Global attractor for the autonomous wave equation in Rn with continuous nonlinearity
We investigate the dynamics of solutions of an autonomous wave equation in ℝn with continuous nonlinearity. A priori estimates are obtained. We substantiate the existence of an invariant global attractor for an m-semiflow.
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| Datum: | 2008 |
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| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2008
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3154 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509195108876288 |
|---|---|
| author | Horban’, N. V. Stanzhitskii, A. N. Горбань, Н. В. Станжицький, О. М. |
| author_facet | Horban’, N. V. Stanzhitskii, A. N. Горбань, Н. В. Станжицький, О. М. |
| author_sort | Horban’, N. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:46:54Z |
| description | We investigate the dynamics of solutions of an autonomous wave equation in ℝn with continuous nonlinearity. A priori estimates are obtained. We substantiate the existence of an invariant global attractor for an m-semiflow. |
| first_indexed | 2026-03-24T02:37:14Z |
| format | Article |
| fulltext |
UDK 517.9
O. M. StanΩyc\kyj, N. V. Horban\ (Ky]v. nac. un-t im. T. Íevçenka)
HLOBAL|NYJ ATRAKTOR
DLQ AVTONOMNOHO XVYL|OVOHO RIVNQNNQ V Rn
Z NEPERERVNOG NELINIJNISTG
We consider the dynamics of solutions of autonomous wave equation in R
n with continuous
nonlinearity. The a priori estimates are obtained. The existence of compact invariant global attractor for
m-semiflow is justified.
Yssledovana dynamyka reßenyj avtonomnoho volnovoho uravnenyq v R
n
s neprer¥vnoj nely-
nejnost\g. Poluçen¥ apryorn¥e ocenky. Dlq m-polupotoka obosnovano suwestvovanye ynva-
ryantnoho hlobal\noho attraktora.
1. Vstup. Teoriq hlobal\nyx atraktoriv neskinçennovymirnyx dynamiçnyx sys-
tem bula zapoçatkovana v 70-x rokax mynuloho stolittq v robotax O.1A.1Lady-
Ωens\ko] po vyvçenng dynamiky dvovymirno] systemy rivnqn\ Nav’[ – Stoksa ta
v robotax J. K. Hale, qki stosuvalysq doslidΩennq qkisno] povedinky funkcio-
nal\no-dyferencial\nyx rivnqn\. Prote burxlyvyj rozvytok ci[] teori], wo
prodovΩu[t\sq i s\ohodni, rozpoçavsq v seredyni 80-x rokiv, koly z’qsuvalosq,
wo na abstraktnomu rivni ti xarakterni rysy, wo dozvolqly z toçky zoru teori]
hlobal\nyx atraktoriv doslidΩuvaty rivnqnnq Nav’[ – Stoksa ta rivnqnnq iz za-
piznennqm, vlastyvi ßyrokomu klasu evolgcijnyx rivnqn\, wo opysugt\ real\-
no isnugçi pryrodni i suspil\ni qvywa: teçig v’qzko] nestyslyvo] ridyny, pro-
cesy ximiçno] kinetyky, riznomanitni xvyl\ovi procesy, fizyçni procesy fazovo-
ho perexodu, kolyvannq obolonok u nadßvydkyx hazovyx potokax, funkcionu-
vannq zamknenyx ekonomiçnyx system towo. Vahomyj vnesok u stanovlennq ta
rozvytok klasyçno] teori] hlobal\nyx atraktoriv neskinçennovymirnyx dyna-
miçnyx system vnesly M. I. Vyßyk, O. A. LadyΩens\ka, V. S. Mel\nyk, I.1Çu[-
ßov, J. M. Ball, J. K. Hale, R.Temam, B. Wang, S. V. Zelik ta ]xni uçni [1 – 19].
Rezul\taty wodo isnuvannq ta vlastyvostej rozv’qzkiv xvyl\ovoho rivnqnnq
z dysypaci[g v obmeΩenij oblasti u vypadku hladkoho za fazovog zminnog neli-
nijnoho dodanka, qk i rezul\taty wodo isnuvannq v c\omu vypadku hlobal\noho
atraktora, [ klasyçnymy i mistqt\sq v [1, 17], dlq neavtonomnyx rivnqn\ z maj-
Ωe periodyçnog zaleΩnistg vid çasovo] zminno] — v [6], dlq vypadku neobmeΩe-
no] oblasti dlq odnoznaçnyx napivhrup — v [19, 5] . Bez dodatkovyx umov wodo
hladkosti nelinijnoho dodanka v avtonomnomu vypadku isnuvannq kompaktnoho
hlobal\noho atraktora dlq vidpovidno] bahatoznaçno] napivhrupy dlq xvyl\ovo-
ho rivnqnnq v obmeΩenij oblasti bulo dovedeno v [8] i pry bil\ß zahal\nyx umo-
vax — v [4]. Isnuvannq tra[ktornoho atraktora bulo dovedeno v [7].
Naßa zadaça polqha[ v doslidΩenni dynamiky rozv’qzkiv xvyl\ovoho rivnqn-
nq v R
n
bez [dynosti rozv’qzku.
2. Isnuvannq ta vlastyvosti rozv’qzkiv. Rozhlqnemo rivnqnnq
utt + γut – ∆u + f x u( , ) + λ0u = h x( ), ( , ) ( , )t x T∈ τ × Rn
, (1)
de γ > 0, λ0 > 0, τ ∈R — poçatkovyj moment çasu, T > τ, n ≥ 3, f — vymirna po
x i neperervna po u funkciq. Nexaj vykonano umovy
h L n∈ 2( )R , ∃ ∈ ( )C L n
1
1
R ∩ L
n2
R( ), C1 0≥ , ∃ ≥c 0:
f x u( , ) ≤ C x1( ) + c u ∀ ∈( , )x u n
R × R ,
∃ ∈
α γ
0
2
, , ∃ ∈( )λ λ0 0, , α γ α( – ) < λ0 – λ, (2)
© O. M. STANÛYC|KYJ, N. V. HORBAN|, 2008
260 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
HLOBAL|NYJ ATRAKTOR DLQ AVTONOMNOHO XVYL|OVOHO RIVNQNNQ… 261
∃ ∈ ( )C Li
n
1 R , Ci ≥ 0, i = 2, 3 , ∀ ∈( , )x u n
R × R :
F x u( , ) : = f x s ds
u
( , )
0
∫ , F x u( , ) ≥ –λ
2
2u – C x2( ),
f x u u( , ) – F x u( , ) ≥ –λ
2
2u – C x3( ).
Dali γ, Ci , i = 1, 2, 3, c, λ, α , λ0 budemo nazyvaty konstantamy zadaçi (1).
Oskil\ky F zadovol\nq[ umovy Karateodori, to
∀ ∈ ×( , )x u n
R R 1: F x u( , ) ≤ C x u1( ) + c u
2
2
. (3)
Budemo poznaçaty çerez ⋅ , ( , )⋅ ⋅ i ⋅ , ( , )⋅ ⋅( ) normu i skalqrnyj dobutok v
L
n2
R( ) i H n1
R( ) vidpovidno. ZauvaΩymo, wo
∀ ∈ ( )u H n, v 1
R 1: ( , )u v( ) = λ0( , )u v +
∂
∂
∂
∂
u
x xi ii
n
, v
=
∑
1
.
Fazovym prostorom zadaçi (1) [ prostir E = H n1
R( ) × L n2
R( ).
Rozv’qzok zadaçi (1) budemo rozumity v sensi nastupnoho oznaçennq.
Oznaçennq 1. Funkcig ϕ ( )⋅ = u ut
T( ), ( )⋅ ⋅( ) ∈ L T E∞( , ; )τ nazyvagt\ roz-
v’qzkom zadaçi (1) na ( , )τ T , qkwo
∀ ∈ ( )ψ H n
0
1
R ∀ ∈ ∞η τC0 ( , )T :
– ( , )u dtt t
T
ψ η
τ
∫ + γ ψ ψ
τ
( , ) ( ,u ut
T
+ ( )( )∫ + f x u( , ), ψ( ) – ( , )h dtψ η) = 0.
Rozhlqnemo klas funkcij WT
τ = C τ, ;T E[ ]( ) . Za umovamy (2), (3) dlq do-
vil\no] funkci] ϕ ( )⋅ = u ut
T( ), ( )⋅ ⋅( ) ∈ WT
τ korektno oznaçeno nastupni funkcio-
naly:
V tϕ( )( ) = 1
2
2u tt ( ) + 1
2
2u t( ) + F x u t( , ( )), 1( ) ,
I tϕ( )( ) = V tϕ( )( ) +
γ
2
( ( ), ( ))u t u tt ,
H tϕ( )( ) = γ F x u t( , ( )), 1( ) –
γ
2
f x u t u t( , ( )), ( )( ) +
γ
2
h u t, ( )( ) + h u tt, ( )( ).
Lema 1. Dlq dovil\noho rozv’qzku ϕ ( )⋅ = u ut
T( ), ( )⋅ ⋅( ) ∈ WT
τ zadaçi1(1)
spravdΩu[t\sq ocinka
∀ ≥t s , t, s T∈[ ]τ, 1:
u tt ( ) 2 + u t( ) 2 ≤ C u s u s et
t s
4
2 2( ( ) ( ) ) ( – )+( −α + h 2 1+ ) ,
de konstanta C4 > 0 zaleΩyt\ lyße vid konstant zadaçi1(1).
Pry c\omu funkci] V ϕ( )⋅( ), I ϕ( )⋅( ), H ϕ( )⋅( ) [ absolgtno neperervnymy na
τ, T[ ] i dlq majΩe vsix t ∈ τ, T[ ]
d
dt
V tϕ( )( ) = – ( )γ u tt
2 + h u tt, ( )( ),
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
262 O. M. STANÛYC|KYJ, N. V. HORBAN|
d
dt
u t u tt ( ), ( )( ) = u tt ( ) 2 – γ ( ( ), ( ))u t u tt – u t( ) 2 –
– f x u t u t( , ( )), ( ))( + h u t, ( )( ) , (4)
d
dt
I tϕ( )( ) = – ( )γ ϕI t( ) + H tϕ( )( ) .
Dovedennq. Nexaj ϕ ( )⋅ = u ut
T( ), ( )⋅ ⋅( ) ∈ WT
τ — dovil\nyj rozv’qzok zada-
çi1(1) na ( , )τ T . Todi na pidstavi (2) f x u( , ) ∈ L T L n2 2τ, ; R( )( ). OtΩe, funkciq
t � u tt ( ) 2 + u t( ) 2
[ absolgtno neperervnog na τ, T[ ] i majΩe skriz\
1
2
2 2d
dt
u ut +{ } = −γ ut
2 – f x u ut( , ),( ) + ( , )h ut . (5)
Dlq toho wob dovesty, wo funkciq t � F x u t( , ( )), 1( ) [ absolgtno nepererv-
nog na τ, T[ ] i majΩe skriz\ na τ, T[ ] vykonu[t\sq rivnist\
d
dt
F x u( , ), 1( ) = f x u ut( , ),( ) , (6)
dostatn\o dovesty ]] neperervnist\ na τ, T[ ] i vykonannq (6) u sensi skalqrnyx
rozpodiliv na ( , )τ T . Dovedennq [ analohiçnym [2, 3].
Rozhlqnemo funkcig
Y t( ) = 1
2
2u tt ( ) + 1
2
2u t( ) + F x u t( , ( )), 1( ) + α u t u tt ( ), ( )( ).
Na pidstavi rivnosti
d
dt
F x u t( , ( )), 1( ) = f x u t u tt( , ( )), ( )( ) i (5) ma[mo
dY
dt
= −( – )γ α ut
2 – α u 2 – αγ ( , )u ut – α f x u u( , ),( ) + α( , )u h + ( , )u ht .
Za umovamy na α isnu[ take ε > 0, wo α γ α( – ) ≤ ( – )λ λ0 1 2– ε
α
. Zvidsy
γ α–
2
2ut + α α λ
λ
ε
2 2 0
2– –
u – α γ α( – ) u ut ≥ 0.
Ce pryvodyt\ do nerivnosti
dY
dt
≤ −αY –
γ α
2
2–
ut – αλ
αλ
ε
0
2+
u – α f x u u( , ),( ) –
– α F x u( , ), 1( ) + α u h,( ) + u ht ,( ) .
Zastosovugçy do ostann\o] umovy (2), otrymu[mo nerivnist\
dY t
dt
( ) ≤ −αY t( ) + C hε 1 2+( ). (7)
Tut Cε = max ( )α C L n3 1 R
; 1
2 2 4
2
0( – )γ α
α
λ ε
+
. Iz (7) na pidstavi umov (2) dlq
T ≥ t ≥ s ≥ τ ma[mo
1
2
2u tt ( ) + 1
2
1
2 0
2– ( )λ
λ
u t + α u t u tt ( ), ( )( ) –
C L n2 1( )R
≤
≤ 1
2
1
2
12 2u s u s F x u s u s u s et t
t s( ) ( ) ( , ( )), ( ), ( ) ( – )+ + ( ) + ( ){ } −α α +
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
263 O. M. STANÛYC|KYJ, N. V. HORBAN|
+
C
h e t sε α
α
1 12+( )( )−– ( – ) .
Todi isnu[ konstanta C4 > 0, qka zaleΩyt\ lyße vid konstant zadaçi (1), taka,
wo dlq T ≥ t ≥ s ≥ τ spravdΩu[t\sq ocinka
u tt ( ) 2 + u t( ) 2 ≤ C u s u s e h et
t s t s
4
2 2 21 1( ) ( ) –( – ) ( – )+( ) + +( )( ){ }− −α α ≤
≤ C u s u s e ht
t s
4
2 2 21( ) ( ) ( – )+( ) + +{ }−α .
ZauvaΩymo, wo
C4 =
max ; ; ; ;( )1
2
2 1 2
2
1
2
2
0
0
1
C
C
C c
L n
R
ε
α
λ α
λ
ε+ + +
∗
.
Tut ε∗ > 0 zadovol\nq[ nerivnist\ α γ α ε( – )( )1 2+ ∗ ≤ λ0 – λ.
Rivnosti (5), (6) dozvolqgt\ oderΩaty (4).
Lemu dovedeno.
Oskil\ky H n1
R( ) neperervno vklada[t\sq v L n2
R( ), to z umov (2) dlq u ∈
∈ L T H n∞ (( ))τ, ; 1
R ma[mo vkladennq f x u( , ) ∈ L T L n2 2τ, ; R(( )). OtΩe, zhidno z
[17] dlq koΩnoho rozv’qzku ϕ( )⋅ zadaçi (1) ma[mo ϕ( )⋅ ∈ C τ, ;T E[ ]( ) , wo i obu-
movlg[ vybir klasu WT
τ . Vkladennq ϕ( )⋅ ∈ C τ, ;T E[ ]( ) dozvolq[ dlq zada-
çi1(1) stavyty zadaçu Koßi vyhlqdu
u t = 0 = u H n
0
1∈ ( )R , ut t = 0 = v0
2∈ ( )L n
R (8)
i ßukaty rozv’qzok lyße u klasi L T E∞( , ; )τ .
Dlq dovedennq rozv’qznosti rozhlqnemo zadaçu Dirixle v obmeΩenij oblasti
utt + γut – ∆u + λ0u + f x u( , ) = h x( ), t > 0, x R∈Ω ,
u
R∂Ω = 0, t > 0, (9)
u t = 0 = u HR R0 0
1
, ( )∈ Ω , ut t = 0 = v0
2
, ( )R RL∈ Ω ,
de ΩR = B R( ; )0 — vidkryta kulq radiusa R ≥ 1 z centrom u nuli, u xR0, ( ) =
= u x xR0( ) ( )ψ , v0, ( )R x = v0( ) ( )x xRψ , ψ R — hladka funkciq,
ψ ξR( ) =
1 1
0 1 1
0
, – ,
( ) , – ,
, .
qkwo 0
qkwo
qkwo
≤ ≤
≤ ≤ ≤ ≤
>
ξ
ψ ξ ξ
ξ
R
R R
R
R
Isnuvannq rozv’qzku zadaçi (9) vstanovlg[t\sq metodom hal\orkins\kyx aprok-
symacij analohiçno do [2, 3] dlq dovil\nyx u R0, ∈1 H R0
1( )Ω , v0, R ∈ 1 L R
2( )Ω (oz-
naçennq rozv’qzku take Ω, qk i v oznaçenni 1, pry c\omu slid R
n
zaminyty na
ΩR). Prypustymo, wo rivnomirno po R > 1 ′ψ R obmeΩena na R+ . Poznaçymo
ER = H R0
1( )Ω × L R
2( )Ω . Budemo poznaçaty çerez ⋅ R, ( , )⋅ ⋅ R i ⋅ R, ( , )⋅ ⋅( ) R
normu i skalqrnyj dobutok v L R
2( )Ω i H R0
1( )Ω vidpovidno. ZauvaΩymo, wo
∀ ∈u H R, ( )v 0
1 Ω : ( , )u Rv( ) = λ0( , )u Rv +
∂
∂
∂
∂
u
x xi ii
n
R, v
=
∑
1
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
264 O. M. STANÛYC|KYJ, N. V. HORBAN|
Teorema 1. Dlq dovil\nyx ϕ0 = ( , )u T
0 0v ∈ E , T > 0 zadaça (1), (8) za
umov1(2) ma[ prynajmni odyn rozv’qzok u klasi WT
0 .
Dovedennq. Nexaj u rj
, rj → + ∞, — poslidovnist\ rozv’qzkiv zadaçi1(9).
ZauvaΩymo, wo
v v0 0
2
– ,rj
= 1
2
0
2– ( )ψrj
n
x dx( )∫
R
v ≤
≤ v0
2
1
0dx
x rj
→
≥
∫
–
pry rj → + ∞.
Analohiçno,
u u rj0 0
2
0– , → pry rj → + ∞.
Povtoryvßy dovedennq lemy11, oderΩymo
d
dt
u tr
r
j
j
( )
2
+ u tr rj
j
( )
2
≤
C e hr r
t s
j j4 0
2
0
2 2 1v u, ,
( – )+
+ +{ }−α
,
de konstanta C4 > 0 zaleΩyt\ lyße vid konstant zadaçi (1). OtΩe, funkciq
ϕrj
( )⋅ = u d
dt
ur r
T
j j
( ), ( )⋅ ⋅
[ obmeΩenog v L T Erj
∞( , ; )τ rivnomirno po rj → + ∞.
ProdovΩymo rozv’qzky zadaç po R
n
. Poklademo
ˆ ( )u xrj
=
u x x B rjr rj j
( ) ( , ),ϕ ( )
v
v inßyx vypadkax,
0
0
ˆ ( )ϕrj
x =
ϕ ψr rj j
x x B rj( ) ( , ),( )
v
v inßyx vypadkax.
0
0
Oskil\ky ϕrj
obmeΩeni v L T Erj
∞( , ; )0 rivnomirno pry rj → ∞, to ϕ̂rj
ta-
koΩ rivnomirno obmeΩeni v L T E∞( , ; )0 . Takym çynom, z toçnistg do pidposli-
dovnosti isnu[ pidposlidovnist\ poslidovnosti ϕ̂rj{ }, qku znovu poznaçymo
çerez ϕrj{ }, dlq qko]
ϕrj
→ ϕ∞ = u d
dt
u
T
∞ ∞
, ∗-slabko v L T E∞( , ; )0 ,
tobto
urj
→ u∞ ∗-slabko v L T H n∞( )0 1, ; ( )R ,
d
dt
urj
→ d
dt
u∞ ∗-slabko v L T L n∞( )0 2, ; ( )R .
Dovedemo, wo ϕ∞ — rozv’qzok zadaçi (1), (8). Dovedennq analohiçne dovedenng
teoremy15 iz [15]. Ideq dovedennq polqha[ u tomu, wob, zafiksuvavßy rk (iz
rj → + ∞ moΩna prypustyty, wo rk ≤ rj – 1), poznaçyvßy çerez ϕk j proekcig
ϕrj
na B rk( , )0 ( )ϕ ϕk j k rL
j
= i znagçy, wo
ϕk j → ϕk∞ = u d
dt
uk k
T
∞ ∞
, ∗-slabko v L T Er k
∞( , ; )0 ,
pereviryty, wo
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
265 O. M. STANÛYC|KYJ, N. V. HORBAN|
Lk ϕ∞ = ϕk,∞ , L
u
tk
rj
∂
∂
=
∂
∂
L u
t
k rj →
∂
∂
u
t
k ∞
slabko v L T L B rk
∞ ( )( )0 02, ; ( , ) ,
f x L uk rj
( , ) → f x uk( , )∞ slabko v L T L B rk
2
20 0, ; ( , )( )( ) ,
L uk rj
→ uk∞ syl\no v L T L B rk
2
20 0, ; ( , )( )( ) .
Dali, vykorystavßy te, wo
∀ ∈ [ ] × ( )( )∞v C T B rk0 0 0, , 1:
L u dtk r tt
T
j
, v( )∫
0
–
γ λ( , ) ( , – )L u L uk r t k r
T
j j
v v v+(∫ ∆ 0
0
–
( ( , ), ) – ( , )f x L u h dtk rj
v v ) = 0
i perejßovßy do hranyci, oderΩymo ßukane tverdΩennq.
Teoremu dovedeno.
Po[dnugçy teoremu11 ta lemu11, oderΩu[mo, wo dlq dovil\nyx ϕ0 =
= ( , )u T
0 0v ∈ E zadaça (1), (8) za umov (2) ma[ prynajmni odyn rozv’qzok u klasi
C E( , );0 +∞( ) ∩ L E∞ +∞( )( ; );0 .
Lema 2. Dlq dovil\noho ( , )u T
0 0v ∈ B ( B � E [ obmeΩenog) i dovil\noho
rozv’qzku ϕ ∈1 C E( , );0 +∞( ) zadaçi (1), (8) za umov (2) dlq dovil\noho ε > 0 i s -
nugt\ T B( , )ε , K B( , )ε taki, wo
∀ ≥t T , k K≥ : ∂
∂
λ ∂
∂
ε
t
u t x u t x
u t x
x
dx
ii
n
x k
( , ) ( , )
( , )2
0
2
1
2
2
+ +
≤
=≥
∑∫ .
Dovedennq vyplyva[ z lemy11 ta z rezul\tativ [15, 2, 3].
Iz lemy12, analohiçno do [15, 2, 3], moΩna oderΩaty take tverdΩennq.
Teorema 2. Nexaj ϕn{ } � WT
τ — poslidovnist\ rozv’qzkiv zadaçi (1), pry-
çomu ϕ τn( ) → ϕτ slabko v E. Nexaj zadano poslidovnist\ tn{ } � τ, T[ ] ta-
ku, wo tn → t0 ∈ τ, T[ ]. Todi isnu[ rozv’qzok ϕ ∈ WT
τ zadaçi (1) takyj, wo
ϕ τ( ) = ϕτ i prynajmni po pidposlidovnosti ϕn
nt( ) → ϕ( )t0 slabko v E.
Qkwo Ω ϕ τn( ) → ϕτ syl\no v E , to prynajmni po pidposlidovnosti
ϕn
nt( ) → ϕ( )t0 syl\no v E.
Poklademo W0
∞ = C E( , );0 +∞( ). Teper dlq dovil\nyx t ≥ 0, ϕ0 ∈ E rozhlq-
nemo mnoΩynu
G t( , )ϕ0 = ϕ ϕ( ) ( )t W⋅ ∈{ ∞
0 — rozv’qzok (1), ϕ ϕ( )0 0= } � E. (10)
Naslidok. MnoΩyna G t( , )ϕ0 — kompakt v E.
3. Pobudova avtonomno] dynamiçno] systemy ta isnuvannq hlobal\noho
atraktora. Nexaj ( , )X ρ — metryçnyj prostir. Dlq neporoΩnix A, B � X
dist ( , )A B = sup inf ( , )
x A y B
x y
∈ ∈
ρ , distH A B( , ) = max ( , ), ( , )dist distA B B A{ },
O Aδ( ) = x X x A∈ <{ }dist( , ) δ , Br = x X x r∈ ≤{ }ρ( , )0 ,
A = clX A — zamykannq A v X, P X( ) — sukupnist\ usix neporoΩnix pidmno-
Ωyn X, β( )X — sukupnist\ usix neporoΩnix obmeΩenyx pidmnoΩyn X, C X( ) —
sukupnist\ usix neporoΩnix zamknenyx pidmnoΩyn v X, K X( ) — sukupnist\ usix
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
266 O. M. STANÛYC|KYJ, N. V. HORBAN|
neporoΩnix kompaktnyx pidmnoΩyn X , � — netryvial\na pidhrupa adytyvno]
hrupy v R, R+ = 0, +∞[ ), � + = � ∩ R+ .
Oznaçennq 2. VidobraΩennq G: � + × X � P X( ) nazyvagt\ bahatoznaç-
nym napivpotokom (m-napivpotokom) na X, qkwo:
1) G( , )0 ⋅ = IX — totoΩne vidobraΩennq X;
2) G t s x( , )+ � G t G s x, ( , )( ) ∀ t s, ∈ � + ∀ x ∈ X.
M-napivpotik nazyvagt\ strohym, qkwo G t s x( , )+ = G t G s x, ( , )( ) ∀ t s, ∈
∈ � + ∀ x ∈ X.
Oznaçennq 3. MnoΩynu A � X nazyvagt\ prytqhugçog mnoΩynog dlq m-
napivpotoku G , qkwo dlq dovil\noho B ∈ β( )X i dovil\noho okolu N A( )
mnoΩyny A v X isnu[ T = T N A B( ),( ) ∈ � + taka, wo G t B( , ) � N A( ) ∀ t ≥
≥ T.
ZauvaΩennq 1. Ostann[ oznaça[, wo
dist G t B A( , ),( ) → 0, t → +∞ ,
tobto dlq vsix ε > 0, B ∈ β( )X isnu[ T = T B( , , )τ ε take, wo G t B( , ) � O Aε( )
∀ t ≥ T.
Dlq fiksovanyx B � X ta s ∈ � + rozhlqnemo taki mnoΩyny:
γ s B( ) = G t B
t s
( , )
≥
∪ , ω( )B = clX s
s
Bγ ( )( )
≥0
∩ .
Oçevydno, wo
γ s B( ) � γ ′s B( ) , s s≥ ′ , ∀ ≥p 0 : ω( )B =
clX s
s p
Bγ ( )( )
≥
∩ .
Oznaçennq 4. MnoΩynu Θ � X nazyvagt\ hlobal\nym atraktorom dlq
m-napivpotoku G, qkwo:
1) Θ — prytqhugça mnoΩyna;
2) dlq dovil\no] prytqhugço] mnoΩyny Y Θ � clXY (minimal\nist\);
3) Θ � G t( , )Θ dlq vsix t ≥ 0 (napivinvariantnist\).
Oznaçennq 5. M-napivpotik G nazyvagt\ asymptotyçno kompaktnym,
qkwo dlq dovil\noho B ∈ β( )X isnu[ A B( ) ∈ K X( ) take, wo
dist G t B A B( , ), ( )( ) → 0, t → +∞ .
ZauvaΩennq 2. M -napivpotik G [ asymptotyçno kompaktnym, qkwo do-
vil\na poslidovnist\ ξn n{ } ≥1, ξn ∈ G t Bn( , ) , t → + ∞, peredkompaktna v X.
Teorema 3 [9]. Nexaj m-napivpotik G zadovol\nq[ nastupni umovy:
1) G [ asymptotyçno kompaktnym;
2) ∃ R0 > 0 ∀ R > 0 ∃T = T R( ) ∀ t > T: G t BR( , ) � BR0
;
3) dlq t ∈ � + vidobraΩennq X � x � G t x( , ) ma[ zamknenyj hrafik.
Todi mnoΩyna Θ =
ω
β
( )
( )
B
B X∈
∪ [ kompaktnym hlobal\nym atraktorom.
Bil\ß toho, qkwo m-napivpotik G [ strohym, to Θ — invariant, tobto
Θ = G t( , )Θ ∀ t ∈ � + .
Osnovnym rezul\tatom wodo analizu qkisno] povedinky rozv’qzkiv zadaçi (1) [
nastupna teorema pro isnuvannq hlobal\noho atraktora.
Teorema 4. Nexaj dlq zadaçi (1) vykonano umovy (2). Todi vidobraΩennq G ,
oznaçene formulog (10), [ m-napivpotokom, dlq qkoho v fazovomu prostori
E = H
n1
R( ) × L
n2
R( ) isnu[ kompaktnyj invariantnyj hlobal\nyj atraktor.
Dovedennq vyplyva[ z teorem11 – 3 i lem11, 2 analohiçno [15].
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
267 O. M. STANÛYC|KYJ, N. V. HORBAN|
Pryklad. Rozhlqnemo rivnqnnq
utt + γut – ∆u +
α βsin u u
x
+
+1 2 + λ0u = h x( ), ( , ) ( , )t x ∈ +∞0 × Rn
,
de γ > 0, λ0 > 0, α, β ∈ R , n ≥ 3, h ∈ L
n2
R( ). Oskil\ky vykonano umovy (2), to
za teoremog14 vidobraΩennq G, oznaçene formulog (10), [ m-napivpotokom,
dlq qkoho u fazovomu prostori E = H
n1
R( ) × L
n2
R( ) isnu[ kompaktnyj inva-
riantnyj hlobal\nyj atraktor.
1. Babyn A. V., Vyßyk M. Y. Attraktor¥ πvolgcyonn¥x uravnenyj. – M.: Nauka, 1989. – 293 s.
2. Kapustqn O. V., Iovane Û. Hlobal\nyj atraktor dlq neavtonomnoho xvyl\ovoho rivnqnnq
bez [dynosti rozv’qzku // Systemni doslidΩennq ta informacijni texnolohi]. – 2006. – # 2. –
S.1107 – 120.
3. Kapustqn O. V. Vlastyvist\ Knezera dlq neavtonomnoho xvyl\ovoho rivnqnnq bez [dynosti
rozv’qzku // Nauk. visti NTUU „KPI”. – 2007. – # 2. – S. 137 – 141.
4. Ball J. M. Global attractors for damped semilinear wave equations // Discrete and Contin. Dynam.
Syst. – 2004. – 10. – P. 31 – 52.
5. Belleri V., Pata V. Attractors for semilinear strongly damped wave equations on R
3 // Ibid. –
2001. – 7, # 4. – P. 719 – 735.
6. Chepyzhov V. V., Vishik M. I. Attractors on non-autonomous dynamical systems and their dimen-
sion // J. math. pures et appl. – 1994. – 73, # 3. – P. 279 – 333.
7. Chepyzhov V. V., Vishik M. I. Evolution equations and their trajectory attractors // Ibid. – 1997. –
76, # 10. – P. 913 – 964.
8. Kapustyan O. V. The global attractors of multi-valued semiflows, which are generated by some
evolutionary equations // Nelynejn¥e hranyçn¥e zadaçy. – 2001. – # 11. – S. 65 – 70.
9. Kapustyan A. V., Melnik V. S., Valero J. Attractors of multivalued dynamical processes generated
by phase-field equations // Int. Bifurcat. Chaos. – 2003. – 13. – P. 1969 – 1983.
10. Kapustyan A. V., Melnik V. S., Valero J. A weak attractor ans properties of solutions for the three-
dimensional Bénard problem // Discrete and Contin. Dynam. Syst. – 2007. – 18. – P. 449 – 481.
11. Melnik V. S. Multivalued dynamics of nonlinear infinite-dimensional. – Kyiv, 1994. – (Preprint /
Acad. Sci. Ukraine. Inst. Cybernetics, # 94 -17).
12. Melnik V. S. Estimates of the fractal and Hausdorff dimensions of sets invariant under multimap-
pings // Math. Notes. – 1998. – 63. – P. 190 – 196.
13. Melnik V. S., Slastikov V. V., Vasilkevich S. I. On global attractors of multivalued semi-processes //
Dokl. Akad. Nauk Ukrainy. – 1999. – # 7. – P. 12 – 17.
14. Melnik V. S., Valero J. On attractors of multivalued semi-flows and differential inclusions // Set-
Valued Anal. – 1998. – 6. – P. 83 – 111.
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rity // Asympt. Anal. – 2005. – 44. – P. 111 – 130.
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J. Math. Anal. and Apll. – 2000. – 252. – P. 790 – 803.
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1988. – 500 p.
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OderΩano 17.10.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 2
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| id | umjimathkievua-article-3154 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:37:14Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/55/a047926ef83bae6308c5dc69f8111e55.pdf |
| spelling | umjimathkievua-article-31542020-03-18T19:46:54Z Global attractor for the autonomous wave equation in Rn with continuous nonlinearity Глобальний атрактор для автономного хвильового рівняння в Rn з неперервною нелінійністю Horban’, N. V. Stanzhitskii, A. N. Горбань, Н. В. Станжицький, О. М. We investigate the dynamics of solutions of an autonomous wave equation in ℝn with continuous nonlinearity. A priori estimates are obtained. We substantiate the existence of an invariant global attractor for an m-semiflow. Исследована динамика решений автономного волнового уравнения в Rn с непрерывной нелинейностью. Получены априорные оценки. Для m-полупотока обосновано существование инвариантного глобального аттрактора. Institute of Mathematics, NAS of Ukraine 2008-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3154 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 2 (2008); 260–267 Український математичний журнал; Том 60 № 2 (2008); 260–267 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3154/3056 https://umj.imath.kiev.ua/index.php/umj/article/view/3154/3057 Copyright (c) 2008 Horban’ N. V.; Stanzhitskii A. N. |
| spellingShingle | Horban’, N. V. Stanzhitskii, A. N. Горбань, Н. В. Станжицький, О. М. Global attractor for the autonomous wave equation in Rn with continuous nonlinearity |
| title | Global attractor for the autonomous wave equation in Rn with continuous nonlinearity |
| title_alt | Глобальний атрактор для автономного хвильового рівняння в Rn з неперервною нелінійністю |
| title_full | Global attractor for the autonomous wave equation in Rn with continuous nonlinearity |
| title_fullStr | Global attractor for the autonomous wave equation in Rn with continuous nonlinearity |
| title_full_unstemmed | Global attractor for the autonomous wave equation in Rn with continuous nonlinearity |
| title_short | Global attractor for the autonomous wave equation in Rn with continuous nonlinearity |
| title_sort | global attractor for the autonomous wave equation in rn with continuous nonlinearity |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3154 |
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