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

Pullback asymptotic behavior and statistical solutions for lattice Klein-Gordon-Schr¨odinger equations with varying coefficient ∗ Caidi Zhao†Rong Zhuang‡Tom´as Caraballob§ aDepartment of Mathematics, Wenzhou University, Wenzhou, Zhejiang Province, 325035, People’s Republic of China bDepartmento de Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, c/Tarfia s/n, 41012-Sevilla, Spain November 21, 2024 Abstract In this article, the authors investigate the pullback asymptotic behavior and statistical solutions for lattice Klein-Gordon-Schr¨odinger equations with varying coefficient. They first prove the global well-posedness of the addressed equations and the existence of a family of time-dependent pullback attractor for the associated process acting on the time-dependent phase spaces. Then they verify that the process possesses a family of invariant Borel probability measures with support contained in the time-dependent pullback attractor. Further, they reformulate the definition of statistical solution for the evolutionary equations on time-dependent phase spaces. As a result, they prove the existence of statistical solution for the lattice Klein-Gordon-Schr¨odinger equations with varying coefficient and show that it satisfies the Liouville theorem. Keywords: Statistical solution; Time-dependent pullback attractor; Invariant Borel probability measures; Varying coefficient; Lattice Klein-Gordon-Schr¨odinger equations. MSC2010: 35B41; 35D99; 76F20 1 Introduction This article is devoted to the study of a non-autonomous lattice dynamical systems (LDSs) corresponding to the initial value problem of the following lattice Klein-Gordon-Schr¨odinger (KGS) equations with varying coefficient: (t)¨um+ν˙um+ (2um−um−1−um+1) + µum−β|zm|2=gm(t), t > τ, m ∈Z,(1.1) i˙zm−(2zm−zm−1−zm+1) + iαzm+zmum=fm(t), t > τ, m ∈Z,(1.2) um(τ) = uτ,m,˙um(τ) = u1τ,m, zm(τ) = zτ,m, τ ∈R, m ∈Z,(1.3) where Zand Rdenote the sets of integer and real numbers, respectively, i=√−1 is the unit of the imaginary numbers, um(·) and zm(·) are the unknown real-valued and complex-valued functions, respectively, and the real-valued function (t)>0 will be assumed to satisfy some conditions. ∗Supported by NSF of China with No.12371245, 11971356 and by Key project of Zhejiang Province’s Natural Science Foundation with No.LZ24A010005. Also supported by FEDER, the Spanish Ministerio de Ciencia e Innovaci´on and AEI under project PID2021122991NB-C21. †Corresponding author E-mail: zhao[email protected] or [email protected] ‡E-mail: [email protected] §E-mail: [email protected] 1 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. Equations (1.1)-(1.2) can be considered to be an approximation of the spatial discretization (x∈R) of the following non-autonomous KGS equations with varying coefficient: (t)utt +νut−∆u+µu −β|z|2=g(x, t).(1.4) izt+ ∆z+iαz +zu =f(x, t).(1.5) KGS equations are used to describe systems of conserved scalar nucleons interacting with neutral scalar mesons [28]. Here, u=u(x, t) represents a real scalar meson field and z=z(x, t) a complex scalar nucleon field. Perturbations of this system including dissipation were introduced by the terms (t)utt,νut,µu,β|z|2,iαz with (t), ν,µ,β,α > 0 and the real-valued driving function g(x, t) and complex-valued function f(x, t). Earlier works on the continuous KGS equations mainly focused on the Cauchy problem and the initial boundary value problems [12,28,45]. With the development of the theory of infinite-dimensional dynamical systems, much attention has been paid to the asymptotic behavior of solutions [2,4,31,36, 50,62]. A popular object used below associated with the long-time behavior of a dynamic system is the attractor. Biler constructed the global finite-dimensional attractor and estimated the dimension of this attractor using the Lyapunov exponents for the flows on compact invariant sets in a domain of Rn(n⩽3) [4]; Guo considered the Cauchy problem in R3and proved the existence of maximal attractors [31]. Recently, the statistical solution and Liouville theorem of equations (1.4)-(1.5) were investigated in [62] via the theories of pullback attractor and generalized Banach limit. The original motivation of this article is to investigate the long-time behavior and probability distribution of solutions for the lattice KGS equations with time-dependent coefficient (t). LDSs form a class of extended systems that are intermediate between partial differential equations and cellular automata [13]. In some situations, LDSs emerge as the spatial discretization of continuous partial differential equations. LDSs are widely used in science and engineering, for instance, electrical engineering [11], propagation of nerve pulses in myelinated axons [33], chemical reaction theory [18], etc. Over the past two decades, various types of attractors were the subject of numerous investigations for LDSs. For example, [7,66] studied the global attractors for the first-order LDSs and retarded LDSs; [1,34,55,67] investigated the global attractors, the singular limiting behavior of pullback attractors, the exponential attractors and their fractal dimension, as well as the uniform attractors for the secondorder LDSs; [8,68] researched the random exponential attractor for the stochastic LDSs. Very recently, the random numerical stability of attractors for nonlinear Schr¨odinger equations on infinite lattices was investigated by [38]. As for lattice KGS equations, [54] established the existence of compact kernel sections and estimated its Kolmogorov -entropy; [56] proved the existence of pullback attractors and invariant measures. However, as far as we know, there is no reference investigating the lattice KGS equations with time-dependent coefficient. The theory of statistical solution comes from Statistical Mechanics [22]. It has long been known that singular trajectories are not as significant or physically relevant as the statistical behavior of some type of dissipative system [15,42]. This is because physically mean numbers of turbulent flow are often well-behaved, while instantaneous quantities (e.g. velocity, kinetic energy, and energy dissipation) tend to change fairly rapidly over time or space. To overcome this difficulty, [19–21] introduced the concept of Foias-Prodi statistical solution, considering a family of invariant probability measures defined on the time-independent phase space of the Navier-Stokes equations, parameterized by the time variable and representing the evolution of the probability distribution of the state of the system; soon after, [48,49] proposed the definition of Vishik-Fursikov statistical solution, considering a single Borel measure in some suitable trajectories space of the incompressible Navier-Stokes equations and representing the probability distribution of the space-time velocity field. Later, Foias, Rosa and Temam considered 2 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. the idea of space-time statistical solution of Vishik and Fursikov with altered hypotheses to make it more analytically tractable [23,24]. The probability distribution of solutions for different types of evolutionary equations is now commonly characterized by statistical solutions and invariant measures. For instance, the existence of invariant measures on the metric space for the continuous autonomous dissipative dynamical systems was investigate by Glatt-Holtz and Chekroun in [14] and Lukaszewicz, Real and Robinson in [39]; [40] generalized the result of [39], constructing a family of probability measures for non-autonomous dissipative dynamical systems; Foias, Rosa and Temam studied systematically the statistical solutions for the 2D and 3D incompressible Navier-Stokes equations in [23–27]; Bronzi, Mondaini and Rosa resented in [5,6] an abstract framework for the theory of statistical solutions and trajectory statistical solutions for general evolution equations, including those with properties similar to the 3D incompressible Navier-Stokes equations. Recently, [58] established sufficient conditions for the existence of trajectory statistical solutions for general autonomous evolution equations and the abstract theory was applied to some models of evolutionary equations (see [32, 57, 59–61]), also the idea of [58] was developed to investigate the invariant sample measures for the 2D stochastic Navier-Stokes equations in [63]. In addition, Fjordholm and Wiedemann proved in [29] a version of Onsager’s conjecture on the conservation of energy for the incompressible Euler equations in the context of statistical solutions. Very recently, Gallenm¨uller, Wagner and Wiedemann gave a survey on probabilistic descriptions of fluid flow in [30], and researched the statistical solutions of the two-dimensional incompressible Euler equations in spaces of unbounded vorticity in [52]; Yang, Han and Zhao proved the existence of statistical solutions for dissipative non-autonomous Zakharov equations in [53]. Very recently, Zhao proved in [65] an essential property of the second-order elliptic equations in half-cylindrical domains, which is that absorbing estimate implies trajectory statistical solutions. Note that all these equations aforementioned are dissipative. However, we notice that there is barely no reference investigating the invariant measure and statistical solution for evolutionary equations on time-dependent phase spaces besides [64]. The first result within the current article is the existence of a time-dependent pullback attractor in a family of time-dependent phase spaces for the process generated by the solution mappings of problem (1.1)-(1.3). The existence of the pullback attractor will play the vital role when we construct the statistical solution. Compared to reference [56], the main difficulty we encounter here comes from the varying coefficient (t). Indeed, the varying coefficient (t) will lead to the fact that the classical theory of pullback attractor is not convenient to be applied. It seems reasonable to settle problem (1.1)-(1.3) in time-dependent phase spaces due to the varying coefficient (t), while the classical theory of pullback attractor (see e.g. [10]) is suitable for the problem addressed in fixed phase space. Fortunately, Temam and his group formulated the theory of time-dependent attractor during the study of non-autonomous oscillon equation and wave equation with varying coefficient [16,17], and this theory was developed by [9,35,37,43,44,47] to investigate the asymptotic behavior of reaction-diffusion equations and wave equations with varying coefficient. However, although we can borrow the theory of time-dependent attractor in our study, there still some difficulties when we estimate the solutions and prove the pullback asymptotically compactness of the process. To handle these difficulties, we first introduce a family of time-dependent phase spaces {Et}t∈R={`2 (t)×`2×L2}t∈Rand endowed them with suitable norms. We then establish that the time-dependent operator H(t) possesses some coercive property. This coercivity is important for estimating solutions. Besides, inspired by [44,56], we provide a sufficient and necessary condition guaranteeing the existence of the time-dependent attractor for dissipative lattice systems with varying coefficients (see Lemma 3.1). Finally, we use the sufficient condition to obtain the existence of the time-dependent pullback attractor by proving the existence of the bounded pullback absorbing set (see Lemma 3.2) and exhibiting the uniform estimates on “Tail End” of solutions (see Lemma 3.3). 3 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. mention Flandoli and Schmalfuss The second result in this article is the existence and Liouville theorem of the statistical solution for problem (1.1)-(1.3). Note that the classical theory of statistical solution was proposed for the evolutionary equations addressed in fixed phase space and seems not suitable for problem (1.1)- (1.3). It seems more reasonable to settle problem (1.1)-(1.3) in time-dependent phase spaces. Here we extend the idea of our recent work [64] to investigate the lattice KGS equations with varying coefficient. Precisely, we first consider a suitable collection of initial data (see (4.1)) and reformulate the definition of τ-continuity (see Definition 4.1) for the associated process. Then we refine the abstract result of [40, Theorem 3.1] such that it can be applied to construct the invariant Borel probability measures for process acting on time-dependent phase spaces. Afterwards, we formulate the definitions of family of test functions and statistical solution for problem (1.1)-(1.3) and prove its existence. Lastly, we point out that the obtained statistical solution satisfies the Liouville type theorem in time-dependent phase spaces. In comparison with the second order lattice systems with varying coefficients investigated in [64], the lattice KGS equations (1.1)-(1.2) are coupling of a lattice nonlinear wave equation with varying coefficients with a lattice nonlinear Schr¨odinger equation. This coupling and the varying coefficient, as well as the nonlinear terms β|zm|2and zmumin equations (1.1)-(1.2), produce some additional difficulties in our investigating. Firstly, it need us to pick an appropriate time-dependent transformation v(t) = ˙u(t) + δ(t) (t)u(t), which enable us to put the addressed lattice systems with varying coefficients into an abstract first-order differential equation with constant coefficients. Secondly, it requires us to choose, in accordance with the structure of the abstract equations (depends on in turn the time-dependent transformation picked above), a family of proper time-dependent phase spaces {Et=`2 (t)×`2×L2}t∈Rand endow them with subtle norms. With these choices, we can verify that the time-dependent operator H(t) (which is the linear and principle part of equations (1.1)-(1.2) has some coercive property. This coercive property plays the key role in both the global existence of solutions and existence of time-dependent pullback attractors. Thirdly, we also need do some meticulous analyses and estimates when verifying the uniform estimates on “Tail End” of solutions due to the nonlinear terms β|zm|2and zmum. The rest of the article is organized as follows. In Section 2, first, we present the mathematical frameworks and demonstrate the global well-posedness of problem (1.1)-(1.3). In Section 3, we prove the existence of a time-dependent pullback attractor for the process {U(t, τ)}t⩾τassociated to problem (1.1)-(1.3). In Section 4, we first construct the invariant Borel probability measures for the process {U(t, τ)}t⩾τon the time-dependent phase spaces. Then we formulate the definitions of a family of test functions and statistical solution for equations (1.1)-(1.2) in the time-dependent phase space and prove that the family of invariant Borel probability measures obtained in Section 4 is its statistical solution. Moreover, we point out that the statistical solution fulfills the Liouville theorem. 2 Global well-posedness In this section, we prove the global well-posedness of problem (1.1)-(1.3) in fixed phase space and time-dependent phase space. We first introduce the mathematical settings and some operators. Set `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 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. For brevity, we use Xto denote `2or L2, and equip Xwith the inner product and norm as (u, v) = X k∈Z uk¯vk,kuk2= (u, u),∀u= (uk)k∈Z, v = (vk)k∈Z∈X, where ¯vkdenotes the conjugate of vk. Obviously, (X, (·,·)) is a Hilbert space. We define on Xthree linear operators 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, the following properties are classical (cf. e.g. [67]):    (B∗Bu, v)=(Bu, Bv)=(Au, v),∀u, v ∈X, kBuk=kB∗uk⩽2kuk,∀u∈X, kAuk⩽4kuk,∀u∈X. Write        u= (um)m∈Z, µu = (µum)m∈Z, (t)u= ((t)um)m∈Z, z= (zm)m∈Z, αz = (αzm)m∈Z, β|z|2= (β|zm|2)m∈Z, zu = (zmum)m∈Z, f(t)=(fm(t))m∈Z, g(t) = (gm(t))m∈Z, uτ= (um,τ )m∈Z, u1τ= (u1m,τ )m∈Z, zτ= (zm,τ )m∈Z. We can now use the above notation and operators to rewrite problem (1.1)-(1.3) as (t)utt +νut+Au +µu −β|z|2=g(t),(2.1) izt−Az +iαz +zu =f(t),(2.2) u(τ) = uτ,˙u(τ) = u1τ, z(τ) = zτ.(2.3) Throughout this article, we will use the following constants that are related to parameters appearing in equations (2.1)-(2.2) σ0=µν pν2+µν(ν+pν2+µν), σ = min{σ0 2,α 4}. In addition, we will use Z+to denote the set of positive integers, and the symbol a.b(and a&b) to mean that a⩽cb (a⩾cb) for a universal constant c > 0 that only depends on the parameters coming from the addressed problem. To guarantee the global well-posedness of problem (2.1)-(2.3), as well as the existence of timedependent pullback attractor, we need some assumptions on the varying coefficients (t) and the external forces f(t) and g(t). (H1) Let (·)∈C1(R) be a decreasing bounded function satisfying lim t→+∞(t)=0, (t)−0(t)⩽ν 4for each t∈R,(2.4) and  0(t) (t)⩽r16µ2 ν2+16µσ ν−4µ ν.(2.5) (H2) Let f(·)=(fm(·))m∈Z∈C(R,L2), g(·)=(gm(·))m∈Z∈C(R, `2). Moreover, we assume that Zt −∞ eσskg(s)k2ds < +∞, t ∈R,(2.6) 5 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. and that there is a certain continuous function J(·) on the real line, bounded on every interval of the form (−∞, t), such that Zt −∞ eσskf(s)k2ds⩽e(σ 2+%)tJ(t)<+∞,for every t∈R,where 0 <%<σ 2.(2.7) We pick (t)=(8 ν+e(q16µ2 ν2+16µσ ν−4µ ν)t)−1, t ∈R. Then (·) satisfies assumption (H1). For the existence of functions f(·) and g(·) satisfying assumption (H2), one can refer to [56, Example 3.1]. Next, set δ(t) = µν(t) ν2+ 4µ(t)(2.8) and take the transformation v(t) = ˙u(t) + δ(t) (t)u(t).(2.9) Then we can write problem (2.1)-(2.3) equivalently as ˙ ψ(t) + H(t)ψ(t) = F(ψ, t),(2.10) ψ(τ) = (u(τ), v(τ), z(τ))T= (uτ, vτ, zτ)T,(2.11) where ψ(t) = (u(t), v(t), z(t))T, v(τ) = vτ=u1τ+δ(τ) (τ)uτ, F(ψ, t) = 0,β|z|2+g(t) (t), izu −if(t)T, H(t) =       δ(t) (t)I−I0 A (t)+µI (t)−δ(t)(ν−δ(t)) 2(t)I+4µ ν2δ0(t)Iν−δ(t) (t)0 0 0 iA +αI       ,(2.12) and Iis the identity on X. Note that H(t) is a time-dependent operator acting on `2×`2×L2. We now introduce some equivalent norms in X, whose main purpose is to motivate the definition of time-dependent phase spaces. First of all, we define a bilinear form on Xas (u, v)µ= (Bu, Bv) + µ(u, v), u, v ∈X. Obviously, µkuk2⩽kuk2 µ= (u, u)µ=kBuk2+µkuk2⩽(4 + µ)kuk2,∀u, v ∈X. (2.13) Thus (·,·)µis an inner product in Xwhich induces the norm k·kµequivalent to k·k. Also, we define (u, v)(t)=−1(t)(u, v)µ=−1(t)(Bu, Bv) + −1(t)µ(u, v),∀u, v ∈X, (2.14) where (t) is the varying coefficient coming from equation (1.1). In term of (2.13), we deduce µ−1(t)kuk2⩽kuk2 (t)= (u, u)(t)=−1(t)kuk2 µ⩽(4 + µ)−1(t)kuk2,∀u, v ∈X, (2.15) 6 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. which implies that, for every t∈R, (·,·)(t)induces a norm k·k(t)in `2which is equivalent to k·k. It is clear that `2 (t)= (`2,(·,·)(t)) is a Hilbert space. Write E=`2×`2×L2, Et=`2 (t)×`2×L2, and equip them with the inner products and norms as: for any two elements ψ(i)= (u(i), v(i), z(i))T∈ Eor Et,i= 1,2, (ψ(1), ψ(2))E= (u(1), u(2))+(v(1), v(2))+(z(1), z(2)) =X m∈Z (u(1) mu(2) m+v(1) mv(2) m+z(1) m¯z(2) m), kψk2 E= (ψ, ψ)E=kuk2+kvk2+kzk2, (ψ(1), ψ(2))Et= (u(1), u(2))(t)+ (v(1), v(2))+(z(1), z(2)) =X m∈Z−1(t)(Bu(1))m(Bu(2))m+−1(t)µu(1) mu(2) m+v(1) mv(2) m+z(1) m¯z(2) m, kψk2 Et= (ψ, ψ)Et=kuk2 (t)+kvk2+kzk2. It is not difficult to deduce from (2.13) and (2.15) that (min{µ−1(t),1}kψk2 E⩽kψk2 Et⩽max{(4 + µ)−1(t),1}kψk2 E,∀ψ∈E, ∀t∈R, kψk2 Eτ⩽kψk2 Et⩽(τ) (t)kψk2 Eτ,∀ψ∈E, ∀t⩾τ∈R.(2.16) Remark 2.1. Note that the spaces Et, for t∈R, are all the same as linear spaces, and the norms k·k2 Eτand k·k2 Etare equivalent for any given τ,t∈R. Notice that we have expressed problem (1.1)-(1.3) as problem (2.10)-(2.11), which is an initial value problem for an abstract first-order ordinary differential equation (ODE). As a result, the classical theory of ODE is applied. Indeed, we have the following result. Lemma 2.1. Let assumptions (H1)-(H2)hold. For every initial data ψτ= (uτ, vτ, zτ)T∈E, there exists an unique local solution ψ(t) = (u(t), v(t), z(t))T∈Eof system (2.10)-(2.11) such that ψ(·)∈ C([τ, T0), E)∩C1((τ, T0), E)for some T0> τ. If T0<+∞, then lim t→T− 0kψ(t)kE= +∞. Proof. For every t∈R, It is clear that operator H(t) : Et→Etis linear. By direct computations, we deduce that kH(t)ψk2 Et=kδ(t) (t)u−vk2 (t)+kAu (t)+µu (t)−δ(t)(ν−δ(t)) 2(t)u+4µ ν2δ0(t)u+ν−δ(t) (t)vk2+kiAz +αzk2 .δ2(t) 2(t)kuk2 (t)+kvk2 (t)+1 2(t)kAuk2+1 2(t)kuk2+δ2(t)(ν−δ(t))2 4(t)kuk2 +|δ0(t)|2kuk2+ν−δ(t)2 2(t)kvk2+kAzk2+kzk2 .δ2(t) 2(t)kuk2 (t)+ ( 1 (t)+δ2(t)(ν−δ(t))2 3(t)+(t)) µ (t)kuk2+1 + (ν−δ(t))2 (t) (t)kvk2+kzk2 .L1(t)kψk2 Et,∀ψ= (u, v, z)T∈Et,(2.17) where L1(t) = max nδ2(t) 2(t),1 (t)+δ2(t)(ν−δ(t))2 3(t)+(t),1 + (ν−δ(t))2 (t) (t)o.(2.18) 7 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. Hence, the linear operator H(t) : Et→Etis bounded. Next we verify the locally Lipschitz property of F(·, t) on Et. In fact, let Bbe a bounded set in Et,ψ(i)= (u(i), v(i), z(i))T∈ B,i= 1,2. Then kF(ψ(1), t)−F(ψ(2), t)k2 Et=kβ (t)(|z(1)|2−|z(2)|2)k2+kiz(1)u(1) −iz(2)u(2)k2 ⩽β2 2(t)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(t)(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 Et 2(t)kz(1) −z(2)k2+2(t) µkz(1)k2ku(1) −u(2)k2 (t)+2(t) µku(2)k2 (t)kz(1) −z(2)k2 ⩽(4β2 2(t)+4(t) µ) sup ψ∈B kψk2 Etkψ(1) −ψ(2)k2 Et.(2.19) Since k·kEand k·kEtare equivalent for every t∈R, we conclude from above analyses that H(t)+F(·, t) is locally Lipschitz from Eto E. By the standard theory of ODE, we deduce the results of Lemma 2.1. We next will establish that the local solution ψ(·) guaranteed by Lemma 2.1 does exist globally, by showing lim t→T− 0kψ(t)kE<+∞for any T0> τ. Firstly, for the boundedness of the component z(·), we have the following result. Lemma 2.2. Let assumptions (H1)-(H2)hold and ψτ= (uτ, vτ, zτ)T∈Ebe the initial value at initial time τ. Suppose ψ(t)=(u(t), v(t), z(t))T∈Ebe the corresponding solution of problem (2.10)-(2.11), then kz(t)k2⩽kzτk2e−α(t−τ)+e−αt αZt τ eαskf(s)k2ds, ∀t⩾τ. (2.20) Proof. Multiplying (2.2) by z(t) in L2and taking the imaginary component of the inner product, we yields 1 2 d dtkz(t)k2+αkz(t)k2=Im(f(t), z(t)) ⩽1 2αkf(t)k2+α 2kz(t)k2,∀t⩾τ. (2.21) Applying Gronwall’s inequality to (2.21) gives (2.20). To verify the boundedness of the solution ψ(·) in space E, we first prove the following coercivity of the time-dependent operator H(t) on space Et. Lemma 2.3. For every t∈Rand ψ(t) = (u(t), v(t), z(t))T∈Et, there holds Re(H(t)ψ, ψ)Et⩾ϑ(t) (t)(kuk2 (t)+kvk2) + ν 2(t)kvk2+αkzk2,(2.22) where ϑ(t) = µν(t) pν2+ 4µ(t)ν+pν2+ 4µ(t)∈(0, δ(t)).(2.23) Proof. By calculation, we have Re(H(t)ψ, ψ)Et=(δ(t) (t)u−v, u)(t)+1 (t)(Bu, Bv) + µ (t)(u, v) + 4µ ν2δ0(t)(u, v) −δ(t)(ν−δ(t)) 2(t)(u, v) + ν−δ(t) (t)kvk2+αkzk2,(2.24) (δ(t) (t)u−v, u)(t)=δ(t) (t)kuk2 (t)−1 (t)(Bv, Bu)−µ (t)(v, u),(2.25) 8 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. and 4µ ν2δ0(t)(u, v)−δ(t)(ν−δ(t)) 2(t)(u, v)⩾−νδ(t) 2(t)kukkvk+ (δ2(t) 2(t)+4µ ν2δ0(t))kukkvk ⩾−νδ(t) 2(t)kukkvk,(2.26) where we have used the fact that δ2(t) 2(t)+4µ ν2δ0(t)⩾0. Thus, we have Re(H(t)ψ, ψ)Et⩾δ(t) (t)kuk2 (t)−νδ(t) 2(t)kukkvk+ν−δ(t) (t)kvk2+αkzk2.(2.27) In view of the facts that kuk⩽((t) µ)1/2kuk(t)and 4(δ(t)−ϑ(t))ν 2−δ(t)−ϑ(t)=ν2δ2(t) µ(t)(2.28) holds for every t∈R, we deduce Re(H(t)ψ, ψ)Et−ϑ(t) (t)(kuk2 (t)+kvk2)−ν 2(t)kvk2−αkzk2 ⩾1 (t)h(δ(t)−ϑ(t))kuk2 (t)+ (ν 2−δ(t)−ϑ(t))kvk2−νδ(t) (t)kukkvki ⩾1 (t)h(δ(t)−ϑ(t))kuk2 (t)+ (ν 2−δ(t)−ϑ(t))kvk2−νδ(t) (t)((t) µ)1/2kuk(t)kvki⩾0.(2.29) This completes the proof. Lemma 2.4. Let assumptions (H1)-(H2)hold and ψτ= (uτ, vτ, zτ)T∈Ebe the initial value at initial time τ. Suppose ψ(·)=(u(·), v(·), z(·))Tbe the corresponding solution of problem (2.10)-(2.11), then kψ(t)k2 E⩽max{(4 + µ)−1(τ),1} min{µ−1(t),1}kψτk2 Ee−σ(t−τ) +L2(t)e−σt min{µ−1(t),1}(t)Zt τ eσs(kf(s)k2+kg(s)k2+kz(s)k4ds, ∀t⩾τ, (2.30) hereinafter L2(t) = max{2β2 ν(t),2 ν(t),2 α}. Proof. Let ψ(·)=(u(·), v(·), z(·))Tbe the solution to problem (2.10)-(2.11). We multiply (2.10) by ψ(t) in Et, and take the real component of the inner product, getting 1 2 d dtkψ(t)k2 Et+1 2 0(t) (t)kuk2 (t)+Re(H(t)ψ(t), ψ(t))Et=Re(F(ψ(t), t), ψ(t))Et,∀t⩾τ. (2.31) Using Cauchy’s inequality, we derive                      Re(F(ψ(t), t), ψ(t))Et⩽(β|z(t)|2 (t), v(t)) + (g(t) (t), v(t)) + Im(f(t), z(t), (β|z(t)|2 (t), v(t)) ⩽β2 ν(t)kz(t)k4+ν 4(t)kv(t)k2, (g(t) (t), v(t)) ⩽1 ν(t)kg(t)k2+ν 4(t)kv(t)k2, Im(f(t), z(t)) ⩽1 αkf(t)k2+α 2kz(t)k2. (2.32) 9 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure 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|&−rσ(t) M,∀τ⩽t0.(3.23) Taking (3.19)-(3.23) into account, we obtain ReH(t)ψ(t), φ(t)Et−ϑ(t) (t)X m∈Z χ(|m| M)1 (t)|(Bu)m|2+1 (t)µu2 m+v2 m −ν 2(t)X m∈Z χ(|m| M)v2 m−αX m∈Z χ(|m| M)|zm|2 &1 (t)X m∈Z χ(|m| M)h(δ(t)−ϑ(t))|um|2 (t)+ν 2−δ(t)−ϑ(t)v2 m−δ(t)(ν−δ(t)) (t)|um||vm|i −ν 4(t)X m∈Z χ(|m| M)v2 m−ν(0(t))2 16µ2(t)X m∈Z χ(|m| M)|ψm|2 Et −1 M(t)(1 + (t) + δ(t))rσ(t),∀τ⩽t0,(3.24) where |um|2 (t)=1 (t)|(Bu)m|2+µu2 m.(3.25) Since 4(δ(t)−θ(t))(ν 2−δ(t)−θ(t)) = ν2δ2(t) µ(t)(see (2.28)), we see for any m∈Zthat (δ(t)−ϑ(t))|um|2 (t)+ν 2−δ(t)−ϑ(t)v2 m−δ(t)(ν−δ(t)) (t)|um||vm|⩾0.(3.26) As a consequence, inequality (3.24) improves to ReH(t)ψ(t), φ(t)Et−ϑ(t) (t)X m∈Z χ(|m| M)1 (t)|(Bu)m|2+µ (t)u2 m+v2 m −ν 2(t)X m∈Z χ(|m| M)v2 m−αX m∈Z χ(|m| M)|zm|2 &−ν 4(t)X m∈Z χ(|m| M)v2 m−ν(0(t))2 16µ2(t)X m∈Z χ(|m| M)|ψm|2 Et−(1 + (t) + δ(t))rσ(t) M(t),∀τ⩽t0.(3.27) For the term ReF(ψ, t), φEtwe have, by exploiting Cauchy’s inequality, that ReF(ψ, t), φEt=( β (t)|z|2, w)+(g, w) + Im(f, l),(3.28) (β (t)|z|2, w)⩽ν 8(t)X m∈Z χ(|m| M)v2 m+2β2 ν(t)X m∈Z χ(|m| M)|zm|4 ⩽ν 8(t)X m∈Z χ(|m| M)v2 m+2β2rσ(t) ν(t)X m∈Z χ(|m| M)|zm|2,(3.29) (g (t), w)⩽ν 8(t)X m∈Z χ(|m| M)v2 m+2 ν(t)X m∈Z χ(|m| M)g2 m,(3.30) Im(f, l)⩽α 2X m∈Z χ(|m| M)|zm|2+1 αX m∈Z χ(|m| M)|fm|2.(3.31) 16 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. Now, using (2.34), (3.15), (3.18), (3.27) and (3.28)-(3.31), we arrive at 1 2 d dtQ(t) + 2σ+0(t) 2(t)−ν 16µ|0(t) (t)|2Q(t) .rσ(t) (t)X m∈Z χ(|m| M)|zm|2+1 (t)X m∈Z χ(|m| M)g2 m+X m∈Z χ(|m| M)|fm|2 +(1 + (t))rσ(t) M(t),∀τ⩽t0,(3.32) where Q(t) = X m∈Z χ(|m| M)|ψm|2 Et. We infer from (2.5) that 2σ+0(t) 2(t)−ν 16µ|0(t) (t)|2⩾σ. (3.33) Hence, the differential inequality (3.32) now reads d dtQ(t) + σQ(t).1 + (t) M(t)rσ(t) + rσ(t) (t)X m∈Z χ(|m| M)|zm(t)|2 | {z } I +1 (t)X m∈Z χ(|m| M)g2 m(t) + X m∈Z χ(|m| M)|fm(t)|2,∀τ⩽t0.(3.34) In order to estimate term I, we take the imaginary part of the inner product (·,·) of equation (2.2) with χ(|m| M)¯zmm∈Zto deduce X m∈Z χ(|m| M)|zm(t)|2.e−αt Zt τ eαsX m∈Z χ(|m| M)|fm(s)|2+rσ(s) Mds +e−α(t−τ)X m∈Z χ(|m| M)|zm(τ)|2,∀τ⩽t0.(3.35) In view of (H2), we find e−αt Zt τ eαs X m∈Z|fm(s)|2ds⩽e(%−σ 2)tJ(t)<+∞,for each t∈R. Thus, for any ε > 0, there exists some M1=M1(t, ε)∈Nsuch that rσ(t) (t)e−αt Zt τ eαs X |m|⩾M1 |fm(s)|2ds < σε2 36 ,∀M⩾M1.(3.36) At the same time, from (3.3) we obtain that e−αt MZt τ eαsrσ(s)ds ⩽e−αt MZt τ eαsds+L2(t)Zt τ e(α−σ)sZt −∞ eσηkf(η)k2+kg(η)k2dηds +L2(t)Zt τ e(α−σ)sZs τ e(σ−2α)ηe(α−σ 2+%)ηJ(η)2dηds .1 M1 + L2(t)Zt −∞ eσηkf(η)k2+kg(η)k2dη+e(2%−σ)t ML2(t)˜ J1(t)<+∞. 17 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. Hence, for any given t(⩾τ) and above ε, there exists some M2=M2(t, ε)∈Z+such that rσ(t) (t)Me−αt Zt τ eαsrσ(s)ds < σε2 36 ,∀M⩾M2.(3.37) It is clear that, for above tand ε, there exists some t1=t1(t, ε, r)⩽tand some M3=M3(t, ε)∈Z+ such that rσ(t) (t)e−α(t−τ)X m∈Z χ(|m| M)|zm(τ)|2⩽rσ(t) (t)e−α(t−τ)r2<σε2 36 ,∀τ⩽t1,(3.38) 1 + (t)rσ(t) M(t)<σε2 12 ,∀M⩾M3.(3.39) Using (3.34)-(3.39) and applying Gronwall’s inequality, we see that for any M⩾max{M1, M2, M3} and τ⩽min{t0, t1}, Q(t).Q(τ)e−σ(t−τ)+e−σt Zt τ eσs1 (s)X m∈Z χ(|m| M)g2 m(s) + X m∈Z χ(|m| M)|fm(s)|2ds+ε2 6.(3.40) Again exploiting assumption (H2), there exists some M4=M4(t, ε)∈Z+such that e−σt Zt τ eσsh1 (s)X m∈Z χ(|m| M)g2 m(s) + X m∈Z χ(|m| M)|fm(s)|2ids ⩽e−σt (t)Zt −∞ eσs X m∈Z χ(|m| M)g2 m(s)ds+e−σt Zt −∞ eσs X m∈Z χ(|m| M)|fm(s)|2ds < ε2 6,∀M⩾M4.(3.41) Since ψτ∈Bτ(r), we conclude that for above given t∈R,ε > 0 and r > 0, there exists some t2=t2(t, r, ε)⩽tsuch that Q(τ)e−σ(t−τ)⩽kψτk2 Eτe−σ(t−τ)⩽r2e−σ(t−τ)<ε2 6,∀τ⩽t2.(3.42) Inserting (3.41) and (3.42) into (3.40) yields sup ψτ∈Bτ(r)X |m|⩾2M∗ |(U(t, τ)ψτ)m|2 Et= sup ψτ∈Bτ(r)X |m|⩾2M∗ |ψm(t)|2 Et⩽2Q(t).ε2,∀τ⩽t∗,(3.43) where M∗= max{M1, M2, M3, M4}, t∗= min{t0, t1, t2}.This completes the proof of Lemma 3.3. In terms of Lemma 3.1, Lemma 3.2 and Lemma 3.3, we obtain the main result of this section as follows. Theorem 3.1. Let assumptions (H1)-(H2) hold. Then the process {U(t, τ)}t⩾τhas a time-dependent pullback attractor A={At}t∈Rfulfilling the three properties of [64, Definition 3.1(4)]. 4 Existence of invariant Borel probability measures and statistical solutions on time-dependent phase spaces In this section we construct a family of invariant Borel probability measures {mt}t∈Rfor the process {U(t, τ)}t⩾τon the time-dependent phase spaces {Et}t∈Rvia generalized Banach limits. Then we propose the concept of statistical solutions for equation (2.12), and prove that the constructed probability measures is its statistical solution and satisfies Liouville theorem in Statistical Mechanics. The idea of the proof of the main results of this section (Theorem 4.1) is based on the framework of Lukaszewicz and Robinson for non-autonomous dissipative dynamical systems [40, Theorem 3.1]. 18 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. In order to emphasize the difference and crux when we construct the invariant Borel probability measures for the dissipative dynamical systems on time-dependent phase spaces, we first introduce the “suitable” initial data collection Γr=n{ϕ(s)}s∈R:ϕ(s) = (p(s)u, v, z)T∈Es, ϕ = (u, v, z)T∈BE(r)o,(4.1) where r > 0 and BE(r) = {ϕ∈E:kϕkE⩽r}. Note that all elements within the collection Γrwould be uniformly bounded, according to [64, Definition 3.1(1)]. As will be seen, the constructed Γrplays the vital role for the following τ-continuous property of {U(t, τ)}t⩾τin {Et}t∈R. Definition 4.1. The process {U(t, τ)}t⩾τis said to be τ-continuous on the time-dependent phase {Et}t∈R, if for every {ϕ∗(τ)}τ∈R∈S r⩾0 Γrand any given t∈R, the Et-valued mapping τ7→ U(t, τ)ϕ∗(τ) (4.2) is continuous and bounded on (−∞, t]. We next establish two auxiliary lemmas concerning a certain continuity of U(t, τ)ψ∗(τ) with respect to the parameters τand t, as suits for our purposes below. Lemma 4.1. Let assumptions (H1)-(H2) hold. Let r > 0,τ∗∈Rand {ϕ∗(s)}s∈R∈Γrbe given. Then for any ε > 0, there exists some ρ=ρ(τ∗, r, ε)>0such that kU(s, τ∗)ϕ∗(τ∗)−ϕ∗(s)k2 Es.ε, ∀s∈[τ∗, τ∗+ρ).(4.3) Proof. Let r > 0, {ϕ∗(s)}s∈R∈Γrwith ϕ∗= (u∗, v∗, z∗)T∈BE(r), and τ∗∈Rbe given. For any s∈[τ∗, τ∗+1], let U(s, τ∗)ϕ∗(τ∗) = (u(s), v(s), z(s))T= (um(s), vm(s), zm(s))T m∈Z∈Esbe the solution of problem (2.10)-(2.11) with initial data ϕ∗(τ∗)=(uτ∗, vτ∗, zτ∗)T= (p(τ∗)u∗, v∗, z∗)T∈Eτ∗at initial time τ∗. Note that kU(s, τ∗)ϕ∗(τ∗)−ϕ∗(s)k2 Es=I1−I2−I3,(4.4) where    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 estimate the three terms in (4.5) separately. For the first one, we have by using (2.33) and the monotonicity of (·) that I1=kU(s, τ∗)ϕ∗(τ∗)k2 Es−kϕ∗(τ∗)k2 Eτ∗=Zs τ∗ dkU(θ, τ∗)ϕ∗(τ∗)k2 Eθ dθdθ .Zs τ∗kf(θ)k2dθ+1 (τ∗+ 1) Zs τ∗kg(θ)k2dθ+1 (τ∗+ 1) Zs τ∗kz(θ)k4dθ. (4.6) In view of (2.20) and (3.9), we obtain Zs τ∗kz(θ)k4dθ.Zs τ∗kzτ∗k4dθ+Zs τ∗e−αθ Zθ −∞ eαηkf(η)k2dη2dθ +kzτ∗k2Zs τ∗ e−αθ Zθ −∞ eαηkf(η)k2dηdθ .r4(s−τ∗) + Zs τ∗ e(2%−σ)θJ2(θ)dθ+r2Zs τ∗ e(%−σ 2)θJ(θ)dθ. (4.7) 19 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. By the fact that g(·)∈C(R, `2), J(·) and J2(·)∈C(R,R), we infer that for any ε > 0 there exists some ρ1=ρ1(τ∗, ε, r)>0 such that I1.Zs τ∗kf(θ)k2dθ+Zs τ∗kg(θ)k2dθ (τ∗+ 1) +Zs τ∗kz(θ)k4dθ (τ∗+ 1) <ε 3,∀s∈(τ∗, τ∗+ρ1).(4.8) For the second term I2, we have I2=kϕ∗(s)k2 Es−kϕ∗(τ∗)k2 Eτ∗ =(kp(s)u∗k2 (s)+kv∗k2+kz∗k2)−(kp(τ∗)u∗k2 (τ∗)+kv∗k2+kz∗k2) =(ku∗k2 µ+kv∗k2+kz∗k2)−(ku∗k2 µ+kv∗k2+kz∗k2)=0.(4.9) For the third term I3, we have 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) Note that the definition of Γrimplies kϕ∗(s)k2 Es=kp(s)u∗k2 (s)+kv∗k2+kz∗k2=ku∗k2 µ+kv∗k2+kz∗k2 ⩽µku∗k2+ 4ku∗k2+kv∗k2+kz∗k2⩽(6 + µ)r2,∀s∈R,(4.11) which means ϕ∗(s)∈Bs(p6 + µ)r) with every s∈R. Thus, similar to (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 .r(τ∗) (τ∗+ 1)Zs τ∗kdU(θ, τ∗)ϕ∗(τ∗) dθk2 Eθdθ1/2(s−τ∗)1/2.(4.12) Next, we prove that there exist constants L5=L5(τ∗, r)>0 and L6=L6(τ∗) such that Zs τ∗kdU(θ, τ∗)ϕ∗(τ∗) dθk2 Eθdθ.L∗(τ∗, r) := L5L6+L2 5+Zτ∗+1 τ∗kg(θ)k2dθ+Zτ∗+1 τ∗kf(θ)k2dθ. (4.13) Indeed, in view of (2.10) and (2.17), we infer 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)−f(θ)k2.(4.14) By (3.4) and using similar derivations to (4.7), we arrive at kU(θ, τ∗)ϕ∗(τ∗)k2 Eθ ⩽kϕ∗(τ∗)k2 Eτ∗e−σ(θ−τ∗)+L2(θ)e−σθ Zθ τ∗ eση(kf(η)k2+kg(η)k2)dη+L2(θ)e−σθ Zθ τ∗ eσηkz(η)k4dη .r2+L2(τ∗+ 1) Zτ∗+1 τ∗kf(η)k2+kg(η)k2dη+L2(τ∗+ 1) Zτ∗+1 τ∗kz(η)k4dη .r2+L2(τ∗+ 1) Zτ∗+1 τ∗kf(η)k2+kg(η)k2dη +L2(τ∗+ 1)Zτ∗+1 τ∗ r4dθ+Zτ∗+1 τ∗ e(2%−σ)θJ2(θ)dθ+Zτ∗+1 τ∗ e(%−σ 2)θJ(θ)dθ = : L5=L5(τ∗, r),∀θ∈[τ∗, τ∗+ 1],(4.15) 20 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. which means kz(θ)k2.L5and ku(θ)k2.L5.(4.16) For any t∈R, there holds δ(t)∈(0, ν/4). By (2.18) we infer that L1(θ)⩽L6=L6(τ∗) = max s∈[τ∗,τ∗+1] L1(s). Inserting (4.15) and (4.16) into (4.14) implies kdU(θ, τ∗)ϕ∗(τ∗) dθk2 Eθ.L5L6+L2 5+kg(θ)k2 2(τ∗+ 1) +L2 5+kf(θ)k2,∀θ∈[τ∗, τ∗+ 1],(4.17) and (4.13) is proved. In light of (4.13) and (4.12), we arrive at |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 .rL∗(τ∗, r)(τ∗) (τ∗+ 1) (s−τ∗)1/2, which implies that, for above ε, there exists some ρ2=ρ2(τ∗, ε, r)>0 such that |2I31|= 2|(U(s, τ∗)ϕ∗(τ∗)−ϕ∗(τ∗), ϕ∗(s))Es|.ε 6,∀s∈[τ∗, τ∗+ρ2).(4.18) For I32 in (4.10), we easily obtain |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.r2|p(τ∗)−p(s)| p(τ∗+ 1) . By the continuity of (·), we have that, for above εthere exists some ρ3=ρ3(τ∗, ε, r)>0 such that 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 obtain |I3|⩽2|I31|+ 2|I32|.ε 3,∀s∈[τ∗, τ∗+ρ4).(4.20) Choosing ρ=ρ(τ∗, r, ε) = min{ρ1, ρ4}, we get (4.3) from (4.4), (4.8), (4.9) and (4.20). Similarly, we have the following lemma. Lemma 4.2. Let assumptions (H1)-(H2) hold, and let r > 0,τ∗∈R, and {ϕ∗(s)}s∈R∈Γrbe given. Then, for any ε > 0, there exists some ρ=ρ(τ∗, r, ε)>0such that kU(τ∗, s)ϕ∗(s)−ϕ∗(τ∗)k2 Eτ∗.ε, ∀s∈(τ∗−ρ, τ∗].(4.21) With above two auxiliary lemmas in hand, we now state and prove the τ-continuous property of the process {U(t, τ)}t⩾τon time-dependent phase spaces {Et}t∈R. Lemma 4.3. Let assumptions (H1)-(H2) hold. Then, for every given t∈Rand {ϕ∗(s)}s∈R∈Γr with some r > 0, the Et-valued mapping τ7→ U(t, τ)ϕ∗(τ)is continuous and bounded on (−∞, t]. 21 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. Proof. Given t∈Rand r > 0, let {ϕ∗(s)}s∈R∈Γrwith ϕ∗= (u∗, v∗, z∗)T∈BE(r). From (3.4) and Lemma 3.2 we deduce for any τ∗∈(−∞, t] that kU(t, τ∗)ϕ∗(τ∗)kEt⩽r2+L2(t)e−σt Zt −∞ eσskf(s)k2+kg(s)k2ds +L2(t)e−σt Zt −∞ e(σ−2α)sZs −∞ eαηkf(η)k2dη2ds < +∞,(4.22) which means that the Et-valued mapping τ7→ U(t, τ)ϕ∗(τ) is bounded on (−∞, t]. Now, for any τ∗∈(−∞, t], it clear that for any ε > 0 there exists some ρ=ρ(ε, t, r)>0 such that |s−τ∗|< ρ =⇒ kU(t, s)ϕ∗(s)−U(t, τ∗)ϕ∗(τ∗)kEt< ε. (4.23) Next we split the proof into two cases. Case I :τ∗⩽s⩽τ∗+ 1. Using the invariance of process and (2.39), we arrive at kU(t, s)ϕ∗(s)−U(t, τ∗)ϕ∗(τ∗)k2 Et=kU(t, s)ϕ∗(s)−U(t, s)U(s, τ∗)ϕ∗(τ∗)k2 Et ⩽exp nL3(r, t, s)(t−s)okϕ∗(s)−U(s, τ∗)ϕ∗(τ∗)k2 Es ⩽exp n˜ L3(r, t, τ∗)(t−τ∗)okϕ∗(s)−U(s, τ∗)ϕ∗(τ∗)k2 Es,(4.24) here we have used the fact that one can choose L3(r, t, τ) to be continuous with respect to τin (2.47), and ˜ L3(r, t, τ∗) = max s∈[τ∗,τ∗+1] L3(r, t, s). In view of Lemma 4.1, we infer that, for any ε > 0, there exists some ρ0=ρ0(τ∗, r, ε)>0 such that kU(t, s)ϕ∗(s)−U(t, τ∗)ϕ∗(τ∗)k2< ε, ∀s∈τ∗, τ∗+ρ0,(4.25) which means U(t, τ)ϕ∗(τ) is right-continuous on τ=τ∗. Case II :τ∗−1⩽s⩽τ∗. Again by the invariance of the process, we have that kU(t, s)ϕ∗(s)−U(t, τ∗)ϕ∗(τ∗)k2 Et =kU(t, τ∗)U(τ∗, s)ϕ∗(s)−U(t, τ∗)ϕ∗(τ∗)k2 Et. ⩽exp nL3(r, t, τ∗)(t−τ∗)okU(τ∗, s)ϕ∗(s)−ϕ∗(τ∗)k2 Eτ∗.(4.26) Applying Lemma 4.2, we conclude that for any ε > 0 there exists some ρ00 =ρ00(τ∗, r, ε)>0 such that kU(t, s)ϕ∗(s)−U(t, τ∗)ϕ∗(τ∗)k2 Et< ε, ∀s∈τ∗−ρ00, τ∗,(4.27) which means U(t, τ)ϕ∗(τ) is left-continuous on τ=τ∗. By the arbitrariness of τ∗, we end the proof. We next update the definition of generalized Banach limit (cf. [22,40]) to construct the invariant Borel probability measures {mt}t∈Rfor the process {U(t, τ)}t⩾τon time-dependent phase spaces {Et}t∈R. Definition 4.2. A generalized Banach limit is any linear functional, which is denoted by LIMt→−∞, defined on the space of all bounded real-valued functions on (−∞,+∞)and satisfying (1) LIMt→−∞h(t)⩾0for nonnegative functions g(·)on (−∞,+∞); (2) LIMt→−∞h(t) = lim t→−∞ h(t)if the usual limit lim t→−∞ h(t)exists. 22 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. Now, we denote by C(Et) the collection of all continuous functionals with real values on Et. Besides, for any given Borel probability measure mton Etand any function Φt∈C(Et), ZEt Φt(ψ)dmt(ψ) denotes the Bochner integral. The main result of this subsection section reads as follows. Theorem 4.1. Let assumptions (H1)-(H2) hold, and let r > 0and {ϕ∗(s)}s∈R∈Γrwith ϕ∗= (u∗, v∗, z∗)T∈BE(r)be given. Then for any given generalized Banach limit LIMτ→−∞, every t∈R and Ψt∈C(Et), there exists a unique Borel probability measure mton Etsuch that LIMτ→−∞ 1 t−τZt τ ΨtU(t, θ)ϕ∗(θ)dθ =ZAt Ψt(ψ)dmt(ψ) = ZEt Ψt(ψ)dmt(ψ) (4.28) =LIMτ→−∞ 1 t−τZt τZEθ ΨtU(t, θ)ψdmθ(ψ)dθ. (4.29) Moreover, the support of the measure mtis contained in At, and the measures {mt}t∈Rsatisfy the following invariant property ZAt Ψt(ψ)dmt(ψ) = ZAτ ΨtU(t, τ)ψdmτ(ψ),∀t⩾τ. (4.30) The proof of Theorem 4.1 is similar with that of [64, Theorem 4.1], with very slightly difference. Here we omit the details. We next propose the concept of statistical solutions for equation (2.10), and prove that the family of invariant Borel probability measures {mt}t∈Rguaranteed by Theorem 4.1 is its statistical solution and satisfies Liouville theorem in Statistical Mechanics. Rewrite equation (2.10) as dψ dt=G(ψ, t) := F(ψ, t)−H(t)ψ, t ∈R.(4.31) To formulate the definition of statistical solution for equation (4.31) on the time-dependent phase spaces {Et}t∈R, we first introduce the family of class {Tt}t∈Rof test functions. Definition 4.3. For every given t, by Ttwe denote the class of real-valued continuous functions Φt on Etthat are bounded on bounded subsets of Etand satisfy the following two conditions. (a) for every ϕ∈Et, the Frech´et derivative Φ0 t(ϕ) exists: for each ϕ∈Etthere exists an element Φ0 t(ϕ)∈Etsuch that |Φt(ϕ+φ)−Φt(ϕ)−Φ0 t(ϕ), φEt| kφkEt−→ 0 as kφkEt→0, φ ∈Et. (b) the mapping ϕ7→ Φ0 t(ϕ) is continuous and bounded from Etto Et. The conditions in Definition 4.3 are sufficient to ensure that if ψ(t) solves equation (4.31) then d dtΦt(ψ(t)) = (G(ψ(t), t),Φ0 t(ψ(t)))Et, t ∈R.(4.32) For every t∈R, the definition of class Ttconsisting of test functions is similar to that of [62, Definition 4.2]. Here we formulate a family of class {Tt}t∈Rof test functions to treat the scenario that the process {U(t, τ)}t⩾τis defined on the time-dependent phase spaces. We want to point out that the family of 23 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. class of functions satisfying Definition 4.3 exists in great profusion. For example, we could take Φtto be a cylindrical test functions (cf. [64, Definition 5.1]) Φt:Et→Rof the form Φt(φ) = $((φ, ψ1)Et,(φ, ψ2)Et,(φ, ψ3)Et), where $is a C1real-valued function on R3with compact support, and ψ1,ψ2,ψ3belong to Et. For such Φt(·), direct computations show that its Frech´et derivative Φ0 tin Ethas the form Φ0 t(φ) = 3 X j=1 ∂j$((φ, ψ1)Et,(φ, ψ2)Et),(φ, ψ3)Et)ψj, where ∂j$is the partial derivative of $with respect to its j-th coordinate. We now specify the definition of statistical solution for equation (4.31) on the time-dependent phase spaces {Et}t∈R. Note that by (2.16) and (4.1), it is not hard to check that Et=[ r⩾0 Bt(r) = [ r⩾0φ(t) : {φ(θ)}θ∈R∈Γr, t ∈R. Definition 4.4. A family {ρt}t∈Rof Borel probability measures with ρton Etis called a statistical solution of equation (4.31) if {ρt}t∈Rsatisfies the following two conditions: (a) for almost t∈R, the function ψ7→ G(ψ, t), φEtis ρt-integrable for every φ∈Et. Moreover, the mapping t7→ ZEtG(ψ, t), φEtdρt(φ) belongs to L1 loc(R)for every φ∈Et. (b) for any {ψ(s)}s∈R∈Sr⩾0Γrand all t⩾τ, the Liouville-type equation ZEt Φt(ψ(t))dρt(ψ(t)) −ZEτ Φτ(ψ(τ))dρτ(ψ(τ)) = Zt τZEθG(ψ(θ), θ),Φ0 θ(ψ(θ))Eθdρθ(ψ(θ))dθ (4.33) holds for any {Φt}t∈R∈ {Tt}t∈R. Theorem 4.2. Let assumptions (H1)-(H2) hold. Then, the family of invariant Borel probability measures {mt}t∈Robtained in Theorem 4.1 is a statistical solution of equation (4.31). Proof. Consider given t∈R. For every φ= (φ1, φ2, φ3)T∈Et, we define Ψt(·) : Et7−→ Ras Ψt(ψ) = G(ψ, t), φEt,∀ψ= (u, v, z)T∈Et.(4.34) We next prove that Ψt(·)∈C(Et). Let ψ∗= (u∗, v∗, z∗)T∈Etbe fixed and consider ψ= (u, v, z)T∈ Etwith kψ∗−ψkEt<1. Then we have |Ψt(ψ∗)−Ψt(ψ)|=G(ψ∗, t)−G(ψ, t), φEt ⩽H(t)(ψ∗−ψ), φEt|+|F(ψ∗, t)−F(ψ, t), φEt.(4.35) By (2.17), (2.18) and the fact that δ(t)∈(0, ν/4) for every t∈R, we arrive at |H(t)(ψ∗−ψ), φEt|.k(H(t)(ψ∗−ψ)kEtkφkEt.L1(t)kψ∗−ψkEtkφkEt.(4.36) Also, we have by (2.19) that |F(ψ∗, t)−F(ψ, t), φEt|⩽kF(ψ∗, t)−F(ψ, t)kEtkφkEt .L8(t)(1 + kψ∗kEt)kψ∗−ψkEtkφkEt,(4.37) 24 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal. where L8(t)=(4β2 2(t)+2+4(t) µ)1/2is a positive constant depending only on t. The continuity of the mapping Ψt(·) : Et7−→ Rfollows from (4.34)-(4.37). Now, Theorem 4.1 tells us that the function ψ7→ G(ψ, t), φEt:= Ψt(ψ) is mt-integrable for every φ∈Et, and that the mapping t7→ ZEtG(ψ, t), φEtdmt(ψ) = ZEt Ψt(ψ)dmt(ψ) belongs to L1 loc(R) for every φ∈Et. Thus property (a) of Definition 4.3 is proved. The proof of property (b) in Definition 4.3 is very similar with that of [ [64], Theorem 5.1] and we omit the details here. We want to remark that in Statistical Mechanics Φ0 t(·) = 0 for all t∈Rimplies that equation (4.31) (and thus system (1.1)-(1.2)) reaches its statistical equilibrium (cf. [22]). In this case, the Liouville type equation (4.33) turns to be ZAt Φt(ϕ∗(t))dmt(ϕ∗(t)) = ZAτ Φτ(ϕ∗(τ))dmτ(ϕ∗(τ)),{ϕ∗(s)}s∈R∈[ r⩾0 Γr,∀t, τ ∈R,(4.38) which reveals that although the shape of the time-dependent pullback attractor A•could change along with the evolution of time from τto t, the “total measures” of Aτand Atalways coincide with each other, that is, given that the system has reached statistical equilibrium, the “total measures” of the time-dependent pullback attractor A•are conservative as time passes. This is exactly the theory of Liouville Theorem in Statistical Mechanics (see e.g. [69, Page19, (1.3.29)]). Therefore, we say that the statistical solutions of the lattice KGS equations with varying coefficient fulfill the Liouville Theorem on the time-dependent phase spaces. We end the article with the issues on the limiting behavior of solution, time-dependent pullback attractor and statistical solution for equations (1.1)-(1.2). Consider the lattice KGS equations with varying coefficients n(t)¨um+ ˙um+ (2um−um−1−um+1) + µum−β|zm|2=gm(t), m ∈Z, i˙zm−(2zm−zm−1−zm+1) + iαzm+zmum=fm(t), m ∈Z,(4.39) and ˙um+ (2um−um−1−um+1) + µum−β|zm|2=gm(t), m ∈Z, i˙zm−(2zm−zm−1−zm+1) + iαzm+zmum=fm(t), m ∈Z,(4.40) corresponding to the case n(·)≡0, n∈N. We have proved that there is a statistical solution {µn t}t∈Rsupported by the time-dependent pullback attractor ˆ An={An(t)}t∈Rwith An(t)⊂ `2 (t)×`2×L2for system (4.39). It is not hard to establish that there also is a statistical solution {µ0 t}t∈Rsupported by the pullback attractor ˆ A0={A0(t)}t∈Rwith A0(t)⊂`2×L2for system (4.40). When lim n→∞ sup t∈R n(t) = 0, some natural and interesting questions are        (a) Does the solution of equation (4.39) tend to that of equation (4.40)? (b) Does the pullback attractor An(t) converge to A0(t)?, (c) Does the statistical solution µn tconverge to µ0 t? (d) In what space and what sense shall we discuss above questions? (4.41) It is also very interesting to investigate the singular limiting behavior as that as [41] for the pullback attractors and statistical solutions for system (4.39). CONFLICT OF INTEREST STATEMENT This work does not have any conflicts of interest. 25 20 Nov 2024 23:13:21 PST 241120-Zhao Version 1 - Submitted to Comm. Pure Appl. Anal.