Phragmen-Lindelof Principle for Some Quasilinear Evolution Equations of the Second Order
We consider the equation $u_{tt} + A (u_t) + B(u) = 0$, where $A$ and $B$ are quasilinear operators with respect to the variable x of the second order and the fourth order, respectively. In a cylindrical domain unbounded with respect to the space variables, we obtain estimates that characterize the...
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| Дата: | 2005 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2005
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3591 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509708122587136 |
|---|---|
| author | Sleptsova, I. P. Shishkov, A. E. Слепцова, И. П. Шишков, А. Е. Слепцова, И. П. Шишков, А. Е. |
| author_facet | Sleptsova, I. P. Shishkov, A. E. Слепцова, И. П. Шишков, А. Е. Слепцова, И. П. Шишков, А. Е. |
| author_sort | Sleptsova, I. P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:59:22Z |
| description | We consider the equation $u_{tt} + A (u_t) + B(u) = 0$, where $A$ and $B$ are quasilinear operators with respect to the variable x of the second order and the fourth order, respectively. In a cylindrical domain unbounded with respect to the space variables, we obtain estimates that characterize the minimum growth of any nonzero solution of the mixed problem at infinity. |
| first_indexed | 2026-03-24T02:45:23Z |
| format | Article |
| fulltext |
UDK 517.946
Y. P. Slepcova (Donec. nac. un-t),
A. E. Íyßkov (Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck)
PRYNCYP FRAHMENA – LYNDELEFA
DLQ NEKOTORÁX KVAZYLYNEJNÁX
∏VOLGCYONNÁX URAVNENYJ VTOROHO PORQDKA
We consider the equation utt + A ut( ) + B u( ) = 0, where A and B are quasilinear operators in the
variable x of second and forth orders, respectively. In the cylindrical domain unbounded in space
variables, we obtain estimates that characterises the minimal grouth of any nonzero solution of the mixed
problem at infinity.
Rozhlqnuto rivnqnnq utt + A ut( ) + B u( ) = 0, v qkomu A i B — kvazilinijni operatory za
zminnog x druhoho i çetvertoho porqdkiv vidpovidno. V neobmeΩenij za prostorovymy zminnymy
cylindryçnij oblasti otrymano ocinky, qki xarakteryzugt\ minimal\nyj rist bud\-qkoho nenu-
l\ovoho rozv’qzku mißano] zadaçi na neskinçennosti.
Pry yzuçenyy kaçestvenn¥x svojstv reßenyj kraev¥x zadaç vaΩnug rol\ yhra-
gt ocenky rosta reßenyj na beskoneçnosty, opredelqem¥e teoremamy typa
Frahmena – Lyndelefa. Dlq lynejn¥x y kvazylynejn¥x πllyptyçeskyx urav-
nenyj v beskoneçnomern¥x oblastqx razlyçnoj struktur¥ klassyçeskaq teore-
ma Frahmena;– Lyndelefa obobwena v rqde rabot (sm., naprymer, [1 – 3]).
Ocenky rosta klassyçeskyx reßenyj smeßann¥x zadaç dlq parabolyçeskyx
uravnenyj poluçen¥ v [4, 5]. Asymptotyçeskye svojstva obobwenn¥x reßenyj
smeßann¥x zadaç dlq lynejn¥x parabolyçeskyx uravnenyj v neohranyçenn¥x
prostranstvenn¥x oblastqx yzuçen¥ v [6, 7]. Dlq nelynejn¥x uravnenyj vtoro-
ho porqdka typa nestacyonarnoj fyl\tracyy v [8] dokazana edynstvennost\
reßenyq zadaçy Koßy v klassax funkcyj, qvlqgwyxsq nelynejn¥m analohom
klassov Tyxonova. V [9, 10] ukazan¥ klass¥ rastuwyx na beskoneçnosty obob-
wenn¥x reßenyj smeßann¥x zadaç dlq kvazylynejn¥x parabolyçeskyx uravne-
nyj v¥sokoho porqdka.
V [11] predloΩenn¥j avtoramy metod vvedenyq parametra yspol\zovan dlq
yzuçenyq πvolgcyonn¥x uravnenyj vyda
∂
∂
−
t
M1
∂
∂
−
t
M u2 = 0, hde M1 y
M2 — lynejn¥e dyfferencyal\n¥e operator¥ po prostranstvenn¥m peremen-
n¥m s hladkymy koπffycyentamy. Edynstvennost\ klassyçeskoho reßenyq
zadaçy Koßy v neohranyçenn¥x prostranstvenn¥x oblastqx ustanovlena v
klassax rastuwyx funkcyj typa Tyxonova – Tπklynda. Dokazatel\stvo edyn-
stvennosty reßenyq v sluçae lynejn¥x uravnenyj ravnosyl\no dokazatel\stvu
al\ternatyvn¥x utverΩdenyj typa teorem Frahmena – Lyndelefa.
V nastoqwej rabote rassmotren¥ kvazylynejn¥e uravnenyq vyda
u A u B u F x ttt t+ + =( ) ( ) ( , ), (1)
qvlqgwyesq obobwenyem uravnenyj yz [11] s operatoramy M1 y M2 vtoroho
porqdka. Dlq odnorodnoj smeßannoj zadaçy v neohranyçennoj po prostran-
stvenn¥m peremenn¥m cylyndryçeskoj oblasty ustanovlen mynymal\n¥j rost
na beskoneçnosty proyzvol\noho nenulevoho reßenyq.
Pust\ G = Ω × (0, T ) , 0 < T < ∞, — neohranyçennaq oblast\ v Rx t
n
,
+1
.
Hranyca ∂ Ω predpolahaetsq dostatoçno hladkoj; Γ = ∂ Ω × (0, T ). V G ras-
smatryvaetsq smeßannaq zadaça
u A u B utt t+ +( ) ( ) ≡
≡
α
α
α
α
=
∑
1
D a x t u D ut t( , , , ) +
α
α α α
=
−∑ ( )
2
2
D D u D u
p
= 0, (2)
© Y. P. SLEPCOVA, A. E. ÍYÍKOV, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2 239
240 Y. P. SLEPCOVA, A. E. ÍYÍKOV
u t = =0 0 , ut t = =0 0 ; (3)
D uα
Γ
= 0 , α ≤ 1. (4)
Zdes\ Dα = D D Dx x xn
n
1
1
2
2α α α… = ∂
∂ …
α
α αx xn
n
1
1
, α = α1 + … + αn
, D um
= D uα{ } ,
α = m, D um 2
= α
α
=∑ m
D u
2
, p > 2, funkcyy a x tα ξ ξ, , , ( )1( ) opredelen¥ y
neprer¥vn¥ dlq vsex ( , )x t G∈ , ξ ∈R, ξ( )1 ∈Rn
y udovletvorqgt neravenstvam
α
α αξ ξ ξ ξ
=
∑ ( ) ≥
1
1 1
0
1a x t a
p
, , , ( ) ( ) ( )
, a0 0> , (5)
a x t a
p
α ξ ξ ξ, , , ( ) ( )1
1
1 1( ) ≤
−
, a1 < ∞ . (6)
Pust\ Ω ′ — lgbaq ohranyçennaq podoblast\ Ω, S ⊂ ∂ Ω ′, Gρ ν, = Ω × (ρ, ν ),
0 ≤ ρ < ν ≤ T, ′Gρ ν, = Ω ′ × (ρ, ν ). Çerez W Sp
m( , )′Ω oboznaçym zam¥kanye v
norme Wp
m( )′Ω mnoΩestva Cm
-hladkyx v Ω ′ funkcyj, obrawagwyxsq v nul\
v okrestnostqx ∂ ′Ω \ S ,
°Wp
m ≡ Wp
m ′(Ω , ∂ ′ ∂ )Ω Ω\ , a çerez Lp ρ ν,( ; W Sp
m( , )′ )Ω —
prostranstvo funkcyj ν( , )x t takyx, çto dlq poçty vsex t ∈ (ρ, ν ) ν( , )x t ∈
∈ W Sp
m( , )′Ω y
ρ
ν
ν∫ ⋅( , )t dt
W
p
p
m < ∞.
Dlq uravnenyj vyda (1) s lynejn¥my πllyptyçeskymy operatoramy A y B
razn¥x porqdkov teoryq razreßymosty smeßann¥x zadaç v ohranyçenn¥x oblas-
tqx xoroßo razvyta (sm. [12, 13]). V sluçae, kohda B — lynejn¥j operator
vtoroho porqdka, a Aut = u ut
p
t
−2
, p > 1, razreßymost\ smeßannoj zadaçy v
klassax obobwenn¥x funkcyj dokazana v [14]. V neohranyçenn¥x oblastqx v
[15] dlq neodnorodnoho uravnenyq s A = B = ∆ dokazano suwestvovanye obob-
wenn¥x reßenyj smeßannoj zadaçy s ohranyçenn¥m yntehralom πnerhyy, v [16]
v sluçae lynejn¥x πllyptyçeskyx operatorov A y B porqdkov 2m y 2m + 2
sootvetstvenno ustanovleno suwestvovanye rastuwyx na beskoneçnosty reße-
nyj. Dlq uravnenyj, v kotor¥x A ut( ) = α
α α
α=∑ −
m
D a( )1 (x , t, ut , D ux t , …
… , D ux
m
t ), operator A udovletvorqet uslovyqm (5), (6) s α = m, a B — ly-
nejn¥j ravnomerno πllyptyçeskyj operator porqdka 2m + 2, v ohranyçennoj
oblasty G suwestvuet reßenye smeßannoj zadaçy yz klassov typa V Gm,
, ( )2
1 0
[17, s.;528]. Podrobnoe dokazatel\stvo πtoho utverΩdenyq budet yzloΩeno v
dal\nejßyx publykacyqx. Dlq odnorodnoj zadaçy v neohranyçenn¥x oblastqx
moΩno poluçyt\ ocenky snyzu rosta na beskoneçnosty nenulev¥x reßenyj yz
takyx Ωe klassov funkcyj.
V dannoj rabote s cel\g uprowenyq texnyçeskoj storon¥ dokazatel\stv
budem rassmatryvat\ reßenyq v sledugwem sm¥sle. Pod obobwenn¥m reßeny-
em zadaçy (2) – (4) budem ponymat\ takug funkcyg u x t( , ) , çto dlq lgboj
ohranyçennoj podoblasty Ω ′ ⊂ Ω y lgb¥x ρ, ν, 0 ≤ ρ < ν ≤ T, u x t( , ) ∈ H =
= u x t( , ){ : u x t( , ) , u x tt ( , ) ∈ Lp ρ ν,( ; Wp
2 ′(Ω , ∂ ′ ∂ ))Ω Ω\ , utt ∈ L
p′(ρ ν, ; Wp
− ′ )}2( )Ω ,
v¥polnen¥ uslovyq (3) y yntehral\noe toΩdestvo
ρ
ν
ν∫ u dttt , +
′ = =
−
∫∫ ∑ ∑( ) +
G
t t
p
a x t u D u D D u D uD dx dt
ρ ν α
α
α α
α
α αν ν
,
, , ,
1 2
2 2
= 0
(7)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
PRYNCYP FRAHMENA – LYNDELEFA DLQ NEKOTORÁX KVAZYLYNEJNÁX … 241
s proyzvol\noj funkcyej ν( , )x t ∈ L Wp pρ ν, ; ( )° ′( )2 Ω . Zdes\ utt — proyzvodnaq
ut v sm¥sle raspredelenyj na (0, T ) so znaçenyqmy v Wp
2 ′ ∂ ′ ∂( )( )Ω Ω Ω, \
*
=
= Wp′
− ′2( )Ω , 〈 w, ν 〉 — znaçenye lynejnoho neprer¥vnoho funkcyonala w ∈
∈ Wp′
− ′2( )Ω na πlemente ν ∈ ° ′Wp
2( )Ω .
Vvedem semejstvo ohranyçenn¥x oblastej Ω( )τ : dlq lgb¥x τ > 0 Ω( )τ =
= Ω ∩ x <{ }τ , Ω( , )τ τ1 2 = Ω Ω( ) \ ( )τ τ2 1 . Dlq lgb¥x ρ , ν , 0 ≤ ρ < ν ≤ T,
Gρ ν τ, ( ) = Ω( )τ × (ρ, ν ), Gρ ν τ τ, ( , )1 2 = Ω( , )τ τ1 2 × (ρ, ν ).
Osnovn¥m rezul\tatom dannoj rabot¥ qvlqetsq sledugwee utverΩdenye.
Teorema (typa Frahmena – Lyndelefa). Dlq proyzvol\noho obobwennoho re-
ßenyq u x t( , ) zadaçy (2) – (4) ymeet mesto al\ternatyva: lybo u ≡ 0 v G ,
lybo u x t( , ) rastet pry x → ∞ tak, çto dlq proyzvol\noj posledovatel\-
nosty { }τi vyda τi +1 = k iτ , k ∈ ∞( , )1 , τi > τ0 > 0, i ≥ 1,
G
i
p
t
p
i
p p
i i
i
u D u dx dt h
( )
( )
τ
γ γτ τ τ τ∫∫ ′ − −+
→ ∞2 1 2
pry i → ∞, (8)
hde γ1 = n + p + p ′ +
2
2
p
p −
, γ 2 = 2
2p −
, h( )τ — proyzvol\naq poloΩytel\naq
monotonno neub¥vagwaq funkcyq, udovletvorqgwaq uslovyqm:
i)
1
∞
∫ = ∞d
h
τ
τ τ( )
;
ii) suwestvuet postoqnnaq ϕ ∈ (0, ∞) takaq, çto h( )τ ≤ ≤ τϕ
dlq vsex
τ > τ 0 .
Zameçanye 1. Sootnoßenyq (8) soderΩat, v çastnosty, ocenky rosta reße-
nyj parabolyçeskyx uravnenyj yz [8, 10].
Dokazatel\stvo. Pust\ ζ( )h ∈ C R2 1( ) — srezagwaq funkcyq: ζ( )h = 1
pry h ≤ 0, ζ( )h = 0 pry h ≥ 1, 0 ≤ ζ( )h ≤ 1 pry 0 < h < 1. Oboznaçym ητ σ, ( )x =
= ζ
τ
σ
p x −
+
3
2 . Oçevydno, ητ σ, ( )x ≡ 0 pry x > 3τ – σ, ητ σ, ( )x = 1 pry
x < 3τ – 2σ, D jητ σ, ≤ dj
jσ− , j = 1, 2; dj = const < ∞ ne zavysqt ot σ. Dlq
proyzvol\noj yzmerymoj funkcyy µ τ( ) > 0 opredelym g tτ( ) = exp ( )−( )µ τ2 t .
Lemma 1. Pust\ u x t( , ) — obobwennoe reßenye zadaçy (2) – (4). Tohda dlq
proyzvol\n¥x ρ, ν, 0 ≤ ρ < ν ≤ T, τ > 0, 0 < σ < 3
2
τ y proyzvol\noj yzmerymoj
funkcyy µ τ( ) > 0 v¥polnqetsq sootnoßenye
g u
p
D u x dxt
p
τ
τ σ
τ σν η
ν
( ) ( )
( )
,
Ω 3
2 21
2
1
−
∫ +
+
+ µ τ η
ρ ν τ σ
τ σ τ
2
3
2 21
2
1( ) ( ) ( )
, ( )
,
Ω −
∫ +
u
p
D u x g t dx dtt
p
+
+ a Du x g t dx dtt
p
0
3Ωρ ν τ σ
τ σ τη
, ( )
, ( ) ( )
−
∫ ≤
≤ c u D u g t dx dtp
t
p p p
1
3 2 3
2
Ωρ ν τ σ τ
τσ σ
, ( , )
( )
−
− − ′∫ +( ) +
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
242 Y. P. SLEPCOVA, A. E. ÍYÍKOV
+ g u
p
D u dxt
p
τ
τ
ρ
ρ
( )
( )Ω 3
2 21
2
1∫ +
. (9)
Zdes\ y dalee c — poloΩytel\n¥e postoqnn¥e, zavysqwye ot yzvestn¥x para-
metrov zadaçy y ne zavysqwye ot τ y σ.
Dokazatel\stvo. PoloΩym v yntehral\nom toΩdestve (7), rassmatryvae-
mom v oblasty ′Gρ ν, = Gρ ν τ, ( )3 , ν( , )x t = u x t( , ) ητ σ, ( )x g tτ( ):
ρ
ν
τ σ τη∫ u u x g t dttt t, ( ) ( ), +
+
G
t t ta x t u D u D u x g t dx dt
ρ ν τ α
α
α α
τ σ τη
, ( )
,, , , ( ) ( )
3 1
∫∫ ∑
=
( ) +
+
G
p
tD u D u D u x g t dx dt
ρ ν τ α
α α
τ σ τη
, ( )
, ( ) ( )
3 2
2 2
∫∫ ∑
=
−
+
+
G
t t ta x t u D u u D
ρ ν τ σ τ σ α
α
α α
τ ση
, ( , )
,, , ,
3 2 3 1− − =
∫∫ ∑ ( )
+
+
α
α
α
α
τ σ τη
=
−
<
−∑ ∑
2
2 2
D u D u D u D g t dx dt
p
i
i
t
i
, ( ) = 0. (10)
Pervoe slahaemoe v levoj çasty ravenstva (10), sohlasno teoreme 1.17 hl. IV
[18], prymenennoj k funkcyy ut , moΩno proyntehryrovat\ s pomow\g formu-
l¥ yntehryrovanyq po çastqm. Poskol\ku D uα ∈ C T Lp0, ; ( )′( )Ω ([14], lem-
ma;1.2 hl. 1), v tret\em slahaemom yntehryrovanye po çastqm takΩe vozmoΩno.
Vtoroe slahaemoe moΩno ocenyt\ snyzu, yspol\zuq uslovye (5), a v yntehrale po
oblasty Gρ ν, (3τ − 2σ, 3τ – σ) prymenym neravenstva Hel\dera y Gnha s ε:
g u
p
D u x dxt
p
τ
τ σ
τ σν η
ν
( ) ( )
( )
,
Ω 3
2 21
2
1
−
∫ +
+
+ µ τ η
ρ ν τ σ
τ σ τ
2
3
2 21
2
1( ) ( ) ( )
, ( )
,
G
t
p
u
p
D u x g t dx dt
−
∫∫ +
+
+ a Du x g t dx dt
G
t
p
0
3ρ ν τ σ
τ σ τη
, ( )
, ( ) ( )
−
∫∫ ≤
≤ ε
ρ ν τ σ τ σ
τ
G
t
pDu g t dx dt
, ( , )
( )
3 2 3− −
∫∫ +
+ c u D u g t dx dt
G
p
t
p p p
1
3 2 3
2
ρ ν τ σ τ σ
τσ σ
, ( , )
( )
− −
− − ′∫∫ +[ ] +
+ g u
p
D u x dxt
p
τ
τ σ
τ σρ η
ρ
( ) ( )
( )
,
Ω 3
2 21
2
1
−
∫ +
. (11)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
PRYNCYP FRAHMENA – LYNDELEFA DLQ NEKOTORÁX KVAZYLYNEJNÁX … 243
Yzbavymsq ot slahaemoho s ε v pravoj çasty (11). Dlq πtoho voz\mem sreza-
gwug funkcyg η
τ σ,
( )
2
x y poluçym analohyçnoe neravenstvu (11) sootnoße-
nye, v levoj çasty kotoroho yntehryrovanye vedetsq po oblastqm Ων τ σ3
2
−
y
Gρ ν τ σ
, 3
2
−
, a v pravoj — po slog Gρ ν τ σ τ σ
, ,3 3
2
− −
y po oblasty
Ωρ τ σ3
2
−
sootvetstvenno. UmnoΩym obe çasty πtoho sootnoßenyq na
σ
2
p
.
Oboznaçym
Aρ σ( ) = σ ρτ
τ σρ
p
t
p
g u
p
D u dx( )
( )Ω 3
2 21
2
1
−
∫ +
,
I a Du g t dx dtp
G
t
p
ρ ν
τ σ
τσ σ
ρ ν
,
( )
( ) ( )
,
=
−
∫∫0
3
,
(12)
I Du g t dx dtp
G G
t
p
ρ ν
τ σ τ σ
τσ σ σ
ρ ν ρ ν
,
( )
( ) \ ( )
( , ) ( )
, ,
0
0
3 3 0
=
− −
∫∫ ,
H0 0( , )σ σ =
= σ σ σ σ σ
ρ ν ρ ντ σ τ σ
τ
p
G G
p
t
p p p
u D u g t dx dt
, ,( ) \ ( )
( ) ( ) ( )
3 3
0 0
2
0− −
− − ′∫∫ − + −
.
Tohda moΩno zapysat\
I I c H Ap
ρ ν ρ ν ρσ ε σ σ σ σ σ
, ,
( ) ( )( ) , ,≤
+
+
2
2 2 2
0
1
0
.
Pust\ ε = − −2 1p
. Yteryruq poslednee neravenstvo j raz, poluçaem
Iρ ν σ, ( ) ≤ 2
2
0−
j
jIρ ν
σ
,
( )
+ c u D u g t dx dt
G
t
p p p p
1
3 3
2
ρ ν τ σ τ
τσ
, ( , )
( )
−
− ′∫∫ +
+
+ σ ρτ
τρ
p
t
p
g u
p
D u dx( )
( )Ω 3
2 21
2
1∫ +
.
Predel\n¥j perexod pry j → ∞ y neravenstvo (11) pryvodqt k utverΩdenyg
lemm¥.
Dalee budet yspol\zovano ynterpolqcyonnoe neravenstvo, qvlqgweesq
çastn¥m sluçaem neravenstva Nyrenberha – Hal\qrdo (sm., naprymer, [19, s. 67]).
Pust\ O — ßar yz Rn
s centrom v naçale koordynat radyusa τ, s ≥ 1, r > 0.
Tohda W Os
1( ) ∩ L Or( ) ⊂ L Oq( ) pry lgbom q, udovletvorqgwem sootnoßenyg
1 1 1 1 1
q s n r
= −
+ −θ θ( ) (13)
s kakym-lybo θ ∈( , )0 1 , a takΩe dlq lgboj funkcyy ν( )x ∈ W Os
1( ) ∩ L Or( )
ymeet mesto neravenstvo
ν τ ν ν νθ θ
L O
n q r
L O L O L Oq r s r
D D D( ) ( ) ( ) ( )( )≤ +
−
−
1
1 1
2
1
, (14)
hde postoqnn¥e 0 < D1, D2 < ∞ ne zavysqt ot ν( )x y ot radyusa ßara τ.
V çastnosty, esly u x t( , ) — reßenye zadaçy (2) – (4) y Ω( )τ ⊂ O , to dlq
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244 Y. P. SLEPCOVA, A. E. ÍYÍKOV
funkcyy u x tt ( , ) pry r = 2, q = s = p yz neravenstva (14) sleduet ocenka
Ω Ω Ω Ω( ) ( ) ( ) ( )
( )
τ τ τ
θ
τ
θ
τ∫ ∫ ∫ ∫
≤
+
−
−
u dx D u dx D Du dx u dxt
p
p
n p
t t
p
p
t
1
1
1 1
2 2
1
2
2
2
1
2
,
θ opredelqetsq yz sootnoßenyq (13): θ =
n p
p n p
( )
( )
−
+ −
2
2 2
.
Vozvedem obe çasty posledneho neravenstva v stepen\ p, umnoΩym na g tτ( )
y proyntehryruem po t ot ρ do ν. Dalee dlq ocenky vtoroho slahaemoho pry-
menym neravenstvo Gnha s ε y neravenstvo (9), v¥brav v nem σ = τ:
G
t
p
u g t dx dt
ρ ν τ
τ
, ( )
( )∫∫ ≤
≤ D c D u dx g t dtp
n p
p p
n p
t
p
1 2
2
2
2
2 2
2
+
− − − −
∫ ∫ε τ
ρ
ν
τ
τ
( ) ( )
( )
( )
Ω
+
+ ε τ τ τ
ρ ν τ τ σ
τD u D u g t dx dtp p
G
p
t
p p p
1
2
2
, ( , )
( )
+
− − ′∫∫ +
+
+ ε ρ ττ
τ σρ
D g u
p
D u dxp p
t
p
1
2
2 21
2
1( )
( )Ω +
∫∫ +
. (15)
Pust\ σ τ τ= −2 1. Oboznaçym
S u
p
D u dxt
p
ρ
τ
τ
ρ
( )
( )
= +
∫∫
Ω
1
2
12 2
,
F s u D u g t dx dt
G
p
t
p p p
sρ ν
τ τ
τ τ σ σ
ρ ν
,
( , )
( , , ) ( )
,
1 2
2
1 2
= +
∫∫ ′
,
˜ ( , ),
( , ),
F u D u dx dt
G
p
t
p p p
ρ ν
τ τ
τ τ σ σ
ρ ν
1 2
2
1 2
= +
∫∫ ′
,
F s u D u g t dx dt
G
p
t
p p p
sρ ν
τ
τ τ τ
ρ ν
,
( )
( , ) ( )
,
= +( )∫∫ ′ 2
, F Fρ ν ρ ντ τ τ, ,( , ) ( )= ,
˜ ( ),
( ),
F u D u dx dt
G
p
t
p p p
ρ ν
τ
τ τ τ
ρ ν
= +
∫∫ ′ 2
.
Tohda neravenstvo (15) moΩno zapysat\ v vyde
G
t
p
u g t dx dt
ρ ν τ
τ
, ( )
( )∫∫ ≤
≤ ε τ τ τ τ ρ τρ ν τ ρ1 1 3 1
2
3D F g Sp p p p− − ′ +
, ( , ) ( ) ( ) +
+ c D D u dx g t dtp
n p
p p
n p
t
p
3 1 1
2
2
2
2 2
2
+
− − − −
∫ ∫ε τ
ρ
ν
τ
τ
( ) ( )
( )
( )
Ω
. (16)
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PRYNCYP FRAHMENA – LYNDELEFA DLQ NEKOTORÁX KVAZYLYNEJNÁX … 245
V sledugwej lemme dokazan¥ ocenky, xarakteryzugwye rost yntehrala
πnerhyy na beskoneçnosty.
Lemma 2. Dlq obobwennoho reßenyq u x t( , ) zadaçy (2) – (4), 0 ≤ ρ < ν ≤ T,
τ > τ 0 > 0 y µ τ( ) > 0 ymeet mesto sledugwee neravenstvo:
Fρ ν τ, ( ) ≤ c c F c
p
n p p p
T
p
p
4 2 5
2
2
0
2
1
6
1 6ε
µ τ
τ τ τ+ ( ) +
− − + + ′ − − − ′
( )
˜ ( )
( )
, ×
× F g Sp p
ρ ν τ ρτ τ ρ τ τ, ( , ) ( ) ( )3 1
2
3+
+ ′
. (17)
Dokazatel\stvo. Analohyçnaq ocenka v sluçae parabolyçeskyx uravnenyj
poluçena v [10]. Vospol\zuemsq predloΩennoj tam sxemoj dokazatel\stva.
Oboznaçym
Φ
Ω
( , ) ( ) ( )
( )
,t i u
p
D u x dx g z dz
t
t
p
i
t
= +
∫ ∫
ρ τ
τ τ τη
2
2 21
2
1
,
voz\mem v yntehral\nom toΩdestve (7) ν( , )x t = u x tt ( , ) Φ( , )t i ητ τ, ( )x . Dejstvuq
analohyçno dokazatel\stvu lemm¥ 1, poluçaem ravenstvo
1
2
2
2
2 2Φ
Ω
( , ) ( )
( )
,ν η
ν τ
τ τi u
p
D u x dxt
p
∫ +
–
– 1
2
2
2
2 2
G
t
p
tu
p
D u t i x dx dt
ρ ν τ
τ τη
, ( )
,( , ) ( )∫∫ +
Φ +
+
G
p
i
i
t
iD u D u D u D x t i dx dt
ρ ν τ τ α
α α
τ τη
, ( , )
, ( ) ( , )
2 2
2 2
0
1
∫∫ ∑ ∑
=
−
=
− Φ =
= − ( )∫∫ ∑ ∑
= =
−
G
t t
i
i
t
ia x t u D u D u D x t i dx dt
ρ ν τ α
α
α α
τ τη
, ( )
,, , , ( ) ( , )
2 1 0
1
Φ .
Neposredstvennoe v¥çyslenye Φ( , )t i t( )′
, oçevydnoe neravenstvo Φ( , )t i ≤
≤ Φ( , )ν i pry ρ ≤ t ≤ ν, uslovye (6), neravenstvo (9) s µ ≡ 1 pryvodqt k soot-
noßenyg
Φ Φ( , ) ( , ) ˜ ( , ) ( )( )
,ν ν τ τ τ τρ ν ρi c i F Sp p+ ≤ +[ ]− + ′
1 3 37 . (18)
Çtob¥ ocenyt\ Φ ν,
p
2
s necel¥m
p
2
, predstavym Φ ν,
p
2
s pomow\g
neravenstva Hel\dera:
Φ ν,
p
2
≤ Φ Φν ν
α α
, ,
p p
2
1
2
1
+
−
,
hde
p
2
— celaq çast\
p
2
, α =
p
2
–
p
2
. Prymenyv k kaΩdomu mnoΩytelg
pravoj çasty rekurrentnoe sootnoßenye (18)
p
2
y
p
2
– 1 raz sootvetstven-
no, poluçym
Φ ν,
p
2
≤ c F Sp p
p
8 2
1
3 3 1τ τ τ τ νρ ν ρ
− + ′ −
+[ ]( )
,
˜ ( , ) ( ) ( , )Φ . (19)
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246 Y. P. SLEPCOVA, A. E. ÍYÍKOV
Yz neravenstva (9) sleduet, çto
Φ( , )
( )
( , , ) ( ) ( )( )
,ν
µ τ
τ τ τ τ ρ τρ ν τ ρ1 3 39
2≤ +( )− + ′c
F g Sp p
.
Dlq ocenky Sρ τ( )3 yspol\zuem neravenstvo (9) na yntervale (0, ρ) s σ = τ,
µ τ( ) ≡ 0 y uçtem uslovyq (2):
Sρ τ( )3 ≤ τ τ τ− + ′( )
,
˜ ( , )p p
TF0 3 6 .
Ocenku (19) moΩno prodolΩyt\ sledugwym obrazom:
Φ ν,
p
2
≤
c
F F g S
p
p p
T
p
p p10
2
2
2
0
2
1
6 3 3
µ τ
τ τ τ τ τ ρ τρ ν τ ρ( )
˜ ( ) ( , ) ( ) ( )
( )
,
( )
,
− − + ′ − − + ′( ) +( ).
(20)
Zametym, çto
ρ
ν
τ
τ∫ ∫
Ω( )
( )u dx g t dtt
p
2
2
≤ Φ ν,
p
2
, y prymenym ocenku (20) k pravoj
çasty neravenstva (16):
G
t
p
u g t dx dt
ρ ν τ
τ
, ( )
( )∫∫ ≤
≤ ε τ
µ τ
ε τ τD
c
D c D Fp p p
n p
p p
p
p
n p p
T
p
1
10
2 1 2
2
2
2
0
2
1
6+ +
( )
− − − − + + ′ −
( )
˜ ( )
( ) ( )
( )
, ×
× τ τ τ ρ τρ ν τ ρ
− + ′ +[ ]( )
, ( , ) ( ) ( )p p F g S3 3 . (21)
Yz neravenstva (9) pry σ = τ sleduet, çto
G
p
D u g t dx dt
ρ ν τ
τ
, ( )
( )∫∫ 2 ≤ c1 2
1
µ τ( )
τ τ τ ρ τρ ν τ ρ
− + ′ +[ ]( )
, ( , ) ( ) ( )p p F g S3 3 . (22)
UmnoΩyv obe çasty neravenstv (21) y (22) na τ ′p
y τ p
sootvetstvenno, a za-
tem sloΩyv poluçenn¥e neravenstva, poluçym utverΩdenye lemm¥.
Lemma 3. Pust\ funkcyq u x t( , ) otlyçna ot toΩdestvennoj postoqnnoj
y udovletvorqet uslovyqm (9) y (17). Tohda suwestvuet posledovatel\nost\
{ }τi : τi → + ∞ takaq, çto
˜ ( ) ( ),F hT i i i0
1 2τ τ τγ γ− − → + ∞ pry i → + ∞, (23)
hde postoqnn¥e γ1 y γ 2 y funkcyq h( )τ opredelen¥ v teoreme 1.
Dokazatel\stvo provedem ot protyvnoho. PredpoloΩym, çto dlq lgboj
posledovatel\nosty { }si , i > 0, y proyzvol\noho çysla c11 > 0
˜ ( ) ( ),F s c s h sT i i i0 11
1 2≤ γ γ
. (24)
UkaΩem takug podposledovatel\nost\ πtoj posledovatel\nosty, çto v¥polne-
nye uslovyq (24) dlq nee vleçet toΩdestvo u x t( , ) ≡ const v G. V¥berem { }si
tak, çtob¥ si +1 = 3si . Yz uslovyq (24) y ocenky (17) dlq v¥brannoj posledova-
tel\nosty { }si sleduet neravenstvo
F siρ ν, ( ) ≤ c
s
h s s
i
i i
p
12 2 1
1ε
µ
+ +( )
− ′
( )
( ) F s s g s S si i s i
p p
iiρ ν ρρ, ( , ) ( ) ( )+
+ ′
++[ ]1 1 .
Zdes\ y dalee h s h s1 6( ) ( )= .
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PRYNCYP FRAHMENA – LYNDELEFA DLQ NEKOTORÁX KVAZYLYNEJNÁX … 247
V sylu monotonnoho neub¥vanyq posledovatel\nosty h si1( ) dlq vsex i,
bol\ßyx nekotoroho i0 0> , si
p− ′ < h si1( ), poπtomu h si1( ) + si
p− ′ ≤ c13 h si1( ).
Tohda yz posledneho neravenstva sleduet ocenka
F siρ ν, ( ) ≤ c
s
c h s
i
i12 2 13 1
1ε
µ
+
( )
( ) F s s g s S si i s i
p p
iiρ ν ρρ, ( , ) ( ) ( )+
+ ′
++[ ]1 1 . (25)
Yz posledovatel\nosty { }si v¥delym podposledovatel\nost\ { }τ j , udovletvo-
rqgwug uslovyg
1
3
2
1
2τ τ τj j j< ≤+ , j = 1, 2, … , (26)
y poloΩym
µ τ εj jh c c2
1 1 13 12
1= ( ) ( )+
−
. (27)
Na yntervale τ τj j−( ]1, dlq vsex j > 1 funkcyg µ τ( ) opredelym ravenstvom
µ τ( ) ≡ µ j −1.
Pust\ 0 < t j < t j −1 ≤ T, j = 1, 2, … . Na yntervale t tj j, −( )1 yz neravenst-
va;(25) y monotonnosty funkcyy h s1( ) sleduet, çto
F s c F s g t s S st t k t t k s j k
p p
t kj j j j k j, ,( ) ( ) ( ) ( )
− −
≤ +[ ]+
+ ′
+1 1
2 12 1 1ε . (28)
PoloΩym ε α= ( )−2 12
1
c , hde α > 3p p+ ′
. Proyteryruem neravenstvo (28) po
k, naçynaq s sk0
= 3 1τ j − , do τ j q = log3
13
τ
τ
j
j −
raz. Dlq posledovatel\nosty
{ }τ j so svojstvom (26) y dlq funkcyy µ τ( ), opredelennoj ravenstvom;(27),
v¥polnena ocenka
F st t kj j, − ( )1 0
= Ft t jj j, ( )
− −1
3 1τ ≤
≤ α τ τα2
1
3
1j t t jF
j j−
−
−
log
, ( ) + 1
2
3
3
1 1
p p
p p j
p p
j t jg t S
j j
+ ′
+ ′ −
+ ′
− −α
τ ττ ( ) ( ) . (29)
V neravenstve (9) pry τ = τ j −1, ρ = t j , ν = t j −1 pravug çast\ ocenym s pomow\g
sootnoßenyq (29):
g t S c F Kg t S
j j j j j jj j j
p p
j t t j j jτ τ
γ
τ ττ τ α τ τ τ
− − − −− − −
− + ′
−
−≤ +
1 1
0
1 11 1 1 1
2
1( ) ( ) ( ) ( ) ( )( )
, , (30)
hde K
p p
p p
= +
−
+ ′
+ ′1 3
3α
, γ α0 3= log .
Oboznaçym δ j = t j −1 – t j , j = 1, 2, … , H j( ) = c j
p p
1
2
1α τ −
− + ′( )
; exp ln−( −γ τ0 1j +
+; µ δj j− − )1
2
1 , Lj = K j jexp µ δ− −( )1
2
1 . V¥berem posledovatel\nost\ { }t j sledug-
wym obrazom:
δ γ β µ τj j j= − −
0
21( ) ln , (31)
parametr¥ β < 1 y γ 0 0> opredelym dalee. Dlq takyx t j H j( ) =
=; c j
p p
1
2
1
0α τ βγ
−
− + ′ −( )
, Lj = K jτγ β
−
−
1
10 ( )
. Yz sootnoßenyq (30) sleduet, çto
S H j F L j St j t t j t jj j j j− −− ≤ +
1 11( ) ( ) ˜ ( ) ( ) ( ),τ τ τ . (32)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
248 Y. P. SLEPCOVA, A. E. ÍYÍKOV
Dlq znaçenyj t j , τ j y µ τ( ), udovletvorqgwyx uslovyqm (31), (26) y (27)
sootvetstvenno, v¥polneno neravenstvo
δ β γ
τ
τ
j c
dz
zh z
j
j
− ≥ − +
∫1
0
14 6
6
1
1
( )
( )
,
pryçem yz uslovyj i), ii) sleduet, çto dlq proyzvol\noho t yz yntervala (0, T )
suwestvuet takoe E = E t( , )τ0 , çto
i
E
i t=
−∑ >
1
1δ ,
i
E
i t=∑ <
1
δ , y St EE
( )τ = 0 . S
uçetom πtoho proyteryruem neravenstvo (32) po j E raz:
S H F H i L j Ft t t
i
E
j
j
t t ii i0 1 0 10 1
2 1
1
1( ) ( ) ˜ ( ) ( ) ( ) ˜ ( ), ,τ τ τ≤ +
= =
−
∑ ∏ −
. (33)
Poskol\ku sohlasno (26) τ τ τ τ τ0 1 1
1
0
1
1
23… ≤−
− −
−i
i
i , to
i
E
j
i
H i L j
= =
−
∑ ∏
2 1
1
( ) ( ) = c j
p p
1
2
1
0α τ βγ
−
− + ′ −( ) Ki
i
−
−
−
…( )1
0 1 1
10τ τ τ
γ β( )
≤
≤ c Ki i
i
p p
1
2 1 1 1
0
1
1
2 1 23 0 0 0α τ τγ β γ β γ β− − − − −
−
− + ′ + −( )( ) ( ) ( ) ( )
. (34)
V¥berem β tak, çtob¥ 2β – 1 = β1 > 1. Oboznaçym c i0( ) = c Ki
1
2 1α − ×
× 3 0 1 1γ β( )( )i − − τi
p p
−
− + ′
1
( )
. Zametym, çto H( )1 = c1
2α τ βγ
0
0− + ′ −( )p p , c0 1( ) =
= c1
2α τ0
− + ′( )p p
. Tohda yz neravenstv (32), (34) y predpoloΩenyq (24) sleduet
ocenka sverxu funkcyy St 0 0( )τ :
St 0 0( )τ ≤
i
E
i t t ic i F
i i
=
− −
−
−∑ −
1
0 0
1
1
20 1 0
1
( ) ˜ ( )( )
,τ τ τγ β β γ ≤
≤
i
E
i i ic i h
=
− −
−
−∑
1
0 0
1
1
20 1 0 1 2( ) ( )( )τ τ τ τγ β β γ γ γ
.
Dlq posledovatel\nosty τi y funkcyy h( )τ , udovletvorqgwyx uslovyqm
(26) y ii) sootvetstvenno, yz posledneho neravenstva sleduet, çto
St 0 0( )τ ≤
i
E
i
pc i
=
− −
− + +
−∑
1
0 0
1
2
20
1 0
( ) ( )τ τγ β
β γ γ ϕ
. (35)
V¥berem parametr γ 0 tak, çtob¥
− + +
−
< <β γ γ ϕ β1 0 2
2
2
0
p
. (36)
Koπffycyent¥ c i0( ) ravnomerno ohranyçen¥ otnosytel\no i y τ0 nekotoroj
konstantoj d0 0> , poπtomu v sylu uslovyq (36) neravenstvo (35) v¥polneno,
esly St 0 0( )τ ≤ d
i i0 0
1
1
0 2τ τγ β β− −
=
∞∑( )
. Rqd
i i=
∞∑ 1
2τβ
sxodytsq, a znaçyt, pry
τ 0 → ∞ St 0 0( )τ → 0. V sylu proyzvol\nosty t0 0> y vozmoΩnosty v¥bora
skol\ uhodno bol\ßoho τ0 sleduet, çto u x t( , ) ≡ const v G.
Lemma dokazana.
Dlq zaverßenyq dokazatel\stva teorem¥ zametym, çto esly v¥byrat\ pod-
xodqwym obrazom srezagwug funkcyg ητ σ, ( )x v opredelenyy probnoj funk-
cyy ν( , )x t , to vse utverΩdenyq lemm 1 – 3 budut spravedlyv¥ dlq proyzvol\-
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
PRYNCYP FRAHMENA – LYNDELEFA DLQ NEKOTORÁX KVAZYLYNEJNÁX … 249
noj posledovatel\nosty { }si : si +1 = ksi , k ∈ ∞( , )1 . ∏to zameçanye y uslovyq (3)
y dokaz¥vagt teoremu.
Zameçanye 2. UtverΩdenye teorem¥ y dokazatel\stvo s sootvetstvugwymy
yzmenenyqmy ostagtsq spravedlyv¥my dlq uravnenyq (2) s operatorom B =
= α
α
=∑ 2
D D D
pα α−
2
.
Zameçanye 3. Sxema, predloΩennaq v¥ße, pozvolqet opredelyt\ klass¥
rastuwyx na beskoneçnosty reßenyj smeßannoj zadaçy dlq uravnenyq (1) s
operatoramy A y B porqdkov 2m y 2l ≤ 2m + 2 sootvetstvenno, ymegwyx
strukturu, analohyçnug rassmotrennoj v teoreme.
1. Lax P. D. Phragman – Lindelef theorem in harmonic analysis and its application in the theory of
elliptic equations // Communs Pure and Appl. Math. – 1957. – 10, # 3. – P. 361 – 389.
2. Landys E. M. O povedenyy reßenyj πllyptyçeskyx uravnenyj v¥sokoho porqdka v neohra-
nyçenn¥x oblastqx // Tr. Mosk. mat. o-va. – 1974. – 31. – S. 35 – 58.
3. Kalantarov V. K., Tahamtani F. Phragmen – Lindelöf principle for a class of fourth order nonlinear
elliptic equations // Bull. Iran. Math. Soc. – 1994. – 20, # 2. – P. 41 – 52.
4. Olejnyk O. A., Yosyf\qn H. A. Analoh pryncypa Sen-Venana y edynstvennost\ reßenyj
kraev¥x zadaç v neohranyçenn¥x oblastqx dlq parabolyçeskyx uravnenyj // Uspexy mat.
nauk. – 1976. – 31, # 6. – S. 142 – 166.
5. Kalantarov V. K. Ob ocenkax reßenyj kvazylynejn¥x parabolyçeskyx uravnenyj v ne-
ohranyçenn¥x oblastqx // Mat. sb. – 1991. – 182, # 8. – S. 1200 – 1210.
6. Íyßkov A. E. Pryncyp Frahmena – Lyndelefa dlq dyverhentn¥x parabolyçeskyx urav-
nenyj // Syb. mat. Ωurn. – 1989. – 30, # 2. – S. 203 – 212.
7. Hlaholeva R. Q. Teorem¥ Frahmena – Lyndelefa y lyuvyllev¥ teorem¥ dlq lynejnoho
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Poluçeno 03.11.2003,
posle dorabotky — 09.08.2004
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 2
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| id | umjimathkievua-article-3591 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:45:23Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/3b/4dee4ca9f545434a26272e1d1e778a3b.pdf |
| spelling | umjimathkievua-article-35912020-03-18T19:59:22Z Phragmen-Lindelof Principle for Some Quasilinear Evolution Equations of the Second Order Принцип Фрагмена - Линделефа для некоторых квазилинейных эволюционных уравнений второго порядка Sleptsova, I. P. Shishkov, A. E. Слепцова, И. П. Шишков, А. Е. Слепцова, И. П. Шишков, А. Е. We consider the equation $u_{tt} + A (u_t) + B(u) = 0$, where $A$ and $B$ are quasilinear operators with respect to the variable x of the second order and the fourth order, respectively. In a cylindrical domain unbounded with respect to the space variables, we obtain estimates that characterize the minimum growth of any nonzero solution of the mixed problem at infinity. Розглянуто рівняння $u_{tt} + A (u_t) + B(u) = 0$, в якому $A$ i $B$ — квазілінійні оператори за змінною x другого і четвертого порядків відповідно. В необмеженій за просторовими змінними циліндричній області отримано оцінки, які характеризують мінімальний ріст будь-якого ненульового розв'язку мішаної задачі на нескінченності. Institute of Mathematics, NAS of Ukraine 2005-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3591 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 2 (2005); 239–249 Український математичний журнал; Том 57 № 2 (2005); 239–249 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3591/3914 https://umj.imath.kiev.ua/index.php/umj/article/view/3591/3915 Copyright (c) 2005 Sleptsova I. P.; Shishkov A. E. |
| spellingShingle | Sleptsova, I. P. Shishkov, A. E. Слепцова, И. П. Шишков, А. Е. Слепцова, И. П. Шишков, А. Е. Phragmen-Lindelof Principle for Some Quasilinear Evolution Equations of the Second Order |
| title | Phragmen-Lindelof Principle for Some Quasilinear Evolution Equations of the Second Order |
| title_alt | Принцип Фрагмена - Линделефа для некоторых квазилинейных эволюционных уравнений второго порядка |
| title_full | Phragmen-Lindelof Principle for Some Quasilinear Evolution Equations of the Second Order |
| title_fullStr | Phragmen-Lindelof Principle for Some Quasilinear Evolution Equations of the Second Order |
| title_full_unstemmed | Phragmen-Lindelof Principle for Some Quasilinear Evolution Equations of the Second Order |
| title_short | Phragmen-Lindelof Principle for Some Quasilinear Evolution Equations of the Second Order |
| title_sort | phragmen-lindelof principle for some quasilinear evolution equations of the second order |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3591 |
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