Cauchy problem for a semilinear Éidel’man parabolic equation
We obtain conditions for the existence and uniqueness of a generalized solution of the Cauchy problem for the equation $$u_1 + \sum_{|\alpha|=|\beta|=2}(-1)^{|\alpha|}D^{\alpha}_x(a_{\alpha \beta}(z, t)D_x^{\beta}u) - \sum_{|\alpha|=|\beta|=1}(-1)^{|\alpha|}D^{\alpha}_y(b_{\alpha \beta}(z, t)D_y^{\...
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| Дата: | 2008 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2008
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3178 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509223960444928 |
|---|---|
| author | Korkuna, O. E. Коркуна, О. Є. |
| author_facet | Korkuna, O. E. Коркуна, О. Є. |
| author_sort | Korkuna, O. E. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:47:45Z |
| description | We obtain conditions for the existence and uniqueness of a generalized solution of the Cauchy problem for the equation
$$u_1 + \sum_{|\alpha|=|\beta|=2}(-1)^{|\alpha|}D^{\alpha}_x(a_{\alpha \beta}(z, t)D_x^{\beta}u) -
\sum_{|\alpha|=|\beta|=1}(-1)^{|\alpha|}D^{\alpha}_y(b_{\alpha \beta}(z, t)D_y^{\beta}u) +$$
$$+ \sum_{|\alpha|=1}c_{\alpha}(z, t) D^{\alpha}_zu + c(z, t, u) =
\sum_{|\alpha|\leq2}(-1)^{|\alpha|}D^{\alpha}_x f_{\alpha}(z, t) -
\sum_{|\alpha|=1}D^{\alpha}_y g_{\alpha}(z, t)$$
in Tikhonov's class. |
| first_indexed | 2026-03-24T02:37:42Z |
| format | Article |
| fulltext |
UDK 517.95
O. {. Korkuna (Nac. lisotexn. un-t Ukra]ny, L\viv)
ZADAÇA KOÍI DLQ NAPIVLINIJNOHO
PARABOLIÇNOHO ZA EJDEL|MANOM RIVNQNNQ
We obtain conditions for the existence and uniqueness of a generalized solution of the Cauchy problem
for the equation
u D a z t D u D b z t D ut x x y y+ − − − +
= = = =
∑ ∑( ) ( , ) ( ) ( , )( ) ( )1 1
2 1
α α
αβ
β
α β
α α
αβ
β
α β
+ ∑ + = ∑ − − ∑
= ≤ =
c z t D u c z t u D f z t D g z tz x yα
α
α
α
α α
α
α
α
α1 2 1
1( , ) ( , , ) ( ) ( , ) ( , )
in Tikhonov’s class.
Poluçen¥ uslovyq suwestvovanyq y edynstvennosty obobwennoho reßenyq zadaçy Koßy dlq
uravnenyq
u D a z t D u D b z t D ut x x y y+ − − − +
= = = =
∑ ∑( ) ( , ) ( ) ( , )( ) ( )1 1
2 1
α α
αβ
β
α β
α α
αβ
β
α β
+ ∑ + = ∑ − − ∑
= ≤ =
c z t D u c z t u D f z t D g z tz x yα
α
α
α
α α
α
α
α
α1 2 1
1( , ) ( , , ) ( ) ( , ) ( , )
v klasse typa Tyxonova.
U 1960 r. S.0D.0Ejdel\man [1] rozhlqnuv uzahal\nennq paraboliçnyx za Petrov-
s\kym system, uvivßy termin „ 2
�
b -paraboliçni systemy”. U cyx systemax dyfe-
rencigvanng za riznymy prostorovymy zminnymy prypysugt\ riznu vahu po vid-
noßenng do dyferencigvannq za zminnog t . Za cej ças bulo dostatn\o
povno0rozrobleno teorig zadaçi Koßi dlq linijnyx system vkazanoho typu
(dyv.0[2 – 21]).
Meta ci[] statti — doslidyty zadaçu Koßi dlq napivlinijnoho dyferen-
cial\noho rivnqnnq z poxidnog perßoho porqdku za çasovog zminnog, v qkomu za
hrupog prostorovyx zminnyx [ dyferencial\nyj operator çetvertoho porqdku,
a za inßog hrupog — druhoho porqdku. OderΩano umovy isnuvannq ta [dynosti
uzahal\nenoho rozv’qzku v klasi typu Tyxonova.
Nexaj x k∈R , y m∈R , z x y n= ∈( , ) R , k + m = n, Qτ = R
n × ( 0, τ ) , 0 < τ ≤
≤ T < ∞ . V oblasti QT rozhlqnemo rivnqnnq
u A ut + ( ) ≡ u D a z t D ut x x+ −
= =
∑ ( ) ( , )( )1
2
α
α β
α
αβ
β –
– ( ) ( , ) ( , ) ( , , )( )− + +
= = =
∑ ∑1
1 1
α α
αβ
β
α β
α
α
αD b z t D u c z t D u c z t uy y z =
= ( ) ( , ) ( , )− −
≤ =
∑ ∑1
2 1
α α
α
α α
α
αD f z t D g z tx y (1)
z poçatkovog umovog
u ( z, 0 ) = u0 ( z ) , z ∈ R
n
, (2)
de
© O. {. KORKUNA, 2008
586 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
ZADAÇA KOÍI DLQ NAPIVLINIJNOHO PARABOLIÇNOHO … 587
Dx
α = ∂
∂ … ∂
α
α αx xk
k
1
1
, α = α α1 + … + k ,
Dy
α = ∂
∂ … ∂
α
α αy ym
m
1
1
, α = α α1 + … + m ,
Dz
α = ∂
∂ … ∂
α
α αz zn
n
1
1
, α = α α1 + … + n.
Prypustymo, wo koefici[nty rivnqnnq (1) i vil\ni çleny [ dijsnoznaçnymy
funkciqmy i dlq nyx vykonugt\sq taki umovy:
A) a L QTαβ ∈ ∞( ), α = β = 2; isnu[ dodatna stala a0 taka, wo majΩe
dlq vsix ( z, t ) ∈ QT i ξ ∈R
N k( )
vykonu[t\sq nerivnist\
α β
αβ α βξ ξ
= =
∑
2
a z t( , ) ≥ a0
2
2
α
αξ
=
∑ , a0 > 0,
de N ( k ) — kil\kist\ vsix k-vymirnyx mul\tyindeksiv z α = 2, a ξα , α = 2,
— koordynaty vektora ξ ;
B) b L QTαβ ∈ ∞( ), α β= = 1; isnu[ dodatna stala b0 taka, wo majΩe dlq
vsix ( , )z t QT∈ i η ∈R
m
vykonu[t\sq nerivnist\
α β
αβ α βη η
= =
∑
1
b z t( , ) ≥ b0
1
2
α
αη
=
∑ , b0 > 0;
C) c L QTα ∈ ∞( ), α = 1; funkciq c z t( , , )⋅ [ neperervnog v R majΩe dlq
vsix ( , )z t QT∈ ; funkciq c( , , )⋅ ⋅ ξ — vymirnog v QT dlq vsix ξ ∈ R ; majΩe
dlq vsix ( , )z t QT∈ i dlq vsix ξ ∈ R vykonugt\sq nerivnosti
c z t c z t( , , ) ( , , ˜) ˜( )ξ ξ ξ ξ−( ) − ≥ 0, c z t( , , )ξ ξ ≥ c r
0 ξ , r ∈ [ 1, 2 ] ,
c z t( , , )ξ ≤ c r
1
1ξ − .
Nexaj R , R — dovil\ni dodatni fiksovani çysla. Poklademo Πx
R =
= { :x k∈R x R< }, Πy
R = { }:x y Rm∈ <R , Ω( ),R R = Π Πx
R
y
R× , Q R Rτ( ), =
= Ω( ), ( , )R R × 0 τ , τ ∈ [ 0, T ] , Ωτ( ),R R = Q R R tτ τ( ), { }∩ = .
Vvedemo prostory
W R R1( ( )),Ω = { ( ( )) }: , ,u D u L R Ry
α α∈ ≤2 1Ω ,
W R R2( ( )),Ω = { ( ( )) }: , ,u D u L R Rx
α α∈ ≤2 2Ω ,
W R R1 0, ( ( )),Ω = u u W R R u
x
R
y
R: , ,( ( ))∈ ={ }×∂1 0Ω Π Π ,
W R R2 0, ( ( )),Ω = u u W R R u
u
x
R
y
R
x
R
y
R
: , , ,( ( ))∈ = ∂
∂
=
∂ × ∂ ×
2 0 0Ω Π Π Π Πν
,
ν — zovnißnq normal\,
W R n
1, ( )loc = u u W R R R R: ,( ( ))∈ ∀ > ∀ >{ }1 0 0Ω ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
588 O. {. KORKUNA
W R n
2, ( )loc = u u W R R R R: ,( ( ))∈ ∀ > ∀ >{ }2 0 0Ω .
Wodo pravo] çastyny (1) i poçatkovo] funkci] u0 prypustymo, wo vyko-
nu[t\sq umova
F) f L T L n
α ∈ 2 20( )( , ); ( )loc R , α ≤ 2 ; g L T L n
β ∈ 2 20( )( , ); ( )loc R , β = 1,
u L n
0
2∈ loc( )R , de L n
loc
2 ( )R = u u L R R R R: ,( ( ))∈ ∀ > ∀ >{ }2 0 0Ω .
Oznaçennq"1. Funkcig u, qka zadovol\nq[ vklgçennq
u ∈ C T L L T W Wn n n( ) ( )[ , ]; ( ) ( , ); ( ) ( ), ,0 02 2
2 1loc loc locR R R∩ ∩
ta intehral\nu rivnist\
R
n
u z z dz u a z t D uD
Q
t x x∫ ∫ ∑+ − +
= =
( , ) ( , ) ( , )τ τ
τ
αβ
α β
β αv v v
2
+
+
b z t D uD u c z t D u c z t u dzdty y zαβ
α β
β α
α
α
α
= = =
∑ ∑+ +
1 1
( , ) ( , ) ( , , )v v =
=
R
n
u z z dz f z t D g z t D dzdt
Q
x y∫ ∫ ∑ ∑+ +
≤ =
0
2 1
0( ) ( , ) ( , ) ( , )v v v
τ α
α
α
α
α
α
(3)
dlq vsix τ ∈ ( 0, T ] i v ∈C T C n1 20( )[ , ]; ( )0 R , nazyvatymemo uzahal\nenym roz-
v’qzkom zadaçi (1), (2).
Nexaj ζ ∈C2( )R , ζ ξ( ) = 1 pry ξ ≤ 0, ζ ξ( ) = 0 pry ξ ≥ 1 i 0 ≤ ζ ( ξ ) ≤ 1
pry ξ ∈ R ,
ψR x( ) = ζ
κ
x R−
, κ > 0, ϕR y( ) = ζ
κ
y R−
, κ > 0.
Todi isnugt\ taki stali µ1, µ2, wo
∂
∂x
x
i
Rψ ( ) ≤ µ κ1
1− , i ∈ { 1, … , k } , ∂
∂y
y
i
Rϕ ( ) ≤ µ κ1
2, i ∈ { 1, … , m } ,
∂
∂ ∂
2
x x
x
i j
Rψ ( ) ≤ µ κ2
2− , i, j ∈ { 1, … , k } .
Krim toho, ψR x( ) = 0 pry x R≥ + κ , ψR x( ) = 1 pry x R≤ , ϕR y( ) = 0
pry y R≥ + κ i ϕR y( ) = 1 pry y R≤ .
Qk u praci [22], dovodymo taku lemu.
Lema. Nexaj u L T W n∈ 2 0( )( , ); ( )2,loc R . Todi
α
α γ ρψ ϕ
=
∑∫ [ ] [ ]
1
2
R
n
D u x y dzx R R( ) ( ) ≤
≤ δ ψ ϕ
α
α γ ρ
R
n
D u x y dzx R R∫ ∑
=
+
2
2 2[ ] [ ]( ) ( ) +
+ k u x y dz
n
R R
1 2
1
2
2
2 2
δ
γ µ
κ
ψ ϕγ ρ+ +
∫ −
R
[ ] [ ]( ) ( ) (4)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
ZADAÇA KOÍI DLQ NAPIVLINIJNOHO PARABOLIÇNOHO … 589
abo
α
α γ ρψ ϕ
=
∑∫ [ ] [ ]
1
2
R
n
D u x y dzx R R( ) ( ) ≤
≤ δ ψ ϕ
α
α γ ρ
R
n
D u x y dzx R R∫ ∑
=2
2
[ ] [ ]( ) ( ) +
+
k u x y dz
k
u x y dz
n n
R R R Rδ
ψ ϕ γ µ
κ
ψ ϕγ ρ γ ρ
R R
∫ ∫+ −2
2
1
2
2
2 2[ ] [ ] [ ] [ ]( ) ( ) ( ) ( ) , (5)
δ ∈ ( 0, 1 ) , γ > 2, ρ > 1.
Dovedennq. Oskil\ky
R
n
i i
u x y dzx x R R∫ [ ] [ ]( ) ( )ψ ϕγ ρ = –
R
n
i
u x y dzx R R∫ 2 [ ] [ ]( ) ( )ψ ϕγ ρ
–
– γ ψ ψ ϕγ ρ
R
n
i i
u u x x y dzx R x R R∫ −
, ( ) ( ) ( )[ ] [ ]1 , i ∈ { 1, … , k } ,
to
R
n
i
u x y dzx R R∫ 2 [ ] [ ]( ) ( )ψ ϕγ ρ ≤ δ ψ ϕγ ρ
2
2 2u x y dzx x R Ri i
n
[ ] [ ]( ) ( )+∫
R
+
+
1
2
1
2
2 2 2
δ
ψ ϕ ψγ ρ γu x y dz u x dzR R x R
n n
i
[ ] [ ] [ ]( ) ( ) ( )−∫ ∫+
R R
+
+
γ µ
κ
ψ ϕγ ρ
2
1
2
2
2 2
2
u x y dzR R
n
[ ] [ ]( ) ( )−∫
R
.
Pidsumovugçy ostanng nerivnist\ po i vid 1 do k, oderΩu[mo (4). Analo-
hiçno dovodymo nerivnist\ (5).
Nexaj R , R — dovil\ni dodatni fiksovani çysla. Rozhlqnemo dopomiΩnu
zadaçu
u A ut + ( ) = F z tR R, ( , ) , ( , ) ,( )z t Q R RT∈ , (6)
u ( z, 0 ) = u zR R
0
, ( ), z R R∈Ω( ), , (7)
u R R T∂Ω ( , ) ( , )× 0 = 0,
∂
∂ ∂ × ×
u
x
R
y
R Tv Π Π ( , )0
= 0, (8)
de ν — zovnißnq normal\,
FR R, = ( ) , ,− −
≤ =
∑ ∑1
2 1
α α
α
α
β
β
β
D f D gx
R R
y
R R ,
f z tR R
α
, ( , ) =
f z t z t Q R R
z t Q Q R R
T
T T
α
α
( , ), ( , ) , ,
, ( , ) , , ,
( )
\ ( )
∈
∈ ≤
0 2
g z tR R
β
, ( , ) =
g z t z t Q R R
z t Q Q R R
T
T T
β
β
( , ), ( , ) , ,
, ( , ) , , ,
( )
\ ( )
∈
∈ =
0 1
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
590 O. {. KORKUNA
u z tR R
0
, ( , ) =
u z z R R
z R Rn
0
0
( ), , ,
, ,
( )
\ ( ).
∈
∈
Ω
ΩR
Oznaçennq"2. Funkcig u, qka zadovol\nq[ vklgçennq
u C T L R R L T W R R W R R∈ ( ( ( ))) ( ( ( )) ( ( )))[ , ]; , ( , ); , ,, ,0 02 2
1 2Ω Ω Ω∩ ∩0 0
ta intehral\nu rivnist\
u dz u a z t D uD
R R
t x x
Q R R
v v v
Ωτ τ
αβ
α β
β α
( , ) ( , )
( , )∫ ∑∫+ − +
= =2
+
+
b z t D uD c z t D u c z t u dzdty y zαβ
α β
β α
α
α
α
= = =
∑ ∑+ +
1 1
( , ) ( , ) ( , , )v v v =
= u dz f z t D g z t D dzdt
R R
k
x y
Q R R
0
2 10
v v v
Ω ( , ) ( , )
( , ) ( , )∫ ∑ ∑∫+ +
≤ =
α
α
α β
β
β
τ
(9)
dlq vsix τ 0∈ ( 0, T ] i dovil\no] funkci] v ∈ (L T2 0( , ): W R R1, ( ),0 Ω( ) ∩
W R R2, ( ),0 Ω( )) tako], wo vt TL Q R R∈ 2( ( )), , nazyvatymemo uzahal\nenym roz-
v’qzkom zadaçi (6) – (8).
Teorema"1. Nexaj vykonugt\sq umovy A, B, C, F. Todi isnu[ [dynyj uza-
hal\nenyj rozv’qzok zadaçi (6) – (8), pryçomu pravyl\nog [ rivnist\
1
2
2
2
u dz a z t D uD u
R R
x x
Q R RΩτ τ
αβ
α β
β α
( , ) ( , )
( , )∫ ∑∫+
= =
+
+
b z t D uD c z t D uu c z t u u dzdty y zαβ
α β
β α
α
α
α
= = =
∑ ∑+ +
1 1
( , ) ( , ) ( , , )v =
= 1
2 0
2
2 10
u dz f z t D u g z t D u dzdtR R
R R
R R
x
R R
y
Q R R
,
( , )
, ,
( , )
( , ) ( , )
Ω
∫ ∑ ∑∫+ +
≤ =
α
α
α β
β
β
τ
(10)
dlq vsix τ ∈ ( 0, T ] .
Dovedennq. Nexaj { ϕ
s
} — baza prostoru W R R W R R2 1, ,( ( )) ( ( )), ,0 0Ω Ω∩ ,
ortonormovana v L R R2( ( )),Ω . Pobudu[mo poslidovnist\
u z tN ( , ) = c t zs
N
s
s
N
( ) ( )ϕ
=
∑
1
, N = 1, 2, … ,
de c cN
N
N
1 , ,… — rozv’qzok zadaçi Koßi
u a z t D u D ut
N s
x
N
x
s
R Rt
ϕ αβ
α β
β α+
= =
∑∫ ( , )
( , ) 2Ω
+
+ b z t D u D c z t D u c z t u dzy
N
y
s s
z
N N s
αβ
α β
β α
α
α
αϕ ϕ ϕ
= = =
∑ ∑+ +
1 1
( , ) ( , ) ( , , ) =
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
ZADAÇA KOÍI DLQ NAPIVLINIJNOHO PARABOLIÇNOHO … 591
=
1
2 2 1
f z t D g z t D dzR
x
s R
y
s
R Rt
α
α
α
β
β
β
ϕ ϕ( , ) ( , )
( , ) ≤ =
∑ ∑∫ +
Ω
, t ∈ [ 0, T ] , (11)
cs
N ( )0 = u s
R R N
0,
, , , s = 1, 2, … , N, (12)
u zR R N
0
, , ( ) = u zR R N s
s
N
0
1
, , ( )ϕ
=
∑ , u uR R N R R
L R R
0 0 2
, , ,
( ( , ))
−
Ω
→ 0 pry N → ∞ .
Zhidno z umovamy A, B, C, F i teoremog Karateodori [23, c. 54] isnu[ abso-
lgtno neperervnyj rozv’qzok zadaçi Koßi (11), (12), vyznaçenyj na deqkomu
promiΩku ( 0, hN ] . Z ocinok, oderΩanyx nyΩçe, vyplyvatyme hN = T. Pomno-
Ωyvßy koΩne rivnqnnq (11) vidpovidno na funkcig c t es
N t( ) −ν , ν > 0, pidsumu-
vavßy po s vid 1 do N i zintehruvavßy po promiΩku [ 0, τ ] , τ ≤ T, oderΩymo
u u a z t D u D u b z t D u D ut
N N
x
N
x
N
y
N
y
N
Q R R
+ +
= = = =
∑ ∑∫ αβ
α β
β α
αβ
β α
α βτ
( , ) ( , )
( , ) 2 1
+
+ c z t u D u c z t u u e dzdtN
z
N N N t
α
α
α ν
=
−∑ +
1
( , ) ( , , ) =
= f z t D u g z t D u e dzdtR
x
N R
y
N t
Q R R
α
α
α β
β
β ν
τ
( , ) ( , )
( , ) ≤ =
−∑ ∑∫ +
2 1
. (13)
Peretvorymo i ocinymo koΩnyj dodanok rivnosti (13). Oçevydno,
J1 ≡ u u e dzdtt
N N t
Q R R
−∫ ν
τ( , )
=
1
2
2
u e dzN
R R
−∫ ντ
τΩ ( , )
+
+ ν ντ
2
1
2
2
0
2
0
u e dzdt u dzN
Q R R
R R N
R RT
−∫ ∫−
( , )
, ,
( , )Ω
.
Na pidstavi umov A, B, C
J2 ≡ a z t D u D u b z t D u D ux
N
x
N
y
N
y
N
Q R R
αβ
α β
α β
αβ
α β
α βτ
( , ) ( , )
( , ) = = = =
∑ ∑∫ +
2 1
+
+ c z t u u e dzdtN N t( , , )
−ν ≥
≥ a D u b D u c u e dzdtx
N
y
N N r t
Q R R
0
2
2
0
2
1
0
α
α
α
α
ν
τ = =
−∑ ∑∫ + +
( , )
.
Zaznaçymo, wo dlq funkcij u W R R∈ 2, ( ( )),0 Ω lehko oderΩaty nerivnist\ (do-
vedennq analohiçne do dovedennq lemy)
D u e dzdtx
t
Q R R
α ν
ατ
2
1
−
=
∑∫
( , )
≤
δ α ν
ατ
1 2
22
D u e dzdtx
t
Q R R
−
=
∑∫
( , )
+
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
592 O. {. KORKUNA
+ 1
2 1
2
δ
ν
τ
u e dzdtt
Q R R
−∫
( , )
, δ1 > 0. (14)
Zhidno z umovog C
J3 ≡ c z t u D u e dzdtN
x
N t
Q R R
σ
σ ν
στ
( , )
( , )
−
=
∑∫
1
≤
≤ D u
C
u e dzdtx
N N t
Q R R
σ
σ
ν
τ
2
1
1 2
4=
−∑∫ +
( , )
,
de C1 = ess sup
QT
c z tα
α
2
1
( , )
=
∑ , α = α α1 + … + n, σ = ( , , )σ σ1 … k ,
J4 ≡ c z t u D u e dzdtN
y
N t
Q R R
ω
ω ν
ωτ
( , )
( , )
−
=
∑∫
1
≤
≤ 1
2 2
2
1
1
2
2
δ
δ
ω
ω
ν
τ
D u
C
u e dzdty
N N t
Q R R =
−∑∫ +
( , )
,
ω = ( ω1 , … , ωm ) , δ2 > 0.
Na pidstavi umovy F i (14)
J5 ≡ f z t D u g z t D u e dzdtR R
x
N R R
y
N t
Q R R
α
α
α β
β
β ν
τ
, ,
( , )
( , ) ( , )
≤ =
−∑ ∑∫ +
2 1
≤
≤ 1
2
1
21
2
2 2
2
1δ δα
ν
α
β
ν
βτ τ
f z t e dzdt g z t e dzdtR R t
Q R R
R R t
Q R R
,
( , )
,
( , )
( , ) ( , )−
≤
−
=
∑∫ ∑∫+ +
+
1
2 2
1
21
1
2 2
2
2
2
1
1 2
2
δ δ δ δ δα
α
α
α
ν
τ
+
+ + + +
= =
−∑ ∑∫ D u D u u e dzdtx
N
y
N N t
Q R R( , )
.
Todi, vraxuvavßy ocinky intehraliv J1 – J5 ta zbiΩnist\ u R R N
0
, ,
do u R R
0
,
v
L R R2( ( )),Ω , z (13) oderΩymo nerivnist\
u e dz
C C
uN
R R
N
Q R R
2
1
1 1
2
1 2
21
2
2 1 2−∫ ∫+ − − − − − −
ντ
τ τ
ν
δ δ
δ δ
Ω ( , ) ( , )
+
+ 2 2
20 1
1
2 2
2
a D ux
N− −
=
∑δ δ α
α
+
+ 2 2 20 0 2
2
1
c u b D u e dzdtn
r
y
N t+ −
=
−∑( )δ α
α
ν ≤ 2 0
2
0
u dzR R
R R
,
( , )Ω
∫ +
+ 1 1
1
2
2 2 1
2
δ δα
α β
βf z t g z t dzdtR R R R
Q R Rt
, ,
( , )
( , ) ( , )
≤ =
∑ ∑∫ +
, τ ∈ ( 0, T ] , (15)
qkwo N > N0 .
Vyberemo δ1 , δ2 , ν z umov
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
ZADAÇA KOÍI DLQ NAPIVLINIJNOHO PARABOLIÇNOHO … 593
2
20 1
1
2
a − −δ δ
= a0 , δ2 =
b0
2
, ν =
1 1
2
1
2 2 2
1 2
1 2δ δ
δ δ+ + + + + .
Todi z (15) matymemo ocinky
u TN
L R R
( , )
( ( , ))
⋅ 2 Ω
≤ M1 , (16)
uN
L T L R R∞( )( , ); ( ( , ))0 2 Ω
≤ M1 , (17)
uN
L T W R R W R R2
1 0 2 00( , ); ( ( , )) ( ( , )), ,Ω Ω∩( ) ≤ M1 , (18)
de stala M1 ne zaleΩyt\ vid N.
Krim toho, zhidno z umovog C i (15)
c z t u dzdtN r
Q R Rt
( , )
( , )
′
∫ ≤ M2 , (19)
de
1 1
r r
+
′
= 1, a stala M2 ne zaleΩyt\ vid N .
Na pidstavi (16) – (19) isnu[ taka pidposlidovnist\ { }uNs ⊂ { }uN
, wo
u TNs ( , )⋅ → χ0 slabko v L R R2( ( , ))Ω ,
u uNs → ∗ -slabko v L T L R R∞(( , ); ( ( , )))0 2 Ω ,
u uNs → slabko v L T W R R W R R2
1 0 2 00( , ); ( ( , )) ( ( , )), ,Ω Ω∩( ) ,
c uNs( ), ,⋅ ⋅ → χ1 slabko v L Q R Rr
T
′( )( , ) pry Ns → ∞ .
Vraxovugçy wil\nist\ mnoΩyny funkcij � = � NN =
∞
1∪ , de
� N =
v vN N
s
N s
s
N
s
Nz t d t z d C T: ( , ) ( ) ( ), ([ , ])= ∈
=
∑ ϕ
1
1 0 ,
u prostori
V Q R RT( )( , ) = v v v: ( , ); ( ( , )) ( ( , )) , ( , ), , ( )∈ ( ) ∈{ }L T W R R W R R L Q R Rt T
2
1 0 2 0
20 Ω Ω∩ ,
standartnym sposobom dovodymo, wo u zadovol\nq[ rivnist\
χ αβ
β α
α β
0
2
v v vdz u a z t D uD
T TR R
t x x
Q R RΩ ( , ) ( , )
( , )∫ ∑∫+ − +
= =
+
+
b z t D uD c z t D u dzdty y zαβ
β α
α β
α
α
α
χ( , ) ( , )v v v
= = =
∑ ∑+ +
1 1
=
=
u dz f z t D u g z t D u dzdtR R
R R
R R
x
R R
y
bQ R RT
0
2 10
,
( , )
, ,
( , )
( , ) ( , )v
Ω
∫ ∑ ∑∫+ +
≤ =
α
α
α
β
β
(20)
dlq dovil\no] funkci] v ∈V R R( ( , ))Ω . Z (20), zokrema, vyplyva[ rivnist\ (u sensi
rozpodiliv)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
594 O. {. KORKUNA
ut = – D a z t D u D b z t D ux x y y
α
αβ
β
α β
α
αβ
β
α β
( , ) ( , )( ) + ( )
= = = =
∑ ∑
2 1
–
– c z t D u z t D f D gz x
R R
y
R R
α
α
α
α α
α
α
β
β
β
χ( , ) ( , ) ( ) , ,− + − −
= ≤ =
∑ ∑ ∑1
1 2 1
1 . (21)
Tomu
ut ∈ L T W R R W R R2
1 0 2 00( , ); ( ( , )) ( ( , )), ,Ω Ω( ) + ( )( )∗ ∗
.
Oskil\ky
u ∈ L T W R R W R R2
1 0 2 00( , ); ( ( , )) ( ( , )), ,Ω Ω∩( ) ,
to na pidstavi teoremy01.17 [24, c. 177] u C T L R R∈ ( )[ , ]; ( ( , ))0 2 Ω .
OtΩe, u ( z, 0 ) = u zR R
0
, ( ) , u ( z, T ) = χ0 ( z ) . Krim toho, zhidno z ti[g samog
teoremog, pravyl\nog [ formula intehruvannq çastynamy
〈 〉∫ u u dtt
t
t
,
1
2
= 1
2
1
2
2 2
2 1
u dz u dz
t tR R R RΩ Ω( , ) ( , )
∫ ∫−
dlq dovil\nyx t1, t T2 0∈[ , ], t t1 2< , de 〈⋅ ⋅〉, poznaça[ znaçennq funkcionala
z prostoru
L t t W R R W R R2
1 2 1 0 2 0( , ); ( ( , )) ( ( , )), ,Ω Ω∩( )( )∗
na elementax prostoru
L t t W R R W R R2
1 2 1 0 2 0( , ); ( ( , )) ( ( , )), ,Ω Ω∩( ) .
Tomu z (21) oderΩu[mo rivnist\
1
2
2
2
u dz a z t D uD u
R R R R
x x
Ω Ωτ τ
αβ
β α
α β( , ) ( , )
( , )∫ ∫ ∑+
= =
+
+ b z t D uD u c z t uD u u dzdty y zαβ
β α
α β
α
α
α
χ( , ) ( , )
= = =
∑ ∑+ +
1 1
1 =
= 1
2 0
0
u dzR R
R R
,
( , )Ω
∫ +
+
Ωτ
α
α
α
β
β
β( , )
, ,( , ) ( , )
R R
R R
x
R R
yf z t D u g z t D u dzdt∫ ∑ ∑
≤ =
+
2 1
, τ ∈ [ 0, T ] . (22)
Dlq dovedennq rivnosti χ1 = c ( ⋅ , ⋅ , u ) vykorystovu[mo monotonnist\ i nepe-
rervnist\ funkci] c ( z, t, ⋅ ) majΩe dlq vsix ( z, t ) ∈ QT ta rivnist\ (22).
Povtorggçy sxemu dovedennq [25, c. 171], oderΩu[mo potribnu rivnist\, a z
neg j isnuvannq uzahal\nenoho rozv’qzku, qkyj zadovol\nq[ rivnist\ (10).
Dlq dovedennq [dynosti prypuska[mo, wo isnugt\ dva uzahal\neni rozv’qzky
u1
i u2
zadaçi (6) – (8). Todi dlq u = u1 – u2
, qk i (10), oderΩymo rivnist\
1
2
2
12
1u e dz a z t D uD u b z t D uD u
R R R R
x x y y
−
= == =
∫ ∫ ∑∑+ +
ν τ
αβ
β α
αβ
β α
α βα βτ τΩ Ω( , ) ( , )
( , ) ( , ) +
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
ZADAÇA KOÍI DLQ NAPIVLINIJNOHO PARABOLIÇNOHO … 595
+ c z t uD u c z t u c z t u u e dzdtzα
α
α
ν τ( , ) ( , , ) ( , , )
=
−∑ + −( )
1
1 2 1 = 0, τ ∈ [ 0, T ] , ν1 > 0.
(23)
Ocinggçy dodanky rivnosti (23), qk i rivnosti (13), vybyragçy dostatn\o velyke
ν1 i vraxovugçy umovu C wodo funkci] c, oderΩu[mo ocinku
u dz
R R
2
Ωτ ( , )
∫ ≤ 0, τ ∈ [ 0, T ] ,
tobto u ( z, t ) = 0 majΩe skriz\ v Q R RT ( , ) .
Teoremu dovedeno.
Teorema"2. Nexaj vykonugt\sq umovy A, B, C, F i dlq dovil\nyx R > 0
i R > 0
u z dz f z t g z t dzdt
R R Q R RT
0
2 2
2
2
1
( ) ( , ) ( , )
( , ) ( , )Ω
∫ ∑ ∑∫+ +
≤ =
α
α
β
β
≤ bea R R( )/4 3 2+ , (24)
de a i b — deqki dodatni stali.
Todi isnu[ take τ0 ∈ ( 0, T ] , wo v oblasti Qτ0
zadaça (1), (2) ma[ uzahal\-
nenyj rozv’qzok u, dlq qkoho pravyl\nog [ ocinka
u z t dzdt
Q R R
2
0
( , )
( , )τ
∫ ≤ b ea R R
1
4 3 2( )/ + , (25)
de stala b1 ne zaleΩyt\ vid R i R.
Dovedennq. Nexaj R = R3 2/ . Rozhlqnemo zadaçu (6) – (8), de R nabuva[
znaçen\ iz mnoΩyny natural\nyx çysel. Todi otryma[mo poslidovnist\ funkcij
{ }us . ProdovΩymo koΩnu funkcig us
nulem na oblast\ QT i zbereΩemo
za0neg te same poznaçennq. Todi, oçevydno, koΩna us, s ∈ N , zadovol\nq[ riv-
nist\0(3) z vil\nymy çlenamy f s
α , α ≤ 2, gs
β , β = 1, i poçatkovog funkci[g
us
0 dlq vsix v ∈C T C n1
0
20([ , ]; ( ))� , supp v ⊂ QT
s . Rozhlqnemo (3) dlq funkcij
us
i ul, vidnimemo vid perßo] druhu i pryjmemo, wo
v = u x y el s
R R
t, [ ] [ ]( ) ( )ψ ϕγ ρ µ− 2
,
de u u ul s s l, = − , γ ≥ 4, ρ ≥ 2, çysla s i l vybyra[mo tak, wob u us l
0 0− = 0 v
Ω( , )R R+ +κ κ i f fs l
α α− = 0, g gs l
β β− = 0 v Q R RT ( , )+ +κ κ , α ≤ 2, β =
= 1. Todi oderΩymo rivnist\
1
2
1
2
2 2 22
u z x y e d u x yl s
R R
l s
R R
Qn
, ,( , ) ( ) ( ) ( ) ( )[ ] [ ] [ ] [ ]τ ψ ϕ τ µ ψ ϕγ ρ µ τ γ ρ
τ
−∫ ∫+
�
+
+ a z t D u D u x yx
l s
x
l s
R Rαβ
β α γ ρ
α β
ψ ϕ( , ) ( ) ( ), , [ ] [ ]( )
= =
∑
2
+
+ a z t D u D u y xy
l s
y
l s
R Rαβ
β α ρ γ
α β
ϕ ψ( , ) ( ) ( ), , [ ] [ ]( )
= =
∑
1
+
+ c z t u x y D ul s
R R z
l s
α
γ ρ α
α
ψ ϕ( , ) ( ) ( ), ,[ ] [ ]
=
∑
1
+
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
596 O. {. KORKUNA
+ c z t u c z t u u x y e dzdtl s l s
R R
t( , , ) ( , , ) ( ) ( ), [ ] [ ]−( ) −ψ ϕγ ρ µ2
= 0, τ ∈ [ 0, T ] . (26)
Na pidstavi umov teoremy ocinymo dodanky v (26). Matymemo
J6 ≡ a z t D u D u x y e dzdtx
l s
x
l s
R R
t
Q
αβ
α β γ ρ µ
α β
ψ ϕ
τ
( , ) ( ) ( ), , [ ] [ ] −
= =
∑∫
2
2
≥
≥ a D u x y e dzdtx
l s
R R
t
QT
0
2
2 2
α
α γ ρ µψ ϕ
=
−∑∫ , [ ] [ ]( ) ( ) ,
J7 ≡ a z t D u D u D x y e dzdtx
l s
x
l s
x R R
t
Q
αβ
β
σ
α σ σ γ ρ µ
α β
ψ ϕ
τ
( , ) ( ) ( ), , ([ ] )[ ]
=
− −
= =
∑∑∫
12
2
≤
≤
M
D u x yx
l s
R R
Q
5
3
2
2
2
δ ψ ϕ
α
α γ ρ
τ =
∑∫
, [ ] [ ]( ) ( ) +
+
k
D u x y e dzdtx
l s
R R
t
3 2
1
2
2
3 1
2 2 2γ µ
κ δ
ψ ϕ
α
α γ ρ µ
=
− −∑
, [ ] [ ]( ) ( ) ,
de δ3 > 0, M5 = max sup ( , )
α β αβ= =2
ess
QT
a z t .
Zhidno z lemog
J7 ≤
M
k
k
D u x yx
l s
R R
QT
5
3
2 4
3 2
1
2
2
3 2
2
2
δ δ γ µ
κ δ
ψ ϕ
α
α γ ρ+
=
∑∫ , [ ] [ ]( ) ( ) +
+
M k
u x y e dzdtl s
R R
t5
4 2
1
2
2
3 4
2
1
2
2
2 4
2
1 2γ µ
κ δ δ
γ µ
κ
ψ ϕγ ρ µ+
− −, [ ] [ ]( ) ( ) , δ4 > 0.
Dali
J8 ≡ a z t u D u D x y e dzdtl s
x
l s
x R R
t
QT
αβ
β α γ ρ µ
α β
ψ ϕ( , ) ( ) ( ), , ([ ] )[ ] −
= =
∑∫
2
2
≤
≤
M k
D u x yx
l s
R R
5 3
2
2
2
2
δ ψ ϕ
α
α γ ρ
=
∑
, [ ] [ ]( ) ( ) +
+
k M
u x y e dzdtl s
R R
t
4
2
2 2 2
5
3
4
2 41
2
2µ γ γ
δ κ
ψ ϕγ ρ µ( )
( ) ( ), [ ] [ ]−
− − ,
J9 ≡ b z t D u D u x y e dzdty
l s
y
l s
R R
t
Q
αβ
β α γ ρ µ
α β
ψ ϕ
τ
( , ) ( ) ( ), , [ ] [ ] −
= =
∑∫
2
1
≥
≥ b D u x y e dzdty
l s
R R
t
Q
0
1
2 2
α
α γ ρ µψ ϕ
τ =
−∑∫ , [ ] [ ]( ) ( ) ,
J10 ≡ b z t D u u x D y e dzdty
l s l s
R y R
t
Q
αβ
β γ α ρ µ
α β
ψ ϕ
τ
( , ) ( ) ( ), , [ ] ([ ] ) −
= =
∑∫
2
1
≤
≤
M
D u x yy
l s
R R
Q
7
5
1
2
2
δ ψ ϕ
α
β γ ρ
τ =
∑∫
, [ ] [ ]( ) ( ) +
+
µ
δ κ
ψ ϕγ ρ µ1
2 2
5
2
2 2 2m
u x y e dzdtl s
R R
t, [ ] [ ]( ) ( ) − −
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
ZADAÇA KOÍI DLQ NAPIVLINIJNOHO PARABOLIÇNOHO … 597
de M7 = max sup ( , )
al Q
b z t
= =β αβ
τ
1
ess , δ5 > 0;
J11 ≡ c z t u x y D u e dzdtl s
R R x
l s t
Q
αβ
γ ρ σ µ
σ
ψ ϕ
τ
( , ) ( ) ( ), ,[ ] [ ] −
=
∑∫
2
1
≤
≤
M
D u k u x y e dzdtx
l s l s
R R
t
Q
7
6
2
2
6
2
2
2
δ
δ
ψ ϕ
α
α γ ρ µ
τ =
−∑∫ +
, , [ ] [ ]( ) ( ) +
+
M k
u x y e dzdt
Q
l s
R R
t7
2
1
2
2
2 2
2
2γ µ
κ
ψ ϕ
τ
γ ρ µ∫ − −, [ ] [ ]( ) ( ) +
+
M k
u x y e dzdt
Q
l s
R R
t7 2
2
2
τ
ψ ϕγ ρ µ∫ −, [ ] [ ]( ) ( ) ,
de M7 = max sup ( , )
α
α
τ
=1
ess
Q
c z t , δ6 > 0, σ = ( σ1, … , σn ) ;
J12 ≡ c z t u x y D u e dzdtl s
R R y
l s t
QT
ω
γ ρ ω µ
ω
ψ ϕ( , ) ( ) ( ), ,[ ] [ ] −
=
∑∫
2
1
≤
≤
M
D u m u x y e dzdty
l s l s
R R
t
Q
7
5
1
2
5
2
2
2
δ
δ
ψ ϕ
ω
ω γ ρ µ
τ =
−∑∫ +
, , [ ] [ ]( ) ( ) ,
ω = ( ω1, … , ωm ) ,
J13 ≡ c z t u c z t u u x y e dzdtl s l s
R R
t
QT
( , , ) ( , , ) ( ) ( ), [ ] [ ]−( ) −∫ ψ ϕγ ρ µ2
≥ 0.
Vraxovugçy ocinky intehraliv J6 – J13
, z (26) oderΩu[mo nerivnist\
Rn
u z x y e dzl s
R R∫ −, ( , ) ( ) ( )[ ] [ ]τ ψ ϕγ ρ µ τ2 2
+
+ 2 20 5 3
2
4
3 2
1
2
2
3
5 3
2
7 6
2
2
a M k
k
M k M D u x yx
l s
R R
Q
− − − −
=
∑∫ δ δ γ µ
κ δ
δ δ ψ ϕα
α
γ ρ
τ
, [ ] [ ]( ) ( ) +
+ ( ) ( ) ( ), [ ] [ ]2 0 7 5 5 7
1
2
b M m M D u x yy
l s
R R− −
=
∑δ δ ψ ϕ
α
α γ ρ +
+ µ
δ δ
ψ ϕγ ρ µ2 7
6
7
7
5
2 2
− − −
−M k
M k
M m
u x y e dzdtl s
R R
t, [ ] [ ]( ) ( ) ≤
≤
M k k M M m M k5
4 2
1
2
2
3 4
2
1
2
2
4
2
2 2 2
5
3
4
7 1
2 2
5
2
7
2
1
2
2
1 1γ µ
κ δ δ
γ µ
κ
µ γ γ
δ κ
µ
δ κ
γ µ
κ
+ +
+ − + +
( )
×
×
Q
l s
R R
tu x y e dzdt
τ
ψ ϕγ ρ µ∫ − − −, [ ] [ ]( ) ( )
2 4 2 2
, τ ∈ [ 0, T ] . (27)
Vyberemo δ3, … , δ6 z umov
2 20 5 3
2
3
3 2
1
2
5
2
3 7 6a M k k M k M− − − −δ δ γ µ δ δ = a0 ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
598 O. {. KORKUNA
2 0 7 5 7 5b M m M− −δ δ = b0 , δ4 = δ κ3
2 2 , µ
2 = µ µ0
2
1
2+ ,
de µ1
2 = M
k
k
m
7
6 5δ δ
+ +
.
Todi z (27) vyplyva[ nerivnist\
u e dz a D u b D u u e dzdtl s
R R
x
l s
y
l s l s
Q R R
,
( , )
, , ,
( , )
2
0
2
2
0
2
1
0
20
2
0
2−
= =
−∫ ∑ ∑∫+ + +
µ τ α
α
α
α
µ τ
τ τ
µ
Ω
≤
≤ M u e dzdtl s
R R
9 4 2
21 1 0
2
κ κ
µ τ
κ κτ
+
−
+ +
∫ ,
( , )Ω
(28)
pry κ ≤ κ , de M9 ne zaleΩyt\ vid κ , κ , R , R , l, s. Zokrema, z (28) ma[mo
ocinku
u e dzdtl s
R R
,
( , )
2
0
2−∫ µ τ
τΩ
≤
M
u e dzdtl s
R R
9
0
2 4 2
21 1 0
2
µ κ κ
µ τ
κ κτ
+
−
+ +
∫ ,
( , )Ω
. (29)
Nexaj p ∈N . Podilymo promiΩky [ ],R R + κ , [ , ]R R + κ na p çastyn i vy-
beremo çysla p, µ0
2, κ, κ z umov
2 9
2
0
2 2
M p
µ κ
≤ e−1,
2 9
4
0
2 4
M p
µ κ
≤ e−1. (30)
Todi, iterugçy (29), qk i u [26], oderΩu[mo ocinky
u dzdtl s
R R
,
( , )
2
Ωτ
∫ ≤ e u dzdtp l s
Q R R
− +
+ +
∫µ τ
κ κτ
0
2 2,
( , )
. (31)
Umovy (30) moΩna zabezpeçyty, vybravßy, zokrema,
p aj
j= +( ) ⋅ +[ ] 1 24 9 , µ λ0
2 42j
j= ⋅ , Rj
j= 22 , Rj
j= 23 , κ j
j= ⋅3 22 ,
κ j
j= ⋅7 23 , λ = max
[ ]
,
[ ]2 1
9
2 1
49
19
9
2 37
9
4
2
M a e M a e+( ) +( )
,
dlq koΩnoho j ∈ N .
Ocinymo elementy poslidovnosti { }u j . Vraxovugçy oznaçennq uzahal\neno-
ho rozv’qzku, ma[mo rivnist\
1
2 2
2 1 2
2
1u e dz u a z t D u D uj
R R Q R R
j
x
j
x
j
j j j j
−
= =
∫ ∫ ∑+ +
ν τ
α β
αβ
β α
τ τ
ν
Ω ( , ) ( , )
( , ) +
+ b z t D u D u c z t u D u c z t u e dzdty
j
y
j j
z
j j t
αβ
α β
β α
α
α
α ν
= = =
−∑ ∑+ +
1 1
1( , ) ( , ) ( , , ) =
= 1
2
0
0
2
Ω ( , )R Rj j
u dz∫ +
+
Q R R
x
j
y
j t
j j
f z t D u g z t D u e dzdt
τ α
α
α
β
β
β ν
( , )
( , ) ( , )∫ ∑ ∑
≤ =
−+
2 1
1 , τ ∈[ , ]0 T , ν1 0> .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
ZADAÇA KOÍI DLQ NAPIVLINIJNOHO PARABOLIÇNOHO … 599
(32)
Zaznaçymo, wo rivnist\ (32) taka sama, qk (13). Tomu dlq u
j
oderΩu[mo neriv-
nist\ (15), z qko] vyplyva[
Q R R
j
j j
u dzdt
τ( , )
∫
2
≤ M u z dz
R Rj j
10 0
2
0Ω ( , )
( )∫ +
+
Q R Rj j
f z t g z t dzdt
τ α
α β
β( , )
( , ) ( , )∫ ∑ ∑
≤ =
+
2
2 2
1
. (33)
OtΩe, na pidstavi umovy teoremy z (33) oderΩymo ocinku
Q R R
j
j j
u dzdt
τ( , )
∫
2
≤ M e
a R Rj j
11
4 3 2( )/ +
, j ∈ N . (34)
Zhidno z (31) i (34)
Q R R
j j
j j
u dzdt
τ( , )
,∫ + +2 1 2
≤ e u dzdt
p j
Q R R
j jj
j j
− + + +
+ +
∫µ τ
τ
0
2
1 1
2 1 2( )
( , )
, ≤
≤ 2 e u dzdt u dzdt
p j
Q R R
j
Q R R
jj
j j j j
− + + +
+ + + +
∫ ∫+
µ τ
τ τ
0
2
1 1 2 2
1 2 2 2( )
( , ) ( , )
≤
≤ 4 0
2
2 2
2
e
p j a R Rj j j− + + ++ +µ τ( ) ( )
. (35)
Oskil\ky
– p j a R Rj j j+ + ++ +µ τ0
2
2 2
2( ) ( ) =
= – [ ] ( )a aj j j j+( ) + + ++ + +1 2 2 2 24 9 4 4 8 4 8λτ =
= – 2 1 24 9j a a[ ] − +( ) −( )λτ ≤ – 2 1 24 9
0
j a−( ) −( ){ } λτ
dlq vsix τ τ∈[ , ]0 0 , de τ0 =
1 29
0−( ) −{ }a α
λ
, 1 29−( ){ }a > α0 > 0, to z (35)
vyplyva[ ocinka
Q R R
j j
j j
u dzdt
τ( , )
,∫ + +1 2 2
≤ 4 0
42e
j−α , τ τ∈[ , ]0 0 , (36)
j — dovil\ne natural\ne çyslo.
Nexaj j ≥ j0 > 1, N — dovil\ne natural\ne çyslo. Todi na pidstavi (36)
u j j N
L Q R Rj j
+ +1
2
0
,
( ( , ))τ
≤
i
N
j j i
L Q R R
u
j j=
−
+ + +∑
1
1
1 1
2
0
,
( ( , ))τ
≤
≤
i
N
j i j i
L Q R R
u
j i j i=
−
+ + +∑
+ − + −1
1
1
2
0 1 1
,
( ( , ))τ
≤ 4
1
1
20
4 1
i
N
e
j i
=
−
−∑
+ −α ( )
=
= 4 0
4
0
4 12
1
2e e
j i
i
−
=
∞
−∑
−α α ( )
= M e
j
12
20
4−α . (37)
Nexaj ε > 0 — dovil\ne fiksovane qk zavhodno male çyslo, R0 , R0 — do-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
600 O. {. KORKUNA
vil\ni fiksovani dodatni çysla. Todi zhidno z (37) isnu[ take j0 ∈ N , wo dlq vsix
j > j0 i natural\nyx N
u j j N
L Q R R
+ +1
2
0 0 0
,
( ( , ))τ
≤ ε ,
tobto poslidovnist\ { }u j
[ fundamental\nog u prostori L Q R R2
0 00
( ( , ))τ .
OtΩe, z (28) vyplyva[ fundamental\nist\ { }u j
u prostorax
L W R R2
0 2 0 00( )( , ); ( ( , ))τ Ω , L W R R2
0 1 0 00( )( , ); ( ( , ))τ Ω , C L R R( )[ , ]; ( ( , ))0 0
2
0 0τ Ω .
Takym çynom, { }u j
syl\no zbiha[t\sq do funkci] u u cyx prostorax.
Vraxovugçy dovil\nist\ R0 i R0 , oderΩu[mo, wo u uj → syl\no v
L W W C Ln n n2
0 0
20 0( ) ( )( , ); ( ) ( ) [ , ]; ( )τ τ1, loc 2, loc locR R R∩ ∩ pry j → ∞ .
Oskil\ky koΩna funkciq u j
zadovol\nq[ rivnist\
R
n
u z z dz u a z t D u Dj
Q
j
t x
j
x∫ ∫ ∑+ − +
= =
( , ) ( , ) ( , )τ τ
τ
αβ
α β
β αv v v
2
+
+ b z t D u D c z t D u c z t u dzdty
j
y z
j j
αβ
α β
β α
α
α
α
= = =
∑ ∑+ +
1 1
( , ) ( , ) ( , , )v v v =
=
R
n
u z z dz f z t dzdtj
Q
j∫ ∫+0 0( ) ( , ) ( , )v v
τ
, (38)
dlq dovil\no] v ∈C C n1
0 0
20( )[ , ]; ( )τ R
supp v ⊂ Q R Rτ0
( , ) i u uj
0 0→ syl\no v
L n
loc
2 ( )R , f fj → syl\no v L L n2
0
20( )( , ); ( )τ loc R , c u c uj( , , ) ( , , )⋅ ⋅ → ⋅ ⋅ slabko
v L L n2
0
20( )( , ); ( )τ loc R , to, perejßovßy v (38) do hranyci pry j → ∞ , oderΩy-
mo, wo u — uzahal\nenyj rozv’qzok zadaçi (1), (2).
Z (37), (33) i (24), zokrema, oderΩymo nerivnist\
u L Q R Rj j
2
0
( ( , ))τ
≤ 2 1
2
0
u j
L Q R Rj j
+
( ( , ))τ
≤ b e
a R Rj j
2
1
4 3
1
2( )/
+ ++
,
de stala b2 ne zaleΩyt\ vid j. Z ci[] nerivnosti vyplyva[ ocinka (25), wo j za-
verßu[ dovedennq teoremy.
Teorema"3. Nexaj vykonugt\sq umovy A, B, C, F. T o d i uzahal\nenyj
rozv’qzok zadaçi (1), (2) u klasi funkcij, qki zadovol\nqgt\ ocinku
u dzdt
Q R RT
2
( , )
∫ ≤ M ea R R
13
4 3 2( )/ +
(39)
dlq dovil\nyx dodatnyx R , R, de a, M13 — dodatni stali, [ [dynym.
Dovedennq. Prypustymo, wo isnugt\ dva uzahal\neni rozv’qzky u1
i u2
zadaçi (1), (2), qki zadovol\nqgt\ (39). Todi, qk i pry dovedenni teoremy02, oder-
Ωu[mo ocinku
u u dzdt
Q R R
1 2 2
−∫
τ ( , )
≤ e u u dzdtp
Q R R
− +
+ +
−∫µ τ
κ κτ
0
2 1 2 2
( , )
, τ ∈ [ 0, T ] . (40)
Zaznaçymo, wo çysla
p = ( )a j[ ] + ⋅ +1 24 5, µ0
2
j = λ ⋅ 24 j , Rj = 22 j ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
ZADAÇA KOÍI DLQ NAPIVLINIJNOHO PARABOLIÇNOHO … 601
Rj = 23 j , κj = 3 22⋅ j , κ j = 7 23⋅ j ,
λ = max ,
( ) ( )2 1
9
2 1
49
9
9
2 17
9
4
2
M a e M a e[ ] + [ ] +
zadovol\nqgt\ umovy (30) dlq dovil\nyx j ∈ N . Tomu z (40) otrymu[mo ocinku
u u dzdt
Q R R
1 2 2
−∫
τ ( , )
≤ M a aj j j j
13
4 5 4 4 4 4 41 2 2 2 2exp ( ) ( )− [ ] + + + +[ ]+ + +λτ . (41)
Vybyragçy τ0 <
32 1( ){ }− a
λ
, z (41) ma[mo
u dzdt
Q R RT
2
( , )
∫ ≤ M e
j
13
20
4−α , α0 > 0.
Nexaj R0 , R0 — dovil\ni fiksovani dodatni çysla, ε — qk zavhodno male
çyslo. Todi isnu[ take j0 ∈ N , wo dlq vsix j > j0
u u dzdt
Q R R
1 2 2
0 0 0
−∫
τ ( , )
≤ ε .
Zvidsy u z t1( , ) = u z t2( , ) majΩe skriz\ v Q R Rτ0 0 0( , ). Vraxovugçy dovil\nist\
R0 , R0 , oderΩu[mo [dynist\ rozv’qzku v oblasti Qτ0
. Todi za skinçenne çyslo
krokiv dovodymo [dynist\ uzahal\nenoho rozv’qzku v oblasti QT .
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ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
|
| id | umjimathkievua-article-3178 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:37:42Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/3b/dc345655d48b5ea22e142d1221b8553b.pdf |
| spelling | umjimathkievua-article-31782020-03-18T19:47:45Z Cauchy problem for a semilinear Éidel’man parabolic equation Задача Коші для напівлінійного параболічного за Ейдельманом рівняння Korkuna, O. E. Коркуна, О. Є. We obtain conditions for the existence and uniqueness of a generalized solution of the Cauchy problem for the equation $$u_1 + \sum_{|\alpha|=|\beta|=2}(-1)^{|\alpha|}D^{\alpha}_x(a_{\alpha \beta}(z, t)D_x^{\beta}u) - \sum_{|\alpha|=|\beta|=1}(-1)^{|\alpha|}D^{\alpha}_y(b_{\alpha \beta}(z, t)D_y^{\beta}u) +$$ $$+ \sum_{|\alpha|=1}c_{\alpha}(z, t) D^{\alpha}_zu + c(z, t, u) = \sum_{|\alpha|\leq2}(-1)^{|\alpha|}D^{\alpha}_x f_{\alpha}(z, t) - \sum_{|\alpha|=1}D^{\alpha}_y g_{\alpha}(z, t)$$ in Tikhonov's class. Institute of Mathematics, NAS of Ukraine 2008-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3178 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 5 (2008); 586–602 Український математичний журнал; Том 60 № 5 (2008); 586–602 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3178/3103 https://umj.imath.kiev.ua/index.php/umj/article/view/3178/3104 Copyright (c) 2008 Korkuna O. E. |
| spellingShingle | Korkuna, O. E. Коркуна, О. Є. Cauchy problem for a semilinear Éidel’man parabolic equation |
| title | Cauchy problem for a semilinear Éidel’man parabolic equation |
| title_alt | Задача Коші для напівлінійного параболічного за Ейдельманом рівняння |
| title_full | Cauchy problem for a semilinear Éidel’man parabolic equation |
| title_fullStr | Cauchy problem for a semilinear Éidel’man parabolic equation |
| title_full_unstemmed | Cauchy problem for a semilinear Éidel’man parabolic equation |
| title_short | Cauchy problem for a semilinear Éidel’man parabolic equation |
| title_sort | cauchy problem for a semilinear éidel’man parabolic equation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3178 |
| work_keys_str_mv | AT korkunaoe cauchyproblemforasemilineareidelmanparabolicequation AT korkunaoê cauchyproblemforasemilineareidelmanparabolicequation AT korkunaoe zadačakošídlânapívlíníjnogoparabolíčnogozaejdelʹmanomrívnânnâ AT korkunaoê zadačakošídlânapívlíníjnogoparabolíčnogozaejdelʹmanomrívnânnâ |