khảo sát phương trình sóng phi tuyến trong không gian sobolev có trọng, chương 4

tài liệu tham khảo khảo sát phương trình sóng phi tuyến trong không gian sobolev có trọng, chương 4

CHUONG 4

KHAO SAT PHUONG TRINH

Uti - (u". + >, ) + F(u, u,) = f(r ,t)

Trong chuang nay, chung ta vfin khao sat st! t6n tC;liva duy nha't nghi~m cua bai toaD (3.1) - (3.4) trong chuang 3, nhung thay nhom gia thi€t thu nha't (HI) - (Hs) bdi nhom gia thi€t thu hai (H~)- (H~) y€u han se du<Jcthanh l~p duoi day Trong chuang nay, chung t6i co sa dl;lngden phuong phap toaD ta don di~u d~ qua giOi hC;lnphi tuy€n IuJi-l Ut. B~ vi~c chung minh khoi l~p lC;li,chung ta sa dl;lng m(>tsf) k€t qua trong chuang 3

Xet bai toaD(3.1)- (3.4)

Ta vi€t lC;libai toaD bi€n phan (3.7) bdi (4.1) sau

~ (ul (t), v) + a(u(t), v)+\ F(u(t), ul (t»), v) +( h(t)UI (l,t)+ get) )v(l)

= (f(t), v), \;IvE VI' a.e t E (O,T).

(4.1)

Ta thanh l~p nhom gia thi€t thu hai nhu sau

(H~)

(H~)

(H~)

(H~)

(H~)

ho>O, O<a<3, O<p~l,

fEL2(O,T;Vo)'

gEHI(O,T),

hECo(JR), h(t)~~>O, Vt~O,

UOEV, ' UIEVo.

Sau day la dinh 19 v~ st! t6n tC;liva duy nha't nghi~m cua bai toaD (3.1) - (3.4) voi nhom gia thi€t thu hai

Binh ly 2. Cho trude T > 0 va (H~)-(H~) tholl Khi do bai loan (3.1) - (3.4) eo it

nhdt mQt nghifm ytu U E L 00(O,T; VI) sao eho

ul ELOO(O,T;Vo)' u(1,.)EHI(O,T),

ra+luELoo(O,T;La+I(Q)), r/3+lul EL/3+I(QT)' Han nila, ne'u 1~ a < 3 thi nghi~m ye'u nay la duy nhdt.

Chung minh Vi~c chung minh dinh ly 2 cling duQc chia lam nhi€u bltoc nhu chung minh dinh ly 1

Bu'oc 1 Xa'p XlGalerkin

Huoc nay dinh ly 1 dff chung minh sl! t6n tc.ti nghi~m x:1p Xl Galerkin

m

um(r,t) = L>mj(t)w/r), tE[O,Tm]c[O,T] cua bai tmln bien phan (4.1) voi nhom

j=1

gii thi€t thu hai (H~)-(H~) Cac danh gia tien nghi~m sail day cha phep ta l:1y

Tm = T, '\1m.

Ta viet lc.ti(3.9) va (3.10) tudng ung bdi (4.2) va (4.3) sail

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

+(h(t) u~ (1,t) + get) )w}(1) =(/(t), WI)' 1 ~ j ~ m.

(4.2)

m

Um(0) = UOm=I am} Wj ~ Uo m{lnh trang ~, khi m~ +00,

j=!

(4.3)

m

U~(0) =Ulm = I PmjWj ~ u! m{lnh trang ~p khi m ~ +00.

j=!

Bu'oc 2 Banh gia tieD nghi~m

Tli chu thkh 3.4 va nhom gii thi€t thu hai (H~)-(H~), ta co th~ ap d1:mglc.tidanh gia 1 trong dinh ly 1

Da v~y, ap d1;1ngdanh gia 1 chung ta cling thu duQc

voi

Xrn (t) = IIU~ (t) W +a( Urn (t),Urn (t)) + 2 fh(T)1 u~ (1,T) 12dT

0

+- frlurn(t)la+Jdr+2ffrlu~(T)1 +Jdrdr.

(4.5)

Tli (4.4), (4.5) va giii thi€t(H~), ta suy ra

(4.6)

Tniong hQp 1 0 < a < 1.

11F;(urn(t))W= frlurn(t)12a dr = frJ-alJ;-urn(t)1 a dr

(4.7)

~K~a Ilurn(t)II~a frJ-adr~ K~a

(

X~(t)

J

a ~ K~a

(

N~J) J

a.

Tniong hQp 2 1~ a < 3

Ap dl;lng b6 de 2.20.i, ta duQc

a

[

N(J)

J

2 1IF;(urn(t))II~K(a)llurn(t)II~~K(a) L .

Co

(4.8)

IIF2 (u~(t))W = IIu~(t)W~ Xrn(t) ~N~l).

Truong hQp2 0 < fJ < 1.

u(;lt P =- > , q =- t 1 - + - = , ta co

J J J

(

J

J

J

~ [(1-jJ)r+jJrlu~(t)12] dr ~ Ii0 +jJXrn(t)~Ii +jJNjl).

Tom l(;li, ta co duQc

IIFrFl(Um)II~2(QT)= JII~(um(r))112 dr.

0

I

IIj;F;(u~(t))l1~2(QT)= III F;(u~(t»112 dr.

0

(4.10)

(4.11 ) Bu'oc 3 Qua giOih~n

Tu (4.4) - (4.11) ta co th€ trich tu day {urn}mQtday con vlin ky hi~u la {urn}'

sao cho

Urn~u trong LOO(O,T;VI) yeu *, u~ ~UI trong LOO(O,T;Vo)yeu *,

00 ,.('

urn(1,.)~u(1,.) trong L (O,T) yeu*,

I ,.('

urn(1,.)~u(1,.) trong H (O,T)yeu,

ra+1Urn~ ra+1u trong L00 (0, T;La+1(Q)) ye'u *,

-r/3+1u~ ~ r/3+1Ul trong L/3+1(QT) ye'u,

j;F; (urn)~ XI trong L2 (QT) yell,

j;F2(u~)~ X2 trong L2(QT)yeU.

Chu yding, tu (4.18), (4.19) ta co

f;(um) ~ J; trong L\O,T;Vo) yell,

F;(u~)~ J; trong L2(0,T;Yo) yell.

Ly lu~n tu'dngtl! nhu'trong dinh ly 1, voi chu y r~ng ham F;(u)= Iu la-Iu,

0 < a < 3, lien t1:1Ctren JR, ta co

Ta viet l~i (3.80) nhu'sau

- I(u~,(t),Wj)q/(t)dt + Ia(urn(t),wj)<p(t)dt + I(F;(urn(t)),wj)<p(t)dt

+ I(F; (u~(t)), Wj) <p(t)dt + I( h(t)u~(1,t)+ get) ) Wj(1)<p(t)dt

T

= f(f(t), Wj)<p(t)dt,v j, V<pED(O,T).

0

(4.12) (4.13) (4.14) (4.15) (4.16) (4.17) (4.18) (4.19)

(4.20) (4.21)

(4.22)

(4.23)

Qua gidi hc.lllkhi m ++00 trong (4.23) bdi (4.12)- (4.15), (4.21) va (4.22), ta du'<;fc

f[ ~ (a'(t), Wi) +a( Get),Wi) +( F, (a(t») + J;, Wi) ]<p(t) dt

T

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

°

T

= f(J(t)'Wj)qy(t)dt, V}, VqyED(O,T).°

(4.24)

Tit (4.24) ta suy ra

:t (uI(t),v)+a(u(t),v)+(F;(U(t))+ Fr'v)

+( h(t)ul (l,t)+ get))v(l) = (J(t), v), VvE VI,

B~ chung minh u la nghi~m ye'u cua bai toan (3.1)- (3.4) ta con phai chung

(4.25)

minh u(O)= UO'ul (0) = UI va F2(ut) = Fr.

a) Chung minh u(O)= Uotu'dng tl! nhu' trong dinh ly 1.

b) Chung minh ul(0)=u, Ta coham

n-~(u~(t),wi) EL2(0,T), Vi, '11m.

Tit (4.2), ta suy ra

t ~ ~(u~(t), w}) = (J(t), Wi) -a( urn(t), Wi)- (F( Urn(t),U~(t)),W})

-( h(t)u~(l,t)+ get) )w/l) EL2(0,T), V},'11m.

(4.26)

(4.27)

Tit (4.26) va (4.27) ta suy ra ham: t~(u~(t),wi) EHI(O,T), Vi, '11m,va do do thuQc v€ CO([0, T]) Tu'dng tl! ta co t ~ (Ul (t), Wi) E CO([0, T]), Vi.

Do phep nhung HI(O,T) c.Co([O,TD la compact va cac ham t ~(u~(t), Wi)'

t ~ (Ul(t), Wi) lien t1,1C,nen ta co

l(u~(O)-ul (0), Wi)1 ~ l(u~(O)-U~(t), wi )1+ I(u~(t)-Ul (t), Wi)1 (4.28)

+I( ul (t)- ul (0), wi )1 + 0 khi m + +00, t + 0+

Tit (4.28), ta co suy ra u~ (0) + ul (0) trong VI ye'u.

Ke't h<;fp (4.3) ta du'<;fc Ul (0) = ul.

) Ch/ . h - %z

c (tng mIll F;(Ut) - J;.

Trudc htt ta chung minh b6 d~ sail

B6 d~ 4.1 GiGsa U ia nghi~m ytu cua hai loan

Uu-(Urr+*Ur) = f(r,t), 0 < r < 1,0 < t <T,

I

lim J; ur(t)

I

< +00,-ur(1,t) = iI(t),

r >O+

u(r,O)=uo' u/(r,O)=ul'

uELOO(0,T;V1), ul ELOO(O,T;Vo)' u(1,.)EH1(0,T),

UoEVI' UI E Va' HE Lz(O,T).

Khi do ta co

t

!II ul (t) W +!II ur(t) W + fiI(T)UI (1,T)dT0

t

~!II U1 W +~ IIUOr W + f(J(T),UI (T))dT, a.e.t E (O,T).

0

Han mia, ntu Uo= UI= 0, thi ta du:r;tc

t

!llul(t)W +!llur(t)W + fiI(T)UI(l,T)dT

0

t

= f(J(T),UI(T»)dT, a.e.tE(O,T).

0

'" (4.29)

(4.30)

(4.31)

Chung minh b6 d~ 4.1 CO'd!nh tl' tz, 0< tl < tz < T va xet ham v(r,t) duQc xac

d!nh giO'ngnhutrong [4] sail

B~t

v(r,t) = [(em(t)Ut(t»)*Pk(t)*Pk(t)]em(t), tE[O,T],

trong do

(i) em(t) la ham lien tlJCtung khuc tren [0,T] duQcd!nh nghia nhu sail

em(t) =

0, neu 0 ~ t ~ tl + l m hoi;ic tz _l m ~ t ~ T,

1 '"'

t+ 2< t < t 2

, neu 1 -m - - z ,m

(

1

)

m t-tl , m neu tl+-~t~ m tl+-, m -m

( t-tz +l ), neu tz -~ ~ t ~ tz _l.

(4.32)

(4.33)

1 2

-[ 1 1]

(iii) (*) la tich ch~p theo bien thai gian t, nghia la

(4.34)

00

-00

L!y tich vo huang cua ham v(r,t) trong (4.32) vai phuong tdnh thu nh!t cua

(4.29), saud6 l!y phantheot tit 0 de'nT , ta duQc

trong d6

0 0

Ymk=- fdt Jg(rur)vdr,

0 0

Zmk= - fdt fr ](r,t)v(r,t)dr.

0 0

(4.37)

Ta Iffn luQt chung minh r~ng

T

limXmk =- fem(t)e~(t)lIul(t)1I2 dt,

k->oo

0

(4.38)

lim Ymk= - fem(t)e~ (t)11ur(t) 112dt + fe~(t)iI(t)ul (l,t)dt ,

k->oo

(4.39)

T

~~Zmk =- fe~(t)(J(t),u/(t))dt.

0

(4.40)

i) Chung minh (4.38) Sa d\:mgtinh cha't cua cac ham em(t) va Pk(t), ta co

Xmk = frdr fern(t)Utt(t)dt f[ (em (s)urCs»)* Pk(S) J pk(t - s)ds

(4.41)

= frdr f[ (em (S)Ut(S»)* Pk (s) Jds f(em (t)Utt(t)) pk(t- s)dt

+00

= f((em(S)Utt(S»)* Pk(S),(em(S)Ut(S»)* Pk (s); ds

-00

T

= f((em (t) Utt (t») * Pk (t), (em (t) Ut (t») * Pk (t); dt

0

= II ~[( em(t)Ut(t») * Pk(t) J,( em (t)Ut(t»)* Pk(t»

)dt

T

- f((e~ (t)Ut(t») * Pk(t),( em(t)Ut(t»)* Pk(t)) dt

()

1 T a

=2"0f atII(em(t)urCt))*Pk(t) 112dt

T

=- f((e~(t)Ut(t»)*Pk(t),(em(t)UrCt))*Pk(t))dt.

()

Cho k ~ +00, tu (4.41) ta thu du'Qc (4.38).

ii) Chung minh (4.39) Ta co

T I

y"'k =- fdt f;(rurCt»)[(em(t)u/(t))*Pk(t)*Pk(t)]em(t)dr

0 0

(4.42)

T I

= - fdtf; [(r ur (t)em (t)) * Pk(t) J [( em (t) ul (t»)* Pk(t) ]dr

0 0

T

=f[(H(t)Bm(t» * Pk(t) ][( em(t)ul (l,t))* Pk(t) ]dt

0

T

+ f((ur(t)em(t))*Pk(t),(em(t)u:(t))*Pk(t))dt.

0

Cho k ~ +00, tu (4.42) ta thu du'Qc

l~Ymk =1 fe;(t)~ II urCt) 112dt + fe;(t)H(t)u/(l,t)dt

= - fem(t)e~(t)11 ur(t) W dt + fe;(t)H(t)ul (l,t)dt.

iii) Chung minh (4.40) Ta co

T

Zmk =- f((em(t)](t)) *Pk(t),(em(t)u!(t))* Pk(t))dt

0

(4.43)

Cho k ~ +00, til (4.43) ta thu duQc (4.40).

D€ qua gioi h<;lnkhi m ~ +00 trong (4.38) - (4.40) ta din b6 d€ sail

B6 d~ 4.2. V6'i ham em(t) dlif,fCcho (J (4.33), ta co

(i)

lim fe;, (t)G(t) dt =fG(t)dt, v6'i mQi GELz(O,T),

m-HOO

(4.44)

(ii)

T

lim fem(t)e~(t)G(t)dt=!G(tl)-!G(tZ), v6'i mQi GEL1(0,T).

Chung minh (i) Ta co

fe;,(t)G(t)dt- fG(t)dt ~ f(1-e;,(t)) IG(t)ldt

,;11 G 11"(0.') 0(1- em(I))' (I + em(I))' dtr ,; 211 G II"(O,T) [:!(I-em (t))' dtr

[

J

~

~ 211G II"(OT) ~ <~ [l-m(t-t, ~ Jr dt+::~[I+m(H, +~ Jr dt

1

(

J

~ 211GII, -+- = ,IIGII, ~O, khl m~+oo.

L «J.T) m 3m 3 ;m L (D.T)

Chung minh (ii) Xem [7].

Qua gioi h<;lnkhi m ~ +00 trong (4.38) - (4.40) b~ng cach ap dl;lng b6 d€ 4.2,

k€t hQp voi (4.36), ta duQc

I,

tllu!(tz)IIZ +tllur(tz)[Iz + fH(t)u!(l,t)dt

I,

(4.46) I,

=tllu/(t[)W +tllur(t[)W +f(J(t),u!(t))dt," a.e.tptz E(O,T),tl <t2.

Til (4.46) chung ta thay t2 =t , sail do qua gioi h<;lnkhi t,~ O+va sa dl;lng tinh ch:1t naa lien tl;lcduoi y€u cua ham IIvW ta thu duQc (4.30)

Truong hQp Uo=UI = 0, chung ta thac tri~n u,], if bdi 0 khi t < O Khi d6 d~ng

thuc (4.46) dung h§u he'"tt2 < T.

Layt2 = tE(O,T), tl < 0 khi d6 tu (4.46) ta duQc

!llul(t2)112 +!llur(t2)W + fif(T)ul(I,T)dt= f(](T),UI(T);dT.

Cho 'I~ 0_, tu (4.47) ta thu duQcd~ng (4.31)

V~y b6 d~ 4.1 dfi duQcchung minh

Trd l~i chung minh F2(ul)= j;.

Ap dt,mgb6 d~ 4.1 voi fl(t) =hou(1,t)+h(t)ul(1,t)+g(t),va ] = f -~(u)- Jr,

ta duQc

-IIUl(t)112+-a( u(t), u(t)) + fh(T)1ul(1,T) 12dT +-11 ra+lu(t) lI~a:~,-,

0

2::!llul W +!a(uo,uo)+~lIra+luo II~u++~(Q)+ f(f(T), UI(T))dT

- fg(T)UI (1, T)dT - f

( Xl- ,ul (T) )

dTo

Lay tich phan hai ve'"(3.36) theo t tu 0 de'"n t, ta duQc

-[llu~(t)112 +a(Um(t), um(t))J+-llra+lum(t)II~':+~(Q)2 a+I

=~ II U1m W + a ~ 111ra+luom II~:+~(Q) + 2 a( uom' uom) + f\f(T)'U~(T) )dT

- fg(T)U~(1,T)dT- f\F;(u~(T)),U~(T))dTo

Tudng tv nhu danh gia trong (3.39), ta suy ra

II

a+l a+l

Il a+l

-f I Ia+ld

r UOm-r Uo LU+I(Q)- r UOm-Un r

0

2Ka+l

:::; L-II UOm- Uo 11~+I~ 0 khi m~ 00.

3-a

Tu (4.50) ta suy ra

II

a+l

Il a+l

II

a+l

Il a+l

kh' ~

(4.47)

(4.48)

(4.49)

(4.50)

(4.51)

Liy liminf hai ve cua (4.49), ket h<Jpvoi (4.3), (4.12)m~oo - (4.16),va (4.51),

ta du'<Jc

-II2 ul (t)W +-a(2 u(t),u(t)) + ()fhef) Iul (1, T) 12dT +-11 a + 1 ra+lu(t) II~:+~n( )

~!llul W +!a(u(puo)+~lIra+luo II~a:~r +f(f(T), UI(T) )dT

0

(4.52)

- fg(T)UI (1, T)dT -li~}~f f(F; (U~(T) ),u~(T))dT.

So sanh (4.48) va (4.52), ta thu du'<Jc

liTj~f.J( F, (U~(T)),U~(T))dT ,; j(J; ,a' (T))dT.

M~t khac, tli (4.13) va (4.21) ta suy ra

(4.53)

li~}~f f( F; (V(T) ),u~(T))dT = f( F; (V(T) ),Ul (T))dT, '\Iv E L2(0,T;Vo)' (4.54)

]i~Jc?f j(F,(U~(T)),V(T))dT ~ j(J"V(T»)dT, VveL2(O,T;V,).

Tli (4.53) - (4.55), ta thu du'<Jc

(4.55)

I

li~}~f f(F; (u~ (T))- F; (V(T)), U~(T) -vCr) )dT

()

(4.56)

,; j(j'; - F, (V(T), u' (T)- V(T))dT, 'tv E L'(O, T; Yo)' a.e t E (O,T).

Tli (4.56) va do Hnhdon di~u tang cua ham F;(x)= IX IP-l x, 0 < fJ ~ 1, ta suy ra

j( 1- f; (V(T)), u' (T) - V(T)}dT '" 0, \Iv E L'(O, T; V,,), a.e t E (0, T). (4.57)

Trong (4.57), cho t = tk ~ T khi k ~ +00, ta du'<Jc

i(7r - F, (V(I»), u' (T) - v(l) )dl? 0, 'Iv E L'(O,T;Yo)' (4.58) Trong (4.58) ta chQn v = Ul +,{ w, voi ,{ > 0 va WE L2(0,T; Va), ta thu du'<Jc

f(7r - F, (u' (I) + 1 w(t)), w(l) )dt os;0, \lw E L'(O,T; Vo)' \11> O. (4.59)

Bay gio ta xet day {,{k}, ,{k > 0, ,{k ~ 0+ khi k ~ +00 Khi d6 ,{k < M, '\Ik.

Ta se chung minh rF;(Ul+~W)w kha tich tren QT' Th~t v~y

Ap d\lng b6 d€ 2.16, ta co

IE;(Ul (t)+Ak W(t») /2=I Ul(t)+Ak W(t) 12fJ ::;Ilul (t)I+ Mlw(t)112fJ

::;M(P)(lul (t)12fJ+ M2fJ IW(t)12fJ),

(4.60)

trong 0

!

'"

0< fl::;-,

1,neu fJ 2

M(P)= 2fJ-1 e'u 1 <P<1.

2 ,n 2 TiY(4.60), va ap d\lng (4.9), ta thu duQc

liE;(Ul (t)+Ak w(t»)W::; M(P)( liE;(Ul (t»)W + M2fJllE;(w(t»)W)

:>M(/i) [Ii +Jillu'(1)11'+ M'PCi+Jillw(t)II' )]

=M(P) [11 (1+M2fJ)+p( Ilul(t)W + M2fJllw(t)W)].

TIT

ffrE;(ul(t)+Ak w(t) )w(t)drdt ::; ~(E;(UI(t)+Ak w(t) ),w(t»)ldt

T

::; filE; (Ul (t)+Ak w(t) )llllw(t)lldt

0

::;1 filE;20 (Ul(t)+Ak w(t»)W dt+t 0fllw(t)W dt

< T (1- P)(1+ M2fJ )M (P) + pM (P) IIul 112,

+ l+P M(P)M2fJ IIwW2

V~y rE; (Ul + AkW )w khii tich tren QT'

Do tinh lien t\lC cua ham E;(x) =IxlfJ-1x, O<P::;l, nen rE;(ul +Akw)w hQi t\l

h~u kha:p ndi tren QT v€ ham rE;(ul)w khi k ~ +00.

Ap d\lng dinh ly hQi t\l bi ch~n, ta thu duQc

f(E;(UI(t)+Ak w(t»),w(t»)dt ~ f(E;(UI(t»),w(t»)dt, khi k~+OO.

(4.61)

Ap d\lng ke't qua (4.61), trong (4.59) cho A ~ O+,ta thu duQc

f(J; - 1', (u' (I)), w(t) )dt =>0, \fw E L'(O, T; Yo)' (4.62)

Trong (4.58) ta l(;li chQn v=ul -AW, voi A>O va WELz(O,T;Vo)' sail d6 cho

A ~ 0+, ta cling thu du<jc

J( J; - F, (u' (t)), wet) )dt;, 0, \1w E L'(O,T; Yo)'

Tli (4.62) va (4.63) ta kC'tlu~n F;(UI)= J;.

(4.63)

V~y c) dff du<jc chung minh

Bl.ioc 4 Chung minh stf.duy nha't nghi~m

Gia sil' 1sa < 3, va ul' Uz la hai nghi<%myC'u cua bai toan (3.1) -(3.4) nhutrong

dinh ly 2 Khi d6 W =UI - Uz la nghi<%myC'ucua bai toan

Wit-(Wrr+~Wr) = f(r,t), 0 < r < 1,0 < t <T,

I

lim.Jrwr(t)

1

<+00, -wr(l,t) = H(t),

r-70+

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

fer, t) = F( uz(t), u~(t)) - F( UI(t),u: (t)),

H(t)=how(l,t)+h(t)w (l,t),

(4.64)

va thoa di€u ki<%n

wELOO(O,T;V[), wi ELOO(O,T;Vo), w(l,.)EL"'(O,T)nHI(O,T),

ra+lw E L 00(0, T;La+1 (0.»), rP+lwl E LP+I(QT).

Ap dl;lng b6 de 4.1 ta thu du<jc (3.89)

BC'n day ta sil'dl;lng l(;likC'tqua trong chung minh sl! duy nha't nghi<%mcua dinh

ly 1, ta thu du<jc UI= uz.

V~y dinh ly 2 dffdu<jcchung minh

Ch6 thich 4.1 Trang chang minh Sl! duy nhd't nghi~m cua dinh ly 2, chung ta

khong the'lam nhl1trang dinh ly 1, nghza la lay v = wi, ma phdi sa dfrlngden b6

d~ 4.1 biJivi chungta khongco diiu ki~nWi ELOO(0, T; VI) .

Dinh ly 2 dii miJ rQngcho dqi ll1(/ngphi tuytn F;(u) = Iu la-Iu, 0 < a < 3 vai

so' mil tang 1 dClnvi so veJimQt trl1ang h(/p trang [3], [7] vai dqi ll1(/ngphi tuyen

Iu IP-2 U, 1< P < 3