Mixed problem for a semilinear ultraparabolic equation in an unbounded domain
We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation $$u_t + \sum^m_{i=1}a_i(x, y, t) u_{y_i} - \sum^n_{i,j=1} \left(a_{ij}(x, y, t) u_{x_i}\right)_{x_j} + \sum^n_{i,j=1} b_{i}(x, y, t) u_{x_i} + b_0(x, y, t, u) =$$ $$= f_0(x, y,...
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| Дата: | 2007 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2007
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3420 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509509975277568 |
|---|---|
| author | Lavrenyuk, S. P. Oliskevych, M. O. Лавренюк, С. П. Оліскевич, М. О. |
| author_facet | Lavrenyuk, S. P. Oliskevych, M. O. Лавренюк, С. П. Оліскевич, М. О. |
| author_sort | Lavrenyuk, S. P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:53:47Z |
| description | We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation $$u_t + \sum^m_{i=1}a_i(x, y, t) u_{y_i} - \sum^n_{i,j=1} \left(a_{ij}(x, y, t) u_{x_i}\right)_{x_j} + \sum^n_{i,j=1} b_{i}(x, y, t) u_{x_i} + b_0(x, y, t, u) =$$ $$= f_0(x, y, t, ) - \sum^n_{i=1}f_{i, x_i} (x, y, t) $$
in an unbounded domain with respect to the variables x. |
| first_indexed | 2026-03-24T02:42:14Z |
| format | Article |
| fulltext |
UDK 517.95
S. P. Lavrengk, M. O. Oliskevyç (L\viv. nac. un-t)
MIÍANA ZADAÇA DLQ NAPIVLINIJNOHO
UL|TRAPARABOLIÇNOHO RIVNQNNQ
U NEOBMEÛENIJ OBLASTI
We obtain some conditions for the existence and uniqueness of a solution of mixed problem for the
ultraparabolic equation
u a x y t u a x y t u b x y t ut
i
m
i y
i j
n
ij x x i x
i
n
i i j i
+ − + +
= = =
∑ ∑ ∑
1 1 1
( , , ) ( ( , , ) ) ( , , )
,
+ = −
=
∑b x y t u f x y t f x y ti x
i
n
i0 0
1
( , , , ) ( , , ) ( , , ),
in a domain unbounded with respect to variables x .
Poluçen¥ uslovyq suwestvovanyq y edynstvennosty reßenyq smeßannoj zadaçy dlq ul\tra-
parabolyçeskoho uravnenyq
u a x y t u a x y t u b x y t ut
i
m
i y
i j
n
ij x x i x
i
n
i i j i
+ − + +
= = =
∑ ∑ ∑
1 1 1
( , , ) ( ( , , ) ) ( , , )
,
+ = −
=
∑b x y t u f x y t f x y ti x
i
n
i0 0
1
( , , , ) ( , , ) ( , , ),
v neohranyçennoj oblasty po peremenn¥m x .
Ul\traparaboliçni rivnqnnq, qki inodi nazyvagt\ paraboliçnymy z bahat\ma ça-
samy abo vyrodΩenymy paraboliçnymy rivnqnnqmy, vynykly qk matematyçna mo-
del\ brounivs\koho ruxu fizyçno] systemy z n stupenqmy vil\nosti [1]. Taki
rivnqnnq vynykagt\ takoΩ pry modelgvanni markovs\kyx dyfuzijnyx procesiv,
rozsigvanni elektroniv, u biolohi], u finansovij matematyci (dyv., napryklad,
[2, 3]). Ci rivnqnnq bahatorazovo uzahal\ngvaly i doslidΩuvaly rizni avtory
(dyv. bibliohrafig v [4]). Zaznaçymo, wo najbil\ß povni rezul\taty dlq linij-
nyx ul\traparaboliçnyx rivnqn\ oderΩaly S.4D. Ejdel\man, S. D. Ivasyßen ta
]xni uçni (dyv., napryklad, [4 – 7]). Okremi rezul\taty dlq nelinijnyx ul\tra-
paraboliçnyx rivnqn\ v neobmeΩenyx oblastqx otrymano v [8 – 13].
U cij praci v neobmeΩenij oblasti rozhlqnuto mißanu zadaçu dlq napivlinij-
noho ul\traparaboliçnoho rivnqnnq, qke, zokrema, mistyt\ nevidomu funkcig zi
stepenem p ∈( , ]1 2 . Hiperboliçna çastyna c\oho rivnqnnq mistyt\ perßi poxidni
za hrupog m + 1, m ≥ 1, nezaleΩnyx zminnyx. Za dopomohog metodu vvedennq
parametra, zaproponovanoho u praci [14], dovedeno isnuvannq ta [dynist\ uza-
hal\nenoho rozv’qzku u klasi zrostagçyx funkcij.
Nexaj Ωx — neobmeΩena oblast\ v R
n
z meΩeg ∂ Ωx ∈ C
1
; Ωy — obmeΩena
oblast\ v R
m
z meΩeg ∂ Ωy ∈ C
1
; Ω = Ωx × Ωy
; Qτ = Ω × ( 0, τ )
, Sτ = ∂Ω ×
× ( 0, τ )
, 0 < τ ≤ T ; Ωτ = QT ∩ { t = τ }
. V oblasti QT rozhlqnemo rivnqnnq
A ( u ) ≡ u a x y t u a x y t u b x y t ut i y
i
m
ij x x
i j
n
i x
i
n
i i j i
+ − +
= = =
∑ ∑ ∑( , , ) ( , , ) ( , , )( )
,1 1 1
+
+ b x y t u0( , , , ) = f x y t f x y ti x
i
n
i0
1
( , , ) ( , , ),−
=
∑ (1)
z poçatkovog umovog
© S. P. LAVRENGK, M. O. OLISKEVYÇ, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12 1661
1662 S. P. LAVRENGK, M. O. OLISKEVYÇ
u x y( , , )0 = u x y0( , ), ( x, y ) ∈ Ω . (2)
Prypuskatymemo, wo dlq rivnqnnq (1) vykonugt\sq taki umovy:
A) a C Qi T∈ ( ), a L Qi y Ti, ( )∈ ∞ , i ∈ { 1, … , m } ; aij , a L Qij y Tk, ( )∈ ∞ , i, j ∈
∈ { 1, … , n } , k ∈ { 1, … , m } ;
a x y tij i j
i j
n
( , , )
,
ξ ξ
=
∑
1
≥ A0
2ξ , A0 > 0,
dlq vsix ξ ∈ R
n
i majΩe vsix ( x, y, t ) ∈ QT ;
B) bi , b L Qi y Tj, ( )∈ ∞ , i = 1, … , n, j = 1, … , m ; funkciq b0( , , , )⋅ ⋅ ⋅ η [ vy-
mirnog v QT dlq vsix η ∈ R ; funkciq b x y t0( , , , )⋅ — neperervnog na R maj-
Ωe dlq vsix ( x, y, t ) ∈ QT ;
b x y t b x y t0 1 0 2 1 2( , , , ) ( , , , ) ( )η η η η−( ) − ≥ 0,
b x y t0( , , , )η ≤ B p
0
1η − ,
b x y ty j0, ( , , , )η ≤ B p
0
1η − , j = 1, … , m,
b x y t0, ( , , , )η η ≤ B0
majΩe dlq vsix ( x, y, t ) ∈ QT i dlq vsix η ∈ R , de B0 — dodatna stala, p4∈
∈ ( 1, 2 ] .
Dlq sprowennq vykladu prypuskatymemo, wo Ωx
R = { }:x x Rx∈ <Ω [ re-
hulqrnog v sensi Kal\derona [15, c. 45] dlq vsix R > R0 > 0.
Poznaçymo çerez ST
1
mnoΩynu tyx toçok poverxni Ωx × ∂Ωy × ( 0, T ) , dlq
qkyx vykonu[t\sq nerivnist\
i
m
i ia x y t y
=
∑
1
( , , ) cos( , )ν < 0,
de ν — zovnißnq normal\ do ST , a çerez ST
2
mnoΩynu toçok poverxni Ωx ×
× ∂Ωy × ( 0, T ) , dlq qkyx
i
m
i ia x y t y
=
∑
1
( , , ) cos( , )ν ≥ 0.
Hovorytymemo, wo dlq rivnqnnq (1) vykonu[t\sq umova S, qkwo
ST
1 = Ωx × Γ1 × ( 0, T ) , ST
2 = Ωx × Γ1 × ( 0, T ) , Γ1 ∪ Γ2 = ∂Ωy , Γ1 ∩ Γ2 = ∅.
Krim poçatkovo] umovy (2) zadamo dlq rivnqnnq (1) krajovi umovy vyhlqdu
u ST
1 = 0, u
x y T∂Ω Ω× × ( , )0 = 0. (3)
Vvedemo prostory
Lloc
2 ( )Ω = { }: ( )u u L R RR∈ ∀ >2
0Ω ,
de ΩR = Ω Ωx
R
y× ;
Lloc
1 0, ( )Ω = u u u L i n uxi x y
: , ( ), { , , },∈ ∈ … ={ }×loc
2 1 0Ω Ω Ω∂ ;
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
MIÍANA ZADAÇA DLQ NAPIVLINIJNOHO UL|TRAPARABOLIÇNOHO RIVNQNNQ … 1663
Lloc
11, ( )Ω = { ∈ ∈ … ∈ …u u u u L i n j mx yi j
: , , ( ), { , , }, { , , }loc
2 1 1Ω ,
u u
x x yΩ Γ Ω Ω× ×= = }1
0 0, ∂ .
Hovorytymemo, wo prava çastyna (1) i poçatkova funkciq zadovol\nqgt\
umovu F, qkwo fi , f L T Li y j, ( )( , ); ( )∈ 2 20 loc Ω ; u0, u Ly j0
2
, ( )∈ loc Ω , i ∈ { 0, 1, … , n } ,
j ∈ { 1, … , m } ; fi TxΩ Γ× ×1 0( , ) = 0
, i ∈ { 0, 1, … , n } ; u
x0 1Ω Γ× = 0
.
Oznaçennq)1. Funkcig u z prostoru C T L L T H( ) ( )[ , ]; ( ) ( , ); ( ),0 02 2 1 0
loc locΩ Ω∩
nazyvatymemo uzahal\nenym rozv’qzkom zadaçi (1) – (3), qkwo vona [ hranyceg u
c\omu prostori poslidovnosti funkcij { }uk
takyx, wo
u C T L L T Hk ∈ ( ) ( )[ , ]; ( ) ( , ); ( ),0 02 2 11
loc locΩ Ω∩ dlq vsix k ∈ N
i zadovol\nq[ rivnist\
u dxdy u a x y t u a x y t uk
Q
k
t i y
k
i
m
ij x
k
x
i j
m
i i j
v v v v
Ωτ τ
∫ ∫ ∑ ∑+ − + +
= =
( , , ) ( , , )
,1 1
+
+ b x y t u b x y t u dxdydti x
k
i
n
k
i
( , , ) ( , , , )v v
=
∑ +
1
0 =
= u dxdy f x y t f x y t dxdydtk
Q
k
i
k
x
i
n
i0 0
10
v v + v
Ω
∫ ∫ ∑+
=τ
( , , ) ( , , ) (4)
dlq vsix τ ∈( , ]0 T i vsix v ∈C T C1 20( )[ , ]; ( )0 Ω , de u uk
0 0→ u prostori
Lloc
2 ( )Ω , f fi
k
i→ , i ∈ { 0, 1, … , n } , u prostori L T L2 20( )( , ); ( )loc Ω , pryçomu
uk
0 , fi
k
zadovol\nqgt\ umovu F.
Teorema)1. Nexaj vykonugt\sq umovy A, B, S i, krim toho, f L Ti ∈ 2 0(( , );
Lloc
2 ( ))Ω , i ∈ { 0, 1, … , n } , u L0
2∈ loc( )Ω ,
u x y dxdy f x y t dxdydt
R RQ
i
i
n
0
2 2
0
( ) +∫ ∫ ∑
=
, ( , , )
Ω τ
≤ aebR2
(5)
dlq dovil\noho R > R0 + 1, de a, b — dodatni stali, QR
τ = Ω
R × ( 0, τ )
,
τ ∈( , ]0 T .
Todi isnu[ uzahal\nenyj rozv’qzok zadaçi (1) – (3) na deqkomu promiΩku [ 0,
t0 ] , t T0 0∈( , ].
Dovedennq. Nexaj k — dovil\ne fiksovane natural\ne çyslo take, wo R =
= R ( k ) = 2
k > R0 + 1, q = λ 22k , de λ — deqke natural\ne çyslo. Rozhlqnemo
poslidovnosti funkcij { }( )fi
s , i ∈ { 0, 1, … , n } , { }( )u s
0 taki, wo elementy cyx
poslidovnostej zadovol\nqgt\ umovu F i
f fi
s
i
( ) → u prostori L T L2 20( )( , ); ( )loc Ω , i ∈ { 0, 1, … , n } ,
u us
0 0
( ) → u prostori Lloc
2 ( )Ω
pry s → ∞ . Todi moΩna vkazaty take çyslo s0 ∈ N , wo dlq vsix s > s0
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
1664 S. P. LAVRENGK, M. O. OLISKEVYÇ
Q
i
s
i
n
s
T
R k R k
f dxdydt u dxdy
( ) ( )
( ) ( )
+ +
∫ ∑ ∫
=
+
3 3
2
0
0
2
Ω
≤ 2 3 2
aeb R k[ ( )]+ , (6)
Q
i
s
i
i
n
s
T
R k R k
f f dxdydt u u dxdy
( ) ( )
( ) ( )∫ ∑ ∫− + −
=
2
0
0 0
2
Ω
≤ e q b R k− + [ ( )]2
. (7)
Oçevydno, s0 zaleΩyt\ vid k . Postavymo u vidpovidnist\ koΩnomu k ∈ N
funkci] fi
k , uk
0 z vidpovidnyx poslidovnostej i ∈ { 0, 1, … , n } , dlq qkyx
vykonugt\sq ocinky (6), (7).
V oblasti QT
R
( R = R ( k )) rozhlqnemo rivnqnnq
A ( u ) = f x y t f x y tk R
i x
k R
i
n
i0
1
,
,
,( , , ) ( , , )−
=
∑ (8)
z poçatkovog umovog
u ( x, y, 0 ) = u x yk R
0
, ( , ), ( , )x y R∈Ω , (9)
i krajovymy umovamy
u S TT x y
1 0∩{ ( , )}Ω Ω× ×∂ = 0, u
x
R
y T∂Ω Ω× × ( , )0 = 0, (10)
de
f x y tk R
0
, ( , , ) =
f x y t x y t Q
x y t Q Q
k
T
R
T T
R
0
0
( , , ), ( , , ) ,
, ( , , ) ,\
∈
∈
f x y ti
k R, ( , , ) = f x y t xi R( , , ) ( )χ , i ∈ { 1, … , n } , u x yk R
0
, ( , ) = u x y xk
R0 ( , ) ( )χ ,
χR
nC∈ 1( )R , χR x( ) = 1 pry x ≤ R – 1, χR x( ) = 0 pry x ≥ R,
0 ≤ χR x( ) ≤ 1 pry x ∈ R
n
.
Vvedemo prostory
H R
0
1 1
1,
, ( )Γ Ω = u u u u L i n j mx y
R
i j
: , , , { , , }, { , , },( )∈ ∈ … ∈ …{ 2 1 1Ω
u u
x
R
y x
R∂Ω Ω Ω Γ× ×= = }0 0
1
, ,
H R
0
1 0, ( )Ω = u u u L i n ux
R
i x
R
y
: , , { , , },( )∈ ∈ … ={ }×
2 1 0Ω Ω Ω∂ .
U [13] dovedeno, wo pry vykonanni umov A, B, S, F isnu[ funkciq
u
k = C T L L T HR R( ( )) ( ( ))[ , ]; ( , ); ,
,0 02 2
0
1 1
1
Ω ΩΓ∩ ,
qka zadovol\nq[ rivnist\
Ωτ τ
R R
i i j
u dxdy u a x y t u a x y t uk
Q
k
t
i
m
i y
k
i j
n
ij x
k
x∫ ∫ ∑ ∑+ − + +
= =
v v v v
1 1
( , , ) ( , , )
,
+
+
i
n
i x
k kb x y t u b x y t u dxdydt
i
=
∑ +
1
0( , , ) ( , , , )v v =
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
MIÍANA ZADAÇA DLQ NAPIVLINIJNOHO UL|TRAPARABOLIÇNOHO RIVNQNNQ … 1665
=
Ω0
0 0
1
∫ ∫ ∑+ +
=
u dxdy f x y t f x y t dxdydtk R
Q
k R
i
n
i
k R
x
R
i
, , ,( , , ) ( , , )v v v
τ
(11)
dlq vsix τ ∈ ( 0, T ] i vsix v ∈L T H R2
0
1 00( ( ))( , ); , Ω takyx, wo vt T
RL∈ 2( )Ω , de
Ωτ
R = Q tT
R ∩ { }= τ .
Krim toho, pravyl\nog [ formula intehruvannq çastynamy
t
t
t
k k
Ru u dt
1
2
∫ 〈 〉, =
Ω Ωt
R
t
R
u dxdy u dxdyk k
2 1
2 2
∫ ∫− (12)
dlq vsix t1 , t2 ∈ [ 0, T ] , t1 < t2 , de 〈⋅ ⋅〉, R poznaça[ znaçennq funkcionala z pro-
storu L T H R2
0
1 10
1
( ( ( ) ))( , ); ,
,
Γ Ω ∗
na elementax iz prostoru L T H R2
0
1 10
1
( ( ))( , ); ,
,
Γ Ω .
Tak pobudovani funkci] u
k
dlq k ≥ k0 ≥ log ( )2 0 1R + prodovΩymo nulem na
oblast\ Q Qt T
R\ i zbereΩemo za nymy ti sami poznaçennq. Todi oderΩymo posli-
dovnist\ funkcij { }uk
k k=
∞
0
.
Nexaj Φ ∈C1( )R , Φ ( η ) = 1 pry η ≤ 0, Φ ( η ) = 0 pry η ≥ 1 i 0 ≤ Φ ( η ) ≤
≤ 1 pry η ∈ R .
Vvedemo
h xR( ) = Φ x R−
κ , κ = 2
k
,
ωR x( ) = h xR( )[ ]γ , γ > 2.
Todi ωR x( ) = 1 pry x R≤ , ωR x( ) = 0 pry x R≥ + κ , 0 ≤ ωR x( ) ≤ 1 pry
x m∈R ,
ωR xi
x, ( ) = d h xRκ
γ( )[ ] −1, x n∈R , i ∈ { 1, … , m } , d = const > 0.
Zapysavßy (11) dlq uk+3
i uk+2, vidnqvßy vid perßo] rivnosti druhu, pryj-
nqvßy
v = u x ek k
R
t+ + −3 2, ( )ω µ ,
de µ > 0, uk k+ +3 2, = u uk k+ +−3 2, i vraxuvavßy formulu (12), oderΩymo riv-
nist\
1
2
1
2
3 2 2 3 2 2
Ωτ τ
ω µ ωµ∫ ∫+ + − + ++
u x e dxdy u xk k
R
t
Q
k k
R
, ,( ) ( ) +
+
i
m
i y
k k k k
Ra x y t u u x
i
=
+ + + +∑
1
3 2 3 2( , , ) ( ), , ω +
+
i j
n
ij x
k k k k
R xa x y t u u x
i j
,
, ,( , , ) ( )( )
=
+ + + +∑
1
3 2 3 2 ω +
+
i
n
i x
k k k k
Rb x y t u u x
i
=
+ + + +∑
1
3 2 3 2( , , ) ( ), , ω +
+ ( )( , , , ) ( , , , ) ( ),b x y t u b x y t u u x e dxdydtk k k k
R
t
0
3
0
2 3 2+ + + + −− ]ω µ =
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
1666 S. P. LAVRENGK, M. O. OLISKEVYÇ
= 1
2
0
0
3 3
0
2 2 2
Ω
∫ + + + +−u u x dxdyk R k k R k
R
, ( ) , ( ) ( )ω +
+
Q
k R k k R k k k
Rf f u x
τ
ω∫ + + + + + +−
( ), ( ) , ( ) , ( )0
3 3
0
2 2 3 2 +
+
i
n
i
k R k
i
k R k k k
R x
tf f u x e dxdydt
i
=
+ + + + + + −∑ −
1
3 3 2 2 3 2( )( ), ( ) , ( ) , ( )ω µ , (13)
τ ∈ ( 0, T ] , R ( k ) = 2
k
.
Na pidstavi umovy A
J1 ≡
Q i
m
i y
k k k k
R
ta x y t u u x e dxdydt
i
τ
ω µ∫ ∑
=
+ + + + −
1
3 2 3 2( , , ) ( ), , =
= 1
2 2
3 2 2
1S
k k t
R
i
m
i iu e x a x y t y dS
τ
µ ω ν∫ ∑+ + −
=
, ( ) ( , , ) cos( , ) –
– 1
2
3 2 2
1Q
k k t
R
i
m
i yu e x a x y t dxdydt
i
τ
µ ω∫ ∑+ + −
=
,
,( ) ( , , ) ≥
≥ –
A
u x e dxdydt
Q
k k
R
t1 3 2 2
2
τ
ω µ∫ + + −, ( ) ,
de
A1 = ess sup ( , , )
Q i
m
i
T
a x y t
=
∑
1
,
J2 ≡
Q
ij
i j
n
x
k k k k
R x
ta x y t u u x e dxdydt
i j
τ
ω µ∫ ∑
=
+ + + + −( , , ) ( )
,
, ,( )
1
3 2 3 2 ≥
≥
Q
k k
RA
nA
u x
τ
δ ω∫ −
∇
+ +
0
2 1 3 2 2
2
, ( ) –
–
n d A
u h x e dxdydtk k
R
t
2 2
2
1
2
3 2 2 2
2δ κ
γ µ+ + − −[ ]
, ( ) , (14)
δ1 > 0, A2 = max sup ( , , )
, { , , }i j n Q
ij
T
a x y t
∈ …1
ess .
Zhidno z umovog B
J3 ≡
Q
i
i
n
x
k k k k
R
tb x y t u u x e dxdydt
i
τ
ω µ∫ ∑
=
+ + + + −( , , ) ( ), ,
1
3 2 3 2 ≥
≥ –
1
2
1
1 1
3 2 2
1
3 2 2
Q
k k k k
R
tB u u x e dxdydt
τ
δ
δ
ω µ∫ ∇ +
+ + + + −, , ( ) ,
de
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
MIÍANA ZADAÇA DLQ NAPIVLINIJNOHO UL|TRAPARABOLIÇNOHO RIVNQNNQ … 1667
B1 = ess sup ( , , )
Q
i
i
n
T
b x y t2
1=
∑ ,
J4 ≡
Q
k k k k
R
tb x y t u b x y t u u x e dxdydt
τ
ω µ∫ + + + + −−( )0
3
0
2 3 2( , , , ) ( , , , ) ( ), ≥ 0.
Krim toho,
J5 ≡
Q
k R k k R k k k
Rf f u x
τ
ω∫ + + + + + +−
( ), ( ) , ( ) , ( )0
3 3
0
2 2 3 2 +
+
i
n
i
k R k
i
k R k k k
R x
tf f u x e dxdydt
i
=
+ + + + + + −∑ −
1
3 3 2 2 3 2( )( ), ( ) , ( ) , ( )ω µ ≤
≤ 1
2 2
3 2 2 1
2
2
3 2 2 2
Q
k k
R
k k
Ru x
d
u h x
τ
δ ω δ
κ
γ∫ + + + + −+ [ ]
, ,( ) ( ) +
+ δ ω µ
1
3 2 2
∇
+ + −u x e dxdydtk k
R
t, ( ) +
+ 1
2
1
2
0
3 3
0
2 2 2
Q
k R k k R kf f
τ
δ∫ + + + +−
, ( ) , ( ) +
+ 2
2 1
3 3 2 2 2
δ
ω µ
i
n
i
k R k
i
k R k
R
tf f x e dxdydt
=
+ + + + −∑ −
, ( ) , ( ) ( ) , δ2 0> .
Vraxuvavßy ocinky intehraliv J1 – J5 , z (13) oderΩymo nerivnist\
Ωτ τ
ω µ δ
δ
ωµ∫ ∫+ + − + ++ − − −
u x e dxdy A u xk k
R
t
Q
k k
R
3 2 2
1 2
1
3 2 21, ,( ) ( ) +
+ ( ) ( ),2 0 2 1 1 1 1
3 2 2
A A n B u x e dxdydtk k
R
t− − − ∇
+ + −δ δ δ ω µ ≤
≤
n A d u h x e dxdydt
Q
k k
R
t
2
2
1
1
2
2
3 2 2 2
δ
δ
κ
τ
γ µ+
[ ]∫ + + − −, ( ) +
+ max ; , ( )1 1 2
2 1δ δ
τ
FR ,
de
FR( )τ =
Ω0
0
3 3
0
2 2 2
∫ + + + +−u u x dxdyk R k k R k
R
, ( ) , ( ) ( )ω +
+
Q i
n
i
k R k
i
k R k
R
tf f x e dxdydt
τ
ω µ∫ ∑
=
+ + + + −−
0
3 3 2 2 2, ( ) , ( ) ( )) .
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
1668 S. P. LAVRENGK, M. O. OLISKEVYÇ
Vyberemo
δ1 =
nA B
A
2 1
0
1+ +
, µ = θ + µ0
, θ > 0,
µ0 > A
A
nA B1
0
2 1 1
+
+ +
, δ2 < µ0 1
0
2 1 1
– A
A
nA B
−
+ +
.
Todi z (14) vyplyva[ ocinka
Ωτ
ω θ∫ + + −u x e dxdyk k
R
t3 2 2, ( ) +
+
Q
k k k k
R
tu u x e dxdydt
τ
θ ω θ∫ + + + + −+ ∇( )3 2 2 3 2 2, , ( ) ≤
≤
M
u h x e dxdydt M F
Q
k k
R
t
R k
1
2
3 2 2 2
2 1κ
τ
τ
γ θ∫ + + − −
+[ ] +,
( )( ) ( ) , (15)
de
M1 =
( )
min{ ; }
n A d e
A
T2
2 1
2 2
1 0
0
1
+ δ
δ
µ
, M2 =
max{ ; ; }
min{ ; }
/ /1 1 1
1
2 1
0
δ δ
A
.
Z (15), zokrema, oderΩymo
Q
k k t
R k
u e dxdydt
τ
θ
( )
,∫ + + −3 2 2
≤
M
u e dxdydt
M
F
Q
k k t
R k
R k
1
2
3 2 2 2
1
1θκ θ
τ
τ
θ
( )
,
( )( )
+
∫ + + −
++ . (16)
Vyberemo
θ = β22k = β R k( )[ ]2,
de β = λ2
1M e .
Oskil\ky
f fi
k R k
i
k R k+ + + +−3 3 2 2 2, ( ) , ( ) ≤ 2 3 3 2 2 2 2
f f f fi
k R k
i i
k R k
i
+ + + +− + −( ), ( ) , ( ) ,
i ∈ { 0, 1, … , n } ,
u uk R k k R k
0
3 3
0
2 2 2+ + + +−, ( ) , ( ) ≤ 2 0
3 3
0
2
0
2 2
0
2
u u u uk R k k R k+ + + +− + −( ), ( ) , ( ) ,
to na pidstavi (7)
FR k( )( )+1 τ ≤ 2 1 2
e q b R k− + +[ ( )] .
OtΩe, z (16) vyplyva[ nerivnist\
Q
k k t
R k
u e dxdydt
τ
θ
( )
,∫ + + −3 2 2
≤ e u e dxdydt
Q
k k t
R k
− + + −
+
∫1 3 2 2
1
τ
θ
( )
, +
+
2 2 1 2M
e q b R k
θ
− + +[ ( )] . (17)
Podilymo vidrizok [ R ( k ) , R ( k ) + κ ] na q çastyn. Todi, qk i v [14], z (17)
oderΩymo ocinku
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MIÍANA ZADAÇA DLQ NAPIVLINIJNOHO UL|TRAPARABOLIÇNOHO RIVNQNNQ … 1669
Q
k k
R k
u dxdydt
τ
( )
,∫ + +3 2 2
≤ e u dxdydt
Q
k k
R k
− + + +
+
∫θ θτ
τ
( )
,
1
3 2 2
+
+
2
1
2 1 2M e
e
e q b R k
θ
θτ
( )
[ ( )]
−
− + + + . (18)
Vykorystavßy (11) pry v = u ek t−ρ , ρ > 0, otryma[mo rivnist\
1
2
1
2
2 2
1ΩR k R k
i
u e dxdy u a x y t u uk
Q
k
i y
k
i
m
k
( ) ( )
( , , )∫ ∫ ∑−
=
+ +
ρτ
τ
ρ +
+
i j
n
ij x
k
x
k
i x
k
i
n
k k k ta x y t u u b x y t u u b x y t u u e dxdydt
i j i
,
( , , ) ( , , ) ( , , , )
= =
−∑ ∑+ +
1 1
0
ρ =
= 1
2
0
0
2
0
1Ω ΩR k R k
i
u dxdy f u f u e dxdydtk R k k R k k
i
n
i
k R k
x
k t
( ) ( )
, ( ) , ( ) , ( )∫ ∫ ∑+ +
=
−
τ
ρ , (19)
τ ∈ ( 0, T ] .
Na pidstavi umov A, B, S z (19) lehko oderΩaty ocinku
Ωτ τ
ρτ ρ
δR k R k
u e dxdy A uk
Q
k
( ) ( )
∫ ∫− + − − −
2
1
3
21
1 +
+ ( )2 0 3 1 3
2
A B u e dxdydtk t− − ∇
−δ δ ρ ≤
≤
Ω0
0
2
0
2
3 1
21
R k R k
u dxdy f f e dxdydtk R k
Q
k R k
i
n
i
k R k t
( ) ( )
, ( ) , ( ) , ( )∫ ∫ ∑+ +
=
−
τ
δ
ρ , δ3 > 0.
Vybravßy u cij nerivnosti
δ3 =
2
1
0
1
A
B +
, ρ = A
B
A1
1
0
2
1
2
+ + +
,
matymemo
Q
k
R k
u dxdy
τ
( )
∫
2
≤ M u dxdy f dxdydt
R k R k
k R k
Q i
n
i
k R k
3 0
2
0
2
0Ω ( ) ( )
, ( ) , ( )∫ ∫ ∑+
=
τ
, (20)
de
M3 = e
B
A
Tρ max ;1
1
2
1
0
+
.
Vraxuvavßy (6), z (20) otryma[mo ocinku
Q
k
R k
u dxdydt
τ
( )
∫
2
≤ 2 3
2
a M eb R k[ ( )] . (21)
Oskil\ky
uk k+ +3 2 2, ≤ 2 3 2 2 2
u uk k+ ++( ) ,
to z (18), (21) vyplyva[
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
1670 S. P. LAVRENGK, M. O. OLISKEVYÇ
Q
k k
R k
u dxdydt
τ
( )
,∫ + +3 2 2
≤
≤ 4 33
2a M q b R kexp [ ( )]− + + +[ ]θτ +
+
2
1
12 2M e
e
q b R k
θ
θτ
( )
exp [ ( )]
−
− + + +[ ] ≤
≤ M q b R k4
23exp [ ( )]− + + +[ ]θτ ,
de
M4 = 4
2
13
2a M
M e
e
+
−θ( )
.
Tomu
Q
k k
R k
u dxdydt
τ
( )
,
+
∫ + +
1
3 2 2
≤
≤ M R k R k b R k4
2 2 21 1 4exp [ ( )] [ ( )] [ ( )]− + + + + +[ ]λ β . (22)
OtΩe, vraxuvavßy (15), (22) i ocinku dlq FR k( )( )+1 τ , oderΩymo nerivnist\
Ωτ τ
R k R k
u dxdy u u dxdydtk k
Q
k k k k
( ) ( )
, , ,∫ ∫+ + + + + ++ + ∇( )3 2 2 3 2 2 3 2 2
≤
≤ M R k R k b R k5
2 2 2 21 1 4exp [ ( )] [ ( )] [ ( )]− + + + + +[ ]λ β τ . (23)
Vyberemo
λ = 2 16 [ ]b +( ).
Todi pravu çastynu (23) moΩna ocinyty tak:
exp [ ( )] [ ( )] [ ( )]− + + + + +[ ]λ β τR k R k b R k1 1 42 2 2 ≤ e
k− +22 2
0α ,
de α0 = ϕ β− −t b0
62 , qkwo
τ = t0 <
λ
β
− b26
.
Nexaj R1 > R0 + 1 — dovil\ne fiksovane çyslo, R ( k ) > R1
. Z (23) vyplyva[
ocinka
u uk k
C t L
k k
L t VR R
+ + + ++3 2
0
3 2
00
2 1 2
0
1
,
([ , ]; ( ))
,
(( , ); ( ))Ω Ω
≤ M e
k
5
20
2 1− +α ,
V R( )Ω 1 =
u u u L ux
R
i x x
R
y
: , ( ), ( )∈ ={ }∂ ∂ ×
2 1
1 0Ω Ω Ω Ω∩ .
Todi
u u u uk l k
C t L
k l k
L t VR R
+ + + + + +− + −2 2
0
2 2
00
2 1 2
0
1([ , ]; ( )) (( , ); ( ))Ω Ω
≤
≤
i
l
k i k i
C t L
k i k i
L t V
u u u uR R
=
−
+ + + + + + + +∑ − + −
0
1
3 2
0
3 2
00
2 1 2
0
1([ , ]; ( )) (( , ); ( ))Ω Ω
≤
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MIÍANA ZADAÇA DLQ NAPIVLINIJNOHO UL|TRAPARABOLIÇNOHO RIVNQNNQ … 1671
≤ M e
i
k i
5
0
20
2 1
=
∞
−∑
+ +α ( )
≤ M e
k
6
20
2 1− +α ,
de M6 ne zaleΩyt\ vid k, l ≥ 1. OtΩe, poslidovnist\ { }uk
[ fundamental\nog u
prostori C t L L t VR R( ) ( )[ , ]; ( ) ( , ); ( )0 00
2 2
0
1 1Ω Ω∩ . Vraxovugçy dovil\nist\ R1, zvid-
sy oderΩu[mo, wo u
k → u u prostori C t L L t H( ) ( )[ , ]; ( ) ( , ); ( ),0 00
2 2
0
1 0
loc locΩ Ω∩ ,
tobto u — uzahal\nenyj rozv’qzok zadaçi (1) – (3).
Teoremu dovedeno.
Teorema)2. Nexaj vykonugt\sq umovy A, B, S, fi ∈ L T L2 20( )( , ); ( )loc Ω ,
u0 ∈ Lloc
2 ( )Ω i ∈ { 0, 1, … , n } . Todi zadaça (1) – (3) ne moΩe maty bil\ße odno-
ho uzahal\nenoho rozv’qzku v klasi funkcij takyx, wo
u dxdydt
QT
R
2∫ ≤ aebR2
∀ R > R0 + 1, (24)
de a, b — dodatni stali.
Dovedennq. Nexaj isnugt\ dva uzahal\neni rozv’qzky u1
i u2
zadaçi (1)4–
–4(3). Zadamo dovil\ne fiksovane çyslo R1 > R0 + 1 i qk zavhodno male çyslo
ε > 0. Nexaj R ( l ) = 2
l > R1 , l ∈ N .
Zhidno z oznaçennqm uzahal\nenoho rozv’qzku isnugt\ poslidovnosti { },ui k
taki, wo u ui k i, → u prostori C T L L T H( ) ( )[ , ]; ( ) ( , ); ( ),0 02 2 1 0
loc locΩ Ω∩ , pryçomu
ui k,
zadovol\nq[ (4) z pravymy çastynamy f j
i k,
i poçatkovymy funkciqmy ui k
0
, ,
de f j
i k, , ui k
0
,
zadovol\nqgt\ umovu F, f fj
i k
j
, → u prostori L t L2
0
20( )( , ); ( )loc Ω ,
u ui k
0 0
, → u prostori Lloc
2 ( )Ω pry k → ∞ , i ∈ { 1, 2 } , j ∈ { 0, 1, … , n } .
Tak samo, qk (15), oderΩymo nerivnist\
Q
k k
R
tu u x e dxdydt
τ
ω θ∫ − −2 1 2, , ( ) ≤
≤
M
u u h x e dxdydt
M
F R
Q
k k
R
t1
2
2 1 2 2 2
θκ θ
τ
τ
γ θ∫ − +− −, , [ ( )] ( , ), (25)
de funkciq ωR x( ) i stali M1, M2 , θ, κ vyznaçeni pry dovedenni teoremy41, a
F R( , )τ =
Ω0
0
2
0
1 2
0
2 1 2
∫ ∫ ∑− + −
=
u u x dxdydt f f x dxdydtk k
R
Q i
n
i
k
i
k
R
, , , ,( ) ( )ω ω
τ
.
Vyberemo q l= λ 22 , κ = 2l , θ β= 22l , de β λ= 2
1M e , λ — deqke natu-
ral\ne çyslo. Oskil\ky
F R( , )τ ≤ 2
0
1
0
2
0
2
0
1
0
2
ΩR l
u u u u dxdyk k
( )
, ,
+
∫ − + −( ) +
+ 2
1 0
2 2 1 2
Q i
n
i
k
i i
k
i
R l
f f f f dxdydt
τ
( )
, ,
+
∫ ∑
=
− + −
,
to, vraxuvavßy zbiΩnosti poslidovnostej { },f j
i k , { },ui k
0 , i ∈ { 1, 2 } , j ∈ { 0, 1, …
… , n } , moΩemo vkazaty take k l0( ), wo
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
1672 S. P. LAVRENGK, M. O. OLISKEVYÇ
F R( , )τ ≤ e q−
dlq vsix k > k l0( ). Todi z (25), qk i pry dovedenni teoremy41, oderΩymo ocinku
u u dxdydtk k
QR l
2 1 2, ,
( )
−∫
τ
≤ e u u dxdydt eq k k
Q
q
R l
− + − +− +
+
∫θτ θτ
τ
2 1 2
1
, ,
( )
(26)
dlq
l ≥ l0 = 1
2
12
2log
M
β
+ .
Oskil\ky u ui k i, → u Lloc
2 ( )Ω , to isnu[ take k1 ( l, ε ) ∈ N , k1 ≥ k0 , wo
u u dxdydti k i
QR l
,
( )
−
+
∫
2
1
τ
≤ ε
16
, i ∈ { 1, 2 }, (27)
dlq vsix k > k1 ( l, ε ) .
Vraxovugçy te, wo
u uk k2 1 2, ,− < 3 2 22 2 2 1 1 2 1 2 2 2
u u u u u uk k, ,− + − + +( ) ,
i umovu (24), z (26) oderΩu[mo ocinku
u u dxdydtk k
QR l
2 1 2, ,
( )
−∫
τ
≤ e aeq b R l− + ++ +
θτ ε3
8
4 11 2[ ( )] ≤
≤ ( ) exp [ ( )]2 4 1 2+ − + + +[ ]a q b R lθτ (28)
pry k > k1 ( l, ε ) . Vyberemo
λ = +4 1([ ] )b , τ = t0 , 0 < t0 <
λ
β
− 4b
, α0 = λ – β t0 – 4 b > 0.
Todi z (28) vyplyva[ ocinka
u u dxdydtk k
Qt
R l
2 1 2
0
, ,
( )
−∫ ≤ ( )2 4 0
22+ −a e
lα .
OtΩe, isnu[ take l1 ∈ N , l1 ≥ l0 , wo
u u dxdydtk k
Qt
R
2 1 2
0
1
, ,−∫ < ε
16
(29)
dlq vsix l > l1 i k > k1 ( l, ε ) .
Oskil\ky
u u2 1 2
− ≤ 3 2 2 2 1 1 2 2 1 2
u u u u u uk k k k− + − + −( ), , , , ,
to na pidstavi (27), (29)
u u dxdydt
Qt
R
2 1 2
0
1
−∫ < ε ,
tobto vnaslidok dovil\nosti ε u x y t2( , , ) = u x y t1( , , ) majΩe skriz\ v Qt
R
0
1 . Os-
kil\ky R1 — dovil\ne çyslo, to u x y t2( , , ) = u x y t1( , , ) majΩe skriz\ v Qt0
. Qk-
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
MIÍANA ZADAÇA DLQ NAPIVLINIJNOHO UL|TRAPARABOLIÇNOHO RIVNQNNQ … 1673
wo t0 < T , to za skinçenne çyslo krokiv dovodymo [dynist\ u vsij oblasti QT .
Teoremu dovedeno.
ZauvaΩennq. Zaznaçymo, wo umovy teoremy41 zabezpeçugt\ [dynist\ uza-
hal\nenoho rozv’qzku zadaçi (1) – (3). Prote [dynist\ rozv’qzku harantu[t\sq na
bud\-qkomu promiΩku [ 0, T ] , todi qk isnuvannq vstanovleno lyße na deqkomu
promiΩku [ 0, t0 ] ⊂ [ 0, T ] , de t0 zaleΩyt\ vid koefici[ntiv rivnqnnq ta sta-
lo]44b .
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1934. – 35. – S. 116 – 117
2. Flemynh U., Ryßel R. Optymal\noe upravlenye determynyrovann¥my y stoxastyçeskymy
systemamy. – M.: Myr, 1978. – 316 s.
3. Lanconelli E., Pascucci A., Polidoro S. Linear and nonlinear ultraparabolic equations of Kolmogo-
rov type arising in diffusion theory and in finance // Nonlinear Problems Math. Phys. and Related
Top. II. In honour of Proff. O. A. Ladyzhenskaya. – New York: Kluwer Acad. Publ., 2002. –
P. 243 – 265.
4. Eidelman S. D., Ivasyshen S. D., Kochubei A. N. Analytic methods in the theory of differential and
pseudo-differential equations of parabolic type. – Birkhäuser, 2004. – 390 p.
5. Dron\ V. S., Ivasyßen S. D. Pro korektnu rozv’qznist\ zadaçi Koßi dlq vyrodΩenyx
paraboliçnyx rivnqn\ typu Kolmohorova // Ukr. mat. visn. – 2004. – # 1. – S.461 – 68.
6. Voznqk O. H., Ivasyßen S. D. Fundamental\ni rozv’qzky zadaçi Koßi dlq odnoho klasu
vyrodΩenyx paraboliçnyx rivnqn\ ta ]x zastosuvannq // Dop. NAN Ukra]ny. – 1996. – # 10. –
S.411 – 16.
7. ∏jdel\man S. D., Malyckaq A. P. O fundamental\n¥x reßenyqx y stabylyzacyy reßenyq
zadaçy Koßy dlq odnoho klassa v¥roΩdagwyxsq parabolyçeskyx uravnenyj // Dyfferenc.
uravnenyq. – 1975. – 11, # 7. – S.41316 – 1331.
8. Polidoro S. On the regularity of solutions to a nonlinear ultraparabolic equation arising in
mathematical finance // Nonlinear Analysis. – 2001. – 47. – P. 491 – 502.
9. Lavrengk S. P., Procax N. P. Mißana zadaça dlq ul\traparaboliçnoho rivnqnnq v
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OderΩano 03.05.06
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 12
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| id | umjimathkievua-article-3420 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:42:14Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d5/d7b0345aed11aa147d5ae69f399928d5.pdf |
| spelling | umjimathkievua-article-34202020-03-18T19:53:47Z Mixed problem for a semilinear ultraparabolic equation in an unbounded domain Мішана задача для напівлінійного ультрапараболічного рівняння у необмеженій області Lavrenyuk, S. P. Oliskevych, M. O. Лавренюк, С. П. Оліскевич, М. О. We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation $$u_t + \sum^m_{i=1}a_i(x, y, t) u_{y_i} - \sum^n_{i,j=1} \left(a_{ij}(x, y, t) u_{x_i}\right)_{x_j} + \sum^n_{i,j=1} b_{i}(x, y, t) u_{x_i} + b_0(x, y, t, u) =$$ $$= f_0(x, y, t, ) - \sum^n_{i=1}f_{i, x_i} (x, y, t) $$ in an unbounded domain with respect to the variables x. Получены условия существования и единственности решения смешанной задачи для ультрапараболического уравнения $$u_t + \sum^m_{i=1}a_i(x, y, t) u_{y_i} - \sum^n_{i,j=1} \left(a_{ij}(x, y, t) u_{x_i}\right)_{x_j} + \sum^n_{i,j=1} b_{i}(x, y, t) u_{x_i} + b_0(x, y, t, u) =$$ $$= f_0(x, y, t, ) - \sum^n_{i=1}f_{i, x_i} (x, y, t) $$ в неограниченной области по переменным x. Institute of Mathematics, NAS of Ukraine 2007-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3420 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 12 (2007); 1661–1673 Український математичний журнал; Том 59 № 12 (2007); 1661–1673 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3420/3583 https://umj.imath.kiev.ua/index.php/umj/article/view/3420/3584 Copyright (c) 2007 Lavrenyuk S. P.; Oliskevych M. O. |
| spellingShingle | Lavrenyuk, S. P. Oliskevych, M. O. Лавренюк, С. П. Оліскевич, М. О. Mixed problem for a semilinear ultraparabolic equation in an unbounded domain |
| title | Mixed problem for a semilinear ultraparabolic equation in an unbounded domain |
| title_alt | Мішана задача для напівлінійного ультрапараболічного рівняння у необмеженій області |
| title_full | Mixed problem for a semilinear ultraparabolic equation in an unbounded domain |
| title_fullStr | Mixed problem for a semilinear ultraparabolic equation in an unbounded domain |
| title_full_unstemmed | Mixed problem for a semilinear ultraparabolic equation in an unbounded domain |
| title_short | Mixed problem for a semilinear ultraparabolic equation in an unbounded domain |
| title_sort | mixed problem for a semilinear ultraparabolic equation in an unbounded domain |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3420 |
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