Pullback asymptotic behavior and statistical solutions for lattice Klein-Gordon-Schrödinger equations with varying coefficient

Pullback asymp o ic beha io and s a is ical solu ions o la ice Klein-Go don-Sch ¨odinge equa ions wi h a ying coe icien ∗ Caidi Zhao†Rong Zhuang‡Tom´as Ca aballob§ aDepa men o Ma hema ics, Wenzhou Uni e si y, Wenzhou, Zhejiang P o ince, 325035, People’s Republic o China bDepa men o de Ecuaciones Di e enciales y An´alisis Num´e ico, Facul ad de Ma em´a icas, Uni e sidad de Se illa, c/Ta ia s/n, 41012-Se illa, Spain No embe 21, 2024 Abs ac In his a icle, he au ho s in es iga e he pullback asymp o ic beha io and s a is ical solu- ions o la ice Klein-Go don-Sch ¨odinge equa ions wi h a ying coe icien . They i s p o e he global well-posedness o he add essed equa ions and he exis ence o a amily o ime-dependen pullback a ac o o he associa ed p ocess ac ing on he ime-dependen phase spaces. Then hey e i y ha he p ocess possesses a amily o in a ian Bo el p obabili y measu es wi h sup- po con ained in he ime-dependen pullback a ac o . Fu he , hey e o mula e he de ini ion o s a is ical solu ion o he e olu iona y equa ions on ime-dependen phase spaces. As a esul , hey p o e he exis ence o s a is ical solu ion o he la ice Klein-Go don-Sch ¨odinge equa ions wi h a ying coe icien and show ha i sa is ies he Liou ille heo em. Keywo ds: S a is ical solu ion; Time-dependen pullback a ac o ; In a ian Bo el p obabili y measu es; Va ying coe icien ; La ice Klein-Go don-Sch ¨odinge equa ions. MSC2010: 35B41; 35D99; 76F20 1 In oduc ion This a icle is de o ed o he s udy o a non-au onomous la ice dynamical sys ems (LDSs) co e- sponding o he ini ial alue p oblem o he ollowing la ice Klein-Go don-Sch ¨odinge (KGS) equa- ions wi h a ying coe icien : ( )¨um+ν˙um+ (2um−um−1−um+1) + µum−β|zm|2=gm( ), > τ, m ∈Z,(1.1) i˙zm−(2zm−zm−1−zm+1) + iαzm+zmum= m( ), > τ, m ∈Z,(1.2) um(τ) = uτ,m,˙um(τ) = u1τ,m, zm(τ) = zτ,m, τ ∈R, m ∈Z,(1.3) whe e Zand Rdeno e he se s o in ege and eal numbe s, espec i ely, i=√−1 is he uni o he imagina y numbe s, um(·) and zm(·) a e he unknown eal- alued and complex- alued unc ions, espec i ely, and he eal- alued unc ion ( )>0 will be assumed o sa is y some condi ions. ∗Suppo ed by NSF o China wi h No.12371245, 11971356 and by Key p ojec o Zhejiang P o ince’s Na u al Science Founda ion wi h No.LZ24A010005. Also suppo ed by FEDER, he Spanish Minis e io de Ciencia e Inno aci´on and AEI unde p ojec PID2021122991NB-C21. †Co esponding au ho E-mail: zhao[email p o ec ed] o [email p o ec ed] ‡E-mail: [email p o ec ed] §E-mail: [email p o ec ed] 1 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. Equa ions (1.1)-(1.2) can be conside ed o be an app oxima ion o he spa ial disc e iza ion (x∈R) o he ollowing non-au onomous KGS equa ions wi h a ying coe icien : ( )u +νu −∆u+µu −β|z|2=g(x, ).(1.4) iz + ∆z+iαz +zu = (x, ).(1.5) KGS equa ions a e used o desc ibe sys ems o conse ed scala nucleons in e ac ing wi h neu al scala mesons [28]. He e, u=u(x, ) ep esen s a eal scala meson ield and z=z(x, ) a complex scala nucleon ield. Pe u ba ions o his sys em including dissipa ion we e in oduced by he e ms ( )u ,νu ,µu,β|z|2,iαz wi h ( ), ν,µ,β,α > 0 and he eal- alued d i ing unc ion g(x, ) and complex- alued unc ion (x, ). Ea lie wo ks on he con inuous KGS equa ions mainly ocused on he Cauchy p oblem and he ini ial bounda y alue p oblems [12,28,45]. Wi h he de elopmen o he heo y o in ini e-dimensional dynamical sys ems, much a en ion has been paid o he asymp o ic beha io o solu ions [2,4,31,36, 50,62]. A popula objec used below associa ed wi h he long- ime beha io o a dynamic sys em is he a ac o . Bile cons uc ed he global ini e-dimensional a ac o and es ima ed he dimension o his a ac o using he Lyapuno exponen s o he lows on compac in a ian se s in a domain o Rn(n⩽3) [4]; Guo conside ed he Cauchy p oblem in R3and p o ed he exis ence o maximal a ac o s [31]. Recen ly, he s a is ical solu ion and Liou ille heo em o equa ions (1.4)-(1.5) we e in es iga ed in [62] ia he heo ies o pullback a ac o and gene alized Banach limi . The o iginal mo i a ion o his a icle is o in es iga e he long- ime beha io and p obabili y dis ibu ion o solu ions o he la ice KGS equa ions wi h ime-dependen coe icien ( ). LDSs o m a class o ex ended sys ems ha a e in e media e be ween pa ial di e en ial equa ions and cellula au oma a [13]. In some si ua ions, LDSs eme ge as he spa ial disc e iza ion o con inuous pa ial di e en ial equa ions. LDSs a e widely used in science and enginee ing, o ins ance, elec ical enginee ing [11], p opaga ion o ne e pulses in myelina ed axons [33], chemical eac ion heo y [18], e c. O e he pas wo decades, a ious ypes o a ac o s we e he subjec o nume ous in es iga ions o LDSs. Fo example, [7,66] s udied he global a ac o s o he i s -o de LDSs and e a ded LDSs; [1,34,55,67] in es iga ed he global a ac o s, he singula limi ing beha io o pullback a ac o s, he exponen ial a ac o s and hei ac al dimension, as well as he uni o m a ac o s o he second- o de LDSs; [8,68] esea ched he andom exponen ial a ac o o he s ochas ic LDSs. Ve y ecen ly, he andom nume ical s abili y o a ac o s o nonlinea Sch ¨odinge equa ions on in ini e la ices was in es iga ed by [38]. As o la ice KGS equa ions, [54] es ablished he exis ence o compac ke nel sec ions and es ima ed i s Kolmogo o -en opy; [56] p o ed he exis ence o pullback a ac o s and in a ian measu es. Howe e , as a as we know, he e is no e e ence in es iga ing he la ice KGS equa ions wi h ime-dependen coe icien . The heo y o s a is ical solu ion comes om S a is ical Mechanics [22]. I has long been known ha singula ajec o ies a e no as signi ican o physically ele an as he s a is ical beha io o some ype o dissipa i e sys em [15,42]. This is because physically mean numbe s o u bulen low a e o en well-beha ed, while ins an aneous quan i ies (e.g. eloci y, kine ic ene gy, and ene gy dissipa ion) end o change ai ly apidly o e ime o space. To o e come his di icul y, [19–21] in oduced he concep o Foias-P odi s a is ical solu ion, conside ing a amily o in a ian p obabili y measu es de ined on he ime-independen phase space o he Na ie -S okes equa ions, pa ame e ized by he ime a iable and ep esen ing he e olu ion o he p obabili y dis ibu ion o he s a e o he sys em; soon a e , [48,49] p oposed he de ini ion o Vishik-Fu siko s a is ical solu ion, conside ing a single Bo el measu e in some sui able ajec o ies space o he incomp essible Na ie -S okes equa ions and ep esen ing he p obabili y dis ibu ion o he space- ime eloci y ield. La e , Foias, Rosa and Temam conside ed 2 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. he idea o space- ime s a is ical solu ion o Vishik and Fu siko wi h al e ed hypo heses o make i mo e analy ically ac able [23,24]. The p obabili y dis ibu ion o solu ions o di e en ypes o e olu iona y equa ions is now com- monly cha ac e ized by s a is ical solu ions and in a ian measu es. Fo ins ance, he exis ence o in a ian measu es on he me ic space o he con inuous au onomous dissipa i e dynamical sys ems was in es iga e by Gla -Hol z and Chek oun in [14] and Lukaszewicz, Real and Robinson in [39]; [40] gene alized he esul o [39], cons uc ing a amily o p obabili y measu es o non-au onomous dis- sipa i e dynamical sys ems; Foias, Rosa and Temam s udied sys ema ically he s a is ical solu ions o he 2D and 3D incomp essible Na ie -S okes equa ions in [23–27]; B onzi, Mondaini and Rosa esen ed in [5,6] an abs ac amewo k o he heo y o s a is ical solu ions and ajec o y s a is ical solu ions o gene al e olu ion equa ions, including hose wi h p ope ies simila o he 3D incom- p essible Na ie -S okes equa ions. Recen ly, [58] es ablished su icien condi ions o he exis ence o ajec o y s a is ical solu ions o gene al au onomous e olu ion equa ions and he abs ac heo y was applied o some models o e olu iona y equa ions (see [32, 57, 59–61]), also he idea o [58] was de eloped o in es iga e he in a ian sample measu es o he 2D s ochas ic Na ie -S okes equa ions in [63]. In addi ion, Fjo dholm and Wiedemann p o ed in [29] a e sion o Onsage ’s conjec u e on he conse a ion o ene gy o he incomp essible Eule equa ions in he con ex o s a is ical solu ions. Ve y ecen ly, Gallenm¨ulle , Wagne and Wiedemann ga e a su ey on p obabilis ic desc ip ions o luid low in [30], and esea ched he s a is ical solu ions o he wo-dimensional incomp essible Eu- le equa ions in spaces o unbounded o ici y in [52]; Yang, Han and Zhao p o ed he exis ence o s a is ical solu ions o dissipa i e non-au onomous Zakha o equa ions in [53]. Ve y ecen ly, Zhao p o ed in [65] an essen ial p ope y o he second-o de ellip ic equa ions in hal -cylind ical domains, which is ha abso bing es ima e implies ajec o y s a is ical solu ions. No e ha all hese equa ions a o emen ioned a e dissipa i e. Howe e , we no ice ha he e is ba ely no e e ence in es iga ing he in a ian measu e and s a is ical solu ion o e olu iona y equa ions on ime-dependen phase spaces besides [64]. The i s esul wi hin he cu en a icle is he exis ence o a ime-dependen pullback a ac o in a amily o ime-dependen phase spaces o he p ocess gene a ed by he solu ion mappings o p oblem (1.1)-(1.3). The exis ence o he pullback a ac o will play he i al ole when we cons uc he s a is ical solu ion. Compa ed o e e ence [56], he main di icul y we encoun e he e comes om he a ying coe icien ( ). Indeed, he a ying coe icien ( ) will lead o he ac ha he classical heo y o pullback a ac o is no con enien o be applied. I seems easonable o se le p oblem (1.1)-(1.3) in ime-dependen phase spaces due o he a ying coe icien ( ), while he classical heo y o pullback a ac o (see e.g. [10]) is sui able o he p oblem add essed in ixed phase space. Fo una ely, Temam and his g oup o mula ed he heo y o ime-dependen a ac o du ing he s udy o non-au onomous oscillon equa ion and wa e equa ion wi h a ying coe icien [16,17], and his heo y was de eloped by [9,35,37,43,44,47] o in es iga e he asymp o ic beha io o eac ion-di usion equa ions and wa e equa ions wi h a ying coe icien . Howe e , al hough we can bo ow he heo y o ime-dependen a ac o in ou s udy, he e s ill some di icul ies when we es ima e he solu ions and p o e he pullback asymp o ically compac ness o he p ocess. To handle hese di icul ies, we i s in oduce a amily o ime-dependen phase spaces {E } ∈R={`2 ( )×`2×L2} ∈Rand endowed hem wi h sui able no ms. We hen es ablish ha he ime-dependen ope a o H( ) possesses some coe ci e p ope y. This coe ci i y is impo an o es ima ing solu ions. Besides, inspi ed by [44,56], we p o ide a su icien and necessa y condi ion gua an eeing he exis ence o he ime-dependen a ac o o dissipa i e la ice sys ems wi h a ying coe icien s (see Lemma 3.1). Finally, we use he su icien condi ion o ob ain he exis ence o he ime-dependen pullback a ac o by p o ing he exis ence o he bounded pullback abso bing se (see Lemma 3.2) and exhibi ing he uni o m es ima es on “Tail End” o solu ions (see Lemma 3.3). 3 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. men ion Flandoli and Schmal uss The second esul in his a icle is he exis ence and Liou ille heo em o he s a is ical solu ion o p oblem (1.1)-(1.3). No e ha he classical heo y o s a is ical solu ion was p oposed o he e olu iona y equa ions add essed in ixed phase space and seems no sui able o p oblem (1.1)- (1.3). I seems mo e easonable o se le p oblem (1.1)-(1.3) in ime-dependen phase spaces. He e we ex end he idea o ou ecen wo k [64] o in es iga e he la ice KGS equa ions wi h a ying coe icien . P ecisely, we i s conside a sui able collec ion o ini ial da a (see (4.1)) and e o mula e he de ini ion o τ-con inui y (see De ini ion 4.1) o he associa ed p ocess. Then we e ine he abs ac esul o [40, Theo em 3.1] such ha i can be applied o cons uc he in a ian Bo el p obabili y measu es o p ocess ac ing on ime-dependen phase spaces. A e wa ds, we o mula e he de ini ions o amily o es unc ions and s a is ical solu ion o p oblem (1.1)-(1.3) and p o e i s exis ence. Las ly, we poin ou ha he ob ained s a is ical solu ion sa is ies he Liou ille ype heo em in ime-dependen phase spaces. In compa ison wi h he second o de la ice sys ems wi h a ying coe icien s in es iga ed in [64], he la ice KGS equa ions (1.1)-(1.2) a e coupling o a la ice nonlinea wa e equa ion wi h a ying coe icien s wi h a la ice nonlinea Sch ¨odinge equa ion. This coupling and he a ying coe icien , as well as he nonlinea e ms β|zm|2and zmumin equa ions (1.1)-(1.2), p oduce some addi ional di icul ies in ou in es iga ing. Fi s ly, i need us o pick an app op ia e ime-dependen ans o - ma ion ( ) = ˙u( ) + δ( ) ( )u( ), which enable us o pu he add essed la ice sys ems wi h a ying coe icien s in o an abs ac i s -o de di e en ial equa ion wi h cons an coe icien s. Secondly, i equi es us o choose, in acco dance wi h he s uc u e o he abs ac equa ions (depends on in u n he ime-dependen ans o ma ion picked abo e), a amily o p ope ime-dependen phase spaces {E =`2 ( )×`2×L2} ∈Rand endow hem wi h sub le no ms. Wi h hese choices, we can e i y ha he ime-dependen ope a o H( ) (which is he linea and p inciple pa o equa ions (1.1)-(1.2) has some coe ci e p ope y. This coe ci e p ope y plays he key ole in bo h he global exis ence o solu- ions and exis ence o ime-dependen pullback a ac o s. Thi dly, we also need do some me iculous analyses and es ima es when e i ying he uni o m es ima es on “Tail End” o solu ions due o he nonlinea e ms β|zm|2and zmum. The es o he a icle is o ganized as ollows. In Sec ion 2, i s , we p esen he ma hema ical amewo ks and demons a e he global well-posedness o p oblem (1.1)-(1.3). In Sec ion 3, we p o e he exis ence o a ime-dependen pullback a ac o o he p ocess {U( , τ)} ⩾τassocia ed o p oblem (1.1)-(1.3). In Sec ion 4, we i s cons uc he in a ian Bo el p obabili y measu es o he p ocess {U( , τ)} ⩾τon he ime-dependen phase spaces. Then we o mula e he de ini ions o a amily o es unc ions and s a is ical solu ion o equa ions (1.1)-(1.2) in he ime-dependen phase space and p o e ha he amily o in a ian Bo el p obabili y measu es ob ained in Sec ion 4 is i s s a is ical solu ion. Mo eo e , we poin ou ha he s a is ical solu ion ul ills he Liou ille heo em. 2 Global well-posedness In his sec ion, we p o e he global well-posedness o p oblem (1.1)-(1.3) in ixed phase space and ime-dependen phase space. We i s in oduce he ma hema ical se ings and some ope a o s. Se `2=nu= (uk)k∈Z:uk∈R,X k∈Z u2 k<+∞o, L2=nu= (uk)k∈Z:uk∈C,X k∈Z|uk|2<+∞o. 4 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. Fo b e i y, we use X o deno e `2o L2, and equip Xwi h he inne p oduc and no m as (u, ) = X k∈Z uk¯ k,kuk2= (u, u),∀u= (uk)k∈Z, = ( k)k∈Z∈X, whe e ¯ kdeno es he conjuga e o k. Ob iously, (X, (·,·)) is a Hilbe space. We de ine on X h ee linea ope a o s A,Band B∗as    (Au)k= 2uk−uk+1 −uk−1, k ∈Z, u = (uk)k∈Z, (Bu)k=uk+1 −uk, k ∈Z, u = (uk)k∈Z, (B∗u)k=uk−1−uk, k ∈Z, u = (uk)k∈Z. Then, he ollowing p ope ies a e classical (c . e.g. [67]):    (B∗Bu, )=(Bu, B )=(Au, ),∀u, ∈X, kBuk=kB∗uk⩽2kuk,∀u∈X, kAuk⩽4kuk,∀u∈X. W i e        u= (um)m∈Z, µu = (µum)m∈Z, ( )u= (( )um)m∈Z, z= (zm)m∈Z, αz = (αzm)m∈Z, β|z|2= (β|zm|2)m∈Z, zu = (zmum)m∈Z, ( )=( m( ))m∈Z, g( ) = (gm( ))m∈Z, uτ= (um,τ )m∈Z, u1τ= (u1m,τ )m∈Z, zτ= (zm,τ )m∈Z. We can now use he abo e no a ion and ope a o s o ew i e p oblem (1.1)-(1.3) as ( )u +νu +Au +µu −β|z|2=g( ),(2.1) iz −Az +iαz +zu = ( ),(2.2) u(τ) = uτ,˙u(τ) = u1τ, z(τ) = zτ.(2.3) Th oughou his a icle, we will use he ollowing cons an s ha a e ela ed o pa ame e s appea ing in equa ions (2.1)-(2.2) σ0=µν pν2+µν(ν+pν2+µν), σ = min{σ0 2,α 4}. In addi ion, we will use Z+ o deno e he se o posi i e in ege s, and he symbol a.b(and a&b) o mean ha a⩽cb (a⩾cb) o a uni e sal cons an c > 0 ha only depends on he pa ame e s coming om he add essed p oblem. To gua an ee he global well-posedness o p oblem (2.1)-(2.3), as well as he exis ence o ime- dependen pullback a ac o , we need some assump ions on he a ying coe icien s ( ) and he ex e nal o ces ( ) and g( ). (H1) Le (·)∈C1(R) be a dec easing bounded unc ion sa is ying lim →+∞( )=0, ( )−0( )⩽ν 4 o each ∈R,(2.4) and  0( ) ( )⩽ 16µ2 ν2+16µσ ν−4µ ν.(2.5) (H2) Le (·)=( m(·))m∈Z∈C(R,L2), g(·)=(gm(·))m∈Z∈C(R, `2). Mo eo e , we assume ha Z −∞ eσskg(s)k2ds < +∞, ∈R,(2.6) 5 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. and ha he e is a ce ain con inuous unc ion J(·) on he eal line, bounded on e e y in e al o he o m (−∞, ), such ha Z −∞ eσsk (s)k2ds⩽e(σ 2+%) J( )<+∞, o e e y ∈R,whe e 0 <%<σ 2.(2.7) We pick ( )=(8 ν+e(q16µ2 ν2+16µσ ν−4µ ν) )−1, ∈R. Then (·) sa is ies assump ion (H1). Fo he exis ence o unc ions (·) and g(·) sa is ying assump ion (H2), one can e e o [56, Example 3.1]. Nex , se δ( ) = µν( ) ν2+ 4µ( )(2.8) and ake he ans o ma ion ( ) = ˙u( ) + δ( ) ( )u( ).(2.9) Then we can w i e p oblem (2.1)-(2.3) equi alen ly as ˙ ψ( ) + H( )ψ( ) = F(ψ, ),(2.10) ψ(τ) = (u(τ), (τ), z(τ))T= (uτ, τ, zτ)T,(2.11) whe e ψ( ) = (u( ), ( ), z( ))T, (τ) = τ=u1τ+δ(τ) (τ)uτ, F(ψ, ) = 0,β|z|2+g( ) ( ), izu −i ( )T, H( ) =       δ( ) ( )I−I0 A ( )+µI ( )−δ( )(ν−δ( )) 2( )I+4µ ν2δ0( )Iν−δ( ) ( )0 0 0 iA +αI       ,(2.12) and Iis he iden i y on X. No e ha H( ) is a ime-dependen ope a o ac ing on `2×`2×L2. We now in oduce some equi alen no ms in X, whose main pu pose is o mo i a e he de ini ion o ime-dependen phase spaces. Fi s o all, we de ine a bilinea o m on Xas (u, )µ= (Bu, B ) + µ(u, ), u, ∈X. Ob iously, µkuk2⩽kuk2 µ= (u, u)µ=kBuk2+µkuk2⩽(4 + µ)kuk2,∀u, ∈X. (2.13) Thus (·,·)µis an inne p oduc in Xwhich induces he no m k·kµequi alen o k·k. Also, we de ine (u, )( )=−1( )(u, )µ=−1( )(Bu, B ) + −1( )µ(u, ),∀u, ∈X, (2.14) whe e ( ) is he a ying coe icien coming om equa ion (1.1). In e m o (2.13), we deduce µ−1( )kuk2⩽kuk2 ( )= (u, u)( )=−1( )kuk2 µ⩽(4 + µ)−1( )kuk2,∀u, ∈X, (2.15) 6 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. which implies ha , o e e y ∈R, (·,·)( )induces a no m k·k( )in `2which is equi alen o k·k. I is clea ha `2 ( )= (`2,(·,·)( )) is a Hilbe space. W i e E=`2×`2×L2, E =`2 ( )×`2×L2, and equip hem wi h he inne p oduc s and no ms as: o any wo elemen s ψ(i)= (u(i), (i), z(i))T∈ Eo E ,i= 1,2, (ψ(1), ψ(2))E= (u(1), u(2))+( (1), (2))+(z(1), z(2)) =X m∈Z (u(1) mu(2) m+ (1) m (2) m+z(1) m¯z(2) m), kψk2 E= (ψ, ψ)E=kuk2+k k2+kzk2, (ψ(1), ψ(2))E = (u(1), u(2))( )+ ( (1), (2))+(z(1), z(2)) =X m∈Z−1( )(Bu(1))m(Bu(2))m+−1( )µu(1) mu(2) m+ (1) m (2) m+z(1) m¯z(2) m, kψk2 E = (ψ, ψ)E =kuk2 ( )+k k2+kzk2. I is no di icul o deduce om (2.13) and (2.15) ha (min{µ−1( ),1}kψk2 E⩽kψk2 E ⩽max{(4 + µ)−1( ),1}kψk2 E,∀ψ∈E, ∀ ∈R, kψk2 Eτ⩽kψk2 E ⩽(τ) ( )kψk2 Eτ,∀ψ∈E, ∀ ⩾τ∈R.(2.16) Rema k 2.1. No e ha he spaces E , o ∈R, a e all he same as linea spaces, and he no ms k·k2 Eτand k·k2 E a e equi alen o any gi en τ, ∈R. No ice ha we ha e exp essed p oblem (1.1)-(1.3) as p oblem (2.10)-(2.11), which is an ini ial alue p oblem o an abs ac i s -o de o dina y di e en ial equa ion (ODE). As a esul , he classical heo y o ODE is applied. Indeed, we ha e he ollowing esul . Lemma 2.1. Le assump ions (H1)-(H2)hold. Fo e e y ini ial da a ψτ= (uτ, τ, zτ)T∈E, he e exis s an unique local solu ion ψ( ) = (u( ), ( ), z( ))T∈Eo sys em (2.10)-(2.11) such ha ψ(·)∈ C([τ, T0), E)∩C1((τ, T0), E) o some T0> τ. I T0<+∞, hen lim →T− 0kψ( )kE= +∞. P oo . Fo e e y ∈R, I is clea ha ope a o H( ) : E →E is linea . By di ec compu a ions, we deduce ha kH( )ψk2 E =kδ( ) ( )u− k2 ( )+kAu ( )+µu ( )−δ( )(ν−δ( )) 2( )u+4µ ν2δ0( )u+ν−δ( ) ( ) k2+kiAz +αzk2 .δ2( ) 2( )kuk2 ( )+k k2 ( )+1 2( )kAuk2+1 2( )kuk2+δ2( )(ν−δ( ))2 4( )kuk2 +|δ0( )|2kuk2+ν−δ( )2 2( )k k2+kAzk2+kzk2 .δ2( ) 2( )kuk2 ( )+ ( 1 ( )+δ2( )(ν−δ( ))2 3( )+( )) µ ( )kuk2+1 + (ν−δ( ))2 ( ) ( )k k2+kzk2 .L1( )kψk2 E ,∀ψ= (u, , z)T∈E ,(2.17) whe e L1( ) = max nδ2( ) 2( ),1 ( )+δ2( )(ν−δ( ))2 3( )+( ),1 + (ν−δ( ))2 ( ) ( )o.(2.18) 7 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. Hence, he linea ope a o H( ) : E →E is bounded. Nex we e i y he locally Lipschi z p ope y o F(·, ) on E . In ac , le Bbe a bounded se in E ,ψ(i)= (u(i), (i), z(i))T∈ B,i= 1,2. Then kF(ψ(1), )−F(ψ(2), )k2 E =kβ ( )(|z(1)|2−|z(2)|2)k2+kiz(1)u(1) −iz(2)u(2)k2 ⩽β2 2( )k(|z(1)|−|z(2)|)k2k(|z(1)|+|z(2)|)k2+kz(1)(u(1) −u(2)) + u(2)(z(1) −z(2))k2 ⩽β2 2( )(2kz(1)k2+ 2kz(2)k2)kz(1) −z(2)k2+ 2kz(1)k2ku(1) −u(2)k2+ 2ku(2)k2kz(1) −z(2)k ⩽ 4β2sup ψ∈B kψk2 E 2( )kz(1) −z(2)k2+2( ) µkz(1)k2ku(1) −u(2)k2 ( )+2( ) µku(2)k2 ( )kz(1) −z(2)k2 ⩽(4β2 2( )+4( ) µ) sup ψ∈B kψk2 E kψ(1) −ψ(2)k2 E .(2.19) Since k·kEand k·kE a e equi alen o e e y ∈R, we conclude om abo e analyses ha H( )+F(·, ) is locally Lipschi z om E o E. By he s anda d heo y o ODE, we deduce he esul s o Lemma 2.1. We nex will es ablish ha he local solu ion ψ(·) gua an eed by Lemma 2.1 does exis globally, by showing lim →T− 0kψ( )kE<+∞ o any T0> τ. Fi s ly, o he boundedness o he componen z(·), we ha e he ollowing esul . Lemma 2.2. Le assump ions (H1)-(H2)hold and ψτ= (uτ, τ, zτ)T∈Ebe he ini ial alue a ini ial ime τ. Suppose ψ( )=(u( ), ( ), z( ))T∈Ebe he co esponding solu ion o p oblem (2.10)-(2.11), hen kz( )k2⩽kzτk2e−α( −τ)+e−α αZ τ eαsk (s)k2ds, ∀ ⩾τ. (2.20) P oo . Mul iplying (2.2) by z( ) in L2and aking he imagina y componen o he inne p oduc , we yields 1 2 d d kz( )k2+αkz( )k2=Im( ( ), z( )) ⩽1 2αk ( )k2+α 2kz( )k2,∀ ⩾τ. (2.21) Applying G onwall’s inequali y o (2.21) gi es (2.20). To e i y he boundedness o he solu ion ψ(·) in space E, we i s p o e he ollowing coe ci i y o he ime-dependen ope a o H( ) on space E . Lemma 2.3. Fo e e y ∈Rand ψ( ) = (u( ), ( ), z( ))T∈E , he e holds Re(H( )ψ, ψ)E ⩾ϑ( ) ( )(kuk2 ( )+k k2) + ν 2( )k k2+αkzk2,(2.22) whe e ϑ( ) = µν( ) pν2+ 4µ( )ν+pν2+ 4µ( )∈(0, δ( )).(2.23) P oo . By calcula ion, we ha e Re(H( )ψ, ψ)E =(δ( ) ( )u− , u)( )+1 ( )(Bu, B ) + µ ( )(u, ) + 4µ ν2δ0( )(u, ) −δ( )(ν−δ( )) 2( )(u, ) + ν−δ( ) ( )k k2+αkzk2,(2.24) (δ( ) ( )u− , u)( )=δ( ) ( )kuk2 ( )−1 ( )(B , Bu)−µ ( )( , u),(2.25) 8 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. and 4µ ν2δ0( )(u, )−δ( )(ν−δ( )) 2( )(u, )⩾−νδ( ) 2( )kukk k+ (δ2( ) 2( )+4µ ν2δ0( ))kukk k ⩾−νδ( ) 2( )kukk k,(2.26) whe e we ha e used he ac ha δ2( ) 2( )+4µ ν2δ0( )⩾0. Thus, we ha e Re(H( )ψ, ψ)E ⩾δ( ) ( )kuk2 ( )−νδ( ) 2( )kukk k+ν−δ( ) ( )k k2+αkzk2.(2.27) In iew o he ac s ha kuk⩽(( ) µ)1/2kuk( )and 4(δ( )−ϑ( ))ν 2−δ( )−ϑ( )=ν2δ2( ) µ( )(2.28) holds o e e y ∈R, we deduce Re(H( )ψ, ψ)E −ϑ( ) ( )(kuk2 ( )+k k2)−ν 2( )k k2−αkzk2 ⩾1 ( )h(δ( )−ϑ( ))kuk2 ( )+ (ν 2−δ( )−ϑ( ))k k2−νδ( ) ( )kukk ki ⩾1 ( )h(δ( )−ϑ( ))kuk2 ( )+ (ν 2−δ( )−ϑ( ))k k2−νδ( ) ( )(( ) µ)1/2kuk( )k ki⩾0.(2.29) This comple es he p oo . Lemma 2.4. Le assump ions (H1)-(H2)hold and ψτ= (uτ, τ, zτ)T∈Ebe he ini ial alue a ini ial ime τ. Suppose ψ(·)=(u(·), (·), z(·))Tbe he co esponding solu ion o p oblem (2.10)-(2.11), hen kψ( )k2 E⩽max{(4 + µ)−1(τ),1} min{µ−1( ),1}kψτk2 Ee−σ( −τ) +L2( )e−σ min{µ−1( ),1}( )Z τ eσs(k (s)k2+kg(s)k2+kz(s)k4ds, ∀ ⩾τ, (2.30) he eina e L2( ) = max{2β2 ν( ),2 ν( ),2 α}. P oo . Le ψ(·)=(u(·), (·), z(·))Tbe he solu ion o p oblem (2.10)-(2.11). We mul iply (2.10) by ψ( ) in E , and ake he eal componen o he inne p oduc , ge ing 1 2 d d kψ( )k2 E +1 2 0( ) ( )kuk2 ( )+Re(H( )ψ( ), ψ( ))E =Re(F(ψ( ), ), ψ( ))E ,∀ ⩾τ. (2.31) Using Cauchy’s inequali y, we de i e                      Re(F(ψ( ), ), ψ( ))E ⩽(β|z( )|2 ( ), ( )) + (g( ) ( ), ( )) + Im( ( ), z( ), (β|z( )|2 ( ), ( )) ⩽β2 ν( )kz( )k4+ν 4( )k ( )k2, (g( ) ( ), ( )) ⩽1 ν( )kg( )k2+ν 4( )k ( )k2, Im( ( ), z( )) ⩽1 αk ( )k2+α 2kz( )k2. (2.32) 9 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. Rei(Bz, Bl) = −Im(Bz, Bl) =Im X m∈Z χ(|m| M)zm+1 ¯zm+Im X m∈Z χ(|m+ 1| M)zm¯zm+1 ⩾−χ0 MX m∈Z|zm¯zm+1|&− σ( ) M,∀τ⩽ 0.(3.23) Taking (3.19)-(3.23) in o accoun , we ob ain ReH( )ψ( ), φ( )E −ϑ( ) ( )X m∈Z χ(|m| M)1 ( )|(Bu)m|2+1 ( )µu2 m+ 2 m −ν 2( )X m∈Z χ(|m| M) 2 m−αX m∈Z χ(|m| M)|zm|2 &1 ( )X m∈Z χ(|m| M)h(δ( )−ϑ( ))|um|2 ( )+ν 2−δ( )−ϑ( ) 2 m−δ( )(ν−δ( )) ( )|um|| m|i −ν 4( )X m∈Z χ(|m| M) 2 m−ν(0( ))2 16µ2( )X m∈Z χ(|m| M)|ψm|2 E −1 M( )(1 + ( ) + δ( )) σ( ),∀τ⩽ 0,(3.24) whe e |um|2 ( )=1 ( )|(Bu)m|2+µu2 m.(3.25) Since 4(δ( )−θ( ))(ν 2−δ( )−θ( )) = ν2δ2( ) µ( )(see (2.28)), we see o any m∈Z ha (δ( )−ϑ( ))|um|2 ( )+ν 2−δ( )−ϑ( ) 2 m−δ( )(ν−δ( )) ( )|um|| m|⩾0.(3.26) As a consequence, inequali y (3.24) imp o es o ReH( )ψ( ), φ( )E −ϑ( ) ( )X m∈Z χ(|m| M)1 ( )|(Bu)m|2+µ ( )u2 m+ 2 m −ν 2( )X m∈Z χ(|m| M) 2 m−αX m∈Z χ(|m| M)|zm|2 &−ν 4( )X m∈Z χ(|m| M) 2 m−ν(0( ))2 16µ2( )X m∈Z χ(|m| M)|ψm|2 E −(1 + ( ) + δ( )) σ( ) M( ),∀τ⩽ 0.(3.27) Fo he e m ReF(ψ, ), φE we ha e, by exploi ing Cauchy’s inequali y, ha ReF(ψ, ), φE =( β ( )|z|2, w)+(g, w) + Im( , l),(3.28) (β ( )|z|2, w)⩽ν 8( )X m∈Z χ(|m| M) 2 m+2β2 ν( )X m∈Z χ(|m| M)|zm|4 ⩽ν 8( )X m∈Z χ(|m| M) 2 m+2β2 σ( ) ν( )X m∈Z χ(|m| M)|zm|2,(3.29) (g ( ), w)⩽ν 8( )X m∈Z χ(|m| M) 2 m+2 ν( )X m∈Z χ(|m| M)g2 m,(3.30) Im( , l)⩽α 2X m∈Z χ(|m| M)|zm|2+1 αX m∈Z χ(|m| M)| m|2.(3.31) 16 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. Now, using (2.34), (3.15), (3.18), (3.27) and (3.28)-(3.31), we a i e a 1 2 d d Q( ) + 2σ+0( ) 2( )−ν 16µ|0( ) ( )|2Q( ) . σ( ) ( )X m∈Z χ(|m| M)|zm|2+1 ( )X m∈Z χ(|m| M)g2 m+X m∈Z χ(|m| M)| m|2 +(1 + ( )) σ( ) M( ),∀τ⩽ 0,(3.32) whe e Q( ) = X m∈Z χ(|m| M)|ψm|2 E . We in e om (2.5) ha 2σ+0( ) 2( )−ν 16µ|0( ) ( )|2⩾σ. (3.33) Hence, he di e en ial inequali y (3.32) now eads d d Q( ) + σQ( ).1 + ( ) M( ) σ( ) + σ( ) ( )X m∈Z χ(|m| M)|zm( )|2 | {z } I +1 ( )X m∈Z χ(|m| M)g2 m( ) + X m∈Z χ(|m| M)| m( )|2,∀τ⩽ 0.(3.34) In o de o es ima e e m I, we ake he imagina y pa o he inne p oduc (·,·) o equa ion (2.2) wi h χ(|m| M)¯zmm∈Z o deduce X m∈Z χ(|m| M)|zm( )|2.e−α Z τ eαsX m∈Z χ(|m| M)| m(s)|2+ σ(s) Mds +e−α( −τ)X m∈Z χ(|m| M)|zm(τ)|2,∀τ⩽ 0.(3.35) In iew o (H2), we ind e−α Z τ eαs X m∈Z| m(s)|2ds⩽e(%−σ 2) J( )<+∞, o each ∈R. Thus, o any ε > 0, he e exis s some M1=M1( , ε)∈Nsuch ha σ( ) ( )e−α Z τ eαs X |m|⩾M1 | m(s)|2ds < σε2 36 ,∀M⩾M1.(3.36) A he same ime, om (3.3) we ob ain ha e−α MZ τ eαs σ(s)ds ⩽e−α MZ τ eαsds+L2( )Z τ e(α−σ)sZ −∞ eσηk (η)k2+kg(η)k2dηds +L2( )Z τ e(α−σ)sZs τ e(σ−2α)ηe(α−σ 2+%)ηJ(η)2dηds .1 M1 + L2( )Z −∞ eσηk (η)k2+kg(η)k2dη+e(2%−σ) ML2( )˜ J1( )<+∞. 17 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. Hence, o any gi en (⩾τ) and abo e ε, he e exis s some M2=M2( , ε)∈Z+such ha σ( ) ( )Me−α Z τ eαs σ(s)ds < σε2 36 ,∀M⩾M2.(3.37) I is clea ha , o abo e and ε, he e exis s some 1= 1( , ε, )⩽ and some M3=M3( , ε)∈Z+ such ha σ( ) ( )e−α( −τ)X m∈Z χ(|m| M)|zm(τ)|2⩽ σ( ) ( )e−α( −τ) 2<σε2 36 ,∀τ⩽ 1,(3.38) 1 + ( ) σ( ) M( )<σε2 12 ,∀M⩾M3.(3.39) Using (3.34)-(3.39) and applying G onwall’s inequali y, we see ha o any M⩾max{M1, M2, M3} and τ⩽min{ 0, 1}, Q( ).Q(τ)e−σ( −τ)+e−σ Z τ eσs1 (s)X m∈Z χ(|m| M)g2 m(s) + X m∈Z χ(|m| M)| m(s)|2ds+ε2 6.(3.40) Again exploi ing assump ion (H2), he e exis s some M4=M4( , ε)∈Z+such ha e−σ Z τ eσsh1 (s)X m∈Z χ(|m| M)g2 m(s) + X m∈Z χ(|m| M)| m(s)|2ids ⩽e−σ ( )Z −∞ eσs X m∈Z χ(|m| M)g2 m(s)ds+e−σ Z −∞ eσs X m∈Z χ(|m| M)| m(s)|2ds < ε2 6,∀M⩾M4.(3.41) Since ψτ∈Bτ( ), we conclude ha o abo e gi en ∈R,ε > 0 and > 0, he e exis s some 2= 2( , , ε)⩽ such ha Q(τ)e−σ( −τ)⩽kψτk2 Eτe−σ( −τ)⩽ 2e−σ( −τ)<ε2 6,∀τ⩽ 2.(3.42) Inse ing (3.41) and (3.42) in o (3.40) yields sup ψτ∈Bτ( )X |m|⩾2M∗ |(U( , τ)ψτ)m|2 E = sup ψτ∈Bτ( )X |m|⩾2M∗ |ψm( )|2 E ⩽2Q( ).ε2,∀τ⩽ ∗,(3.43) whe e M∗= max{M1, M2, M3, M4}, ∗= min{ 0, 1, 2}.This comple es he p oo o Lemma 3.3. In e ms o Lemma 3.1, Lemma 3.2 and Lemma 3.3, we ob ain he main esul o his sec ion as ollows. Theo em 3.1. Le assump ions (H1)-(H2) hold. Then he p ocess {U( , τ)} ⩾τhas a ime-dependen pullback a ac o A={A } ∈R ul illing he h ee p ope ies o [64, De ini ion 3.1(4)]. 4 Exis ence o in a ian Bo el p obabili y measu es and s a- is ical solu ions on ime-dependen phase spaces In his sec ion we cons uc a amily o in a ian Bo el p obabili y measu es {m } ∈R o he p ocess {U( , τ)} ⩾τon he ime-dependen phase spaces {E } ∈R ia gene alized Banach limi s. Then we p opose he concep o s a is ical solu ions o equa ion (2.12), and p o e ha he cons uc ed p obabili y measu es is i s s a is ical solu ion and sa is ies Liou ille heo em in S a is ical Mechanics. The idea o he p oo o he main esul s o his sec ion (Theo em 4.1) is based on he amewo k o Lukaszewicz and Robinson o non-au onomous dissipa i e dynamical sys ems [40, Theo em 3.1]. 18 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. In o de o emphasize he di e ence and c ux when we cons uc he in a ian Bo el p obabili y measu es o he dissipa i e dynamical sys ems on ime-dependen phase spaces, we i s in oduce he “sui able” ini ial da a collec ion Γ =n{ϕ(s)}s∈R:ϕ(s) = (p(s)u, , z)T∈Es, ϕ = (u, , z)T∈BE( )o,(4.1) whe e > 0 and BE( ) = {ϕ∈E:kϕkE⩽ }. No e ha all elemen s wi hin he collec ion Γ would be uni o mly bounded, acco ding o [64, De ini ion 3.1(1)]. As will be seen, he cons uc ed Γ plays he i al ole o he ollowing τ-con inuous p ope y o {U( , τ)} ⩾τin {E } ∈R. De ini ion 4.1. The p ocess {U( , τ)} ⩾τis said o be τ-con inuous on he ime-dependen phase {E } ∈R, i o e e y {ϕ∗(τ)}τ∈R∈S ⩾0 Γ and any gi en ∈R, he E - alued mapping τ7→ U( , τ)ϕ∗(τ) (4.2) is con inuous and bounded on (−∞, ]. We nex es ablish wo auxilia y lemmas conce ning a ce ain con inui y o U( , τ)ψ∗(τ) wi h espec o he pa ame e s τand , as sui s o ou pu poses below. Lemma 4.1. Le assump ions (H1)-(H2) hold. Le > 0,τ∗∈Rand {ϕ∗(s)}s∈R∈Γ be gi en. Then o any ε > 0, he e exis s some ρ=ρ(τ∗, , ε)>0such ha kU(s, τ∗)ϕ∗(τ∗)−ϕ∗(s)k2 Es.ε, ∀s∈[τ∗, τ∗+ρ).(4.3) P oo . Le > 0, {ϕ∗(s)}s∈R∈Γ wi h ϕ∗= (u∗, ∗, z∗)T∈BE( ), and τ∗∈Rbe gi en. Fo any s∈[τ∗, τ∗+1], le U(s, τ∗)ϕ∗(τ∗) = (u(s), (s), z(s))T= (um(s), m(s), zm(s))T m∈Z∈Esbe he solu ion o p oblem (2.10)-(2.11) wi h ini ial da a ϕ∗(τ∗)=(uτ∗, τ∗, zτ∗)T= (p(τ∗)u∗, ∗, z∗)T∈Eτ∗a ini ial ime τ∗. No e ha kU(s, τ∗)ϕ∗(τ∗)−ϕ∗(s)k2 Es=I1−I2−I3,(4.4) whe e    I1=kU(s, τ∗)ϕ∗(τ∗)k2 Es−kϕ∗(τ∗)k2 Eτ∗, I2=kϕ∗(s)k2 Es−kϕ∗(τ∗)k2 Eτ∗, I3= 2U(s, τ∗)ϕ∗(τ∗)−ϕ∗(s), ϕ∗(s)Es. (4.5) We es ima e he h ee e ms in (4.5) sepa a ely. Fo he i s one, we ha e by using (2.33) and he mono onici y o (·) ha I1=kU(s, τ∗)ϕ∗(τ∗)k2 Es−kϕ∗(τ∗)k2 Eτ∗=Zs τ∗ dkU(θ, τ∗)ϕ∗(τ∗)k2 Eθ dθdθ .Zs τ∗k (θ)k2dθ+1 (τ∗+ 1) Zs τ∗kg(θ)k2dθ+1 (τ∗+ 1) Zs τ∗kz(θ)k4dθ. (4.6) In iew o (2.20) and (3.9), we ob ain Zs τ∗kz(θ)k4dθ.Zs τ∗kzτ∗k4dθ+Zs τ∗e−αθ Zθ −∞ eαηk (η)k2dη2dθ +kzτ∗k2Zs τ∗ e−αθ Zθ −∞ eαηk (η)k2dηdθ . 4(s−τ∗) + Zs τ∗ e(2%−σ)θJ2(θ)dθ+ 2Zs τ∗ e(%−σ 2)θJ(θ)dθ. (4.7) 19 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. By he ac ha g(·)∈C(R, `2), J(·) and J2(·)∈C(R,R), we in e ha o any ε > 0 he e exis s some ρ1=ρ1(τ∗, ε, )>0 such ha I1.Zs τ∗k (θ)k2dθ+Zs τ∗kg(θ)k2dθ (τ∗+ 1) +Zs τ∗kz(θ)k4dθ (τ∗+ 1) <ε 3,∀s∈(τ∗, τ∗+ρ1).(4.8) Fo he second e m I2, we ha e I2=kϕ∗(s)k2 Es−kϕ∗(τ∗)k2 Eτ∗ =(kp(s)u∗k2 (s)+k ∗k2+kz∗k2)−(kp(τ∗)u∗k2 (τ∗)+k ∗k2+kz∗k2) =(ku∗k2 µ+k ∗k2+kz∗k2)−(ku∗k2 µ+k ∗k2+kz∗k2)=0.(4.9) Fo he hi d e m I3, we ha e I3=2U(s, τ∗)ϕ∗(τ∗)−ϕ∗(s), ϕ∗(s)Es =2U(s, τ∗)ϕ∗(τ∗)−ϕ∗(τ∗) + ϕ∗(τ∗)−ϕ∗(s), ϕ∗(s)Es =2U(s, τ∗)ϕ∗(τ∗)−ϕ∗(τ∗), ϕ∗(s)Es+ 2ϕ∗(τ∗)−ϕ∗(s), ϕ∗(s)Es:= 2I31 + 2I32.(4.10) No e ha he de ini ion o Γ implies kϕ∗(s)k2 Es=kp(s)u∗k2 (s)+k ∗k2+kz∗k2=ku∗k2 µ+k ∗k2+kz∗k2 ⩽µku∗k2+ 4ku∗k2+k ∗k2+kz∗k2⩽(6 + µ) 2,∀s∈R,(4.11) which means ϕ∗(s)∈Bs(p6 + µ) ) wi h e e y s∈R. Thus, simila o (4.6), |I31|=U(s, τ∗)ϕ∗(τ∗)−ϕ∗(τ∗), ϕ∗(s)Es=Zs τ∗ dU(θ, τ∗)ϕ∗(τ∗) dθdθ, ϕ∗(s)Es ⩽Zs τ∗kdU(θ, τ∗)ϕ∗(τ∗) dθkEsdθkϕ∗(s)kEs⩽Zs τ∗ (θ) (s)kdU(θ, τ∗)ϕ∗(τ∗) dθkEθdθkϕ∗(s)kEs . (τ∗) (τ∗+ 1)Zs τ∗kdU(θ, τ∗)ϕ∗(τ∗) dθk2 Eθdθ1/2(s−τ∗)1/2.(4.12) Nex , we p o e ha he e exis cons an s L5=L5(τ∗, )>0 and L6=L6(τ∗) such ha Zs τ∗kdU(θ, τ∗)ϕ∗(τ∗) dθk2 Eθdθ.L∗(τ∗, ) := L5L6+L2 5+Zτ∗+1 τ∗kg(θ)k2dθ+Zτ∗+1 τ∗k (θ)k2dθ. (4.13) Indeed, in iew o (2.10) and (2.17), we in e kdU(θ, τ∗)ϕ∗(τ∗) dθk2 Eθ.kH(θ)U(θ, τ∗)ϕ∗(τ∗)k2 Eθ+kF(U(θ, τ∗)ϕ∗(τ∗)), θ)k2 Eθ .L1(θ)kU(θ, τ∗)ϕ∗(τ∗)k2 Eθ+k|z(θ)|2+g(θ)k2 2(θ)+kz(θ)u(θ)− (θ)k2.(4.14) By (3.4) and using simila de i a ions o (4.7), we a i e a kU(θ, τ∗)ϕ∗(τ∗)k2 Eθ ⩽kϕ∗(τ∗)k2 Eτ∗e−σ(θ−τ∗)+L2(θ)e−σθ Zθ τ∗ eση(k (η)k2+kg(η)k2)dη+L2(θ)e−σθ Zθ τ∗ eσηkz(η)k4dη . 2+L2(τ∗+ 1) Zτ∗+1 τ∗k (η)k2+kg(η)k2dη+L2(τ∗+ 1) Zτ∗+1 τ∗kz(η)k4dη . 2+L2(τ∗+ 1) Zτ∗+1 τ∗k (η)k2+kg(η)k2dη +L2(τ∗+ 1)Zτ∗+1 τ∗ 4dθ+Zτ∗+1 τ∗ e(2%−σ)θJ2(θ)dθ+Zτ∗+1 τ∗ e(%−σ 2)θJ(θ)dθ = : L5=L5(τ∗, ),∀θ∈[τ∗, τ∗+ 1],(4.15) 20 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. which means kz(θ)k2.L5and ku(θ)k2.L5.(4.16) Fo any ∈R, he e holds δ( )∈(0, ν/4). By (2.18) we in e ha L1(θ)⩽L6=L6(τ∗) = max s∈[τ∗,τ∗+1] L1(s). Inse ing (4.15) and (4.16) in o (4.14) implies kdU(θ, τ∗)ϕ∗(τ∗) dθk2 Eθ.L5L6+L2 5+kg(θ)k2 2(τ∗+ 1) +L2 5+k (θ)k2,∀θ∈[τ∗, τ∗+ 1],(4.17) and (4.13) is p o ed. In ligh o (4.13) and (4.12), we a i e a |I31|=U(s, τ∗)ϕ∗(τ∗)−ϕ∗(τ∗), ϕ∗(s)Es .(τ∗) (τ∗+ 1)Zs τ∗kdU(θ, τ∗)ϕ∗(τ∗) dθk2 Eθdθ1/2(s−τ∗)1/2 . L∗(τ∗, )(τ∗) (τ∗+ 1) (s−τ∗)1/2, which implies ha , o abo e ε, he e exis s some ρ2=ρ2(τ∗, ε, )>0 such ha |2I31|= 2|(U(s, τ∗)ϕ∗(τ∗)−ϕ∗(τ∗), ϕ∗(s))Es|.ε 6,∀s∈[τ∗, τ∗+ρ2).(4.18) Fo I32 in (4.10), we easily ob ain |I32|=(ϕ∗(τ∗)−ϕ∗(s), ϕ∗(s))Es=p(τ∗)−p(s)u∗,p(s)u∗(s) =p(τ∗)−p(s) p(s)ku∗k2 µ. 2|p(τ∗)−p(s)| p(τ∗+ 1) . By he con inui y o (·), we ha e ha , o abo e ε he e exis s some ρ3=ρ3(τ∗, ε, )>0 such ha 2|I32|=2ϕ∗(τ∗)−ϕ∗(s), ϕ∗(s)Es<ε 6,∀s∈[τ∗, τ∗+ρ3).(4.19) Taking ρ4= min{ρ2, ρ3}and using (4.10), (4.18) and (4.19), we ob ain |I3|⩽2|I31|+ 2|I32|.ε 3,∀s∈[τ∗, τ∗+ρ4).(4.20) Choosing ρ=ρ(τ∗, , ε) = min{ρ1, ρ4}, we ge (4.3) om (4.4), (4.8), (4.9) and (4.20). Simila ly, we ha e he ollowing lemma. Lemma 4.2. Le assump ions (H1)-(H2) hold, and le > 0,τ∗∈R, and {ϕ∗(s)}s∈R∈Γ be gi en. Then, o any ε > 0, he e exis s some ρ=ρ(τ∗, , ε)>0such ha kU(τ∗, s)ϕ∗(s)−ϕ∗(τ∗)k2 Eτ∗.ε, ∀s∈(τ∗−ρ, τ∗].(4.21) Wi h abo e wo auxilia y lemmas in hand, we now s a e and p o e he τ-con inuous p ope y o he p ocess {U( , τ)} ⩾τon ime-dependen phase spaces {E } ∈R. Lemma 4.3. Le assump ions (H1)-(H2) hold. Then, o e e y gi en ∈Rand {ϕ∗(s)}s∈R∈Γ wi h some > 0, he E - alued mapping τ7→ U( , τ)ϕ∗(τ)is con inuous and bounded on (−∞, ]. 21 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. P oo . Gi en ∈Rand > 0, le {ϕ∗(s)}s∈R∈Γ wi h ϕ∗= (u∗, ∗, z∗)T∈BE( ). F om (3.4) and Lemma 3.2 we deduce o any τ∗∈(−∞, ] ha kU( , τ∗)ϕ∗(τ∗)kE ⩽ 2+L2( )e−σ Z −∞ eσsk (s)k2+kg(s)k2ds +L2( )e−σ Z −∞ e(σ−2α)sZs −∞ eαηk (η)k2dη2ds < +∞,(4.22) which means ha he E - alued mapping τ7→ U( , τ)ϕ∗(τ) is bounded on (−∞, ]. Now, o any τ∗∈(−∞, ], i clea ha o any ε > 0 he e exis s some ρ=ρ(ε, , )>0 such ha |s−τ∗|< ρ =⇒ kU( , s)ϕ∗(s)−U( , τ∗)ϕ∗(τ∗)kE < ε. (4.23) Nex we spli he p oo in o wo cases. Case I :τ∗⩽s⩽τ∗+ 1. Using he in a iance o p ocess and (2.39), we a i e a kU( , s)ϕ∗(s)−U( , τ∗)ϕ∗(τ∗)k2 E =kU( , s)ϕ∗(s)−U( , s)U(s, τ∗)ϕ∗(τ∗)k2 E ⩽exp nL3( , , s)( −s)okϕ∗(s)−U(s, τ∗)ϕ∗(τ∗)k2 Es ⩽exp n˜ L3( , , τ∗)( −τ∗)okϕ∗(s)−U(s, τ∗)ϕ∗(τ∗)k2 Es,(4.24) he e we ha e used he ac ha one can choose L3( , , τ) o be con inuous wi h espec o τin (2.47), and ˜ L3( , , τ∗) = max s∈[τ∗,τ∗+1] L3( , , s). In iew o Lemma 4.1, we in e ha , o any ε > 0, he e exis s some ρ0=ρ0(τ∗, , ε)>0 such ha kU( , s)ϕ∗(s)−U( , τ∗)ϕ∗(τ∗)k2< ε, ∀s∈τ∗, τ∗+ρ0,(4.25) which means U( , τ)ϕ∗(τ) is igh -con inuous on τ=τ∗. Case II :τ∗−1⩽s⩽τ∗. Again by he in a iance o he p ocess, we ha e ha kU( , s)ϕ∗(s)−U( , τ∗)ϕ∗(τ∗)k2 E =kU( , τ∗)U(τ∗, s)ϕ∗(s)−U( , τ∗)ϕ∗(τ∗)k2 E . ⩽exp nL3( , , τ∗)( −τ∗)okU(τ∗, s)ϕ∗(s)−ϕ∗(τ∗)k2 Eτ∗.(4.26) Applying Lemma 4.2, we conclude ha o any ε > 0 he e exis s some ρ00 =ρ00(τ∗, , ε)>0 such ha kU( , s)ϕ∗(s)−U( , τ∗)ϕ∗(τ∗)k2 E < ε, ∀s∈τ∗−ρ00, τ∗,(4.27) which means U( , τ)ϕ∗(τ) is le -con inuous on τ=τ∗. By he a bi a iness o τ∗, we end he p oo . We nex upda e he de ini ion o gene alized Banach limi (c . [22,40]) o cons uc he in a ian Bo el p obabili y measu es {m } ∈R o he p ocess {U( , τ)} ⩾τon ime-dependen phase spaces {E } ∈R. De ini ion 4.2. A gene alized Banach limi is any linea unc ional, which is deno ed by LIM →−∞, de ined on he space o all bounded eal- alued unc ions on (−∞,+∞)and sa is ying (1) LIM →−∞h( )⩾0 o nonnega i e unc ions g(·)on (−∞,+∞); (2) LIM →−∞h( ) = lim →−∞ h( )i he usual limi lim →−∞ h( )exis s. 22 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. Now, we deno e by C(E ) he collec ion o all con inuous unc ionals wi h eal alues on E . Be- sides, o any gi en Bo el p obabili y measu e m on E and any unc ion Φ ∈C(E ), ZE Φ (ψ)dm (ψ) deno es he Bochne in eg al. The main esul o his subsec ion sec ion eads as ollows. Theo em 4.1. Le assump ions (H1)-(H2) hold, and le > 0and {ϕ∗(s)}s∈R∈Γ wi h ϕ∗= (u∗, ∗, z∗)T∈BE( )be gi en. Then o any gi en gene alized Banach limi LIMτ→−∞, e e y ∈R and Ψ ∈C(E ), he e exis s a unique Bo el p obabili y measu e m on E such ha LIMτ→−∞ 1 −τZ τ Ψ U( , θ)ϕ∗(θ)dθ =ZA Ψ (ψ)dm (ψ) = ZE Ψ (ψ)dm (ψ) (4.28) =LIMτ→−∞ 1 −τZ τZEθ Ψ U( , θ)ψdmθ(ψ)dθ. (4.29) Mo eo e , he suppo o he measu e m is con ained in A , and he measu es {m } ∈Rsa is y he ollowing in a ian p ope y ZA Ψ (ψ)dm (ψ) = ZAτ Ψ U( , τ)ψdmτ(ψ),∀ ⩾τ. (4.30) The p oo o Theo em 4.1 is simila wi h ha o [64, Theo em 4.1], wi h e y sligh ly di e ence. He e we omi he de ails. We nex p opose he concep o s a is ical solu ions o equa ion (2.10), and p o e ha he amily o in a ian Bo el p obabili y measu es {m } ∈Rgua an eed by Theo em 4.1 is i s s a is ical solu ion and sa is ies Liou ille heo em in S a is ical Mechanics. Rew i e equa ion (2.10) as dψ d =G(ψ, ) := F(ψ, )−H( )ψ, ∈R.(4.31) To o mula e he de ini ion o s a is ical solu ion o equa ion (4.31) on he ime-dependen phase spaces {E } ∈R, we i s in oduce he amily o class {T } ∈Ro es unc ions. De ini ion 4.3. Fo e e y gi en , by T we deno e he class o eal- alued con inuous unc ions Φ on E ha a e bounded on bounded subse s o E and sa is y he ollowing wo condi ions. (a) o e e y ϕ∈E , he F ech´e de i a i e Φ0 (ϕ) exis s: o each ϕ∈E he e exis s an elemen Φ0 (ϕ)∈E such ha |Φ (ϕ+φ)−Φ (ϕ)−Φ0 (ϕ), φE | kφkE −→ 0 as kφkE →0, φ ∈E . (b) he mapping ϕ7→ Φ0 (ϕ) is con inuous and bounded om E o E . The condi ions in De ini ion 4.3 a e su icien o ensu e ha i ψ( ) sol es equa ion (4.31) hen d d Φ (ψ( )) = (G(ψ( ), ),Φ0 (ψ( )))E , ∈R.(4.32) Fo e e y ∈R, he de ini ion o class T consis ing o es unc ions is simila o ha o [62, De ini ion 4.2]. He e we o mula e a amily o class {T } ∈Ro es unc ions o ea he scena io ha he p ocess {U( , τ)} ⩾τis de ined on he ime-dependen phase spaces. We wan o poin ou ha he amily o 23 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. class o unc ions sa is ying De ini ion 4.3 exis s in g ea p o usion. Fo example, we could ake Φ o be a cylind ical es unc ions (c . [64, De ini ion 5.1]) Φ :E →Ro he o m Φ (φ) = $((φ, ψ1)E ,(φ, ψ2)E ,(φ, ψ3)E ), whe e $is a C1 eal- alued unc ion on R3wi h compac suppo , and ψ1,ψ2,ψ3belong o E . Fo such Φ (·), di ec compu a ions show ha i s F ech´e de i a i e Φ0 in E has he o m Φ0 (φ) = 3 X j=1 ∂j$((φ, ψ1)E ,(φ, ψ2)E ),(φ, ψ3)E )ψj, whe e ∂j$is he pa ial de i a i e o $wi h espec o i s j- h coo dina e. We now speci y he de ini ion o s a is ical solu ion o equa ion (4.31) on he ime-dependen phase spaces {E } ∈R. No e ha by (2.16) and (4.1), i is no ha d o check ha E =[ ⩾0 B ( ) = [ ⩾0φ( ) : {φ(θ)}θ∈R∈Γ , ∈R. De ini ion 4.4. A amily {ρ } ∈Ro Bo el p obabili y measu es wi h ρ on E is called a s a is ical solu ion o equa ion (4.31) i {ρ } ∈Rsa is ies he ollowing wo condi ions: (a) o almos ∈R, he unc ion ψ7→ G(ψ, ), φE is ρ -in eg able o e e y φ∈E . Mo eo e , he mapping 7→ ZE G(ψ, ), φE dρ (φ) belongs o L1 loc(R) o e e y φ∈E . (b) o any {ψ(s)}s∈R∈S ⩾0Γ and all ⩾τ, he Liou ille- ype equa ion ZE Φ (ψ( ))dρ (ψ( )) −ZEτ Φτ(ψ(τ))dρτ(ψ(τ)) = Z τZEθG(ψ(θ), θ),Φ0 θ(ψ(θ))Eθdρθ(ψ(θ))dθ (4.33) holds o any {Φ } ∈R∈ {T } ∈R. Theo em 4.2. Le assump ions (H1)-(H2) hold. Then, he amily o in a ian Bo el p obabili y measu es {m } ∈Rob ained in Theo em 4.1 is a s a is ical solu ion o equa ion (4.31). P oo . Conside gi en ∈R. Fo e e y φ= (φ1, φ2, φ3)T∈E , we de ine Ψ (·) : E 7−→ Ras Ψ (ψ) = G(ψ, ), φE ,∀ψ= (u, , z)T∈E .(4.34) We nex p o e ha Ψ (·)∈C(E ). Le ψ∗= (u∗, ∗, z∗)T∈E be ixed and conside ψ= (u, , z)T∈ E wi h kψ∗−ψkE <1. Then we ha e |Ψ (ψ∗)−Ψ (ψ)|=G(ψ∗, )−G(ψ, ), φE  ⩽H( )(ψ∗−ψ), φE |+|F(ψ∗, )−F(ψ, ), φE .(4.35) By (2.17), (2.18) and he ac ha δ( )∈(0, ν/4) o e e y ∈R, we a i e a |H( )(ψ∗−ψ), φE |.k(H( )(ψ∗−ψ)kE kφkE .L1( )kψ∗−ψkE kφkE .(4.36) Also, we ha e by (2.19) ha |F(ψ∗, )−F(ψ, ), φE |⩽kF(ψ∗, )−F(ψ, )kE kφkE .L8( )(1 + kψ∗kE )kψ∗−ψkE kφkE ,(4.37) 24 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal. whe e L8( )=(4β2 2( )+2+4( ) µ)1/2is a posi i e cons an depending only on . The con inui y o he mapping Ψ (·) : E 7−→ R ollows om (4.34)-(4.37). Now, Theo em 4.1 ells us ha he unc ion ψ7→ G(ψ, ), φE := Ψ (ψ) is m -in eg able o e e y φ∈E , and ha he mapping 7→ ZE G(ψ, ), φE dm (ψ) = ZE Ψ (ψ)dm (ψ) belongs o L1 loc(R) o e e y φ∈E . Thus p ope y (a) o De ini ion 4.3 is p o ed. The p oo o p ope y (b) in De ini ion 4.3 is e y simila wi h ha o [ [64], Theo em 5.1] and we omi he de ails he e. We wan o ema k ha in S a is ical Mechanics Φ0 (·) = 0 o all ∈Rimplies ha equa ion (4.31) (and hus sys em (1.1)-(1.2)) eaches i s s a is ical equilib ium (c . [22]). In his case, he Liou ille ype equa ion (4.33) u ns o be ZA Φ (ϕ∗( ))dm (ϕ∗( )) = ZAτ Φτ(ϕ∗(τ))dmτ(ϕ∗(τ)),{ϕ∗(s)}s∈R∈[ ⩾0 Γ ,∀ , τ ∈R,(4.38) which e eals ha al hough he shape o he ime-dependen pullback a ac o A•could change along wi h he e olu ion o ime om τ o , he “ o al measu es” o Aτand A always coincide wi h each o he , ha is, gi en ha he sys em has eached s a is ical equilib ium, he “ o al measu es” o he ime-dependen pullback a ac o A•a e conse a i e as ime passes. This is exac ly he heo y o Liou ille Theo em in S a is ical Mechanics (see e.g. [69, Page19, (1.3.29)]). The e o e, we say ha he s a is ical solu ions o he la ice KGS equa ions wi h a ying coe icien ul ill he Liou ille Theo em on he ime-dependen phase spaces. We end he a icle wi h he issues on he limi ing beha io o solu ion, ime-dependen pullback a ac o and s a is ical solu ion o equa ions (1.1)-(1.2). Conside he la ice KGS equa ions wi h a ying coe icien s n( )¨um+ ˙um+ (2um−um−1−um+1) + µum−β|zm|2=gm( ), m ∈Z, i˙zm−(2zm−zm−1−zm+1) + iαzm+zmum= m( ), m ∈Z,(4.39) and ˙um+ (2um−um−1−um+1) + µum−β|zm|2=gm( ), m ∈Z, i˙zm−(2zm−zm−1−zm+1) + iαzm+zmum= m( ), m ∈Z,(4.40) co esponding o he case n(·)≡0, n∈N. We ha e p o ed ha he e is a s a is ical solu ion {µn } ∈Rsuppo ed by he ime-dependen pullback a ac o ˆ An={An( )} ∈Rwi h An( )⊂ `2 ( )×`2×L2 o sys em (4.39). I is no ha d o es ablish ha he e also is a s a is ical solu ion {µ0 } ∈Rsuppo ed by he pullback a ac o ˆ A0={A0( )} ∈Rwi h A0( )⊂`2×L2 o sys em (4.40). When lim n→∞ sup ∈R n( ) = 0, some na u al and in e es ing ques ions a e        (a) Does he solu ion o equa ion (4.39) end o ha o equa ion (4.40)? (b) Does he pullback a ac o An( ) con e ge o A0( )?, (c) Does he s a is ical solu ion µn con e ge o µ0 ? (d) In wha space and wha sense shall we discuss abo e ques ions? (4.41) I is also e y in e es ing o in es iga e he singula limi ing beha io as ha as [41] o he pullback a ac o s and s a is ical solu ions o sys em (4.39). CONFLICT OF INTEREST STATEMENT This wo k does no ha e any con lic s o in e es . 25 20 No 2024 23:13:21 PST 241120-Zhao Ve sion 1 - Submi ed o Comm. Pu e Appl. Anal.