Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation
We prove a statement on the averaging of a hyperbolic initial-boundary-value problem in which the coefficient of the Laplace operator depends on the space $L^2$-norm of the gradient of the solution. The existence of the solution of this problem was studied by Pokhozhaev. In a space domain in $ℝ^n,\;...
Gespeichert in:
| Datum: | 2006 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2006
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3449 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509541850939392 |
|---|---|
| author | Sidenko, N. R. Сиденко, Н. Р. Сиденко, Н. Р. |
| author_facet | Sidenko, N. R. Сиденко, Н. Р. Сиденко, Н. Р. |
| author_sort | Sidenko, N. R. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:54:47Z |
| description | We prove a statement on the averaging of a hyperbolic initial-boundary-value problem in which the coefficient of the Laplace operator depends on the space $L^2$-norm of the gradient of the solution. The existence of the solution of this problem was studied by Pokhozhaev. In a space domain in $ℝ^n,\; n ≥ 3$, we consider an arbitrary perforation whose asymptotic behavior in a sense of capacities is described by the Cioranesku-Murat hypothesis. The possibility of averaging is proved under the assumption of certain additional smoothness of the solutions of the limiting hyperbolic problem with a certain stationary capacitory potential. |
| first_indexed | 2026-03-24T02:42:45Z |
| format | Article |
| fulltext |
UDK 517.946
N. R. Sydenko (Yn-t matematyky NAN Ukrayn¥, Kyev)
USREDNENYE ZADAÇY DYRYXLE DLQ SPECYAL|NOHO
HYPERBOLYÇESKOHO URAVNENYQ KYRXHOFA
We prove assertion on homogenization of a hyperbolic initial boundary-value problem, in which the
coefficient of the Laplace operator depends on the space L2-norm of a solution gradient. The problem
of the existence of a solution of this problem is investigated by S. I. Pokhozhaev. In the spatial domain
in R
n, n ≥ 3, we consider an arbitrary perforation whose asymptotic behaviour in the capacity sense is
described by the D. Cioranesku – F. Murat hypothesis. The possibility of the homogenization is proved
under the assumption that solutions of the limit boundary-value hyperbolic problem with the capacity
stationary potential possess some additional smoothness.
Dovedeno tverdΩennq pro userednennq hiperboliçno] poçatkovo-krajovo] zadaçi, u qkij koefi-
ci[nt pry operatori Laplasa zaleΩyt\ vid prostorovo] L 2-normy hradi[nta rozv’qzku. Pytannq
isnuvannq rozv’qzku ci[] zadaçi doslidΩene S. I. PoxoΩa[vym. U prostorovij oblasti v R
n,
n ≥ 3, rozhlqda[t\sq dovil\na perforaciq, asymptotyçna povedinka qko] v [mnisnomu sensi
opysana hipotezog D. Çioranesku – F. Mgra. MoΩlyvist\ userednennq dovedeno za prypuwen-
nqm deqko] dodatkovo] hladkosti rozv’qzkiv hranyçno] hiperboliçno] zadaçi z pevnym [mnisnym
stacionarnym potencialom.
Hyperbolyçeskym uravnenyem Kyrxhofa naz¥vaetsq uravnenye vyda
u x t a u t u x ttt L( , ) ( , ) ( , )( )− ∇ ⋅( )2
2
Ω ∆ = f ( x, t ) , x n∈ ⊂Ω R , t ∈ R, (1)
hde x = ( x1, … , xn ) , ∇ = ( / / ), ,∂ ∂ … ∂ ∂x xn1 , ∆ = ∂ ∂=∑ 2 2
1
/ xii
n
, s poloΩytel\noj
neprer¥vnoj funkcyej a : R R
+ +→ . Dlq πtoho uravnenyq rassmatryvalas\
[1, 2] naçal\no-kraevaq zadaça Dyryxle v cylyndre Q = Ω × ( , )0 T , hde Ω —
ohranyçennaq oblast\ v R
n, n ≥ 3, s hranycej ∂Ω , T — lgboe fyksyrovannoe
poloΩytel\noe çyslo, s hranyçn¥my uslovyqmy
u t( , )⋅ ∂Ω = 0, t T∈ ( , )0 , u ( x, 0 ) = ϕ ( x ) , ut ( x, 0 ) = ψ ( x ) , x ∈ Ω . (2)
Dlq proyzvol\noj funkcyy a ( t ) , udovletvorqgwej uslovyqm
a t C( ) ( )∈ +1
R , a ( t ) ≥ α0 > 0 ∀ ∈ +t R , α0 = const, (3)
razreßymost\ zadaçy v celom ustanovlena v [1] dlq specyal\noho klassa besko-
neçno dyfferencyruem¥x zadann¥x funkcyj f , ϕ, ψ y ∂Ω klassa C
∞. V ra-
bote [2] dlq konkretnoj funkcyy
a ( t ) = a0 ( t ) : = ( )C t C1 2
2+ − , Ci = const > 0, i = 1, 2, (4)
ustanovlena razreßymost\ zadaçy v celom dlq klassa dann¥x, ymegwyx summy-
ruem¥e s kvadratom proyzvodn¥e do vtoroho porqdka vklgçytel\no, y hladkoj
hranyc¥ ∂Ω klassa C
2. Pry πtom okaz¥vaetsq [3], çto funkcyq (4) qvlqetsq
edynstvennoj v klasse
a C∈ +2( )R , a ≥ 0, (5)
pry kotoroj zadaça (1), (2) razreßyma v celom vo mnoΩestve dann¥x, ymegwyx
lyß\ proyzvodn¥e do vtoroho porqdka, summyruem¥e s kvadratom.
V dannoj rabote rassmatryvaetsq ohranyçennaq oblast\ Ω ⊂ R
n
y oblast\
Ω( )s ⊂ Ω s mnohosvqznoj hranycej
© N. R. SYDENKO, 2006
236 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
USREDNENYE ZADAÇY DYRYXLE DLQ SPECYAL|NOHO … 237
∂Ω( )s =
∂
∂
=
Ω Ωi
s
i
N s
( )
( )
1
∪ ∪ ,
naz¥vaemaq perforyrovannoj oblast\g v Ω . Zdes\ s ∈ N — parametr, N s( )
—
peremennoe çyslo mnoΩestv Ωi
s( ) ⊂ Ω , qvlqgwyxsq zam¥kanyqmy oblastej
˙ ( )Ωi
s
s hladkymy odnosvqzn¥my hranycamy ∂Ωi
s( )
razmernosty n – 1, pryçem
Ω Ωi
s
j
s( ) ( )∩ = ∅, i ≠ j, Ω Ωi
s( ) ∩ ∂ = ∅, i = 1, ( )N s ,
F s( ) =
Ωi
s
i
N s
( )
( )
=1
∪ , Ω( )s = Ω \ ( )F s .
Oboznaçaq Q s( ) = Ω( ) ( , )s T× 0 , hde T = const > 0,
v( )s = v( )
( )( )
s
L s2 Ω
yly v = v L2 ( )Ω ,
çto budet qsno po sm¥slu, rassmotrym na Q s( )
zadaçu vyda (1), (2), (4) dlq ve-
westvennoj funkcyy u ts( )( ) = u x ts( )( , ) :
u x t a u t u x ttt
s s s( ) ( ) ( )( , ) ( , ) ( , )− ∇ ⋅( )0
2
∆ = f x ts( )( , ) , ( , ) ( )x t Q s∈ ,
(6)
u ts
s
( )( , ) ( )⋅
∂Ω
= 0, t T∈ ( , )0 , u xs( )( , )0 = ϕ( )( )s x , u xt
s( )( , )0 = ψ( )( )s x , x s∈ Ω( ),
y ukaΩem uslovyq, pry kotor¥x obespeçyvaetsq sxodymost\ reßenyq u s( )
pry
s → ∞ k nekotoroj predel\noj funkcyy u ( x, t ) , opredelennoj na Q , kohda
perforacyq F s( )
beskoneçno yzmel\çaetsq y uplotnqetsq.
Dlq rassmatryvaemoho obæekta — perforyrovannoj oblasty — ne xarakter-
na hladkost\ hranyc perforacyy ∂Ω( )s . Poπtomu, nemnoho perefrazyruq rabo-
tu [2], budem predpolahat\, çto ∂Ω( )s
qvlqetsq lypßycevoj y v¥polnen¥ us-
lovyq
ϕ
( ) ( ) ( )( ), ( ) ( )s s sD L H∈ ∆ Ω Ω2 1∩
�
, ψ
( ) ( )( )s sH∈
�
1 Ω ,
(7)
f L T D L Hs s s( ) ( ) ( )( ( )), ; , ( ) ( )∈ 2 2 10 ∆ Ω Ω∩
�
,
v kotor¥x
D L s( ), ( )( )∆ Ω2 = v v∈ ∈{ }L Ls s2 2( ) : ( )( ) ( )Ω ∆ Ω ,
pryçem norm¥
f fs
L Q
s
L Qs s
( )
( )
( )
( )( ) ( )2 2+ ∆ , ∇ +ϕ ϕ( ) ( )s s∆ , ∇ψ( )s ≤ K0 ∀ s (8)
ohranyçen¥ ravnomerno otnosytel\no s. Tohda zadaça (6), kak y v [2], ymeet
edynstvennoe reßenye s takymy svojstvamy:
u C T Hs s( ) ( )( )[ , ]; ( )∈ 0 1
�
Ω , u L T Ht
s s( ) ( )( ), ; ( )∈ ∞ 0 1
�
Ω ,
(9)
∆ Ωu L T Ls s( ) ( )( ), ; ( )∈ ∞ 0 2 , u L Qtt
s s( ) ( )( )∈ 2 ,
pryçem v¥polnqgtsq ocenky
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
238 N. R. SYDENKO
max ( )
[ , ]
( )
t T t
su t
∈ 0
2
≤ K1 , max ( )
[ , ]
( )
t T
su t
∈
∇
0
2
≤ K2 ,
(10)
ess sup
t T
t
su t
∈
∇
( , )
( )( )
0
2
≤ C K2
1
2
− , ess sup
t T
su t
∈( , )
( )( )
0
2
∆ ≤ K C K C2 1 2 2( )+ ,
hde postoqnn¥e K1 , K2 ne zavysqt ot peremennoj s ∈ N , a zavysqt tol\ko ot
C1 , C2 , T, K0 .
Opyßem rehulqrnost\ povedenyq oblasty Ω( )s
pry s → ∞ . Dlq πtoho v¥-
berem sledugwug hypotezu D. Çyoranesku – F. Mgra [4, 5] o malosty y kom-
paktnosty raspoloΩenyq otverstyj F s( ) .
Hypoteza (A): pust\ suwestvuet posledovatel\nost\ vewestvenn¥x funkcyj
w xs( ), s ∈ N , so svojstvamy:
1) w x H Ls( ) ( ) ( )∈ ∞1 Ω Ω∩ ;
2) w xs( ) = 0 , x F s∈ ( )
;
3) w xs( ) → 1 slabo v H1( )Ω , slabo* v L∞( )Ω y dlq poçty vsex x ∈ Ω pry
s → ∞ ;
4) – ∆w xs( ) = µ γs sx x( ) ( )− , hde µs , γ s H∈ −1( )Ω , pryçem µ µs → syl\no
v H −1( )Ω pry s → ∞ , ( , )γ s v Ω = 0 dlq lgboj v ∈
�
H1( )Ω takoj, çto v = 0
na F s( ) .
Zdes\ y nyΩe oboznaçeno
( , )u v Ω =
u x x dx( ) ( )v
Ω
∫ .
Sledstvye [5]. Pry uslovyqx 1 – 4 ymeem
5) 0
1 1≤ ∈ −µ ( ) ( ) ( )x H LΩ Ω∩ , tak çto µ ( x ) poroΩdaet koneçnug radono-
vu meru na Ω .
Zameçanye�1. Netrudno dokazat\, çto uslovyq B1 , B2 , C yz monohrafyy
[6] (hl.Q9) y rabot¥ [7] s plotnost\g mer¥ ν( ) ( )x Lr∈ Ω , r n> /2 , sformuly-
rovann¥e dlq operatora Laplasa na H1( )Ω , dostatoçn¥ dlq v¥polnenyq
hypotez¥ (A) s funkcyej µ ∈C( )Ω y µ ∈ Lr ( )Ω , r n> /2 , sootvetstvenno y
w xs( ) ≥ 0.
Oboznaçym çerez ̂ ( )( )v s x prodolΩenye na Ω funkcyy v
( )( )s x , zadannoj na
Ω( )s , putem doopredelenyq ee nulem na F s( ) . Prymem takΩe sledugwye sokra-
wenn¥e oboznaçenyq dlq otnoßenyj dvojstvennosty vektorn¥x funkcyj
�
u x t( , ) = ( )( , ), , ( , )u x t u x tn1 … :
〈 〉
� �
u s, ( )v =
� �
u x t x t dx dt
Q s
( , ) ( , )
( )
⋅∫ v , 〈 〉
� �
u, v =
� �
u x t x t dx dt
Q
( , ) ( , )⋅∫ v
y analohyçn¥e oboznaçenyq dlq otnoßenyj dvojstvennosty skalqrn¥x funk-
cyj, a takΩe oboznaçym
N ts( ) = ∇ ⋅u ts( )( , ) .
UmnoΩaq uravnenye (6) na ws v, hde v ∈ ∞C Q0 ( ), y yntehryruq zatem po
Q s( ) , poluçaem yntehral\noe toΩdestvo
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
USREDNENYE ZADAÇY DYRYXLE DLQ SPECYAL|NOHO … 239
〈 〉f ws
s
s( ) ( ), v = 〈 〉 − 〈 〉u w a N u wtt
s
s
s
s
s
s
s( ) ( ) ( ) ( ), ,( )v v0
2 ∆ –
– 2 0
2
0
2〈 ∇ ⋅∇ 〉 + 〈 − 〉a N u w a N u ws
s
s
s
s
s
s
s( ) ( )( ) ( ) ( ) ( ), , ( )v v ∆ ,
yly, s uçetom svojstva 4, toΩdestvo dlq prodolΩenyj
〈 〉ˆ ,( )f ws
sv = 〈 〉 − 〈 〉ˆ , ˆ ,( ) ( )( )u w a N u wtt
s
s s
s
sv v0
2 ∆ –
– 2 0
2
0
2〈 ∇ ∇ 〉 + 〈 〉a N u w a N us
s
s s
s
s( ) ( )ˆ , ˆ ,( ) ( )v v µ . (11)
S cel\g usrednenyq (11) predpoloΩym dopolnytel\no k (8), çto pry s → ∞
ymegt mesto sxodymosty
ˆ( )f s → f slabo v L Q2( ),
(12)
ˆ ( )ϕ s → ϕ, ˆ ( )ψ s → ψ slabo v
�
H1( )Ω .
Pry πtom uçyt¥vaem, çto vsledstvye (10), (8) dlq ˆ( )u s
spravedlyv¥ takye
ocenky:
max ˆ ( )
[ , ]
( )
t T
t
su t
∈ 0
2
≤ K1 , max ˆ ( )
[ , ]
( )
t T
su t
∈
∇
0
2
≤ K2 ,
(10 ′ )
ess sup
t T
t
su t
∈
∇
( , )
( )ˆ ( )
0
2
≤ C K2
1
2
− , ˆ( )
( )
utt
s
L Q2 ≤ C TK C K C K2
2
2 1 2 2
1 2
0
− + +[ ]( ) / .
Krome toho, yz neravenstva
u ts( )( ) ≤ ϕ τ τ( ) ( )( )s
t
s
t
u d+ ∫
0
, t T∈ [ , ]0 ,
y ocenok (8), (10) sleduet ocenka
max ˆ ( )
[ , ]
( )
t T
su t
∈ 0
2
≤ K3 , (13)
hde K3 zavysyt tol\ko ot C1 , C2 , T, K0 . Poπtomu moΩno v¥brat\ yz N ta-
kug posledovatel\nost\, oboznaçaemug { s } , çto ymegt mesto sxodymosty
ˆ( )u s → u slabo v H Q1( ) y syl\no v L Q2( ),
∇ ˆ( )ut
s → ∇ut slabo* v L T L∞ ( ), ; ( )0 2 Ω , (14)
ˆ( )utt
s → utt slabo v L Q2( ).
Pry πtom ymeem takΩe
ˆ( )u s → slabo* v W T H∞
1 10( ), ; ( )
�
Ω . (15)
Yz (15) y tret\ej sxodymosty v (14) sleduet [5] sxodymost\
ˆ( )u s → u v C T Hsc
1 10( )[ , ]; ( )
�
Ω , (16)
hde dlq banaxova prostranstva V oboznaçaem çerez C T Vsc ([ , ]; )0 prostranstvo
skalqrno neprer¥vn¥x funkcyj yz [ 0, T ] v V [8] (hl.Q3, 8.4). Znaçyt, predel\-
naq funkcyq u ( x, t ) udovletvorqet naçal\n¥m uslovyqm
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
240 N. R. SYDENKO
u t =0 = ϕ, ut t =0 = ψ . (17)
Pust\ ustanovleno (πto poka hypoteza), çto pry nalyçyy (14)
N ts( ) → N t( ) syl\no v C ( [ 0, T ] ) . (18)
Tohda s uçetom (14), (18), svojstv 3, 4 y sxodymostej
ws v → v, ws ∆ v → ∆ v syl\no v L Q2( )
perexodym k predelu v ravenstve (11) po v¥brannoj posledovatel\nosty { s } , v
rezul\tate çeho poluçaem
〈 f, v 〉 = 〈 〉 − 〈 〉 + 〈 〉u a N u a N utt, ( ) , ( ) ,v v v0
2
0
2∆ µ ,
yly v dyfferencyal\noj forme s uçetom (17)
u x t a N t u x t a N t x u x ttt( , ) ( ( )) ( , ) ( ( )) ( ) ( , )− +0
2
0
2∆ µ = f ( x, t ) , ( x, t ) ∈ Q ,
(19)
u t( , )⋅ ∂Ω = 0, t ∈ ( 0, T ) , u ( x, 0 ) = ϕ ( x ) , ut ( x, 0 ) = ψ ( x ) , x ∈ Ω .
Yz uravnenyq (19) vydno, çto veroqtno opredelenye
N t2( ) = u t V( ) 2 : = ∇ +u t u t L dx( ) ( ) ( ; )
2 2
2 Ω µ ,
(20)
V =
�
∩H L dx1 2( ) ( ; )Ω Ω µ .
Zameçanye�2. V sluçae uslovyj zameçanyq 1 ymeem lybo µ ∈C( )Ω , lybo
µ ∈ ∞L ( )Ω , esly ν( ) ( )x L∈ ∞ Ω . Pry πtom L L dx2 2( ) ( ; )Ω Ω⊂ µ , V =
�
H1( )Ω .
V posledugwem m¥ dokaΩem, çto predpoloΩenyq (18), (20) vern¥ y zadaça
(19) dejstvytel\no qvlqetsq predel\noj dlq zadaçy (6) v tom sm¥sle, çto yme-
gt mesto sxodymosty (14), (15) reßenyj (6) k predel\noj funkcyy u ( x, t ) , qv-
lqgwejsq reßenyem zadaçy (19). Toçnee, spravedlyvo takoe utverΩdenye.
Teorema. PredpoloΩym, çto hranyca ∂Ω( )s
lypßyceva, v¥polnen¥ hypote-
za (A) s predel\noj funkcyej µ ( ) ( )x L∈ 2 Ω y uslovyq (7), (8). Pust\ pry
s → ∞ ymegt mesto sxodymosty (12) y sledugwye:
ˆ( )
( , ; ( ))
f fs
L T L
− 1 20 Ω
→ 0, ∇ −( ˆ )( )ϕ ϕs
sw → 0. (21)
PredpoloΩym takΩe, çto lgboe reßenye zadaçy (19), (20) ymeet sledugwye
svojstva:
u C T W Ln∈
∞[ , ]; ( ) ( )0 1
�
∩Ω Ω , ∇ ∈ ( )∞u L T L1 0, ; ( )Ω , ∆ Ωu L T L∈ ( )∞ 0 2, ; ( ) ,
∂2u = ∂ ∂ ∂ =( )2 1u x x i j ni j/ : , , ∈ L T Ln1 0, ; ( )Ω( ), (22)
u L T H Lt ∈
∞ ∞0 1, ; ( ) ( )
�
∩Ω Ω , ∇ ∈ ( )u L T Lt
n1 0, ; ( )Ω .
Tohda dlq polnoj posledovatel\nosty s ∈ N ymegt mesto sxodymosty (14) –
(16), (18), (20) y dopolnytel\n¥e sxodymosty (43) k reßenyg zadaçy (19), (20),
edynstvennomu v klasse (22).
Dokazatel\stvo. Otmetym, çto vsledstvye (10 ′ ) v¥polnqgtsq neravenstva
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
USREDNENYE ZADAÇY DYRYXLE DLQ SPECYAL|NOHO … 241
0 < k0 ≤ a N ts0
2( ( )) ≤ C2
2− , t ∈ [ 0, T ] , s ∈ N , k0 = ( )C K C1 2 2
2+ − . (23)
Pust\ u ( x, t ) — reßenye zadaçy (19), (20). Opredelym vspomohatel\nug funk-
cyg
˜ ( , )u x ts = w x u x ts( ) ( , ).
Dlq nee ymeem sledugwye ravenstva (v dal\nejßem oboznaçaem v ′ = ∂ ∂v/ t ,
v ″ = ∂ ∂2 2v/ t ):
˜ ( ) ˜′′ −u a N us s s0
2 ∆ = w u a N w u w u us s s s s s′′ − + ∇ ⋅∇ + −0
2 2( )( ( ))∆ γ µ =
= w u a N u a N a N w u a N w u us s s s s s s( ( ) ) [ ( ) ( )] ( )[ ( )]′′ − + − − ∇ ⋅∇ + −0
2
0
2
0
2
0
2 2∆ ∆ γ µ =
= w f a N u a N a N w u a N u w us s s s s s s( ( ) ) [ ( ) ( )] ( )[ ( )]− + − − ∇ ⋅∇ + −0
2
0
2
0
2
0
2 2µ γ µ∆ .
Dlq funkcyy
vs = u us
s
( ) ˜−
ymeem uravnenye
′′ −v vs s sa N0
2( )∆ = f w f a N uws
s s
( ) ( )− + 0
2 µ +
+ [ ( ) ( )] ( ) ( ) ( )a N a N w u a N u w a N us s s s s s s0
2
0
2
0
2
0
22− + ∇ ⋅∇ + −∆ γ µ .
UmnoΩaq πto uravnenye na 2 ′vs y zatem yntehryruq proyzvedenye po cylyndru
Ω( ) ( , )s t× 0 s uçetom ravenstva
2 0
2a Ns s s s( )( , ) ( )∇ ∇ ′v v Ω =
d
dt
a N t ts s0
2 2( ( )) ( )∇[ ]v –
–
∇ ′ ∇ ∇( )vs s
s
t
st a N t u t u t s( ) ( ( )) ( ), ( )( ) ( )
( )
2
0
2 2
Ω
,
dlq prodolΩenyq ̂ ˆ ˜( )vs
s
su u= − poluçaem neravenstvo
ˆ ( ) ( ( )) ˆ ( )′ + ∇v vs s st a N t t2
0
2 2 ≤ ˆ ( ( )) ( ˆ )( ) ( )ψ ψ ϕ ϕs
s s
s
sw a N w− + ∇ −
2
0
2 2
0 +
+
2 2
0
0
2 2
01 2
t
s
s
t
s
s
s
s L t L
a N u u d f w f∫ ′ ∇ ∇( ) ∇ + −( ( )) ˆ ( ), ˆ ( ) ˆ ( ) ˆ( ) ( ) ( )
( , ; ( ))
τ τ τ τ τ
Ω Ω
v ×
× ˆ ( ) ( ) ( ) ˆ ( )([ , ]; ( ))′ + − ′− ∫v vs C t L
t
s sC C M N N u d0 1 2
3
0
0
2 2
2 4Ω ∆τ τ τ τ τ +
+
4 2
0
0
2
0
0
2
t
s s s
t
s sa N u w d a N u w d∫ ∫∇ ⋅ ∇ ′( ) + ′( )( ( )) ( ) , ˆ ( ) ( ( )) ( ) , ˆ ( )τ τ τ τ τ µ τ τ τv vΩ Ω +
+
2 2
0
0
2
0
0
2
t
s s s
t
s s sa N u d a N u d∫ ∫′( ) − ′( )( ( )) , ( ) ˆ ( ) ( ( )) , ( ) ˆ ( )τ γ τ τ τ τ µ τ τ τv vΩ Ω , (24)
hde
M0 = sup ( )
s
s Lw ∞ Ω < + ∞ , max ( )
t
a t
≥
′
0 0 = 2 1 2
3C C− . (25)
Sohlasno svojstvu 4 pqt¥j yntehral v (24)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
242 N. R. SYDENKO
γ τ τ τ τs
t
s sa N u d, ( ( )) ( , ) ˆ ( , )
0
0
2∫ ⋅ ′ ⋅
v
Ω
= 0, t ∈ [ 0, T ] . (26)
Dalee ocenyvaem kaΩd¥j yz ostavßyxsq v (24) yntehralov. Dva perv¥x ynteh-
rala sohlasno (10 ′ ) y (22), (23) ocenyvaem tak:
I ts
1
( )( ) ≤
4 1 2
7 2
2
0
2C C K d
t
s
− ∫ ∇/ ˆ ( )v τ τ ,
(27)
I ts
2
( )( ) ≤
4 1 2
3
0 0
0
2 2
2C C M u N N dL T L
t
s s
−
∞ ∫ − ′∆ Ω( , ; ( )) ( ) ( ) ˆ ( )τ τ τ τv .
V poslednem yntehrale
N Ns
2 2( ) ( )τ τ− = ( ( ) ( )) ( ) ( )N N N Ns sτ τ τ τ+ − ≤
≤ ( )/ / ( )ˆ ( ) ˜ ( ) ˜ ( ) ( )K K u u u Ns
s s2
1 2
4
1 2+ ∇ − ∇ + ∇ −( )τ τ τ τ ≤
≤ ( )/ / ˆ ( ) ( )K K s s2
1 2
4
1 2+ ∇ +( )v τ δ τ ,
hde
K4 = max ( )
[ , ]t T
N t
∈ 0
2 , δs t( ) = ∇ −˜ ( ) ( )u t N ts , (28)
tak çto ymeem neravenstvo
I ts
2
( )( ) ≤ 2 1 2
3
0 0 2
1 2
4
1 2
2C C M u K KL T L
−
∞ +( )∆ Ω( , ; ( ))
/ / ×
×
0
2
0
2
0
22
t
s
t
s
t
sd d d∫ ∫ ∫∇ + ′ +
ˆ ( ) ˆ ( ) ( )v vτ τ τ τ δ τ τ . (29)
Tretyj yntehral v (24) zapyßem tak:
I ts
3
( )( ) = 4 ( ), ( )∇w g ts s
�
Ω ,
�
g ts( ) =
0
0
2
t
s sa N u d∫ ′ ∇( ( )) ˆ ( ) ( )τ τ τ τv .
Dlq lgboj ∂ = ∂ ∂k kx/ , k = 1, n , s uçetom (10 ′ ) ymeem
∂k sg t
�
( ) ≤ C u u M u u w
t
L t
s
L s2
2
0
0
− ∫ ∇ ∇ + ∇ ′ + ′ ∇( )[ ∞ ∞( ) ˆ ( ) ( ) ( )( )
( )
( )τ τ τ τΩ Ω +
+ ˆ ( ) ( ) ( ) ( )( )
( ) ( ) ( )/( )u u M u u dt
s
L L Ln n nτ τ τ τ τ2 2
2
0
2
− ∞∂ + ′ ∂ ]Ω Ω Ω ≤
≤ C u C K M u M uL T L L T L L Q2
2
0 2
1
2 0 0 11 2
− −∇ + ∇ ′ + ′( )∞ ∞ ∞( , ; ( )) ( , ; ( )) ( )Ω Ω +
+ c C K u M u u
L T L L Q L T Ln( ) /
( , ; ( )) ( ) ( , ; ( ))2
1
2
1 2 2
0 0
2
01 1 2
− ∂ + ′ ∂∞
Ω Ω
=
= K5 < + ∞ ∀ s, t ∈ [ 0, T ] ,
hde
M1 = sup
s
sw∇ < + ∞ (30)
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
USREDNENYE ZADAÇY DYRYXLE DLQ SPECYAL|NOHO … 243
y yspol\zovana ocenka
ess sup ˆ ( )( )
( )/( )
t
t
s
L
u t n n2 2− Ω
≤ c u t
t
t
sess sup ˆ ( )( )∇ ≤ c1 = c C K( ) /
2
1
2
1 2− , (31)
c = 2 1 2( ) ( )/n n− − .
Pry πtom
� �
g ts( ) ∂ =Ω 0. Znaçyt, mnoΩestvo { }( ),
�
g t ss ∈N kompaktno v L2( )Ω
∀ t ∈ [ 0, T ] . Sledovatel\no, I ts
3 0( )( ) → pry s → ∞ ∀ t ∈ [ 0, T ] . Dalee, ymeem
d
dt
I ts
3
( )( ) = 4 0
2a N t w u t u t w u ts s t
s
s( )( ) , ˆ ( ) ( ) ( )( )∇ − ′( )∇( )Ω
,
d
dt
I ts
3
( )( ) ≤ 4 2
2
1 1
1 2
0C M K u t M u t u tL L
− ∇ + ∇ ′( )∞ ∞
/
( ) ( )( ) ( ) ( )Ω Ω ≡
≡ ψ ( t ) ∈ L T1 0( , ) ∀ s.
Sledovatel\no, I ts
3 0( )( ) → pry s → ∞ ravnomerno otnosytel\no t ∈ [ 0, T ] ,
t. e. suwestvuet
lim ( )
([ , ])s
s
C T
I
→∞ 3 0
= 0. (32)
Teper\ rassmotrym yntehral
I ts
4
( )( ) = 2 1
0
0
2
t
s t
s
sa N w u u u w d∫ − − ′( )( ( )) , ( ) ( ) ˆ ( ) ( )( )( )τ µ τ τ τ τ
Ω
=
= 2 1
0
0
2
t
s t
sa N w u u d∫ −( )( ( )) , ( ) ( ) ˆ ( )( )τ µ τ τ τ
Ω
–
– 2 1
0
0
2
t
s sa N w u u w d∫ − ′( )( ( )) , ( ) ( ) ( )τ µ τ τ τΩ ≡ I t I ts s
4 1 4 2,
( )
,
( )( ) ( )+ .
V yntehrale I ts
4 2,
( ) ( ) funkcyq
v ( t ) =
0
0
2
t
a N u u d∫ ′( ( )) ( ) ( )τ τ τ τ ∈ C T L([ , ]; ( ))0 ∞ Ω
v sylu uslovyj u C T L∈ ∞([ , ]; ( ))0 Ω , ′ ∈ ∞u L Q( ); pry πtom verna ocenka
( ) ( ) ( )w w ts s L− ∞1 v Ω ≤ ( ) ( ) ( )M M t L0 01+ ∞v Ω ,
y vvydu vklgçenyq µ ∈L1( )Ω , sohlasno svojstvu 5 funkcyj ws, pry s → ∞
ymeet mesto sxodymost\ I ts
4 2 0,
( ) ( ) → ∀ t ∈ [ 0, T ] . Krome toho,
d
dt
I ts
4 2,
( ) ( ) = – 2 10
2a N t w w u t u ts s( ( )) , ( ) ( ) ( )µ − ′( )Ω ,
d
dt
I ts
4 2,
( ) ( ) ≤ ψ ( t ) = 2 12
2
0 0 01C M M u u tL C T L L
− + ′∞ ∞µ ( ) ([ , ]; ( )) ( )( ) ( )Ω Ω Ω ∀ s,
hde ψ ( t ) ∈ L T∞( , )0 . Znaçyt, suwestvuet
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
244 N. R. SYDENKO
lim ,
( )
([ , ])s
s
C T
I
→∞ 4 2 0
= 0. (33)
V yntehrale I ts
4 1,
( )( ) funkcyq
v1
( )( )s t = ( ) ( ( )) ( ) ˆ ( )( )w a N u u ds
t
t
s− ∫1
0
0
2 τ τ τ τ
s uçetom (31) ocenyvaetsq tak:
v1 2 2
( )
( )
( ) / ( )
s
L
t n n− Ω
≤ ( ) ([ , ]; ( ))M C c t u C T L0 2
2
1 01+ −
∞ Ω ∀ s .
Tohda pry s → ∞ ymeem
v1 2 2
( )
( )
( ) / ( )
s
L
t n n− Ω
→ 0 ∀ t ∈ [ 0, T ] ,
I ts
4 1,
( )( ) ≤ 2 2 2 2 21µ L
s
L
n n n nt/( ) /( )( )
( )
( )
( )+ −Ω Ω
v → 0 ∀ t ∈ [ 0, T ] ,
d
dt
I ts
4 1,
( )( ) ≤ 2 12 2 0 2
2
1 0µ L C T Ln n M C c u/( ) ( ) ([ , ]; ( ))( )+ ∞+ −
Ω Ω ∀ s, t ∈ [ 0, T ] .
Sledovatel\no, suwestvuet
lim ,
( )
([ , ])s
s
C T
I
→∞ 4 1 0
= 0. (34)
Takym obrazom, v (24) ostalos\ rassmotret\ yntehral
I ts
6
( )( ) =
2
0
0
2
0
2
t
s s s sa N u a N u d∫ ′ − ′[ ]( ( ))( , ( ) ˆ ( )) ( ( ))( , ( ) ˆ ( ))τ µ τ τ τ µ τ τ τv vΩ Ω =
= 2
0
0
2
0
2
t
s s sa N a N u d∫ −[ ] ′( ( )) ( ( )) ( , ( ) ˆ ( ))τ τ µ τ τ τv Ω +
+
2
0
0
2
t
s s sa N u d∫ − ′( ( ))( , ( ) ˆ ( ))τ µ µ τ τ τv Ω ≡ I t I ts s
6 1 6 2,
( )
,
( )( ) ( )+ .
Dlq I ts
6 1,
( )( ) s uçetom (25), (10 ′ ), (28) y uslovyq na µ ( x ) zapys¥vaem sledugwye
neravenstva:
I ts
6 1,
( )( ) ≤ 4 1 2
3
0
0C C N N N N u
t
s s C T L
− ∫ + − ∞( ( ) ( )) ( ) ( )( ) ([ , ]; ( ))τ τ τ τ µ Ω
ˆ ( )′vs dτ τ ≤
≤
4 1 2
3
2
1 2
4
1 2
0
0
C C K K u dC T L
t
s s s
− + ∇ +( ) ′∞ ∫( ) ˆ ( ) ( ) ˆ ( )/ /
([ , ]; ( ))µ τ δ τ τ τΩ v v ≤
≤ 2 1 2
3
2
1 2
4
1 2
0C C K K u C T L
− + ∞( )/ /
([ , ]; ( ))µ Ω ×
×
0
2
0
2
0
22
t
s
t
s
T
sd d d∫ ∫ ∫∇ + ′ +
ˆ ( ) ˆ ( ) ( )v vτ τ τ τ δ τ τ . (35)
Dalee, yspol\zuq (10 ′ ), (31) y svojstvo 4 funkcyj ws , poluçaem ocenku dlq
I ts
6 2,
( ) ( ) :
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
USREDNENYE ZADAÇY DYRYXLE DLQ SPECYAL|NOHO … 245
I ts
6 2,
( ) ( ) =
2
0
0
2
t
s s sa N
d
d
u d∫ −( ( )) ( , ( ) ˆ ( ))τ
τ
µ µ τ τ τv Ω –
–
2
0
0
2
t
s s sa N u d∫ − ′( ( ))( , ( ) ˆ ( ))τ µ µ τ τ τv Ω =
2 0
2
−a N t u t ts s s( ( ))( , ( ) ˆ ( ))µ µ v Ω –
– a N w a N u us s
s
s
t
s
s
t
s
0
2
0
0
20 2( ( ))( , ( ˆ )) ( ( ))( ˆ ( ), ˆ ( ))( ) ( ) ( )µ µ ϕ ϕ ϕ τ τ τ− − − ′ ∇ ∇∫Ω Ω ×
×
( , ( ) ˆ ( )) ( ( ))( , ( ) ˆ ( ))µ µ τ τ τ τ µ µ τ τ τ− − − ′
∫s s
t
s s su d a N u dv vΩ Ω
0
0
2 ,
I ts
6 2,
( ) ( ) ≤
2 2
2
1C t u t u t ts H s s
− − ∇ + ∇( )−µ µ ( )
ˆ ( ) ( ) ( ) ˆ ( )Ω v v +
+ 2 2
2
1C w ws H
s
s
s
s
− − − ∇ + ∇ −( )−µ µ ϕ ϕ ϕ ϕ ϕ ϕ( )
( ) ( )( ˆ ) ( ˆ )Ω +
+
8 1 2
3
2
1 2
2
1
2
1 2
0
1C C K C K u u ds H
t
s s
− − − ∇ + ∇( )− ∫/ /
( )( ) ˆ ( ) ( ) ( ) ˆ ( )µ µ τ τ τ τ τΩ v v +
+ 2 2
2
0
1C u us H
t
s s
− − ∇ ′ + ′ ∇( )− ∫µ µ τ τ τ τ( )
ˆ ( ) ( ) ( ) ˆ ( )Ω v v dτ ≤
≤ 2 2
2
0 01C c u u ts H C T L C T L sn
− −
∇ +( ) ∇− ∞µ µ ( ) ([ , ]; ( )) ([ , ]; ( ))
ˆ ( )Ω Ω Ω v +
+ c w C C K c uL L
s
s
t
Ln n∇ +( ) ∇ − + ∇(∞
− ∫ϕ ϕ ϕ ϕ τ( ) ( )
( ) /
( )( ˆ ) ( )Ω Ω Ω4 1 2
3 2
2
0
+
+
u d c u u dC T L s
t
L L Q sn([ , ]; ( )) ( ) ( )
ˆ ( ) ( ) ˆ ( )0
0
∞ ∞) ∇ + ∇ ′ + ′( ) ∇
∫Ω Ωv vτ τ τ τ τ ≤
≤ ν µ µ ν1
1
2
4 2
0 0
2
1
2
1
− − − ∇ +( ) + ∇− ∞C c u u ts H C T L C T L sn( ) ([ , ]; ( )) ([ , ]; ( ))
ˆ ( )Ω Ω Ω v +
+ 2 2
2
1C w cs H
s
s L Ln
− − ∇ − ∇ +( )− ∞µ µ ϕ ϕ ϕ ϕ( )
( )
( ) ( )( ˆ )Ω Ω Ω +
+ 16 2
1
1
2
2
7
2
2 2 2
0 0
2
1ν µ µ− − − ∇ +( )− ∞C C K T c u us H C T L C T Ln( ) ([ , ]; ( )) ([ , ]; ( ))Ω Ω Ω +
+
2 2 0
2
2
1
2
4 2
0
2
2 1 1ν ν µ µ∇ + − ∇ ′ + ′( )− −
− ∞ˆ
([ , ]; ( )) ( ) ( , ; ( )) ( )vs C t L s H L T L L QC c u T unΩ Ω Ω ,
(36)
t ∈ [ 0, T ] , νi = const > 0, i = 1, 2.
Takym obrazom, yz (24), uçyt¥vaq (23), (26), (27), (29), (32) – (36), poluçaem
ˆ ( ) ˆ ( )′ + ∇v vs st k t2
0
2 ≤ ˆ ( ˆ )( ) ( )ψ ψ ϕ ϕs
s
s
sw C w− + ∇ −−2
2
2 2
+
+
ν ν τ τ0 0
2
0
1
0
2
1 2
7 2
2
0
2
2 1 2 4ˆ ˆ ˆ ( )([ , ]; ( ))
( )
( , ; ( ))
/′ + − + ∇− − ∫v vs C t L
s
s L T L
t
sf w f C C K dΩ Ω
+
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
246 N. R. SYDENKO
+ 2 1 2
3
2
1 2
4
1 2
0 0 02C C K K M u uL T L C T L
− + +( )∞ ∞( )/ /
( , ; ( )) ([ , ]; ( ))∆ Ω Ωµ ×
×
0
2
0
2
0
2
3 0 4 0
2
t
s
t
s
T
s
s
C T
s
C T
d d d I I∫ ∫ ∫∇ + ′ +
+ +ˆ ( ) ˆ ( ) ( ) ( )
([ , ])
( )
([ , ])
v vτ τ τ τ δ τ τ +
+ ν ν µ µ ϕ ϕ1
2
2 0 2
22 22 1∇ + ∇ + − ∇ −−
−ˆ ( ) ˆ ( ˆ )([ , ]; ( )) ( )
( )v vs s C t L s H
s
st C wΩ Ω ×
× c C C C K TL L s Hn∇ +( ) + + −∞ −
− − − −ϕ ϕ ν ν µ µ( ) ( ) ( )( )Ω Ω Ω2
4
1
1
2
1
1
2
2
3
2
2 2 216 1 ×
× c u uC T L C T Ln∇ +( )∞([ , ]; ( )) ([ , ]; ( ))0 0
2
Ω Ω +
+ ν µ µ2
1
2
4 2
0
2
1 1
− − − ∇ ′ + ′( )− ∞C c u T us H L T L L Qn( ) ( , ; ( )) ( )Ω Ω , t ∈ [ 0, T ] , (37)
hde ν0 , ν1 , ν2 — proyzvol\n¥e poloΩytel\n¥e postoqnn¥e. Polahaq v (37)
ν0 = 1 2/ , ν1 = k0 2/ , ν2 = k0 16/ y oboznaçaq
x ts( ) = ̂ ( )′vs t ,
�
y ts( ) = ∇ ˆ ( )vs t ,
ymeem neravenstvo
x t
k
y ts s( ) ( )2 0 2
2
+
�
≤
∆s t s t sx
k
y t+ +
∈ ∈
1
2 80
2 0
0
2max ( ) max ( )
[ , ] [ , ]τ τ
τ
�
+
+ B x y d
t
s s
0
2 2∫ +( )( ) ( )τ τ τ
�
, (37 ′ )
v kotorom
∆s = ˆ ( ˆ ) ˆ( ) ( ) ( )
( , ; ( ))
ψ ψ ϕ ϕs
s
s
s
s
s L T L
w C w f w f− + ∇ − + −−2
2
2 2
0
2
2 1 2 Ω
+
+ D I Is L T
s
C T
s
C T
δ 2 0
2
3 0 4 0( , )
( )
([ , ])
( )
([ , ])
+ + +
+ 2 2
2
1C w cs H
s
s L Ln
− − ∇ − ∇ +( )− ∞µ µ ϕ ϕ ϕ ϕ( )
( )
( ) ( )( ˆ )Ω Ω Ω +
+ 2 1 20
1
2
4 7
2
3
1 2
2
0 0
2
k C C C K T c u uC T L C T Ln
− − −+( ) ∇ +( )
∞( ) ([ , ]; ( )) ([ , ]; ( ))Ω Ω +
+ 8 1 10
2 2c u T uL T L L Q s Hn∇ ′ + ′( )
−∞ −( , ; ( )) ( ) ( )Ω Ωµ µ ,
B = max ,/4 21 2
7 2
2C C K D D− +( ) ,
D = 2 1 2
3
2
1 2
4
1 2
0 0 02C C K K M u uL T L C T L
− + +( )∞ ∞( )/ /
( , ; ( )) ([ , ]; ( ))∆ Ω Ωµ .
Yz (37 ′ ) sleduet
x t
k
y ts s( ) ( )2 0 2
2
+
�
≤ 4 4
0
2 2∆s
t
s sB x y d+ +( )∫ ( ) ( )τ τ τ
�
, t ∈ [ 0, T ] .
Otsgda v sluçae, naprymer, k0 2≤ ymeem
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
USREDNENYE ZADAÇY DYRYXLE DLQ SPECYAL|NOHO … 247
x t y ts s( ) ( )2 2+
�
≤
8 80
1
0
1
0
2 2k k B x y ds
t
s s
− −+ +( )∫∆ ( ) ( )τ τ τ
�
, t ∈ [ 0, T ] ,
sledovatel\no (πto neravenstvo Hronuolla),
max ( ) ( )
[ , ]t T s sx t y t
∈
+( )
0
2 2�
≤ 8 0
1 8 0
1
k e k BT
s
− −
∆ . (38)
V sylu sxodymostej (12), svojstva 3 funkcyj ws y uslovyj (21) try perv¥x
slahaem¥x v ∆s pry s → ∞ sxodqtsq k nulg. Dlq ocenky δs L T2 0( , ) v (28)
ymeem
∇ ˜ ( )u ts
2 = w u t u t w w u t u t ws s s s∇ + ∇( ) + ∇ ∇( )( ) ( ), ( ), ( )2 2 2 2
Ω Ω ≡
≡ J t J t J ts s s
1 2 3
( ) ( ) ( )( ) ( ) ( )+ + .
Yz vklgçenyq ∇ ∈u t C T L( ) ([ , ]; ( ))0 2 Ω sleduet, çto pry s → ∞ norma
J t u ts
1
2( )( ) ( )→ ∇ ravnomerno otnosytel\no t ∈ [ 0, T ] . Poskol\ku sohlasno
svojstvu 4 ∇ →w x xs( ) ( )2 µ slabo kak mer¥ na { }( ) :v v∈ =∂C Ω Ω 0 , a v
rassmatryvaem¥x uslovyqx dlq poçty vsex t ∈ ( 0, T )
u t W Wn n( , ) ( ) ( )⋅ ∈ 2 1Ω Ω∩
�
⊂ Cα( )Ω , α ∈ ( 0, 1 ) ,
krome toho,
u C T L∈ ∞α ([ , ]; ( ))0 Ω , α ∈ ( 0, 1 ) ,
y norm¥ ∇ws ravnomerno ohranyçen¥, funkcyy J ts
2
( )( ) ravnomerno ohrany-
çen¥ y ravnostepenno neprer¥vn¥ na [ 0, T ] , y dlq poçty vsex t ∈ ( 0, T ) suwe-
stvuet
lim ( )( )
s
sJ t
→∞ 2 = ( )( ) ,u t2 µ Ω . (39)
Otsgda sleduet, çto predel (39) qvlqetsq ravnomern¥m na [ 0, T ] . Yz uslovyj
(22) na u ( t ) y svojstva 4 funkcyj ws v¥tekaet, çto pry s → ∞
max ( ) ( )
[ , ]t T sw u t
∈
− ∇
0
1 → 0, u t ws( )∇ → 0 v C T Lsc([ , ]; ( ))0 2 Ω , (40)
sledovatel\no, J ts
3
( )( ) → 0 ravnomerno otnosytel\no t ∈ [ 0, T ] . Tohda dlq
norm¥ (20) ymeem sxodymost\ velyçyn¥ (28):
δs C T([ , ])0 → 0. (41)
Dva sledugwyx slahaem¥x v ∆s stremqtsq k nulg sohlasno (32) – (34). Po-
slednyx dva slahaem¥x v ∆s stremqtsq k nulg v sylu svojstva 4 funkcyj ws
y uslovyq (21). Takym obrazom, ustanovleno, çto v (38) velyçyna ∆s → 0 pry
s → ∞ , çto dokaz¥vaet sxodymosty
max ( )
[ , ]t T sy t
∈ 0
�
= max ˆ ( ) ˜ ( )
[ , ]
( )( )
t T
s
su t u t
∈
∇ −
0
→ 0,
(42)
max ( )
[ , ]t T sx t
∈ 0
= max ˆ ( ) ˜ ( )
[ , ]
( )( )
t T
s
su t u t
∈
− ′
0
→ 0.
Otsgda y yz (41) sleduet spravedlyvost\ hypotez¥ (18), (20):
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
248 N. R. SYDENKO
max ( ) ( )
[ , ]t T sN t N t
∈
−
0
≤ max ˆ ( ) ˜ ( )
[ , ]
( )
([ , ])t T
s
s s C Tu t u t
∈
∇ − ∇ +
0 0δ ≤
≤ max ˆ ( ) ˜ ( )
[ , ]
( )
([ , ])t T
s
s s C Tu t u t
∈
∇ −( ) +
0 0δ → 0 ( s → ∞ ) ,
a vmeste s nej y utverΩdenyq, çto u ( t ) qvlqetsq reßenyem zadaçy (19), (20).
Yz (40), (42) v¥tekagt takΩe dopolnytel\n¥e k (14), (15) sxodymosty pry s →
→ ∞ :
∇ = ∇ + → ∇ˆ ˜( )u u y us
s s
�
v C T Lsc([ , ]; ( ))0 2 Ω ,
(43)
ˆ ˜( )u u x ut
s
s s t= ′ + → syl\no v C T L([ , ]; ( ))0 2 Ω ,
tak kak yz uslovyj teorem¥ y (19) sleduet, çto u C T Lt ∈ ([ , ]; ( ))0 2 Ω y, takym ob-
razom, ˜′ = →u w u us s t t syl\no v C T L([ , ]; ( ))0 2 Ω .
Pust\ u1, u2 — reßenyq zadaçy (19) takye, çto
ui ∈
�
∩ ∩C T H L L Q( )[ , ]; ( ) ( ) ( )0 1 4Ω Ω ∞ , ∆ui ∈ L T L∞( ), ; ( )0 2 Ω ,
(44)
′ui ∈
�
∩L T H C T L∞( ) ( ), ; ( ) [ , ]; ( )0 01 2Ω Ω , i = 1, 2.
Tohda yz (19) s uçetom (20) dlq v = u2 – u1 ymeem πnerhetyçeskoe ravenstvo
′ +v v( ) ( ) ( )( )t a N t t V
2
0 2
2 2 = 2 0 2
2
2 2
2
0
′ ∇ ∇ ′∫ a N u u dV
t
( )( )( ) ( ), ( ) ( )τ τ τ τ τΩ v +
+
2 0 2
2
0 1
2
1 1
0
a N a N u u d
t
( ) ( ) ( )( ) ( ) ( ) ( ), ( )τ τ τ τ τ τµ− −[ ] ′∫ ∆ Ωv , t T∈ [ , ]0 , (45)
hde N ti( ) = u ti V( ) . V oboznaçenyqx
′K1 = max max ( )
, [ , ]i t T iN t
= ∈1 2 0
, ′K2 = ∆ Ωu L T L1 0 2∞ ( , ; ( )), ′K3 = u L Q1 ∞ ( ),
′K4 = ∇ ′ ∞u L T L2 0 2( , ; ( ))Ω , ′k0 = ( )C K C1 1 2
2′ + −
yz (45) poluçaem neravenstvo Hronuolla
′ + ′v v( ) ( )t k t V
2
0
2 ≤ 4 1 2
3
1 4
2
0
C C K K dV
t
− ′ ′ ∫ v( )τ τ +
+
8 1 2
3
1 2 3
0
C C K K K d
t
− ′ ′ + ′( ) ∇ ′∫µ τ τ τv v( ) ( ) ≤
′ ′ + ′( )∫B k dV
t
v v( ) ( )τ τ τ2
0
2
0
,
t T∈ [ , ]0 , ′B = const,
yz kotoroho sleduet ′ + ′v v( ) ( )t k t V
2
0
2 ≡ 0, t. e. v( )t ≡ 0. Takym obrazom, v
klasse funkcyj (44), a tem bolee v klasse (22), reßenye zadaçy (19), (20) edyn-
stvenno. Znaçyt, vse ukazann¥e v teoreme sxodymosty ymegt mesto dlq polnoj
posledovatel\nosty s ∈ N.
Teorema dokazana.
Zameçanye�3. Dlq toho çtob¥ v¥polnqlos\ uslovye sxodymosty (21) dlq
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
USREDNENYE ZADAÇY DYRYXLE DLQ SPECYAL|NOHO … 249
ˆ ( )ϕ s , dostatoçno [5], çtob¥ pry v¥polnenyy hypotez¥ (A) funkcyq ϕ( )s
b¥la
reßenyem zadaçy
– ∆ϕ( )( )s x = g xs( )( ), x s∈ Ω( ), ϕ( )s ∈
�
H s1( )( )Ω , g s( ) ∈ H s−1( )( )Ω ,
pryçem pry nekotoroj g H∈ −1( )Ω suwestvoval predel
lim ˆ( )
( )s
s
H
g g
→∞
− −1 Ω
= 0,
a predel\naq v sm¥sle (12) funkcyq ϕ, qvlqgwaqsq pry πtom edynstvenn¥m
reßenyem zadaçy
– ∆ϕ µ ϕ( ) ( ) ( )x x x+ = g x( ), x ∈ Ω, ϕ ∈ V ,
prynadleΩala y C ( )Ω .
Avtor hluboko blahodaren Stanyslavu Yvanovyçu PoxoΩaevu za soobwenye
o svoyx rabotax, peredannoe v ygne 2004 h. çerez nezabvennoho Yhorq Vladymy-
rovyça Skr¥pnyka.
1. PoxoΩaev S. Y. Ob odnom klasse kvazylynejn¥x hyperbolyçeskyx uravnenyj // Mat. sb. –
1975. – 96(138), # 1. – S.Q152 – 166.
2. PoxoΩaev S. Y. Ob odnom kvazylynejnom hyperbolyçeskom uravnenyy Kyrxhofa // Dyf-
ferenc. uravnenyq. – 1985. – 21, # 1. – S.Q101 – 108.
3. PoxoΩaev S. Y. Kvazylynejn¥e hyperbolyçeskye uravnenyq Kyrxhofa y zakon¥ soxrane-
nyq // Tr. Mosk. πnerh. yn-ta. – 1974. – V¥p. 201. – S.Q118 – 126.
4. Cioranesku D., Murat F. Un terme étrange venu d’ailleurs // Res. Notes Math. – 1982. – 60. –
P. 93 – 138.
5. Cioranesku D., Donato P., Murat F., Zuazua E. Homogenization and corrector for the wave equa-
tion in domains with small holes // Ann. Scuola norm. super. Pisa. Sci. fis. e mat. – 1991. – 18, # 2.
– P. 251 – 293.
6. Skr¥pnyk Y. V. Metod¥ yssledovanyq nelynejn¥x πllyptyçeskyx hranyçn¥x zadaç. – M.:
Nauka, 1990. – 448Qs.
7. Dal Maso G., Skrypnik I. V. Asymptotic behaviour of nonlinear Dirichlet problems in perforated
domains. – Trieste, 1995. – 65 p. – Ref. SISSA 162|95|M (December 95).
8. Lyons Û.-L., MadΩenes ∏. Neodnorodn¥e hranyçn¥e zadaçy y yx pryloΩenyq. – M.: Myr,
1971. – 372Qs.
Poluçeno 06.09.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
|
| id | umjimathkievua-article-3449 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:42:45Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/68/429c524c1cc8fb4aadfd0b51ad33f268.pdf |
| spelling | umjimathkievua-article-34492020-03-18T19:54:47Z Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation Усреднение задачи Дирихле для специального гиперболического уравнения Кирхгофа Sidenko, N. R. Сиденко, Н. Р. Сиденко, Н. Р. We prove a statement on the averaging of a hyperbolic initial-boundary-value problem in which the coefficient of the Laplace operator depends on the space $L^2$-norm of the gradient of the solution. The existence of the solution of this problem was studied by Pokhozhaev. In a space domain in $ℝ^n,\; n ≥ 3$, we consider an arbitrary perforation whose asymptotic behavior in a sense of capacities is described by the Cioranesku-Murat hypothesis. The possibility of averaging is proved under the assumption of certain additional smoothness of the solutions of the limiting hyperbolic problem with a certain stationary capacitory potential. Доведено твердження про усереднення гіперболічної початково-крайової задачі, у якій коефіцієнт при операторі Лапласа залежить від просторової $L^2$-норми градієнта розв'язку. Питання існування розв'язку цієї задачі досліджене С. I. Похожаєвим. У просторовій області в $ℝ^n,\; n ≥ 3$, розглядається довільна перфорація, асимптотична поведінка якої в ємнісному сенсі описана гіпотезою Д. Чіоранеску - Ф. Мюра. Можливість усереднення доведено за припущенням деякої додаткової гладкості розв'язків граничної гіперболічної задачі з певним ємнісним стаціонарним потенціалом. Institute of Mathematics, NAS of Ukraine 2006-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3449 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 2 (2006); 236–249 Український математичний журнал; Том 58 № 2 (2006); 236–249 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3449/3636 https://umj.imath.kiev.ua/index.php/umj/article/view/3449/3637 Copyright (c) 2006 Sidenko N. R. |
| spellingShingle | Sidenko, N. R. Сиденко, Н. Р. Сиденко, Н. Р. Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation |
| title | Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation |
| title_alt | Усреднение задачи Дирихле для специального гиперболического уравнения Кирхгофа |
| title_full | Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation |
| title_fullStr | Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation |
| title_full_unstemmed | Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation |
| title_short | Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation |
| title_sort | averaging of the dirichlet problem for a special hyperbolic kirchhoff equation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3449 |
| work_keys_str_mv | AT sidenkonr averagingofthedirichletproblemforaspecialhyperbolickirchhoffequation AT sidenkonr averagingofthedirichletproblemforaspecialhyperbolickirchhoffequation AT sidenkonr averagingofthedirichletproblemforaspecialhyperbolickirchhoffequation AT sidenkonr usredneniezadačidirihledlâspecialʹnogogiperboličeskogouravneniâkirhgofa AT sidenkonr usredneniezadačidirihledlâspecialʹnogogiperboličeskogouravneniâkirhgofa AT sidenkonr usredneniezadačidirihledlâspecialʹnogogiperboličeskogouravneniâkirhgofa |