Boundary-Value Problem for Linear Parabolic Equations with Degeneracies
In spaces of classical functions with power weight, we prove the correct solvability of a boundary-value problem for parabolic equations with an arbitrary power order of degeneracy of coefficients with respect to both time and space variables.
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| Datum: | 2005 |
|---|---|
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| Format: | Artikel |
| Sprache: | Russisch Englisch |
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2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509724187820032 |
|---|---|
| author | Pukalskyi, I. D. Пукальский, И. Д. Пукальский, И. Д. |
| author_facet | Pukalskyi, I. D. Пукальский, И. Д. Пукальский, И. Д. |
| author_sort | Pukalskyi, I. D. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:59:42Z |
| description | In spaces of classical functions with power weight, we prove the correct solvability of a boundary-value problem for parabolic equations with an arbitrary power order of degeneracy of coefficients with respect to both time and space variables. |
| first_indexed | 2026-03-24T02:45:39Z |
| format | Article |
| fulltext |
UDK 517.946
Y. D. Pukal\skyj (Çernovyc. nac. un-t)
KRAEVAQ ZADAÇA DLQ LYNEJNÁX
PARABOLYÇESKYX URAVNENYJ S VÁROÛDENYQMY
In the spaces of classical functions with power weight, we prove the correct solvability of a boundary-
value problem for parabolic equations with an arbitrary power order of degeneration of coefficients both
in time and space variables.
U prostorax klasyçnyx funkcij zi stepenevog vahog dovedeno korektnu rozv’qznist\ krajovo]
zadaçi dlq paraboliçnyx rivnqn\ z dovil\nym stepenevym porqdkom vyrodΩennq koefici[ntiv qk
za çasovog, tak i za prostorovymy zminnymy.
V sovremenn¥x prykladn¥x y teoretyçeskyx yssledovanyqx oçen\ çasto vstre-
çagtsq zadaçy s razn¥my v¥roΩdenyqmy. V çastnosty, uravnenye Íredynhera,
opredelqgwee sostoqnye kvantovomexanyçeskoj system¥, ymeet stepenn¥e oso-
bennosty v koπffycyentax [1]. Kraev¥e zadaçy s nelokal\n¥m uslovyem po
vremennoj peremennoj dlq parabolyçeskyx uravnenyj vtoroho porqdka so ste-
penn¥my osobennostqmy v koπffycyentax yssledovalys\ v [2, 3].
Yzuçenyg fundamental\n¥x matryc reßenyj parabolyçeskyx system, yx
prymenenyg dlq yssledovanyq korrektnoj razreßymosty zadaçy Koßy posvq-
wena monohrafyq [4]. V [5] postroena teoryq klassyçeskyx reßenyj zadaçy
Koßy y kraevoj zadaçy dlq ravnomerno parabolyçeskyx uravnenyj, kotor¥e
ymegt stepenn¥e osobennosty ohranyçennoho porqdka na hranyce oblasty v ko-
πffycyentax pry mladßyx proyzvodn¥x.
Zdes\ v prostranstvax klassyçeskyx funkcyj so stepenn¥m vesom yzuçaetsq
kraevaq zadaça dlq parabolyçeskyx uravnenyj bez ohranyçenyq na stepennoj
porqdok v¥roΩdenyq koπffycyentov uravnenyq.
Postanovka zadaçy y osnovnoj rezul\tat. Pust\ D — ohranyçennaq v¥-
puklaq oblast\ v R
n
s hranycej ∂ D . Rassmotrym v oblasty Q = ( 0, T ] × D
kraevug zadaçu
( L u ) ( t, x ) ≡ D A t x D u t xt
k b
k x
k−
≤
∑
2
( , ) ( , ) = f0 ( t, x ) , (1)
u ( 0, x ) = ϕ ( x ) , (2)
k r
k
i
x
k
i
b t x D u t x
≤
∑
( )( , ) ( , ) Γ = fi ( t, x ) , (3)
hde Γ = ( 0, T ] × ∂ D, 0 ≤ ri ≤ 2b – 1, i ∈ { 1, … , b } , k = ( k1, … , kn ) , Dx
k =
= D D Dx
k
x
k
x
k
n
n
1
1
2
2 … , k = k k kn1 2+ + … + .
Pust\ q( )1 , q( )2 — proyzvol\n¥e vewestvenn¥e çysla, t T( ) ( , )0 0∈ , x D( )0 ∈ ,
min ( )
y D
y x
∈∂
− 0 ≥ e > 0, e = const , y x− ( )0 = ( )( )
/
y xl ll
n −[ ]=∑ 0
1
1 2
. Osobennosty
koπffycyentov dyfferencyal\noho v¥raΩenyq L xarakteryzugtsq sledug-
wymy funkcyqmy: s q t1
1( , )( ) = t t
q
− ( )
( )
0
1
pry t t− ( )0 ≤ 1, s q t1
1( , )( ) = 1 pry
t t− ( )0 ≥ 1; s q x2
2( , )( ) = x x
q
− ( )
( )
0
2
pry x x− ( )0 ≤ 1, s q x2
2( , )( ) = 1 pry
x x− ( )0 ≥ 1.
© Y. D. PUKAL|SKYJ, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3 377
378 Y. D. PUKAL|SKYJ
Pust\ Q = [ 0, T ] × D , D = D ∪ ∂ D, a P ( t, x ) , P t x1
1 1( )( ) ( ), , H t xl( )( ) ( ),1 2 ,
Nl t x( )( ) ( ),2 2 , l ∈ { 1, … , n } , — toçky yz Q , x( )1 = ( )( ) ( ), ,x xn1
1 1… , x( )2 =
= ( ( ) ( ) ( ), , , ,x x xl l1
1
1
1 2… − x xl n+ …1
1 1( ) ( ), , ). Oboznaçym çerez β ν
l
( ), µ ν
0
( ) , µ ν
kl
( ) , ν ∈ { 1, 2 } ,
vewestvenn¥e çysla takye, çto β ν
l
( ) ∈ ( – ∞ , ∞ ) , µ ν
0
( ) ≥ 0, µ ν
kl
( ) ≥ 0. PoloΩym
( , )k kµ =
l
n
l kk
l
=
∑
1
µ ν( ), s q P( ; ) = s q t s q x1
1
2
2( ) ( )( ) ( ), , ,
s k P( )( , );γ β− = s k t s k x1
1 1
2
2 2( ) ( )( , ), ( , ),( ) ( ) ( ) ( )γ β γ β− − ,
( ), ( ) ( )k γ βν ν− =
l
n
l lk
=
∑ −
1
( )( ) ( )γ βν ν , γ ν( ) = const ≥ 0.
Pust\ r — nekotoroe neceloe çyslo, [ r ] — celaq çast\ r, { r } = r – [ r ] .
Opredelym funkcyonal\n¥e prostranstva, v kotor¥x yssleduetsq zadaça.
C q Qr ( , ; ; )γ β — prostranstvo funkcyj u, ( t, x ) ∈ Q , ymegwyx çastn¥e pro-
yzvodn¥e v oblasty
Q( )0 ≡ Q t x x D t x t T\ ( , ), ( , ), [ , ]( ) ( )0 0 0∈( ) ∈( ){ }∪
vyda D D ut
j
x
k , 2bj + | k | ≤ [ r ] , y koneçnoe znaçenye velyçyn¥
u q Q r; , ; ;γ β = u q Q u q Qr r; , ; ; ; , ; ;[ ]γ β γ β+ [ ] =
= sup (( ) ( ; ); ) ( )
[ ]P Q bj k r
t
j
x
ks bj q k P D D u P
∈ + ≤
∑ + + −
2
2 γ γ β +
+
l
n
P H Q bj k r
l l l
r
l
s bj q k r P x x
= ⊂ + =
−∑ ∑ + + − + − −
1 2
1
1 2
1
2sup ( ) ( ; ) { }( ); ˜
{ , } [ ]
( ) ( ) { }
( )γ γ β γ β ×
× D D u P D D u H s bj q r k Nt
j
x
k
t
j
x
k
l
H N Q bj k r
l
l l
( ) ( ) sup ( { }) ( ; ); ˜
{ , } [ ]
( )1
2
2− + + + + −
⊂ + =
∑ γ γ β ×
× t t D D u H D D u N
r b
t
j
x
k
l t
j
x
k
l
( ) ( ) { / }
( ) ( )1 2 2
− −
−
,
u Q; , ; ;γ β 0 0 = sup ( )
P Q
u P
∈
= u 0.
Zdes\
s q P( ; ˜ )1 = min( ( , ), ( , ))s q P s q Hl1 , s q Nl( ; ˜ ) = min( ( , ), ( , ))s q N s q Hl l .
C Qr
k( , )µ — mnoΩestvo funkcyj vk, ( t, x ) ∈ Q , ymegwyx çastn¥e proyz-
vodn¥e v Q( )0
vyda Dx k
λ v , | λ | ≤ [ r ] , dlq kotor¥x koneçna norma
vk k rQ; ;µ =
λ
λµ δ µ λ
≤ ∈
∑ + +
[ ]
sup ( , ) ; ( )( )
r P Q
k
k
x ks k P D P0 0 v +
+
l
n
r P H Q
k
k
l l
r
l
s k P s r x x x
= = ⊂
−∑ ∑ + +
−
1
0 0 1 2
1 2
1λ
µ δ µ λ
[ ] { , }
( ) ( ) { }
sup ( , ) ; ˜ { }, ˜( ) ( ) ×
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
KRAEVAQ ZADAÇA DLQ LYNEJNÁX PARABOLYÇESKYX URAVNENYJ … 379
×
D P D H s k Nx k x k l
N H Q
k
k
l
l l
λ λ µ δ µv v( ) ( ) sup ( , ) ; ˜
{ , }
( )1 0 0− + +
⊂
×
×
s
r
b
t t t H N
r b
k l k l1
1 2 2
2{ }
− −
−
, ˜ ( ) ( )( ) ( ) { / }
v v +
+ sup ( , ) ; ˜
{ , }
( )
N P Q
k
k
l
l l
s k N
⊂
+ +µ δ µ λ0 0 ×
× s
r
b
t t t D H D N
r b
x k l x k l1
1 2 2
2
{ }
, ˜ ( ) ( )( ) ( ) { }/
− −
− λ λv v ,
hde λ = ( λ1, … , λn ) , δ0
k
— symvol Kronekera, | λ | = λ1 + λ2 +…+ λn .
Otnosytel\no zadaçy (1) – (3) predpolahaem v¥polnenye uslovyj:
a) koπffycyent¥ uravnenyq A C Qk k∈ α µ( , ) pry k b≤ −2 1, A0 0< ,
A C Qk ∈ α β( , ) pry k b= 2 , b Ck
i b ri i( ) ( )∈ − +2 α Γ pry k ri< , b C Qk
i b ri( ) ( , )∈ − +2 α β
pry k ri= , αi ∈( , )0 1 , ∂ αD C b∈ +2 y kraevaq zadaça
D s k P A P D u t xt
k b
k x
k−
=
∑
2
(( , ); ) ( ) ( , )β = g ( t, x ) ,
u ( 0, x ) = ϕ ( x ) ,
k r
k
i
x
k
i
s k P b P D u
=
∑ (( , ); ) ( )( )β
Γ
= gi ( t, x ) ,
ravnomerno parabolyçeskaq [4];
b) funkcyy
f0 ∈ C b Qα γ β( , ; ; )2 , ϕ ∈ C Db2 0+α γ β( ˜ , ˜; ; ), γ̃ = ( , )( )0 2γ , β̃l = ( , )( )0 2βl ,
l ∈ { 1, … , n } , fi ∈ C rb r
i
i i2 − +α γ β( , ; ; )Γ , ( )( )B xi ϕ Γ = fi ( 0, x ) ,
γ ν( ) = max , max
( , )
,
{ , , }
( )
( ) ( ) ( )
j n j k b
kk
b k b∈ … <
+
−
−
1 2
01
2 2
β
µ β µν
ν ν ν
, ν ∈ { 1, 2 } .
Sformulyruem osnovnoj rezul\tat o razreßymosty zadaçy (1) – (3).
Teorema,1. Pust\ dlq zadaçy (1) – (3) v¥polnen¥ uslovyq a), b). Tohda su-
westvuet edynstvennoe reßenye zadaçy (1) – (3) v klasse C Qb2 0+α γ β( , ; ; ) y
dlq neho spravedlyva ocenka
u Q b; , ; ;γ β α0 2 + ≤ C f b Q0 2; , ; ;γ β α
+ ϕ γ β
α
; ˜ , ˜; ;0
2
D
b+
+
+
i
b
i i b rf r
i i=
− +∑
1
2; , ; ;γ β αΓ . (4)
Dlq dokazatel\stva teorem¥L1 postroym podposledovatel\nost\ reßenyj
kraev¥x zadaç s hladkymy koπffycyentamy, predelom kotoroj budet reßenye
zadaçy (1) – (3).
Ocenka reßenyj kraev¥x zadaç s hladkymy koπffycyentamy. Pust\
Qm = Q t x Q s t m s x m∩ ( , ) ( , ) , ( , )∈ ≥ ≥{ }− −
1 1
1
2 2
11 1 ,
m = ( m1, m2 ) , m1, m2 — natural\n¥e çysla, m1 > 1, m2 > 1, — posledova-
tel\nost\ oblastej, kotoraq pry m1 → ∞ , m2 → ∞ sxodytsq k Q,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
380 Y. D. PUKAL|SKYJ
Dm = x D s x m∈ ≥{ }−
2 2
11( , ) , Γm = ( 0, T ) × ∂ Dm .
Rassmotrym kraevug zadaçu dlq parabolyçeskoho uravnenyq
( )( , )L u t xm1 ≡ D a t x D u t xt
k b
k x
k
m−
≤
∑
2
( , ) ( , ) = Ψm ( t, x ) , (5)
um ( 0, x ) = Φm ( x ) , (6)
( )( ) ( , )L u t xi m
0
Γ
≡
k r
k
i
x
k
m
i
b t x D u t x
≤
∑
( )( , ) ( , ) Γ = fi ( t, x ) . (7)
Zdes\ koπffycyent¥ ak , funkcyy Ψm , Φm opredelen¥ sledugwym obrazom.
Esly ( , ) ( , ]t x T Dm∈ ×0 y k b≤ −2 1, ( ), ( )k β 1 0≥ pry k b= 2 , to
ak ( t, x ) = min ( , ), ,( )A t x A m xk k 1
1−{ } pry t m( ) ( ],0
1
10∈ −
y
ak ( t, x ) = min ( , ),
( )
,
( )
( )( )A t x
m t t
A t m xk k
1
1 0
0
1
11
2
−
−− + −
+
+
m t t
A t m xk
1
1 0
0
1
11
2
−
−− + +
( )
,
( )
( )( ) pry t m( )0
1
1≥ − .
V sluçae, kohda ( ), ( )k β 1 0< , pry k b= 2 v¥berem
ak ( t, x ) = max ( , ), ,( )A t x A m xk k 1
1−{ } pry t m( ) ( ],0
1
10∈ −
y
ak ( t, x ) = max ( , ),
( )
,
( )
( )( )A t x
m t t
A t m xk k
1
1 0
0
1
11
2
−
−− + −
+
+
m t t
A t m xk
1
1 0
0
1
11
2
−
−− + +
( )
,
( )
( )( ) pry t m( )0
1
1≥ − .
Funkcyy
Ψm ( t, x ) = min ( , ), ,( )f t x f m x0 0 1
1−( ) pry t m( ) ( ],0
1
10∈ −
y
Ψm ( t, x ) = min ( , ),
( )
,
( )
( )( )f t x
m t t
f t m x0
1
0
0
0
1
11
2
− + −
− +
+
m t t
f t m x1
0
0
0
1
11
2
( )
,
( )
( )( )− + +
−
pry t m( )0
1
1≥ − .
Pry x ∈ Dm funkcyy Φm ( x ) = ϕ ( x ) .
Dlq ( t, x ) ∈ Q T Dm\ ( , ){ }0 × koπffycyent¥ ak y funkcyy Ψ m est\ reße-
nyq vnutrennej zadaçy
Dt u = ∆ u, u ( 0, x ) = 0,
∂
∂
u
n m
�
Γ
= ψ ( t, x ) ,
hde, naprymer, dlq ak berem ψ = ak mΓ ,
�
n — normal\ k Γm . Dlq x D Dm∈ \
funkcyq Φm qvlqetsq reßenyem vnutrennej zadaçy Dyryxle
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
KRAEVAQ ZADAÇA DLQ LYNEJNÁX PARABOLYÇESKYX URAVNENYJ … 381
∆ u = 0, u Dm∂ = ϕ ( )x Dm∂ .
Vvedem v prostranstve C Qb2 +α( ) normu u q Qm b; , ; ;γ β α2 + , πkvyvalentnug
pry kaΩdom fyksyrovannom m1, m2 hel\derovoj norme, kotoraq opredelqetsq
tak Ωe, kak u q Q b; , ; ;γ β α2 + , tol\ko vmesto funkcyj s q t1 1( , ), s q x2 2( , ) berem
sootvetstvenno d q t1 1( , ), d q t2 2( , ), hde d q t1 1( , ) = s q t1 1( , ) pry t t m− ≥ −( )0
1
1,
d q t1 1( , ) = m q
1
1−
pry t t m− ≤ −( )0
1
1, d q x2 2( , ) = s q x2 2( , ) pry x x m− ≥ −( )0
2
1,
d q x2 2( , ) = m q
2
2−
pry x x m− ≤ −( )0
2
1; d k P(( , ); )γ β− = d k t1
1 1(( , ), )( ) ( )γ β− ×
× d k x2
2 2(( , ), )( ) ( )γ β− .
Pry nalahaem¥x uslovyqx na hladkost\ koπffycyentov dyfferencyal\-
n¥x v¥raΩenyj L1, Li
( )0 , i ∈ { 1, … , b } , suwestvuet edynstvennoe reßenye
zadaçy (5) – (7), kotoroe prynadleΩyt prostranstvu C Qb2 +α ( ) y ymeet pry
kaΩdom fyksyrovannom m1, m2 koneçnug normu u Qm b; , ; ;γ β α0 2 + [5, c. 83]
(teoremaL7.1). Ustanovym ocenku norm¥ u Qm b; , ; ;γ β α0 2 + .
Teorema,2. Esly v¥polnen¥ uslovyq a), b), to dlq reßenyq zadaçy (5) – (7)
spravedlyva ocenka
u Qm b; , ; ;γ β α0 2 + ≤ C b QmΨ ; , ; ;γ β α2
+ Φm b
D; ˜ , ˜; ;γ β
α
0
2 +
+ um 0 +
+
i
b
i i b rf r
i i=
− +∑
1
2; , ; ;γ β αΓ . (8)
Postoqnnaq C ne zavysyt ot m.
Dokazatel\stvo. Yspol\zuq opredelenye norm¥ y ynterpolqcyonn¥e ne-
ravenstva [4, c. 176], ymeem
u Qm b; , ; ;γ β α0 2 + ≤ ( ) ; , ; ;1 0 2+ [ ] +ε γ βα
αu Qm b + C um( )ε 0 . (9)
Poπtomu dostatoçno ocenyt\ u Qm b; , ; ;γ β α0 2[ ] + . Yz opredelenyq polunorm¥
u Qm b; , ; ;γ β α0 2[ ] + sleduet suwestvovanye v Q toçek P1, Hl, Nl, dlq kotor¥x
v¥polneno odno yz neravenstv
1
4
0 2u Qm b; , ; ;γ β α[ ] + ≤ E1 =
=
l
n
bj k b
l l ld k bj P x x
= + =
−∑ ∑ − + + −( ) −
1 2 2
1
1 22( , ) ( ); ˜ ( ) ( )γ β γ α γ β
α
×
× D D u P D D u Ht
j
x
k
m t
j
x
k
m l( ) ( )1 − , (10)
1
4
0 2u Qm b; , ; ;γ β α[ ] + ≤ E2 =
=
l
n
bj k b
b
d k bj P t t
= + =
−∑ ∑ − + +( ) −
1 2 2
2
1 2 2
2( , ) ( ) ; ˜ ( ) ( ) /
γ β α γ
α
×
× D D u H D D u Nt
j
x
k
m l t
j
x
k
m l( ) ( )− . (11)
Esly
x xl l
( ) ( )1 2− ≥ ρ γ βd Pl( ); ˜− 1 ≡ T1, ρ ∈ ( 0, 1 ) ,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
382 Y. D. PUKAL|SKYJ
to, yspol\zuq ynterpolqcyonn¥e neravenstva, ymeem
E1 ≤ ε γ β εα
αu Q c um b m; , ; ; ( )0 2 0[ ] ++ .
V¥byraq ε = 16 1− /α
yz neravenstva (10), naxodym
u Qm b; , ; ;γ β α0 2[ ] + ≤ c um 0 . (12)
Yspol\zuq ynterpolqcyonn¥e neravenstva y neravenstvo (11) v sluçae
t t( ) ( )1 2− ≥ ρ βγ2 2b
ld N( ); ˜ ≡ T2,
takΩe poluçaem (12).
Pust\ x x Tl l
( ) ( )1 2
1− ≤ yly t t T( ) ( )1 2
2− ≤ . Budem sçytat\, çto
min ( , ˜ ), ( , ˜ ){ }d P d Nlγ γ1 = d P( ; )γ 1 . PredpoloΩym, çto x( )ν ξ− ≥ 2 T1, ν ∈ { 1,
2 } , ξ ∈ ∂ D. Zapyßem zadaçu (5), (6) v vyde
D u a P D ut m
k b
k x
k
m−
=
∑
2
1( ) =
k b
k k x
k
ma P a P D u
=
∑ −[ ]
2
1( ) ( ) +
+
k b
k x
k
ma P D u
<
∑
2
( ) + Ψm ( t, x ) ≡ F1 ( t, x ) , (13)
um ( 0, x ) = Φm ( x ) . (14)
Pust\ V1 ∈ Q,
Vr = ( , ) , , , { , , }( ) ( )t x Q t t r T t x x rT l nb
l l∈ − ≤ ≥ − ≤ ∈ …{ }1 2
2
1
10 1 .
V zadaçe (13), (14) v¥polnym zamenu
um ( t, x ) = vm ( t, y ) ,
yl = d t d x xl l l1
1 1
2
2 1( , ) ( , )( ) ( ) ( ) ( )β β ≡ d P xl l( ; )β 1 ,
l ∈ { 1, … , n } .
Oblast\ opredelenyq vm ( t, y ) oboznaçym çerez Q1.
Tohda vm ( t, y ) budet reßenyem zadaçy
( L2 vm ) ( t, y ) ≡
D d k P a P D t yt
k b
k y
k
m−
=
∑
2
1 1(( , ); ) ( ) ( , )β v =
= F t d P y1
1
1( , ( ; ) )− β ≡ F2 ( t, Y ) ,
(15)
vm ( 0, y ) = Φm ( Y ) ,
hde Y ≡ ( )( ; ) , , (( ; )d P y d P yn n
− −…1
1 1 1
1
1β β .
Oboznaçym yl
( )1 = d P xl l( ; ) ( )β 1
1 , P t x1
1 1( )( ) ( ), = R t y1
1 1( )( ) ( ), ,
Nr
( )2 = ( , ) , , ( ; ), { , , }( ) ( )t y Q t t r T t y y r d P l nb
l l∈ − ≤ ≥ − ≤ ∈ …{ }1 2
2
1
10 1ρ γ ,
y voz\mem funkcyg η ( t, y ) , 2b + 1 raz dyfferencyruemug y udovletvorqg-
wug uslovyqm
η ( t, y ) =
1 0 1
0 2
1 4
2
3 4
2 1
1
, ( , ) , ( , ) ,
, ( , ) , ( , ) ; .
/
( )
/
( )
t y N t y
t y N D D t y C d b j k Pt
j
y
k
jk
∈ ≤ ≤
/∈ ≤ +( )( )
−
η
η γ
Tohda funkcyq ωm ( t, y ) = vm ( t, y ) η ( t, y ) budet reßenyem zadaçy Koßy
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KRAEVAQ ZADAÇA DLQ LYNEJNÁX PARABOLYÇESKYX URAVNENYJ … 383
( L2 ωm ) ( t, y ) =
k b
k
k
k y
k
y md k P a P C D D
= ≤ −
−∑ ∑
2
1 1
1
(( , ); ) ( )β η
λ
λ λ λ v +
+ vm tD F t Y t yη η+ 2( , ) ( , ) ≡ F3 ( t, y ) , (16)
ωm ( 0, y ) = Φm ( Y ) η ( 0, y ) ≡ ϕ1 ( y ) .
Zametym, çto v sylu uslovyq a) koπffycyent¥ uravnenyq (16) ohranyçen¥
postoqnn¥my, ne zavysqwymy ot toçky P1 . Poπtomu v sylu teorem¥ 4.1 yz [5,
c. 41] dlq proyzvol\n¥x toçek { M1, M2 } ⊂ N1 4
2
/
( )
v¥polnqetsq neravenstvo
d M M D D M D D Mt
j
z
k
m t
j
z
k
m
− −α ω ω( , ) ( ) ( )1 2 1 2 ≤
≤
C F C N C N tb3 1 03 4
2 2
3 4
2α αϕ( ) ( { })/
( )
/
( )+( )+ =∩ , (17)
hde d ( M1, M2 ) — parabolyçeskoe rasstoqnye meΩdu toçkamy M1 y M2, a 2bj +
+ k = 2b.
Uçyt¥vaq svojstva funkcyy η ( t, y ) y neravenstvo d ( γ ; M ) ≥ ρ γd P( ; )/1 4
dlq toçek M ∈ N3 4
2
/
( ) , ymeem
F C N3 3 4
α( )/
≤ cd b P− +( )( )1
12 α γ; ×
× F b N Nm b m2 3 4
2
3 4
2
2 00 2 0 0; , ; ; ; , ; ;/
( )
/
( )γ γ
α
+ +( )v v ,
(18)
ϕ α1 02
3 4C N tb+ =( ( ))/ ∩ ≤ cd b P− +( )( )1
12 α γ; Φm b
N t; ˜, ; ; { }/
( )γ
α
0 0 03 4
2
2
∩ =
+
.
Yz opredelenyq prostranstva C Qb2 0+α γ β( , ; ; ) sleduet v¥polnenye neravenstv
c Nm b1 3 4
2
2
0 0v ; , ; ; /
( )γ
α+
≤ u Vm b; , ; ; /γ β α0 3 4 2 + ≤
c Nm b2 3 4
2
2
0 0v ; , ; ; /
( )γ
α+
.
Podstavlqq (18) v (17) y vozvrawaqs\ k peremenn¥m ( t, x ) , poluçaem nera-
venstvo
Eν ≤ c F b V u V um b m1 3 4 3 4 2 02 0; , ; ; ; , ; ;/ /γ β γ βα + +( +
+ ϕ γ β
α2 3 4 2
0 0; ˜, ˜; ; { }/V t
b
∩ = )+
, ν ∈ { 1, 2 } . (19)
Ustanovym ocenku norm¥ F b V1 3 42; , ; ; /γ β α . Uçyt¥vaq ynterpolqcyonn¥e
neravenstva, dostatoçno ocenyt\ polunormu kaΩdoho slahaemoho v¥raΩenyq
F t x1( , ). Naprymer, dlq
a D u b Vk x
k
m; , ; ; /γ β
α
2 3 4[ ] ≡ T3
pry k ≤ 2b – 1 poluçaem
T3 ≤
l
n
B R V
l l l z
k
m
l
d k B z z D u B
= ⊂
−∑ − + −[{ −
1
1
1 2
1
1 3 4
sup (( , ) ( ); ˜ ) ( )
{ , }
( ) ( )
/
γ β α γ β
α
–
– D u R a B d b k B a B a Rz
k
m l k k k l( ) ( ) ( ( , ); ˜ ) ( ) ( )] − −[ ] + −[1 1 12 γ γ β ×
× z z d b k B d k B D u Bl l l z
k
m
( ) ( ) ( ( , ) ( ); ˜ ) (( , ); ˜ ) ( )1 2
1 12− − − + − ] −[ ]}−α
γ γ β α γ β γ β +
+
l
n
R B V
b
z
k
m l z
k
m l
l l
d k B D u B D u R
= ⊂
−∑ − + − −[ ]{
1
2
1 2 2
3 4
sup (( , ) ; ˜ ) ( ) ( )
{ , }
( ) ( ) /
/
γ β αγ τ τ
α
×
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384 Y. D. PUKAL|SKYJ
× a R d b k B a R a Bk l l k l k l
b
( ) ( ( , ); ˜ ) ( ) ( ) ( ) ( ) /
2 1 2 2
γ γ β τ τ
α
− −[ ] + − −[ −
×
× d b k B d k B D u Bl l z
k
m l( ( , ) ; ˜ ) (( , ); ˜ ) ( )2 γ γ β αγ γ β− − + ] −[ ]
≤
≤ c u V um k m; , ; ;γ β α0 1 0[ ] +( )+ .
Analohyçno ustanavlyvagtsq ocenky ostal\n¥x slahaem¥x v¥raΩenyq
F t x1( , ). Sledovatel\no,
F b V1 3 42; , ; ; /γ β α ≤ c u Vm bε γ β α1 1 20; , ; ; + + c u b Vm m0 12+( )Ψ ; , ; ;γ β α ,
(20)
hde ε1 = n b b2 2ρ εα+ — fyksyrovann¥e vewestvenn¥e çysla, ε ∈ ( 0, 1 ) ,
ρ ∈ ( 0, 1 ) .
Podstavlqq (20) v (19), naxodym
Eν ≤ c u V um b mε γ β α1 1 2 00; , ; ; + +( +
+ Ψ Φm m b
b V V t; , ; ; ; ˜, ˜; ; { }γ β γ βα α
2 0 01 1 2
+ = )+
∩ .
Otsgda, yspol\zuq neravenstva (10), (11) y v¥byraq ρ y ε dostatoçno mal¥my,
poluçaem
u Qm b; , ; ;γ β α0 2 + ≤ c b Q D um m b mΨ Φ; , ; ; ; ˜, ˜; ;γ β γ βα α
2 0
2 0+ +( )+
. (21)
Pust\ x( )ν ξ− ≤ 2 T1 , ξ ∈ ∂ D. Rassmotrym ßar K ( r, P ) radyusa r, r ≥
≥ 4 T1 , soderΩawyj toçky P1 , Hl , Nl , s centrom v nekotoroj toçke P ∈ Γ .
Yspol\zuq ohranyçenyq na hladkost\ poverxnosty ∂ D, moΩno rasprqmyt\
∂ D ∩ K ( r, P ) s pomow\g vzaymno odnoznaçnoho preobrazovanyq x = ψ ( y ) [6 ,
c. 126], v rezul\tate çeho oblast\ Π = Q ∩ K ( r, P ) perejdet v oblast\ Π1, dlq
toçek kotoroj yn ≥ 0, t ≥ 0. Esly poloΩyt\ um ( t, x ) = ζm ( t, y ), Pν ≡ Bν, ν ∈
∈ { 1, 2 } , Hl = Rl, d ( γ ; P1 ) ≡ h ( γ , B1 ) y koπffycyent¥ operatorov zadaçy (5)
– (7) pry πtom preobrazovanyy oboznaçyt\ çerez λ k ( t, y ) , l t yk
i( )( , ), to ζm v Π1
budet reßenyem zadaçy
D B D t yt
k b
k y
k
m−
=
∑
2
1λ ζ( ) ( , ) =
k b
k k y
k
mt y B D t y
=
∑ −[ ]
2
1λ λ ζ( , ) ( ) ( , ) +
+
k b
k y
k
m mt y D t y t y
<
∑ +
2
λ ζ ψ( , ) ( , ) ( , ( ))Ψ ≡ F4 ( t, y ) , (22)
ζm ( 0, y ) = Φm ( ψ ( y ) ) ≡ ϕ2 ( y ) , (23)
k r
k
i
y
k
m y
i
n
l B D t y
=
=∑ [ ]( )( ) ( , )1 0ζ =
k r
k
i
k
i
y
k
m
i
l t y l B D
=
∑ −[ ]
( ) ( )( , ) ( )1 ζ +
+
k r
k
i
y
k
m i
yi n
l t y D f t y
< =
∑ +
( )( , ) ( , ( ))ζ ψ
0
≡ g t ym
i
yn
( )( , )
=0
. (24)
V zadaçe (22) – (24) v¥polnym zamenu ζm ( t, y ) = ωm ( t, z ) , zl = h B yl l( ; )β 1 , l ∈
∈ { 1, … , n } . Oblast\ opredelenyq ωm ( t, z ) oboznaçym çerez Π2 . Tohda ωm ( t, z )
budet reßenyem zadaçy
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KRAEVAQ ZADAÇA DLQ LYNEJNÁX PARABOLYÇESKYX URAVNENYJ … 385
( L3 ωm ) ( t, z ) ≡ D h k B B D t zt
k b
k z
k
m−
=
∑
2
1 1(( , ); ) ( ) ( , )β λ ω = F4 ( t, Z ) , (25)
ωm ( 0, z ) = ϕ2 ( Z ) , (26)
( )( ) ( , )L t zi m
0 ω ≡
k r
k
i
z
k
m z
i
n
h k B l B D
=
=∑ [ ]( )( , ); ( )( )β ω1 1 0
= g t Zm
i
zn
( )( , )
=0
, (27)
hde Z ≡ ( )(( ; ) , (( ; ) , , (( ; )h B z h B z h B zn n
− − −…1
1 1 1
1
2 1 2
1
1β β β .
Vvedem oboznaçenyq
zl
( )1 = h B yl l( ; ) ( )β 1
1 ,
Nν
( )1 = ( , ) ( ; ),( )t z t t h b Bb b∈ − ≤{ Π2
1 2 2
12ν ρ γ
z z h B l n z tl l n− ≤ ∈ … ≥ ≥ }( ) ( ; ), { , , }, ,1
1 1 0 0νρ γ
y voz\mem funkcyg η1( , )t z , 2b + 1 raz dyfferencyruemug y udovletvorqg-
wug uslovyqm
η1( , )t z =
1 0 1
0 2
1 4
1
1
3 4
1
1
1
1
, ( , ) , ( , ) ,
, ( , ) , ( , ) ; .
/
( )
/
( )
t z N t z
t z N D D t z C h b j k Bt
j
z
k
jk
∈ ≤ ≤
/∈ ≤ +( )( )
−
η
η γ
Tohda funkcyq w t zm( , ) = η ω1( , ) ( , )t z t zm budet reßenyem kraevoj zadaçy
( L3 wm ) ( t, z ) =
k b
k
p k
k
p
z
k p
z
p
mh k B B C D D
= ≤ −
−∑ ∑
2
1 1
1
1(( , ); ) ( )β λ η ω +
+ ω η ηm tD F t Z z1 4 1 0+ ( , ) ( , ) ≡ F5 ( t, z ) ,
w zm( , )0 = ϕ η2 1 0( ) ( , )Z z ≡ ϕ3( )z ,
(28)
( )( ) ( , )L w t zi m zn
0
0=
=
k r
k
i
p k
k
p
z
k p
z
p
m
i
h k B l B C D D
= <
−∑ ∑
(( , ); ) ( )( )β η ω1 1 1 +
+ g t Z t zm
i
zn
( )( , ) ( , )η1
0
=
≡ Gm
i( ) .
Koπffycyent¥ uravnenyq y kraev¥x uslovyj zadaçy (28), sohlasno vvedenn¥m
predpoloΩenyqm, ohranyçen¥ postoqnn¥my, ne zavysqwymy ot toçky B1 . Po-
πtomu sohlasno teoremeL7.1 yz [5, c. 83] dlq proyzvol\n¥x toçek { M1 , M2 } ⊂
⊂ N1 4
1
/
( )
v¥polnqetsq neravenstvo
d M M D D M D D Mj
z
k
m
j
z
k
m
− −α
τ τ( , ) ( ) ( )1 2 1 2v v ≤
≤
C F GC N C N t
i
b
m
i
c N t
b b ri i3 3 0
1
03 4
1 2
3 4
1 2
3 4
1α α αϕ( ) ( { })
( )
( { })/
( )
/
( )
/
( )+ +
+ − +=
= =∑∩ ∩
. (29)
Povtorqq rassuΩdenyq, pryvedenn¥e pry dokazatel\stve ocenky (21), y uçyt¥-
vaq (12), poluçaem neravenstvo (8).
Ustanovym ocenku norm¥ um 0. Dlq πtoho dokaΩem sledugwug teoremu.
Teorema,3. Pust\ um — edynstvennoe klassyçeskoe reßenye zadaçy (5) –
(7) y v¥polnen¥ uslovyq a), b). Tohda dlq um v¥polnqetsq neravenstvo
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386 Y. D. PUKAL|SKYJ
um 0 ≤ C L u b Q u Dm m b1 2
2 0; , ; ; ; ˜, ˜; ;γ β γ βα α
+
+
+
+
i
b
i m i b r
L u r
i i= − +∑
1
0
2
( ) ; , ; ;γ β
α
Γ . (30)
Postoqnnaq C ne zavysyt ot m.
Dokazatel\stvo. Vospol\zuemsq rassuΩdenyqmy, yspol\zovann¥my pry
dokazatel\stve zameçanyqL2 yz [7, c. 75]. PredpoloΩym, çto neravenstvo (30) ne
v¥polnqetsq. Tohda suwestvuet posledovatel\nost\ funkcyj V n ∈ C Qb2 +α( )
takyx, çto Vn 0 = 1, y V xn( , )0 , L Vn1 , L Vi n
( )0 , stremqtsq k nulg dlq sootvet-
stvugwyx Vn , kohda n → ∞ . Yz (8) sleduet, çto norm¥ V Qn b; , ; ;γ β α0 2 +
ravnomerno ohranyçen¥. Poπtomu suwestvuet podposledovatel\nost\ Vn j( ) ,
kotoraq pry j → ∞ sxodytsq k reßenyg V ∈ C Qb2 +α( ) odnorodnoj kraevoj
zadaçy. Poskol\ku reßenye kraevoj zadaçy (5) – (7) edynstvennoe, to V = 0,
çto protyvoreçyt ravenstvu V 0 = 1.
Dokazatel\stvo teorem¥,1. Otmetym, çto
Ψm b Q; , ; ;γ β α2 ≤ c f b Q0 2; , ; ;γ β α ,
Φm b
D; ˜, ˜; ;γ β
α
0
2 +
≤ c D
b
ϕ γ β
α
; ˜, ˜; ;0
2 +
.
Yz πtyx sootnoßenyj y neravenstv (30), (8) sleduet ocenka
u Qm b; , ; ;γ β α0 2 + ≤ c f b Q Dm b0 2
2 0; , ; ; ; ˜, ˜; ;γ β ϕ γ βα α
+
+
+
+
i
b
i i b rf r
i i=
− +∑
1
2; , ; ;γ β αΓ . (31)
Sledovatel\no, pravaq çast\ neravenstva (31) ne zavysyt ot m1 , m2 , a po-
sledovatel\nosty
{ }( )Wkj
m = d bj k P D D u P P Qt
j
x
k
m( ( , ); ) ( ) ,2 γ γ β+ − ∈{ } , 2bj k+ ≤ 2b,
ravnomerno ohranyçen¥ y ravnostepenno neprer¥vn¥. Sohlasno teoreme Arçe-
la suwestvugt podposledovatel\nosty { }( ( )),W lkj
m l ≥ 1 , ravnomerno sxodqwye-
sq pry l → ∞ k Wkj . Perexodq k predelu pry l → ∞ v zadaçe (5) – (7), poluça-
em, çto u = W00 — edynstvennoe reßenye zadaçy (1) – (3), u ∈ C Qb2 0+α γ β( , ; ; )
y spravedlyva ocenka (4).
Teorema,4. PredpoloΩym, çto v¥polnen¥ uslovyq a), b), f0 ∈ C Qα γ β( , ; ; )0 ,
fi ∈ C b ri i2 0− +α γ β( , ; ; )Γ . Tohda edynstvennoe reßenye zadaçy (1) – (3) v pro-
stranstve C Qb2 0+α γ β( , ; ; ) opredelqetsq yntehralamy Styl\t\esa s bore-
levskoj meroj
u ( t, x ) =
Q D
t x d d f t x d∫ ∫+Γ Γ1 0 2( , ; , ) ( , ) ( , ; ) ( )τ ξ τ ξ ξ ϕ ξ +
+
i
b
i it x d d S f
=
∑ ∫
1 Γ
Γ ( , ; , ) ( , )τ τ ξξ . (32)
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KRAEVAQ ZADAÇA DLQ LYNEJNÁX PARABOLYÇESKYX URAVNENYJ … 387
Dokazatel\stvo. V sylu toho, çto C Qα γ β( , ; ; )0 ⊂ C b Qα γ β( , ; ; )2 y
C b ri i2 0− +α γ β( , ; ; )Γ ⊂ C rb r
i
i i2 − +α γ β( , ; ; )Γ , dlq f0 ∈ C Qα γ β( , ; ; )0 y f i ∈
∈ C b ri i2 0− +α γ β( , ; ; )Γ v¥polnqgtsq neravenstva
f b Q0 2; , ; ;γ β α ≤ C f Q0 0; , ; ;γ β α ,
f ri i b ri i
; , ; ;γ β αΓ 2 − + ≤ C fi b ri i
; , ; ;γ β α0 2Γ − + .
Poπtomu s uçetom teorem¥L1 dlq reßenyq zadaçy (1) – (3) ymeet mesto ocenka
u Q b; , ; ;γ β α0 2 + ≤ C f Q D
b0 2
0 0; , ; ; ; ˜, ˜; ;γ β ϕ γ βα α
+
+
+
+
i
b
i b rf
i i=
− +∑
1
20; , ; ;γ β αΓ . (33)
Budem rassmatryvat\ u ( t, x ) pry fyksyrovannom ( t, x ) kak lynejn¥j nepre-
r¥vn¥j funkcyonal na normyrovannom prostranstve
Cα ≡ C Q C D Cb b rα α αγ β γ β γ β( , ; ; ) (˜, ˜; ; ) ( , ; ; )0 0 02 2 1 1× ×+ − + Γ × …
… × C b rb b2 0− +α γ β( , ; ; )Γ
s normoj, ravnoj pravoj çasty neravenstva (33). V sylu vloΩenyq Cα ⊂ C y
teorem¥ Ryssa moΩno sçytat\, çto u ( t, x ) poroΩdaet borelevskug meru Γ ( t, x,
Z ) , kotoraq opredelena na σ-alhebre podmnoΩestv Ω oblasty Q , vklgçaq
Q y vse ee otkr¥t¥e podmnoΩestva takye, çto znaçenyq funkcyonala oprede-
lqgtsq formuloj (32).
1. Landau L. D., Lyfßyc E. M. Kvantovaq mexanyka. – M.: Fyzmathyz, 1963. – 702 s.
2. Pukal\s\kyj I. D. Odnostoronnq nelokal\na krajova zadaça dlq synhulqrnyx paraboliç-
nyx rivnqn\ // Ukr. mat. Ωurn. – 2001. – 53, # 11. – S. 1521 – 1531.
3. Pukal\skyj Y. D. Zadaça s kosoj proyzvodnoj dlq neravnomerno parabolyçeskoho uravne-
nyq // Dyferenc. uravnenyq. – 2001. – 37, # 12. – S. 1521 – 1531.
4. ∏jdel\man S. D. Parabolyçeskye system¥. – M.: Nauka, 1964. – 445 s.
5. Matijçuk M. I. Paraboliçni synhulqrni krajovi zadaçi. – Ky]v: In-t matematyky NAN Ukra-
]ny, 1999. – 176 s.
6. Kam¥nyn L. Y., Maslennykova V. N. Hranyçn¥e ocenky ßauderovskoho typa reßenyq zadaçy
s kosoj proyzvodnoj dlq parabolyçeskoho uravnenyq v necylyndryçeskoj oblasty // Syb.
mat. Ωurn. – 1966. – 7, # 1. – S. 83 – 128.
7. Ahmon S., Duhlys A., Nyrenberh L. Ocenky reßenyj πllyptyçeskyx uravnenyj vblyzy hra-
nyc¥. – M.: Yzd-vo ynostr. lyt., 1962. – 208 s.
Poluçeno 17.02.2004
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
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| id | umjimathkievua-article-3605 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:45:39Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/70/9ff67476e3cb65cd4d23f02ae57b7b70.pdf |
| spelling | umjimathkievua-article-36052020-03-18T19:59:42Z Boundary-Value Problem for Linear Parabolic Equations with Degeneracies Краевая задача для линейных параболических уравнений с вырождениями Pukalskyi, I. D. Пукальский, И. Д. Пукальский, И. Д. In spaces of classical functions with power weight, we prove the correct solvability of a boundary-value problem for parabolic equations with an arbitrary power order of degeneracy of coefficients with respect to both time and space variables. У просторах класичних функцій зі степеневою вагою доведено коректну розв'язність крайової задачі для параболічних рівнянь з довільним степеневим порядком виродження коефіцієнтів як за часовою, так і за просторовими змінними. Institute of Mathematics, NAS of Ukraine 2005-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3605 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 3 (2005); 377–387 Український математичний журнал; Том 57 № 3 (2005); 377–387 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3605/3941 https://umj.imath.kiev.ua/index.php/umj/article/view/3605/3942 Copyright (c) 2005 Pukalskyi I. D. |
| spellingShingle | Pukalskyi, I. D. Пукальский, И. Д. Пукальский, И. Д. Boundary-Value Problem for Linear Parabolic Equations with Degeneracies |
| title | Boundary-Value Problem for Linear Parabolic Equations with Degeneracies |
| title_alt | Краевая задача для линейных параболических уравнений с вырождениями |
| title_full | Boundary-Value Problem for Linear Parabolic Equations with Degeneracies |
| title_fullStr | Boundary-Value Problem for Linear Parabolic Equations with Degeneracies |
| title_full_unstemmed | Boundary-Value Problem for Linear Parabolic Equations with Degeneracies |
| title_short | Boundary-Value Problem for Linear Parabolic Equations with Degeneracies |
| title_sort | boundary-value problem for linear parabolic equations with degeneracies |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3605 |
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