Solonnikov parabolic systems with quasihomogeneous structure
We consider a new class of systems of equations that combine the structures of Solonnikov and Éidel’man parabolic systems. We prove a theorem on the reduction of a general initial-value problem to a problem with zero initial data and a theorem on the correct solvability of an initial-value problem i...
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| Date: | 2006 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2006
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3550 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509661834248192 |
|---|---|
| author | Ivasyshen, S. D. Ivasyuk, H. P. Івасишен, С. Д. Івасюк, Г. П. |
| author_facet | Ivasyshen, S. D. Ivasyuk, H. P. Івасишен, С. Д. Івасюк, Г. П. |
| author_sort | Ivasyshen, S. D. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:57:29Z |
| description | We consider a new class of systems of equations that combine the structures of Solonnikov and Éidel’man parabolic systems. We prove a theorem on the reduction of a general initial-value problem to a problem with zero initial data and a theorem on the correct solvability of an initial-value problem in a model case. |
| first_indexed | 2026-03-24T02:44:39Z |
| format | Article |
| fulltext |
UDK 517.956.4
S. D. Ivasyßen (Nac. texn. un-t Ukra]ny „KPI”, Ky]v),
H. P. Ivasgk (Çerniv. nac. un-t)
PARABOLIÇNI ZA SOLONNYKOVYM SYSTEMY
KVAZIODNORIDNO} STRUKTURY
We consider a new class of systems of equations combining structures of Solonnikov-parabolic and
Eidelman-parabolic systems. We prove a theorem on reducing a general initial problem to a problem
with null initial data and a theorem on the correct solvability of initial problem in the model case.
Rozhlqda[t\sq novyj klas system rivnqn\, qki po[dnugt\ u sobi struktury system, paraboliçnyx
za Solonnykovym i Ejdel\manom. Dovedeno teoremy pro zvedennq zahal\no] poçatkovo] zadaçi do
zadaçi z nul\ovymy poçatkovymy danymy ta pro korektnu rozv’qznist\ poçatkovo] zadaçi v mo-
del\nomu vypadku.
U 1938 r. I.8H. Petrovs\kyj [1] uviv dosyt\ ßyrokyj klas paraboliçnyx system
linijnyx dyferencial\nyx rivnqn\ iz çastynnymy poxidnymy, qkyj na danyj ças
[ najhlybße i najpovniße doslidΩenym. Oznaçennq I.8H. Petrovs\koho uzahal\-
ngvalos\ u riznyx naprqmkax. Zokrema, v 1960 r. S.8D. Ejdel\man [2] rozhlqnuv
novyj klas system, qkyj uzahal\ngvav klas system, paraboliçnyx za Petrov-
s\kym. U cyx systemax dyferencigvannq za riznymy prostorovymy zminnymy
magt\, vzahali kaΩuçy, riznu vahu vidnosno dyferencigvannq za çasovog zmin-
nog, tobto systemy magt\ vektornu paraboliçnu vahu 2b : = ( , , )2 21b bn… . Tomu
taki systemy nazvani 2b -paraboliçnymy. DoslidΩennqm zadaçi Koßi dlq nyx
prysvqçeno praci [3 – 5]. U 1964 r. V.8O. Solonnykov [6] zaproponuvav we odne
uzahal\nennq paraboliçnyx za Petrovs\kym system. U cyx systemax porqdok
operatora, qkyj di[ na nevidomu funkcig uj u rivnqnni z nomerom k, moΩe zale-
Ωaty qk vid k, tak i vid j . Teorig krajovyx zadaç i zadaçi Koßi dlq takoho kla-
su system detal\no rozrobleno v fundamental\nij praci [7] (dyv. takoΩ [8]).
U danij statti rozhlqdagt\sq systemy, qki pryrodno uzahal\nggt\ systemy,
paraboliçni za S. D. Ejdel\manom, i systemy, paraboliçni v rozuminni V.8O.8So-
lonnykova (taki systemy my nazyva[mo paraboliçnymy za Solonnykovym syste-
mamy kvaziodnoridno] struktury). Vyvçennq takyx system u model\nomu vypad-
ku rozpoçato druhym spivavtorom pid kerivnyctvom perßoho. U praci [9] dlq
c\oho vypadku opysano strukturu ta vlastyvosti fundamental\no] matryci roz-
v’qzkiv (FMR) i navedeno formuly dlq rozv’qzkiv poçatkovo] zadaçi.
Z metog pobudovy dlq novoho klasu system teori] rozv’qznosti poçatkovyx
zadaç u prostorax Hel\dera qk obmeΩenyx funkcij, tak i zrostagçyx na neskin-
çennosti, u cij statti my rozhlqda[mo dva vaΩlyvyx pytannq: zvedennq
zahal\no] poçatkovo] zadaçi do zadaçi z nul\ovymy poçatkovymy danymy i
vstanovlennq korektno] rozv’qznosti poçatkovo] zadaçi v model\nomu vypadku.
1. Oznaçennq paraboliçno] systemy. Nexaj n, N , b bn1, ,… — zadani na-
tural\ni çysla, b — najmenße spil\ne kratne çysel b bn1, ,… , m : =
= ( , , )m mn1 … , m b0 2:= , m b bj j: ( )/= 2 2 , j n∈ …{ , , }1 , α α:= =∑ mj jj
n
0
,
qkwo α α α α: ( , , , )= … ∈ +
+
0 1
1
n
n
Z , α α:= =∑ mj jj
n
1
, qkwo α α α: ( , , )= … ∈1 n
∈ Z+
n , M mjj
n
:= =∑ 0
, i — uqvna odynycq, A t x A t xt x kj t x k j
N
( , , , ) : ( , , , )
,
∂ ∂ = ∂ ∂( ) =1
,
u u uN: ( , , )= …col 1 i f f fN: ( , , )= …col 1 — nevidoma ta zadana vektor-funkci],
ΠH
n nt x t H x: ( , ) ,= ∈ ∈ ∈{ }+
R R
1
, qkwo H ⊂ R , T — zadane dodatne çyslo.
Prypustymo, wo isnugt\ taki çysla sk i tj iz Z , wo stepin\ vidnosno λ
mnohoçlena A t x p ikj
m m( , , , )λ σλ0 , σλ σ λ σ λm m
n
mn: ( , , )= …1
1 , ne perevywu[ sk + tj
© S. D. IVASYÍEN, H. P. IVASGK, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11 1501
1502 S. D. IVASYÍEN, H. P. IVASGK
(qkwo sk + tj < 0, to Akj := 0 ) i ( )s t brk kk
N + ==∑ 1
2 , de r — stepin\
det ( , , , )A t x p iσ qk mnohoçlena vid p.
Nexaj A Akj k j
N0 0
1: ( ) ,= = — holovna çastyna A , tobto
A t x p ikj
m m0 0( , , , )λ σλ = λ σs t
kj
k j A t x p i
+ 0 ( , , , ).
Oznaçennq.1. Systemu rivnqn\
A t x u t x f t xt x( , , , ) ( , ) ( , )∂ ∂ = , ( , ) [ , ]t x T∈Π 0 , (1)
budemo nazyvaty rivnomirno paraboliçnog za Solonnykovym z kvaziodnoridnog
strukturog na mnoΩyni Π[ , ]0 T , qkwo isnu[ taka stala δ > 0, wo dlq bud\-
qkyx ( , ) [ , ]t x T∈Π 0 i σ ∈R
n
p -koreni rivnqnnq det ( , , , )A t x p i0 0σ = zado-
vol\nqgt\ nerivnist\
Re ( , , )p t x σ ≤ – δ σ σ1
2 21b
n
bn+ … +( ) .
Çastynnymy vypadkamy oznaçenyx vywe system [ systemy, paraboliçni za
Petrovs\kym ( mk = 1, k n∈ …{ , , }1 , sj = 0 i t bnj j= 2 , nj ∈N, j N∈ …{ , , }1 ) ,
2b -paraboliçni za Ejdel\manom ( mk > 1 dlq prynajmni odnoho k n∈ …{ , , }1 ,
sj = 0 i t bnj j= 2 , j N∈ …{ , , }1 ) i paraboliçni za Solonnykovym odnoridno]
struktury ( mk = 1, k n∈ …{ , , }1 ) .
Dali vvaΩatymemo, wo vykonu[t\sq umova
A) systema (1) [ rivnomirno paraboliçnog v Π[ , ]0 T zi stalog δ > 0 zhid-
no z oznaçennqm81.
2. Poçatkovi umovy. Dlq system (1) zadavaty poçatkovi umovy tak, qk dlq
system Petrovs\koho, vzahali kaΩuçy, ne moΩna. Zadavatymemo ]x tak samo, qk
dlq system Solonnykova z odnoridnog strukturog [7].
Nexaj B x B xt x kj t x k
r
j
N( , , ) : ( ( , , )) ,
,∂ ∂ = ∂ ∂ = =1 1 — matryçnyj dyferencial\nyj vy-
raz, ϕ ϕ ϕ: ( , , )= …col 1 r — zadana vektor-funkciq. Prypustymo, wo isnugt\ ta-
ki cili çysla pk , wo stepin\ vidnosno λ mnohoçlena B x p ikj
m m( , , )λ σλ0
ne pe-
revywu[ p tk j+ , a qkwo p tk j+ < 0 , to Bkj := 0. Tut tj — ti sami, wo j u sys-
temi (1). Holovnog çastynog vyrazu B nazvemo vyraz B Bkj k
r
j
N0 0
1 1: ( ) ,
,= = = , de
B x p i B x p ikj
m m p t
kj
k j0 00( , , ) ( , , )λ σλ λ σ= +
.
Poçatkovi umovy dlq systemy (1) zadamo u vyhlqdi
B x u t xt x t( , , ) ( , )∂ ∂ = 0 = ϕ( )x , x n∈R . (2)
Dlq zabezpeçennq korektnosti zadaçi z umovog (2) matryçnyj vyraz B povynen
zadovol\nqty taku umovu dopovnql\nosti typu vidomo] umovy Lopatyns\koho:
rqdky matryci
C x p B x p A x p( , ) : ( , , ) ( , , , )= 0 00 0 0
Ò
,
d e
Ò
A A A0 0 0 1: det ( )= −
— matrycq, vza[mna dlq A0 , linijno nezaleΩni za
modulem odnoçlena p
r
u koΩnij toçci x n∈R .
Qk i v [7], cq umova dozvolq[ dlq koΩno] systemy (1) vyznaçyty çysla pk i z
toçnistg do deqkyx alhebra]çnyx peretvoren\ pobuduvaty matrycg B x p0 0( , , ).
Matrycq
′ = −B x p i B x p i B x p( , , ) : ( , , ) ( , , )σ σ0 0 0
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
PARABOLIÇNI ZA SOLONNYKOVYM SYSTEMY … 1503
vidihra[ rol\ molodßoho çlena, dlq ]] elementiv povynna vykonuvatysq lyße
odna umova: stepin\ vidnosno λ mnohoçlena ′B x p ikj
m m( ), ,λ σλ0 dorivng[ p tk j+ .
Tak samo, qk u praci [7], dovodqt\sq taki tverdΩennq:
1) umova dopovnql\nosti rivnosyl\na umovi
det ( )( )H xp′ ≠ 0, x n∈R , (3)
dlq bud\-qko] matryci H p( )′ , vyznaçeno] v [7, c. 15, 24];
2) qkwo komponenty uj vektor-funkci] u, koefici[nty dyferencial\nyx
vyraziv A i B, a takoΩ komponenty fk i ϕ s vektor-funkcij f i ϕ [ dosyt\
hladkymy funkciqmy svo]x arhumentiv, to pry vykonanni umovy (3) systema (1) i
poçatkova umova (2) dagt\ moΩlyvist\ za dopomohog operaci] dyferencigvannq
ta rozv’qzuvannq linijnyx alhebra]çnyx system vyznaçyty pry t = 0 znaçennq
bud\-qko] poxidno] vid koΩno] iz svo]x funkcij uj çerez fk i ϕs ta ]x poxidni.
Prypuskatymemo, wo vykonu[t\sq rivnomirnyj variant umovy (3), tobto umova
V) isnu[ taka stala δ1 > 0, wo dlq vsix matryc\ H p( )′
i toçok x n∈R
spravdΩu[t\sq nerivnist\
det ( )( )H xp′ ≥ δ1 .
3. Fundamental\na matrycq rozv’qzkiv model\no] systemy. Rozhlqnemo
u prostori R
n+1
systemu rivnqn\, paraboliçnu za Solonnykovym kvaziodnorid-
no] struktury zi stalymy koefici[ntamy, qka mistyt\ lyße hrupu starßyx çle-
niv,
A ut x
0( , )∂ ∂ = f . (4)
Zhidno z rezul\tatamy [9] FMR Z Zkj k j
N: ( ) ,= =1 systemy (4) vyraΩa[t\sq çe-
rez fundamental\nyj rozv’qzok (FR) Γ rivnqnnq
det ( , )A ut x
0 ∂ ∂ = 0 (5)
takymy formulamy:
Z t xkj ( , ) =
Ò
A t xkj t x
0 ( , ) ( , )∂ ∂ Γ ,
(6)
t > 0, x n∈R , { , } { , , }k j N⊂ …1 ,
de
Ò
A kj
0
— element vza[mno] matryci
Ò
A0 , tobto alhebra]çne dopovnennq ele-
menta Ajk
0
matryci A0 .
Rivnqnnq (5) [ 2b -paraboliçnym rivnqnnqm uzahal\nenoho porqdku 2br za
prostorovymy zminnymy x i porqdku r za çasovog zminnog t . Vlastyvosti joho
FR vyvçeno v [2 – 4]. Zokrema, dlq Γ spravdΩugt\sq ocinky
∂t x t x, ( , )α Γ ≤ C t E t xr M b
cα
α− +( )/( ) ( , )2 ,
(7)
t > 0, x n∈R , α ∈ +
+
Z
n 1,
de ∂ = ∂ ∂ … ∂t x t x xn
n
, :α α α α0
1
1 , Cα i c — dodatni stali,
E t x c t xc
q
j
q
j
n
j j( , ) : exp= −
−
=
∑ 1
1
, q b bj j j: ( )/= −2 2 1 .
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
1504 S. D. IVASYÍEN, H. P. IVASGK
Z (6), (7) vyplyvagt\ taki ocinky dlq elementiv matryci Z :
∂t x kjZ t x, ( , )α ≤ C t E t x
M s t b
c
j k
α
α− + − −( )/( )
( , )
2
,
(8)
t > 0, x n∈R , α ∈ +
+
Z
n 1, { , } { , , }k j N⊂ …1 .
Rozhlqnemo porodΩenyj FR Γ ob’[mnyj potencial
V t x d t x h dh
t
n
( , ) : ( , ) ( , )= − −
−∞
∫ ∫τ τ ξ τ ξ ξ
R
Γ , ( , )t x n∈ +
R
1, (9)
vvaΩagçy, dlq prostoty, wo h — dosyt\ hladka i finitna funkciq. Vykorys-
tovugçy vlastyvosti FR Γ, oderΩu[mo, wo funkciq Vh [ rozv’qzkom rivnqnnq
det ( , )A Vt x h
0 ∂ ∂ = h . (10)
Zvidsy vyplyva[, wo funkci]
u t xj
k
N
( , ) :=
=
∑
1
Ò
A V t xkj t x fk
0 ( , ) ( , )∂ ∂ , ( , )t x n∈ +
R
1, j N∈ …{ , , }1 ,
dlq dovil\nyx dosyt\ hladkyx i finitnyx funkcij fk , k N∈ …{ , , }1 , [ kompo-
nentamy rozv’qzku systemy (4) z f f fN: ( , , )= …col 1 .
Navedeni vywe rezul\taty [ pravyl\nymy takoΩ dlq model\nyx system vy-
hlqdu
A y ut x
0( , , , )β ∂ ∂ = f, (11)
qki oderΩugt\sq iz systemy (1), qkwo vidkynuty molodßi çleny, a koefici[nty
holovno] çastyny zafiksuvaty v toçci ( , ) [ , ]β y T∈Π 0 . Zokrema, dlq FR
Γ( , , , )⋅ ⋅ β y rivnqnnq
det ( , , , )A y ut x
0 β ∂ ∂ = 0
ta elementiv Z ykj ( , , , )⋅ ⋅ β FMR systemy (11) spravdΩugt\sq ocinky (7) i (8), v
qkyx stali Cα i c, vzahali kaΩuçy, zaleΩat\ vid toçky ( , )β y . Ale na pidstavi
umovy A) ta umovy C), qka bude navedena v p.85, ci stali moΩna vybraty tak,
wob vony ne zaleΩaly vid toçky ( , ) [ , ]β y T∈Π 0 . OtΩe, moΩna vvaΩaty pravyl\-
nymy taki ocinky:
∂t x t x y, ( , ; , )α βΓ ≤ C t E t xr M b
cα
α− +( )/( ) ( , )2 ,
(12)
t > 0, x n∈R , ( , ) [ , ]β y T∈Π 0 , α ∈ +
+
Z
n 1.
4. Prostory funkcij. Navedemo oznaçennq potribnyx prostoriv Hel\dera
obmeΩenyx i zrostagçyx funkcij. Krim uvedenyx u pp.81 i 3 poznaçen\ budemo
vykorystovuvaty we j taki: c a an0 1, , ,… — zadani çysla taki, wo 0 < c0 < c , de
c — stala z ocinok (12), aj ≥ 0, j ∈ { 1, … , n } , T c a
j
j
bj< −
min ( )/0
2 1
;
�
a a an: ( , , )= …1 ,
� �
k t a k t a k t an n( , ) : ( ( , ), , ( , ))= …1 1 ,
de k t a c a c a tj j j
b
j
b qj j j( , ) : ( )= −− − −
0 0
2 1 2 1 1
, j ∈ { 1, … , n } ;
Ψ ( , ) : exp ( , )t x k t a xj j j
q
j
n
j=
=
∑
1
, ( , ) [ , ]t x T∈Π 0 ;
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
PARABOLIÇNI ZA SOLONNYKOVYM SYSTEMY … 1505
p x y x y
j
n
j j
mj( ; ) :
/
/
= −
=
∑
1
2
1 2
— 2b -paraboliçna vidstan\ miΩ toçkamy x i y iz R
n;
∆ t f t f t fβ β( , ) : ( , ) ( , )⋅ = ⋅ − ⋅ , ∆x
y f x f x f y( , ) : ( , ) ( , )⋅ = ⋅ − ⋅
— pryrosty funkci] f ; l i λ — zadani çysla vidpovidno z mnoΩyn Z+ i ( , )0 1 ;
l l t bj j: ( ) ( )/= +[ ]2 , de [ ]a — cila çastyna çysla a. Zaznaçymo [5, c. 97], wo
� � �
k a a( , ) :0 = i
E t x t xc0
( , ) ( , ) ( , )− − ≤τ ξ τ ξΨ Ψ , 0 ≤ τ < t ≤ T , { },x nξ ⊂ R . (13)
Budemo vykorystovuvaty taki prostory:
Hl
k a
+
⋅
λ
� �
( , )
— prostir funkcij u T: [ , ]Π 0 → C, qki magt\ neperervni poxidni
∂t xu,
α , α ≤ l , i skinçennu normu
u u ul
k a
l
k a
j
l
j
k a
+
⋅
+
⋅
=
⋅= 〈〈 〉〉 + 〈 〉∑λ λ
� � � � � �
( , ) ( , ) ( , ):
0
.
Tut
〈〈 〉〉 = 〈 〉 + 〈 〉+
⋅
+
⋅
+
⋅u u ul
k a
l x
k a
l b t
k a
λ λ λ
� � � � � �
( , )
,
( , )
( )/( ),
( , ): 2 ,
〈 〉 = ∂+
⋅
≤ − <
− +
⋅∑ 〈 〉u ul b t
k a
l b
t x l b t
k a
( )/( ),
( , )
, ( )/( ),
( , ):λ
α
α
α λ2
0 2
2
� � � �
,
〈 〉 = ∂+
⋅
=
⋅∑ 〈 〉u ul x
k a
l
t x x
k a
λ
α
α
λ,
( , )
, ,
( , ):
� � � �
,
〈 〉 = − +( )( )⋅
⊂
≠
− −u u t x t t x xt
k a
t x x
t
t
T
λ
β
β
β λβ β,
( , )
{( , ),( , )}
: sup ( , ) ( , ) ( , )
[ , ]
� �
Π
∆ Ψ Ψ
0
1 ,
〈 〉 = ( ) +( )( )⋅
⊂
≠
− −u u t x p x y t x t yx
k a
t x t y
x y
x
y
T
λ
λ
,
( , )
{( , ),( , )}
: sup ( , ) ( ; ) ( , ) ( , )
[ , ]
� �
Π
∆ Ψ Ψ
0
1 ,
〈 〉 = ∂ ( )( )⋅
= ∈
−∑u u t x t xj
k a
j t x
t x
T
� �
( , )
( , )
,: sup ( , ) ( , )
[ , ]α
α
Π
Ψ
0
1 ;
Cl
a
+λ
�
— prostir funkcij v : R C
n → , dlq qkyx isnugt\ neperervni poxidni
∂x
α v , α ≤ l , i [ skinçennog norma
v v vl
a
l
a
j
l
j
a
+ +
=
= + 〈 〉∑λ λ
� � �
: [ ]
0
,
de
[ ] : ,v vl
a
l
x x
a
+
=
= ∂∑ 〈 〉λ
α
α
λ
� �
,
〈 〉 = ( ) +( )( )
⊂
≠
− −v vλ
λ
,
{ , }
: sup ( ) ( ; ) ( , ) ( , )x
a
x y
x y
x
y
n
x p x y x y
�
R
∆ Ψ Ψ0 0 1 ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
1506 S. D. IVASYÍEN, H. P. IVASGK
〈 〉 = ∂ ( )( )
= ∈
−∑v vj
a
j x
x
n
x x
�
: sup ( ) ( , )
α
α
R
Ψ 0 1 ,
H Hl l
k
+ +
⋅=λ λ: ( , )
� �
0 , C Cl l+ +=λ λ:
�
0 ,
�
0 0 0: ( , , )= … ;
0
Hl
k a
+
⋅
λ
� �
( , )
— pidprostir prostoru Hl
k a
+
⋅
λ
� �
( , )
, elementy qkoho razom z usima svo]my po-
xidnymy dorivnggt\ nulg pry t = 0;
Hr
k a
j
N
j +
⋅
=∏ λ
� �
( , )
1
,
0
1
Hr
k a
j
N
j +
⋅
=∏ λ
� �
( , ) i
Cr
a
j
r
j
�
=∏ 1
— dekartovi dobutky vidpovidnyx
prostoriv z indeksamy rj ∈ +Z .
5. Zvedennq zahal\no] poçatkovo] zadaçi do zadaçi z nul\ovymy poçat-
kovymy danymy. Rozhlqnemo zahal\nu poçatkovu zadaçu (1), (2), prypuskagçy
vykonannq umov A) i B), a takoΩ umov
C) koefici[nty dyferencial\nyx vyraziv Akj i Bsj naleΩat\ vidpovidno
do prostoriv Hl sk− +λ i Cl ps− +λ , { , } { , , }k j N⊂ …1 , s r∈ …{ , , }1 ;
D)
f Hl s
k a
j
N
j
∈ − +
⋅
=∏ λ
� �
( , )
1
, ϕ λ∈ − +=∏ Cl p
a
s
r
s
�
1
.
Na pidstavi tverdΩennq 2 z p.82 ta vlastyvostej prostoriv Hel\dera dovo-
dyt\sq nastupna lema, analohiçna lemi84.5 iz [7].
Lema.1. Qkwo vykonugt\sq umovy A) – D), to funkci]
ϕ α α
j t j t
u( ) :0 0
0
= ∂
=
, α0 0∈ …{ , , }lj , j N∈ …{ , , }1 , (14)
qki znaxodqt\sq iz systemy (1) ta poçatkovo] umovy (2), naleΩat\ do prosto-
riv Cl t b
a
j+ − +2 0α λ
�
i
ϕ ϕα
α λ λ λj l t b
a
j
N
j l s
k a
s
r
s l p
a
j j s
C f( ) ( , )
0
02
1 1
+ − +
=
− +
⋅
=
− +≤ +
∑ ∑
� � � �
. (15)
ZauvaΩymo, wo funkci] (14) odnoznaçno vyznaçagt\sq koefici[ntamy i pra-
vymy çastynamy zadaçi (1), (2).
Lema.2. Nexaj ϕ α
j
( )0
— funkci] iz lemy 1. Todi isnugt\ funkci] vj8∈
∈
Hl t
k a
j+ +
⋅
λ
� �
( , ) , j N∈ …{ , , }1 , dlq qkyx vykonugt\sq poçatkovi umovy
∂ =
=t j t jt x xα αϕ0 0
0
v ( , ) ( )( ) , x n∈R , α0 0∈ …{ , , }lj , (16)
ta nerivnosti
v j l t
k a
j
N
j l s
k a
s
r
s l p
a
j j s
C f
+ +
⋅
=
− +
⋅
=
− +≤ +
∑ ∑λ λ λϕ
� � � � �( , ) ( , )
1 1
. (17)
Dovedennq. Funkcig v j moΩna pobuduvaty qk rozv’qzok zadaçi Koßi z po-
çatkovymy umovamy (16) dlq 2b -paraboliçnoho rivnqnnq
∂ + − ∂
=
+
∑t
b
x
b
k
n l
j
k
k
k
j
( )1 2
1
1
v = 0. (18)
Vidomo [4, 9, 10], wo rozv’qzok zadaçi (16), (18) vyznaça[t\sq formulog
v j
l
j jt x G t x d
j
n
( , ) ( , )( ) ( )( )= −
=
∑ ∫
α
α αξ ϕ ξ ξ
0
0 0
0 R
, ( , ) [ , ]t x T∈Π 0 ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
PARABOLIÇNI ZA SOLONNYKOVYM SYSTEMY … 1507
de
G t x P t xj j t x j
( ) ( )( , ) : ( , ) ( , )α α0 0= ∂ ∂ Γ , P p i P p ij
m m b l
j
j( ) ( ) ( )( ), ( , )α α αλ σλ λ σ0 0 0 02= −
,
Γj — FR rivnqnnq (18).
Z ci[] formuly ta ocinok intehrala Puassona, oderΩanyx v [11], vyplyva[
ocinka
〈〈 〉〉 ≤+ +
⋅
=
+ − +∑v j l t
k a
l
j l t b
a
j
j
j
Cλ
α
α
α λϕ
� � �
( , ) ( )[ ]
0
0
0
0
2 , (19)
a z ne] ta poçatkovyx umov (16) — ocinka
v j l t
k a
l
j l t b
a
j
j
j
C
+ +
⋅
=
+ − +
≤ ∑λ
α
α
α λ
ϕ
� � �
( , ) ( )
0
0
00
2
. (20)
Spravdi, zhidno z oznaçennqmy vidpovidnyx norm dlq oderΩannq ocinky (20) do-
syt\ vstanovyty nerivnosti
〈 〉 ⋅v j s
k a
� �
( , ) ≤
C
l
j l t b
aj
jα
α
α λ
ϕ
0
0
00
2
=
+ − +∑ ( )
�
, s l t j∈ … +{ , , }0 . (21)
Za dopomohog formuly Tejlora ta umov (16) dlq α ∈ +
+
Z
n 1, α ≤ +l t j , ot-
rymu[mo
∂t x j t x, ( , )α v =
k
k
k
x j
t
x j
t
k
x t x d
=
−
=
−∑ ∫∂ ∂ +
−
− ∂ ∂
0
1
0 0 0
1
0
0 01
1
β
τ τ
α
τ
β
τ
β
τ
ατ
β
τ τ τ
!
( , )
( )!
( ) ( , ), ,v v =
=
k
k
x j
k
t
x j
t
k
x t x d
=
+ − +∑ ∫∂ +
−
− ∂ ∂
0 0 0
1 0
0
0 0 0 01
1
β
α α β
τ τ
β α αϕ
β
τ τ τ
!
( )
( )!
( ) ( , )( ) ∆ v ,
de β α α0 02: ( ) ( )/= + −[ ] −l t bj . Zastosuvavßy dlq ocingvannq pryrostu
∆τ τ
β α α0 0 0∂ ∂+
x jv ocinku (19), z ostann\o] formuly oderΩymo nerivnosti (21).
Z (15) i (20) vyplyva[ ocinka (17) i, zokrema, te, wo
v j l t
k aH
j
∈ + +
⋅
λ
� �
( , ) .
Nastupna teorema, qka lehko dovodyt\sq za dopomohog lem81 i 2, mistyt\
odyn z osnovnyx rezul\tativ statti.
Teorema.1. Nexaj vykonugt\sq umovy A) – D) i ϕ α
j
( )0 , v j — funkci] iz
lem81 ta82. Dlq isnuvannq [dynoho rozv’qzku u Hl t
k a
j
N
j
∈ + +
⋅
=∏ λ
� �
( , )
1
zadaçi (1) i (2)
neobxidno i dosyt\, wob vektor-funkciq
w u:= − v, v v v: ( , , )= …1 N ,
bula [dynym rozv’qzkom iz prostoru
0
1
Hl t
k a
j
N
j+ +
⋅
=∏ λ
� �
( , )
systemy
A t x w t x g t xt x( , , , ) ( , ) ( , )∂ ∂ = , ( , ) [ , ]t x T∈Π 0 , (22)
de vektor-funkciq
g t x f t x A t x t xt x( , ) : ( , ) ( , , , ) ( , )= − ∂ ∂ v , ( , ) [ , ]t x T∈Π 0 ,
naleΩyt\ do prostoru
0
1
Hl s
k a
j
N
j− +
⋅
=∏ λ
� �
( , )
.
Qkwo dlq w spravdΩu[t\sq ocinka
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
1508 S. D. IVASYÍEN, H. P. IVASGK
j
N
j l t
k a
w
j=
+ +
⋅∑
1
λ
� �
( , )
≤
C g
j
N
j l s
k a
j=
− +
⋅∑
1
λ
� �
( , )
, (23)
to dlq u [ pravyl\nog ocinka
j
N
j l t
k a
u
j=
+ +
⋅∑
1
λ
� �
( , )
≤
C f
j
N
j l s
k a
s
r
s l p
a
j s
=
− +
⋅
=
− +∑ ∑+
1 1
λ λϕ
� � �( , )
. (24)
Oznaçennq.2. Zadaçu pro znaxodΩennq rozv’qzku w ∈
0
1
Hl t
k a
j
N
j+ +
⋅
=∏ λ
� �
( , )
syste-
my (22), v qkij g ∈
0
1
Hl s
k a
j
N
j− +
⋅
=∏ λ
� �
( , )
, nazyvatymemo zadaçeg z nul\ovymy poçatko-
vymy danymy dlq paraboliçno] za Solonnykovym systemy kvaziodnoridno] struk-
tury u prostorax Hel\dera zrostagçyx funkcij.
6. Korektna rozv’qznist\ poçatkovo] zadaçi v model\nomu vypadku. Roz-
hlqnemo model\nyj vypadok zadaçi (1), (2), tobto zadaçu
A u t x f t xt x
0( , ) ( , ) ( , )∂ ∂ = , ( , ) [ , ]t x T∈Π 0 ,
(25)
B u t x xt x t
0
0
( , ) ( , ) ( )∂ ∂ =
=
ϕ , x n∈R ,
u prypuwenni, wo vykonugt\sq umovy paraboliçnosti ta dopovnql\nosti.
Teorema.2. Qkwo vektor-funkci] f i ϕ zadovol\nqgt\ umovu D), to is-
nu[ [dynyj rozv’qzok
u Hl t
k a
j
N
j
∈ + +
⋅
=∏ λ
� �
( , )
1
zadaçi (25), dlq qkoho spravdΩu[t\sq
ocinka (24).
Dovedennq. Zhidno z teoremog81 dosyt\ dovesty isnuvannq [dynoho rozv’qz-
ku w ∈
0
1
Hl t
k a
j
N
j+ +
⋅
=∏ λ
� �
( , )
systemy
A w t x g t xt x
0( , ) ( , ) ( , )∂ ∂ = , ( , ) [ , ]t x T∈Π 0 , (26)
qkyj zadovol\nq[ nerivnist\ (23), u prypuwenni, wo g ∈
0
1
Hl s
k a
j
N
j− +
⋅
=∏ λ
� �
( , )
.
Zhidno z vykladenym u p.83 komponenty rozv’qzku systemy (26) z dosyt\ hlad-
kog i finitnog vektor-funkci[g g vyznaçagt\sq formulamy
w t xj
k
N
( , ) =
=
∑
1
Ò
A V t xjk t x gk
0 ( , ) ( , )∂ ∂ , ( , ) [ , ]t x T∈Π 0 , j N∈ …{ , , }1 , (27)
de Vh — ob’[mnyj potencial (9). Dlq funkci] h, qka vyznaçena v Π[ , ]0 T i do-
rivng[ nulg pry t = 0, vin nabuva[ vyhlqdu
V t x d t x h dh
t
n
( , ) ( , ) ( , )= − −∫ ∫
0
τ τ ξ τ ξ ξ
R
Γ , ( , ) [ , ]t x T∈Π 0 . (28)
Formuly (27) vyznaçagt\ potribnyj rozv’qzok systemy (26), pryçomu [dynyj,
i u vypadku, koly g ∈
0
1
Hl s
k a
j
N
j− +
⋅
=∏ λ
� �
( , ) . Wob u c\omu perekonatysq, dosyt\ doves-
ty, wo funkciq (28) [ [dynym rozv’qzkom 2b -paraboliçnoho rivnqnnq (10) z h8∈
∈ Hs
k a
+
⋅
λ
� �
( , ), s ∈ +Z , qkyj naleΩyt\ do prostoru
0
2Hs br
k a
+ +
⋅
λ
� �
( , )
i dlq qkoho [ pra-
vyl\nog ocinka
V C hh s br
k a
s
k a
+ +
⋅
+
⋅≤2 λ λ
� � � �
( , ) ( , ) . (29)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
PARABOLIÇNI ZA SOLONNYKOVYM SYSTEMY … 1509
Dovedennq toho, wo dlq h8∈ Hs
k a
+
⋅
λ
� �
( , )
funkciq Vh [ rozv’qzkom rivnqnnq (10),
analohiçne dovedenng vidpovidnoho tverdΩennq dlq 2b -paraboliçnoho riv-
nqnnq perßoho porqdku vidnosno zminno] t u [5].
Dovedemo pravyl\nist\ ocinky (29). U praci [11] vstanovleno ocinku
〈〈 〉〉 ≤ 〈〈 〉〉+ +
⋅
+
⋅V C hh s br
k a
s
k a
2 λ λ
� � � �
( , ) ( , ) .
We slid dovesty, wo
〈 〉 ≤⋅
+
⋅V C hh j
k a
s
k a
� � � �
( , ) ( , )
λ , j s br∈ … +{ , , }0 2 ,
a dlq c\oho dosyt\ vstanovyty nerivnosti
∂t x hV t x, ( , )α ≤ C h t xs
k a
+
⋅
λ
� �
( , ) ( , )Ψ , ( , ) [ , ]t x T∈Π 0 , α ≤ +s br2 . (30)
Oskil\ky ∂ =
∂t x h h
V V
t x
,
,
β
β , β ≤ s , to dovedennq nerivnostej (30) zvodyt\sq do
dovedennq takyx Ωe nerivnostej pry s = 0.
Qkwo α < 2br , to
∂ = ∂ − −∫ ∫t x h
t
t xV t x d t x h d
n
, ,( , ) ( , ) ( , )α ατ τ ξ τ ξ ξ
0 R
Γ , ( , ) [ , ]t x T∈Π 0 .
Zvidsy za dopomohog nerivnostej (7) i (13) oderΩu[mo
∂t x hV t x, ( , )α ≤
C h d t E t xk a
t
r M b
c
n
〈 〉 − − −( )⋅ − +∫ ∫0
0
2
0
� �
( , ) ( )/( )( ) ( , ) ( , )τ τ τ ξ τ ξα
R
Ψ ×
× E t x dc c− − −
0
( , )τ ξ ξ ≤
C h t x t dk a
t
r b〈 〉 −⋅ − −∫0
0
1 2
� �
( , ) /( )( , ) ( )Ψ τ τα ×
×
R
n
t E t x dM b
c c∫ − − −−
−( ) ( , )/( )τ τ ξ ξ1 2
0
=
= C h t x tk a r b〈 〉 ⋅ −
0
2
� �
( , ) /( )( , )Ψ α ≤ C h t xk a
λ
� �
( , ) ( , )⋅ Ψ , ( , ) [ , ]t x T∈Π 0 .
Pry α = 2br pravyl\nym [ zobraΩennq
∂ = + ∂ − −∫ ∫t x h r
t
t x
xV t x h t x d t x h d
n
, ,( , ) ( , ) ( , ) ( , )α
α
α
ξδ τ τ ξ τ ξ ξ
0
0 R
Γ ∆ , ( , ) [ , ]t x T∈Π 0 ,
de δα0 r — symvol Kronekera. Vykorystovugçy cg formulu, nerivnosti (7) i
(13), a takoΩ nerivnist\
p x E t x Ct E t xc
b
c( ; ) ( , ) ( , )/( )ξ ξ ξλ λ( ) − ≤ −′
2 , t > 0, { , }x nξ ⊂ R ,
de c c c0 < ′ < , ma[mo
∂t x hV t x, ( , )α ≤
〈 〉 + 〈 〉 − ( )⋅ ⋅ −∫ ∫h t x C h d t p xk a
x
k a
t
M b
n
0
0
2
� � � �
( , )
,
( , ) /( )( , ) ( ) ( ; )Ψ λ
λτ τ ξ
R
×
× Ψ Ψ( , ) ( , ) ( , )τ ξ τ τ ξ ξ+( ) − −x E t x dc ≤ 〈 〉 + 〈 〉⋅ ⋅h t x C hk a
x
k a
0
� � � �
( , )
,
( , )( , )Ψ λ ×
×
0
1 2 1 2
0
t
b M b
ct d t E t x t x
n
∫ ∫− − − − +( )− + −( ) ( ) ( , ) ( , ) ( , )/( ) /( )τ τ τ τ ξ τ ξλ
R
Ψ Ψ ×
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
1510 S. D. IVASYÍEN, H. P. IVASGK
× E t x dc c′ − − −
0
( , )τ ξ ξ ≤
C h t d t xk a
t
b
λ
λτ τ
� �
( , ) /( )( ) ( , )⋅ − ++ −
∫1
0
1 2 Ψ ≤
≤ C h t xk a
λ
� �
( , ) ( , )⋅ Ψ , ( , ) [ , ]t x T∈Π 0 .
Z oderΩanyx ocinok vyplyva[ vykonannq nerivnostej (30) s = 0 i, otΩe, dlq
dovil\noho s .
Dlq dovedennq [dynosti rozv’qzku rivnqnnq (10) u prostori
0
2Hs br
k a
+ +
⋅
λ
� �
( , )
moΩna
skorystatysq formulog (20) iz [4] dlq rozv’qzkiv zadaçi Koßi dlq zahal\noho
2b -paraboliçnoho rivnqnnq. Cq formula oderΩana v [4] dlq dosyt\ hladkyx i
finitnyx rozv’qzkiv. Ale qkwo detal\no proslidkuvaty ]] vyvid, to moΩna pere-
konatysq, wo vona [ pravyl\nog i dlq rozv’qzkiv iz vidpovidnyx prostoriv typu
Hl
k a
+
⋅
λ
� �
( , ). Zastosuvannq vkazano] formuly do rozv’qzku rivnqnnq (10) iz prostoru
0
2Hs br
k a
+ +
⋅
λ
� �
( , )
pryvodyt\ do joho zobraΩennq u vyhlqdi (28), a zvidsy vyplyva[, wo
rozv’qzok Vh iz (28) [dynyj.
ZauvaΩennq. Teorema, analohiçna teoremi82, [ pravyl\nog i dlq zahal\no]
poçatkovo] zadaçi (1), (2), qkwo vykonano umovy A) – D). Dovedennq tako] teo-
remy, dodatkovo do navedenyx vywe rezul\tativ, potrebu[ we pobudovy ta de-
tal\noho doslidΩennq vlastyvostej rehulqryzatora vidpovidno] zadaçi z nul\o-
vymy poçatkovymy danymy. Cym pytannqm budut\ prysvqçeni nastupni publi-
kaci].
1. Petrovskyj Y. H. O probleme Koßy dlq system lynejn¥x uravnenyj s çastn¥my proyzvod-
n¥my v oblasty neanalytyçeskyx funkcyj // Bgl. Mosk. un-ta. Matematyka y mexanyka. –
1938. – 1, # 7. – S. 1 – 72.
2. ∏jdel\man S. D. Ob odnom klasse parabolyçeskyx system // Dokl. AN SSSR. – 1960. – 133,
# 1. – S. 40 – 43.
3. Yvasyßen S. D., ∏jdel\man S. D. 2b -Parabolyçeskye system¥ // Tr. sem. po funkc. analy-
zu. – Kyev: Yn-t matematyky AN USSR, 1968. – V¥p.81. – S. 3 – 175, 271 – 273.
4. Ivasyßen S. D., Kondur O. S. Pro matrycg Hrina zadaçi Koßi ta xarakteryzacig deqkyx
klasiv rozv’qzkiv dlq 2b -paraboliçnyx system dovil\noho porqdku // Mat. studi]. – 2000. –
14, # 1. – S. 73 – 84.
5. Eidelman S. D., Ivasyshen S. D., Kochubei A. N. Analytic methods in the theory of differential and
pseudo-differential equations of parabolic type // Operator Theory: Adv. and Appl. – 2004. – 152. –
390 p.
6. Solonnykov V. A. O kraev¥x zadaçax dlq obwyx parabolyçeskyx system // Dokl. AN SSSR.
– 1964. – 157, # 1. – S. 56 – 59.
7. Solonnykov V. A. O kraev¥x zadaçax dlq lynejn¥x parabolyçeskyx system dyfferency-
al\n¥x uravnenyj obweho vyda // Tr. Mat. yn-ta AN SSSR. – 1965. – 83. – S. 3 – 163.
8. Lad¥Ωenskaq O. A., Solonnykov V. A., Ural\ceva N. N. Lynejn¥e y kvazylynejn¥e uravne-
nyq parabolyçeskoho typa. – M.: Nauka, 1967. – 7368s.
9. Ivasgk H. P. Poçatkova zadaça dlq model\nyx paraboliçnyx za Solonnykovym system neod-
noridno] struktury // Nauk. visn. Çerniv. un-tu. – 2005. – Vyp.8269. – S.849 – 52.
10. ∏jdel\man S. D. Parabolyçeskye system¥. – M.: Nauka, 1964. – 4438s.
11. Ivasgk H. P. Pro vlastyvosti potencialiv model\noho 2b -paraboliçnoho rivnqnnq dovil\-
noho porqdku // Nauk. visn. Çerniv. un-tu. – 2006. – Vyp.8288. – S. 51 – 56.
OderΩano 16.06.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 11
|
| id | umjimathkievua-article-3550 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:44:39Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0d/f0be052c910a1fad20de5c474878980d.pdf |
| spelling | umjimathkievua-article-35502020-03-18T19:57:29Z Solonnikov parabolic systems with quasihomogeneous structure Параболічні за Солонниковим системи квазіоднорідної структури Ivasyshen, S. D. Ivasyuk, H. P. Івасишен, С. Д. Івасюк, Г. П. We consider a new class of systems of equations that combine the structures of Solonnikov and Éidel’man parabolic systems. We prove a theorem on the reduction of a general initial-value problem to a problem with zero initial data and a theorem on the correct solvability of an initial-value problem in a model case. Розглядається новий клас систем рівнянь, які поєднують у собі структури систем, параболічних за Солонниковим і Ейдельманом. Доведено теореми про зведення загальної початкової задачі до задачі з нульовими початковими даними та про коректну розв'язність початкової задачі в модельному випадку. Institute of Mathematics, NAS of Ukraine 2006-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3550 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 11 (2006); 1501–1510 Український математичний журнал; Том 58 № 11 (2006); 1501–1510 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3550/3835 https://umj.imath.kiev.ua/index.php/umj/article/view/3550/3836 Copyright (c) 2006 Ivasyshen S. D.; Ivasyuk H. P. |
| spellingShingle | Ivasyshen, S. D. Ivasyuk, H. P. Івасишен, С. Д. Івасюк, Г. П. Solonnikov parabolic systems with quasihomogeneous structure |
| title | Solonnikov parabolic systems with quasihomogeneous structure |
| title_alt | Параболічні за Солонниковим системи квазіоднорідної структури |
| title_full | Solonnikov parabolic systems with quasihomogeneous structure |
| title_fullStr | Solonnikov parabolic systems with quasihomogeneous structure |
| title_full_unstemmed | Solonnikov parabolic systems with quasihomogeneous structure |
| title_short | Solonnikov parabolic systems with quasihomogeneous structure |
| title_sort | solonnikov parabolic systems with quasihomogeneous structure |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3550 |
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