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

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...