Luận văn thạc sĩ toán Giải Tích-Chuyên đề :Phương trình Parabolic phi tuyến trong miền hình cầu
Trang 1Khao sat phuong trinh parabolic phi tuyin
trong mi€n hinh cdu
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CHUONG3
sV TON T~I vA DUY NHAT NGHlt:M CUA PHUONG TRINH
NHIET Val tUEU KIEN DAu
Trong chuang mlY,chung toi nghien Clmbai toan gia trt bien va ban d~u (1.1)-(1.3) nhu sau:
(3.1)
(3.2)
(3.3)
(3.4)
Ut-a(t{urr+~Ur)+F(U)=f(r,t),O<r<l, O<t<T,
I
limrUr(r,t)
1
<+00, ur(l,t)+h(t)(u(l,t)-lIo)=O'
r-+O+
u(r,O) = uo(r),
IIP-2
F(u) =u u,
trong do 2~ p < 3, lIo la cac h&ng s6 cho truac, aCt),her), f(r,t), uo(r) la cac ham s6 cho truac thoa cac diSu ki~n sau:
Nghi~m ySu cua bai toan gia trt bien va ban d~u (3.1)-(3.4) duQ'c thanh l~p nhu sau:
Tim u E L2(0,T;V)nLoo(0,T;H) sao cho u(t) thoa bai toan biSnphan
sau
d
-(u(t), v) + a(t)(ur(t),vr) + a(t)h(t)u(l,t)v(l) + (F(u(t)),v) dt
= (f(t), v) + lIoa(t)h(t)v(l),VvE V, a.e.,t E (O,T),
va diSu ki~n d~u
(3.5)
(3.6) u(O)=UO'
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(HI) Uo EH,
(HJ lIo E JR,
(HJ a, hE W1,oo(0,T), aCt) ao > 0,
(H4) f E L2(0,T;H).
Trang 2Khao sat phuang trinh parabolic phi tuyin
trong mi~n hinh cdu
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Khi do ta co dinh ly sau
Dinh ly 3.1 ChoT>O va (H)-(H4) i/ung.Khi i/o, bat toim (3.1)-(3.4)co duy nhdt mr}t nghi?m yiu UELZ(O,T;V)nL"'(O,T;H),sao cho
(3.7) tu E Loo(o,r; v), tu' E L2 (0, r; H), r2/ P E LP(Qr).
Chung minh ChUng minh g6m nhiSu bu6c.
Bmyc 1 PhuO'ngphap Galerkin Ky hi~u bai {wJ j =1,2, la mQt ca So'
tr\Ic chu~n trong kh6ng gian Hilbert tach duQ'CV Ta tim Um (t) theo d~ng
(3.8)
m
Um (t) = 2:>m/t)Wj'
j=l
trong do Cmj (t), 1~ j ~ m thoa h~ phuang trinh vi phan phi tuySn
(u~ (t), Wj) + a(t)(umr, Wjr) + a(t)h(t)um (1,t)w/1)
+ (F(um (t)), Wj) = (J(t), Wj) + uoa(t)h(t)wj (1), 1 ~ j ~ m,
(3.9)
(3.10) um(O)=uOm'
trong do
DS thfiy r~ng v6i mQi m t6n t~i mQt nghi~m um(t) theo d~ng (3.8) thoa (3.9)
va (3.10) hAu kh~p nai tren 0~ t ~ Tm v6i mQt Tm' 0 < Tm~ T Cac danh gia
tien nghi~m sau day cho phep ta lfiyTm= T v6i mQi m.
BU'O'c2 Danh gia tieD nghi~m.
Ta se IAnluQ'tthiSt l~p hai danh gia tien nghi~m du6i day Kho khan chinh
trinh, do do vi~c danh gia tinh bi ch~n va sau do qua gi6i h~n cua s6 h~ng
phi tuySn nay la mQt kho khan
a) Danh giG thu nhdt Nhan phuang trinh thu j cua h~ (3.9) bai Cmj(t)va t6ng theo j, ta co
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Trang 3Khao sat phuong trinh parabolic phi tuyin
trong miJn hinh cdu
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1
~llum(t)112 +2a(t)llumr(t)112+2u;(1,t)+2 fr2Ium(t)IP dr
= 2(1- a(t)h(t)u; (l,t) + 2(f(t),um (t)) + 2uoa(t)g(t)um(l,t).
Tir b~t ding thuc (2.9), ta sur ra dng
2a(t)lIumr (t)112+ 2u; (l,t) ~ 2aollumr(t)1I2+ 2u; (l,t)
(3.13) ~ 2 mill {I, ao}~Iumr(t)112+ u; (1,t) )~ allium(t)II~,
val al =-mm 1,aa
3
Ta sur tir (2.6)-(2.8), (3.12), (3.13) r~ng
a
~ 211- a(t)h(t)1 ~llumr (t)112 + (3 + 1/ fJ ~Ium (t)II2 ]
+ 211f(t)llllum (t)11+ 4Iuaa(t)h(t)1 IIum(t)llv
~ 2(1 + IlahllJ~llumr (t)II~ + (3 + 1/ fJ~lum (t)112]
+11/(t)112 +Ilum (t)II2 +~IuDnahll: + 2fJllum (t)II~
~ ~ /uanah!l: + Ilf(t)112 + 2fJ(2 + liGht )llum(t)II~
+ [1+ 2(3 + 1/ fJ Xl + Ilahll", )]llum (t)112, V fJ > 0, trong do ky hi~uIHI",= II-llroo(a,T) dS chi chu~n trong L"'(O,r).
(3.14)
ChQn fJ > 0 sao cho
(3.15) 2fJ(2+lIahIIJ~ ~al'
Do do, tir (3.14), (3.15) ta thu duQ'c
I
~llum(t)1I2 +~alllum(t)II~ +2 fr2Ium(r,t)IP dr
~ ~IUanah!l: +IIf(t)112+[1+2(3+1/ fJX1+IIahIIJ]llum(t)112.
Tir (3.16), l~y tich phan theo t, va su d\lng (3.10), (3.11) ta duQ'c
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Trang 4Khiw sat phuong trinh parabolic phi tuyin
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(3.17)
Ilum (t)112 + -al filum(s)ll~ds + 2 Ids fr21um(r,s)IP dr
~lluomll' + ~luol'llahll:+ ~lf(S)II'ds
t
+ [1 + 2(3 + 1 / fJ Xl + lIaht)] filum (s )112ds
0
t
~ M}2) + M~) filum (s)112ds,
0
trong doM j1) , M i2) la cac h~ng s6 chi ph\! thuQc vao T va duQ'cchQn nhu
sail
Mil) =1 + 2(3 + lIfJX1 +Ilaht),
M;" "llao.II'+(~ lu,I'llahll: )T+ pli(s)II'ds, ';1m.
Ap d\!ng b6 dS Gronwall, ta thu duQ'ctir (3.17) r~ng
Ilum(t)112+-al fllum(s)ll~ds+2 Ids fr2Ium(r,sf dr
~Mi2) exp(tMil»)~ M T'
\;fm, \;ft, O~t~Tm ~T, i.e., Tm=T.
b) Danh gia thu halo Nhan (3.9) b6i t2C~j(t)va t6ng theo j, ta co
21[tu~(t)112+~
[ a(t)lltumr(t)112 +a(t)h(t)t2u;(1,t)+~t2 Jr2IUm(r,t)IPdr ]
= Ilu r(t)112~~2a(t)]+u;(1,t)~~2a(t)h(t)] m dt dt
I
+~t fr21um(r,t)IP dr +2(tf(t),tu~ (t))
p 0
+ 2uo~ ~2a(t)h(t)um (1,t)]- 2uoum (1,t)~ ~2a(t)h(t) J
Tich phan (3.19) d6i v6'i biSn thai gian tir 0 dSn t, sail do s~p xSp l~i cac s6
h~ng, ta thu duQ'c
(3.19)
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Trang 5Khao sat phuong trinh parabolic phi tuyin
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2 fllsu~ (s)112ds + a(t)lltumr (0112+ t2u~ (l,t) + 3 t2 fr21um (r,tf dr
=[1- a(t)h(t) ]t2u~ (l,t) + f[S2 a(s) J IJumr(S)112 ds
0 (3.20)
t
J
+ f[s2a(s)h(s) u~(l,s)ds+i fsds fr2Jum(r,sf dr
t
+ 2 f(sf(s),su~ (s))ds + 2uot2a(t)h(t)um(l,t)
0
t
-2uo f[s2a(s)h(s)J um(l,s)ds.
0
Dung b~t d~ng thuc (2.9), ta co
(3.21) a(t)/itumr(t)112 +t2u~(l,t)~~/ltum(t)II~,2 VtE[o,rl Vm.
Dung cae b~t d~ng thuc (2.6), (2.8), (2.9) va v6i 13 > 0 nhu trong (3.15), ta
danh gia khong kho khan cae s6 h~ng a vS phai cua (3.20) nhu sau
[1- a(t)h(t)] t2u~ (1, t) ::; (1 + Ilahll ) (.alltumr (0112 + (3 + 1/ 13~Itum (0112)
::; (1 + lIGht )~lltum (Oll~ + (3 + 1/ fJ)t2 M T)
f[S2a(s)]'/lumrCs)1I2 ds+ f[S2a(s)h(s)Ju~(l,s)ds
(3.23) ~[(t2a) ro +4&2ah) J}IUm(S)II~£i,
2
[
]
~-MT (t2a) +4 (t2ah) ,
(3.24)
21uofiB' a(s)h(s) J u (l,s)ds ,; ~uolll{t'ahfII ~Iu (S)II, ds
,; 4juolllV'ahfll ~Olu.(s)ll~dS r
';~UoIIIV'ahfll JT ~,
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Trang 6KhilO sat phuong trinh parabolic phi tuyin
trong mi~n hinh cdu
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(3.26)
- fsds fr2Ium(r,s)IPdr 5,-t Ids fr2Ium(r,s)IPdr
2Iuot2a(t)h(t)um(1,t)/5, 2IuoltllahIIJtum(1,t)/
5, 41uoItllaht Iitum(t)llv
5, plltum (t)II~+;(uotjjaht)2,
(3.27) 12f(sf(s),su~ (s»)ds 5, fllsf(s)112ds + fllsu~ (s)112 ds. .
(3.28)
Do do, ta suy tu (3.20)-(3.27) f~ng
t
~Isu~ (s)112 ds + ~al11tum(t)II~
+~M,[11&2aill. +411&'ahill.J+2T;,
+ Fls!(s)1I"b + 41u,IIIV'ahfII § ~: MT
4
+ pT2 uollhlL 5,Mp trong do if r la cac h~ng sa chi ph\! thuQCvao T.
M~itkhac tu (3.18), ta co danh gia
Ids ]r2/P'F(um(r,s)f dr = Ids fr21Ium(r,s)IP-r dr
=Ids fr2Ium(r,sf dr 5,-Mr 5,Mr,
BU'o-c3 Qua gio-i h~n.
Do (3.18), (3.28), (3.29) ta suy fa f~ng, tan t~i illQt day con cua day {uJ, v~n ky hi~u la {um}sao cho
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Trang 7KhilO sat phu:ang trinh parabolic phi tuyin
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(3.30)
(3.31)
(3.32)
Urn~ U trong L2(O,T;V) ySu,
turn ~ tu trong Loo(O,T;V) ySu *,
(3.33) (tuJ ~ (tu) trong L2(O,T;H)yell,
(3.34) r2/ PUrn~ r2/ Pu trong Y (Qr) ySu.
Dung b6 dS 2.11 vS tinh compact cua Lions [3], ap d\mg vao (3.32), (3.33)
ta co thS trich fa tu day {urn} mQtday con van ky hi~u la {urn} sao cho
(3.35) turn~ tu m(;lnh trong L2(O,T;H).
Theo dinh ly Riesz-Fischer, tu (3.35) ta co thS trich ra mQt day con cua day
{urn} van ky hi~ula {urn}sao cho
(3.36) Urn (r,t) ~ u(r,t) a.e (r,t) trong Qr = (O,l)x(O,T).
Do F(u) = lujP-2u lien t1,1C,ta co
(3.37) F(urn(r,t)) ~ F(u(r,t)) a.e (r,t) trong Qr'
Ap d1,1ngb6 dS 2.12 vS S\l'hQi t1,1ySu trong Lq(Qr) v6i
I I
1I
P-2
N = 2, q = p, Grn= r PF(urn)= r P urn urn' G=r P F(u) = r P u U.
Ta suy tu (3.29), (3.37) r~ng
Grn~ G trong LP'(Qr) ySu,
hay
(3.38) r2/P'lurnIP-2urn~r2/P'luIP-2u trong LP'(Qr) ySu.
Gia sir rpE C1([O,TJ),rp(T)= o Nhan phuong trinh (3.9) v6i rp, r6i tich phan hai vS theo biSn t, ta duqc
Trang 8KhilO sat phuang trinh parabolic phi tuyin
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(3.39)
- (Uorn,wi )rp(O) - f(Urn(t), Wi )rp'(t)dt + fa(t)( Urnr (t), Wir )rp(t)dt
+fa(t)h(t)urn (l,t)w/l)rp(t)dt + f(F(urn (t)), Wi)rp(t)dt
= f(/(t), Wi)rp(t)dt + fa(t)h(t)w/l)rp(t)dt, 1:5,j :5,m.
DS qua gi6i h~n cua sf>h~ng phi tuySn F(urn (t)) = Iurn(t)IP-2 Urn(t) trong (3.39),
ta su d\lng b6 dS sail
nBd~3.1
l~~oof(F(urn (t)), wi )rp(t)dt = f(F(u), Wi )rp(t)dt.
ChUngminh b6 d~3.1.
Chuy r~ng(3.38) tUOTIgdUOTIgv6i
f fr2/P'IUrn(t)IP-2Urn(t)<1>(r,t)dtdr~ f fr2/P'lu(t)IP-2u(t)<1>(r,t)dtdr,
V<1> E (U' (QT))' = U(QT)'
M~t khac, ta co
f(F(urn (t)), Wi )rp(t)dt = ffr21urn (t)IP-2 Urn(t)Wi (r)rp(t)drdt
= ff~2/ p'IUrn(tW-2 Urn(t) ~r2/ Pwi(r)rp(t) }irdt.
0 0
Do (3.40), b6 dS 3.1 se duQ'cchung minh nSu ta nghi~m l~i duQ'cr~ng
<1>(r,t) = r2/Pwi(r)rp(t) ELP(QT)'Th~t v~y, do b~t d~ng thuc (2.7), ta co
(3.42)
ffl<1>(r,t)IPdrdt =ffr2Iwi(r)rp(tf drdt
=fr2-plrwi(rf dr ]rp(t)IPdt
:5,(FsIIWillv! fr2-Pdr flrp(tW dt
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Trang 9KhilO sat phuO'ng trinh parabolic phi tuyin
trong miJn hinh cdu
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T
~ 3 ~ p (~IIWjllv r flqJ(t)la+ldt < +00.
V~y bE>d@3.1 duQ'c chung minh ho~m t~t
Cho m ~ +00 trong (3.39), ta sur ra tir (3.11), (3.30), (3.31) va bE>d@3.1, r&ng u thoa phuang trinh biSn phan
(3.43)
- (uo, Wj }qJ(O)- f(u(t), Wj }qJ'(t)dt + fa(t)(ur(t), Wjr }qJ(t)dt
+ fa(t)h(t)u(l,t)wj(1)qJ(t)dt + f(F(u(t)), Wj}qJ(t)dt
= f(f(t), Wj}qJ(t)dt + lio fa(t)h(t)wj (l)qJ(t)dt,
1 ~ j ~ m, \t qJE c1 ([0, T]), qJ(T)= O.
Do do ta co
(3.44)
- (uo, v)qJ(O)- f(u(t), v)qJ'(t)dt + fa(t)(ur (t), Vr )qJ(t)dt
+ fa(t)h(t)u(l,t)v(l)qJ(t)dt + f(F(u(t)), v)qJ(t)dt
= f(f(t), v)qJ(t)dt.+ lio fa(t)h(t)v(l)qJ(t)dt,
\t qJE C1([0, T]), qJ(T) = 0, \tv E V.
f- [(u(t), v)}P(t)dt + fa(t)(ur (t), Vr)qJ(t)dt
+ fa(t)h(t)u(l,t)v(l)qJ(t)dt + f(F(u(t)), v)qJ(t)dt
= f(f(t), v)qJ(t)dt+ liofa(t)h(t)v(l)qJ(t)dt,
\tqJED(O,T), \tVEV.
Do do, ta co
(3.45)
Trang 10KhilO sat phuong trinh parabolic phi tuyin
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(3.46) ~ [(u(t),v)]+ a(t)(ur (t), vr) + a(t)h(t)u(l, t)v(l) + (F(u(t)), v)
= (J(t), v)+ uaa(t)h(t)v(l), Vv E V, dung trong D(O,T) va do do hAllhSt trong (O,T).
Cho q>E C1([0,TJ),q>(T) = o Nhan phuang trinh (3.46) cho q>,sau do tich
phan hai vS theo biSn thai gian ta thu duQ'c
(3.47)
- (u(O), v)q>(O)- f(u(t), v)q>'(t)dt + fa(t)(ur (t), Vr )q>(t)dt
+ fa(t)h(t)u(l,t)v(l)q>(t)dt + f(F(u(t)), v)q>(t)dt
= f(J(t), v)q>(t)dt + ua fa(t)h(t)v(l)q>(t)dt,
v q>Eel ([O,TJ), q>(T) = 0, Vv E V.
SO sanh (3.44) v6i (3.47), ta thu duQ'c
(3.48) - (u(O), v)q>(O)= -(ua, v)q>(O), V q>Eel ([O,T]), q>(T) = 0, Vv E V,
ma (3.48) tuang duang v6i diSu ki~n dAti
Ta chu yr~ng, tiT(3.30)-(3.34) ta co
u E L2(0,T;V)nD"(0,T;H), tu E L"'(O,T;V),
tu' E L2 (0, T; H), r21pu E LP(QT}
V~y SlJt6n t~i nghi~m duQ'c chung minh
BtrO'c 4 Tinh duy nh§t nghi~m
Tru6c hSt ta cAn b6 dS sau day
BB d~ 3.2 GiGsir w Ia nghi?m yiu Gilabai toim sau
(3.50) Wt - a(t)( Wrr + ~Wr) = f(r,t), 0 < r < 1, 0 < t < T,
(3.51) I;~~ rwr(r,t)! < +00,-Wr(l,t) = h(t)w(l,t),
Trang 11Khao sat phuong trinh parabolic phi tuyin
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(3.52) w(r,O)= 0,
(3.53) WE L2(0,T;V)nL"'(0,T;H), twE L"'(O,T;V), tw' E L2(0,T;H).
Khi d6
(3.54)
~llw(t)112 + fa(s)~IWr (s)112+ h(S)W2 (1,s) ]ds
t
- f(J(s), w(s))ds = 0, a.e t E (O,T).
0
Chti thich 2 B6 dS 3.2 Ii mQt S\ft6ng quat hoa cua b6 dS trong cu6n sach
cua Lions [3] cho truemg hQ'Pkhong gian Sobolev co trQng ChUng minh b6
dS 3.2 co thS tim thfiy trong [2] Bay gia, ta se chUng minh tinh duy nhfit nghi~m
Gicisiru vi v Ii hai nghi~m ySu cua bii toan (3.1 )-(3.4) Khi do w = u - v Ii
nghi~m ySu cua bii toan (3.50)-(3.52) v6i vS phcii cua (3.50) Ii
J(r,t) = -lu(t)IP-2u(t) + Iv(t)IP-2V(t).Dung b6 dS 3.2, ta co ding thuc sau
(3.55)
~llw(t)112 + fa(s)~lwr(s)112 +h(S)W2(1,S)]ds
t
=- f(lu(sf-i u(s) -lv(s)la-i v(s), w(s) )ds ~O.
0
Do tinh chfit dan di~u tang cua him s6 th\fc u ~ lulP-2 u TIT(3.55) ta suy ra
r&ng w = o Tinh duy nhfit duQ'c chung minh
V~y dinh Iy 3.1 duQ'c chUng minh xong
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