Phương trình Parabolic phi tuyến trong miền hình cầu 6

Luận văn thạc sĩ toán Giải Tích-Chuyên đề :Phương trình Parabolic phi tuyến trong miền hình cầu

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KhilO sat phuong trinh parabolic phi tuyin

trang mi~n hinh cdu

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CHUONG4

sV TON T~I, DUYNHA.TvA ON DJNH NGHl~M T-TUANHoAN

Trong chuang nay, chung t6i nghien Clm nghi~m T-tu~n hoan cua

bai toan gia tri bien phi tuySn sau day

(4.1)

(4.2)

(4.3)

(4.4)

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

lim rur (r,t)

!

< +00, ur (l,t) + h(t){u(l,t) - uo) = 0,

r~O+

u(r,O) = u(r,T),

IIP-z

F(u) =u u,

trong d6 2~ p < 3, Uo lel cac h&ngs6 cho truac, h(t), f(r,t) lelcac ham s6

cho truac T-tu~n hoan theo t, thoa cac giii thiSt sau

(Hz)

(H;)

(H~)

UoE JR,

hE wi"" (O,T), h(O)= her), h(t) ~ ho > 0,

f E CO([0,T}H), f(r,O) = f(r,T).

Nghi~m ySu cua bai toan (4.1)-(4.4) duQ'cthanh l~p nhu sau tu bai toan biSn phan sau:

Tim u E LZ(O,T;V)n L'"(O,T;H) sao cho u' E LZ(o,T;H) va u(t) thoa phuang

trinh biSnphan sau:

(4.5)

f(u'(t), vet))dt + J[(ur(t), vr(t)) + h(t)u(l, t)v(l,t)}it

T + f(F(u(t)), v(t))dt

°

= f(f(t), v(t))dt + uo fh(t)v(l,t)dt, '\IvE LZ(O,T;V),

va di@uki~n T-tu~n ho~m

H9C vien Nguyln Vii Dzilng

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Khao sat phuong trinh parabolic phi tuyin

trong mi~n hinh cdu

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(4.6) U(O)=u(T).

Trong ph~n mlY, chung toi se chung minh bai tmln (4.1 )-(4.4) co

duy nhat mQtnghi~m y~u T-tu~n hoan va nghi~m nay cling emdinh d6i vai

I, h, uo.

4.1 S1}'tAn t~i va duy nhAt cua nghi~m y~u T-tuftn hoan

Lien quail d~n s\l't6n t~i va duy nhat nghi~m y~u T-tu~n hoan cua

bai tmln (4.1)-(4.4) ta co dinh ly sau.

Dinh If 4.1 Cho T > 0 va (H), (H~), (H~) dung Khi do, hai loan (4.1)-(4.4) co duy nhdt mr}t nghi?m yiu T-tudn hoan U E L2(0,T;V)nL"'(0,T;H), saD cho

u'EL2(0,T;H), rI/PUELP(Qr).

Chung minh Chung minh g6m nhiSu buac

Bmyc 1 PhU'O'ng phap Galerkin Ky hi~u bai {Wj},j = 1,2, la mQt cO' sa tr\l'c chu~n trong khong gian Hilbert tach duQ'CV Ta tim Urn (t) theo d~ng

(4.7)

rn

urn(t) = LCmj(t)Wj'

j=l

trong do Crnj (t), 1::;j ::;m thoa h~ phuO'ng trinh vi phan phi tuy~n

(u~ (t), Wj) + (urnr, Wjr) + h(t)urn (1,t)w/1) + (F(urn (t)), Wj)

= (/(t), Wj) + uoh(t)wj(l), 1::;j::; m,

(4.8)

va diSu ki~n T-tu~n hoan

(4.9) Urn (0) = Urn(T).

f)~u tien, ta xet h~ phuO'ng trinh (4.8) va diSu ki~n d~u

( 4.9' ) Urn(0) = UOrn'

Wj' j = 1,2, Khi do, ta thu duQ'c mQt h~ m phuO'ng trinh vi phan thuemg

H9C vien Nguyln Vii Dziing

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trong miJn hinh cdu

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phi tuySn v6i cac ~n ham Cmj(t),1<5,j <5,m, va cac diSu ki~n d~u (4.9') DS

th~y r~ng t6n t~i um(t) co d~ng (4.7) thoa (4.8) va ( 4.9') v6i h~u khAp nO'i

tren 0<5,t <5,Tm' v6i mQtTm' 0 < Tm<5,T Cac danh gia tien nghi~m sau day cho

phep ta l~y Tm= T v6i mQi m.

Bmyc 2 Danh gia tH~nnghi~m.

Nhan phuO'ng trinh thu j cua h~ (4.8) b6i Cmj (t), va sau do l~y tfmg theo j,

ta duQ'c

1

~llum(t)112 +21IUmr(t)112+2h(t)u;(l,t)+2 Jr2Ium(r,t)IP dr

= 2(f(t),um (t)) + 2uoh(t)um(l,t).

Tir giii thiSt (H~ ) va b~t d~ng thuc (2.9), ta suy ra r~ng

(4.11) 211umr (t)112+ 2h(t)u~ (l,t) ~ CiliUm(t)II~,

v6i CI = mill {I,ho}.

Do do, ta suy tir (4.10), (4.11) r~ng

:tllum(t)112 +CIIIUm(t)II~ +2fr2Ium(r,t)IP0 dr

<5,2(f(t),um (t)) + 2uoh(t)um(l,t)

<5, ~llf(t)112 +51IUm(t)112+ ;Iuonh[ +51IUm(t)II~

= ~llf(t)112+ ;Iuonhll: +251IUm(t)II~,'15>0

(4.12)

ChQn 5 > 0 sao cho

(4.13) CI-25=C2>0

Do do, tir (4.12), (4.13) ta thu duQ'c

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~llum dt (t)112 + C211um(t)112

I

~ ~llum(t)112+C21Ium(t)II~+2 fr2Jum(r,tW dr0

~ ;luol21lhll: + ~Ilf(t)112 = ~(t).

Nhan b~t d~ng thuc (4.14) b6i eCz! va sau do l~y tich phan theo t ta thu

(4.14)

duQ'c

!

(4.15) Ilum(t)112 ~lIuomI12e-Cz! +e-Cz! f~(s)eczsds.

0

Cho T > 0, ta xet ham s6 sau

~

{

(eel! -It f~(s)eczSds,O<t~T,

hi (0) / C2, t = O.

Khi do RE Co[O,T].Ta d~t R = max~R(t) Ta thu duQ'c tir (4.15), (4.16) r~ng o,;,!,;,r

nSu lIuomll ~ R, khi do

(4.17) Ilum(t)11~ R, i.e., Tm = T v6i mQi m.

GQi Bm(O,R)la qua c~u dong tam 0, ban kinh R trong khong gian m chiSu sinh b6i cac ham Wj' j = 1,2, ,d6i v6i chu~n 11.11.

Xet anh XI;!Fm : BJO,R) ~ BJO,R) cho b6i cong thuc

(4.18) Fm (Uom) = Um (T).

Ta se chung minh dng Fm lamQt anh XI;!co.

Giasu UOm' VOmEBJO,R) vad~t <1>m(t)=um(t)-vm(t),trongdo um(t) va vm(t)

la cac nghi~m cua h~ (4.8) tren [O,T]thoa cac diSu ki~n d~u um(O)=UOmva Vm(0) = VOm'l~n luQ't Khi do, <1> m(t) thoa h~ phuang trinh vi phan sau day

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«D~ (t), Wj ) + «D mr(t), Wjr) + h(t)<D m (1, t)Wj (1)

=-\lum(t)1 um(t)-lvm(t)1 vm(t),Wj/' l~J~m,

va diSu ki~n d~u

(4.20) <Dm(O)= Uam-v am'

B~ng cach tinh toan gi6ng nhu 6 chuang 3, ta thu duQ'c

~II<Dm(t)llz + 211<Dmr (t)llz + 2h(t)l<D m(l,t)lz

(4.21) dt

= -2(lum (t)IP-z Um(t) -Iv m(t)IP-z Vm(t),um (t) - v m(t)) ~ 0

Nha vao (4.11), ta sur tir (4.21) r~ng

(4.22) ~11<Dm(t)llzdt +C111<Dmr(t)II~ ~ O

Tich phan b~t d~ng thuc (4.22), ta thu duQ'c

-.!.TC[

(4.23) IIUm(T)-vm(T)II~eZ IIUam-vamll,

i.e., Fm : Bm(O,R)~ BJO,R) la anh x'ilco Do do tan t'iliduy nh~t Uam E Bm(O,R)

sao cho Uam = Fm (uam) = Um (T).

Do do, v6i mQi m, tan t'ili m9t ham UamE Bm(O,R) sao cho nghi~m cua bai

toan gia trj ban d~u (4.8), (4.9') la m9t nghi~m T-tu~n hoan cua h~ (4.8) Nghi~m nay cling thoa b~t d~ng thuc (4.17) v6i h~u hSt tE [o,T] va nha

(4.14), ta sur ra

(4.24) IIUm(t)llz +Cz filum(s)lI~ds + 2 fds frzlum (r,s)IP dr ~ C3,

trong do C3 la m9t h~ng s6 d9C l~p v6i m.

M~t khac, b~ng cach nhan phuang trinh thu j cua h~ (4.8) b6i c~, l~y tlmg

theo j va sau do l~y tich phan d6i v6i biSn thai gian tir 0 dSn T, ta thu

duQ'c

H(Jc vien Nguyin Vfl Dzflng

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KhilO sat phlfO'ng trinh parabolic phi tuyin

trang33

fllu~(t)112dt+- f-llumr(t)112dt+- fh(t)-u~(l,t)dt

T

(4.25) +0f(lum (t)IP-2 Um(t),u~ (t) }dt

= f(J(t),u~ (t))dt + Uofh(t)u~ (l,t)dt.

Tir (4.9') ta th~y r~ng

Td

f-Ilumr (t)112dt = 0,

0 dt

f(

)

= ~ fr2 ~um(r,T)IP -Ium (r,O)IP ~r = O.

Po

Do do, d~ng thuc (4.25), nha itch phan tUng ph~n, ta thu duQ'c

(4.26) fllu~(t)112dt= f(J(t),u~(t))dt+- fh'(t)-u~(l,t)dt-uofh'(t)um(I,t)dt.

Sail cling, nha VelO(4.24), (4.26), ta suy ra b~t d~ng thuc sail

2 fllu~ (t)112dt ~ fIIJ(t)112 dt +fllu~ (t)112dt +IIh'll", fu~ (I,t)dt

T

(4.27) + 21uolllh't]um(l,t)~t

0

~ fllu~ (t)1I2dt +]IJ(t)1I2dt+411h't filum (t)II~ dt

T

+41UolIlh'IL filum (t)lIvdt

0

~ fllu~ (t)1I2dt +fIlJ(t)112dt +411h't filum (t)II~ dt

+4v'Tlil, Illh'll.(~~m (')II~dtr

T

~ fllu~ (t)112 dt + C4 ,

0

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trong do C4 la ill9t h~ng s6 d9C l~p v6'i m.

V~y

T

(4.28) ~Iu~ (t)112dt ~ C4, v6'i illQi m.

0

M~t khac, tir (4.24) ta co danh gia

(4.29) Ids flr2! p'IUrn (r,s)IP-2 Urn(r,sf' dr = Ids fr21urn(r,s)IP dr ~ !C3'

BU'o-c3 Qua gio-i h~n.

Do (4.24), (4.28), (4.29) ta sur ra r~ng, tfm t<;liill9t day con cua day {uJ, v~n ky hi~u la {urn}sao cho

(4.30) Urn~ U trong LOfO(O,T;H)ySu *,

(4.31) Urn~u trong L2(0,T;V) ySu,

(4.32) u~ ~u' trong L2(0,T;H) ySu,

(4.33) r2!Purn~r2!pu trong Y(QT) ySu.

Tru6'c hSt, ta nghi~ill l<;lir~ng

(4.34) u(O) = u(T).

V 6'i illQi v E H, ta co tir (4.9) r~ng

T

(4.35) f(u~ (t), v)dt = (urn(T)- Urn (0), v) = o.

0

Ta sur tir (4.32) va (4.35) r~ng

(4.36) f(u~ (t), v)dt ~ f(u'(t), v)dt = 0, khi m~ +00,

Tinh toan tuong t\1'nhu (4.35), ta cling co d~ng thuc

T

(4.37) (u(T)-u(O),v) = f(u'(t),v)dt=O, VvEH,

0

va do do (4.34) dung.

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Dung b6 dS 2.11 vS tinh compact cua Lions [3], ap d\1ng vao (4.31), (4.32)

ta co thS trich ra tir day {urn} mQtday con v§,nky hi~u la {urn} sao cho

(4.38) Urn -) U m~nh trong L2(O,T;H).

Theo dinh ly Riesz-Fischer, tir (4.38) ta co thS trich ra mQt day con cua day

{urn} v§,nky hi~ula {uJ sao cho

(4.39) Urn (r,t) -) u(r,t) a.e (r,f) trong QT = (0,1)x (O,T).

Do Uf-7IUIP-2 u lien t\1C,ta co

(4 40) 21'I I

I I

P-2

r P Urn(r,f) Urn(r,f) -) r P u(r,f) u(r,f) a.e (r,f) trong QT'

Ap d\1ngb6 dS 2.12 vS S\l hQit\1ySu trong Lq(QT) v6i

II

P-2

Tir (4.29), (4.40) r~ng

(4.41) r2IP'lurnIP-2urn -)r2IP'luIP-2utrong y'(QT) ySu.

Ky hi~u g;(f) = ~ sinC:). i= 1,2, la mQt ccysa tf\lC chu~n trong khong

gian Hilbert th\lc L2(0,T) Khi do t~p {g;Wj:i, i=1,2, } cfing thanh l~p mQt ccysa tr\lc chu~n trong khong gian L2(0,T;V).

Nhan phucyng trinh thil i cua (4.8) cho g; (f), va sail do l~y tich phan d6i v6i biSn thai gian f, 0 ~ f ~ T, ta thu duQ'c

f( u~ (f), Wj ;g; (f)<if + f( urnr(f), Wjr ;g; (f)df

(4.42)

+ fh(t)urn (1, f)w/1)g; (f)df + f(lurn (f)IP-2 Urn(f), Wj )g; (f)df

= f(f(f), Wj ;g;(f)df + fuoh(f)W/1)g;(f)df, Vi = 1,2, ,m, Vi E N.

DS nghien cUu vS vi~c qua gi6i h~n cua s6 h~ng phi tuySn lurn (f)IP-2 Urn(f)

trong (4.42), ta su d\1ng b6 dS sail

H9C vien Nguyin Vfi Dzfing

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BBd~4.1.

J~~oof([urn (t)IP-2 Urn(t), wi )gi(t)dt = f\lu(t)IP-2 U(t), Wi )gi(t)dt, Vi, j = 1,2,

ChungminhbBd~4.1.

Chuy r~ng (4.41)tuO'ng duO'llg v6i

fdt fr2/P'[urn(t)IP-2urn(t)<l>(r,t)dr~ f fr2/P'lu(t)la-lu(t)<l>(r,t)dt

V<l>E (U' (QT))= LP(QT).

M?t khac, ta co

f(lurn (t)IP-2 Urn(t), Wi )gi(t)dt = f fr21urn (t)IP-2Urn (t)w/r)gi(t)drdt

(

= ffr2/p'IUrn (t)IP-2Urn (t)Xr2/PWi(r)g;Ct)}irdt.

0 0

Do (4.44), b6 dS4.1 seduQ'cchUngminh nSu ta khAngdinh duQ'cr~ng

<l>(r,t)= r21 PWj (r)q:>(t) E U (QT)'Th~t v~y, do bfit dAng thuc (2.7), ta co

f f[<l>(r,t)IP drdt = f ~r2w/r)q:>(tf drdt

= fr2-P Irw/rf dr flq:>(t)IP dt

(4.45)

~ (FsIIWillvr fr2-Pdr ]q:>(tWdt

T

~ ~(Fsllwill 3 r nq:>(t)la+J dt<+00.

Cho m~ +00 trong (4.42), ta suy ra tll (4.30), (4.31), (4.32) va b6 dS 4.1,

r~ng u th6a phuO'ng trinh biSn phan

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(4.46)

f(U'(t), wi )g;Ct)dt + f(u,(t), Wi' )g;Ct)dt

+ fh(t)u(l,t)Wia (l)gj(t)dt + f(lu(t)IP-2 u(t), Wi )gj(t)dta

= f(/(t), Wi )gj(t)dt + Uafh(t)w/l)g;Ct)dt, Vi, j E N.

V~y, ta suy tu (4.46) r&ngphuang trinh sail day dung

f(u'(t), v)dt + f(u, (t), v, )dt + fh(t)u(l,t)v(l)dt

T

+ f\lu(t)IP-2 u(t), v)dta

(4.47)

= f(/(t), v(t))dt+ua fh(t)v(l,t)dt, Vv E L2 (O,T;V}

V~ys\l' t6n t~i nghi~m duQ'c chUng minh xong

Bmyc 4 Tinh duy nh~t nghi~m.

Gia su u va v la hai nghi~m ySu cua bai tmin (4.1)-(4.4) Khi do w=u-v

thoa bai toan biSn phan sail day

f(w'(t), lp(t) )dt + f[( W, (t), lp, (t)) + h(t)w(l, t)lp(1,t)}it

T

(4.48) + f(lu(t)IP-2u(t) -lv(t)IP-2vet), lp(t))dt =0,

a

Vlp E L2(0,T; V),

(4.49) w(O)= weT),

v6i u, VE L2(0, T; V)nD'(O,T;H), u', V'E L2(0, T; H), r2lpu, r21pvE H(QT}

T

L~y lp=w trong (4.48) va chu y r&ng f(w'(t),w(t))dt= o Khi do su d\lng

a (4.11) va (4.49), ta thu duQ'c

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Khao sat ph Lfang trinh parabolic phi tuyin

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(4.50)

2C11Iw(t)II~2(o,r;v)~ fllwr ° (t)112dt +0fh(t)w2 (1, t)dt

r

= - f(lu(t)1 p-2 u(t) -lv(t)IP-2 vet), u(t) - vet) )dt ~ O.

0

DiSunay dfin dSn w= 0, i.e.,u = v Dinh ly 4.1 dugc chUngminh hoan t&t.

4.2 S1}'An dinb cua ngbi~m y~u T-tuftn boan

Trong phan ll<lYchung t6i se kh~lOsat tinh 6n dinh d6i v6i f, h, Uo cua nghi~m ySu T-tuan hoan cua bai tmin (4.1 )-(4.4).

Tuang ung v6i f, h, uo, Ian lugt thoa cac gia thiSt (H2), (H~), (H~), bai

loan (4.1)-(4.4) co duy nh&t mQt nghi~m ySu T-tuan hoan

ph\! thuQc vao u=u(f, h,uo) Ta se chung minh nghi~mnay 6n dinh d6i v6i

f, h, Uotheo illQt nghla ma ta se qui dinh sau.

Tru6c hSt ta d~t

H = {h E wi"" (O,T), h(O) = her), h(t)? ho > a},

y = {JE CO([O,T];H), f(r,O) = f(r,T)}.

Khi do, ta co dinh ly sau day lien quail dSn tinh 6n dinh cua nghi~m ySu

Binb Iy 4.2.

Nghi<?mu = u(f, h, uo) 6n dtnh dr5ivai f, h, uo, theo ngh'ia

Niu (fk,hk,uOk)' (f,h,uo)EYxHxJR, saocho

fk ~ f trong CO([O,T];H),

(4.51) hk ~h trong w1""(0,T),

UOk~ u trong JR,

thi

(4.52) Uk~u trong L2(0,T;V) va r21puk~r2lpu trong LP(Qr),

trongdo Uk =U(fk,hk,uOk)' u =u(f,h,uo).

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Chung minh.

Truac hSt ta d~t them cac ky hi~u

Vk =Uk -u, Jk =fk - f, hk =hk -h, UOk=UOk-uo.

Cho v E L2(O,T;v) tllYy, tru hai d~ng thuc sau:

(4.53)

f(u~ (t), vet) )dt + f[(Ukr(t), Vr (t)) + hk (t)Uk (1, t)v(1, t)]dt + f(F(Uk (t)), vet) )dt

= f(fk (t), v(t))dt + UOkfhk (t)v(l,t)dt,

Uk(0) = Uk(T),

(4.54)

f(U'(t), vet) )dt + f[(ur (t), Vr(t)) + h(t)u(l,t)v(l, t)]dt + f(F(u(t)), vet) )dt

= f(f(t), v(t))dt+UOk fh(t)v(l,t)dt,

U(O) = u(T),

ta thu duQ'c

(4.55)

f(v~ (t), vet) )dt + f[( Vkr(t), Vr (t)) + (hk (t)Uk (l,t) - h(t)u(l, t) )v(1, t)]dt

T + f(F(Uk(t))-F(u(t)),v(t))dt

0

=f((Jk(t)}v(t))dt+ f(UOkhk(t)-uoh(t))v(1,t)dt.

Ch<;>nv = Vk' trong (4.55) va sau khi chu y r~ng

(4.56) f((v~(t)), Vk(t))dt = - f-lIvk (t)lldt= -llvk (T)II llvk (0)11= 0,

ta thu duQ'c

Trang 13

Khao sat phuang trinh parabolic phi tuyin

trang 40

(4.57)

fllvkr (t)112dt + f(hk (t)Uk (l,t) - h(t)U(l,t))vk (l,t)dt

T + f(F(Uk (t))- F(u(t)), Vk(t))dt

0

= f(Jk (t), Vk(t) }dt + f(UOkhk(t) - uOh(t))vk (l,t)dt,

hay

flhr (t)112dt + f hk (t)vi (l,t)dt + fhk (t)u(l,t)vk (l,t)dt

T

+ f(F(Uk (t))-F(u(t)),Vk (t))dt

0

= f(Vk (t)) Vk (t) }dt + UOkfhk (t)Vk (l,t)dt

T +uo fhk(t)vk(l,t))dt.

0

Chu y r&ng

(4.59) ]IVkr(t)112dt + fhk (t)vi (l,t)dt ~ C) ]h(t)II~dt = c)llvkll~2(O,T;V)'

trong d6 C) = mill{1,ho} Dung bfit d~ng thuc

(4.60) '\Ip~2, 3Cp >O:~XIP-2X-lxIP-2XXX-y)~cplx-yIP '\Ix, yeIR,

ta suy ra

(4.61) f(F(Uk (t))- F(u(t)), Vk(t))dt ~ Cp fdt fr21uk (t) - u(t)IP dr =Cp IIr2/ PVk II;p(Qr)'

B&ng each sir d\lng bfit d~ng thuc (4.60), ta suy ra tll (4.58)-(4.61) r&ng

c)llvkll~2(O,T;V) +Cpllr2/Pvkll;p(Qr)

(4.62)

~ - fhk (t)u(l, t)v k (1, t)dt + f\Jk (t), Vk (t) }dt

+UOkfhk (t)Vk (l,t)dt + Uo fhk(t)Vk (l,t)dt

H9C vien Nguyln Vii Dzilng

Trang 14

trang 41

S 411hkt fllu(t)llv Ilvk(t)llv dt + Illkllco([O,T];H) fllvk (t)llv dt

+ 21uOk Illhk t ~h (t)llv dt + 211hk t Iuol fllvk (t)IIv dt

S 411hk IIJuIIL2(O,T;V) Ilvk IIL2(O,T;V)+ Illk Ilco([o,T];H)Frllvk t2(O,T;V) + 21uOk Illhk t Frllvk t2(O,T;V) + 211hk t IUDIFrllvk IIL2(O,T;V)

=[2(21Iut2(O'T;V)+ Frluolllhkt + Frlllkllco([O,T];H)+2FrllhkIUuOkl]lvkt2(O,T;V)

== 8k IIVkt2(O,T;V)'

tfong do

(4.63) 8k =2(21IuIIL2(O'T;V) + Frluol)llhkt + Frlllkllco([O,T];H)+2FrllhktluOkl.

Ta Suy fa tu (4.62) dng

1 (4.64) Ilvk(t)112( .

) S-8k'

L O,T,V C)

II

21

li

(4.65) c)llvkIIL2(O,T;V)+Cpr PVkLP(QT)SC) k'

11

21p

21pV

II

(4.67) Ih L2(O,T;V) + r k LP(QT)- C) ~C)Cp

Tu gia thiSt (4.51), ta co

Il hkII ~ 0,

00 CO([O,T];H)

va day s6 ~Ihk t } bi ch~n, nen 8k ~ O.

V~y

I II II

21pV

IVk L2(O,T;V) + r k LP(QT) - c) ~C)Cp

Dinh ly 4.2 duQ'c chung minh hoan t~t

Tiêu đề	Khảo sát phương trình parabolic phi tuyến trong miền hình cầu
Tác giả	Nguyễn Văn Dũng
Trường học	Học Viện
Thể loại	Luận văn

Định dạng
Số trang	14
Dung lượng	3,24 MB