TAP CHI KHOA HOC
Khoa hpc Tir nhien va Cong nghe, S6 16 (6/2019) tr.54 - 61
NGHIEM HAU TUAN HOAN CUA PHU'GTNG TRINH PARABOLIC v o l TRE H C U H A N
Le Van K i e n ' , N g u y i n Hiiu T r i 'Tnc&ng Dai hoc Tdy Bdc, -Trutmg THPT Trung Vdn. Hd Npi
Tom tdt: Trong bai bm nay, cluing toi ngliien cuu sir ton lai nghiem hiiu tuan hoan doi vdi phirong trinli parahohc vdi tie liHii han. Sir dung ket giia da c6 doi v&i cdc phuong uinh vi phdn ham trong khong gian Banach vo hgn chieu. chiing lot dua ra dicorc dieu kien ton tai duy nhat nghiem hdu tudn hoan doi vol cac top phuong trinh tren
Tir khoa; Phuong trinh purabolic, Nghiem hdu tudn hodn. Pho Clia hilm so.
1. Mffdau
Cho Q. la mien vcri bien Oil trcm, xet bai toan
du(x,l) ^ Au(x,t) + Fu,(x)+ f(x,t),xe^iXt>(i, dt
"U=0,(>0,
u(x,s) = :p{s)(x),x e 0,s G[-i-,0]
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d 46, Ala toan tti Laplace, r>0 la so thuc duong cho triroc, Fu,=ftt(.,t + s)dii(.s),u, €C([-r,0],X),u,(s):=u{t+s),Ti:l~r,0]^L(X),X = L,(n), la ham CO biSn phan bi ehan, ipeC([-r,Q],X),f(t)=f(.,t)GX la ham hSu takn hoan theo biSn (. Bang each xet u(t):M.^ X = L,(Q),w(')(.v) =«(',-!:) , va A = A: X - » X vdi
D(A) = Hl(Q)nH-(ft), ta dua bai toan tren ve dang tong quat du(t)
dt -- Ait(t) + Fu, + f(t), tt(t) eX,t>Q, 2 u(s) = ipis),se[-r,0].
Ham lien tuc bi ehan deu u:[-r,+oo)^X dugc goi la nghiem nhe (nghiem tich phan) Clia bai toan (1.2) niu u„=ip va
1,(1) = T(t)Lp(Q) + fr(t- .?)[Fu, + f{s)]ds, t > 0, voi {r(0},>o 1^ nvra nhom lien tuc manh sinh boi A.
Nhu vay, viec nghien cuu su ton tai duy nhat nghiem hau tuan hoan ciia Bai toan (1) tuong img vol su t6n tai duy nhit nghiem hiu tuin hoan cua bai toan (2). Bai toan (2) da dugc Ngay nhan bai: 14/04/2019. Ngay nhan dang: 24/05/2019.
Lien lac: Le Van Kien, e-mail: mi.kienoan@gmail,com 54
nghien ciru bai V.T. Luong va N.V. Minh gan day trong bai bao [2], trong bai bao nay chung toi su dung kk qua trong bai bao [2] de nghien ciiu sir ton tai duy nhat nghiem hiu tukn hoan cua Bai toan (1). Han th8 niia, chiing toi dua ra nhimg vi du cu th§ th6 minli hoa cho k8t qua thu dugc.
Truac tien chiing ta di tong h a p mot so cong cu va ket qua ban dau co lien quan dugc trinh bay trong cac bai bao [2, 3, 5].
Phuang trinh thuan nhat hen ket vai {1.2) sinh ra mgt mia nhom hen tuc manh K(r),^o tren khong gian C = C{[—r, 0], X) That vay,
V(t):C^C,^^u,, vai u la nghiem duy nhat ciia bai toan:
u(t) = T(t)<f(0) + J^' T(t - Ofi',^^. t > 0,
Tiep theo, chiing ta di xet toan tii sinh cua K(/),-,g. Ta dinh nghia toan tu nhu sau:
4 , : C —»• C , 9 H-*• A^,<.p iA,.^){e) = ^{0),-r<0<Q.
vai
d& dO Bo de L l . V&i ^ , dup'c dinh nghia a tren, mien xdc dinh D{A^,) trii mat trong $C$, la todn tusinh cua nira nhom lien tuc manh V{t),^Q.
Bo de L 2 . Neu A sinh ra nica nhom lien tuc manh compact T{t),^Q tren X, thi V(t),^^
Id nica nhom lien tuc manh compact tren C.
Ta gia sii A sinh ra nvra nhom lien tuc manh compact r(0,>o tren X. Khi do, ta co cac ket qua sau:
Menh de 1.3. Voit >r. thi cr(K(0) la tap dim duac vd compact v&i diem tu duy nhdt Id0,vd neu /j e a(V(t))\{0} thi^ s Pcr(K(/)).
Menh de 1.4. V&it>0, thiP<7iV(t)) = e"'''^''''\{0}. Ddt biet, niu jU = M(t)^PcT(V(t)) voi t>rvdfj^O, khi do ton tai Xe-Pcr(4,) sao cho fx - e ' "
Vod moi X ta dinh nghTa toan tii bai:
A(/i)x = ^ - 2.x + j " e^^dr}(0)x, x e D(A).
Khi do, nlu AGPcr(.<,)thi t6n tai mpt ham 0 ? i ^ G D { 4 , ) , - ^ - A ^ = 0. Digu nay co da
nghia la:
(p{0) - e ' V ( 0 ) , ^(0) ^ 0, dtp- (0) / de = Atp{0) + J" e''^dT](&)(piO).
Suyra, A ( / l ) ^ ( 0 ) - 0 .
Ngugc lai, neu ton tai x^^OeD(A) sao cho:
A(X)x = Ax- Ax +1" e^''dTj(0)x = 0.
Khi do, dat ^(0) = e'-^x, 6 e [-/-, 0 ] , thi ^ = Xep, G e [-/-, 0] va
^(0) - X ;t 0, £/<p'(0) / de = A(p(0) + f e'-^dr](0)(piO).
Dieu do co nghTala^eD(y^,,)$,S4,^ = /l(5,hayAe/'cr(/^^,).
Nhu vay, X & PCT(A^, ) khi va chi khi ton tai x ?t 0 e D(A) sao cho
A{X)x = Ax-Xx + f e^^d?](e)x = 0. 3 2. Ket qua chinh
Trong muc nay chiing ta sii dung cac ki hieu sau day:
(i) ri] , X ) , 5 C ( i ,X),BUCi\ , ^ ) , ^ 4 P ( X ) h r a n g ling la cac khong gian cua cac ham nhan gia tri trong X bi ehan hau khap nai, lien tiic bi ehan, bi ehan hen tue deu tren i va khong gian cac ham hau tuan hoan (theo nghia eiia Bohr). Ta co moi quan he
APiX)<^BUC(\ ,X)ciBC(\ ,X)cr(i ,X), (ii) r la ducmg tron dan vi trong mat phang phiic £ .
(in) Toan tu dich chuyen 5'(r) dugc xae dinh boi S(T)g(t) = g(t-\-T) vai mpi te] ,geZ,^'(i , X ) . D a c b i e t t a k i h i e u 5 : = 5 ( l ) .
2.1. Phd tron cua hdm so
Trong muc nay chiing ta di xay dung mgt bien doi ciia hamg'GZr'(i ,X) tren duang thang thuc, cai dan den mgt khai niem pho cua ham so.
Pho nay trimg vai tap e"'''^' neu them dieu kien g^ la ham lien tuc deu, a day sp(g) ki hieu la pho Beurling cua g . Cac ket qua nay dugc de cap duai day ve pho tron ciia ham s6 dupc trinh bay trong [3].
Cho g- e Z," (i , X) . Xet ham phiic Sg(X) vai Xe £\r dugc xae dinh nhu sau
Sg(X) := R{X, S)g, A e £ \ r . 4 Vi $S$ la toan tu dich chuyen, nen bien doi nay la ham giai tieh theo /I e £ \ r .
Dinh nghta 2.1. Pho tron cita g e JT{\ ,X) duac dinh nghia Id tap tdt cd ^^ E r sao cho Sg{X) khong co mot thde trien gidi tich ndo trong bdt ki Idn can cita 4 ^''^n mat phdng phuc. Pho tron cua g duac ki hieu boi o-{g) vd de ngdn gpi ta gpi Id pho cua g. Chung ta kihieu pig)la tap r\cr{g).
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Menh 6e 2.2. Ldy {g.X^^ c L ^ ( i , X ) sao cho g„ -^gGL'd , X ) , va Idy Aid tap con dong cua duong Iron don vi. Khi do ta co cdc khdng djnh sau day:
(i) a(g) la tap dong.
(ii) Niu o-(g„) Cl A v&i mpi H s ¥ , thi a{g) a A.
(iii) cT{Ag)(^a{g)v&i mpi todn tit tuyen tinh bi chgn A tren BUC(\ ,X) cdi md giao hodn vdi S.
(iv)Niu<jig)=0,thig = O.
Tiep theo ta nhae lai khai niem pho Beurling ciia mgt ham. Ki hieu F la bien doi Fourier, eo nghTa la:
(F:n(s):=]e-'"fit)dt,ssi J^L'U).
Khi do, Pho Beurling eiia ham u e BUC( ] , X) dupc dinh nghTa la tap sau day:
sp(u):={t^e] : V e > 0 3 / e i ' ( i \suppFfc{i-e,^ + e),f*ui=0}
a d o , f*u(s):=^^f{s-t)ii{t)dt.
Dinh li 2.3. Vtri cdc ki hieu a tren, sp(u) triing v&i tap gom cdc E, e \ sao cho bien doi Fourier-Carleman ciia u, xdc dinh h&i:
U" e-''u{t)dt, ReX>^, U | " e-'-'u(t)dt, ReX<0, khong the thde trien gidi tich trong mpi Idn can cua i^ .
Dinh li dupc chiing minh trong [4, Proposition 0.5, p.22].
Nhu mgt he qua cua dinh li anh xa pho yeu ta co moi lien he giua pho tron cua ham so va pho Beurling nhu sau:
Menh de 2.4. [3,Corollary 3 . 8 ] i 4 ) ' geBUCd ,X). Khido a{g)^7^\
Vi du 2.5. Ham g G ir{\ ,X), la mot ham tuSn hoan vdi chu ki r tren i Khi do de dang ta tinh dupc bien doi Fourier-Carleman
/(A) = (1 -e"'^)-' J' e-^'f{t)dt,ReX ^ 0,
CO nghTa la /(X) co the md rpng mgt ham phan hinh vdi cac cue diem don X^ = 2mn/T,ne^ .
Thang du tai cac cue diem dan -^„ cho bdi:
Res[f; A„ ] = / , = - \l e-""""'f{t)dt, n&i:. 1 ,
la he so trong khai trien Fourier cua / . Dodo:
"Pif) = {2OT / r : « e )f ,/„ 7! 0}.
Ta nhae lai dinh nghla ve ham hiu tuin hoan (theo nghia ciia Bohr), mgt ham f eBC(\ ,X) dugc ggi la hiu tuin hoan theo nghia ciia Bohr nSu moi E > 0 ton tai mgt r > 0 sao cho mgi khoang co do dai T cua j chira it nhit mgt dilm r it.
SiXJ,\fit + T)-f(t}\<C.
NSu / la ham hau tuin hoan, thi (Dmh li xip xi [1, Chap. 2] nd co th8 xap xi deu tren i boi mgt day cac da thiic lugng giae, cu thS la, mgt day cac ham ctia te] co dang:
P„(t).= '^a„,e""'. n = l,2,...;\,G] , u „ , e X , r e i . (0.5)
i=l
Hien nhien la mgi ham dugc xip xi boi mgt day ham cac cac da thuc lugng giae deu la ham hau tuan hoan.
Vidu2.S.Ham /(() = cos/ + cos%/2(,/e j la mgt ham thuc hiu tuin hoan tren i Dac biet, so mu cua cac da thuc lugng giae (cac so thuc A,, ^ trong (5)) co the chgn tu tap cua cac so thttc A (so mu Fourier) sao cho cac tich phan sau day (he so Fourier).
a(A,/) := lim— f"^ f(t)e-''-'dt
la khae 0 . Nhu ta da biet, nhirng so thuc X nhu tren la dem dugc /l^, -ij,..., no dugc ki hieu (T^ ( / ) = {/^i, /^^,...} va dirge ggi la phg Bohr cua / .
Do do, vdd / e^P(X) xae dinhbdfi chuoi Fourier ^a„e~'^"',a„ =«(/!„,/).
Liy ham / rtong Vi du 2.6 co a^(f) = {-1,-~h, 1,42\.
Trong bai bao nay chung ta co moi lien he sau day (xem [4, Proposition 7])
sp(f)='^jr\f<iAP{x).
Menh de 2.7. ia> g & AP(X). Khi do:
CT(g) = 7 ^ .
2.2. Su ldn tgi duy nhdi nghiem hdu tudn hodn
Mgt s6 kit qua hen qua den su ton tai duy nhat nghiem hau tuan hoan d6i vdi phuong trinh (1.2) sau day dugc chung minh trong [2].
Kihieu M = r(l) va cr^{M) = !j(M)n{X:\M=\).
Dinh li 2.8. V&i cdc ki hieu & tren gid su rdng:
<Tr(/W)ncr(/) = 0
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duoc thoa mdn Khi do co dm nhat nghiem hdu tudn hodn u cua phuang trinh (1.2) .•iao cho
<7iu)^aif).
Ta biet rang, /^Ga-p(Af) khi va chi khi /./ = e'\^&i , a do i^ la nghiem ciia phuang trinh (1.3). Pho cua A gom cac gia tri rieng 0>X^,X2 '^„,---, khi do, toan tii
A=A sinh ra mgt C„ -nua nhom comppact co va giai tich {Tit)},-^^ trong 1 , (fi), D ( ^ ) = - H ; (O) n / / ' ( Q ) c 4 (O).
Dinh li 2.9. Gid sir phuong trinh
X^=X-^\^-^df](0)J = ],2,...,n,...
CO cdc nghiem doX,, =1^^ ^ = 1,2,...,»,.... Khi do neu e'^* ^ c r ( / ) thi phuong trinh (1.2) co duy nhdt nghiem hdu tudn hodn u sao cho ciu) cz cri f).
Cu the ta xet hai vi du sau day.
Vi du 2.10.
Xet trudng hgp Q = (0, / ) , va
[fl/, ^ = 0, ri{e) = lo, ~r<0<O.
[-bl,
0 = -r,
d d d / la toan tii dong nhat t r e n L , ( Q ) , va a,be\
Khi dd Bai toan (1.1) trd thanh
^^^^'^^ -^'"^""'^^ -au(x,t)-bu(x,t-r) + f(x,t),xe(0,l),t>0, 6 dt 8x'
»(0,f) = u ( / , 0 = 0 , r > 0
U{X,S) = <P(S)(X),XGCI,S G [ - r , 0 ]
d^ kit Khi do, cac gia tri rieng ciia A - —j la A^ = - ( — f ,k>\.
dx I Xet phuang trinh:
X + a-vbe-'^=-{^f,k>\. 1
Neu eae phuang trinh (7) khdng cd nghiem ao thi Bai toan (6) co duy nhat nghiem hau tuan hoan u vdi (T(ZO <= c r ( / ) .
Ta xet trudng hgp cu thd hem vdi a = - 1 , 6 = TT / 2, / = ;r, r = 1, khi dd phuang trinh (7) chi cd hai nghiem ao la/;7 / 2,-;;r / 2 . Do dd neu i , - / ^ fT(/) thi thi bai toan Bai toan (6) cd duy nhat nghiem hau tuan hoan u vdi criit) a a(f).
V i d u 2.11. Xet trudng h g p Q - 5 ( 0 , 1 ) = { 0 < i ^ < l , 0 < ^ < 2 ; r } c i ' , v a
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-/, e=o, ti(e) = \ 0, - i < e < o ,
- ; r / 2 / , 0 = -\, o do, / la toan tu dong nhat trenZ.3(n) .
Trong trucmg hgp nay cac gia tri rieng ctia toan tu A la:
g do a„,„ la cac khong diem cua ham Bessel loai mgt cap n vai 0<a„^<a„^<L < a,, „, < «„.,.+[ <L
Khi do, cac phuong trinh:
.i-l-l-;T/2e"'' =(«„,„)',n,me¥,
khong CO nghiem ao, do vay Bai toan (1.6) co duy nhat nghiem hau tuan hoan u vcri
<T(«) C cr{f).
Nghien cim ndy dieoc tdi tro bdi Bo Gido due vd Bdo tao thudc de tdi khoa hoc cong nghe cdp Bo, Md sd: B2018-TTB-IJ.
TAI LIEU THAM KHAO
[1] B. M. Levitan, V. V. Zhikov (1982), "Almost Periodic Functions and Differential Equations", Moscow Univ. Publ. House 1978. English translation by Cambridge University Press.
[2] Vu Trong Luong, Nguyen Van Minh (2019), Almost periodic solutions of periodic linear partial functional differential equations. Funkcialaj Ekvacioj. To appear.
[3] Nguyen Van Minh, G. N'Guerekata, S. Siegmund (2009), Circular spectrum and bounded solutions of periodic evolution equations, J. Differential Equations 246 (2009), No 8, 3089-3108.
[4] J. Pruss, (1993), "Evolutionary Integral Equations and Applications", Birkhauser, Basel.
[5] CC. Travis, G.F. Webb (1974), Existence and stability for partial functional differential equations. Trans. Amer. Math. Soc., 200 394-418.
ALMOST PERIODIC SOLUTIONS FOR PARABOLIC WITH FINITE DELAY
Le Van Kien\ Nguyen Huu Tri^
Tay Bac University
^Trung Van High school, Hanoi
Abstract: In this paper, we study the existence of almost periodic solutions for parabolic equations with finite delav Basing on the existing results for functional differential ecpiations in infinite-dimensional Banach space, we establish the unique existence of almost periodic solution for this class equtions.
Key words: Parabolic equations, almost periodic solutions, spectrum of functions
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