Khảo sát phương trình sóng phi tuyến trong không gian Sobolev có trọng 5

Luận văn thạc sĩ -ngành Toán Giải Tích-Chuyên đề :Khảo sát phương trình sóng phi tuyến trong không gian Sobolev có trọng

CHUONG 3

KHAO SA T PHUONG TRINH

Uti -( u" + ~ u,)+ F(u,u,) = f(r,t\

Xet bai tmin (3.1)- (3.4)sail

uu-( Urr+~Ur )+F(U,Uf)= f(r,t), O<r<l, O<t<T,

I

lill J;ur (r, t)

1

< +00, - u,(l, t) = hou(l, t) + h(t) Uf(1,t) + g(t),

r~O+

(3.1)

(3.2)

u(r,O) = uo(r), uJr,O) = ul(r), F(u,uf) =f;(u)+F2(Uf) = lula-l u+luf IP-l Up

(3.3)

(3.4)

111.1 Du'a v~ bili tmin bitn philo

Xet bai tmin (3.1)- (3.4)

Til (3.1) ta suy ra

r I

T I

f fr Ull-(urr +.lur)+ F(u,ul) wdrdt = f frfivdrdt, VwE D(O,T;V1) (3.5)

ChQn w(r,t)=~(t).v(r), trong d6 ~ED(O,T), VEVp k~t h<jp (2.12) va (3.2), ta

vi~t l~i (3.5) nhu' sail

ref!{( ul (t), v) + a(u, v)+ (F( u(t),ul (t»), v; ] ~(t)dt

JLdt T

+ f[h(t)ul(l,t)+g(t)Jv(1)~(t)dt

0

(3.6)

r

= f(!(t), v)~(t)dt, V; E D(O,T), Vv E VI.

0

Tli (3.6) ta thu duQc

~ \ul (t), v) + ău(t), v)+ (F(u(t), ul (t)), v) +( h(t)UI (l,t) + g(t) )v(l)

= (J(t), v), 'v'v E Ví ạẹ t E (O,T).

(3.7)

111.2 811t6n t~i va duy nhát nghĩm yéu

Ta thanh l~p nh6m gici thiét thu nhát nhu sail

(HI) ho > 0, 1::;a < 3, 1::;fJ < 3,

(H4) hEC2(~), h(t)"20, 'v't>O, h(O)=O,

/

va t6n ti;lih~ng sÓ8 E (O,ho) saDcho hI(t)"2 -8, 'v't"2 0,

(H5) Uo E V2 ' UI E VI

Chti thich 3.1 ho ia hang s{f duang xudt hĩn trang b6 d~ 2.5.

voi nh6m gici thiét thu nhát

Binh ly 1 Cho truac T>O va (HI)- (Hs) thoG Khi d6 bai loan (3.1) - (3.4)

ul E Loo(0, T; VI), Ull E LOO(O,T;Vo)'

Chung minh Vĩc chung minh dinh 19 1 duQc chia lam nhi€u buoc

Búocl Xáp Xl Galerkin

ặ,.) nhu trong b6 de 2.6.

Ta Hmnghĩm xáp xi cua bai loan bién phan (3.7) duai d~ng

m

um(r,t) = L>mj(t)Wj(r),

j=l

(3.8)

trong do cac ham s6 cmJt), j = 1,m thml h~ phuong trlnh vi phan thuong

(u~(t), WI)+a(um(t), WI)+(F( Um(t),u~(t)), Wi)

=(f(t),wJ,l'!;}'!;m,

(3.9)

cling vdi di€u ki~n d~u

m

um(O)= UOm= Lamiwi ~ Uo mqnh trang V2, khi m ~ 00,

J=I

(3.10)

m

u~(O)=Ulm =LPmJwJ ~ UI mqnhtrang~, khi m ~ 00.

J=I

Vdi m6i T > 0 cho trudc, ta se stt dl;lng dinh ly diem ba't d9ng Schauder de

chung minh h~ (3.9), (3.10) co nghi~m cm= (cm!"'" cmm) tren [O,Tm]c [O,T].

Ta co b6 d€ sail day v€ slf t6n t(;linghi~m cua h~ (3.9), (3.10)

nghifm cm = (cml"'" cmm) tren [O,Tm]c[O,T].

Chung minh b6 d~ 3.1.

H~ (3.9), (3.10) duQcvie't l(;linhusau

c~/t) + AJCmJ(t)

=II~\2 [( F( um(t),u~(t)), wJ) +( h(t)u~(1,t)+ g(t) )WJ(1)-(f(t), Wi)1 J

Cm/O) = amJ' C~/O) = PmJ' I'!; j '!;m,

hay

cm;(t)= amicostA t)+ Jx; sin(At) (3.11)

1

II sinA (t-T)

-II Wi W 0 jx; [(F(Um(T),U~(T)), WJ)+ h(T)U~(1, T)Wi(l) JdT

- 1 II sinA (t-T)

IlwJW 0 A [g(T)w/l)-(f(T),wJ)]dT,l'!;j'!;m.

Bo qua chI s6 m, khi do h~ (3.11) duQcvie't l(;linhu sail

trong do

c = (cp , cm), UC= (U c\, ,(U c)m)' (3.13)

t

(U e)/t) =G/t)+ fN;U-r)(Ve)/r)dr,

0

(3.14 )

GU) =aNI (t)+ f3N(t)- 2fNU-r) [g(r)w ,(l)_1 fer), Wj )J dr,

(3.15)

(Ve)/t) =- 1 2

[

/F [ i>iU)wpIe:U)Wi ] ,W;)+h(t)Ie:(t)w;(1)Wj(1) } , (3.16)

sin(At) 1~ j ~ m .

Voi m6i 0 < Tm~ T, M > 0, ta d~t

S={CEC1([O,Tm];IRm): IleI11~M}, Ilelll=llcllo+lle/llo'

m

Ilello= sup leU)I\' leU)11= Ile;(t)I

Thl S Ia t~p con 16i dong va bi ch~n cua y = C1([o,Tm];IRm).

Sa dl;mg dinh ly diem bcft dQng Schauder, chung ta se chung minh r~ng anh X(;l

dQngnay chinh Ia nghi~m cua h~ (3.9), (3.10)

a) Chung minh U(S)cS.

Ta co

GIU)1 =-XaN. III ( t) + f3 N/(t)11 (3.18)

- 2 fN;U-r) [g(r)w/l)- <fer), Wj >] dr,

II Wj II 0

t

(U e)~(t) = G~U)+ fN;U-r)(Ve);(r)dr , 1~ j ~ m.

0

(3.19) Tli (3.14)- (3.19) ta suy ra anh X(;lU: Y +Y xac dinh.

Cho e E S, tli (3.14) - (3.19) ta suy ra

1 t T

(3.20)

.

1a; 1+ f1 II /3; 1+ f1 rT + f1 /3(M,T),

t

I(Ue)/(tH~ IG/(tH + fl(Ve)('Hd,~ IIG1 110+Tm IIVello

0

(3.21)

~ KII a;l+ II /3;1+rT+Tm/3(M,T),

trong do

m 1 T

rT =~IIwi W l(lg(t)w/l)l+I\i(t)'Wi)1 )dt ,

(3.22)

~ /3;CM,T)

/3;(M,T)=SUP

{

/F

Ch1ithich 3.2 Tit b6 di 2.20.i ta suy ra rling

F( ~e,(t) w"~e: (t)W,) EVo, 'Ie E S.

(3.23)

Do d6 (3.24) luon tbn tc;zi.

Tli (3.20) va (3.21) ta suy ra

(3.25) trong do

va /3(M,T) du'QCxac dinh bdi (3.23) va (3.24).

Chon M va 0 <1~ < T sao cho M?: 2"" va To(I + J;., ]P(M,T)'; ~.

Tli (3.25) ta thu du'QcIIUeIII ~ M, voi mQie ES

b) Chung minh U lien tl,lCtren S.

Cho C E S, {Ck} C S va Ck ~ C trong Y, ta co

t

(U Ck);(t) - (U c)/t) = fN;(t-T)[(VCk)/T)-(VC)/T)]dT,

0

t

(U Ck)~(t) - (U c)~(t) = fN;(t- T)[ (VCk)/T)-(V C)/T) ]dT,

0

Tm IIUCk - Ucllo::; JX: IIVCk - Vcllo,

II(UCk)/ - (Uci 110::;Tm IIVCk - Vcllo,

( fck;(t)w;,fc~;(t)W; )

-F

( fc/t)w;,fc:(t)w; ) 'Wj)

j=1 II Wj II \ ;=1 ;=1 ;=1 ;=1

(

~

)[

~lw/l)I

J

( (

m

m

)

)

+ F; LCk;(t)W; -F; LC/t)W;,L 2

1=1 1=1 J=1 II Wj II

( (

) (

m /

)

)

+ F; LCk;(t)W; -F; LC;(t)W;,L 2.

m

Dịit R = MIll w,IIÍKhi do tli (3.28) va b6 d€ 2.20.iii, ta dúQC

1=1

I (V c, )(1) - (V c)( t) I,S K,' IIhilL"ÍT) (tII w, "l ~ IIII:~ II:; J II c~ - C'II,

+

(fIIWIIII)( f 1 2

J [ ~KR(a)llck-cllỡKR(j3)llc~-c/lloJ.

;=1 J=l IIwi II

Tli (3.27) va (3.29) ta suy ra U lien tl,lCtren S.

(3.27)

(3.28)

(3.29)

c) Chung minh US compact trong CI ([O,Tm];IRm).

Do US c S, nen hQ cac ham us = {Ue: e E S} bi ch~n d€u theo chu~n 11.111trong

d6ngb~cd6i voichu~n 11.111trongkhonggian CI([O,Tm];IRm).

Cho eES, t,tl E[O,TmLta co

=G/t)-Gj(t/)+ IN/t-r)(Vc)lr)dr- INj(tl-r)(Vc)/r)dr

(

=G/t)-G/tl)+(, aI[N/t-r)-N/tl -r)]cVC)/r)d~

- IN/tl -r)(Vc)j(r)dr,

(

G(t)-G(t/)1 1 =a.1[NI (t)-NI j j (tl) J + {3.1[N.(t)-N.(tl) j j J

- 2 I [N.(t-r)-N.(tl -r) J [g(r)w ,.(1)_1 f(r),w j .

)J dr

('

+II W: 112fNj(t1 -r) [g(r)w/l)-\f(r), Wj) Jdr.

(3.31)

M~t khac ta co

f INj(t)- N/tl) I:::;It- tl I,

lIN~(t)-N~(t/)1 :::;[X:lt-tll, Vt,tl E[O,Tm],l:::;j:::;m.

(2.32)

Tli (3.23), (3.30) - (3.32), ta suy ra

m

I(Ue)(t) - (Ue)(tl) II= LI(Ue), (t) - (Ue)/tl) I

,;[tlaj 1+F.tIP, I+r, }-tll+p(M,r{r + k )It-tl

Danh gia tu'dng tt,I'nhu' (3.33), ta co

m

I(Uei (t)-(Ue)1 (t/)11=LI(Ue)~(t)-(Ue)~(t/)1

j=1

(3.34)

,; K[ Ktl aj 1+ ~Ilij I+r,] 1t-t' 1+1i(M,T)(KTm +1)1 I-I' I.

Tli (3.33) va (3.34) ta suy ra hQ cac ham us Ia lien t1;lCd6ng b~c d6i voi chuffn 11.111trong khong gian C\[O,Tm];IRm).V~y theo dinh ly Arzela - Ascoli thl

US compact Tli cac ket qua a), b), c) va dinh ly di€m beltdQngSchauder, U co

di€m beltdQngtrong S

Nhu v~y bai toan (3.9), (3.10) t6n t(;linghi<$mUmtren [O,Tm].V~y b6 d~

3.1 duQc chung minh xong

Cac danh gia tien nghi<$msan day cho phep ta lelyTm = T, Vm.

Chli thich 3.3. Trang chang minh Sl! tbn tf;li nghi~m xfip xl Galerkin cua bili

{

hO>0, 0<a<3, 0<{J<3, fEL\O,T;Vo),

gEL\O,T), hECo(IR), UOEVI' ul EVo'

V6'i gid thift UoE VI, UIE Vo thi (3.10) du(fc thay thf bili

m

um(O)=uom = IamjWj ~uo mf;lnh trang~, khi m~oo,

j=1

(3.35)

m

u~ (0) = Ulm=I PmjWj ~ UI mf;lnh trang ~p khi m ~ 00.

j=1

Bu'oc 2 Danh gia tit~n nghi~m

Danh gia 1 Nhan phuong trlnh thu j cua h<$(3.9) voi C~j(t) va lely t6ng theo j

tli 1 den m, ta duQc

Id

I

1

+ frlu~(t)IJJ+l dr+h(t)lu~(1,t)12 +g(t)u~(1,t)

0

=(f(t),u~(t)).

f)~t

t

Xm(t) = Ilu~(t)W +a(um(t), um(t)) + 2 fh(r)lu~(1,r)12 dr

0

+- fr Ium(t) Ia+ldr+2 f fr Iu~(r) IJJ+ldrdr.

(3.37)

La'y tich phan hai vfS(3.36) theo t tu 0 dfSn t, ta du(jc

Xm(t) =Xm (0) + 2 f(J('l"),u~(r»)dr - 2 fg(r)u~(l, r)dr

~Xm(O)+ fIIJ(r)112dr+ fllu~(r)112dr

I

+21g(t)um(l,t) I+2 IiI (r)um(l, r) Idr

0

(vi g(O)=0)

I

~ Xm(O)+ IIJ 11~2(o,T;Vo) + fXm(r)dr

0

+-l(t)+-.JLu~(l,t)+ - fll(r)12 dr+ ho fu~(l,r)dr.

a+1o

2 I I-a

a+1

a+10

2Ka+1 I I-a

~ II Ulm W + CIII UOm II~ + 2 IIUOmII~+I fr 2 dr

4Ka+1

(a + 1)(3-a)

Tu (3.10), va (3.39) ta suy ra

Xm(O)~M?), '\1m,

trong do Mil) dQc l~p vdi ffi Tu (3.38) - (3.40) ta suy ra

X (t)~ gl (t)+-X (t)+2 fX (r)dr,

hay

I

0

trong do gj (t) =MT + II f IIL2(O,T;Vo) +-,;-10 g(t) I +-,;-110 g IIL2(o,T).

Tu giii thifSt (H3) va do H'(O,T) c.Co([O,T]) Den ta suy ra

gl (t) ~ Mi2) a.e t E [0, T]

Ap dl;lng b6 d€ Gronwall vao (3.41), ta du(jc

Xm(t) S 2M?)e41 ~ M?) , a.e t E [O,T].

(3.38)

(3.39)

(3.40)

(3.41)

(3.42)

(3.43)

Ch6 thich 3.4. Trang danh gia 1, ta chua sit df:lng htt nhom gid thitt thri nh{{t, thq.m chi co thi thay m(Jt s{;'gid thitt bJi cac gid thitt ytu han tuang ring, ch~ng hqn

(Hi) dur;c thay bJi ho > 0, 0 < a < 3, 0 < fJ < 3;

(H3) dur;c thay bJi g E Hi(O,T) Trang tru(jng hr;p nay, ta chi vifc c(Jng

them VaGvt phdi cua (3.38) dqi lur;ng 21g(O)Uom (1)\ I hi chq.n; .

Nhom gid thitt thri nh{[t nay th1;lcs1;lcan den trang danh gia 2 sau day"va trang cac burJc sau db.

Danh gia 2 L1y d<;loham hai v€ cua phuong trlnh thuj cua h~ (3.9) d6i voi t,

ta du<;lc

(u:;(t),Wj) + a(u~(t),Wj)+(:tF(Um(t),U~(t)),Wj)

+(hi (t)UI (1, t) + h(t)UII (l,t) + gl (t) )w/l)

(3.44)

=(Ir(t),Wj)' l~j~m.

Nhan phuong trlnh thu j cua h~ (3.44) vOi c~/t) va l1y t6ng theo j tit 1 d€n m,

ta du<;lc

+-hl (t)-I u~(l,t) 12+h(t) Iu~(1,t) 12

+l (t)U~(1,t)+(:t F;(um(t)), U~(t))

=(Ir(t),u~(t)), l~j~m.

Bi;it

Ym(t) = II u~ (t) W +a (u~ (t), u~(t)) + ( hi (t) + e) Iu~ (1, t) 12 (3.46)

L1y tich phan (3.45) theo t tit 0 d€n t, ta du<;lc

Ym(t)=Ym(o)+elu~(1,T)121~:;) + fhll(T)lu~(1,T)12 dT

0

(3.47)

(

d

)

t

+2 f\ft(T),U~(T))dT-2 f -F; (Um(T)),U~(T) dT-2 fl(T)U~(1,T)dT

t

::;;ym(o)+eKj21Iulm II~+elu~(1,t)12 +llhll Ilc'(O,T) flu~(1,T)12dT

0

I

+2 fllft(T)llllu~(T)lldT+2 fll~F;(Um&))IIIIU~(T)lldT

t

+ 21l (O)Ulm(1) I+ 21l (t)u~(1, t) I+ 2 fill (T)U~(1,T) IdT.

0

Ta co

ho I u~(1,t) 12::;;a( u~(t),u~(t))::;; Ym(t), (3.48)

(3.49)

2 +~Y (t)

h - £I0 2h0 m '

2fll/(T)U~(1,T)ldT::;;~llgIIW L 2 T +fYm(T)dT,

h (0, )

(3.50)

IIU~(t)II~::;; ~a( u~(t),u~(t))::;;~ Ym(t).

(3.51)

Til (3.37), (3.43), (3.51), va b6 d€ 2.20.ii, ta duQc

1

II ~FJum(t))112=a2 frlum(t)12a-2Iu~(t)12 dr

0

(3.52)

1

::;; a2 K; II U~(t) II~ fl Um(t) 12a-2dr::;; a2 K; Ym(t) k\(a) II Um(t)ll~a-2

a2 K2 K (a )

(

M(3)

)

a-l a2 K2K(a )(M(3) )a-l

Til (3.47)- (3.50), (3.52) ta thuduQc

Ym(t) s Ym(0) + 8 KJz II uJm II~ + :0 Ym(t) + II it IIL2(0,T;Vo)

+ fYmCr)dT+ L"'(O,T) fYm(T)dT

-( (3)

aZK;K1(a) MT fYm(T)dT+ fYm(T)dT

+ Il (0) IZ +KJZII UJm II~+~ Igl (t) IZ

ho-8

h - 8 II gll 11~2 T

f

l +~Ym(t)+ (0, ) + Ym(T)dT,

hay

t

Ym(t) sgzm(t)+ Mi4) fYm(T)dT,

0

(3.53) trang do

gZm(t) = h2~)8[Ym(0)+KIZ(8+1)IIUlm0 II~+llit 1I~2(O,T;Vo)J

+~

[

ll(O)IZ + 2Il(t)lz +llglllI~2(0'T)

]

,

(3.54)

Mi4) =~

[

3+ IIhIIIILoo(O,T) + aZK;KJ(a)(M?)r-l

]

ho -8 h0 Co a. // (3.55)

Nhan phuong trlnh thli j cua h~ (3.9) vOi c~ (t) va la"y t6ng theo j tli' 1 de"n m, J

ta duQc

Ilu~(t)IIZ +(Aum(t), u~(t»)+(F(um(t),U~(t»), u~(t»)

+( h(t) u~(1,t) + get) )u~ (1,t) = (J(t), u~ (t»).

Trang (3.56), cho t = 0, va chuyding g(O)=h(O)= 0, ta duQc

Ilu~(O)llz +(Auom' u~(O»)+(F(uom,uJm)' u~(O») = (J(O),u~(O»).

(3.56)

Suy ra

Ilu~(O)11s IIAum(O)II+11 F{uom,uJm) II+IIJ(O)II.

Ap d\lng b6 d€ 2.20.i, ta duQc

s K(a) II UOm II~ +K (,8) II U1m Iii.

(3.57)

(3.58)

Tli (3.46), (3.57) va (3.58), ta thu du'Qc

Ym(O)= Ilu~(O)W+ a(Ulm'ulm)+(h/(O)+B)u~m(1)

(3.59)

+[CI + KI2(e+IIhi Ilco(o,TJJllulrn II~.

Tli (3.10), (3.54), (3.59), gii thi€t (H3), va do HI (0, T) C.CO([0, Tn nen ta suy ra

Tli (3.53), (3.60) va b6 d~ Gronwall, ta du'Qc

Yrn(t)~MT e T ~MT, a.e tE[O,T]. (3.61)

Tu'dng tlj (3.58), ta co du'Qc

IIF(um(t),u~(t))II:S; K(a) IIum(t) lit +K(jJ)llu~(t)llf

(

M(3)

J

~

(

M(6)

J

~

(3.62)

Tli (3.62), ta suy ra

IIF(urn,u~)lloo .

) ~Mr),a.e.tE[O,T],

va

I

(3.64)

Bu'oc 3 Qua gioi h~n

Tli (3.37), (3.43), (3.46), (3.61), va (3.64) ta co th€ trich tli day {urn} mQt day

con v~n ky hi~u la {urn}'sao cho

Urn~ u trong D$J(O,T;VI)y€u *,

u~ ~ul trong DJ(O,T;VI) y€u *,

uti ~Ull rn tron g L"' ( O T V ), , 0 Y€u *,

(3.65)

(3.66)

urn(l,t) ~ u(l,t) trong WI""(O,T) y€u *,

(3.67)

(3.68)

ra+IUm ~ ra+lu trong LOO(O,T;La+l(o.)) ye'u *, (3.69)

(3.71)

Ap dl;lng b6 d€ v€ tinh compact cua Lions vao (3.65) - (3.68), va do phep

con v§n ky hi~u la {um}, sao cho

um ~ u m(;lnh trongL2(0,T;Vo),

U~ ~ UI m(;lnh trong L2(0,T;Vo),

(3.72) (3.73)

Um(1,t) ~ u(l, t) m(;lnh trong CO([0, T]) (3.74)

mQtday con v§n ky hi~u Ia {um}, sao cho

um(r,t) ~ u(r,t), a.e (r,t)EQT'

~.

u~(r,t) ~ ul (r,t), a.e (r,t)EQT'

(3.75)

(3.76)

Do F(u,ut) = Iu Ia-I U+1 Ut 113-1Ut lien tl;lc, nen tu (3.75) va (3.76) ta suy ra

Ap dl;lng b6 d€ 2.15 vOi q=n=2, Q=Qp Gm=FrF(um,u~), G=FrF(u,ul), tu (3.64) va (3.77), ta du'Qc

Tu (3.71) va (3.78) ta suy ra

F(um,u~)~ F(u,uJ trong L2(0,T;Vo) ye'u. (3.79)

Nhan (3.9) voi rpE D(O,T) tuy y, r6i lfiy tich phan theo t tuO de'n T, ta du'Qc

f(u~(t),w/)rp(t)dt + fa(um(t),wJrp(t)dt

+ f(F (um (t), u~ (t) ), WI)rp(t) dt + f(h(t) u~ (1,t) + get) )wi (1) rp(t)dt

T

= f(f(t), wJrp(t)dt, VI

o

(3.80)

Do (3.67) ta co

f(u~(t),wJqy(t)dt~ f(UII(t),Wj)qy(t)dt, khi m~+oo.

(3.81)

Qua giOi h(;ln khi m ~ +00 trang (3.80) bdi (3.65), (3.68), (3.79) va (3.81), ta

duQc

[( Ull(t), Wi) + a( u(t), Wj) +( F( u(t),ul (t)), Wi) Jqy(t)dt

T

+ f( h(t)ul (l,t)+ get) )WJ(l)qy(t)dt

0

T

= f(f(t)'Wj)qy(t)dt, Vi, VqyED(O,T).

0

(3.82)

Tu (3.82) ta thu duQc(3.7)

D€ chung minh u la nghi<%myeu cua bai tmin (3.1) - (3.4) ta con ph,U chung minh u(O)=u(p ui (O)=u]'

a) Chung minh u(O)=uo'

f

urn(t)=uOrn+ urn(s)ds.

0

(3.83)

Nhan (3.83) voi r Wi r6i l§y tich phan theo r tu 0 den 1, ta duQc

t (urn(t),wJ = (uorn,wi)+ f(u~(s),wJ)ds.

0

(3.84)

L§y tich phan theo t tu 0 den T trang (3.84), ta duQc

f(urn(t),Wj) dt = T(uorn,wj)+ f(u~(s),(T-s)wj)ds.

(3.85)

Qua gioi h(;ln khi m ~ 00 trang (3.85) bdi (3.10), (3.65) va (3.66) ta duQc

f(u(t),Wi)dt = T(uo,wj)+ f(u\s),(T-s)wj)ds, Vi=1,2,

(3.86)

Trang (3.85) thay Urn bdi u, ta duQc

f(u(t),wi)dt = T(u(O),Wj)+ f(UI(S),(T-s)wj)ds, Vj=1,2,

(3.87)

So sanh (3.86) va (3.87) ta duQc

(u(O),v) = (uo,v), Vv E VI' nghla la u(O)= UO'

b) Chungminhul(O)=Ul'

Tit (3.10), (3.66), (3.67) va ly lu~n tu'dng tv nhu' ph:1n a) ta cling thu du'Qc

Ul (0) = Ul '

M~it khac tU (3.67), (3.79) va gia thi€t (Hz), ta suy ra rflng

Au = f -uti -F(u,ut) E LZCO,T;Vo), nghla Ia u E Lz(O,T;Vz)'

V~y sv t6n t<;linghi<%mdu'Qc chung minh

Bu'O'c4 Chung minh sl! duy nha't nghi~m

Gia sa Up Uz la hai nghi<%my€u cua bai toan (3.1) -(3.4) nhu'trong dinh ly 1

Khi do w=Ul - Uz la nghi<%my€u cua bai toan

(Wll(t), v) + a( wet),v) + h(t) Wi(l,t)v(1)

+( F( Ul(t),u{ (t))- F( uz(t),ui(I)), v) = 0, \Iv E VI' a.e.t E (O,T),

WE LOO(O,T;V1)nLz(O,T;Vz)'

wi E LOO(0,T;V1), Wll E LOO(O,T;Vo), w(1,.) E W1,OO(O,T),

(3.88)

ra+lw E LOO(O,T;La+l(Q)), r,B+lwl E L,B+l(Qr),

w(r,O) = wi (r,O) = 0,

VI WiELOO(O,T;V1),nenl1y v=wl, tit (3.88) ta du'Qc

t

t

=- f( F(Ul(T),U{ (T))- F( uz(T),ui(T)), wi (T)}dT.

0

(3.89)

f)~t

t

O"(t)=IIwi (t) W +a( w(t), w(t)) + 2 fh(T) IWi (1, T)IzdT.

0

(3.90)

Tit (3.89) ta suy ra

t

O"(t)=-2 f\F(Ul(T),U{ (T)) - F( uz(T),ui(T)), Wi (T) }dT

0

t

=-2 f(~(Ul(T)) -~(Uz(T)),WI(T))dT

0

(3.91)

t

-2 f(lu{(T)I,B-l U{(T) -lui(T)I,B-l Ui(T))(U{(T)-ui(T))dT

0

~ -2 f\F; (Ul ( T») - F; (uz ( T) ), Wi ( T ))d T

0

~ fllF;(u,(T») -F;(Uz(T»)W dT + fiiwl(T)W dr.

Ap d\:mg b6 d€ 2.20.iii ta duQc

IIF; (UI(t») - F; (uz U») III ~ KR (a)11 wU) II~~ KR(a) aU),

trong d6 R =~nII Ui IIC (O,T;VI).

Til (3.91) va (3.92), ta thu duQc

aU)~

(

1+ KR(a)

] fa(T)dT.

(3.93)

Ap dt,mg b6 d€ Gronwall vao (3.93), ta duQc

aU) = 0, nghla 1a UI= uz.

V~y dinh 1y 1 dff duQc chung minh