Khảo sát phương trình sóng phi tuyến trong không gian Sobolev có trọng4

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

CHUaNG 2

" K ~ "

MQT SO KY HI~U,

Trang chuang nay chung toi trlnh bay cac khong gian Sobolev co trQng va cac tinh chilt v€ cac phep nhung compact giii'a cac khong gian ham co lien

quail; Cac khong gian ham LP(0, T; X), 1~ p ~ 00; Cac ky hi~u va cac bilt d~ng

thlic se duQc sli' dvng trang cac chuang sail cling duQc giai thi~u d chuang nay

11.1 Cae khong gian ham

f)~t Q = (0,1), ta bo qua dinh nghla cac khong gian ham thong dvng nhu

Cm(Q), Y(Q), Hm(Q), Wm,P(Q).

f)~t

Vo~{ u Iu: (0,1) -+ 111, do du'<)c,sao cho 1rl u(r)I' dr< -}, (2.1)

thl Vola khong gian Hilbert vai tich vo huang

I

(u, v) = fru(r)v(r)dr.

0

(2.2) f)~t

thl VI Ia khong gian Hilbert vai tich vo huang

cac d~o ham d day theo nghla phan bO'

Chufin tuang ling tren Vo' VI ky hi~u la 11.11va 11.1It

Ta co cac b6 d€ sail day v€ phep nhung giii'a cac khong gian Vo' VI,

B6 d~ 2.1. VI nhung lien t~c va ndm trit mqt trang Vo.

B6 d~ 2.2. Ta d6ng nh{l't Vo WYi V~ (V~ la d(J'ingdu cua Vo).

Khi do ta co VI C Vo C V: ' vai cdc phep nhung lien t~c va ndm trit mqt.

Chung minh b6 d~ 2.1 va b6 d~ 2.2 du'QCtlm tha'y trong [3]

Chti thich 2.1 Chung ta ciing co thi dtnh nghfa khong gian VI nhu la day du

hod cua cl (Q) d6i vai chudn

1

(xem [1])

B6 d~ 2.3. Vai m9i v E c1 (Q), ta co

(i)

(ii)

IlvW::; IIvi W +v2(1),

Iv(1)1 ::=;K11IvI11,

(iii) Jrlv(r)1 ::=;K211vllpVrE[O,l],

. Kl ="h+J2, K2=~1+J5.

trangdo

Chung minh (i) Voi mQi v EC1(Q), ta co

1

r

Nhan hai ve cua (2.6) voi r r6i 1a'ytich phan d6i voi r tir 0 den 1, ta du'Qc

-v2(1) =IIvW +fS2V(S)vl (s)ds,

(2.7)

hay

1 v2(1r~21IvW - fs(lv(s)12+lv/(s)nds=llvW -11/ W.

0 V~y ta co (i)

1

v2(1)::=;21IvW +2 fslv(s)llv/(s)lds

0 ::=;211vW+21Ivll.llvl II::=;(A+2)llvW +lllviA W, VA >0.

Chon A > 0 sao cho 1,+ 2= 1, do do A = v'2 -1.

V~y ta co (ii) vdi Kj = / 1,+2 = Jl+.Ji

Chung minh (iii) Ta co

Id

r Iv(r )12=v2(1)- rf-ds(s I v( s ) 12)ds

=v2(1)- fl v(s) 12ds - 2 fsv(s)vl (s)ds

:<:;(211vii' +2 IS21v(S)IIv'(S)IdS) +2IS'V(S)"V'(s)jdS

I

:5:211vW+2 f(s2+s)lv(s)llv/(s)lds

0

1

:5: 211 vW +4fsI v(s) II vi (s) I ds

0

,;211 vii' +2(rllvII' + ~ IIvIII'), Iiy>O.

ChQn r > 0 sao cho r +1 = r' dodo r = 2

V~y ta co (iii) vdi K2 = ~2r+2 = ~l+J5

B6 d~ 2.4 Phep nhung VI C Vo La compact. (xem [6])

Chu thich 2.2.

lim vir vCr) = 0, \iv E VI.

r->O+

(xem [1, B6 d~ 5.40, p.I28])

MiJt khdc, do Hl(t:,I) C Co([t:,I]), 0 < t: < 1, va

-rc: IIv IIHl(c,l)::;; IIV III' \iv E vI' 0 < t: < 1.

Ta suy ra rling

vlrvl[c,ljECo([t:,I]), \it:, 0<t:<1.

Tu (2.8) va (2.10), ta kit LutJnrling

FrvE Co([O,I]), V'vE VI,

(2.8)

(2.9)

(2.10)

(2.11 )

B6 d~ 2.5 Cho tru(Jc ~) > ọ EJ(it

I

ău, v) = frulvldr + hou(l)v(l), u, v Ẽ.

0

(2.12)

Thl ặ,.) la dqng song tuyén tinh d6'i xring, lien tl:lCtren VIx VI va cuiJng bric tren VI, nghla ia

(i) lău,v)1 ~clilulllllvllí VU,VEV"

(ii) collull~~ău,u), VUEV"

vJi Co= ~ min {l, ho}, CI = 1 + hoKI2.

~-Chung minh (i) Voi mQi u, v E VI, ta co

lău,v)I=lfrulvldr+hou(1)v(1) ~ frlulv/ldr+lhou(1)v(1)1

~llulllll/ II+hoK~ Ilull,llvll,~c,llull,llvll,

Chung minh (ii) Voi mQi u E VI, ta co

ău,u) = IIull12 +110I u(1)12

~ CO[II Ul W +(11 Ul 112+I u(1) 12)]+ ~ 110 Iu(1) 12

~ co(llul 112+lluI12)=co Ilull~.

Chli thich 2.3.

Tic btJ d€ 2.3 va btJ d€ 2.5 ta suy ra ba chudn sau tuang duang tren VI

~ăv,v),)llvIW+v2(1) vallvllp

va ta co

CoIlvll~~ ăv,v) ~ c,llvll~,

~ IIv IĨ~ II vi W +v2(1) ~ (1+K12) IIvIĨ,

(2.13)

(2.14 )

vJi mQi VEVỊ

B6 d~ 2.6 T6n t(li c(/ sa tr1!cchudn Hilbert {WI}' j = 1,2, cua Vo g6m cac ham rieng WI tuang ling vai cac gia tri rieng Aj sao cho

{

0 < Al :::;~ :::; :::; Ạ :::;"', lim A I ' = +00,

ăWj'v) = Aj(Wj'v), 'VVEVÍ 'Vj=1,2,

(2.15)

huang tuang ling la ặ,.).

M(it khac, cac ham rieng WI thod bai loan gia tri bien

(

dwj

J

-Awj= ;dr r dr = AjWj' O<r<l,

I

dw,

lim-vr-d J <+00, - d J (l)+hOwj(1)=Ọ

I

\\

(2.16)

Toan tit A: VI + v/ du(fc xac dinh duy nhd'tbai b6 d~ Lax- Milgram, ngh'iala

ă u, v) = (Au, v), Vu, VE VI,

Chung minh b6 d€ 2.6 duQctill thffytrong [8, Binh ly 6.2.1, p.137]

B~t

(2.17)

V2 = {uIu E VI : Au E Vo},

thi V2 la khong gian Hilbert vai tich vo huang

(u,v) +(ul ,Vi)+(Au,Av),

(2.18)

(2.19)

va chũn tudng ung

I

Ch6 thich 2.4 Khang gian ham V2 tily thuQc vao hó

B6 d~ 2.8 Vai m6i u E V2, ta co

(ii) Ilullll:::;#IIAUII,

(iii) IIu 11;'(0) :; (211uII+J-z II Au II) IIu II,

Chung minh b6 d€ 2.7 va b6 d€ 2.8 duQc till thffy trong [2]

11.2 Khong gian ham U(O,T;X), l~p~oo

Cho x la khong gian Banach thlfc d6i vOi chu£n 11.llx.

Ky hi~u U(O,T;X), 1~ p ~ 00, Ia khong gian cac lOp tu'dng du'dng chua ham

T

u: (0, T) ~ X do du'Qc, sao cho IIi u(t) IIi dt < +00, voi 1 ~ p < 00,

0

hay

t6n t(;li M > 0 sao cho Ilu(t)1Ix~ M, a.e t E (0, T), voi p = 00.

Ta dinh nghla chu£n trong U (0,T;X), 1~ p ~00 nhu' sau

I

HI "".TX) =( piu(1) II: dt J' vol I:; p<00,

(2.21) va

IIU IILoo(o,T;X)=ess sup O<t<TIIu(t) Ilx

(2.22)

= inf{M > 0:11u(t) Ilx~ M,a.e.t E (O,T)}, voi P = 00.

Ta co cac b6 d~ sau day ma chung minh chung co th€ Hmthffytrong [4]

B6 d~ 2.10 GQi Xl la d{fi ng(iu cua X Ta co

1 1

vCfl - + - =, < P <00 .

H{fn mia, ntu X phdn XlJthi U (0, T; X) ciing phdn XlJ.

H{fn mia, cac khong gian L1(0, T; X) va L~(O,T;xl) khong phdn XlJ.

11.3 Phan b6 co ghi tri vectd

Dinh nghia 2.1 Cha x la khong gian Banach thl;tc.Mfjt phtin bo' co gia trj

trang X la mfjt anh XlJ tuytn link lien tf:lCtit D(O,T) vaa X T~p cac phtin b{f co gia trj trang X dur;c ky hifU la

DI(O,T;X) = L(D(O,T);X) = {f: D(O,T) ~ X, f tuye'n tinh, lien tl,lc}.

Djnh nghia 2.2 Ta djnh nghla dC)oham : theo nghla phdn b5 cua f bili cong thac

Cae tinh eha't

(i) Cho v E U(O,T;X), (1< p < +00).

Ta lam tLtong zing veJi v m(jt anh xc) ~ : D(O, T) ~ X dLt(lc xac djnh nhLt sflu

T

(~,cp) = fv(t) cp(t)dt, VcpED(O,T).

0

(2.24)

(ii) Anh xq v H ~ la m(jt don anh, tuyen tinh lien t1;tCtit U(O,T;X) vao

D/ (0, T;X), do do ta co thi dbng nhd't Tv ==v

Ta co ke't qua sau day

B6 d~ 2.12 (Lions [4]) U(O,T;X) C D/ (O,T;X) vai phep nhung lien t1;tc.

11.4 D~o ham trong U (0, T;X)

Do b6 d6 2.12, voi fEU (0, T;X) ta co th€ coi f va do do df Ia cae phffn ttt cua

dt D/ (0, T;X) Ta co cae ke't qua sau day.

B6 d~ 2.13 (Lions [4])

(i) Neu f,f/ E L1(0,T;X) thi f bling hdu het vai m(jt ham lien t1;tCtit

[0,T] ~ x

(ii) Neu f,f E U(O,T;X) thi f bling hdu het vai mo.LhiUll lien~ tit

(~H.I<H.TLJNHIEt~

[0,T] ~ x

11.5 HQi !t,l trong U (0, T; X), 1 S P S00 \ 001205

(i) Urn ~ U trong LP (O,T;X) mC)nh neu IIurn - U tP(O,T;X)~ O (2.25)

(ii) Urn~U trang LP(O,T;X), (1<p<oo) yeu

neu

f(urn(t),g(t))dt ~ f(u(t),g(t))dt, Vg E LP!(O,T;X/),

(2.26)

f(um(t),v)cp(t)dt + f(u(t), v) cp(t)dt, \lVEXI, cpELP/(O,T).

ne'u

f(um(t),g(t))dt ~ f(u(t), g(t)) dt, \lg E L1(0,T;X),

hay

f(um(t), v)cp(t)dt ~ f(u(t), v) cp(t)dt, \Iv E X, cpE L1(0,T).

11.6 B6 d~ v~ tinh compact cua Lions [4]

Cho 3 khong gian Banach XO,X,X1 vdi Xoc.X c.X1 sao cho XO,X1 la phan Xc;l

va phep nhung XoC.X la compact.

Voi 0< T < +00, 1< Pi < +00, i = 0,1 Ta d~t

WeT) ={ v E LPo(O,T;Xo) :/ E LP' (O,T;X1)}.

Ta trang bi cho W (T) bdi chu~n

(2.28)

IIvIIW(T)=II V IILPo(o,r;xo)+ II/ IILPl(o,r;x1). (2.29)

Khi do W(T) la khong gian Banach.

Ta co b6 d€ sail lien quail de'n phep nhung compact

B6 d~ 2.14 (B6 d€ v€ tinh compact cua Lions [4])

Phep nhung WeT) c.Uo (O,T;X) Lacompact.

(xem [4], trang 57)

11.7 B6 d~ v~ sf! hQi tQ ye'u trong Lq(Q), 1<q<oo

B6 d~ 2.15 (xem [4], trang 12) Cho Q La mQt tfjp mCt, hi chfjn cua ~n va

Gm, GELq(Q), l<q<+oo, saocho

(ii) Gm +G a.e (r,t) trong Q.

Khi do Gm~ G trong Lq(Q) ye'u.

11.8 MQt s6 bitt diing thuc thúCingdung

Bfl d~ 2.16 Vai mri X,YEIR., &>0, l~p<+oo, O<q~l, ta luon co

(i)

Ix+yIP';(I+s)'-llxIP +(1+ ~fl,y,P.

Dijc bift, ntu chrn £ = 1, ta du:(/c

(ii)

Ix+ YIP~2P-l(lxIP +lyIP),

Ix+ylq~lxlq +IYlq~i-q(lxl+IYIY,

(iii) Ilxlq-IYlql~lx-Ylq,

IlxlP-l x-I YIP-lyl ~p(lxl+IY Iy-llx-YI,

(iv)

(v) II X Iq-l x-I Y Iq-l y I ~ i-q I x- y Iq.

Chung minh (i).

Nh~n xet ding n€u x = 0, hõc Y = 0, hõc p = 1 (q = 1) thi (i)- (v) dung

Do do ta chi xet X,Y:;t:0,1 < p < +00,0 < q < 1.

Trúong h<jp 1 x> 0, Y> Ọ

Xet ham s6 Jet) = tP, t> 0, co III (t) = pep -1)tp-2 > 0, Vt > 0, nen I la ham 16i tren (0,+00) Do do ta co

(x+ f)' ~[Ẵ)+(I-A)C~A)J';ẴJ +(H)C~J, 1;10d <1.

Chon A = ~, ta thu dúdc (i)

Trúong h<jp 2 x,y:;t: Ọ

Taco Ix+YIP~(lxl+IYIY, apdl,mgtrúongh<jp l,taciingthudú<jc(i).

Chung minh (ii).

Ly lũn túdng tl! nhú (i), nen ta chi xet trúong h<jp x> O,y > Ọ

Xet ham s6

Ta co

I(t)=tq +(1-t)q, O<t<1.

II(t)=q[tq-l_(I-t)q-1J, II(t)=Õt= ~,

(2.30)

II (t) > 0 voi 0 < t <!, va fl (t) < 0 voi ! < t < 1,

f(O) = f(l) = 1, f(lI 2) = i-q.

V~y ta co

l<f(t)<i-q, VtE(O,I).

ChQn t = ~ E (0,1), tu (2.30) va (2.31) ta thu duQc (ii)

x+y

(2.31)

Chung minh (iii).

Ap dl;lng (ii), ta co Ix Iq=1(x - y) + y Iq~ Ix - y Iq+ 1y Iq,

hay

TucJng tl!, ta cling co

Iylq -lxlq~lx-ylq.

Tu (2.32)va (2.33)ta thu duQc(iii).

(2.33)

IlxlP-l x-I YIP-lyl= IJ(x)- J(y)l= If~J(y+s(X-Y))dS0

1

=plx-yl flsx+(I-s)YIP-lds~p(lxl+IYlr1Ix-YI.

0

Chung minh (v).

Truong hQp 1 xy < O Ta co

Ilxlq-lx-lylq-l yl=lxlq+lylq,va Ix-Ylq=(lxl+IYlt.

Do do ap dt,mg(ii) ta thu duQc(iv)

Truong hQp 2 xy > O.

Ap dl;lng (iii), ta co

Ilxlq-l x-Iylq-l yl = Ilxlq -IYlql~lx-Ylq~21-q Ix-ylq.

V~y b6 d€ 2.16 dff duQc chung minh

Cac bc1td~ng thuc thong dl;lngdudi day co th€ tlm thc1ytrong cac sach va duQc phat bi€u bdi cac b6 d€ sail

B6 d~ 2.17 (Bit d~ng thuc Young) Cho a, b ~ 0, E:> 0, p,/ > 1,l+~ = 1,

ta co

ab<-E:P- a P +-E:-PI b p .

B6 d~ 2.18 (Bit d~ng thuc Cauchy) Cho a,b ~ 0, E:> 0, ta co

(2.34)

~

B6 d~ 2.19 (Bit d~ng thuc Gronwall) Gid sa f, g : [O,T]~ R, la cac ham khd

rich, khong am tren [0,T] va thod ba't dang thac

t

f(t)5,C+ jf(T)g(T)dT, a.e tE[O,T],

0

trang doC la hang so' khong am.

Khi do ta co

1(1)'; C exp lfg(T) dT J, a.e t E [O,T], noi rieng, khi C = 0 thi f(t) = 0, a.e t E [O,T].

(2.36)

11.9 Dinh ly Schauder va djnh ly Ascoli

Dinh ly Schauder Cho K la mQt t(ip lbi, dong trang khong gian Banach X va T: K +K la anh xt;llien tf:lC,compact Khi do T co dilm bat dQngtrang K.

Dinh ly Ascoli Cho x la mQt t(ip compact trang khong gian dinh chuiln

(E,II.II)va A la mQtt(ip con cua Co(X,IR).

Khi do A la mQtt(ip compact trang Co(X,IR)ntu va chi ntu A co hai tinh cha't

(i) A bi ch(in tang diim, nghla fa: vai m6i x E X, t(ip {lex) If E A} bi ch(in trang

JR.,

(ii) A lien tf:lCdbng b(ic, nghla la

V E:> 0, :J17> 0: Vx,Y E x, IIx- yII < 17 ~ If(x) - fey) I< E:, Vf E A.

11.10 M(}t s6 ky hi~u

Ta dung cae ky hit;:u u(t), ul(t) = Ut(t),UII(t)= Utt(t),ur(t), urr(t) Hin luQt d€ chI

u(r,t), -(r,t),at ~(r,t),at -(r,t),ar arz(r,t); QT= Qx(O,T).

Sail day la b6 d€ v€ cac bfft d~ng thuc trong Va' V, duejc sa d\mg nhi€u l~n trong chuang 3 va chuang 4

B6 d~ 2.20 Cho trU(1ca E[1,3),ta co cac beltdang thac sau

(i) III u la-l u II:::;K(a) IIu II~' vu E v"

(ii)

,

Jlu(r)12a-2 dr:::;K,(a)llull~a-2, vu E V"

a

(iii) VR > 0, :3KR(a)> 0 saD cho

III u,la-' u, -I u2Ia-, u211:::;JKR(a) IIu, -u2111'

Vu" U2 E VI' IIU, II,:::; R, IIu211,:::; R,

trang do cac hdng srI K(a), K,(a) va KR(a) lan lu(/t xuitt hifn trong (i), (ii) va

(iii) co c6ng thac Cf:lthi duai day

1

1, ne'u a = 1,

(

2a )

a;, 3a-'

1

1 < <

- _2 ' neu - a - _

2 '

K ( a ) - -a

K,2a-2 + (a -1) 82a (K(a» a , ne'u "2 < a < 3,

va

1, ne'u a = 1,

KR(a)=~2a2R2a-2K;K,(a), ne'u I<a:::;~,

a2(2R)2a-2 K;K,(a), ne'u ~< a < 3.

Chuyrdng F;(u)=lula-'uEVo' VUE V, vai O<a<1.

Chung minh (i).

Truong hejp 1 a = 1 Ta co

11F;(u)W=llull:::; Ilull,

Truong hejp2 1< a < 3 Ta co

IIFJu)W= Jrlu(r)12a dr

0

(2.37)

="21 u(l) 12a - a Jr21 u(r) 12a-2 u(r) ur(r)dr

0

::;; -'-llull~a + a Jr2Iu(r)12a-l (Frlur(r)l)dr "

}(2a

(

1

J

1 ::;;-t-Ilull~a+ allurll fr3Iu(r)14a-2dr

Di,it P = a+1 > 1,P = I -a+1 > 1, thl,1- + /1 =1

Ta co

Jr3Iu(r)14a-2 dr= Jr2Iu(r)14a-4 [Frlu(r)1r dr

(2.38)

[

2

J

::;;}(~llull~Jr2Iu(r)14a-4dr= }(~llull~Jri rPlul4a-4 dr

(

1

J

;I

(

1

J

~

::;;}(~llull~ fr2 dr jr2Iu(r)12a+2dr

,; Killull;( ty(frlu(r)I'" (""lu(r)l)' dr)'

,; K;lIull;(tY (K;lIull; )* Orlu<r)!'" dr)'

(

1 )

::;; }(211u111 p 3 11~(u)IIP

Tu (2.37) va (2.38) ta du'Qc

1

11F;(U)W::;;}(rIlull~a+allulll(}(21Iulll r+~ (~)2PI 11F;(u)ll~ (2.39)

[

1

]

2~~1

::;; t-Ilull~a+ ~p &allulll(}(2I1ulll) p (3) p +2j;IIF;(u)W.

Tn (2.39) ta chQn &> 0 sao cho &2p= p hay &=p2P , ta duQc

2p

II F;(uJII' :5K,'" II ull;" +2pp-r[ap;; II uII,(K, IIull,)"~ G)';' r'

\

a-3 a+1 a-I J

]2 K:~-llIull;a.

a+l a+l

Dodo

[

aB

]

Chung minh (ii).

Truong hQp 1 1~ a ~ ~ Ta co

flu(r) 12a-2dr = frl-a (.Jrl u(r) I) a- dr

~K;a-2 II u II;a-2 frl-adr0 = 2-a2 II u II;a-2.

Truonghd p 2 i < a < 3 Bat

p = > 1, p = - > 1, thl - + / = 1

Ta co

flu(r) 12a-2 dr = Iu(l) 12a-2 - 2(a -1) fr Iu(r) 12a-4 u(r) uJr)dr

I

~ KI2a-21Iull;a-2+ 2(a-l) f.Jr1 u(r) 12a-3 (v'"rlur(r)l)dr

0

(2.40)

I

,; K,'.-'lIull;a-'+ 2(a-1Jllu,1I (fr,u(r)l'"-" dr )'

(

I

J

frlu(r)14a~6dr= frl rPlu(r)14a-6 dr

(2.41 )

,; Ordr J? Or 1u(r) I'" drr ~ 2;: IIFMII~.

Tli (2.40) va (2.41) ta thu duQc

flu(r)12a-2dr::; KI2a-21Iull~a-2+ (a-I) 2 2/ IlulllllF;(u)IIP

0

::; KI2a-21Iull~a-2+ (a-I) 22a Ilulll(K(a)lIullnp

[

a-I 2a-3

]

/

::; KI2a-2+(a-I)82a(K(a))a- lIull~a-2

V~y (ii) dii duQc chung minh

Chung minh (iii).

Truong hQp 1 a = 1 Ta co

II F; (ul) - F; (u2) 112=IIul -u2 112::; II ul -u2 II~

Truong hQp 2 1 < a ::;%

Ap dlJng (ii) va b6 d~ 2.16, ta duQC

1

IIF;(u1)-F;(u2) W= Ir 1F;(uI(r))-F;(u2(r)) Idr

0 I

::; fr(al~(r)-U2(r)I(luI(r)1 +lu2(r)lr-lf dr

0

I

::;aK21Iul-u2111 luI(r)I+lu2(r)1 dr

0

I

::;a2K~ IluI-U211~ f(luI(r)12a-2 +lu2(r)12a-2)dr

0

::;a2K~KI(a)(lluIII12a-2 +llu21112a-2)lluI-U211~

::; 2a2R2a-2K~KI(a)11 ul -u211~

Truong hQp 3 % < a < 30 Ta co

I

11F;(uI)-F;(U2)1I ~aK21Iul-u2111 lul(r)I+lu2(r)1 dr

0

I

::; 22a-3 a2 K~ II UI - U2 II~ f( IUI (r) 12a-2 +IU2(r) 12a-2) dr

0 2

( )2a-2 2

2

::;a 2R K2KI(a) UI-U2 10 V~y (iii) dii duQc chung minh