Global attractor for non-autonomous wave equation without uniqueness of solution

In the paper the non-autonomous wave equation with non-smooth right-hand side is considered. It is proved that all its weak solutions generate multi-valued non autonomous dynamical system, which has invariant global attractor in the phase space. Досліджується неавтономне хвильове рівняння за умов, я...

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Veröffentlicht in:	Системні дослідження та інформаційні технології
Datum:	2006
Hauptverfasser:	Iovane, G., Kapustyan, O.V.
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Sprache:	English
Veröffentlicht:	Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України 2006
Schlagworte:	Нові методи в системному аналізі, інформатиці та теорії прийняття рішень
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spelling	Iovane, G. Kapustyan, O.V. 2013-03-11T12:50:12Z 2013-03-11T12:50:12Z 2006 Global attractor for non-autonomous wave equation without uniqueness of solution / G. Iovane, O.V. Kapustyan // Систем. дослідж. та інформ. технології. — 2006. — № 2. — С. 107–120. — Бібліогр.: 11 назв. — англ. 1681–6048 https://nasplib.isofts.kiev.ua/handle/123456789/42181 517.9 In the paper the non-autonomous wave equation with non-smooth right-hand side is considered. It is proved that all its weak solutions generate multi-valued non autonomous dynamical system, which has invariant global attractor in the phase space. Досліджується неавтономне хвильове рівняння за умов, які не гарантують єдиності розв’язку задачі Коші. Доведено, що всі слабкі розв’язки утворюють неавтономну многозначну динамічну систему, для якої в фазовому просторі існує глобальний атрактор. Исследуется неавтономное волновое уравнение при условиях, которые не гарантируют единственности решения задачи Коши. Доказано, что все слабые решения порождают неавтономную многозначную динамическую систему, для которой в фазовом пространстве существует глобальный аттрактор. en Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України Системні дослідження та інформаційні технології Нові методи в системному аналізі, інформатиці та теорії прийняття рішень Global attractor for non-autonomous wave equation without uniqueness of solution Глобальний атрактор неавтономного хвильового рівняння без єдиності розв’язку Глобальный аттрактор неавтономного волнового уравнения без единственности решения Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Global attractor for non-autonomous wave equation without uniqueness of solution
spellingShingle	Global attractor for non-autonomous wave equation without uniqueness of solution Iovane, G. Kapustyan, O.V. Нові методи в системному аналізі, інформатиці та теорії прийняття рішень
title_short	Global attractor for non-autonomous wave equation without uniqueness of solution
title_full	Global attractor for non-autonomous wave equation without uniqueness of solution
title_fullStr	Global attractor for non-autonomous wave equation without uniqueness of solution
title_full_unstemmed	Global attractor for non-autonomous wave equation without uniqueness of solution
title_sort	global attractor for non-autonomous wave equation without uniqueness of solution
author	Iovane, G. Kapustyan, O.V.
author_facet	Iovane, G. Kapustyan, O.V.
topic	Нові методи в системному аналізі, інформатиці та теорії прийняття рішень
topic_facet	Нові методи в системному аналізі, інформатиці та теорії прийняття рішень
publishDate	2006
language	English
container_title	Системні дослідження та інформаційні технології
publisher	Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
format	Article
title_alt	Глобальний атрактор неавтономного хвильового рівняння без єдиності розв’язку Глобальный аттрактор неавтономного волнового уравнения без единственности решения
description	In the paper the non-autonomous wave equation with non-smooth right-hand side is considered. It is proved that all its weak solutions generate multi-valued non autonomous dynamical system, which has invariant global attractor in the phase space. Досліджується неавтономне хвильове рівняння за умов, які не гарантують єдиності розв’язку задачі Коші. Доведено, що всі слабкі розв’язки утворюють неавтономну многозначну динамічну систему, для якої в фазовому просторі існує глобальний атрактор. Исследуется неавтономное волновое уравнение при условиях, которые не гарантируют единственности решения задачи Коши. Доказано, что все слабые решения порождают неавтономную многозначную динамическую систему, для которой в фазовом пространстве существует глобальный аттрактор.
issn	1681–6048
url	https://nasplib.isofts.kiev.ua/handle/123456789/42181
citation_txt	Global attractor for non-autonomous wave equation without uniqueness of solution / G. Iovane, O.V. Kapustyan // Систем. дослідж. та інформ. технології. — 2006. — № 2. — С. 107–120. — Бібліогр.: 11 назв. — англ.
work_keys_str_mv	AT iovaneg globalattractorfornonautonomouswaveequationwithoutuniquenessofsolution AT kapustyanov globalattractorfornonautonomouswaveequationwithoutuniquenessofsolution AT iovaneg globalʹniiatraktorneavtonomnogohvilʹovogorívnânnâbezêdinostírozvâzku AT kapustyanov globalʹniiatraktorneavtonomnogohvilʹovogorívnânnâbezêdinostírozvâzku AT iovaneg globalʹnyiattraktorneavtonomnogovolnovogouravneniâbezedinstvennostirešeniâ AT kapustyanov globalʹnyiattraktorneavtonomnogovolnovogouravneniâbezedinstvennostirešeniâ
first_indexed	2025-11-24T16:27:56Z
last_indexed	2025-11-24T16:27:56Z
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fulltext	© G. Iovane, O.V. Kapustyan, 2006 Системні дослідження та інформаційні технології, 2006, № 2 107 TIДC НОВІ МЕТОДИ В СИСТЕМНОМУ АНАЛІЗІ, ІНФОРМАТИЦІ ТА ТЕОРІЇ ПРИЙНЯТТЯ РІШЕНЬ УДК 517.9 GLOBAL ATTRACTOR FOR NON-AUTONOMOUS WAVE EQUATION WITHOUT UNIQUENESS OF SOLUTION G. IOVANE, O.V. KAPUSTYAN In the paper the non-autonomous wave equation with non-smooth right-hand side is considered. It is proved that all its weak solutions generate multi-valued non autonomous dynamical system, which has invariant global attractor in the phase space. Introduction. One of the main directions to investigate the asymptotic behaviour of solutions of non-linear problems is given by the mathematical physics through the theory of minimal attracting sets (global attractors). The topic methods of this theory and a great number of applications are described in [1–3]. This theory pre- sents some generalizations in the cases of non-uniqueness of solutions [4–7] and also non-autonomous problems [8–11]. From this point of view, non-linear wave equation is difficult for studying because under conditions of global resolvebility it does not generate compact semigroup ( even with smooth non-linearity). Different variants of additional conditions on non-linear term, which provide the existence of global attractor in spite of non-compactness of semigroup are discussed in [1, 2]. In [7] it is suggested a new idea of verifying Ladyzheuskaya’s condition ( or asymptotic semi-compactness condition ) in order to prove the existence of global attractor for wave equation without the restrictive conditions imposed in the non- linearity for uniqueness of solution. In this paper we use a similar approach in situations of non-autonomous problem. Setting of the problem We consider the problem ⎪ ⎩ ⎪ ⎨ ⎧ +∆−+ Ω∂ ,)(=\|),(=\| ,0=\| ,0=),( == xuxuu u utfuuu ttt ttt ττττ ν γ )2( )1( where 0>γ is constant, nR⊂Ω is bounded domain with smooth boundary, 3≥n , R∈τ and non-linear term f satisfies the following condition ,>),( infinflim),(, 1 \|\| 2 λ−∈′ ∈∞→ u utfCff tu t R R uttutfuCutf tn n )()(),(,1),( 2 βα +≤′⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +≤ − , (3) G. Iovane, O.V. Kapustyan ISSN 1681–6048 System Research & Information Technologies, 2006, № 2 108 where 0>C is constant, 0>1λ is the first eigenvalue of ∆− in ),(1 0 ΩH 0)( ≥⋅α , 0)( ≥⋅β are given continuous functions from ).(1 RL We denote by ⋅ , ),( ⋅⋅ and ( )( )⋅⋅⋅ ,, the norm and scalar product in )(2 ΩL and )(1 0 ΩH respectively. Our aim is to study the asymptotic behaviour of ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ )( )( =)( tu tu t t ϕ in the phase space )()(= 21 0 Ω×Ω LHE on ∞→t by the methods of the theory of global at- tractors of multivalued non-autonomous dynamical systems. Definition 1. Function ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ )( )( =)( tu u ϕ is called solution of (1) on ( )T,τ , if ( ) ( ))(;, 1 0 Ω∈⋅ ∞ HTLu τ , ))(;,()( 2 Ω∈⋅ ∞ LTLut τ and )(1 0 Ω∈∀ Hψ ∈∀η ( )T,0 τ∞∈C ( ) 0=)),,(()),((),(),( ηψψψγηψ ττ utfuuu t T tt T +++− ∫∫ , (4) where tu denotes the distributional derivative with respect to t of .u Note, that since )(1 0 ΩH is continuously embedded in )(2 2 Ω−n n L , by (3) for every ))(;,( 1 0 Ω∈ ∞ HTLu τ we have ( ) )).(;,(, 22 Ω∈ LTLutf τ Then for each so- lution ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ )( )( =)( tu u ϕ of (1) from [2] we have ))(];,([)( 1 0 Ω∈⋅ HTu τC , ∈⋅)(tu ))(];,([ 2 Ω∈ LTτC , )(1 0 Ω∈∀ Hψ ),()),(( 1 Tut τψ C∈⋅ and ),( Tt τ∈∀ 0=))),,(()),((),(),( ψψψγψ utfuuu dt d tt +++ . (5) Firstly we prove that under conditions (3) the problem (1), (2) τ>T∀ E u ∈⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∀ τ τ τ ν ϕ = has at least one solution on [ ],,Tτ and each solution of (1), (2) (independently from the method of finding) satisfies certain energy equality (Lemma 5). Note that there is no Lipschitz's condition on f with respect to variable u , so the problem (1), (2) is not necessary uniquely resolved. Since f depends on t , solutions of (1), (2) do not generate semigroup, but under additional condition on f as a function of t we can construct non- autonomous analogue of semigroup. For this purpose, following by [9], we consider the space );(= 2RRCM of continuous vector-functions ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ )( )( =)( 2 1 p p p and equip it with a uniform conver- gence topology on each segment [ ] R⊂21,νν , that is Global attractor for non-autonomous wave equation without uniqueness of solution Системні дослідження та інформаційні технології, 2006, № 2 109 [ ] 0)()(sup ], ,in 2 21[ 21 →−⊂∀⇔→ ∈ Rνν ν νν νν pppp nn RM . It is known that with such topology M is a complete metric space. Further we consider the space );( MRC of continuous functions ),(tg R∈t with values in M . It is also equipped with a uniform convergence topology on each segment [ ] R⊂21,tt that is [ ] 0))(),((sup ], ,);(in 21[ 21 →⊂∀⇔→ ∈ tgtg t ttgg n tt n MRMRC ρ . It is known that with such topology );( MRC is a complete metric space. For every );( MRC∈g we put { }RMRC ∈+ hhtggH \|)(cl=)( );( . The function );( MRC∈g is called translation-compact (tr.-c.) in );( MRC if the set )(gH is compact in );( MRC . Our additional condition on function f , which we use to construct the non- autonomous dynamical system is the following: );(inc.tr.is MRC−⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′tf f . (6) As an example of the function f which satisfies (3), (6), we can consider )(=),( 2 uhueutf t +− , where )(RC∈h (but not smooth), ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +− − ∞→ 21 1)(and>)(inflim n n u uCuh u uh λ . Then ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + −21~),( n n uCutf 1 2 >)( inflim=)( infinflim λ−⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ∞→ − ∈∞→ u uh u uhe u t tu R , uetuteutf tt t 22 22=),( −−−′ and ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′tf f is obviously tr. -c. in ).;( MRC We note, that in this example f and tf ′ are not almost-periodic in Bohr sense. We denote ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′ Σ tf f H= . (7) From [9] we have that continuous shift group { } R∈Σ→Σ hhT :)( , )(=)()( htthT +σσ acts on Σ . Now we need the following Lemma. Lemma 1. Each function Σ∈σ has the form ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′tg g =σ , and functions g , tg ′ satisfy the following conditions: uttutguCutg u utg ' tn n tu )()(),(,1),(,>),( infinflim 21 σσ βαλ +≤⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +≤− − ∈∞→ R , G. Iovane, O.V. Kapustyan ISSN 1681–6048 System Research & Information Technologies, 2006, № 2 110 where dttdtt )()( αασ ∫∫ +∞ ∞− +∞ ∞− ≤ , dttdtt )()( ββσ ∫∫ +∞ ∞− +∞ ∞− ≤ . Proof . For each Σ∈⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ l g =σ according to (6) there exists sequence { }nh such that [ ] R⊂∀ 21, tt [ ] R⊂∀ 21,νν [ ] [ ] ( ) ∞→→−+′+−+ ∈∈ nvtlvhtfvtgvhtf t ntn vtt 0,),(),(),(),(sup , sup , 2121 νν . From this we can easy obtain ).,(=),( vtgvtl t′ Since ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +≤+ −21),( n n n vCvhtf , we have ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +≤ −21),( n n vCvtg . Choosing 0>ε such that >),( infinflim v vtf tv R∈∞→ ελ +−> 1 , we have ελ +− + ≥∀∈∀≥∀∃ 1> ),( 10> v vhtf ntRvR nR . So ελ +−≥ 1 ),( v vtg and we obtain .>),( infinflim 11 λελ −+−≥ ∈∞→ v vtg tv R Since vhthtvhtf nnnt )()(),( +++≤+′ βα , we have for ∞→nh 0=),( vtgt′ and for 0hhn → vhthtvtgt )()(),( 00 +++≤′ βα , where dttdtht )(=)( 0 αα ∫∫ +∞ ∞− +∞ ∞− + , dttdtht )(=)( 0 ββ ∫∫ +∞ ∞− +∞ ∞− + . Lemma is proved. Now we dip the problem (1), (2) into the family of similar problems: ⎪ ⎩ ⎪ ⎨ ⎧ +∆−+ Ω∂ ,)(=\|),(=\| ,0=\| ,0=),( == xuxuu u utguuu ttt ttt ττττ ν γ σ σ (2) (1) where .= Σ∈⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′tg g σ As functions g , tg ′ satisfy the conditions (3), for each Σ∈σ the problem ,(1)σ σ(2) is globally resolved for all E u ∈⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ τ τ τ ν ϕ = . The main object which we consider in this paper is a family of multivalued maps { } Σ∈→× σσ E d EU 2:R , { }ττ ≥∈ ttd \|),(= 2RR ( ) ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ τστσ ϕτϕϕϕϕτ =)(,(1)ofsolutionis )( )( =)(\|)(=,, tu u ttU . (8) For the family (8) our goal is to prove the existence in phase space E of minimal invariant uniformly attracting set — global attractor. Global attractor for non-autonomous wave equation without uniqueness of solution Системні дослідження та інформаційні технології, 2006, № 2 111 Elements of abstract theory of global attractors for multivalued non- autonomous dynamical systems. Let ( )ρ,X be a complete metric space. We denote by ))()(( XXP β the set of all non-empty (non-empty bounded) subsets of X , XBA ⊂∀ , ( ) ),(infsup=,dist yxBA ByAx ρ ∈∈ , { }δδ <),(dist\|=)( AxXxAO ∈ , { }rxXxBr ≤∈ )0,(\|= ρ . Let Σ be some complete metric space, →Σ:)({ hT R∈Σ→ h} be some continuous group acting on Σ . Definition 2. The family of multivalued maps { } Σ∈→× σσ )(: XPXU dR is called family of multivalued processes )(MP or non-autonomous multivalued dynamical system, if Σ∈∀σ , Xx∈∀ : 1) ( ) xxU =,,ττσ R∈∀τ ; 2) ( ) ( )( ) τττ σσσ ≥≥∀⊂ stxsUstUxtU ,,,,,, ; 3) ( ) ( ) ( ) R∈∀≥∀⊂++ htxtUxhhtU hT ,,,,, τττ σσ . The family of MP is called strict, if in conditions 2), 3) equality takes place. We denote ( ) ( )xtUxtU ,,=,, ττ σ σ ∪ Σ∈ Σ . Definition 3. The set X⊂ΘΣ is called global attractor of the family of { } Σ∈σσUMP , if X≠ΘΣ and 1) ΣΘ is uniformly attracting set, that is ( )XB β∈∀ R∈∀τ ( )( ) 0,,,dist →ΘΣΣ BtU τ , ∞→t ; 2) ΣΘ is minimal uniformly attracting set, that is for arbitrary uniformly at- tracting set Y we have YXcl⊂ΘΣ . Global attractor ΣΘ is called semi-invariant (invariant) if ( ) dt R∈∀ τ, ( )ΣΣΣ Θ⊂Θ ,,τtU , ( )( )ΣΣΣ ΘΘ ,,= τtU . Lemma 2. 1) If the family of { } Σ∈σσUMP satisfies the following condi- tions: ( ) ( ) ( ) ( )XBtUBTTXB Tt ββ ∈∃∈∀ Σ ≥ ,0,= ∪ , (9) ( ) { } ( ){ } { } ,inprecompactissequencethe ,0,\|\| X BtUttXB n nnnnn ξ ξξβ Σ∈∀∞→∀∈∀ (10) then there exists global attractor ΣΘ , ( ) ( )0== ΣΣΣ ΘΘΘ τ τ ∪ , (11) where ( ) ( ) ( )B XB ,= τωτ β Σ ∈ ΣΘ ∪ , ( ) ( )BtUB sts ,,=, ττω τ Σ ≥≥ Σ ∪∩ is compact in X ; 2) if, additionally, 0≥∀t the map ( ) ( )xtUxX ,0,, σσ →∋Σ× (12) has closed graph, then ΣΘ is semi-invariant; G. Iovane, O.V. Kapustyan ISSN 1681–6048 System Research & Information Technologies, 2006, № 2 112 3) if, additionally, the family of { } Σ∈σσUMP is strict, then ΣΘ is invariant. Proof. The properties 1), 2) directly derived from the result of [11]. Now we prove 3). From [11] we have the embedding ( )⊂Σ B0,ω ( )( )BtU 0,,0, ΣΣ⊂ ω ( )XB β∈∀ 0≥∀t . So 0≥∀p ( )( )⊂+ ΣΣ BtptU 0,,, ω ( ( )( ))BtUtptU 0,,0,,, ΣΣΣ +⊂ ω ( )( ).0,,0,= BptU ΣΣ + ω Then ( )( ) =ΣΣ BpU 0,,0,ω ( ) ( )( ) ( )( ) ( )( ).0,,0,0,,,0,,0,= BptUBtptUBpU tT ΣΣΣΣΣΣ +⊂+= ωωω . From this for all 0≥p , for all p≥τ ( )( ) ( )( ) ( )( )BkUBkUBpU kk 0,,0,0,,0,0,,0, ΣΣ ≥ ΣΣ ≥ ΣΣ ⊂⊂ ωωω ττ ∪∪ . So, ( )( ) ( )( ) ( )( ) ΣΣΣΣΣ ≥≥ ΣΣ Θ⊂⊂ BBkUBpU tkp 0,0,=0,,0,0,,0, ωωωω τ ∪∩ . Therefore, 0≥∀ p ( ) ΣΣΣ Θ⊂Θ,0,pU . Then R∈∀τ ( ) ( ) ( ) ΣΣΣΣΣΣΣ Θ⊂ΘΘΘ+ ,0,=,0,=,, )( pUpUpU T τττ and Lemma is proved. Properties of solutions of the problem (1), (2). We put ( )=,utF ( )∫= u dsstf 0 , , ( ) ( )∫ ′′ u tt dsstfutF 0 ,=, . Then F , ( )2RC∈′tF and according to (3) there exist constants 1< λλ , 0>1C , R∈2C which only depend on 0>C , 3≥n and 0>1λ such that ( ) 2, R∈∀ ut ( ) ( ) ( ) ( ) ( ) . 2 , , 2 ,,1, 2 2 22 22 1 utututF CuutFuCutF t n n β α λ +≤′ +−≥⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +≤ − − (13) In view of (13) for every function ( ) [ ]( )ET u u t ;, )( )( = τϕ C∈⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ we can cor- rectly define the following functionals: ( )( ) ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( ) ( )( )( ) ( )( ) ( )( ).,, 2 ,1,,1,=, ,),( 2 ,=, ,,1,)( 2 1)( 2 1=, 22 tututftutFtutFttH tututtVttI tutFtututtV ' t t t γγϕ γϕϕ ϕ −+ + ++ Lemma 3. The following properties take place: 1) functions ( )( )( ),1,, ⋅⋅ uF ( )( )( ) ( )( ) ( )( ),,,,1,, ⋅⋅⋅⋅⋅ uufuF' t ( )( ) ( )( )∈⋅⋅⋅ tuuf ,, [ ]( )T,τC∈ ; 2) if ( ){ } [ ] ( )( )Ω⊂⋅ 1 0;, HTn τρ C and [ ]Tt ,τ∈∀ ( ) ( )tutn →ρ in ( )Ω1 0H , then [ ]Tt ,τ∈∀ ( )( )( ) ( )( )( ) ( )( )( ) ( )( )( ),1,,1,,,1,,1, tutFttFtutFttF ' tn ' tn →→ ρρ , Global attractor for non-autonomous wave equation without uniqueness of solution Системні дослідження та інформаційні технології, 2006, № 2 113 ( )( ) ( )( ) ( )( ) ( )( )tututftttf nn ,,,, →ρρ . If, additionally, ( ){ } [ ] ( )( )Ω⊂⋅ 1 0 1 ;, HTn τρ C and [ ]Tt ,τ∈∀ ( ) ( )tut tn →′ρ in ( )Ω2L , then ( )( ) ( )( ) ( )( ) ( )( )tututftttf tnn ,,,, →′ρρ . Proof. In the proof of this Lemma and in all results, given below, we use the following version of the dominated convergence Lebesgue's Theorem: if for measurable functions { } 1≥nnξ , ξ we have ξξ →n a.e., nn ηξ < a.e. and ηη →n in 1L , then ξξ →n in 1L . We consider the function ( )( ) ( )( )⋅⋅⋅ tuuf ,, (for others one can apply the same arguments). Let 0ttn → . Then ( ) ( )0tutu n → in ( )Ω1 0H , ( ) ( )0tutu tnt → in ( )Ω2L , so ( ) ( )xtuxtu n ,, 0→ a.e., ( ) ( )xtuxtu tnt ,, 0→ a.e. Since ( )2RC∈f , we obtain ( )( ) ( ) ( )( ) ( )xtuxtutfxtuxtutf tntnn ,,,,,, 000→ a.e. Moreover, in view of (3) ( )( ) ( ) ( ) ( ) ( )xtuxtuCxtuCxtuxtutf ntn n nntntnn ,,,,,, 2−+≤ . As ( )⊂Ω1 0H ( )Ω⊂ −2 2 n n L , we have ( ) ( )xtuxtu n ,, 0→ in 2 2 −n n L . Since →),( xtu nt ),( 0 xtut in ( )Ω2L , we easy obtain ( ) ( ) ( ) ( )xtuxtuxtuxtu tn n ntn n n ,,,, 0202 −− → in ( )Ω1L . Applying Lebesgue's theorem, we have ( )( ) ( )→xtuxtutf ntnn ,,, ( )( ) ( )xtuxtutf t ,,, 000→ in ( )Ω1L and thus ( )( ) ( ) [ ]( )Tuuf t ,, τC∈⋅⋅⋅ . Statement 2 can be proved in the same way. Lemma is proved. As a consequence of Lemma 3 we immediately obtain that ( )=⋅∀ϕ [ ]( )ET u u t ;, )( )( τC∈⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ = functions ( )( ) ( )( ) ( )( )⋅⋅⋅⋅⋅⋅ ϕϕϕ ,,,,, HIV belong to [ ]( )T,τC . Lemma 4. For every ( ) [ ] ( )( )Ω∈⋅ 1 0;, HTu τC , ( ) [ ] ( )( )Ω∈⋅ 2;, LTut τC function ( )( )( ),1, ⋅⋅ uF belongs to ( )T,1 τC and ( )Tt ,τ∈∀ ( )( )( ) ( )( )( ) ( )( ) ( )( )tututftutFtutF dt d t ' t ,,,1,=,1, + . (14) Proof. From Lemma 3 it suffices to show that [ ] ),(, 10 Ttt τ⊂∀ ( )100 , tt∞∈∀ Cη ( )( )( ) ( )( )( ) ( )( ) ( )( )( )∫∫ +− 1 0 1 0 ,,1,,=,1, t t t ' t t t t tututftutFtutF ηη . (15) We can mollify u with respect to t to obtain a sequence ( ){ } [ ] ( )( )Ω⊂⋅ 1 010 1 ;, Httn Cρ with un →ρ in [ ] ( )( )Ω1 010 ;, HttC , tn u→′ρ in [ ] ( )( )Ω2 10 ;, LttC . Equality (15) obviously holds for ( )⋅nρ . Using Lemma 3 and boundness of ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′n n ρ ρ in [ ]( )Ett ;, 10C we can apply Lebesgue’s theorem and obtain (15) by passing to the limit in the same identify for nρ . Lemma is proved. G. Iovane, O.V. Kapustyan ISSN 1681–6048 System Research & Information Technologies, 2006, № 2 114 Lemma 5. Under conditions (3) R∈∀τ τ>T∀ E u ∈⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∀ τ τ τ ν ϕ = problem (1), (2) has at least one solution on ( )T,τ . Moreover, for each solution ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ )( )( = tu u ϕ of problem (1) on ( )T,τ the functions ( ))(),( ⋅⋅ uut , ( )( )⋅⋅ ϕ,V , ( )( )⋅⋅ ϕ,I belong to ( )T,1 τC and ( )Tt ,τ∈∀ we have ( )( ) ( ) ( )( )( ),1,=, 2 ttFtuttV dt d ' tt ϕγϕ +− , (16) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( )tututftutututututu dt d ttt ,,,=, 22 −−− γ , (17) ( )( ) ( )( ) ( )( )ttHttIttI dt d ϕϕγϕ ,,=, +− . (18) Proof. We construct solution of (1),(2) using the Faedo-Galerkin method. Let { }∞ 1=jjω be a complete system of functions in ( )Ω1 0H and ( )=tum ( )( ) i m i m i tg ω∑= 1= be the Galerkin approximation, satisfying the following ordinary differential system ( ) ( ) ( )( ) ( )( ) mjutfuu dt du dt d jmjmjmjm ,...,1=,0=,,,,,2 2 ωωωγω +++ (19) with the initial conditions ( ) ( ) ,=,= m' m m m uuu ττ νττ where ∞→→ muu m ,ττ in ( ) ∞→→Ω mH m ,,1 0 ττ νν in ( )Ω2L . Local exis- tence of ( )⋅mu is obvious. Existence on [ ]T,τ will be guaranteed by following a priori estimates: ( ) ( )( ) ( )( ) 0=,,,, 2 mmmmmmm uutfuuuuu ′+′+′+′′′ γ , ( )( ){ } ( )( ) 0=,1,22,1,2 222 mmmmmm utFuutFuu dt d ′−′+++′ γ . From this equality and (13) we deduce that τ≥∀t ( ) ( ) ( ) ( ) ( )⎜⎜ ⎝ ⎛ ++++′≤+′ − − 12 2222 3 22 n n mmmmm uuuCtutu τττ ( ) ( )( ) ( ) ( )( ) ⎟ ⎟ ⎠ ⎞ +′++ ∫ dssususs mm t 22βα τ , (20) where constant 0>3C depends only on 30,>0,>1 ≥nCλ . Using Gronwall inequality, we obtain: ( ) ( ) ( ) ( )( ++′≤+′ 22 3 22 ττ mmmm uuCtutu Global attractor for non-autonomous wave equation without uniqueness of solution Системні дослідження та інформаційні технології, 2006, № 2 115 ( ) ( ) ( )( )∫ + ⎟⎟ ⎠ ⎞ ++ − − t dssseu n n m τ βα τ 12 22 . (21) From (21) we deduce that ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′m m u u is bounded in ( )ETL ;,τ∞ . So we can extract a subsequence, still denoted m , such that ( )( ) starweak;,in 1 0 −Ω→ ∞ HTLuum τ , ( )( ) starweak;,in 2 −Ω→′ ∞ LTLuu tm τ . Thanks to a classical compactness theorem ( )( ) strongly;,in 22 Ω→ LTLuum τ . Hence on some subsequence ( ) ( )xtuxtum ,, → a.e. and so ( )( )→xtutf m ,, ( )( )xtutf ,,→ a.e. From (21) ( ){ }tum is bounded in ( )( )Ω∞ 1 0;, HTL τ , so ( )( ){ }tutf m, is bounded in ( )( )Ω22 ;, LTL τ . Then in a standard way we obtain ( )( ) ( )( )tutftutf m ,, → in ( )( )Ω22 ;, LTL τ weakly. It allows us to pass to the limit in (19) and find that ( ) ( )ETL u u t ;, )( )( = τϕ ∞∈⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ and satisfies (4). Thus ( )⋅ϕ is a solution of (1), ( ) [ ]( )ET ;,τϕ C∈⋅ . Moreover, as { }mu ′′ is bounded in ( )( )Ω−12 ;, HTL τ , from compactness theorem we have [ ] ( ) ( ) ( ) ,inweakly, 2 Ω→∈∀ LtutuTt mτ [ ] ( ) ( ) ( )Ω→∈∀ −1inweakly, HtutuTt t ' mτ and, again applying (21), ( ) ( ) ( ) ( )t tu tu t m m m ϕϕ →⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′ = weakly in E . In particular, ( ) ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ →⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ τ τ τ τ υ τϕ υ τϕ uu m m m == in E and existence is proved. Now let ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ )( )( = tu u ϕ is an arbitrary solution of (1), (2) on ( )T,τ . Since ( )( ) ( )( )Ω∈ 22 ;,, LTLtutf τ , from [2] we deduce that in the sense of scalar distributions on ( )T,τ ( ) ( ) ( )( ) ( )( )tututftuuu dt d ttt ,,= 2 1 22 −−+ γ . (22) Similarly to the proof of Lemma 4 we can obtain in the sense of distributions ( ) 2,=, tttt uuu dt duu − , (23) G. Iovane, O.V. Kapustyan ISSN 1681–6048 System Research & Information Technologies, 2006, № 2 116 where ⋅⋅, is the scalar product between ( )Ω1 0H and ( )Ω−1H . From (23) and (1) we have equality (17) in the sense of distributions on ( )T,τ . According to ( ) [ ]( )ET u u t ;, )( )( = τϕ C∈⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ we deduce that functions ( ))(),( ⋅⋅ uut , ( ) ( ) 22 ⋅+⋅ uut belong to ( )T,1 τC and so identities (17), (22) take place in classical sense ( )Tt ,τ∈∀ . Then using the result of Lemma 4 and (17), (22) we can easily obtain (16)-(18). Lemma is proved. Remark 1. As τ>T is arbitrary, we can state a global resolvebility of (1), (2), that is we say that ( ) [ ]( )E u u t ;, )( )( = +∞∈⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ τϕ C is a solution of (1), (2), if ( ) τϕτϕ = and ( )⋅ϕ satisfies (4) τ>T∀ . Remark 2. It is easy to see that if (16)-(18) hold, then for each solution ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ )( )( = tu u ϕ of (1) we can repeat arguments, using in proof of Lemma 5 and ob- tain (21). Hence, for arbitrary solution ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ )( )( = tu u ϕ of (1), for which ( ) ( ) Ruu t ≤+ 22 ττ , we have ( ) ( ) ( )RKtutut t ≤+≥∀ 22τ , (24) where constant ( ) 0>RK depends only on constants 30,>0,>0,> 1 ≥nCR λ and values of ( )dttα∫ +∞ ∞− , ( )dttβ∫ +∞ ∞− . Main results. For every Σ∈⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′tg g =σ we consider the problem σ(1) , σ(2) . In view of Lemmas 1, 5 for every R∈τ , E u ∈⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ τ τ τ υ ϕ = the problem σ(1) , σ(2) has at least one solution on ),( +∞τ and for all solutions of σ(1) , σ(2) the equali- ties (16)–(18) take place, if we change HIV ,, on σσσ HIV ,, respectively. Lemma 6. Let ( )⋅nϕ be a solution of nσ (1) , where ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′ →⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′ g g g g n n n == σσ in Σ and ( ) τϕτϕ →n weakly in E . Then τ>T∀ [ ]Tt ,τ∈∀ ( ) ( )ttn ϕϕ → weakly in E , where ( )⋅ϕ is solution of σ(1) , ( ) τϕτϕ = and ( )( )( ) ( )( )( ),1,,1,, tutFtutF nn σσ → ( )( )( )→′ 1,, tutF nnσ ( )( )( ),1, tutFσ′→ ( )( ) ( )( )tututf nnn ,,σ ( )( ) ( )( )tututf ,,σ→ where nn gf =:σ ; gf =:σ . Global attractor for non-autonomous wave equation without uniqueness of solution Системні дослідження та інформаційні технології, 2006, № 2 117 Proof. Thanks to Lemma 1, (16)-(18) and boundness of ( ){ }τϕn in E we can in the same way as in Lemma 5 obtain for ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅′ ⋅ ⋅ )( )( = n n n u u ϕ : ( ) ( ) ( ) ( )( ++′≤′+≥∀ 22 3 22 τττ nnnn uuCtutut ( ) ( ) ( )( )∫ +∞ ∞− + ⎟⎟ ⎠ ⎞ ++ − − dttteu n n n βατ 12 22 . (25) So using the compactness theorem we can extract a subsequence such, that starweak);,(in −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ =→ ∞ ETL u u t n τϕϕ , ( ) ( ) [ ]TtEttn ,weaklyin τϕϕ ∈∀→ , (26) ( ) strongly);,(in 22 Ω→ LTLuun τ ( ) ( ) a.e.,, xtuxtun → From Lemma 1 and (25) ( ){ }nn utg , is bounded in ))(;,( 22 ΩLTL τ . Accord- ing to convergence σσ →n in Σ we have 0>R∀ [ ] ( ) ( ) ∞→→− ≤∈ nvtgvtgn RvTt 0,,,supsup ,τ . Hence ( )( ) ( )( )xtutgxtutg nn ,,,, → a.e. and from Lions Lemma we obtain ( ) ( )utgutg nn ,, → in )(;,( 22 ΩLTL τ weakly. It allows us to pass to the limit in (4), wrote for ( )⋅nϕ , and we deduce that ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ )( )( = tu u ϕ is solution of σ(1) , ( ) τϕτϕ = . Now we prove that [ ]Tt ,τ∈∀ ( )( )( ) ( )( )( ),1,,1, tutFtutF nn σσ → (other statements can be proved by similar arguments). Firstly ( )( )→xtutF nn ,,σ ( )( )xtutF ,,σ→ for a.a. Ω∈x and from Lemma 1 and (13) ( )( ) ≤xtutF nn ,,σ ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +≤ − − 2 22 1 ,1 n n n xtuC . As [ ]Tt ,τ∈∀ ( ) ( ) ( ) ( ) ( ) ( ) ≤−⋅−− − Ω − − Ω ∫∫ dxxtuxtuxtuxtudxxtuxtu n n nnn n n 22 22 ,,,,=,, ( ) ( ) ( ) ( ) 2−−⋅−≤ n n nn tutututu , and ( ) ( )tutun → in ( )Ω2L strongly, from (25) we deduce that ( ) →− − 2 22 , n n n xtu ( ) 2 22 , − − → n n xtu in ( )Ω1L . So we can apply Lebesgue theorem and obtain that [ ]Tt ,τ∈∀ ( )( ) ( )( )xtutFxtutF nn ,,,, σσ → in ( )Ω1L . Lemma is proved. G. Iovane, O.V. Kapustyan ISSN 1681–6048 System Research & Information Technologies, 2006, № 2 118 Remark 3. From Lemma 6 we have that [ ]Tt ,τ∈∀ ( )( )→ttH nn ϕσ , ( )( )ttH ϕσ ,→ and the following estimate holds: [ ] ( )( ) 5 , ,sup CttH nnTtt ≤ ∈ ϕσ , (27) where constant 0>5C dependes only on 4C from ( ) .\|\| 4Cn ≤τϕ Theorem. Under conditions (3), (6) the family of maps, constructed in (8), is a strict family of { } Σ∈→× σσ )(: EPEUMP dR , for which there exists an invari- ant global attractor in the phase space E . Proof. Let us prove that the family (8) satisfies Definition 2 with equalities in 2), 3). Condition 1) is obvious. Let ).,,( τσ ϕτξ tU∈ Then ( ) ( )⋅ϕϕξ ,= t is solu- tion of σ(1) on ),( +∞τ , ( ) τϕτϕ = . Then [ ]Ts ,τ∈∀ ( ) ),,( τσ ϕτϕ sUs ∈ . We put ( )pp ϕψ =)( , sp ≥ . Then ( )⋅ψ is solution of σ(1) on ( )+∞,s , ( )ss ϕψ =)( . So ( )∈tψξ = )),,(,,())(,,( τσσσ ϕτϕ sUstUsstU ⊂ . Let )),,(,,( τσσ ϕτξ sUstU∈ . Then ),,( ηξ σ stU∈ , ),,( τσ ϕτη sU∈ . Hence ( )tϕξ = , ( )⋅ϕ is solution of σ(1) on ( )∞+,s , ( ) ηϕ =s , )(= sψη , ( )⋅ψ is solution of σ(1) on ( )∞+,τ , τϕτψ =)( . We put [ ] ( )⎩ ⎨ ⎧ ∈ = spp spp p >, ,),( )( ϕ τψ θ . Then ( ) ),(== tt θϕξ )(⋅θ is solution of σ(1) on ( )∞+,τ , τϕτψτθ =)(=)( . Thus ),,( τσ ϕτξ tU∈ Let ),,( τσ ϕτξ hhtU ++∈ . Then ( )ht +ϕξ = , ( )⋅ϕ is solution of σ(1) on ( )+∞+ ,hτ , ( ) τϕτϕ =h+ . We put ( )hppv +ϕ=)( , τ≥p . Then )(⋅v is solution of σ)((1) hT on ),( +∞τ , τϕτ =)(v , so ),,()(= )( τσ ϕτξ tUtv hT∈ Let ).,,()( τσ ϕτξ tU hT∈ Then ( )tϕξ = , ( )⋅ϕ is solution of σ)((1) hT on ( )+∞,τ , ( ) τϕτϕ = . We put ( )hppv −ϕ=)( , hp +≥τ . Then τϕτ =)( hv + , )(⋅v is solution of σ(1) on ( )∞++ ;hτ , that is ),,()(= τσ ϕτξ hhtUhtv ++∈+ . So, { } Σ∈σσU is a strict family of MP . Now we verify conditions 1)–3) of Lemma 2. From estimate (24) with 0=τ we immediately obtain property (9). Let ( )nnn tU ηξ σ ,0,∈ , ξξ →n , ηη →n in E . Since Σ is compact, we can claim σσ →n in Σ . Then ( )tnn ϕξ = , ( )⋅nϕ is solution of nσ (1) , ( )=0nϕ ηη →= n . From Lemma 6 we deduce that 0≥∀ s ( ) ( )ssn ϕϕ → weakly in E , where ( ) ( )ηϕ σ ,0,sUs ∈ . Thus ( ) ( ) ( )ηξϕϕξ σ ,0,== tUttnn ∈→ and property 2) is proved. To finish the proof we should check the property (10). Let ∈nξ ( )nnn tU ησ ,0,∈ , ( ),EBn βη ∈∈ ,∞→nt σσ →n . Then ( )nnn tϕξ = , ( )⋅nϕ is solution of nσ (1) , ( ) nn ηϕ =0 . Using (24) we have that ( ){ }nn tϕ is bounded in E . Hence there exists E∈θ such that on some subsequence ( ) θϕξ →nnn t= weakly in E . In the same way 0≥∀M ( ) Mnn Mt θϕ →− weakly in E . Global attractor for non-autonomous wave equation without uniqueness of solution Системні дослідження та інформаційні технології, 2006, № 2 119 Moreover 0≥∀ t ( ) ( ) =−−+−∈+− )(,, MtMttMtUtMt nnnnnnn ϕϕ σ ( ) ( )( )MttMU nnnntT −− ϕσ ,0,= . It follows that ( ) ( )ttMt nnn νϕ =+− , ( )⋅nν is a solution of ( ) ( ) nntT M σ−1 , ( ) ( )Mtnnn −ϕν =0 . Since ( ) →−= nnn MtT σσ :~ σ~→ in Σ , from Lemma 6 we obtain that 0≥∀ t ( ) ( )ttn νν → weakly in E , where ( ) ( )MtUt θν σ ,0,~∈ . In particular, ( ) ( ) ( )Mnn MUMM θθνξν σ ,0,= ~∈=→ weakly in E . From equality (18) writed for ( )⋅nν we have 0≥∀ t ( )( ) ( )( ) ( ) ( )( )dpppHeeIttI nn tp t t nnnn ννν σ γγ σσ ,00,=, ~ 0 ~~ −− ∫+ and with Mt = ( ) ( )( ) ( ) ( )( )dpppHeeIMI nn Mp M M nnnn ννξ σ γγ σσ ,00,=, ~ 0 ~~ −− ∫+ . Hence ( ) ( )( ) +≤ − ∞→∞→ M nn n nn n eIMI γ σσ νξ 00,suplim,inflim ~~ ( ) ( )( )dpppHe nn Mp M n νσ γ ,suplim ~ 0 − ∞→ ∫+ . (28) Thanks to (24) ( )( ) 6~ 00,suplim CI nn n ≤ ∞→ νσ ,where constant 0>6C does not depend on n and M . Moreover, from Remark 3 we conclude that ( ) ( )( ) ( ) ( )( )dpppHedpppHe Mp M nn Mp M n νν σ γ σ γ ,=,suplim ~ 0 ~ 0 −− ∞→ ∫∫ . (29) If we write equality (18) for function ( )⋅ν and for Mt = , then ( )( ) ( )( ) ( ) ( )( )dpppHeeIMMI Mp M M ννν σ γγ σσ ,00,=, ~ 0 ~~ −− ∫+ . (30) Moreover, if we denote ( ) ( ) ( )⎟⎟⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ tω ω ν = , then ( )nn n MI ξσ ,inflim ~ ∞→ ( ) ( )( ) ( )( )( ),1,, 2 inflim 2 1 ~ 2 MMFMMtEn n ωωωγξ σ++ ∞→ . (31) From (28)–(31) we obtain ( ) ( )( ) ( )( )( ),1,, 2 inflim 2 1 ~ 2 MMFMMtEn n ωωωγξ σ++ ∞→ ( )( ) ( )( ) =00,, ~~6 M nn M eIMMIeC γ σσ γ νν −− −− G. Iovane, O.V. Kapustyan ISSN 1681–6048 System Research & Information Technologies, 2006, № 2 120 ( )( ) ( ) ( )( ) ( )( )( ),1,, 22 100,= ~ 2 ~6 MMFMMeIeC tE MM ωωωγθν σ γ σ γ +++− −− . So ( )( ) 2 ~6 2 2 100,inflim 2 1 E MM En n eIeC θνξ γ σ γ +− −− ∞→ . (32) From (24) ( ) ( )BKMt Enn 2 −ϕ , where constant ( ) 0>BK does not de- pend on n , M . As ( ) Mnn Mt θϕ →− weakly in E , we have 2 EMθ ( ) ( )BKMt Enn n 2 inflim − ∞→ ϕ . Since ( )0=νθM , then we can pass to limit in (32) for ∞→M and obtain 22 2 1inflim 2 1 EEn n θξ ∞→ . In view of weak convergence nξ to θ in E we have inverse inequality, so θξ →n strongly in E . Theorem is proved. REFERENCES 1. Babin A.V., Vishik M.I. Attractors of evolution equations. — Moscow: Nauka, 1989. — 284 p. 2. Temam R. Infinite-dimensional dynamical systems in mechanics and physics. — New York: Springer, 1997. — 615 p. 3. Chueshov I.D. Introduction to the theory of infinite-dimensional dissipative systems. — Harkiv: ACTA, 1999. — 415 p. 4. Kapustyan O.V., Melnik V.S. Attractors of multivalued semidynamical systems and their approximations // Report of NAS Ukraine. — 1998. — № 1. — P. 21–25. 5. Kapustyan O.V., Valero J. Attractors of multivalued semiflows generated by differ- ential inclusions and their approximations // Abstract and Applied Analysis. — 2000. — 5, № 1. — P. 33–45. 6. Ball J.M. 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Received 05.10.2005 From the Editorial Board: The article corresponds completely to submitted manuscript.

Global attractor for non-autonomous wave equation without uniqueness of solution

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