . Do CA.6), + d111gCt)/iq 1,q (1.61) )fCUntCt»untCt)dx ~ 0 SL L-,I h '" ' 1 .,., ) 'e ,,--, ) ,.,;,J.\.et up \. .):; -\-L.O.c , ta QuOC (1.62 ) t fll ur\ (s)1Ii ds0 ,p I ~ ',~ , trong do C khong tHY tnuoc theo n va t. -:- ,-' I ., ~h~n m6i phJ6ng trinh trong (1.58) vdi \kc~k(t) v~ t5ng thee k'cho < C (1.63) ';1. Cl/2)~ t (ilun ..(t;:lii? + IIUn Cdil?) + J0'CUn t CL))uY) t eL/axa x '" xx J2..' X xx + Jf'(un t (t»un t (t)ZdX =fg (t)un t (t)dxJL x J2.x x rJ (1.63) va (1.64), ta BUY ra (1.64) II un (t)!j2 + /Iun (t)1/2xt xx < c (1.65) I - ,,~ " trong do C khong tuy thuoe thee n va t. ~ ~" I ? ... Nhan moi phuong trinh t~ong (1.53) vdi t-e~k(t) va tong thee k eho t f s2j1un..t (S)U2dS < c('r)° '" 'riep t':1-enhu t~ong chung mint cua cU,nhif 1.1, ta BUY ra s 11tont?i c~a m~tnghi~mGUYnha't u c~a (1.1)-(1.3) tren [O,T}, vdi. mih 'r> 0. I ~I, ,-~i ~ - , + Do tint duy nhat, u co tne mu r?ng len toan Irt : d dt(Ut(t),v) + (ux(t),vx) +(~(uxt(t»,vx)+(f(ut(t»,v) = (g(t),v) ',.. h t) 0 ' . '\11,pVOl. n. ,. va mol. vE'" 0 ... ) Nay, gia su IIg(t)lI lq :{. ah(t),q khi t .~'r flat n ( .2 n 2 4>(t) =I!u t t)U + Uu (t)1In x xx J, (t) =IIlin t (t )112 + II un (t) H2n x rich phan (1.59) tren [t,t+4J eho t+4 Cl f II Unt(s)iIi ds~t ,P \ t+4 ~(t)- 4>(t+4)+ 2dl J Ii g(s)l! lq ds ~~(t) .n n t ,q n (1.66), , td do suy ra t+4 J il unxt ( t )Ij Z ::;;.t dZ '\fJ (t )2/pn (1.67) lieh phan (1.64) tren [t,t+4} eho t+4 I ')(("/n ) n 2_,.. , ( " ' ( " ' / --'~ ( 'L) j P \U, t u t aXelS ~ <PC) - 'f' bit) + (1 C:)1p t) + t JZ. x xx n n n t+4 +d 3 !lIg(s)ili dst ' q ==1V (t)n (1.68) ~, I ., Nh~n rubi phd6ng trinh trang (1.58) val ~kCnk(t) v~ tang thee k eha d d - t CUn (t),un ,(t» -II un t (t)i\2 + II un Ct)il2+ fA.'(un t )un t Un (t)dxx xt x xx t' x xx xx .JL f~I ( n ) n n (t)d+ ~ u t u ~u x.J2. x" x ra eo = (g un (t) dx) x x J2. C 1.69) !(un (t),un t (t» - (un (t+4),un t (t+4»)!~liun (t)l\ ilun t (t)/Ix x x x x x + + 1\un (t+4)\( II un t (t+4)iI ~ (1/2)( II un t (t)jJ2 + Hun (t)Jl2) +x x x xx + (1/2)(i\unx)t+4)112 + l1unxt(t+4)il2) ~ ruax<Pn(s) t ~"s{;t+4 (1. 70) Tieh ph~n (1.69) tren It,t+4] eha t+4 J II un (t)!i2ds ~, xxt max4>,(s) +n t ~s ~t+4 t+4 .2J Ii un t (s) II ds +t x t+4 + f ~~'(un.t )un- t Un \dxds + t ~ x xx xx t+4 f r Ifl(unt)un tUn Idxds t..Q x x + + t+4 f fig un Idxds t x x, ..12. (1.71) Do (1.67) v,~ (1.71), to~ tal t ELt,t+4] sao ehon 2/ t+4 ~'-,(t ) <:: (d.j2)'\j) (t) P + (1/4)max<2(s) +(1/4) if Ig un IdxdsIn n -. c:: n n x x t :::"s ~ t +4 t J2. t+4 t+4. n n n 4 II n n n+ (1/4) f lip-leu )u tu \dxds + (1/) !f'(U)U u Idxds t J-' xt xx xx t ..fL t xt x ~'L ( 1. '!2 ) c.) TJ (1.64) v~ (1.72), ta dU~c / t+4 max9.(s) ~ (d2/2)~ (t)2 P + (1/4)maxcp (s) +(1/4) f fig un jdxds n n n tJ2. x x t~s~t+4 t~s(t+4 t+4 t+4 + (1/4) f ~e:.'(unt )un tUn \dxds + (1/4) f f Ifl(un t )un t Un Idxds t ii, x xx xx t JL x x + 'ljJ (t) +n t+4 t+4 2 if fl(Un t )un t2dXdS + 2 fflg un t ldxds t.JL x tJ2. x x (1.73) J 'ra co f '£ n n . f In n 2IB (u' t )u' .u (t)!dx ~ (1/3c) r:\ (u ...)u (t) dxr x xxt xx ~ x~ xx 2 ~ + r ' n n 2 + 2C p, (u - t )u t (t) dx~ x xx -'1. (1.74) I ,I' / Do dinh 1;;- gia tr~ trung binh va bat dang thuG Schwarz, \ n .. [I 'nn' max ~(u .xt(t»I~ P (u xt)U xxt(t)j dx XfJl JL . S ' n n 2;h 'n ;h. " 'n ~ ( p, (u t )u . t ( t) dx) ( max ~(u '-( t ) ) ~ (1/2 L.)max f3 (u t ( t) )r x xx Xl, X JL xE.52. xe:R \ f In n 2 .+ (C/2) F (u xt)u xxt(t) dx JL (1.75) Do , ~ ~' (x) ~ C(1 +i l~ (x)! ), ta suy ra tu (1. 75) I I max} (unxt(t» ~ xE5l.. eel + maxl~eunxt(t»1 ) x<:S2. 'n' 2 S ' n n 2 ~ (1/2) max g (u ..\,t» + (e /2) p., (1.1' t )u t Ct) dx + er xv ,- x xx x Eo.J'l .;'1. I tu do BUY ra maxr'eun (t»:::;- xt XESL 2C + e2 f~'cunt Ct»un ...(t)2dx. x xx~.JL ( 1. 76) Do (1.65) va (1.76), ( 1/8e) f p' ( un x t ) u n ~ x: ( t ) 2 dx ~ S2.. (1/4)/1 un (t)U2 +xx f 'n n 2 + d4 ? (u xt)u xxt(t) dx 5L Do (1.65)va d~nh ly nh~ng Sobolev, ( 1. 77 ) flfl(Unt)unxtUnx(t)j dx ~ -"1. 0.5 Slunxt(t)unx(t)! o.x ->'1.. ~ (1/3) II un (t)1/2 +xx d6 it Unxt(t)/l2 (1.78) ...va 2 f f ' ( un (t» un t C t ) 2 dx ~t x ."-1.., , Do b~~ ding th~e Young, d7 11un (t)!i2xt ( 1. 79 ) )lgx(t)unxtCt)! dx ~ ..JL (l/p)/I Un t ( t)UPl + (l/q)ii g( t)ilq l,p ,q (1.80) . Do (1.65), ( I g Cdun Ct) I dx ~j x xJL C iI un ( t)1I \I g( t)1I 1 ~ (1/3) Ii un ( t )112 +xx , q xx + q d8i1g(t)1I1,q (1.31) ri (1.65),(1.66),(1.63),(1.73),(1.74),(1.77)-(1.81), ta suy,ra max(s) ~ d][(~ Ct)/2)2/p + '\j! (t)2/p + (J~~g(s)lI lq ds)2/p 1 , n, 7 n n t ,q 1,; .::; S ~ t; +<+ ~ dl0. (~ (t)/2 + ~ (t) +n n t+4 J II g(s)l\q ds)2/p t 1,q t+4 /= dl 0C -I} (t) - -~1'1(t+4) + (0. 3 + 20.1) J il g(s)/l'1 . o.s)2 pn.. t l,q trong db ~ (t) = <p (t) + ~ (t)n n n (1. 82) Do to~ntai [') 0 sao eho "" \' I <p(t) -<0 cP (t) 'lei moi t ) 0n 11 ta BUY ra '7n( t) ~ (l+S')<tn(t) V?y do (1.82), max°1r"(s) p/2 ~ t ::;s ~ t+4 t+4 dll (7j (t) - 'J (t+4) + (d3 + 2dl ) fil g(s)/l lQ ds)(1.83) n n t,q C?ng (1.59),(1.64)va (1.83) eho ~'(t) + ::'l3X~(s)p/2 + dll (~ (t+4) - '? (t» ~ d12h(t), t 1 Tn r. n n t ~ s { t+4 Dat k (t) = 'J (t) +n n t+4 all f ~,.,<s)dst U , 1 'I'hi k t hoa n k'(t) + dk (t)p/2 6 d12h(t), t ~ Tn n (1.84) A' , .;- ( 0", ) - ~"' ~.,£ieu c<= 0 'chi ta s uy ra tu. 1.:se+ va bo de 1.2 <? (t) :;: k (t) ~oCe-"70t,t). orn n (1.85) , voi mot 'h > O. . °'0 Nay ta giB. sQ ex) O. ra c~; ~-",~- .'? 1" A'.". A +..,'bO ae 1.3. Gla SU K la mot ham kna Vl tren I~ ~noa--"i 1 l et) , (t o ) 1+8 <: t -(l+l/e) ,'. t ..., mK + aoK ~ al VOl ~ ~,va k(r):;: a2 I ., I trong do ai(i=O,1,2) 18. cae hang so dtlonp;,e > O., Thi k(t) ~ a t -l/e3 I ,~-' -"", ~., ., troD;;; do a3 la mot han?'.,so chi tuy thuoe vaG e ,f ~ ai,i=O,1,2. Chtl~fi.lminh bo"de' 1. 3. / -1j,.:< I Xet k1(t) = a,t v trong do a3 Quae chon sac cho .1.. :J l-r-fJ ~ 1/0 a 0a 3 - d. 3/ e ) a1 ' a3 ~ a2T , Thi k 1'(t) + a k1Ct)1+e = (a a 1+&- a /e)t-C1+1/e)~ t -(1+1/G)0 0 3 3 "/ al kl (r) ~. a2 - ,? vat VI =k - kl. Thi w thoa w'Ct) + p(t)w(t) ~ 0 C1. 86) weT) ~ 0 / C ) ,~ ,~ ~ + trong do p t la mot ham lien tue tren ~ . 'I'tf (1.%), ta BUYra +-, "Q - dt(exp( ~pCs)ds)w(t»)~ 0c',J. / voi t ) T '/ , I tu do BUY ra t exp( ~ p(s)ds)w(t) ~. r I W (T) ~ 0 voi t ~ 'r I ," , , V,ay wet) .{; 0 vdi t ~ T va bo de 1.3 611,oechung minh. / ." - I Ap dung bo de 1. 3 vao (1.84)voi e =0</2, ta duae c? (t) ~ k (t) ~ Ct-2/ex vdi t ;;- Tn n (1.87) I Ket hop (1.85) va (1\87), ta BUY ra Ilu ..(t)1!2 + ii u (t)!l2 ~ Ch Ct)x~ xx 0 I -2/0<.' ' ( - "h t I ~ ~ I trong do h (t)= t neuO 0 neu0 0 /0 ct =O. I I Dinh 1y 1.4 duoc chung mint. Cu;i cling ta x~t tr~~ng h~p trong db rCUt) kh3ng nh~~thi~~., phai don di~u hay Lipschitz. Dinh ly 1.5. Gi~ sJ (A.5) duac th~a. Gi~ sJ N =1, u E H 1nH20 0 , p 1 T q(IR+ 1\11, q ) L 2 ( ,,+ L 2 ) G. .., "',ti l Ed ,gE.LJ' jn (\ 1 bj .1.3.SU0 0 oc 27 ( ) ~ lo 1,.. ~( ) , A.7 f thuoc lvp C tren IR, 1 a =a va d: I f'(x) ~ -C3\X\ vdi Ixi<' K trong d~a a. , , 'I J 'fhi ton tai mot tlang so 6) a saD eha V6i ? 2 00 0 IIu 1 \1- + 1\ U 11 + 511g( s)i1 { ds <: C ,x a,xx: a -,q / ,,' , thi bai to~n bien vdi dieu kien ban dati (1.1)-(1.3) co ~uyn$t , 'f' ~ + I , " ~ mot nsh~~::l'ye~ u ~ L::Z vc3i C3.C tinh crJ.it sau day u E Loo(IR+;H 1(\H2)0 1 0"0 (1':>+." l )n LP(,.,,+.T 1,p )Ut E 11' ,n li1. ,do o ~ I non nua, r.eu IIg(t}lIlq =a(h(d) ~<hi t~ 00,q ; ) -(1+2/c<)_' ' ( ) -l1t " trongdoh(t =t ~>a~ht =e J,~?)a,neuc<=O , thi 2 2 lIu t (t)!\ + lIu (t)1!x xx =O(h (t» kni t~c-:)° i -2/0< ~ I " -'!1 t trODi?', do h (t) =t neue<) a va h (t) =e /0 voi :lot", )0a - - a )0 , I lieu (X=O. , ..,") .., Chung minh. Khong giam tong qu~t, ta gia su ~ = (0,1). ~."):; ( n ) , I,' '; ,.'" '.,1 ( )( ' ) " Gla su u la cae xap Xl Ga1erKln cua bal ~oan 1.1 - ~.3 nnu trong (1.53). D~t 4' (t) =lIun t (t)1I2 + Ii un (t)ii2n 'x xx \ q; (t) =i1unt (t)!i2 + Hun (t)h2n x Nhan m6i phuong trinh trang (1.53) vdi c', (t) va to;g tieo k ehan,<;: ( ""'Co,f ) (,,(n ,n 1"- ) " ,(.~ ( ,n\on[,, ) ,o l/d+ ,t + )r')I.U--- t)U ' t 'l" llX J.L U -"/u' ,,\" c.xn 0 X x l" '-' ..>1. vL ~, ' Nhan moi phuong trinh trong (1.53) yoi \C','<t) vaK Dr. - (.J \ "-\o,n (-" ) -1X -)b";"t'v,-<, J2.. (1. 33) ... 1;ong thee k cha r Inn ( ) 2 r n n 2 (1/2) LI ( +:) ~ ,R (" /)1 1 t- i!y ~;~ f ' ( " ) 'u (t ) l-1v -- C,n' .,) r ,~- xt - xxt -, -"- . ~ t xt ' ~ ," Jl Jl.. = fg (t) un 1-( t) dxX x~ I Jl..., J Do bat d~ng thuc Young, (1.89) Jlg(t)un t (t)!dX + fIg (t)un t(t)!dx ~ (1/2)(C l - ~l Cs)ilUnt(t)iii JL JL x x p./ ,p + dlilg(t)iI.; -,:]. ~~ ." ~!',ay gl.a Su cp (0) +n 2 OOq < K fllg(t)111,qdt 2max(1,dl)0 " rhi q:>,(0) +n d1 Df g( t ) 1\ q dt < !:.2 0 l,q 2 ., T9. kiem <p.(t) <: K2n I vdi m?i t ~ 0 (1. 90) ~ .," J " That v~y, gia su trai lai thi ton t~i T > 0 sao cho ;;:> I " 2 cp(t) <: }:- vdi t~[O,T) V9.4 (T) =K .n n V~y I sup\ur\(t)! ~llunxt(t)il<K vdi te[o,T) x E.)1.. tJ do suy ra, do (A.?), Si'(U!\)U'\(t)dX 1- -(C3/p_1)l!u"\(t)lIr ~ ,p" va ~f'(U~t)unyt(t)2dX ~ -C3 JIUnt(t)lunxt(t)2dX ~ ,JL J'l do supJun (t)k(\u!1 ..(t)\ dx ~ lIu!1t (t)/l l~ J Xl" ,p x E S2. ..f1- C?ng (1.83) vao (1.89), ~a d~ac A,I(t) + q,1(t) +(Cl - -L 1C3)l1unt (t)\lPl ~ 2dl llg(t)ljql'Tn np- ,p ,q . n ( )" P-C~llu 1- t 11;)1" ,p (1.91) c:..'j DO '? \ti ~ '-1')(d, (1.91) keG tileDr:. n C ( '1') <.n. " 0'0 2( 4' (0) + d1 Jlig(t)!\ lq dt) < K2n 0 ,q Di~u jau thuin nay chu~gto (1.90). -rich ~h3.r.(1.83) tren [o,d, ta dudc t t ~~(:) -;- (Cl - ~ (;3) )\iur:...(s)iii !)ds ~~r,(O) ... 2d1 ~Hg(s)ii~ dsu - 1;'-"- 0 " ,~ .. -0 ,:{ , , tu do BUY ra 00 ( ii un (t )I!Pi dt < C oj t ,p (1.92) "',.,' :-', '~,', .'.. ( ~ 3) '. 2 II ( l,_,an ::'Ol pnuong tr1.nh "rang 1.:; vch t cnk t) ., tong thee k va ,I h ... ,tlC., :;::1a,. cno r .2" n ( ),\2d ('T1) jtlUtttl t<Cl 0 (1.93) vdi mJi l' ) O. ru' ciG danh gi~ (1.90),(1.92),(1.93), ta EUY ra nhu t:'ong chu'ng ., "',.<" l J ll ,1 t ~'" .,~.. ,.- ' (11)(13)t ~ mlnn cua ulnn ~Y . Su on t~l cua mo~ ngnlem cua . -. rea L-r,m l "".";.."",.;...,""' Ov, J 10... uv-"- 1 , . , I , Chun2 3ir:.h tint duy nhat. ,">. ' ~ 1, \'" ., Aul3. EU u,v a ,"3.l nghlem ~"/ ( )( ) ,,'1 \ yeu Qua 1.1 - 1.3 va gla su IV =U - v . , Thi Ntt - Vixx ;) 01 ) , (J 0/, \ ) , '" C . \--,J\U r -- .)\V I T.i. U ) 2'X' xt eX' xt t fCv) ::: 0 t (1.94) NCe) = IV (0) = 0t , , "' ) ~' ,.',. ~ hl'oil",r1 ,l.;:d VOl 'ii, VEt tlcn p:1a" c 0t - (1/2)(\\WtCt)/ .. /iwxCt)I\2) ~. }ilf(Ut(S» - f(VtCs»jjilWt(s)li dsv ::; t ;J.C l!iw..(s)j\-ds 0" " do B don (Jieu ~ay ao b& d~ Gronwall, Ilw(t)ll=iiw(t)II=O~ x en/. ,),.,") '.'~ "I' ,j' ., ,-' "j.? ~ln~ on ~~nn cua ngnl~m-QUOc cnung ~lnn nnu trong cnung mlnh cua ';,.", 1 I, 1 !,d_.c.. -;J ,'-t, ) , Dinh 1y 1.5 dQoc chung mint . "'"., ti-.-i,.". l' 7\,~- 'r>'~',- ';' \, rt",,' 1i. 1 5 t .,,;' .3' ;(-v..U "._-~., ~.-'-. .ioUe..u.n", ~..nn cu~ U_HI: J ., a t y r-~u.. - 00, ..' :- or':'.. 1;' :'. ~ir' 1 I 1 5 d' ~ ,.,~ . ,.,~ ~' d;;' ,..; t1..~;' ~,\..n1 l~e ~ _U:::ln eu.;:. ~-:..n -y -, uno :::l 1:..0.;'0 Cctn ~n !Ol.. lLJ..~ t 'Ie I 'circa ? - r"o r.u." ,. '1- - a. J:;' '"0' "'I ct o' A' P::Ari 2 . ,-, I , , I'rong ph~n nay, ta xet bai toaD (1.1)-(1,3) vdi ~ = 1 v~ f chi tuy t.huoc vao u..' \.. ? ;, I'a gi3. su I ? ( " "J ) (\ ,.' ,:::,- ,', .~- -r;, n 6(r. ) - ", . .., 6 ' .., ,.. >0,"-, ,j '"' ..'1 \..V - ~°.9 v ,,~~n 1-" v - v 'l a {i vI , 1 (A,g) 8 t~u3c 1d~ c- tr~n B, 8(0) = 0 v~ , ~ , ~ iX' , , "2 \ x! f:' P I. x) ~ ex 0,/1 + ixi ) ..J ,. '; ,., ,; :; -r"'r,:::" d ' ., 0 c .?, (' 1, ',.".:',., ",-' '11 "":::" .0- X E-I.., ,,~~u::o o!X~ , 2 "Ii.. '-'3 a ca~ na..g _0 G."-o.",, (.-\ , 10) , 1 f th~0C Id~ C- tren 12, f(O) = 0 \ ' , 1 D~t .9 =~ + 2, q = p/(p-1). Thi ta co , ) '., ':' 1 ' ~.::~~(, ') "' 1"' ) (:'-;>.;"- 1 ",1,,2, H I 'l?n !.I , ,0. 'JJ.,c' oL, ",,-..) -\.'"1.. \.J . "lEJ. "'u 1, - ,U t:: h II n , L' l e .---'- 0 () 0 "'J,"~",:l'J ) ,.2 ("."c2 ) ..': ,'. ,-" ("\1,1.") '.,' .~ gfJ..J.\V,.Io';';~ - ()J..J J,l;.u ,Tru. VOl. .l)v 1<:[130n::o, bal. toaD b1.er1v ' ,'. ..-' .-. -~' I ) 1 ~ ) I. ,~1 ~. ..- '",Val ale~ ~l~~ O&D Q3U \1.1 -\l'J co auy nnat mot n~nle~ yeti u I -' - tr:':;" [J,l'}',,"::ci de :ir,":-l e!c,t Rau day - GO . H 1 .,2 u E '-' (0,'1';11 11:1)0 ' ,. " (}() (01' . H 1 )(, f P (0 m. "j 1, p) Utl::..J '-"0"'" ,.L"o Q ( ) ,"1 I i-'\U . E 1'-1 (" !,,;,..-,q )1 x: \.J,- ,.I n.) ., ~"1 r' ~' '? " V~~,€L!G.!:1:7~~'E.':.l.?S.1. ~E-~a p ~~Q2:l~.£Y.~ ".l ( ) ' J , ,-'" ~.1 \ Gl.a Sl.t W k la cae ham rleng cua - A tren H .0..1, -1:.W, = A, Wk '0 ~ K vat .0. unCi) = L c ,(t)w, 1 D.K K I I ,'" - - iron;;: do cae c. inca he sau day ~ D.K . (un" ("C),w~) ... (6(11.0. (t»),w, ) +"C"C K X K,X (f3(un t (t»,w, ) + x K,X + U(un t (t»,w,) =(g(t),w,)K K n ( 0) . T. 1 ..2 u =u ---" u tron;;: Ii (\Ii cn 0 ~ 0 n (0) ~ ~ 1u t =u, ~ Ul ~rong Ii. J..n 0 l~k~n (1.95) " " j", '~ Tu cae gia thi9t cua d~nh ly, taillY ra he (1,95) co nghi~m tren " t , "."'-r- -Ln.., ]mo ..,"°"-,",,,v, - ,n 'K="? ~":,..'--.;:" i-", ""'.-~,-,uoc -, .;.1::1..'0'"-~'" t_e." n_file"" V; o',..,.,.-l'~'" -~...,.,- q1,"'~ t '" ""l'; So'j n - (0 1)"'"" ":0 b ~... "'~."::: '"'~~, ~ 0 ~ ...H...- , , -, ' ~ l;hirl !J.oi p}-""::o:-,;:trinh iron£: (1.95) voi >-, c', (t) vaton~ theo k cho- - - K nK - 0/2 )c?~(t) +,. j I ? J II .-' 3 (un ~)ur- t (t)-dx = (1/2) 6 (un )u.o.tUn (t)cdx: xv xx X x xx ~ ~ r ""l ( n ) .o. () 2, ~ u' u " t ex + "C xt J1.. tron;: db ( ~ un t ( t )dxJ -x x ..fL \ (1.95) ctJt) =nun r(t)1\2 + , x~ (' n n 2 )6 (u')u (t) dxx xx ..Q " n (-' n (, ) '1 ' -', ( 0 " )DO U ~\'-',s):= U t \J..,s ::: v, 'C3.3J.y ra 'Can tal x f \ ,-'- sao ehoG S n ( ) -u x 5 =u xt 5' Vay do c,;rlg tt..lie 'C::,ung binG lun (x d i Pxt ' ~. p f 'un (t)! p-ljun (t)idx ~- xt xxi ...>1.. ) ' p r<"1 2 1/2 ~ ' n ' ul /2 - ?..U( 1'.1.,(t)\u. .(t) dx) (\u ,(t)\~c.:d- VXESL-.;:~ x~ xx~ xt Jl. J1 - I tti do suy ra r::axlun ,( t) 1 ~ Xi: xE 5L d1(r,BI('.1n ,)UD ,(t )2~y ) 1/pJ xt XXi: ....-. ...'1 (1.97) " J Do (1.97) va oat dangthuG Young, (Ig un ..Ct)!dx { (1/4) f f>l(un ..)un ,.(t)2dX) X x~ ' x~ xx~ -,1. .Q + d?lIg(t)ii1- 1,1 (1. 93) fl ',<:"( n ) r. r: C . ) 2.-1./ I " ( ) ' 1 I n ( )1" n ( ) 1 ' 1 2 '.) u u y"U t I...x .::: sup!j s me.xu .. t 1\u . tX --~ xx x " xx TL \S j~ 1\un ( t H x E ..R.xx ('" " 2 ~' (1/2)\p (u~' ,)u" ,(t) dx. xt xxt J2.. + " ~ , ( i () ' II ,n () IC:'~ (13 sup 16 s lu xx t II ) 'i \S~Jiun (t)I I ' xx ~ (1/2)CSICun ,)UD ,(t)2dx J xt xxt SL + .11 (4:r:(:)) (1.99) t ,/ - ( , ) , (c -I t I tI c ) ' 1 ) J ,..."',,-,.. " .. - ~ "". r. - . ~ v." 2') ~0 - 1 ~ '1 S u.~ v 0> . ~ J... iSI~(t/C1)~ ra c6 2 \If I C UL th/ t Cd 1ax:~ 2 sap if' (s) \Iiun ..( dii 2~ F2(cp( t) )"x" x" n ...)'1.. IS\~I\Un t (t)1I (1.100)x \ troDg do F,,(d =2tsup lfl(s)\ c. ~h is I~t" /~].. r;"; ( ,- ) ..,,; [ " ,-T ,.; ,.:,~- ( " 3\ ( , ", ) '-- .~'~r.l.V., ,:L.",D _.90 ,,~~n v,~J L, ~...n6 1.j /- .L.1vv , ~c:.cuOv t., 'r? 0<0 cj: (t) + ~),3Cu""..)uD ... -c.xds ~ 4. (0) + 2d2 \ il g(s)!I lq ~dsn ~ ,." xx" D ",', '-I. U ~ U + ~ t fF(4-Js)):is ~ C + )F(4:: (s»)ds 0.. 0 n (1. 101) '-"nr- r~' ;;,r.. ) - ? ( ,- ) , ~ C '-\ ~~v..::, ,~O . \" - ~'l ~ -r l' 2 ~J T::~ c -~... " ( . . '- , ) " ,~, -.' , ,.-" ~o d5 Blha~l V~ ~~ngennop 0la SU 1 1a ~ot ~a~ duong, Khong . '~I' ?" ~., .~, ~ -ro,' .' glaD :ren L;""OO)' GlE:. SU 4='13.mot rlar:l llen -cue -cren Lv,'l'j:no.:1. c?(t)~ C + .. Lo f:'(4>(s) )ds 0 , " C>O l1'~trans GO . lnl -1 ) - <:p(t) ::; G (I:. , tfLO,r1) L. t ,.or- ,.<,~,'-\ (";'/ s) -l d- 'r -; n('l" CT11 ) n ( ",-,-- ) L - O " )."::,v.,."ul.<.-/=,,J.;.I. ~, 1='-"'" '2,uLoJ,V"..J = ,i.,v - "')...:',.,- ,.~ "'0"'" ',..; ";.' r-' ,..,;;"-" t '" n- -l3JV"'oL.o, .~-,-n.. '-'u'-" ..,oJ J.- H='-Yco Lon, X,=,,-, ~0"5 . A~ dung b6 de Biiari V3 Langenhop via (1.101), ta dl)c 4>r,(t)~ G-1Cr) (l.lO.~) 0{) c>i' . Y ,- -} t d' 0 0 on .ce' f ( ) -1"v¥lr:olr.~I~.,t6L'),l' rong 0 <1 J!S "'s C , I, , I\;han ::;02. Dhu.or.;o: t~inh tron a' (1.95) vc3i c', (d , 6n;: ::-,80:Z Vo;' t; c\. ~ - r.K ~ - " pie.n eho un (t) "', ? . X C1/2)i\°.l'-..Ct)I\- + I f 6(s)dsdx + 1.0 . J'L 0 t r f~(un ,)un ..dxds 0 .' Xt x" S'L. = ,.., u t. t (-, ,c:. -'O:lX, _",no n=(l/dnUl.ll + J J 'o-(s)asdx - f J rCu ,)U ..dxds + f) gu dxd:s~r. ,- n- t" 1"'\- t.J1..v v ~'- '-' ..;<.- (1.10\, ) Do CL 3) , 1t (;:;) ,0 x J f' 6(s) dsdx ~, 0 S"-0 (1.10:.. ) Do (A.9), \.0 t f r,3(uD..)u'! t dxds ? C2 ~'llun..(s)ii lP ds 01' A" x ol.o.P (1.10') ". ( 1 In? ) ." ,~;r' ,.!. '/,,- c: 1 'uO -._v- \~ ~~~n ~J nnu~5 uO..,O e/, ". t 2 j]lf(ul"...)uD..ldxQS ~ au. [ilun..(s)!! dsC .;'"1.. Lo \.0 '0 Lo , tE[C,TJ (1.10u) Do I 1ll, \ ~~iiu ,I\:<;a,tacoun,x vo on,xx u a \f for:';~(s)dsdx I ~ JIGCs)! is \j n toiN SL0 -a (1.107 ) 1:11'(1.103)-C1.107), ia s uy ra ~ t I r: C ) c:. r, t r: ( ) "D IU t t11 + ]IU t si!; ds <:0 ~,p c (1.103) I vOi tloi n, tE:[0, r} . Do b~t d~r:g chic Yeur:g v~(1.101),(1.102),(1,103) .. r J ' I '? r I -, -'"-C " )1 " , - oJ U ..... 0 ~1..eX xt 'axds:; ,. ,':f ' ' ( n )q/2 ' ( r. ) q/2 n J ,q \ \,b U t p U t U 'Id:ds,...," X ' X xxt v ...,1 ,-r';..(' n n 2 :~(' n D/:Y. ..,..,~::J.5l \, J ,° (u . ~ ) U - dxd S T \ J ~ ( U t ) '- dxds \ ~ c.-' x" xx" ,...", x - (j _n.. i..; .)(.. ,I Leu :x ) 0 va ~ ' i f[~'_pCun,) j'1r1X0:"' 0:.' rj :) c x' x:; "'-..~'" -'"' ->... t, ~ , , J f~( n ) "' 2. d < 'd .. U .. U ,dX S " v r:eu ex =0;) o x" xxc-,1.. V3Y ~ .... CQ(r.. ) ,. ,- t , ,..,-~'i( , )--- fJ\ U ' 01. C::3.D "-'.'6.u "(.rex' Xt (1.109) ..r~I, 1 , ::. ,.. ,." t (1 ""- ) a " t2 "C) ""'"~.=>.,--"'., ,,- "'".. ,- "'cr~ ~ a"'" .. ,..0' .,- H- "-'--~, IL.::1.n,",0- .t'"",,-on5t.~,_.. ,ona .'j./ v 1. cnk'" tv'~a ...Dee k ',.I. ,~ ,va t1.cn p:lar. cno r, 'I' " " .., u",(t) T , \,<:,,", C , ) ",-" 'TIL I ' J 'xc:. (J. C) ' d (,Ll fC D ) D () ' dJ t Ii U H' t Ii at + 1 P L. at; x =- ) t) U, U ,.. t ax t 0 v" "0 t ~~..;'1. 0 .n.. n T u (d ,',[( XI; ' (-','-' d ' c:. I:'" :0 L) j. (..) xc.:- O' .;'L ,"".l. (.2( f ,rn,n n C , ) , ) ,j " \ 0 \ U ) U ,- U r t t CtX Ct:-x AX ~ v ->1.. + n ;- 2(( '1 ) t "\ gu" .,.'"( d ax) dt.. v" li 5/- C1.110) ././ ~i do suy ra, do (1.102) va d~nh 1y nh~~g Sobo1ev r !'t21Iuntt(t)1\2dt< c0 (1.111) Cuoi cung, ~ ( n ) ,. ,- t _2 ( ,""\ ): u. Ol cnan ~ong L ~-t. . -T ( L 112) , I DUO: 3. }u~ ~ioi'han Do (1.102),(1.108),(1.109),(1.111) va (1.112), (un) co motday con '. ~ . , e n ) h~a ~~ van g?l 1a u , sao c~o n +- L oo (0 TI >41 __2) ~, * u ---;>u ~rong "";"0 nrt yeti n .I- T2 ( ) "', ~u :;>1.1...rono:1' 1J ::)~ ;':ianh va n.h trenj;x x "1'. - (1.113) (1.114) n , _::>0 ( ~'T1T_1 ) ~'*' -D (O .""l,D ) ~' (1 )u ,~u ~rong L u,qri yeti va trong 1J~ ,1.;.'. - yeti .115t t 0 - 0 . ?In ) y -~( ,", ) . ' h "'~"6~u .'>/1..trOn o""L 't- r1annva .n",:-enJ<...x --; "1' (1.116 ) 3(unxt)~suong LQ(0,r;W1'1) ye~ (1.117) ,( n ) .2 ( ) ,I :,u +-.-.1-7 trong ~." yeti... ... n i-u:u .. -? v t... 2 ( ) , - trone; L:t., ille.nhva h. h. tren..'t1 (1.118) (1.119) Do (1.119), r, U . ..-:;,ut t h.h. Viy ~2 dtn~ l~ h3i tu bi ch~n v2 (1.102), , ( n ) , ( ) .. T2 ( ) , ;: U t ---1 I Ut '"rong 1J ~r mann .,' . , tu co suy ~a ,/=f(u) t Do (1.102) v~ (1.114), 6(un ) ."" 6\u ) tron o"" 12(1,) !1B.n:"1x --} x ~1' , ~... I -en GO suy ra x = 6(u )x Ihay t a c),tlnf'; miLD s =~(uxt) ~ " - .")"::'" ,..- , ,."D:> no.:::>..::;g~a=, t r(R(un -) - ~(v) ,un - - v)as~a~j r x ~ x ~v V'IE LP(Qr) (1.120) 1'3. c6 t ,'~ t :) J (B(un -) ,u'- ..)(s)ds =- ~ C~:-~Cunt ) ,Unt )(s):is- I x~ x" ,,"\ eX XU v !lei tu 'Ie' t "t t - ((-~'~(s),\l..Cs»ds= fCs(s),u t Cs»ds C~ eX t,. 0 ){ do (1.2.15) va (1.117). Ch:::>!l~O-:> tron§; (1.1Z0), t::;, dlloc Chaf"v = t:. ..Xv t (~-$(v),u - - v)(s)ds ~ o 'V'IE LP(~,)) , , x" .I.0 -.A'fi,>")O,WEi.PC~r,) va. cho).~O, ta d~oc.I. " ,'/t.. ~' ) )( ) ,, \)- . \" , ,w s ~sr: "'Xl. 1,.' = o VWELP(~.,,)1 'lav. " \ f-= b (U ) ) I xt Cho n ..-:::.>°0l:.O'O!l§;(1.9.5), t3. d:J.,oc ~-(u~(t),v) + (6(u (~»,v ) + (~(u .L(t»,v ) + (f(U t (t»,v) =c.'. " x x x" x ::: (~-(t),v) ,',. ' - L ..,} " 0', . .,..l,pV0l:-..h.tE-O,.L vaV'l::lOlVC'/ -.0- 1'a~~ie= u(O) = u0 va u.(G) = ul'" /. Do (1.113) va (1.11), n (O) f O) t ., 1 ~,u --7 u\' rang no yeti Vs.y u(e) =u0 Do (1.115), (ur:..(~),w )~(u,(t),w,) tror:g LP(O,T) ye~~ ~ ~ ~ Do (1.95), d r, d . 'J ~,- d~(u ,.(t),w.)~ -dt-(u,.(t),w,) trong lJ>(o,r) yetiv v ~ v v K v;'y:-" u (C) = u1t ::<'-"C I. "",,'~~ "'~"'n "~'~h duv roh:::'+-~.,~ "T. --"".,",(', '--.~".. ...~.l v .'."'-v -' -:.12- - .o~ ~I ~ Gl~ SU U ,u 1~ t~l nsnle~ yeti cua (1.1)-(1.3). Do B(ui. ..)f ~~(O,?;~l,q), ta suy ra , x~ ; - 1u- ..(t)e c(~) ,tE[O,r , i =1,2x" Do Ult(C,t) = Uit(l,t) = 0, ta suy ra ti cong th~c trung tint 0 '1 \1 '" 0 c;{. - {' 1. (.. C\ +- <' ci l l..1:=:.X,Iu t ",)1 ) '- ",,-(Iu ,(t)!u t (t»]::1X- x . c,X xt X XE:.5L Jl . ( j ' i i I ' jot i i q~ «~+1)/C 2) \ ,E;(u .)u ,(:) dx ~ «C\+l)/Cz)( I~(u t )u t (t)\ dx...,' xt xxt X xx ~ ~ + I ) V~y ::~x(\uiy,-(t)i)E Ll(O,1:), i = 1,2..\" x Co~'? I .J - ~ D~tw =u~ - u-. Thi w thoa ;:' ( ~ ( 1 ) ( 2 » ; (R( 1 ) R( 2 » ( 1 ) ~ 2 )w - -=:--0 u - (5 u -,..-,- r u - r u + f U -I (u =ott c X X X eX' xt xt t t w(0) =w (0) =o t ( 1.121) ./v ;, I ; I Lay tic~ vo huong (1.121) voi Wt va tich phan eha , 2 t 1 2 t 1 .., ~iI';;..( t) II + ~ (G(u )-<J(u ) ,w ..)( s) as + )(f (u t ) -f (u c:t ) ,1ft (s) ds ~0c:" .., x x x~ ~v v (1. 122) do ? d3:l dieu. J'-lr~gecr.g th~c t!'ung bir:h v:i tieh phan t~g pha~'1, ta dude ~ ( I 1 ( » 12 ( ) ' ( » .) '.J\ U x s - 6 \ U X S ), Wx t S 0.5 I.J = = t 1 , 1 2\ \ W (5)'.'1 ..(5)( }C"(eu (5) + (l-e)u (s»de)dxds"," x x-, .., x x v ,01... u 2 ~, 1 2 = (1/2)(v; (t), J 15(8u (5) + (l-e)u (s»de)- x 0 x x t, 1 II 1 2 1 2 2- (1/2) f J ( f 6 (Gu + (l-S)u . )(eu t + (l-e)u y..h d$dxds 0 -~ x x X --I. XJl. u Vay f (6 ( u1 (s) - 6(l (s», 'tVt (5» as ~ (::;1/2) It w (t) 112-o' x x x x ':;\ ;;> 1 I 2.C \ j'w -(\u- + \u \)dxds.., x xt xt iJ SL ( 1.123 ) -n .c2i. (;: t i(:(u\.(s)- fcl tCs»,WtCS»\dS :f:,' C SIiWt Cs)iI2dSc " 0 (1.124) I ~--~,- '_.~_. ( 11,):::» (, 12:- ) t - r1,~'>',.,'--:"'-"".:- :: ,-.-'i', ::.c..,u,-, II-,'; (:H2 + Ilw (t)1I2 ~ C ~\(S)(i\w..(s)1I2 + Ilw (5)i\2)ds t x 0." x , 1 / 2 Itro::;:: GO '0(5) = r:.:ax(\U~t (S)1 + \U ,(~;) ) + 1~ - - x xt xESl. Jo f ELl (-:), 1'), t 3. guy ra /7 \\WtCt)\\ =Hwx(t)\i = 0 .. Co-L" l' T, ...~u,- J I I Dinh 1y 1.6 dloe chung minh. - .., .., Dinh ly 1.7. Gi~ s~ (A.8)-(A.1O) dtiac thoa va f'~ c > O.Gia su ~ =1, ,- 1 _2 -- 1, ~J r';;:+ '" 1, q)f)L 2 ( ...,+ - 2 ) "',: t -' .- UoE.t:o n.t: , '-'16:10 ~ gElJ'~L. jio ' loc 1.'- jlJ . 'l::'~(:)!'1__ta~mot. [' , c:A '.)~ ~ -" ',-, - ;~O Qu~n5 ' =_0 CuO vd- 2 f I 2 OQ \'1\U l II + 6 (u )u dx + III g( t) /Iq dt <: (;,x o,X o,xx 0 l,qSL ,'. / ..A '".-',." .- ( )( ) I ~'- .,A oa~ toar. D~en VOl GleU Klen Dan dati 1.1 - 1.3 co Guy nhat mot n~nle? , " yeti u tren G~ t:-"oa -1. ' , ?- "-V ('~'" -. 1. -.- )UeL I.. ill f).::0 -,0 L' 1. ,r J (i- T .:; 1.)r\ T P ( ') +0 P ~,p )U~E...J < ,", :IJ.J I.:; ,.1'- 0 0 Q ( ) ~ J ( ' + 1 .., r' " , t:.. 1.1' I~ 0',/-' "1), ""xt '- loe'" 'f1,-\r y,': a ,.,;:.,-,,", ,c.. . ""vel II - ( .. \ '1'1' =. ... II 1~ ,1 ,', " a ' , :- VOl CiO:;_~) '. :;,~::. =O(e-~t) khi t ---4 0-0 ? iiu ..(t):i~ .;. Xi" r , A '1, , \ 6 (u (t»u (t)C:dx =aCe-lot) khi t~ 00; x xx J1.. vdi mot "') 0) o. \ . "A...,'. ( n ) ,'~' ~r' .. ' ( )( ) , CnuTIg5lnn. ~la~ U la cae xap Xl 0alerKln cua 1.1 - 1.3 nhu Ll oneS (1. ~<)) . '" , , .., ,I lihs.n r:oi ph15ng trir,h tror:g (1.9.5) vdi c', (t), tong thee k va tichr,,-:: A rhEin eho u" (t) I li A ) 1' n C ' )\ ',:::.. \ ' \' X 6 ( ) - d\, c:.!lU..:' \ + ,5 C1S X +... . - J1.G t f S ~(un .. )un t dxdSa x... XJ1- -;. v ur, (a) t -;. rr fCun,Ju""dxd5 =(1/2)i!u l rI\2 + f ) X 6Cs)dsdx.;. )(guntd~ds O' -,2... ~ ~ -, J2. a 0 ..,1. (1. 125 ) Do bat d3ng thuG Young va (A.9), t t 00 f Jlgun t ldxas~' 2~ S f~(un, t )un t axes + dl J Ii gCs)lI~ ds 0 0 x x -0 - t qJl. .fL (1.126) Do (1.125),(1.126) va (A.8)-(A.9), ta suy ra 'Uun(t)1!2 +t t fIIunt ( s) IIi ds <0 ,p c J ~ ~ - " trong do C khong tuy thu~c vao n va t. ~, I .., I~han r:oi :p.";uong trinh trong (1. 95) vch /\ c" (t) V3. tong thee k chaK I:K (1/2)~ t (Hun t (t)i!2 + f6'(un (t»un (t/dx) + f~'(un t )un t (t)2axCl x j x xx . x xx JL SL r 2 ) ". ? J . + fl(un t )un t (t) dx = (1/2) 6 (un )un tUn (t)-dx + g un t Ct)dx, x x x xx x x .J1.. J1. JL ( 1.127) .-J " I .., l~h3.n Il'loi Dhlion9: t rinh trong (1. 95) yoi >-kc ,Ct). ve. to::::.;;:thee k cho~ ~ - TIK - d ( n ( ) n ( » r ' ( n ( ) n 2 "::t U t"u t t + 6 u t)u (t)dxQ x x X xx JL = H ul: . (t) Ii2 Xi; rflCUn t )un .un . (t)dxJ Xi; X .::1. + (g un (t)dx - (B' (un. )Ull ..un (t):ixJ x x J' xi; xx~ xx Jl. 5t \ C1.123) ,I " I ~ / Do Gong thUG trung binh va bat dang thUG Schwarz, r max_IuJxtCt)\p ~ x E J?.. , p \ lun t (t)j p-1jull t (t)\ ax; x xx Jl. ~ D ( rIun. C t ) run t ( t )2dx )1/2 ( r\un t ( t ) \Pdx)1/2c) xc xx J x Jl J1. , I tV do BUY ra max lun t (t)1 ~ d2( (~' (un ;.)Ull . (t)2ax)1/px Jr x" xx;; x L.r~ 51. (1.129) -"" I ~I Do bat d~ng thuG so cap, , , r 1 2 (ip,(1Jn )un . i/i (t)\ dx {: (C"/gc?) \ $ (un i )un (t) dx i 'J xc XXi:, xx -'- .J J I xc, xx A ~ + . \ ' n n 2+ Ci~ B('.). ..>u .(t) dx;; . i x" :oct Jl. Jo CL 9) , (1. 130) f -1 f r. ) r: ( )2, " c " 11 ;. ( " ) ' 11 2 ,., f'l r: 'ID. n ( )2 ( ):> \u t t;. t G..o'{'::::: ~ t;. t + "' 3 U .. 11 t dx 1.131J x xx ;; xx x~ xx ~ ~ . ~,~~ I , Do ~~t d~r:~ t~uc Yot:.ngva (1.129), (c,/3) f lu!: ..\ ur. (t)2dx ~ (1/3)( f 6'(un )un (t)2dx)p/2- x~ xx x xx J1- ~ + -;. r ' ? Cli, \ 3 en ..)ur..o...( t ) - dx.,. . I x~ x~ .,'1. . t;;'.. . .:<) fI " 3 ' ( n ) :l n ( ) Id <. ( 1/ ~) r Fe' ( ~ ) n r " )2~. u.u.u t x ...; vU U \,tu.X I xt xxt xx x xx R ~ + -;. (1/3)(Co'Cur..)ur. (t)2dx)p/2 + cL(0'(un t )un.o (t)2dXj x xx ,) X AX~ ~ ~ II ~~t >11=s:.;~16 (8)\ , i'i2 = sup\f'(s)j \s I~1 ( 1.132 ) isl~ 1 "n; ""' I , "~~ ;,.,- ':'},.01' v., <;"l_l~ ,,~ co ao :J""L- d.::;..::, ~.!UC ~ou.no r r'" "::1 /u" '.1" (tf:: !dx <xt xx .... j2. \ , . ? l>:, \ur. t (t)\ II ur. (t;)II-- x ~ 0-::> xx , va -,1. \" n n ,2. Q+ d, ( 6 (u )u (t) ax)-::> ~ x xx '.oft (1.133):; (1/4) (3'C'.lny )ur t(t/dX,\' ..t X ~ I';) (\"~n un (t)i dx ~ ;'12 1\un ..(t)iiliun ('c.)!i- j x: x x~ xx Jl ~ ~ /" r,., ;;>" :) ...1 - \'1./) (5eu )u (t) dxx xx J1. + d71i Unxt(t)1\2 (1.134) 'tC: I ~ Do be,t dang tnuc Young, rig un t (t)\ dx ~ Iun t (t)\ 1\g(t)lI lJ x x x 00 ,q SL { (1/4))~'(unxt)unxxt(t)2dX -J1.. + d8 Iig(t)l\q1,q (1.135) (\ g un (t) \dx <Itg( t) II I II un (t) i1J x x ,q xx Jl ~ (1/3)«( 6' (un )un (t)2dx)p/2 +) x xx: JI.. d9I1g(t)1!i,q (1.136 ) Chon k ;> ftlax(1/Cl,1,2d5,(1+d7)/C) I ~han (1.127) voik'va c$ng vao (1.128), ta daoe d~..( 2~(I\un t (tHl2 + (6' (un )un (t)2dx) + (un (t) ,un- t (t» +'" x j x xx x x .fL + «3k/4) - d_)((2,' (un t )un t (t)2dX + (kc - l)i\un t (t)\l2) J I' x xx x SL + + (7/3) ( 6' (un) un ( t )2dx ~J x xx SL f " n n n 2 1(k/2) J 6" (u )u tu ( t) dx +x x xx Jl. \+ flf'(Unt)unxtunx(t)1 dx + (1/4)(fo'(unx)unxxCt)2dX)P/2 ~ ~ + + (k<13 + u9)JigCt)!ii,q (1.137) -Bat k..n ,\2 f ' n ) n 2 n n 4' (t) = 2-( IIu t ( t ) I + Q (u u ( t) dx) + CU ( t ) ,u t ( t) )Ii X X xx x x SL Do j(un (t),un t (t»\ {iiur. Ct)lll\un .t (t)!lx x x x. ~ ~ ' n n 2 (1/2Cl) 0 (u )u (t) dx. x xx JL + (1/2)l\uIl (t)I\2xt -r.J ta BUY ra d l ,,(l!uD ..(t)i!2 + f 6'(Un )un (t)2dx).::; <? (t) ~ dll (lIuD t (t)ji2+v X~ X XX n X S2.. (0' (uD);;n (t)2dx)J X XX .J'l. (1.133) J trong ~o dlO =(1/2)(~- maxCl,1/C1», dll =(1/2)(k+ tlax{1,1/C1» ~"~:. ',.-." ( ~' ) ., '2 ~ ,u1.a su Y 13 n;::rae:n. QUanE; duy nhat cua Phuun;;: tr1.nn0 - 0 - - 1-1kL Y - =1/3 0 0 "), I 1'a ki8:!l rang neu 00 9 (0) + (;~d3 + do) \llg(t)l!o l. dt < S'n / o' ,q 0 (1.133) I 0 ' , rI \ .' (1 ) ' C ) .h " trong ~O Go =QIOm1.n-, 0' 1 ' ~ 1. 9(t)< [;n 0 , vdi r:Ji nfil\, -: ~ o. ""' t ~ ,") -::(01' 1 ' ":-'~' t ' T 0 h'It.a vay, g:.a 5U tra1. a1., tn.1. ton a1.' "/ saa c. 0. 0 n 9 (s) < r"n -0 , , \' vdi sE[O,r ) va 9 (1' ) = 0n n n 0 Do (1.133), Bur- ,Cs)1\2 + (o'Cun )un Cs)2dx~ <P (s)/d lO ~ rninCl,y,C l ),s~IO,I'}xt J x xx non ..!(. Do iu;'t Cs)[ ::; IjuL t (s)11~ 1C~J X S f [0 ,r -1n ta BUY ra t~ (1.13~) ("" I r ~ ~ :1.'1( )u" ..-' (-)1-'-- ./ I~ \u t ' xt" x:" ~A:::; " ('In D ()j '~~ 1'.1 tu s ax ~ J X x .)1-- II "D 2 ~ (1/3)\6(u"")u (s) dxJ x xx -'"!.. + d7 1i UD (s)1\2xt ' sELo,Tr) (1.140) Do [un (5)\ ~HUD (s)I\~ (l/c l f«(o'(uD )un (s)2dx)1/2~ 1x 0-J xx J X xx->'1.. Lf-Lf- " va ., ( r \ ' ( n ) n () 2 d) q< ' d yq-l( ' ( TJ. ) n () 2d}:(1.- <5 u u s x -.; K- 0 \ () u u s x0 . x xx 0 0 , x xx ~ ~ / ~ , , 2 (1/8) \6' (un )un (s) d.x. x xx J'.:.. sf[o,r} n .. I ta suy ra t,.l (1.133) voi Sf'[O,'rnJ \r " ~ n n 2'~ \r5 (u" )u ..U 1 ( 0) Idx, x Xv xx Jl f", n 2'1~ H1k lu"..u (s) dxxv xx: 31. ~ (F../4)\,S'Cur: t )un . (s)2dx + (1/8) ( 6' (un )un (s)2dxj , x xxt j X xx .-"1 .JL 1'a eo (1.141) r t >c n ;) p/2. (, n n ? (1/4)( \6 (u",)u ,(s)-dx) ~ (1/4»)6 (u )u (s)-dx. j( xx 'x xx .J1.. ..;2. do D ~. 2 va \'51Cun) un (s )2dx ~ 1- J X xx C1.142) jl, K ...' " ( 1" , ) ( 1 1',, ) ( 1 l' 2 ) t ',~ '.e: LJP _.~)(, . ~v- ~.-~ , a Q~OC 9'(t) + «1:/2)- d_) fp'(un t )un ..(t)2dx + (ke - 1-d7)!!un (t)\i2r, :; x xx... . xt .Q + (1/4) \O'CUXl, )un (t)2dx:( (I:d g + d9)ilg(t)!(ql. x xx ,~.:1.. (1.143) I - v3i tfLo,r 1\Xl , n;',,~ ~'-~~, (1 11.- ) - ;;.,., [ () m I" ,.J.~v.l ...l;;<l. ~ ,) ...r_.l u,l J CliOn 0-0 6,.,U,) { ep(O) + (kd. 3 + d9 ) flig(t)lI lq dt <' th.. no' q 0 :;3.u -::,u~1""r.?.:I c'c.lI}~lg t;) ,1:>(t ,J <' r' , r ..::) , voi GlOi t '>O. , ., 1 l~?.Y 2~:"35ll ') II \L n- + ~,x r , 2\6 (U )u dx ~L.. 0 ,X 0 ,xx 0-:) + flJg(t)ii lq dt < ['0 ,1 trong GO f = ~/r:1:;x(dll ,1-:d8+ d9) he:.'/ V3.:l ? Iiu1 n- + .dl,X ( I 2 00 Jv (u )u dx + fHg(t)lii ds < [ -':.. on,x on,xx 0 ' q I vc3i n ~, I;. Do (1.133), OD 9,.,<0) + (kd~ + d) 511 g(t)i! 11 at < m3.xCd11,kd~ + d9)S'=-. ~ 7 0 'q - 6 [' 0 , " ~ I t11 GO suy r3. 6 C:;) <: S'" vai ::loi Il:?,t;, t 1- 0'n 0 . Vay do (1.133), jilin (t)i!2 + \6(UIl )un (t)2dx':; cp(d/d lO <' S'/d 10x:; .n. x xx n 0 , v3i r:oi n ~:;, 1;'1 0 '-';;;~' n;r',.',.,' +~ r,- ...,i- ""1'1b 2 d ' , 1:' 1 ,- "'" ~ ;.~'.J._~" -,~ n..u v.o.,:, c..U!l", = CU3. ~r,n J .0, ta s y r~ su ...on " I ,'1 ~ "~ ~, '~ t t ~ r.o "" J a' '- aDt~l. cua :~ot ng::::.~:r.yeu auy nna u ren ,I ,v l. mol. T I . I 0 t "r' "". .,,::::'.. "":"""';"',~~"'- 1;;'" :;: r' 1 + -,-_on w.UJ To-,:;: , .. ~'"' ; ~ r:o ro.-,-E, ~H tr~n voan :ct . Do (1.143), ,-J.--,'( ,- ) , ' ('i..Il C , )( , [2 , (",I C n ) .n Cr ) 2, ) -" C'ri ri )j! Ct) i \1 y. ... .,. Gi?\\"" ,. t -r ) v U U v doX:::- K-'8 + "'--, g I, n -- x v 0'1. x xx ':}.L, =1 +"'0 "'- ;at...,:; - ",',.,(:, _ I -" 1/4) > 0v~ "'6 ,-,~, "1;;; - _l.i\..C ""'7'. .-- \ 'T~' f'l~~ ) ' <:-y GO ~..:.._)5 , , q (.-,) "'" (c:,.,/i,,)cf. (t) {n ..:.<: -'..1. n I ' II , " ,I,.. I ... , ' 1 ' 1 q \K.G.;:< .,. Ct9)lio\~) 1J -,1 (1. :;.4:+) ... ". ,..." 1,="Y ;'1.3. su ,,\ _1- ) ,,::) . -"'It-:,.,\c 1\; .::; Ce IV -'-.q khi t ) 'i'*:' - vch 'c.,3t r')') c. .."...,.'..- ~ ri ',-."'"":;;;" 3 ; (1 11./ )1._l\.o::<'CO,:...o<.;~,-,-0-'..' v~ .~~'+ 9~(:) ~ Ce- '>0t.. khi t >.T */ I ,~ rc~,-, " 0V 0-,- .~J~ J I .0 Do n 00 1 A' li t ~ lit trong L (o,TjHo) yeti. , va I n 1/2 n' 1/26' (u ) u ~ 6 (u) ux xx x xx I voi moi T ~ 0, ta suy ra 0{) / , trong L (O,fj1-) y~u* 2 (I 2 \I u t (t)\\ + .)6 (u)u (t) dxx S2.. x xx Dinh ly 1. 7 du,oe ehu~g minh, ., ., Binh 11. 1.8. Gia s11 (A.8)-(A.l0) dude thoa va f' ~ 0, o~ cr;< 3A<-1. ~ -1'\ tCe '0 I' t ~ r * ,... . ~ ?, N" 1 H InH2 H 1, Lq( "'~+ "1 1, q) ~ 2 ( ~+ 12)iJ~a s u = ,u E t1 , Ul Eo vag G ~,j . n1.> , " L. j .0 0 0 - 0 .LO: Thi ton t ai I!lgtso'_dU_Q!lg S' sac eho vai Ii u 1 n2 + f 0'(u )u, 2dx +-,x O,x O,xx J1. bai t03n bien vdi dieu kien ban dati 00 fllg(t)jj':ldt <. S ° l,q (1.1)-(1.3) eo duJ nha't ::lot . - -' '" + .., ll!<,h~ernyeu u ireD 12 thca, UE10{)OR+jH lna2)0 Ut b LO{)(IR+jH l)nLP(iR+jW 1,p)0 ° ~(u t ) ~ 1q (iR+ .wl,q )x loe' Eall nual ne'ti \ 11<7,(t)i! ~ = O(h( t» khi t -) o{)~,q - , -~t At , -(1...;::>/<x),I tron!'; do net) =e ,":)'> O~_CO, thi / j-, 2il u ..(t) Ii-' + 6 (u)u (t) dx x l, _no x xx I -'1 t ItronR do h (t) =e 0 voi mot ~ )-, 0 /0 _I neu ex) O. = O(h (t» 0 khi t +oo ~I , -::>/c<0 neu eX = 0 va h (t) = t -0 J ') "'1 I'" ~ Chang minh. Uia au. lu") la cae xap xi Ualerkin cua (1.1)-(1.3) nhu trong (1.95). ~, "", I Nhan mot phuong t !'inh trong (1. 95) vat c " (t), tong thee k va tichnK phan cho 1\ ur\(t)ii2 + t fliunt(s)IIi ds < Ca ,p (1.145) trong do C doc l~p doi vai n va t l'ihu troll':; ehu~g mint eua d~nh ly 1.7, ta co maxiun, t (t)\ ~ dl «(p,I(un )un (t)2dx)1/p- x .\ r xt xxt x E SL J/" (1. 146) -.I t., Khan mat phJdng trfnh trong (1.95) vdi ~kel k (t) va tang thee k ehe. n. 'I (' n n 2 f n n 2 (1/2)4>n(t) + J t:> (u xt)u xxt(t) dx + fl(U t)u xt(t) dx ~ o~ :: f " n n n 2 f n :: (1/2) G (u' )u - t u (t) dx + g u t ( t) dxx x xx xx oJ/. .f1- t ( ) n ( ) 2 f ' ( n ) n 2t rong do q, t :: 1\ U t t II + 6 u u (t) dxn x x xx .fl- " J "~ ' ( "- )Do bat dang tnue Young va l.l~o, (1.147) r \g un - t (t)! dx ~ (l/Z) f J5' (un ",)un " t (t)2dX .\ x x X I.. xx Jl- Jl ru' (L1Lt?) v~ (1.148), tamy ra + (dZ /2) /I g ( t) II i, q (1.148) I \ ' n n ;:J O f " n n n 2 ) q <to(t) + p, (u t )u' t ( t) - d x ~ f 6 (u ) u tu ( t ) dx + dz lI g ( t ) II In . I' x xx x x xx , q 5L JL (1. 1,1+9) 3~t T* :: 8max(2C3/Cl,1,1/Cl,dlP). l{eh phSn (1.149) trgn [t,t+r*J ehe t n* 00 + J.., ,-J J ( n I' L ) . , I) (u ~)u' t dxds'; (F>(t - cp (t+'r*) t ' oj [ Xl.. xx n n JL. + + t+T* t+T* f J '[d' (un )un tUn 2\ dxds + d;:J J il g(s)/! ql ds =='tJ (t) t .JL. x x xx - t ' q n (1.150) Ket h?p (1.146) va (1.150), ta duQC ,,+ ~:',k J II un t (s)1j2dS ~ T*d "0/(t)2/p t x 3 n ( 1.151) ~ ' , I ., I Nhan moi phuong trinh trong (1.95) voi A. C k(t), tong thee k va tichK n. phan eho t+'T*.. f ' n n 2 j 6 (u )u .. dxds ~ d;, max4>.(s) +x xx ~ n t ...>7.. t <. ~ t 'T *s - + t+T* f i! un t (s)H2dS t x + t+r* .. 'n n n J J If(u" t)u t u !dxds + t .J1.. x xx xx t+T* f !Ifl(un t )un t Un Idxds t ..52.. x x + + t+T* J fig un \dxds t ..r1.. x x ( 1.152 ) trong db d4 =max(l,l/Cl)' Do (1.151) v~ (1.152), ton tai t E[t,t+T*}. n sac eho t+'T* ~ (t ) ~' 2d 3"1j) (t)2/p + (1/8)maxc:p, (s) + (l/T*) If I g un jdxdsnn n n xx t ~ s~ t+r* t ..J2. t+T* t+T* f Ool n n n . n n n + (l/T*) t -'~Ip(u xt)U xxtU xx\dxds + (l/T*) tf..L ~f'(U t)U xtU xldxds ( 1. 153 ) Do (1.147),(1.143) va (1.150), t+T* r l p'(un t)un t 2dXdS + max<f,,(s) ~ 2d3'\}J(t)2/p + (1/3)maxq" (8) t) Q. xv xx . r. n n .J t ~ s ~ t +T* t ::;8 ::;t +T* t+f* t+T* J ' fl ' n n n. If n n n + (l/T*) F (u t )u tu Idxds + (l/r*) If'(U t )U" t U Idxds t .J"L x xx xx t J2. x x t+T* t+T*. + (l/T*) f r jg un Idxds + 4lJ.1.(t) + 2 f f !fl(un t )un t2jdXds t xx n t x...,1.. J2. + t n"+i " 2 J f ' n n n I+ I Q (u" )u tu dxds +x x xx t .IL t+T* d2 f II g(8)l\q l dst ,q (l ~154) '.I , ~, :J o' I "I Do (A.9) va bat dang thuc so cap, \ '' n n n \ . f In n 2 Ie, (u. ...)u. tU. (t) dx ~ (2G3/Cl ) R (u )u t (t) dx. r Xv xx xx r xt xx Q ft (' n n 2 (' n n 2 + (C1/8C3) j ~ (u 'xt)u x)t) dx ~ (r*/3) j ~ (u xt)u xxt(t) dx R ~ + (C1/3)i1un (t)jj2 + (Gl /8) f lun t t un (t)2dXxx R x xx "" , Do (1.146) va bat dang thuc Young, (1.155 ) Clflunxttunx/t)2dX ~ <Pn(t)PI2 + dlP f~'(unxt)u~xt(t)2dX (1.156) R ft II / Bat N =sup \6 (s)l. 'Thi ta co isj ~1 ~ n I n 2 I r~ ' ( n ) n 2 q5H lu t u (t) dx :; (1 4) u t u t (t) dx + (d5/T*)<p (s)x xx x xx n -.I'/.. .Jl.. (1.157) -Bat Hl =sup \f'(s)i isi~1 do = 2d3(M1T* +X12/Cl + 1) 0 1h ,- Do O~. ex < 3' -1, ta co t + T *\ .0 . Y' Y' ;-:>, ? 1 , d,..(i'.l J J jU~- tU" ~-ldxds)~/.i:' ~ I:) t.:,1.. X xx t+r* ?" ,I 1'1 "'. '- (1/4) j ) i3(u.~xt)uhxxtdxds t .ft + d7 maxc?n(s)'lo t~s~t+r* (1.158 ) ") ~ I 2 I( '- 2) 0 troll,": 00 0 :::: 1) D -- ;;> 1. ~ .0 ,,~ ;" fa co (Ig tt (t)i dx ~ (liS),*, (s)pI2 + J -x x n Jl vat q* = min(q,q ).. 0 dsT*ljg(t)1! ql,q (1.159 ) Gis. 61.1"6'la r.g;hiem duong cila phuong trinh (ri .- ) ::].*-1 1:3.X\'-'5,(17 x ",-, (' , 11 ro V'\ val; C =:!l~r.1. ,vl,~J0 ~ =1/8 "1 ,.~" .La K1.em ( * ) ~:~' ( ) \'.."'"he u r:.ax,? S 0:::.G l-h~ +-.(~n..",* 0" - '"'" l-T.. ct (t+I'*) :; Co(t) +f. 'n t+I'* d~Jlig(s)(jq ds "t 1, q irons dbd9=3(d2 + dB). . ~ ,,' i 1';",::>t '/=t v d~,:o-;::> tr.; e t ~..~ ~ , ol' - ",...~ v..k ~ :!lax IuD (5)\ ::; 1x I vdi t ~ s £: t+I'*- x E.:1. do; m3..Xc2(s)1./ r: : <fs ~ t+1'* ~ (1/3baxq> (s)n I;<fs~t+r* d 7 max6 (s)10 t{ 0/3)::3.X6 (s)'r, 'n t~s~t+I'" t~S:::'t+:r* Vay do (1.152.), t+1'* ~ t+T*- I ~ ( i ~ , ( c. ) n c:. [ , , / " \' f '0 n ( ) ..2, < 2" ~ * ' ( )2/0c'. , I I U - U - t aXiS.:::- <::"1 \I U +-S \I c.s" "'11' a3'tJ t ~oj ~ "x Xl- n t R t (1.160) \ va t+I* , t+T* C/ ) ,t" C n ) n n" /}; ('"r.. C) "ln C) " I r* J J I:' u , U ,U laxds ~ 1) II U t s J\!!u silas, l: Xl: X -:;;- , x xx t .L ... t { (l/ 3) !":ox'?"( EO) (2~f I 2Ie )d '-"'C ... )2/p , 1 ~ 3 'n " ( 1.161) t ,:; s ::; t+T* I,eu cb (:+1:*» Co C"C) 'n 'n t+:!.'* d ! " I ) ' 1 1 ' ' h ~ d (1 1 r'~ )a 1\gl.S i 1 ' as 1;.2. 0 .-./ I"...' ,1" :+1'* 11 t+T*' 4y Ct) + ( J'lrs Cun )uD un 2I dx,.1~ .::: 5ivI f) ' J' UD t UD xx 21cLXdSD J x xt I '" X t S'L. . xx t .J2. 51 ~ t'"*,-r.:. r I r. '" 2 (1/4) J B (u' )-.;.h . dxds, , i ' xt xxt t -,1. + (1/3) max9,.,(S ) (1.162) t{;s~t+T" Dc; (1. 153), A , " ( :J/ '" '," q,' -) - ,,;0 E" - ~ -+"* d,( J ]16"(Un )ur" un 21dx:J.s)2/p0 , x xt xx t J1. n* .-rL, 2 :J/< -1 "( ,,,r f ":-' ,r. \ ' d ' )-'P/" u-") \u ,..u.. xds ~0 ~ A" XX I., ..,(.. t+1'" I Cl/4) f f ~(un +)un ..2dX:J.S, ' x" xx" t J2. + (1/3):::iXC (s)'r (1.163) t{s~t+I'" rdt hop(1.15:0.)-(1.156) va (1.160)-(1.163), ta Quae t+T" , t+T" (1/3)max9 (s) + (1/4) f f f3 (un t )un" ;-2dxds ~ (d2+:13) !lIg(S)jjq1 ds-<:_f,_..,,- t .f2. x xx. ,'~'"' ;:: ~ 1.,+1 t ""u)' -::" ~"" "'-" '-'v ou... ..0:1. t+r* L'l3.Xq,(S)~ d- J ii g(s)i! lq ds n ~\ ,q t~s~t+r* m&~ttu~n . Vay(-) dung. , 1;a;/ ta kie::J (**, '\ 6 (t) -:-'r t+l'* d~ J II g(s)i! lq ~ds < (7/3) S' /t ,~ 0, t-\-.;~..-'- :::iX4' (5) < S- 'n 0 ~~(s~ r-d'" -,',' "'" ,: -"""" 1' "'- J '012. sU t~a1 1?'1, tal ten tall E t,t+l sac encn 9 (s)< ~\n G I ' n 'j di t { s <r va ct (1' ) = 2 .n n n 0 riC!'l philrl (1.14;1) tren [t,s} va dung (1.157), ta dlioe s<pr(s) ~- (7/3)~' + (d5/r*) J 9,,(L:)1dt... 0 t". t.{s~r n , , tti do suy rEi 9,Jl') ~ (7/3)2'" [1 - (d_/r*)(:r-l)['q-l(T -t))1-1/(q-l\ [;.. -- 0 ) 0 n ~ 0 ~ 0;:: .,- ( *,, ) ";>' »' .,mati t~uaL. .ay aU0C cnung mlnn. , ") r;ay ts. gia SD. 6 (C) 'n + 0-;:> d~ [It g( t ) J!lq dt < (7/3) S'-' 0 ' '1 0 ( 1.164) ... 1'a kie::l kI* <t (kI") ~ <?(0)n n + d9 IiI g(d!i{ "dt0 ~'"i Y kE II; C1.165) ~:::; r;"'- (1 1.' ) - ) d."...~' C - - 0~~J--"0 \-.-,-J ~.'O 101 .<- . ,,;' _.:. C 1 i'::'O:: ) -~11 ~/. '.r~:IJ_a "'-' -'-'-~./ -""U"Q Vv1 ..Eh,. Do (1.16~) va (1.165), cf Cd"') n (k+l)r" Ck+l)r" .,. dj J it g(s)H; ds f; <t.(0) + d9 l' ii g(s)ili dskl'" ~,q - n 0 ,q < (7/3)[ 0 t.-,)' Q.;., ~", ", -10 C *,, )u. v "'...: ~ ~, u maxejo(s) .( ;)'n 0 kr*o{s ~(;<::+l)r* V~y CJ Cz), . '(, ' ) ~ ) . C9~\'<+J. lZ ~ 0 k;"')n I,.. - (l<;:+1)'* + d9 / il g( s)iI ~ ds~r" L,q ~ 4- (0) +n Ck+1)1'" d 9 f Ii gCs)ili dst"\ ,qV Vay, (1.165) d{ng voi ill?i k G iN, do quy n~p. 53 .., l;ay L 6~c). :.>:;. Ii U1 H2 + J 0' (U )u 2dx + Df g( t )!!lq dt .( d,x O,x O,xx 0 ,q~~ trong do [' = 7~/3xax(1,d9) , Thi ern( \) + o{) d 9 [II g(t)l\<ll dt < (7/8) [0 ,1. 0 khi n ~ N tJ do BUY ra, do (1.164),(1.165) va (**) <Pn(s) < ~° (1.166 ) , vdi 1TI?i S1 0, n ~ r~. TJ (1.145) va (1.165), ta suy ra S~ ton t?i cua m?t nghi~mye~duy -' ., )( ) ~ + nhat u cua (1.1 - 1.3 tren ~ . " ., Gia s11 IIg(t)1\ ll=1 = OCtet»~ khi t.~ 00,q .' . -,t ~I ' ) .-(1+2/cx)~'trong 00 aCt) =e '1) 0 neu IX = 0 va h(t =t neuO() O. Do (1.154) va (1.166), t+I'* In9.xc:?(s),:::; dl0[ (,-b (t) - rh (t+T*) + d j 1/ c-(s)IIQ ds)2/pn Tn ~ 9 a l,q t ~ s ~ t+T* t + + t.+T * . ..,;', -, ( J ii g( s )jj'-i . ds )'-I;;J j t 1,q " I tu do BUY 1'a t+r* 2d 5max<p,(s)P/2~ dll r9 (t) - cp (t+I'*) + d12 f Ii g(s)il 1 . dsJ (1.167). n. n n t ,q t (; 5 {: t+T* C?ng (1.147) VaG (1.167) eha I . "0/2 . t+I'* ctn(t) + d(s)" + d11 (q, (t+T*)- 4>(t» ~ d13 f it g(s)lIi ds~- ~ n -,* - n n t ,1,,~s~t+l I vdi t ~ T .a 51.i D3:t :.c (t) = 6 (t) +- . ~"" " t+T* d11 f 9n( s ) d st " T~i , ~ (t) r.. d.14kIl(t)p/2 ~ d13h(t) t ~. r / 0 V~y do ba"\d~' 1.2 va 1.3, c (t) ~ k (t) ~ . r: r: dl~h (t) ,-:J 0 1"1 t 4 .0 tron5 do h (t) = t-2/~0 ~I , -~ J t ' ~ neu C(, '> 0 VB. h (t) =e 0 voi :no: '? / C,0 0 ~, neu IX = O. ='"' i' l ;~.";~'~ "'~::- ;Y' b ;-,; .'",.;.; nfl -Y . J '-'UO" cnc<u:::: ffi_.." ..O~l1toa...

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