Nonlocal Dirichlet problem for linear parabolic equations with degeneration
In the spaces of classical functions with power weight, we prove the correct solvability of the Dirichlet problem for parabolic equations with nonlocal integral condition with respect to the time variable and an arbitrary power order of degeneration of coefficients with respect to the time and space...
Saved in:
| Date: | 2007 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2007
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3294 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509356224675840 |
|---|---|
| author | Pukalskyi, I. D. Пукальський, І. Д. |
| author_facet | Pukalskyi, I. D. Пукальський, І. Д. |
| author_sort | Pukalskyi, I. D. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:50:22Z |
| description | In the spaces of classical functions with power weight, we prove the correct solvability of the Dirichlet problem for parabolic equations with nonlocal integral condition with respect to the time variable and an arbitrary power order of degeneration of coefficients with respect to the time and space variables. |
| first_indexed | 2026-03-24T02:39:48Z |
| format | Article |
| fulltext |
UDK 517.946
I. D. Pukal\s\kyj (Çerniv. nac. un-t)
NELOKAL|NA ZADAÇA DIRIXLE DLQ LINIJNYX
PARABOLIÇNYX RIVNQN| Z VYRODÛENNQM
In the spaces of classical functions, we prove the correct solvability of the Dirichlet problem for
parabolic equations with nonlocal integral condition for a time variable and with arbitrary power order of
the degeneration of coefficients with respect to the time variable and space variables.
V prostranstvax klassyçeskyx funkcyj so stepenn¥m vesom dokazana korrektnaq razreßymost\
zadaçy Dyryxle dlq parabolyçeskyx uravnenyj s nelokal\n¥m yntehral\n¥m uslovyem po vre-
mennoj peremennoj y proyzvol\nomu stepennomu porqdku v¥roΩdenyq koπffycyentov kak po
vremennoj, tak y po prostranstvenn¥m peremenn¥m.
U pracqx [1, 2] rozhlqdalos\ zastosuvannq pryncypu ekstremumu dlq linijnyx
eliptyko-paraboliçnyx rivnqn\ 2-ho porqdku z nevid’[mnog xarakterystyçnog
formog, koefici[nty qkyx magt\ stepenevi osoblyvosti obmeΩenoho porqdku
na meΩi oblasti. Metodom bar’[rnyx funkcij vstanovleno apriorni ocinky i
strohyj pryncyp maksymumu.
U praci [3] pobudovano teorig klasyçnyx rozv’qzkiv zadaçi Koßi i krajovyx
zadaç dlq rivnomirno paraboliçnyx rivnqn\, qki magt\ stepenevi osoblyvosti
obmeΩenoho porqdku na meΩi oblasti v koefici[ntax pry molodßyx poxidnyx.
Za dopomohog special\nyx funkcional\nyx prostoriv u praci [4] dlq paraboliç-
nyx rivnqn\ z nevid’[mnog kvadratyçnog formog, qka vyrodΩu[t\sq na meΩi
oblasti, vstanovleno rozv’qznist\ zadaçi Koßi. Vyvçennq krajovo] zadaçi dlq
system zi stalymy koefici[ntamy ta intehral\nog nelokal\nog umovog za çaso-
vog zminnog provedeno u [5].
Vstanovlenng korektno] rozv’qznosti zadaçi z skisnog poxidnog ta odnosto-
ronn\o] krajovo] zadaçi z nelokal\nog umovog za çasovog zminnog dlq parabo-
liçnyx rivnqn\, qki vyrodΩugt\sq na meΩi oblasti za sukupnistg zminnyx ste-
penevym çynom, prysvqçeno praci [6, 7].
Tut za dopomohog apriornyx ocinok i pryncypu maksymumu vyvça[t\sq zadaça
Dirixle dlq paraboliçnyx rivnqn\ zi stepenevymy osoblyvostqmy v koefici[ntax
na meΩi oblasti ta intehral\nog nelokal\nog umovog za çasovog zminnog. V
hel\derovyx prostorax zi stepenevog vahog vstanovleno isnuvannq i [dynist\
rozv’qzku nelokal\no] zadaçi Dirixle.
Postanovka zadaçi ta osnovnyj rezul\tat. Nexaj D — obmeΩena opukla
oblast\ v R
n
z meΩeg ∂D. Rozhlqnemo v oblasti Q = ( 0, T ] × D zadaçu zna-
xodΩennq funkci] u ( t, x ), qka pry t > 0, t ≠ t(
0
)
, t(
0
) ∈ ( 0, T ) zadovol\nq[ riv-
nqnnq
( L u ) ( t, x ) ≡ ∂ − ( )∂ ∂ − ( )∂ − ( )
( )
= =
∑ ∑t ij x
ij
n
x i x
i
n
A t x A t x A t x u t x
i j i
, , , ,
1 1
0 = f ( t, x )
(1)
i nelokal\nu umovu
u x q t u x d
T
( ) + ( ) ( )∫0
0
, , ,τ τ τ = ϕ ( x ), (2)
a na biçnij meΩi Γ = ( 0, T ] × ∂D — krajovu umovu
u | Γ = ψ ( t, x ) | Γ . (3)
© I. D. PUKAL|S|KYJ, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1 109
110 I. D. PUKAL|S|KYJ
Nexaj l( )1
, l( )2
— dovil\ni dijsni çysla, D = D ∪ ∂ D, | x – ξ | =
= inf
/
ξ
ξ
∈∂ =
( − )
∑
D
i i
i
n
x 2
1
1 2
, x ∈ D, Q(
0
) = Q \ { ( t, x ) ∈ Q | t = t(
0
)
, x ∈ D }.
Osoblyvosti koefici[ntiv dyferencial\noho vyrazu L budut\ xarakteryzu-
vaty taki funkci]: s1 ( l
(1), t ) = | t – t(
0
)
| l
(1)
pry | t – t(
0
)
| ≤ 1, s1 ( l
(1), t ) = 1 pry
| t – t( 0
)
| ≥ 1, s2 ( l
(2
), x ) = | x – ξ | l(2)
pry | x – ξ | ≤ 1, s2 ( l
(2
), x ) = 1 pry | x – ξ | ≥ 1.
Nexaj Q = [ 0, T ] × D , a P ( t, x ), P1 ( t(1), x(1)
), Bk ( t(1), x
(2
)
) i Pk
( )2 ( t(2
), x
(2
)
),
k ∈ { 1, … , n }, — toçky iz Q , x (1) = ( … )x xn1
1 1( ) ( ), , , x
(2
) = ( … −x xk1
1
1
1( ) ( ), , , xk
( )2 ,
x xk n+ … )1
1 1( ) ( ), , . Poznaçymo çerez β ν
k
( )
, γ ν( )
, µ ν
i
( )
, α dijsni çysla, taki, wo
β ν
k
( ) ∈ ( – ∞, ∞ ), γ ν( ) ≥ 0, µ ν
i
( ) ≥ 0, i ∈ { 0, 1, … , n }, α ∈ ( 0, 1 ), ν ∈ { 1, 2 }. Po-
klademo s ( l; P ) = s1 ( l
(1), t ) s2 ( l
(2
), x ). Oznaçymo funkcional\ni prostory, v qkyx
doslidΩu[t\sq zadaça.
C2
+
α
( γ, β; l; Q ) — prostir funkcij u, ( t, x ) ∈ Q , qki magt\ neperervni ças-
tynni poxidni v oblasti Q(
0
)
vyhlqdu ∂ ∂t
k
x
ju, 2k + | j | ≤ 2, dlq qkyx [ skinçen-
nog norma
|| u; γ, β; l; Q || 2 + α =
j=
∑
0
2
|| u; γ, β; l; Q || j + ||−
−
u; γ, β; l; Q ||−
−2 + α ,
de, napryklad,
|| u; γ, β; 0; Q || 0 = sup
P Q
u P
∈
( ) ≡ || u; Q || 0 ,
||−
−
u; γ, β; l; Q ||−
−2 + α =
i j k
n
P B Q
i j k
k
s l P
, , ,
sup ; ˜
= { }⊂
∑
[ ( + ( + ) − − − )
1
1
1
2 α γ β β αβ ×
× x x u P u B s l Pk k x x x x k
P B Q
ki j i j
k
( ) ( )
,
sup ; ˜1 2
1 1
1
2− ∂ ∂ ( ) − ∂ ∂ ( ) ] + [ ( + ( + ) − )
−
{ }⊂
α
α γ αβ ×
× x x u P u B s l Pk k t t k
P B Q
i j
k k
( ) ( )
,
sup ; ˜
( )
1 2
1 2
2
2− ∂ ( ) − ∂ ( ) ] + [ ( + ( + ) − − )
−
{ }⊂
α
α γ β β ×
× t t u P u B s l Px x k x x k
P B Q
i j i j
k k
( ) ( ) / ( )
,
sup ; ˜
( )
1 2 2 2
2
2
2− ∂ ∂ ( ) − ∂ ∂ ( ) ] + [ ( + ( + ) )
−
{ }⊂
α
α γ ×
× t t u P u Bt k t k
( ) ( ) / ( )1 2 2 2− ∂ ( ) − ∂ ( ) ]
−α
.
Tut s l P( ); 1̃ = min ( s ( l; P1 ); s ( l; Bk ) ), s l P( ); 2̃ = min ( s ( l; Pk
( )2 ); s ( l; Bk ) ).
Cr
( µj
; Q ) — mnoΩyna funkcij vj
, ( t, x ) ∈ Q , qki magt\ çastynni poxidni v
Q(
0
)
vyhlqdu ∂x
k
jv , | k | ≤ [ r ], dlq qkyx [ skinçennog norma
v vj j r
P Q
j x
k
j
k r
Q s k P P; , sup ;µ µ= +( ) ∂ ( )[ ]
∈≤[ ]
∑ +
+
i
n
k r P B Q
j i i
r
i
s k P s r x x x
= =[ ] { }⊂
−{ }∑ ∑
+( ) { } −( )
1
1 2
1 2
1
sup ; ˜ , ˜
,
( ) ( )µ ×
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
NELOKAL|NA ZADAÇA DIRIXLE DLQ LINIJNYX … 111
× ∂ ( ) − ∂ ( ) + ( )
−
{ }⊂
−{ }
x
k
j x
k
j i
B P Q
j
r
P B s P s r t t t
i i
v v1 2 1
1 2 2
2 2
sup ; ˜ , ˜
,
( ) ( ) /
( )
µ ×
×
v vj i j i
B P Q
jB P s k P
i i
( ) − ( ) + +( )
{ }⊂
( )
,
sup ; ˜
( )
2
2
2
µ ×
×
s r t t t B P
r
x
k
j i x
k
j i1
1 2 2 2
2
{ }
− ∂ ( ) − ∂ ( )
−{ }
, ˜ ( ) ( ) / ( )v v ,
de [ r ] — cila çastyna çysla r, { r } = r – [ r ], s l t1
1( )( ); ˜ = min ( s1 ( l
(1), t(1)
),
s1 ( l
(1), t(2
)
) ), s l x2
2( )( ); ˜ = min ( s2 ( l
(2
), x(1)
), s2 ( l
(2
), x
(2
)
) ).
Prypustymo, wo dlq zadaçi (1) – (3) vykonugt\sq taki umovy:
1°) koefici[nty Ai ∈ C
α
( µj
; Q ), i ∈ { 0, 1, … , n }, A0 ≤ K < + ∞, K — stala,
Aij ∈ C
α
( βi + βj
; Q ) i dlq dovil\noho vektora ξ ∈ { ξ1 , … , ξn } vykonu[t\sq ne-
rivnist\
c s P A P c
ij
n
i j ij i j1
2
1
2
2ξ β β ξ ξ ξ≤ ( + ) ( ) ≤
=
∑ ; ,
c1
, c1 — fiksovani dodatni stali;
2°) funkci] f ∈ C
α
( γ, β; µ0
; Q ), ϕ ∈ C2
+
α
( γ̃ , β̃ ; 0; D ), ψ ∈ C2
+
α
( γ, β; 0; Q ),
γ ν( ) = max max , max ,
i
i
i
i i( + ) ( − )
( ) ( ) ( )1
2
0β µ β µν ν ν
ν
, ν ∈ { 1, 2 }, i ∈ { 1, 2, … , n },
γ̃ = ( 0, γ
(2
)
), β̃ = ( 0, β
(2
)
);
3°) meΩa ∂D naleΩyt\ klasu C 2
+
α
, funkciq q ( t, x ) ∈ C2
+
α
( Q ),
sup ,
Q
T
q x e d( ) −∫ τ τλτ
0
≤ λ0 < 1, de λ — dovil\ne çyslo, qke zadovol\nq[ neriv-
nist\ λ < inf ,
Q
A t x( )− ( )0 , ψ τ ψ τ τ ϕ( ) + ( ) ( ) − ( )
∫
∂
0
0
, , ,x q x x d x
T
D
= 0.
Teorema 1. Nexaj dlq zadaçi (1) – (3) vykonano umovy 1° – 3° . Todi isnu[
[dynyj rozv’qzok zadaçi (1) – (3) u prostori C2
+
α
( γ, β; 0; Q ) i dlq n\oho sprav-
dΩu[t\sq ocinka
|| u; γ, β; 0; Q || 2 + α ≤ c ( || f; γ, β; µ0
; Q || α + || ϕ; γ̃ , β̃ ; 0; D || 2 + α +
+ || ψ; γ, β; 0; Q || 2 + α ). (4)
Dlq dovedennq teoremy 1 pobudu[mo poslidovnist\ rozv’qzkiv krajovyx za-
daçMz hladkymy koefici[ntamy, hranyçnym znaçennqm qko] bude rozv’qzok zadaçi
(1)M– (3).
Nexaj Qm = Q ∩ { ( t, x ) ∈ Q | s1 ( 1, t ) ≥ m1
1− , s2 ( 1, x ) ≥ m2
1−
} — poslidovnist\
oblastej, qka pry m1 → ∞, m2 → ∞ zbiha[t\sq do Q, D m = { x ∈ D | s2 ( 1, x ) ≥
≥ m2
1−
}, ∂Dm = { x ∈ D | s2 ( 1, x ) = m2
1−
}, Γm = ∂Dm × ( 0, T ], de m = ( m1
, m2 ),
m1
, m2 — natural\ni çysla, m1 > 1, m2 > 1.
Rozhlqnemo v oblasti Q krajovu zadaçu
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
112 I. D. PUKAL|S|KYJ
( L1 um ) ( t, x ) = ∂ − ( )∂ ∂ − ( )∂ − ( )
( )
= =
∑ ∑t ij x
ij
n
x i x
i
n
ma t x a t x a t x u t x
i j i
, , , ,
1 1
0 =
= fm ( t, x ), (5)
u x q x u x dm m
T
( ) + ( ) ( )∫0
0
, , ,τ τ τ = ϕm ( x ), (6)
um ( t, x ) | Γ = ψm ( t, x ) | Γ . (7)
Koefici[nty aij
, ai
, a0
, funkci] fm , ϕm , ψm vyznaçagt\sq takym çynom.
Qkwo ( t, x ) ∈ ( 0, T ] × Dm i β βi j
( ) ( )1 1+ ≥ 0, to aij ( t, x ) = min ( Aij ( t, x ), Aij (m1
1− ,
x ) ) pry t(
0
) ∈ ( 0, m1
1− ] i
aij ( t, x ) =
= min , , , ,
( )
( )
( )
( )A t x
m t t
A t m x
m t t
A t m xij ij ij( ) ( − )+ ( − ) + ( − )+ ( + )
− −1
0
0
1
1 1
0
0
1
11
2
1
2
pry t(
0
) > m1
1−
. U vypadku β βi j
( ) ( )1 1+ < 0 vyberemo aij ( t, x ) = max ( Aij ( t, x ),
Aij (m1
1− , x ) ) pry t(
0
) ∈ ( 0, m1
1− ] i
aij ( t, x ) =
= max , , , ,
( )
( )
( )
( )A t x
m t t
A t m x
m t t
A t m xij ij ij( ) ( − )+ ( − ) + ( − )+ ( + )
− −1
0
0
1
1 1
0
0
1
11
2
1
2
pry t(
0
) > m1
1−
.
Koefici[nty ai ( t, x ) = min ( Ai ( t, x ), Ai (m1
1− , x ) ) pry t(
0
) ∈ ( 0, m1
1− ] i
ai ( t, x ) =
= min , , , ,
( )
( )
( )
( )A t x
m t t
A t m x
m t t
A t m xi i i( ) ( − )+ ( − ) + ( − )+ ( + )
− −1
0
0
1
1 1
0
0
1
11
2
1
2
pry t(
0
) ≥ m1
1−
, i ∈ { 0, 1, … , n }.
Funkci] fm ( t, x ) = min ( f ( t, x ), f (m1
1− , x ) ) pry t(
0
) ∈ ( 0, m1
1− ] i
fm ( t, x ) =
= min , , , ,
( )
( )
( )
( )f t x
m t t
f t m x
m t t
f t m x( ) ( − )+ ( − ) + ( − )+ ( + )
− −1
0
0
1
1 1
0
0
1
11
2
1
2
pry t(
0
) ≥ m1
1−
. Pry x ∈ Dm funkci] ϕm ( x ) = ϕ ( x ).
Dlq ( t, x ) = Q \ { ( 0, T ) × Dm } koefici[nty aij , ai
, a0 i funkci] fm , ψm [
rozv’qzkamy zovnißn\o] zadaçi
∂t u = ∆ u, u ( 0, x ) = 0, u
mΓ
= g ( t, x ),
de, napryklad, dlq ai g = ai | Γm
,
�
n — normal\ do Γm
. Dlq x ∈ D \ Dm funkciq
ϕm [ rozv’qzkom zovnißn\o] zadaçi Dirixle
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
NELOKAL|NA ZADAÇA DIRIXLE DLQ LINIJNYX … 113
∆ v = 0, v | ∂Dm
= ϕ | ∂Dm
.
U zadaçi (5) – (7) vykona[mo zaminu
um ( t, x ) = vm ( t, x ) e–
λ
t + ψm ( t, x ),
de λ zadovol\nq[ umovu 3°. OderΩymo
( L2 vm ) ( t, x ) ≡
∂ − ( )∂ ∂ − ( )∂ − ( ) −
( )
= =
∑ ∑t ij x
ij
n
x i x
i
n
ma t x a t x a t x t x
i j i
, , , ,
1 1
0 λ v =
= fm ( t, x ) e–
λ
t – ( L ψm ) ( t, x ) ≡ F ( t, x ), (8)
v vm m
T
x q x e x d( ) + ( ) ( )−∫0
0
, , ,τ τ τλτ =
=
ϕ ψ τ ψ τ τm( ) − ( ) − ( ) ( )∫x x q x x dm m
T
0
0
, , , ≡ Φm ( x ), (9)
vm | Γ = 0. (10)
Znajdemo ocinku rozv’qzkiv krajovyx zadaç (8) – (10).
Teorema 2. Nexaj vm — klasyçnyj rozv’qzok zadaçi (8) – (10) v oblasti Q
i vykonano umovy 1° – 3°. Todi dlq vm vykonu[t\sq nerivnist\
| vm | ≤ max , ; , ;Φm
T
q t e d D F a Q1
0
1
0
0
1
0
− ( )
(− − )
−
−
−∫ τ τ λλτ
. (11)
Dovedennq. MoΩlyvi try vypadky: rozv’qzok vm [ nedodatnym v Q, abo
najbil\ße dodatne znaçennq vm dosqha[t\sq na D , abo ce najbil\ße znaçennq
dosqha[t\sq v toçci P1 ∈ Q.
U perßomu vypadku max
Q
vm ( t, x ) ≤ 0, u druhomu — 0 ≤ max
Q
vm ( t, x ) =
= max
D
vm ( t, x ) = vm ( 0, x(3)
). Todi z nelokal\no] umovy (9) ma[mo
Φm ( x(3)
) ≥ vm ( 0, x(3)
) 1 3
0
1
− ( )
−
−
∫ q x e d
T
τ τλτ, ( )
.
Tomu
vm ( 0, x(3)
) ≤ max , ( )
D
m
T
x q x e dΦ ( ) − ( )
−
−
∫1 3
0
1
τ τλτ
.
U tret\omu vypadku max
Q
vm ( t, x ) = vm ( P1 ), pryçomu v toçci P1 vykonugt\-
sq spivvidnoßennq
∂t vm ≥ 0, ∂xi
vm = 0, − ( )∂ ∂ ( )
=
∑a P Pij x
ij
n
x mi j1
1
1v ≥ 0. (12)
Nerivnist\ (12) ma[ misce, oskil\ky v toçci maksymumu druhi poxidni
∂ ∂y y mk k
v za bud\-qkym naprqmkom
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
114 I. D. PUKAL|S|KYJ
yk = α βki i i i
i
n
s P x x( )( − )
=
∑ ; ( )
1
1
1
, det || αki || ≠ 0,
nedodatni, a
α β β α αki x x m
ik
n
lj
n
i k ik kl ji
ik
n
y y mP P s P a P P
i k l j
( )∂ ∂ ( ) = ( + ) ( )
∂ ∂ ( )
= = =
∑ ∑ ∑1 1
1 1
1 1
1
1v v; =
=
λl y y m
l
n
l l
∂ ∂
=
∑ v
1
< 0.
Tomu, zhidno z obmeΩennqm 1°, xarakterystyçni çysla λ1
, … , λn kvadratyçno]
formy dodatni. Z uraxuvannqm (12) i rivnqnnq (8) u toçci P1 vykonu[t\sq ne-
rivnist\
vm ( P1 ) ≤ F ( P1 ) ( – a0 ( P1 ) – λ )
–
1
.
Analohiçno, rozhlqdagçy toçku najmenßoho vid’[mnoho znaçennq funkci] vm ,
ma[mo
vm ≥ min , min , , min0 1
0
1
0
1
D
m
T
Q
q x e d F aΦ − ( )
(− − )
−
−
−∫ ( )τ τ λλτ
.
OtΩe, dlq rozv’qzku zadaçi (8) – (10) spravdΩu[t\sq ocinka (11).
Rozhlqnemo odnoridnu zadaçu Dirixle
( L2 vm ) ( t, x ) = 0, vm ( 0, x ) = g ( x ), vm | Γ = 0. (13)
Nexaj Em ( t, x, τ, ξ ) — funkciq Hrina zadaçi (13) [8, s. 469].
ZauvaΩennq 1. Dlq funkci] Em ( t, x, τ, ξ ) vykonugt\sq nerivnosti
Em ( t, x, τ, ξ ) ≥ 0, 0 ≤
D
∫ Em ( t, x, 0, ξ ) dξ ≤ 1.
Vstanovymo isnuvannq rozv’qzku zadaçi (8) – (10).
Teorema 3. Qkwo vykonano umovy teoremy 1, to isnu[ [dynyj rozv’qzok za-
daçi (8) – (10), dlq qkoho spravdΩu[t\sq ocinka (11).
Dovedennq. Rozv’qzok zadaçi (8) – (10) ßuka[mo u vyhlqdi
vm ( t, x ) =
D
∫ Em ( t, x, 0, ξ ) vm ( 0, ξ ) dξ + ωm ( t, x ), (14)
de
ωm ( t, x ) =
D
∫ Em ( t, x, 0, ξ ) Φm ( ξ ) dξ + d
t
D
τ
0
∫ ∫ Em ( t, x, τ, ξ ) F ( τ, ξ ) dξ
— rozv’qzok zadaçi Dirixle (8) – (10) z poçatkovog umovog
ωm ( 0, x ) = Φm ( x ).
Zhidno z teoremog 2, dlq ωm ( t, x ) ma[ misce ocinka
| ωm | ≤ max ; ; ;Φm D F a Q
0 0
1
0
(− − )( )−λ .
Zadovol\nqgçy nelokal\nu umovu (9), ma[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
NELOKAL|NA ZADAÇA DIRIXLE DLQ LINIJNYX … 115
v vm
T
m
D
mx q x e d E x d( ) + ( ) ( ) ( )−∫ ∫0 0 0
0
, , , , , ,τ τ τ ξ ξ ξλτ =
= − ( ) ( )−∫ q x e x dm
T
τ ω τ τλτ, ,
0
≡ F2 ( x ). (15)
Rozv’qzok intehral\noho rivnqnnq ßuka[mo metodom poslidovnyx nablyΩen\.
Rekurentni spivvidnoßennq dlq poslidovnyx nablyΩen\ magt\ vyhlqd
v vm
k
T
m
D
m
kx F x q x e d E x d( ) − ( − )( ) = ( ) + ( ) ( ) ( )∫ ∫0 0 02
0
1, , , , , ,τ τ τ ξ ξ ξλτ
,
vm x( )( )0 0, = F2 ( x ).
Vraxovugçy zauvaΩennq 1, otrymu[mo
q x e d E x d q x e d
T
m
D
T
( ) ( ) ≤ ( )− −∫ ∫ ∫τ τ τ ξ ξ τ τλτ λτ, , , , ,
0 0
0 ≤ λ0 < 1.
Tomu, ocinggçy riznyci miΩ poslidovnymy nablyΩennqmy, oderΩu[mo
v vm
k
m
k kx x F Q( ) ( − )( ) − ( ) ≤0 01
0 2 0, , ;λ .
OtΩe, rozv’qzok intehral\noho rivnqnnq (15) zobraΩu[t\sq rivnomirno zbiΩ-
nym funkcional\nym rqdom
vm ( 0, x ) = F2 ( x ) + ( )( ) ( − )
=
∞
( ) − ( )∑ v vm
k
m
k
k
x x0 01
1
, ,
i dlq n\oho spravdΩu[t\sq ocinka
| vm ( 0, x ) | ≤
λ
λ
0
0
2 01 −
F Q; . (16)
Pidstavlqgçy znaçennq vm ( 0, x ) u (14), oderΩu[mo rozv’qzok zadaçi (8) –
(10).
Znajdemo ocinky poxidnyx rozv’qzku krajovo] zadaçi
( L0 v ) ( t, x ) ≡
∂ − ( + ) ( )∂ ∂
( )
=
∑t i j ij x
ij
n
xs P A P t x
i j
β β ; ,1 1
1
v = F3 ( P ),
v v( ) + ( ) ( )−∫0
0
, , ,x q x e x d
T
τ τ τλτ = ϕ0 ( x ), (17)
v | Γ = 0.
Koefici[nty dyferencial\noho vyrazu L0 , zhidno z nakladenymy umovamy,
obmeΩeni stalymy, ne zaleΩnymy vid toçky P1 . Tomu isnugt\ taki stali c, cjk ,
wo dlq funkci] Hrina odnoridno] zadaçi Dirixle
( L0 v ) ( t, x ) = F3 ( P ), v ( 0, x ) = ϕ0 ( x ), v | Γ = 0
spravdΩu[t\sq ocinka [8, s. 469]
∂ ∂ ( ) ≤ ( − ) − −
−
−( + ) −
t
j
x
k
jk
n k jt x c t c
x
t
Γ , , , exp/τ ξ τ ξ
τ
2
2
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
116 I. D. PUKAL|S|KYJ
Ma[ misce taka teorema.
Teorema 4. Nexaj F3 ∈ C
α
( Q ), ϕ 0 ∈ C2
+
α
( D ) i vykonano umovy 1° , 3° .
Todi isnu[ [dynyj rozv’qzok zadaçi (17) u prostori C 2
+
α
( D ) i dlq n\oho vyko-
nu[t\sq ocinka
v C Q C Q C Dc F2 23 0+ +( ) ( ) ( )≤ +( )α α αϕ . (18)
Dovedennq. Rozv’qzok zadaçi (17) ßuka[mo u vyhlqdi
v ( t, x ) = Γ Γ( ) ( ) + ( ) ( )∫ ∫ ∫t x d d t x F d
D
t
D
, , , , , , , ,0 0
0
3ξ ξ ξ τ τ ξ τ ξ ξv +
+ Γ( ) ( )∫ t x d
D
, , ,0 0ξ ϕ ξ ξ. (19)
Zadovol\nqgçy nelokal\nu umovu zadaçi (17), oderΩu[mo intehral\ne riv-
nqnnq
v v( ) + ( ) ( ) ( ) = ( )−∫ ∫0 0 0
0
1, , , , , , ,( )x q x e d t x d t x
T
D
τ τ ξ ξ ξ ωλτ Γ , (20)
de
ω(
1
)
( t, x ) =
= − ( ) ( ) ( ) + ( ) ( )
−∫ ∫ ∫ ∫q x e d d t x F d t x d
T T
D D
τ τ β β ξ β ξ ξ ξ ϕ ξ ξλτ, , , , , , , ,
0 0
3 00Γ Γ .
Rozv’qzok rivnqnnq (20) ßuka[mo metodom poslidovnyx nablyΩen\. Povto-
rggçy mirkuvannq z dovedennq teoremy 3, ma[mo
| v ( 0, x ) | ≤ c F Q DC Q C D3 0 2; ;α αϕ( ) ( )+( )+ .
Vykorystovugçy ocinku funkci] Hrina i rivnist\ (20), znaxodymo
v( ) ≤ +( )+ +( ) ( ) ( )0 2 23 0, x c FC D C Q C Dα α αϕ . (21)
Vraxovugçy vlastyvosti funkci] Hrina Γ ( t, x, τ, ξ ), ocinku (21) i rivnist\ (19),
oderΩu[mo nerivnist\ (18).
Isnuvannq rozv’qzku nelokal\no] zadaçi Dirixle dlq rivnqnnq z vyrod-
Ωennqm. Vvedemo u prostori C2
+
α
( Q ) normu || vm ; γ, β; l; Q || 2 + α
, ekvivalent-
nu pry koΩnomu fiksovanomu m 1
, m2 hel\derovij normi, qka vyznaça[t\sq qk
|| u; γ, β; l; Q || 2 + α
, til\ky zamist\ funkcij s1 ( l
(1), t ), s2 ( l
(2
), x ) beremo vidpovid-
no d1 ( l
(1), t ), d2 ( l
(2
), x ), de d1 ( l
(1), t ) = max ( s1 ( l
(1), t ), m l
1
1− ( )
) pry l
(1) ≥ 0 i
d1 ( l
(1), t ) = min ( s1 ( l
(1), t ), m l
1
1− ( )
) pry l
(1) < 0; d2 ( l
(2
), x ) = max ( s2 ( l
(2
), x ),
m l
2
2− ( )
) pry l
(2
) ≥ 0 i d2 ( l
(2
), x ) = min ( s2 ( l
(2
), x ), m l
2
2− ( )
) pry l
(2
) < 0, d ( l; P ) =
= d1 ( l
(1), t ) d2 ( l
(2
), x ).
Teorema 5. Qkwo vykonano umovy 1° – 3°, to dlq rozv’qzku zadaçi (8) – (10)
spravdΩu[t\sq ocinka
|| vm ; γ, β; 0; Q || 2 + α ≤
c F Q D Qm m; , ; ; ; ˜ , ˜; ; ;γ β γ γ βα α
2 0
2 0+ +( )+
Φ v .
(22)
Stala c ne zaleΩyt\ vid m.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
NELOKAL|NA ZADAÇA DIRIXLE DLQ LINIJNYX … 117
Dovedennq. Vykorystovugçy oznaçennq normy j interpolqcijni nerivnosti
iz [9, s. 176], ma[mo
|| vm ; γ , β; 0; Q || 2 + α ≤ ( 1 + εα ) ||−
−
vm ; γ , β; 0; Q ||−
−2 + α + c ( ε ) || vm ; Q || 0 .
Tomu dosyt\ ocinyty pivnormu ||−
−
vm ; γ, β; 0; Q ||−
−2 + α
.
Iz vyznaçennq normy vyplyva[ isnuvannq v Q toçok P1
, Br
, Pr
( )2
, dlq qkyx
vykonu[t\sq odna z nerivnostej
1
2
||−
−
vm ; γ, β; 0; Q ||−
−2 + α ≤ Ek
, k ∈ { 1, 2, 3, 4 }, (23)
E1 ≡ d P x xi j r
i j r
n
r r( )− − + ( − ) −
=
−∑ 2 1
1
1 2γ β β α γ β
α
; ˜
, ,
( ) ( ) ×
×
∂ ∂ ( ) − ∂ ∂ ( )x x m x x m ri j i j
P Bv v1 ,
E2 ≡ d P t ti j
i j r
n
( )− − + −
=
−∑ 2 2
1
1 2 2
γ β β αγ
α
; ˜
, ,
( ) ( ) /
×
×
∂ ∂ ( ) − ∂ ∂ ( )x x m r x x m ri j i j
B Pv v ( )2
,
E3 ≡ d P x x P Br
r
n
r r t m t m r( )+ ( − ) − ∂ ( ) − ∂ ( )
=
−∑ 2 1
1
1 2
1γ α γ β
α
; ˜ ( ) ( ) v v ,
E4 ≡ d P t t B P
r
n
t m r t m r( )( + ) − ∂ ( ) − ∂ ( )
=
−∑ 2 2
1
1 2 2 2α γ
α
; ˜ ( ) ( ) / ( )v v .
Qkwo t t( ) ( )1 2− ≥ d P( )2
16
2
γ ρ
; ˜ ≡ T1
, ρ — dovil\na stala, ρ ∈ ( 0, 1 ), d ( γ, P̃ ) =
= min ( d ( γ, P̃1), d ( γ, P̃2) ), to
Ek ≤ 2ρ–
α
|| vm ; γ , β; 0; Q || 2 , k ∈ { 2, 4 }.
Vraxovugçy interpolqcijni nerivnosti, ma[mo
Ek ≤ εα ||−
−
vm ; γ , β; 0; Q ||−
−2 + α + c ( ε ) || vm ; Q || 0 . (24)
Vybyragçy ε dostatn\o malym ( ε = 4–
α
/
2
), z nerivnostej (23) znaxodymo
||−
−
vm ; γ , β; 0; Q ||−
−2 + α ≤ c || vm ; Q || 0 . (25)
Qkwo x xi i
( ) ( )1 2− ≥ n d Pi
− ( − )1
4
γ β ρ
; ˜ ≡ T2
, to
Ek ≤ 2ρ–
α
|| vm ; γ , β; 0; Q || 2 , k ∈ { 1, 3 }.
Vykorystovugçy interpolqcijni nerivnosti, otrymu[mo ocinku (25) i u vypadku
k ∈ { 1, 3 }.
Nexaj x xi i
( ) ( )1 2− ≤ T2 i t t( ) ( )1 2− ≤ T1
. Budemo vvaΩaty, wo d ( γ, P̃ ) ≡
≡ d ( γ, P1
). Zapyßemo zadaçu (8) – (10) u vyhlqdi
( L3 vm ) ( t, x ) ≡
∂ − ( )∂ ∂
= ( ) − ( ) ∂ ∂
= =
∑ ∑( )t ij x
ij
n
x m ij ij x
ij
n
x ma P a t x a P
i j i j1
1
1
1
v v, +
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
118 I. D. PUKAL|S|KYJ
+
a t x a t x F t xi x
i
n
m mi
( )∂ + ( ) + + ( )
=
∑ ( ), , ,
1
0v vλ ≡ F4 ( t, x ),
v vm m
T
x q x e x d( ) + ( ) ( )−∫0
0
, , ,τ τ τλτ = Φm ( x ), (26)
vm | Γ = 0.
Nexaj V1 ∈ Q, V1 — kub iz centrom u toçci P1
, Vr = { ( t, x ) ∈ Q | | t – t(1)
| ≤
≤ 16r2 T1
, t(1) ≥ 0, | xi
( )1 – xi | ≤ 4r T2
, i ∈ { 1, … , n } }.
U zadaçi (26) vykona[mo zaminu vm ( t, x ) = ωm ( t, y ), yi = d ( βi
, P1
) xi , i ∈ { 1, …
… , n }. Oblast\ vyznaçennq ωm ( t, y ) poznaçymo çerez Q0
. Todi funkciq
Wm ( t, y ) = ωm ( t, y ) η ( t, y ) zadovol\nq[ krajovu zadaçu
∂ − ( + ) ( )∂ ∂
=
∑t i j ij y y
ij
n
md P a P W
i j
β β ; 1 1
1
=
= d P a Pi j ij y m y y m y
ij
n
i j j i
( + ) ( ) ∂ ∂ + ∂ ∂[ ]
=
∑ β β ω η ω η; 1 1
1
+
+ ω β β η η ηm i j ij y y t
ij
n
d P a P F t Y
i j
( + ) ( )∂ ∂ − ∂
+ ( )
=
∑ ; ,1 1
1
4 ≡ F5 ( t, y ),
W y q Y e W y dm m
T
( ) + ( ) ( )−∫0
0
, , ,τ τ τλτ =
= q Y e y y y d Ym
T
m( ) ( ) ( ) − ( ) + ( )− [ ]∫ τ ω τ η τ η τλτ, , , ,0
0
Φ ≡ Ψm ( y ), (27)
Wm | Γ = 0,
de
η ( τ, y ) =
1 2
0 0 1
1 4
1
1
3 4
, , , , ; ,
, , , , ,
/
/
( ) ∈ ∂ ∂ ( ) ≤ ( + )
( ) ∉ ≤ ( ) ≤
− ( )t y H t y c d j k P
t y H t y
t
j
y
k
kjη γ
η
Hr =
( t, y ) ∈ Q0 | | t – t(1)
| ≤ r T1
, | yi – yi
( )1
| ≤ r d ( γ, P1
) ρ
4
n–
1, yi
( )1 =
= d ( βi
, P1
) xi
( )1
, Y = ( d
–
1
( β1
, P1
) y1 , … , d
–
1
( βn
, P1
) yn
).
Koefici[nty rivnqnnq (27) obmeΩeni stalymy, ne zaleΩnymy vid P1
. Tomu
na pidstavi teoremy 4 dlq dovil\nyx toçok M1 ( t(1), ξ(1)
) ∈ H1 / 4 i M2 ( τ(2
), ξ
(2
)
) ∈
∈ H1 / 4 vykonu[t\sq nerivnist\
d M M M Mj k
m
j k
m
− ( ) ∂ ∂ ( ) − ∂ ∂ ( )α
τ ξ τ ξω ω1 2 1 2, ≤
≤ c F C H m C H t5 03 4
2
3 4
α α( ) ( { = })+( )+
/ /
Ψ ∩ , (28)
de d ( M1, M2 ) — paraboliçna vidstan\ miΩ toçkamy M1
, M2 , 2j + | k | = 2.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
NELOKAL|NA ZADAÇA DIRIXLE DLQ LINIJNYX … 119
Vykorystovugçy vlastyvosti funkci] η ( t, y ), oznaçennq prostoru C2
+
α
( γ,
β; 0; Q ) i povertagçys\ do zminnyx ( t, x ), znaxodymo
Ek ≤
c V F V Vm m( + +v v; , ; ; ; , ; ; ;/ / /γ β γ β γ
α
0 23 4 2 4 3 4 3 4 0
+
+
Φm V t; ˜ , ˜; ; /γ β
α
0 03 4 2
∩ { = } )
+
. (29)
Znajdemo ocinku F V4 3 42; , ; ; /γ β γ
α
. Vraxovugçy interpolqcijni nerivnos-
ti, dosyt\ ocinyty pivnormu koΩnoho dodanka vyrazu F4 ( t, x ). Napryklad,
||−
−
( ai ( t, x ) – aij ( P1 ) ) ∂xi
∂xj
vm ; γ, β; 2γ, V3 / 4 ||−
− α ≤
≤
k
n
A B A V
i j m
k k
i j
d A A
= { }⊂
−∑
(( − − ) ∂ ∂ ( )
−
1
1
1 2 2
1
2
3 4
2sup ; ˜
, ,
( ) ( ) /
( )
/
γ β β τ τξ ξ
α
v ×
× d A a A a Bi j ij k ij k( + + ) ( ) − ( )β β αγ; ˜ ( )2 +
+
l
n
i j l l l ij ij ld A a A a B
=
−∑ ( )+ + ( − ) − ( ) − ( )
1
1 2
1β β α γ β ξ ξ
α
; ˜ ( ) ( ) +
+
k
n
A B A V
i j ij ij k
k k
d A a A a B
= { }⊂
∑ ( + ) ( ) − ( )
1
1
1
2
3 4
sup ; ˜
, , ( )
/
β β ×
×
− − + ( − ) ) − ∂ ∂ ( ) − ∂ ∂ ( )
=
−∑ ( )
l
n
i j l l l m m ld A A B
i j i j
1
1 2
12γ β β α γ β ξ ξ
α
ξ ξ ξ ξ; ˜ ( ) ( ) v v +
+
d A B Ai j m l m li j i j
( − − + ) − ∂ ∂ ( ) − ∂ ∂ ( )
−
2 1 2 2 2γ β β αγ τ τ
α
ξ ξ ξ ξ; ˜ ( ) ( ) / ( )v v ≤
≤ c ρα ||−
−
vm ; γ, β; 0; V3 / 4 ||−
− 2 + α + c1
|| vm ; γ, β; 0; V3 / 4 || 2
.
OtΩe, dlq normy || F4 ; γ, β; 2γ; V3 / 4 || α dista[mo ocinku
|| F4 ; γ, β; 2γ; V3 / 4 || α ≤
≤
c F V V Vm m( + ) + +
; , ; ; ; ; , ; ;/ / /γ β γ ε γ β
α α
2 03 4 3 4 0 1 3 4 2
v v , (30)
de ε1 = n2 ρα + εα, ε ∈ ( 0, 1 ), ρ ∈ ( 0, 1 ), ρ, ε — dovil\ni fiksovani çysla.
Pidstavlqgçy (30) v (29), znaxodymo
Ek ≤
c F Q V Dm m( + + )
+
; , ; ; ; ; ˜ , ˜; ;/γ β γ γ βα α
2 03 4 0 2
v Φ +
+
ε γ β α1 20vm Q; , ; ; + , k ∈ { 1, 2, 3, 4 }. (31)
Vykorystovugçy nerivnosti (23), (25), (31) i vybyragçy ρ i ε dosyt\ maly-
my, otrymu[mo nerivnist\ (22).
Teper dovedemo teoremu 1, vykorystavßy teoremy 2, 5.
Oskil\ky
|| F ; γ, β; 2γ; Q || α ≤ c f Q Q( + )+; , ; ; ; , ; ;γ β µ ψ γ βα α0 20 ,
(32)
Φm D c D Q; ˜ , ˜; ; ; ˜ , ˜; ; ; , ; ;γ β ϕ γ β ψ γ β
α α α0 0 0
2 2 2+ + +≤ +( ) ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
120 I. D. PUKAL|S|KYJ
to na pidstavi nerivnosti (22) ma[mo
vm Q c f Q; , ; ; ; , ; ;γ β γ β µα α0
2 0+ ≤ ( +
+ ϕ γ β ψ γ βα α; ˜ , ˜; ; ; , ; ;0 02 2D Q+ ++ ). (33)
Prava çastyna nerivnosti (33) ne zaleΩyt\ vid m i poslidovnosti { }Vm
( )0 =
= { | vm ( P ) | }, { }Vm
( )1 = { d ( γ – β i
, P ) | ∂xi
vm ( P ) | }, { }Vm
( )2 = { d ( 2γ – βi – β j ,
P ) | ∂xi
∂xj
vm ( P ) | }, { }Vm
( )3 = { d ( 2γ, P ) | ∂t vm ( P ) | }, P ∈ Q, rivnomirno obmeΩeni i
odnostajno neperervni. Za teoremog Arçela isnugt\ pidposlidovnosti { }( )
( )Vm r
k ,
k ∈ { 0, 1, 2, 3 }, rivnomirno zbiΩni v Q. Perexodqçy v zadaçi (8) – (10) do hranyci
pry r → ∞, oderΩu[mo, wo u = ve–
λ
t + ψ — [dynyj rozv’qzok zadaçi (1) – (3), u
∈ C2
+
α
( γ, β; 0; Q ), i spravdΩu[t\sq ocinka (4).
ZobraΩennq rozv’qzku zadaçi (1) – (3).
Teorema 6. Nexaj vykonano umovy 1° – 3°, f ∈ C
α
( γ, β ; 0; Q ). Todi [dynyj
rozv’qzok zadaçi (1) – (3) u prostori C2
+
α
( γ, β; 0; Q ) vyznaça[t\sq intehrala-
my Stil\t\[sa z borelivs\kog mirog
u ( t, x ) = u1 + u2 + u3 ≡ Γ1( ) ( )∫ t x d d f
Q
, ; , ,τ ξ τ ξ +
+ Γ Γ
Γ
2 3( ) ( ) + ( ) ( )∫ ∫t x d t x d d S
D
, ; , ; , ,ξ ϕ ξ τ ψ τ ξξ (34)
i dlq komponent ( Γ1 , Γ2 , Γ3 ) vykonugt\sq nerivnosti
0 ≤ Γ1 0
1
0
( ) ≤ (− ( ) − )∫ −t x d d e A t x Q
Q
t, ; , , ;τ ξ λλ
,
0 ≤ Γ
Γ
3( ) ≤∫ t x d d S e T, ; ,τ ξ
λ
, (35)
0 ≤ Γ2
0
1
0
1( ) ≤ − ( )
∫ ∫ −
−
t x d q x e d D
D
T
, ; , ;ξ τ τλτ
.
Dovedennq. Oskil\ky C
k
( γ, β; 0; Q ) ⊂ Ck
( γ, β; µ0
; Q ), to dlq f ∈ C
α
( γ, β;
0; Q ) vykonu[t\sq nerivnist\
f Q c f Q; , ; ; ; , ; ;γ β µ γ β α0 0
0≤ .
OtΩe, na pidstavi teoremy 1 dlq rozv’qzku zadaçi (1) – (3) spravdΩu[t\sq
ocinka
u Q c f Q; , ; ; ; , ; ;γ β γ βα α0 02+ ≤ ( + ϕ γ β ψ γ βα α; ˜ , ˜; ; ; , ; ;0 02 2D Q+ ++ ).
(36)
Rozhlqdatymemo u ( t, x ) pry fiksovanyx ( t, x ) qk linijnyj neperervnyj
funkcional Φ ( f, ϕ, ψ ) na normovanomu prostori Cα ≡ C α
( γ, β; 0; Q ) × C2
+
α
( γ̃ ,
β̃ ; 0; D ) z normog, wo dorivng[ pravij çastyni nerivnosti (36).
Beruçy do uvahy vklgçennq Cα ⊂ C i teoremu Rissa, moΩna vvaΩaty, wo u ( t,
x ) porodΩu[ borelivs\ku miru Γ ( t, x, Z ), qka vyznaçena na σ-alhebri pidmno-
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
NELOKAL|NA ZADAÇA DIRIXLE DLQ LINIJNYX … 121
Ωyn Z oblasti Q , vklgçagçy Q i vsi ]] vidkryti pidmnoΩyny taki, wo
znaçennq funkcionala vyznaça[t\sq fomulog (34).
Z teoremy 2 vyplyva[ vykonannq dlq rozv’qzkiv zadaçi (1) – (3) nerivnostej
0 ≤ u1 ≤ fe A Qtλ λ(− − )−0
1
0
; , 0 ≤ u3 ≤ ψ λe t ; Γ
0
,
(37)
0 ≤ u2 ≤ ϕ τ τλτ1
0
1
0
− ( )
−
−
∫ q x e d D
T
, ; ,
de u1 — rozv’qzok krajovo] zadaçi (1) – (3) pry ϕ ≡ 0, ψ ≡ 0, u2 — rozv’qzok
krajovo] zadaçi (1) – (3) pry f ≡ 0, ψ ≡ 0 i u3 — rozv’qzok zadaçi (1) – (3) pry f ≡
≡ 0, ϕ ≡ 0.
Pidstavlqgçy v nerivnosti (37) vidpovidno f ( t, x ) ≡ 1, ϕ ( x ) ≡ 1 i ψ ≡ 1,
oderΩu[mo nerivnosti (35).
1. Kam¥nyn L. Y., Xymçenko B. N. Ob apryorn¥x ocenkax reßenyq parabolyçeskoho uravnenyq
2-ho porqdka vblyzy nyΩnej kr¥ßky parabolyçeskoj hranyc¥ // Syb. mat. Ωurn. – 1981. –
22, # 4. – S. 94 – 113.
2. Kam¥nyn L. Y., Xymçenko B. N. O pryncype maksymuma dlq πllyptyko-parabolyçeskoho
uravnenyq vtoroho porqdka // Tam Ωe. – 1972. – 13, # 4. – S. 777 – 789.
3. Matijçuk M. I. Paraboliçni synhulqrni krajovi zadaçi. – Ky]v: In-t matematyky NAN Uk-
ra]ny, 1999. – 176 s.
4. Babyn A. V., Kabakbaev S. Û. O hladkosty vplot\ do hranyc¥ reßenyj parabolyçeskyx
uravnenyj s v¥roΩdagwymsq operatorom // Mat. sb. – 1994. – 185, # 7. – S. 13 – 38.
5. Borok V. M., Perel\man M. A. O klassax edynstvennosty reßenyq mnohotoçeçnoj kraevoj
zadaçy v beskoneçnom sloe // Yzv. vuzov. Matematyka. – 1973. – # 8. – S. 29 – 34.
6. Pukal\s\kyj I. D. Nelokal\na zadaça Nejmana dlq paraboliçnoho rivnqnnq z vyrodΩennqm
// Ukr. mat. Ωurn. – 1999. – 51, # 9. – S. 1232 – 1244.
7. Pukal\s\kyj I. D. Odnostoronnq nelokal\na krajova zadaça dlq synhulqrnyx paraboliçnyx
rivnqn\ // Tam Ωe. – 2001. – 53, # 11. – S. 1521 – 1531.
8. Lad¥Ωenskaq O. A., Solonnykov V. A., Ural\ceva N. N. Lynejn¥e y kvazylynejn¥e uravne-
nyq parabolyçeskoho typa. – M.: Nauka, 1967. – 736 s.
9. ∏jdel\man S. D. Parabolyçeskye system¥. – M.: Nauka, 1964. – 444 s.
OderΩano 23.05.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 1
|
| id | umjimathkievua-article-3294 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:39:48Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b4/5ebd937a5b67871574add6e13971c0b4.pdf |
| spelling | umjimathkievua-article-32942020-03-18T19:50:22Z Nonlocal Dirichlet problem for linear parabolic equations with degeneration Нелокальна задача Діріхлє для лінійних параболічних рівнянь з виродженням Pukalskyi, I. D. Пукальський, І. Д. In the spaces of classical functions with power weight, we prove the correct solvability of the Dirichlet problem for parabolic equations with nonlocal integral condition with respect to the time variable and an arbitrary power order of degeneration of coefficients with respect to the time and space variables. В пространствах классических функций со степенным весом доказана корректная разрешимость задачи Дирихле для параболических уравнений с нелокальным интегральным условием по временной переменной и произвольному степенному порядку вырождения коэффициентов как по временной, так и по пространственным переменным. Institute of Mathematics, NAS of Ukraine 2007-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3294 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 1 (2007); 109–121 Український математичний журнал; Том 59 № 1 (2007); 109–121 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3294/3333 https://umj.imath.kiev.ua/index.php/umj/article/view/3294/3334 Copyright (c) 2007 Pukalskyi I. D. |
| spellingShingle | Pukalskyi, I. D. Пукальський, І. Д. Nonlocal Dirichlet problem for linear parabolic equations with degeneration |
| title | Nonlocal Dirichlet problem for linear parabolic equations with degeneration |
| title_alt | Нелокальна задача Діріхлє для лінійних параболічних рівнянь з виродженням |
| title_full | Nonlocal Dirichlet problem for linear parabolic equations with degeneration |
| title_fullStr | Nonlocal Dirichlet problem for linear parabolic equations with degeneration |
| title_full_unstemmed | Nonlocal Dirichlet problem for linear parabolic equations with degeneration |
| title_short | Nonlocal Dirichlet problem for linear parabolic equations with degeneration |
| title_sort | nonlocal dirichlet problem for linear parabolic equations with degeneration |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3294 |
| work_keys_str_mv | AT pukalskyiid nonlocaldirichletproblemforlinearparabolicequationswithdegeneration AT pukalʹsʹkijíd nonlocaldirichletproblemforlinearparabolicequationswithdegeneration AT pukalskyiid nelokalʹnazadačadíríhlêdlâlíníjnihparabolíčnihrívnânʹzvirodžennâm AT pukalʹsʹkijíd nelokalʹnazadačadíríhlêdlâlíníjnihparabolíčnihrívnânʹzvirodžennâm |