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In this paper we investigate the dynamics of solutions of the semilinear wave equation, perturbed by additive white noise, in sense of the random attractor theory. The conditions on the parameters of the problem do not guarantee uniqueness of solution of the corresponding Cauchy problem. We prove th...
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The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
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System research and information technologies| _version_ | 1867334253509869568 |
|---|---|
| author | Iovane, G. Kapustyan, O. V. Paliichuk, L. S. Pereguda, O. V. |
| author_facet | Iovane, G. Kapustyan, O. V. Paliichuk, L. S. Pereguda, O. V. |
| author_institution_txt_mv | [
{
"author": "G. Iovane",
"institution": "associate professor at University of Salerno, Salerno, Italy"
},
{
"author": "O. V. Kapustyan",
"institution": null
},
{
"author": "L. S. Paliichuk",
"institution": null
},
{
"author": "O. V. Pereguda",
"institution": null
}
] |
| author_sort | Iovane, G. |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2018-03-30T15:12:23Z |
| description | In this paper we investigate the dynamics of solutions of the semilinear wave equation, perturbed by additive white noise, in sense of the random attractor theory. The conditions on the parameters of the problem do not guarantee uniqueness of solution of the corresponding Cauchy problem. We prove theorem on the existence of random attractor for abstract noncompact multi-valued random dynamical system, which is applied to the wave equation with non-smooth nonlinear term. A priory estimate for weak solution of randomly perturbed problem is deduced, which allows to obtain the existence at least one weak solution. The multi-valued stochastic flow is generated by the weak solutions of investigated problem. We prove the existence of random attractor for generated multi-valued stochastic flow. |
| first_indexed | 2025-07-17T10:19:43Z |
| format | Article |
| fulltext |
© G. Iovane, O.V. Kapustyan, L.S. Paliichuk, O.V. Pereguda, 2013
Системні дослідження та інформаційні технології, 2013, № 1 87
TIДC
НОВІ МЕТОДИ В СИСТЕМНОМУ АНАЛІЗІ,
ІНФОРМАТИЦІ ТА ТЕОРІЇ ПРИЙНЯТТЯ РІШЕНЬ
УДК 517.9
ON RANDOM ATTRACTOR OF SEMILINEAR
STOCHASTICALLY PERTURBED WAVE EQUATION
WITHOUT UNIQUENESS
G. IOVANE, O.V. KAPUSTYAN, L.S. PALIICHUK, O.V. PEREGUDA
In this paper we investigate the dynamics of solutions of the semilinear wave equa-
tion, perturbed by additive white noise, in sense of the random attractor theory. The
conditions on the parameters of the problem do not guarantee uniqueness of solution
of the corresponding Cauchy problem. We prove theorem on the existence of ran-
dom attractor for abstract noncompact multi-valued random dynamical system,
which is applied to the wave equation with non-smooth nonlinear term. A priory es-
timate for weak solution of randomly perturbed problem is deduced, which allows to
obtain the existence at least one weak solution. The multi-valued stochastic flow is
generated by the weak solutions of investigated problem. We prove the existence of
random attractor for generated multi-valued stochastic flow.
INTRODUCTION
In [1], [2], [3] as an adequate mathematical apparatus for describing the dynamics
of a stochastically perturbed evolution systems has been proposed concept of
random attractor, which has been applied to the stochastically perturbed reaction-
diffusion system and 2D Navier-Stokes system, white noise-driven Burgers
equation and to nonlinear wave equation with smooth nonlinear term. Recently
many results concerning various properties of random attractors have appeared
(see [4], [5] and the references therein). In particular, random attractors for
stochastic damped nonlinear wave equation were investigated in [6], [7], [8].
Beginning with pioneering work [9], the ideas and methods of classical theory of
global attractors systematically applied in the case of non-uniqueness of Cauchy
problem. Modern research in this field with many applications contained in the
monograph [10]. Theory of random attractors has been generalized for multi-
valued case in [11], [12], [13] for the systems with attracting random compact set,
and in [14], [15], [16] for the compact systems, which are dissipative in
probability. In this paper we obtain a result on the existence and properties of
random attractor for abstract asymptotically compact multi-valued random
dynamical system (MRDS), which made it possible to prove the existence of
random attractor for the semilinear wave equation with non-smooth nonlinear
term, perturbed by additive white noise.
G. Iovane, O.V. Kapustyan, L.S. Paliichuk, O.V. Pereguda
ISSN 1681–6048 System Research & Information Technologies, 2013, № 1 88
SETTING OF THE PROBLEM
In bounded domain nRQ ⊂ , 3≥n we consider the problem:
⎩
⎨
⎧ +∆−+
∂ 0,=|
,=))((
Q
tt
u
dwdtufuudu φβ
(1)
where w is Wiener process, 0>β and )()( 1
0
2 QHQH ∩∈φ are given,
nonlinear term f is continuous (not necessary smooth) function, which satisfies
conditions:
),||(1|)(| 2
1
−+≤ n
n
uCuf (2)
1),|(|)(1),)(()( 2
22
32 −≥−≥ −
−
n
n
uCuFuFCuuf (3)
where .)(=)(
0
dssfuF
u
∫
In [2], [6], [7] under additional conditions smoothness of f and
)||(1|)(| 1
4
−+≤′ puCuf ,
2
<
−n
np authors proved the existence and investigated
dimension of random attractor for stochastic flow, generated by the problem (1).
Without restrictions on derivative of f two difficulties appear: absence of
uniqueness of weak solution of Cauchy problem and impossibility of
decomposition of the flow on compact and decaying parts. In deterministic case
)0=(φ in [17], using the apparatus of energy equations, the existence of global
attractor for corresponding multi-valued semigroup was proved only under
condition (2) and sign condition 1
||
>)(inflim λ−
∞→ u
uf
u
. Using energy equations
approach, in [8] was proved existence of random attractor for stochastic wave
equation with critical exponents on .3R For non-autonomous case, when
,),(= utff and function ),( utf is smooth only on the first variable, theorems
about existence of global attractor for multi-valued dynamical processes were
obtained in [18].
The aim of this paper is to prove the existence of random attractor for multi-
valued stochastic flow, generated by the problem (1), under conditions (2), (3).
For this purpose we prove result about existence of random attractor for abstract
non-compact MRDS system, which made it possible to investigate dynamics of
solutions of the problem (1).
MULTI-VALUED RANDOM DYNAMICAL SYSTEMS
Let )||||,( ⋅X be separable Banach space with Boreal σ -algebra )(Xσ , )(XC is
the set of all non-empty closed subsets of X , for XBA ⊂, we denote:
;inofclosureis XAA
On random attractor of semilinear stochastically perturbed wave equation without uniqueness
Системні дослідження та інформаційні технології, 2013, № 1 89
;infsup=),(dist yxBA
ByAx
−
∈∈
};<),(dist|{=)( δδ AyXyAO ∈
.sup=},|||{= |||||||||| aArxXxB
Aa
r
∈
+≤∈
Let ),,( PΦΩ be a probability space, Φ is P -completion of Φ ,
Rtt ∈ΩΩ }:{θ is metric dynamical system [4], that is a measure preserving
group of transformations in Ω such that the map ωθω tt ),( is measurable,
where parameter t takes valued in R endowed with Boreal σ -algebra )(Rσ .
The map )(: XCF Ω is called random set )(ωF , if F is measurable,
that is the function ))(,(dist ωω Fy is measurable.
Definition 1. The map )(: XCXRG ×Ω×+ is called MRDS, if
1) Xx∈∀ the map xtGt ),(),( ωω is measurable;
2) Xxst ∈∀≥∀Ω∈∀ 0,,,ω
.),(),(),(,=)(0, xsGtGxstGxxG s ωωθωω ⊆+
Note, that it will be enough to assume that condition 2) takes place on
tθ -invariant set of full measure.
Definition 2. The random set )(ωA is called random attractor of MRDS ,G
if for P -almost all ( P -a.a.) :Ω∈ω
1) )(ωA is compact;
2) ;0)(),()( ≥∀⊂ tAtGA t ωωωθ
3) 0>r∀ .0,))(,),((dist ∞+→→− tABtG rt ωωθ
Theorem 3. Let assume that MRDS G satisfies the following conditions:
1) there exists P -almost everywhere ( P -a.e.) bounded random set )(ωB
such, that for P -a.a. Ω∈ω , 0>r∀
;0,))(,),((dist ∞+→→− tBBtG rt ωωθ (4)
2) Ω∈∀≥∀ ω0,t the map )(:),( XCXtG ω has compact valued and is
upper semicontinuous;
3) 0>r∀ the map rBtGt ),(),( ωω is measurable;
4) for P -a.a. Ω∈ω , 0>r∀ , ∞+↑∀ nt arbitrary sequence ∈nξ
rntn BtG ),( ωθ−∈ is precompact in .X
Then the set
BtGA t
TtT
BrB
r
),(=)(where,)(=)(
0>0>
ωθωωω −
≥
ΛΛ ∪∩∪ (5)
is random attractor of MRDS .G It is P -a.e. unique and is a minimal among
closed sets, satisfying (4).
Remark 4. From condition (4) we have existence of P -a.a. bounded
random set )()( ωω BD ⊃ such, that for P -a.a. Ω∈ω , 0>r∀
G. Iovane, O.V. Kapustyan, L.S. Paliichuk, O.V. Pereguda
ISSN 1681–6048 System Research & Information Technologies, 2013, № 1 90
).(),(),(= ωωθω DBtGTtrTT rt ⊂≥∀∃ − (6)
Remark 5. In the paper [11] theorem 3 was proved under condition of
compactness of )(ωB (which is stronger than condition 4)), in paper [14] it was
proved under condition of precompactness of rBtG ),( ω , which is also stronger
than condition 4), because
.)()(1,)1,()(1,),( 111 ωθωθωθωθωθξ −−−−− ⊂−⊂∈ DGBtGGBtG rntnrntnn
Proof. It is well known [5], that
.:),()( yyBtGyty nntnnnB →∈∃∞+↑∃⇔Λ∈ − ωθω
So from 4) 0Ω∃ , 1=)( 0ΩP such that 0Ω∈∀ω , 0>r∀ .)( ∅≠Λ ω
rB
Assuming that 0Ω∈∀ω condition 4) takes place, we have that ,0Ω∈∀ω
0>r∀ )()( ωω B
rB ⊂Λ . Further ,0Ω∈∀ω )(}{ ω
rBnz Λ⊂∀ ,+∞↑∃ nt
rntnn BtG ),( ωθξ −∈∃ such that
n
znn
1
≤− |||| ξ . Then due to 4) the sequence
}{ nz is precompact, so ,0Ω∈∀ω 0>r∀ )(ω
rBΛ is compact. Let us prove
that ,0Ω∈∀ω 0>r∀ )(ω
rBΛ attracts rB in the sense of (4). If not, then
0>δ∃ ,+∞↑∃ nt rntnn BtGy ),( ωθ−∈∃ such that .))(,( δω ≥Λ
rBnydist
But on some subsequence )(ω
rBn yy Λ∈→ , and we have contradiction. So
0Ω∈∀ω the set )(=)(
0>
ωω
rB
r
A Λ∪ satisfies condition 3) of Definition 2.
Moreover, )()( ωω BA ⊂ , so )(ωA is bounded P -a.e. Let us prove that )(ωA is
compact P -a.e. Let us put n
n
ΩΩ
∞
∩
1=
0 = , where .}|{= 00 Ω∈Ω∈Ω − ωθω nn
Then 1=)( 0ΩP . For 0Ω∈ω let us define the set )(=)(
0>
ωω
rB
r
K Λ∪ . For every
sequence )(}{ ωξ Kn ⊂ and for arbitrary 1≥n we can find 0>nr such that
)(ωξ
nr
Bn Λ∈ . Then
).(),(),(=1 0 ωθωθωω nnrt DBtGTtnTTn −− ⊂≥∀∃Ω∈∀≥∀
For 1≥n
nrt
nTt
n BtG ),( ωθη −
+≥
∈∃ ∪ such that
nnn
1
≤− |||| ξη . So 1≥∀ n
Tm ≥∃ such that
⊂⊂+∈ −−−−− nrnmnnrnmn BmGnGBmnG ),(),(),( ωθθωθωθη
).(),( ωθωθ nn DnG −−⊂
Let us consider for arbitrary 0>R sets:
On random attractor of semilinear stochastically perturbed wave equation without uniqueness
Системні дослідження та інформаційні технології, 2013, № 1 91
,})(|{=)( RDR ≤Ω +|||| ωω
}.manyinfinitelyfor)(|{=)( nRR n Ω∈Ω −∞ ωθω
Due to Poincare’s recurrence theorem ))(())(( RPRP Ω≥Ω∞ . As the set
)(ωD is bounded P -a.e., we have 1))(( →Ω∞ RP , ∞→R . For ∩Ω∈ 0ω
)(R∞Ω∩ we can find subsequence ∞
1=)}({ kkn ω such that for every 1≥k
Rknkkn BnG ),( ωθη −∈ . So ,0>R∀ )(0 R∞Ω∩Ω∈∀ω )(ωK is precompact
in X and we deduce that )(ωA is compact P -a.e.
From 3) one can easy obtain (see [11]) that the map
nt
ktkn
BtG ),(
=0=1=
ωθω −
∞∞∞
∪∩∪
is Φ -measurable. So from [3] there exists compact random set )(~ ωA such that
)(=)(~ ωω AA for P -a.a. Ω∈ω . Thus the set )(~ ωA is compact random set,
which satisfies condition 3) of definition 2. Then from results of [11] MRDS G
has random attractor, which coincides with )(ωA P -a.e. and it is a minimal
among closed sets, satisfying (4). Theorem is proved.
RANDOM ATTRACTOR FOR STOCHASTIC FLOW, GENERATED
BY THE PROBLEM (1)
We consider problem (1) under conditions (2),(3), w is two-sided real Wiener
process. Let us consider canonical Wiener probability space ),,( PΦΩ . Then we
have )(=),( ttw ωω Rt ∈∀ and formula )()(=)( sstts ωωωθ −+ defines metric
dynamical system Rss ∈ΩΩ }:{θ [4].
Following to [2], we define ),(= ωtWW as a solution of the problem
⎩
⎨
⎧ +
.0=(0)=(0)
,=
t
tt
WW
dwWdW β
(7)
We make the change of variable )()(=)( tWtutv φ− . Equation (1) turns into
⎩
⎨
⎧ ∆++∆−+
∂ .0=|
),(=))((
Q
ttt
v
tWtWvfvvv φφβ
(8)
Now we deduce a priory estimate for weak solution ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
tv
v
=ϕ of randomly
perturbed problem (8) in the phase space )()(= 21
0 QLQHX × with usual norm
X||||ϕ , || ⋅ is a norm in )(2 QL , |||| ⋅ is a norm in .)(1
0 QH
Multiplying (8) by vvv t η+=~ for 0>η small enough, we have [2]
G. Iovane, O.V. Kapustyan, L.S. Paliichuk, O.V. Pereguda
ISSN 1681–6048 System Research & Information Technologies, 2013, № 1 92
+−−−+++ ),~)((|~|)()|~(|
2
1 2222 vvvvvv
dt
d ηβηηβη ||||||||
),,~(=)~),(( Wvvuf φ∆+
where for sufficiently small :0>η
),|~|(
2
),~)((|~|)( 2222 vvvvvv +≥−−−+ |||||||| ηηβηηβη
,|)(||~|
4
|),~(| 22 tWCvWv η
ηφ +≤∆
)),(),((||),1)((),1)(()~,)(( 12 WWufQCuFCuF
dt
dvuf t ηφηη +−−+≥
).||(1),1)((
2
|||)),((| 2
22
2 −
−
+++≤+ n
n
tt WWCuF
C
WWuf η
η
ηφ η
From the above inequalities for some small 0>δ we obtain
),,(),1))((|~(|),1))((|~(| 2222 ωδ tguFvvuFvv
dt
d
≤+++++ |||||||| (9)
where
),|),(||),(|(1=),( 2
22
2
22
−
−
−
−
++ n
n
t
n
n
tWtWCtg ωωω
and 0>C is some constant, which does not depend on ω .
So from Gronwall’s lemma st ≥∀
.),()),1))((()(|)(~(|
)),1))((()(|)(~|
)(22)(
22
τωττδδ dgesuFsvsve
tuFtvtv
t
t
s
st −−−− ∫+++≤
≤++
||||
||||
(10)
From (10) we deduce final estimate: 0>C∃ st ≥∀ Ω∈∀ω
).),()((1)( )(2
22
)(2 τωτϕϕ τδδ dgeseCt t
t
s
n
n
X
st
X
−−−
−
−− ∫++≤ |||||||| (11)
From estimate (11) we can claim [18] that Ω∈∀ω Rs∈∀ Xs ∈∀ϕ there
exists at least one (weak) solution of (8) ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅
⋅
⋅
)(
)(
=)(
tv
v
ϕ on ),[ ∞+s , ss ϕϕ =)( . In
further arguments we denote it by .),,,( sst ϕωϕ Moreover, every solution of (8)
on ),[ ∞+s belongs to ));,([ XsC ∞+ , satisfies (11) and the following equality:
)),(,(=))(,())(,( ttHttIttI
dt
d ϕϕβϕ ωωω + (12)
where
),,(
2
),1)((||||
2
1=))(,( 22 vvuFvvttI tt
βϕω +++ ||
On random attractor of semilinear stochastically perturbed wave equation without uniqueness
Системні дослідження та інформаційні технології, 2013, № 1 93
+−∆+∆ )),((
2
))(,(
2
))(,(=))(,( vuftWvtWvttH t
βφβφϕω
).()(=)(),1),(()),(( tWtvtuuFWuf t φβφ +++
Moreover, we have the following result.
Lemma 6. [18] Let 0ωω →n in Ω , sttn ≥→ 0 , )(⋅nϕ is solution of (8) on
),[ ∞+s with random parameter nω , sn s ϕϕ →)( weakly in .X Then there exists
)(⋅ϕ — solution of (8) on ),[ ∞+s with random parameter 0ω , ss ϕϕ =)( such
that on some subsequence
,inweakly)()( 0 Xttnn ϕϕ →
.))(,())(,( 000
ttHttH nnnn
ϕϕ ωω →
If, moreover, sn s ϕϕ →)( strongly in X , then )()( 0ttnn ϕϕ → strongly in .X
Let us put ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
tW
W
W
φ
φ
= and define the maps:
,)(: XPXRS d ×Ω×
,)},()),(,,,({=),,( 00 ωωϕωϕϕω tWsWststS +− (13)
,)(: XPXRG ×Ω×+
.)},(),0,,({=),( 00 ωϕωϕϕω tWttG + (14)
It is easy to show [2], that for every Rs∈
,),,(=),( xsstSxtG sωθω −+ (15)
and the map S for every Ω∈ω generates multi-valued process [18], that is
,=),,( xxSR ωτττ ∈∀ .),,(),,(),,( xsrSrtSxstSsrt ωωω ⊂≥≥∀
The main result of the paper is the following.
Theorem 7. The formula (14) defines MRDS, which has random attractor in
the phase space .)()(= 21
0 QLQHX ×
Proof. Let us prove condition 2) of definition 1. From (14) Ω∈∀ω
Xx∈∀ xxG =)(0,ω . For 0, 21 ≥tt we have
⊂+++ − xssttSxttG s ),,(=),( 2121 ωθω
=++++⊂ −− xsstSststtS ss ),,(),,( 2221 ωθωθ
.),(),(=),,(),,( 221222221 xtGtGxsstSststtS tstst ωωθωθωθθ −−− ++++=
Now let us verify conditions of theorem 3 (from conditions 2), 3) we, in
particular, obtain, that the map G has closed values and is measurable). Let
G. Iovane, O.V. Kapustyan, L.S. Paliichuk, O.V. Pereguda
ISSN 1681–6048 System Research & Information Technologies, 2013, № 1 94
nn tG ηωξ ),(∈ . Then ),(),0,,(= ωηωϕξ tWt nnn + and from lemma 6 we have
condition 2). Let us consider for fixed Ra∈ , 0≥r , Xy∈ the set
}.)),(,(dist|),{(= aBtGytC r ≤ωω
If Ct nn ∈),( ω , ),(),( 00 ωω tt nn → , then rn B∈∃η , 0ηη →n weakly in
X , nnnn tGy ηω ),(∈∃ , ayyn ≤− |||| . As ),(),0,,(= nnnnnn tWty ωηωϕ + , so
due to lemma 6 ),(),0,,(= 0000 ωηωϕ tWtzyn +→ weakly in .X Thus
rBtGz ),( 00 ω∈ and ayyyz n ≤−≤− |||||||| liminf . Therefore Ct ∈),( 00 ω and
condition 3) takes place.
According to (15)
))}.(,,(0,{=),(0,=),( 000 tWttStG t −−−−− ϕωϕϕωϕωθ
So from (11) we deduce
).),())(((1),(
0
2
22
2 τωτωθ δτδ dgetWreCBtG
t
n
n
X
t
rt ∫
−
−
−
−
+− +−++≤ |||||||| (16)
It means that the conditions (6) and 1) take place with ))((2=)( ωω rCBD + ,
where
..<),(=)(
0
eaPdger −∞∫
∞−
τωτω δτ
Let us verify condition 4). If nntnn tG ηωθξ ),( −∈ , where +∞↑nt , ηη →n
weakly in ,X then from (15) )),(,,(0,= ωηωϕξ nnnnn tWt −−− and from
estimate (16) ξξ →n weakly in .X For 0>M we consider
⊂−−∈−+−−−− nnnnnnn tMtSMtWtWtMttz ηωωωηωϕ ),,(),()),(,,,(=)(
)).2())(,,,2()(,,( MWtWtMMMtS nnnn −+−−−−−−⊂ ηωϕω
So )())(,,,(~=)( MtWMWMMttz n
Mnn −+−−−− γωϕ , where
.inweakly)2())(,,,2(= XMWtWtM Mnnnn
n
M γηωϕγ →−+−−−−
Then due to lemma 6 ][0, Mt∈∀
.inweakly)())(,,,(~=)()( XMtWMWMMttztz Mnn −+−−−−→ γωϕ
From equality (12), applying to function nϕ
~ , we obtain
.)(~,())(,(=)(0,
0
dpppHeMWMIeI n
p
M
n
M
M
n ϕγξ ω
β
ω
β
ω ∫
−
− +−−−
As
=∫∫
−−∞→
dpppHedpppHe p
M
n
p
Mn
)(~,(=)(~,(lim
00
ϕϕ ω
β
ω
β
On random attractor of semilinear stochastically perturbed wave equation without uniqueness
Системні дослідження та інформаційні технології, 2013, № 1 95
)),(,((0))~(0, MWMIeI M
M −−−−= − γϕ ω
β
ω
so
+−−−+≤ − |))(,(|
2
1inflim
2
1 22 MWMIe M
M
XXn γξξ ω
β||||||||
.|))(,(|suplim MWMIe n
M
M −−−+ − γω
β
From the last inequality, passing to the limit when ∞→M , we have
inequality XXn |||||||| ξξ ≤inflim , which means, that the sequence }{ nξ is
precompact in .X Theorem is proved.
CONCLUSIONS
For semilinear wave equation, perturbed by additive white noise, in sense of the
random attractor theory the dynamics of solutions is investigated.
In particular, the existence of random attractor for abstract noncompact
multi-valued random dynamical system is proved. The abstract theory allows to
apply this result to the wave equation with non-smooth nonlinear term. A priory
estimate for weak solution of randomly perturbed problem in the phase space is
deduced, which contributes to obtain the existence of the weak solutions. The
existence of random attractor for generated multi-valued stochastic flow is proved.
Thus for the class of mathematical models with non-smooth dependencies
between determining parameters of the problem, controlled by nonlianerized pie-
zoelectric and viscoelasticity theory with nonlinear stochastic perturbations, the
opportunity of long-time forecasts for state functions is obtained. As a result, it
became possible to direct the state functions to the desired asymptotic level.
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Received 13.12.2012
From the Editorial Board: the article corresponds completely to submitted manu-
script.
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| id | journaliasakpiua-article-57547 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:19:43Z |
| publishDate | 2013 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/5e/0d1d46e3ef826e81d0bb08eafbf96e5e.pdf |
| spelling | journaliasakpiua-article-575472018-03-30T15:12:23Z On random attractor of semilinear stochastically perturbed wave equation without uniqueness Случайный аттрактор полулинейного стохастического возмущенного волнового урав-нения без единственности решения Випадковий атрактор напівлінійного стохастичного збуреного хвильового рівняння без одиничності розв’язку Iovane, G. Kapustyan, O. V. Paliichuk, L. S. Pereguda, O. V. In this paper we investigate the dynamics of solutions of the semilinear wave equation, perturbed by additive white noise, in sense of the random attractor theory. The conditions on the parameters of the problem do not guarantee uniqueness of solution of the corresponding Cauchy problem. We prove theorem on the existence of random attractor for abstract noncompact multi-valued random dynamical system, which is applied to the wave equation with non-smooth nonlinear term. A priory estimate for weak solution of randomly perturbed problem is deduced, which allows to obtain the existence at least one weak solution. The multi-valued stochastic flow is generated by the weak solutions of investigated problem. We prove the existence of random attractor for generated multi-valued stochastic flow. Исследована динамика решений полулинейного волнового уравнения, возмущенного аддитивным белым шумом, с точки зрения теории случайных аттракторов. Условия на параметры задачи не гарантируют единственности решения соответствующей задачи Коши. Доказано теорему о существовании случайного аттрактора для абстрактной некомпактной многозначной случайной динамической системы, которая была применена к волновому уравнению с негладким нелинейным слагаемым. Установлена априорная оценка для слабого решения случайно возмущенной задачи, которая позволила получить существование, по крайней мере, одного слабого решения. На слабых решениях исследованной задачи построен многозначный стохастический поток. Доказано существование случайного аттрактора для построенного многозначного стохастического потока. Досліджено динаміку розв’язків напівлінійного хвильового рівняння, збуреного адитивним білим шумом, із точки зору теорії випадкових атракторів. Умови на параметри задачі не гарантують єдності розв’язку відповідної задачі Коші. Доведено теорему про існування випадкового атрактора для абстрактної некомпактної багатозначної випадкової динамічної системи, що була застосована до хвильового рівняння з негладким нелінійним доданком. Встановлено апріорну оцінку для слабкого розв’язку випадково збуреної задачі, що дозволило отримати існування принаймні одного слабкого розв’язку. На слабких розв’язках досліджуваної задачі побудовано багатозначний стохастичний потік. Доведено існування випадкового атрактора для побудованого багатозначного стохастичного потоку. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2013-03-19 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/57547 System research and information technologies; No. 1 (2013); 87-96 Системные исследования и информационные технологии; № 1 (2013); 87-96 Системні дослідження та інформаційні технології; № 1 (2013); 87-96 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/57547/53830 Copyright (c) 2021 System research and information technologies |
| spellingShingle | Iovane, G. Kapustyan, O. V. Paliichuk, L. S. Pereguda, O. V. Випадковий атрактор напівлінійного стохастичного збуреного хвильового рівняння без одиничності розв’язку |
| title | Випадковий атрактор напівлінійного стохастичного збуреного хвильового рівняння без одиничності розв’язку |
| title_alt | On random attractor of semilinear stochastically perturbed wave equation without uniqueness Случайный аттрактор полулинейного стохастического возмущенного волнового урав-нения без единственности решения |
| title_full | Випадковий атрактор напівлінійного стохастичного збуреного хвильового рівняння без одиничності розв’язку |
| title_fullStr | Випадковий атрактор напівлінійного стохастичного збуреного хвильового рівняння без одиничності розв’язку |
| title_full_unstemmed | Випадковий атрактор напівлінійного стохастичного збуреного хвильового рівняння без одиничності розв’язку |
| title_short | Випадковий атрактор напівлінійного стохастичного збуреного хвильового рівняння без одиничності розв’язку |
| title_sort | випадковий атрактор напівлінійного стохастичного збуреного хвильового рівняння без одиничності розв’язку |
| url | https://journal.iasa.kpi.ua/article/view/57547 |
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