Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 16, pp. 1–16. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.16 UNIFORM ATTRACTORS OF NON-AUTONOMOUS SUSPENSION BRIDGE EQUATIONS WITH MEMORY LULU WANG, QIAOZHEN MA Abstract. In this article, we investigate the long-time dynamical behavior of non-autonomous suspension bridge equations with memory and free boundary conditions. We first establish the well-posedness of the system by means of the maximal monotone operator theory. Secondly, the existence of uniformly bounded absorbing set is obtained. Finally, asymptotic compactness of the process is verified, and then the existence of uniform attractors is proved for non-autonomous suspension bridge equations with memory term. 1. Introduction In this article, we focus on the long-time dynamical behavior of solutions for the following non-autonomous suspension bridge equations with memory in Ω = (0, π)× (−l, l) ⊂ R2, utt + α∆2u+ βut − ∫ ∞ 0 µ(s)∆2u(t− s)ds+ f(u(x, y, t)) = g(x, y, t), (x, y) ∈ Ω, t ≥ τ, τ ∈ R, (1.1) with the boundary conditions u(0, y, t) = uxx(0, y, t) = u(π, y, t) = uxx(π, y, t) = 0, y ∈ (−l, l), t ≥ τ, uyy(x,±l, t) + σuxx(x,±l, t) = 0, x ∈ (0, π), t ≥ τ, uyyy(x,±l, t) + (2− σ)uxxy(x,±l, t) = 0, x ∈ (0, π), t ≥ τ, (1.2) and initial conditions u(x, y, t) = uτ0(x, y), ut(x, y, t) = vτ0 (x, y), (x, y) ∈ Ω, t ≥ τ, τ ∈ R, (1.3) where α, β are positive constant, β is the damping coefficient, 0 < σ < 1 2 is the Poisson ratio, f is the nonlinear term, g is the external force. Since we have in mind a long narrow rectangle, that is l � π, it is reasonable to assume that the forcing term g does not depend on y, see [9] for details. So, we now assume that g(x, t) = g(x, y, t) and g ∈ L2 loc(R+;L2(Ω)). The assumptions on µ(s), f(u) will be given in details in the next section. 2020 Mathematics Subject Classification. 35B40, 37B55, 37L30. Key words and phrases. Uniform attractor; non-autonomous bridge equation; maximal monotone operator; memory term. ©2024. This work is licensed under a CC BY 4.0 license. Submitted November 20, 2023. Published February 10, 2024. 1 2 L. WANG, Q. MA EJDE-2024/16 We know that the earlier suspension bridge equations are derived from the math- ematical model of a one-dimensional hinged beam suspended by hangers, which describes the deflection of the roadbed in the vertical plane, see [15, 17]. As a new problem in the field of nonlinear analysis in 1990, Lazer and McKenna [16] introduced the following one-dimensional suspension bridge equation utt + EIuxxxx + δut + ku+ = W (x) + εf(x, t), (x, t) ∈ (0, L)× R+, u(0, t) = u(L, t) = uxx(0, t) = uxx(L, t) = 0, t ≥ 0. (1.4) In 1998, Ahmed and Harbi [1] made a rigorous mathematical analysis for the cou- pled suspension bridge equations, they studied the dynamical behavior of system under the different conditions, which are clamped, hinged and mixed boundary condition (one end clamped and the other one hinged), respectively, and gave the relevant numerical simulation and physical interpretation. A series of important works have investigated around the existence of a global attractor for suspension bridge equations, see for example [3, 9, 14, 17, 18, 19, 20, 24, 25, 27, 28, 29, 30] and the references therein. Ma and Zhong [18] first obtained the global attractor of the weak solution for coupled suspension bridge equations in 2005, and they further studied the existence of strong solution and strong global attractor for beam-string coupling system in [30]. Bochicchio, Giorgi and Vuk [3] proved the existence and regularity of the global attractor with finite fractal dimension for the extensible suspension bridge equation. Park and Kang [25] studied existence of global attractor for suspension bridge equation with nonlinear damping in 2011. Recently, Wang and Ma surveyed the long-term behavior of solutions for the suspension bridge equation with either time delay or state delay, see [28, 29]. When µ = 0 in (1.1), Ferrero and Gazzola [9] introduced the following model of suspension bridges utt(x, y, t) + ∆2u(x, y, t) + αut(x, y, t) + f(x, y, u) = g(x, y, t), for (x, y) ∈ Ω and t > 0. The above model regards the suspension bridges as a rectangular plate of length π with the same boundary value conditions as (1.2). They obtained the well-posedness of the system and analyzed several other boundary value problems. For further details on mathematical models for suspension bridge, we refer the reader to the new book [11] published by Gazzola. More much work related to the above-mentioned rectangular plate models for suspension bridge can be found in [2, 4, 5, 10, 12, 13, 22, 23, 27] and reference therein. For example, Messaoudi et al. [23] considered the suspension bridge prob- lem with memory under the above-mentioned boundary conditions and initial data in 2016, and established the well-posedness of the system and the existence of global attractors. Al-Gwaiz et al. [2] studied the bending and stretching energy about the rectangular plate model proposed in [9]. Berchio et al. [5] investigated the struc- tural instability of nonlinear plate modeling suspension bridges. In 2019, Wang and Ma [27] paid attention to the following nonlinear plate modeling suspension bridges with time delay in Ω = (0, π)× (−l, l) under the same conditions (1.2) as in [9], ∂ttu+ ∆2u+ γ1∂tu+ γ2∂tu(x, y, t− h) + f(u(x, y, t)) = g(x, y, t), (x, y) ∈ Ω, t ≥ τ, τ ∈ R, (1.5) EJDE-2024/16 SUSPENSION BRIDGE EQUATIONS WITH MEMORY 3 where γ1 > 0 is the damped coefficient, γ2 ∈ R. ∂tu(x, y, t − h) is the delay term, h > 0 represents the time delay. The existence of uniform attractors was achieved for (1.5) and (1.2). To the best of our knowledge, we do not find any results of non-autonomous suspension bridge equations with history memory, so we focus on the long-time dynamical behavior of problem (1.1)-(1.3). For this purpose, as in [8], we shall add a new variable ηt to the system, which corresponds to the relative displacement history, that is, ηt = ηt(x, y, s) = u(x, y, t)− u(x, y, t− s), (x, y) ∈ Ω, s ∈ R+, t ≥ τ, (1.6) then differentiating with respect to t, it is easy to see that ηtt(x, y, s) = −ηts(x, y, s) + ut(x, y, t), (x, y) ∈ Ω, s ∈ R+, t ≥ τ. (1.7) Thus, taking α− ∫∞ 0 µ(s)ds = 1, problem (1.1)-(1.3) is equivalent to utt + ∆2u+ βut + ∫ ∞ 0 µ(s)∆2ηt(s)ds+ f(u(x, y, t)) = g(x, t), (x, y) ∈ Ω, t ≥ τ, τ ∈ R, ηtt(x, y, s) = −ηts(x, y, s) + ut(x, y, t), (x, y) ∈ Ω, s ∈ R+, t ≥ τ, (1.8) with boundary conditions u(0, y, t) = uxx(0, y, t) = u(π, y, t) = uxx(π, y, t) = 0, y ∈ (−l, l), t ≥ τ, uyy(x,±l, t) + σuxx(x,±l, t) = 0, x ∈ (0, π), t ≥ τ, uyyy(x,±l, t) + (2− σ)uxxy(x,±l, t) = 0, x ∈ (0, π), t ≥ τ, (1.9) ηt(0, y, s) = ηtxx(0, y, s) = ηt(π, y, s) = ηtxx(π, y, s) = 0, y ∈ (−l, l), s ∈ R+, ηtyy(x,±l, s) + σηtxx(x,±l, s) = 0, x ∈ (0, π), s ∈ R+, ηtyyy(x,±l, s) + (2− σ)ηtxxy(x,±l, s) = 0, x ∈ (0, π), s ∈ R+, (1.10) and initial conditions u(x, y, τ) = uτ0(x, y), (x, y) ∈ Ω, τ ∈ R, ut(x, y, τ) = vτ0 (x, y), (x, y) ∈ Ω, τ ∈ R, ητ (x, y, s) = ητ0 (x, y, s), (x, y) ∈ Ω, s ∈ R+, ηt(x, y, 0) = 0, (x, y) ∈ Ω. (1.11) We denote z(t) = (u(t), ut(t), η t(s)), z0 = (uτ0 , v τ 0 , η τ 0 ). The rest of this article is organized as follows. In Section 2, we present some basic concepts and abstract conclusion. After that we establish the well-posedness of the system by means of the maximal monotone operator theory, and further obtain the existence of the uniformly bounded absorbing set and asymptotical compactness; ultimately, the existence of uniform attractors to (1.8)-(1.11) is proved in Section 3. All C throughout the paper represent real positive numbers, each C is not exactly the same in the same line, and C(·) denotes a positive constant depending on the quantities in parentheses. 4 L. WANG, Q. MA EJDE-2024/16 2. Preliminaries In this section, we will give some preliminaries on the existence and uniqueness of solutions to our problem (1.8)-(1.11), and recall some definitions and results concerning the existence of uniform attractors. Firstly, let us introduce the phase space as in [9] H2 ∗ (Ω) = {w ∈ H2(Ω) : w(0, y) = w(π, y) = 0, ∀y ∈ (−l, l)}, equipped with the inner product and norm (u, v)H2 ∗ = ∫ Ω [∆u∆v + (1− σ)(2uxyvxy − uxxvyy − uyyvxx)] dx dy, ‖u‖H2 ∗ = [ ∫ Ω [(∆u)2 + 2(1− σ)(u2 xy − uxxuyy)] dx dy ]1/2 . It has been proven that ‖ · ‖H2 ∗ is a norm on H2 ∗ which is equivalent to the usual H2(Ω)-norm in [9, Lemma 4.1]. Moreover, H2 ∗ is a Hilbert space endowed with the scalar product (·, ·)H2 ∗ . For the new variable ηt, we introduce the weighted L2-space M = L2 µ(R+;H2 ∗ (Ω)) = { ξ : R+ → H2 ∗ (Ω) : ∫ ∞ 0 µ(s)‖ξ(s)‖2H2 ∗ ds <∞ } , which is a Hilbert space endowed with inner product and norm (ξ, ζ)M = ∫ ∞ 0 µ(s)(ξ(s), ζ(s))H2 ∗ ds, ‖ξ‖2M = ∫ ∞ 0 µ(s)‖ξ(s)‖2H2 ∗ ds, respectively. Now, the phase space is defined as H = H2 ∗ (Ω)× L2(Ω)×M, equipped with the inner product and norm (U, V )H = (u, ũ)H2 ∗(Ω) + (v, ṽ) + (w, w̃)M, ‖U‖2H = (U,U)H = ‖u‖2H2 ∗(Ω) + ‖v‖2 + ‖w‖2M, respectively. where ‖ · ‖ = ‖ · ‖L2(Ω), and U = (u, v, w)T , V = (ũ, ṽ, w̃)T ∈ H. Next, we assume that the memory and nonlinear term satisfy the following con- ditions: (H1) The memory kernel µ(·) ∈ C1(R+) ∩ L1(R+) and satisfies µ(s) ≥ 0, µ′(s) ≤ 0, ∀s ∈ R+, (2.1)∫ ∞ 0 µ(s)ds = k0 > 0, ∀s ∈ R+, (2.2) µ′(s) + k1µ(s) ≤ 0, for some k1 > 0, ∀s ∈ R+. (2.3) (H2) The nonlinear function f ∈ C1(R) and satisfies |f(s1)− f(s2)| ≤ C(|s1|p + |s2|p)|s1 − s2|, ∀s1, s2 ∈ R, p > 0, (2.4) −c ≤ F (s) ≤ sf(s), ∀s ∈ R, (2.5) where F (s) = ∫ s 0 f(ν)dν. EJDE-2024/16 SUSPENSION BRIDGE EQUATIONS WITH MEMORY 5 Lemma 2.1 ([9]). Let u ∈ H2 ∗ (Ω) and suppose that 1 ≤ p < +∞. Then there exists a positive constant c∗ = c∗(Ω, p) > 0 such that ‖u‖Lp(Ω) ≤ c∗‖u‖H2 ∗(Ω). (2.6) To obtain the existence of uniform attractors corresponding to (1.8)-(1.11), we also need the following definitions and abstract results. Definition 2.2. Let E be a metric space, Σ be a parameter set, and σ ∈ Σ be a time symbol. A family of two-parameter operators {Uσ(t, τ)} = {Uσ(t, τ)|t, τ ∈ R, t ≥ τ} is called a process acting on E, if (i) Uσ(t, s)Uσ(s, τ) = Uσ(t, τ) for all t ≥ s ≥ τ and τ ∈ R; (ii) Uσ(τ, τ) = I for all τ ∈ R. Let {T (h)|h ≥ 0} be the translation semigroup on Σ. We say that a family of processes {Uσ(t, τ)}σ∈Σ satisfies the translation identity if T (h)Σ = Σ, (2.7) Uσ(t+ h, τ + h) = UT (h)(t, τ), ∀σ ∈ Σ, t ≥ τ, τ ∈ R, h ≥ 0. (2.8) Definition 2.3 ([10, 7]). Let E be a Banach space, and B be a bounded subset of E and Σ be a symbol space. We call a function φ(·, · ; ·, ·) defined on (E×E)×(Σ×Σ) be a contractive function on B×B, if for any sequence {xn}∞n=1 ⊂ B and {σn}∞n=1 ⊂ Σ, there are subsequence {xnk}∞k=1 ⊂ {xn}∞n=1 and {σnk}∞k=1 ⊂ {σn}∞n=1 such that lim k→∞ lim l→∞ φ(xnk , xnl ;σnk , σnl) = 0. We denote the set of all contractive functions on B×B×Σ×Σ by C(B,B; Σ,Σ). Theorem 2.4 ([26]). Let {Uσ(t, τ)}σ∈Σ be a family of processes satisfying the translation identity (2.8) on a Banach space E and having a bounded uniformly (w.r.t.σ ∈ Σ) absorbing set B0 ⊂ E. Moreover, assume that for any ε > 0 there exist T = T (ε) and φT ∈ C(B0,B0; Σ,Σ) such that ‖Uσ1(T, τ)x− Uσ2(T, τ)y‖ ≤ ε+ φT (x, y;σ1, σ2), ∀x, y ∈ B0, ∀σ1, σ2 ∈ Σ. (2.9) Then {Uσ(t, τ)}σ∈Σ is uniformly (with respect to σ ∈ Σ) asymptotically compact in E. Theorem 2.5 ([10, 26]). Let E be a complete metric space, {Uσ(t, τ)}σ∈Σ be a fam- ily of processes satisfying the translation identity (2.8) on E. Then {Uσ(t, τ)}σ∈Σ has a compactly uniform (with respect to σ ∈ Σ) attractor AΣ in E if and only if (i) {Uσ(t, τ)}σ∈Σ has a bounded uniformly (with respect to σ ∈ Σ) absorbing set B0 ⊂ E; (ii) {Uσ(t, τ)}σ∈Σ is uniformly (with respect to σ ∈ Σ) asymptotically compact in E. Let X be a Banach space with space. Then Lploc(R+;X) denotes all functions with spatial values in Banach space X and time variable locally p-power integrable in the Bochner sense; that is, the norm ∫ t2 t1 ‖ · ‖pXds < ∞ for any time interval [t1, t2] ⊂ R+. Moreover, the space L2 b(R+;X) denotes all translation bounded functions in L2 loc(R+;X) satisfying ‖σ‖2L2 b(R+;X) = sup t∈R+ ∫ t+1 t ∥∥σ(s) ∥∥2 X ds < +∞, ∀σ ∈ L2 b(R+;X) 6 L. WANG, Q. MA EJDE-2024/16 Now, we define the symbol space so as to obtain the asymptotic behavior of the solutions to problem (1.8)-(1.11). For an arbitrary function g0 ∈ L∞(R+;L2(Ω))∩ W 1,r b (R+;Lr(Ω))(r > 1), then we define the symbol space H(g0) as H(g0) = [g0(x, t+ r)|r ∈ R+]L2,w loc (R+;L2(Ω)), where L2,w loc (R+;L2(Ω)) denotes the space L2,w loc (R+;L2(Ω)) endowed with local weak convergence topology, and [ ] denotes the closure of a set in a topological space L2,w loc (R+;L2(Ω)). Thus, for any g ∈ H(g0), (1.8)-(1.11) with g0 instead of g pos- sesses a corresponding process {Ug0(t, τ)} acting on H. The translation semigroup {T (r)|r ≥ 0} satisfies (2.7) and (2.8), namely, T (r)H(g0) = H(g0), Ug(t+ r, τ + r) = UT (r)g(t)(t, τ), for all g ∈ H(g0), t ≥ τ , τ ∈ R, r ≥ 0. Proposition 2.6 ([7]). Let E be reflexive separable Banach space. Then the fol- lowing statements hold: (i) ‖g‖L2 b(R+;E) ≤ ‖g0‖L2 b(R+;E), for all g ∈ H(g0); (ii) the translation group T (t) is weakly continuous on H(g0); (iii) T (r)H(g0) = H(g0), for all r ∈ R. Proposition 2.7 ([26]). Let σ ∈ L∞(R+;L2(Ω))∩W 1,r b (R+;Lr(Ω))(r > 1). Then there exists M > 0 such that sup t∈R+ ‖σ(x, t+ s)‖L2(Ω) ≤M, ∀s ∈ R+. Proposition 2.8 ([26]). Let σ ∈ L∞(R+;L2(Ω)) ∩W 1,r b (R+;Lr(Ω))(r > 1), si ∈ R(i = 1, 2, · · ·), {un(t)|t ≥ 0, n = 1, 2, · · ·} be bounded in H2(Ω) ∩ H1 0 (Ω), and {unt(t)|n = 1, 2, · · ·} be bounded for any T1 > 0 in L∞(0, T1;L2(Ω)). Then there exist subsequence {unk}∞k=1 ⊂ {un}∞n=1 and {snk}∞k=1 ⊂ {sn}∞n=1, such that lim k→∞ lim l→∞ ∫ T 0 ∫ t s ∫ Ω (σ(x, τ + snk)− σ(x, τ + snl))∂t(unk − unl)(τ)dxdτds = 0. 3. Well-posedness and uniformly bounded absorbing set In this section, we will establish the well-posedness of problem (1.8)-(1.11). To achieve this, we set U = (u, v, ηt)T , where v = ut, initial data Uτ = (uτ0 , v τ 0 , η τ 0 )T . Then problem (1.8)-(1.11) is transformed into Ut +AU = F (U), U(τ) = Uτ , (3.1) where AU =  −v ∆2u+ βv + ∫∞ 0 µ(s)∆2ηt(s)ds ηts(s)− v  , F (U) =  0 −f(u) + g(x, t) 0  , Uτ = uτ0vτ0 ητ0  , EJDE-2024/16 SUSPENSION BRIDGE EQUATIONS WITH MEMORY 7 and the domain of A is D(A) = { (u, v, ηt) ∈ H : u ∈ H4(Ω), v ∈ H2 ∗ (Ω), ηt ∈ L2 µ(R+, H4(Ω)), and (1.9)-(1.10) hold } . To obtain the well-posedness of problem (1.8)-(1.11), we first need to prove the following statement. Lemma 3.1. The operator A : D(A) ⊂ H → H is maximal monotone. Proof. Letting U = (u, v, ηt)T we have (AU,U)H = (−v, u)H2 ∗(Ω) + (∆2u, v) + (βv, v) + (ηt, v)M + (ηts, η t)M + (−v, ηt)M = β‖v‖2 + (ηts, η t)M, From (H1), we infer that (ηts, η t)M = ∫ ∞ 0 µ(s)(ηts(s), η t(s))H2 ∗ ds = −1 2 ∫ ∞ 0 µ′(s)‖ηt(s)‖H2 ∗ ds ≥ k1 2 ‖ηt‖M. (3.2) Using (3.2), we arrive at (AU,U)H ≥ β‖v‖2 + k1 2 ‖ηt‖M ≥ 0, (3.3) thus, A is monotone. Next, we prove that A is maximal, so we need to prove that R(I +A) = H. We prove that there exists Ũ = (ũ, ṽ, η̃t)T ∈ H such that U +AU = Ũ (3.4) has a solution U = (u, v, ηt)T ∈ D(A). Equation (3.4) can be written u− v = ũ, v + ∆2u+ βv + ∫ ∞ 0 µ(s)∆2ηt(s)ds = ṽ, ηt + ηts − v = η̃t. (3.5) Inserting (3.5)1 into (3.5)2, we obtain u+ ∆2u+ βv + ∫ ∞ 0 µ(s)∆2ηt(s)ds = ũ+ ṽ, ηt + ηts − v = η̃t, (3.6) then, for any U = (u, v, ηt) ∈ V = H4(Ω)×H2 ∗ (Ω)×L2 µ(R+, H4(Ω)), problem (3.6) is equivalent to L1(U ,U) = L2(U), ∀U = (u, v, ηt) ∈ V, where L1 : V × V → R is the bilinear operator, L2 : V → R is the linear operator with the following forms, respectively, L1(U ,U) = (u, u) + (∆2u, u)H2 ∗ + (βv, v) + ∫ ∞ 0 µ(s)(η, u)H2 ∗ ds + (ηt, ηt)M + (ηts, η t)M − (v, v), 8 L. WANG, Q. MA EJDE-2024/16 L2(U) = ((ũ+ ṽ), u)H2 ∗ + (η̃t, ηt)M. Obviously, L1 is a bilinear and continuous from on V × V , L2 is a linear and continuous from on V . Moreover, for some C1 > 0, we have L1(U ,U) ≥ C1‖U‖2V . Furthermore, there exist C2, C3 > 0 such that |L1(U ,U)| ≤ ‖u‖‖u‖+ ‖u‖H2 ∗ ‖u‖H2 ∗ + β‖v‖‖v‖+ ‖η‖M‖u‖H2 ∗ + ‖ηt‖M‖ηt‖M + ‖ηts‖M‖ηt‖M + ‖v‖‖v‖ ≤ C2‖U‖V ‖U‖V , |L2(U)| ≤ ‖ũ+ ṽ‖H2 ∗ ‖u‖H2 ∗ + ‖η̃t‖M‖ηt‖M ≤ C3‖U‖V . By the Lax-Milgram theorem, equation (3.6) admits an unique (weak) solution U ∈ V . In addition, from (3.5)-(3.6), we deduce that v = ut = u− ũ ∈ H2 ∗ (Ω), ∆2u = ũ+ ṽ − u− β(ũ− u)− ∫ ∞ 0 µ(s)∆2ηt(s)ds ∈ L2(Ω), ηt − η̃t = −ηts(s) + ut(t) ∈ L2 µ(R+, H4(Ω)). Then (u, v, ηt) ∈ D(A). Hence, R(I +A) = H, which completes the proof. � Theorem 3.2. Assume (H1) and (H2) and Uτ ∈ H. Then problem (3.1) has a unique global solution U = (u, ut, η t) ∈ C([τ,+∞];H). Proof. From Lemma 3.1, we know that the operator A is monotone and maximal, and F obviously satisfies locally Lipschitz from (2.4). Therefore, by the Hille-Yosida theorem, we obtain the existence of a unique weak local solution for (1.8)-(1.11); that is, U = (u, ut, η t) ∈ C([τ, Tmax],H), for all Tmax > 0. Next, we prove that the solution is global, namely, Tmax = ∞. For this purpose, we need to prove that ‖U(t)‖H is uniformly bounded with respect to time. For simplicity, from now on we set d$ = dx dy. Multiplying the first equation of (1.8) by ut and integrating over Ω, we have d dt (1 2 ‖u‖2H2 ∗ + 1 2 ‖ut‖2 + ∫ Ω F (u)d$ ) + β‖ut‖2 + (ηt, ut)M = (g(t), ut), (3.7) multiplying the second equation of (1.8) by ηt and integrating over M, we obtain 1 2 d dt ‖ηt‖2M + (ηts, η t)M = (ηt, ut)M. (3.8) Then, by (3.2) and (3.7)-(3.8), we obtain d dt E(t) = −β‖ut‖2 − (ηts, η t)M + (g(t), ut), (3.9) where E(t) = 1 2 ‖u‖2H2 ∗ + 1 2 ‖ut‖2 + 1 2 ‖ηt‖2M + ∫ Ω F (u)d$. (3.10) By Hölder’s inequality and Young’s inequality, for 0 < ξ ≤ 2β, we have (g(t), ut) = ∫ Ω g(t)utd$ ≤ ‖g(t)‖‖ut‖ ≤ 1 2ξ ‖g(t)‖2 + ξ 2 ‖ut‖2. (3.11) EJDE-2024/16 SUSPENSION BRIDGE EQUATIONS WITH MEMORY 9 Using (3.9)-(3.11), we deduce that d dt E(t) ≤ −(β − ξ 2 )‖ut‖2 − k1 2 ∫ ∞ 0 µ(s)‖ηt(s)‖H2 ∗ ds+ 1 2ξ ‖g(t)‖2, (3.12) integrating (3.12) over (τ, t), it is easy to see that E(t) ≤ E(τ) + 1 2ξ ∫ t τ ‖g(s)‖2ds. (3.13) By Proposition 2.6, we know that ‖g‖2 L2 b(Rτ ;L2(Ω)) ≤ ‖g0‖2L2 b(Rτ ;L2(Ω)) . Then E(t) ≤ E(τ) + 1 2ξ ‖g0‖2L2 b(Rτ ;L2(Ω)), (3.14) and by (2.5), we obtain E(t) ≥ 1 2 ‖u‖2H2 ∗ + 1 2 ‖ut‖2 + 1 2 ‖ηt‖2M − c|Ω|. (3.15) Thus, for each t ≥ τ , we obtain E(t) ≥ C4 ∥∥(u(t), ut(t), η t(t)) ∥∥2 H − C5. (3.16) This and (3.14) imply∥∥(u(t), ut(t), η t(t)) ∥∥2 H ≤ 1 C4 ( E(τ) + 1 2ξ ‖g0‖2L2 b(Rτ ;L2(Ω)) + C5 ) ≤ C6 (3.17) for all t ≥ τ , which completes the proof. � Remark 3.3. From Theorem 3.2, we deduce that problem (1.8)-(1.11) generates a family of processes {Ug(t, τ)}, g ∈ H(g0) in the space H. Then we define a family of two-parameter operators Ug(t, τ) : H → H given by Ug(t, τ)(uτ0 , v τ 0 , η τ 0 ) = (u(t), ut(t), η t(s)). (3.18) where (u(t), ut(t), η t(s)) is the unique global solution of (1.8)-(1.11) corresponding to initial data (uτ0 , v τ 0 , η τ 0 ), and {Ug(t, τ)}, g ∈ H(g0) satisfies Definition 2.2. More- over, for all initial data zτ0 = (uτ0 , v τ 0 , η τ 0 ) and zτ1 = (uτ1 , v τ 1 , η τ 1 ), we let zτ = zτ0 − zτ1 . Then there exists a positive constant C depending on zτ0 and zτ1 , such that ‖Ug(t, τ)zτ0 − Ug(t, τ)zτ1‖H ≤ eCT (‖zτ‖2H + ‖g1(t)− g2(t)‖2L2 b(Rτ ;L2(Ω))), (3.19) for τ ≤ t ≤ T . This shows that solutions of (1.8)-(1.11) depend continuously on the initial data. Next, we prove the existence of a uniformly absorbing set in H. We need to introduce a Lyapunov functional L(t) = PE(t) +QΦ(t), (3.20) where P,Q are positive constants, which will be defined later, and Φ(t) = (ut, u). Lemma 3.4. Let Q be small enough and P be large enough. Then there exist θ1 and θ2 > 0 such that θ1‖(u(t), ut(t), η t(t))‖2H − c1 ≤ L(t) ≤ θ2‖(u(t), ut(t), η t(t))‖2H + Pc2. (3.21) 10 L. WANG, Q. MA EJDE-2024/16 Proof. Firstly, we prove the left inequality of (3.21). Choosing Q small enough and then P large enough such that P−Q 2 > 0, P−Qc∗ 2 > 0. By (3.15), we obtain L(t) ≥ P 2 ‖u‖2H2 ∗ + P 2 ‖ut‖2 + P 2 ‖ηt‖2M − cP |Ω| − Qc2∗ 2 ‖u‖2H2 ∗ − Q 2 ‖ut‖2 ≥ (P −Qc2∗ 2 ) ‖u‖2H2 ∗ + (P −Q 2 ) ‖ut‖2 + P 2 ‖ηt‖2M − cP |Ω| ≥ θ1‖(u(t), ut(t), η t(t))‖2H − c1. (3.22) where θ1 = min{P−Qc 2 ∗ 2 , P−Q2 , P2 }, c1 = cP |Ω|. Moreover, by (2.4), (2.6), (3.17), and H2(Ω) ↪→ Lp(Ω), (1 ≤ p ≤ ∞), we arrive at∫ Ω F (u(t))d$ ≤ ∫ Ω |u||f(u)|d$ ≤ ∫ Ω |u||f(u)− f(0)|d$ + ∫ Ω |u||f(0)|d$ ≤ C ∫ Ω |u|2|u|pd$ + 1 2 ∫ Ω |u|2d$ + 1 2 |Ω||f(0)|2 ≤ c2∗C ( ‖u‖pL∞(Ω) + 1 ) ‖u‖2H2 ∗(Ω) + c2. (3.23) Applying Young’s inequality, Sobolev’s embedding Theorem and (3.23), we deduce that L(t) ≤ P 2 ‖u‖2H2 ∗ + P 2 ‖ut‖2 + P 2 ‖ηt‖2M + P ∫ Ω F (u(t))d$ + Qc2∗ 2 ‖u‖2H2 ∗ + Q 2 ‖ut‖2 ≤ (P +Qc2∗ 2 ) ‖u‖2H2 ∗ + (P +Q 2 ) ‖ut‖2 + P 2 ‖ηt‖2M + P ∫ Ω |u||f(u(t))|d$ ≤ (P + c2∗[Q+ 2PC(‖u‖pL∞(Ω) + 1)] 2 ) ‖u‖2H2 ∗ + (P +Q 2 ) ‖ut‖2 + P 2 ‖ηt‖2M + Pc2 ≤ θ2‖(u(t), ut(t), η t(t))‖2H + Pc2. This completes the proof. � Lemma 3.5. The function Φ(t) = (ut, u) satisfies Φ′(t) ≤ ( 1 + β 2ζ ) ‖ut‖2 + ( (βζ + 2ζ) c2∗ 2 − 1 ) ‖u‖2H2 ∗ + 1 2ζ ‖ηt‖2M + 1 2ζ ‖g(t)‖2 + c|Ω|. (3.24) Proof. By (1.8)1, we have Φ′(t) = (utt, u) + ‖ut‖2 = ‖ut‖2 − ‖u‖2H2 ∗(Ω) − β(ut, u)− (ηt, u)M − (f(u), u) + (g(t), u). (3.25) Using Young’s inequality, Hölder’s inequality, and (2.6), for each ζ > 0, we have −β(ut, u) ≤ β 2ζ ‖ut‖2 + βζ 2 ‖u‖2 ≤ β 2ζ ‖ut‖2 + c2∗βζ 2 ‖u‖2H2 ∗ , (3.26) EJDE-2024/16 SUSPENSION BRIDGE EQUATIONS WITH MEMORY 11 −(ηt, u)M ≤ ζ 2 ‖u‖2 + 1 2ζ ‖ηt‖2M ≤ c2∗ζ 2 ‖u‖2H2 ∗ + 1 2ζ ‖ηt‖2M, (3.27) (g(t), u) ≤ 1 2ζ ‖g(t)‖2 + ζ 2 ‖u‖2 ≤ 1 2ζ ‖g(t)‖2 + c2∗ζ 2 ‖u‖2H2 ∗ . (3.28) From (2.5), it is easy to see that − (f(u), u) ≤ c|Ω|. (3.29) Inserting (3.26)-(3.29) into (3.25), we arrive at (3.24), which completes the proof. � Theorem 3.6. Under the assumption of Theorem 3.2, a family of processes {Ug(t, τ)}, g ∈ H(g0) corresponding to (1.8)-(1.11) possesses a bounded uniformly (with respect to g ∈ H(g0)) absorbing set B in H. Proof. From (3.12), (3.20) and (3.24), for P,Q > 0 it follows that L′(t) = PE′(t) +QΦ′(t) ≤ −Q ( 1− (βζ + 2ζ) c2∗ 2 ) ‖u‖2H2 ∗ − ( (β − ξ 2 )P − (1 + β 2ζ )Q ) ‖ut‖2 − (k1P 2 − Q 2ζ ) ‖ηt‖2M + Pζ +Qξ 2ξζ ‖g(t)‖2 + cQ|Ω|. (3.30) where ζ, ξ > 0. Choosing first ζ, ξ small enough such that 1− (βζ + 2ζ) c2∗ 2 > 0, β − ξ 2 > 0, after that, choosing again Q small enough, and then P large enough such that k1P 2 − Q 2ζ > 0, (β − ξ 2 )P − (1 + β 2ζ )Q > 0. Thus, there exist positive constants ϕ1, ϕ2, ϕ3, ϕ4, and ϕ5 such that d dt L(t) ≤ −ϕ1‖u‖2H2 ∗ − ϕ2‖ut‖2 − ϕ3‖ηt‖2M + ϕ4‖g(t)‖2 + ϕ5, (3.31) choosing ϕ = min{ϕ1, ϕ2, ϕ3} yields d dt L(t) ≤ −ϕ ( ‖u‖2H2 ∗ + ‖ut‖2 + ‖ηt‖2M ) + ϕ4‖g(t)‖2 + ϕ5. (3.32) By Lemma 3.4, we claim that d dt L(t) + %L(t) ≤ c3‖g(t)‖2 + c4, (3.33) where % = ϕ/θ2. Using Gronwall Lemma for (3.33), leads to L(t) ≤ L(τ)e−%(t−τ) + c3 ∫ t τ e−%(t−s)‖g(s)‖2ds+ c4 ∫ t τ e−%(t−s)ds ≤ L(τ)e−%(t−τ) + c3 1− e−% sup t≥τ ∫ t+1 t ‖g(s)‖2ds+ c6 ≤ L(τ)e−%(t−τ) + c5 1− e−% ‖g0‖L2 b(Rτ ;L2(Ω)) + c6. (3.34) 12 L. WANG, Q. MA EJDE-2024/16 If for any bounded set B ⊆ H, and the initial data (uτ0 , v τ 0 , η τ 0 ) ∈ B, there exists a constant CB > 0 such that L(τ) ≤ CB , then we deduce from (3.34) that L(t) ≤ CBe−%(t−τ) + c5 1− e−% ‖g0‖L2 b(Rτ ;L2(Ω)) + c6, (3.35) for any t ≥ t0. It follows that ‖(u, ut, ηt)‖2H ≤ 1 θ1 (L(t) + c1) = R2. This means that a family of processes {Ug(t, τ)} generated by (1.8)-(1.11) has an uniformly absorbing ballB = B(0, R) = {(u, ut, ηt) ∈ H : ‖(u, ut, ηt)‖2H ≤ R2} ⊆ H for any g ∈ H(g0), which completes the proof. � Next, we prove the asymptotic compactness of a family of processes {Ug(t, τ)}, g ∈ H(g0) associated with (1.8)-(1.11) in H. Our main results are as follows. Theorem 3.7. Assume that (H1) and (H2) hold and g ∈ H(g0). Then a family of processes {Ug(t, τ)}, g ∈ H(g0) corresponding to (1.8)-(1.11) is uniformly (with respect to g ∈ H(g0)) asymptotically compact in H. Proof. Let z1 = (u1, u1 t , η 1) and z2 = (u2, u2 t , η 2) be two solutions of (1.8)-(1.11) with the initial data z1 0 = (u1τ 0 , v1τ 0 , η1τ 0 ), z2 0 = (u2τ 0 , v2τ 0 , η2τ 0 ) and the symbols g1, g2, respectively. Set z = z1 − z2 = (u, ut, η t), the initial data z0 = z1 0 − z2 0 = (uτ0 , v τ 0 , η τ 0 ). Then (u, ut, η t) satisfies the equations utt + ∆2u+ βut + ∫ ∞ 0 µ(s)∆2ηt(s)ds+ f(u1)− f(u2) = g1(t)− g2(t), ηtt(x, y, s) = −ηts(x, y, s) + ut(x, y, t), (3.36) with boundary conditions (1.9)-(1.10). We denote Ẽ(t) = 1 2 ‖u‖2H2 ∗ + 1 2 ‖ut‖2 + 1 2 ‖ηt‖2M, (3.37) L̃(t) = P1Ẽ(t) +Q1Φ̃(t), (3.38) where Φ̃(t) = (ut, u). Obviously, Ẽ(t) and L̃(t) are equivalent. Then there exist two positive constants γ1 and γ2 depending on P1, Q1 such that γ1Ẽ(t) ≤ L̃(t) ≤ γ2Ẽ(t), (3.39) where P1 > 0 large enough and Q1 > 0 small enough. First, multiplying (3.36)1 by ut and integrating over Ω, multiplying (3.36)2 by ηt and integrating over M, then adding them, we have d dt Ẽ(t) ≤ −β‖ut‖2 − k1 2 ‖ηt‖2M + ∫ Ω (f(u2)− f(u1))utd$ + ∫ Ω (g1(t)− g2(t))utd$, (3.40) according to the proof of Lemma 3.5, there exists ζ > 0 such that Φ̃′(t) ≤ (1 + β 2ζ )‖ut‖2 + ( (βζ + ζ) c2∗ 2 − 1 ) ‖u‖2H2 ∗ + 1 2ζ ‖ηt‖2M + ∫ Ω (f(u2)− f(u1))ud$ + ∫ Ω (g1(t)− g2(t))ud$, (3.41) EJDE-2024/16 SUSPENSION BRIDGE EQUATIONS WITH MEMORY 13 combining with (3.40) and (3.41), it is easy to see that L̃′(t) = P1Ẽ ′(t) +Q1Φ̃′(t) ≤ −Q1 ( 1− (βζ + ζ) c2∗ 2 ) ‖u‖2H2 ∗ − ( P1β − (1 + β 2ζ )Q1 ) ‖ut‖2 − (k1P1 2 − Q1 2ζ ) ‖ηt‖2M + P1 ∫ Ω (f(u2)− f(u1))utd$ +Q1 ∫ Ω (f(u2)− f(u1))ud$ + P1 ∫ Ω (g1(t)− g2(t))utd$ +Q1 ∫ Ω (g1(t)− g2(t))ud$. (3.42) First of all, taking ζ small enough such that 1− (βζ + ζ) c2∗ 2 > 0. After that, choosing Q1 small enough and then P1 large enough, such that k1P1 2 − Q1 2ζ > 0, P1β − ( 1 + β 2ζ ) Q1 > 0. Thus, there exists ψ > 0 such that d dt L̃(t) ≤ −ψẼ′(t) + P1 ∫ Ω (f(u2)− f(u1))utd$ +Q1 ∫ Ω (f(u2)− f(u1))ud$ + P1 ∫ Ω (g1(t)− g2(t))utd$ +Q1 ∫ Ω (g1(t)− g2(t))ud$, thanks to (3.39), we have d dt L̃(t) + χL̃(t) ≤ P1 ∫ Ω (f(u2)− f(u1))utd$ +Q1 ∫ Ω (f(u2)− f(u1))ud$ + P1 ∫ Ω (g1(t)− g2(t))utd$ +Q1 ∫ Ω (g1(t)− g2(t))ud$, (3.43) where χ = ψ γ2 . Integrating (3.43) over [τ, t], we conclude that L̃(t) ≤ L̃(τ)e−χ(t−τ) + P1 ∫ t τ ∫ Ω e−χ(t−s)(f(u2)− f(u1))utd$ds +Q1 ∫ t τ ∫ Ω e−χ(t−s)(f(u2)− f(u1))ud$ds + P1 ∫ t τ ∫ Ω e−χ(t−s)(g1(s)− g2(s))utd$ds +Q1 ∫ t τ ∫ Ω e−χ(t−s)(g1(s)− g2(s))ud$ds. (3.44) 14 L. WANG, Q. MA EJDE-2024/16 For each ε > 0, there exists T > τ , such that L̃(τ)e−χ(t−τ) ≤ ε for t ≥ T . Then, by (3.39) and (3.44), we deduce that Ẽ(t) ≤ ε+ P1 ∫ t τ ∫ Ω e−χ(t−s)(f(u2)− f(u1))utd$ds +Q1 ∫ t τ ∫ Ω e−χ(t−s)(f(u2)− f(u1))ud$ds + P1 ∫ t τ ∫ Ω e−χ(t−s)(g1(s)− g2(s))utd$ds +Q1 ∫ t τ ∫ Ω e−χ(t−s)(g1(s)− g2(s))ud$ds := ε+ φT ( (u1τ 0 , v1τ 0 , η1τ 0 ), (u2τ 0 , v2τ 0 , η2τ 0 ); g1, g2 ) . (3.45) Now, we prove φT (·, ·; ·, ·) ∈ C(B,B; Σ,Σ) for every fixed T > τ . By Theorem 3.6, we know that ∪g∈H(g0) ∪t∈[τ,T ] Ug(t, τ)B is bounded in H. Let the sequence (uτ0n, v τ 0n, η τ 0n) ∈ B, gn ∈ H(g0), n = 1, 2, · · ·. Because B is bounded, the corresponding sequence of solutions (un, vn, η t n) asso- ciated with the system (1.8)-(1.11) is uniformly bounded in H. Without loss of generality, we assume that (i) um → u weak star in L∞(τ, T ;H2 ∗ (Ω)), (ii) umt → ut weak star in L∞(τ, T ;L2(Ω)), (iii) um → u in L2(τ, T ;L2(Ω)), (iv) um(τ)→ u(τ), um(T )→ u(T ) in Lk(Ω), k <∞. Then, applying Proposition 2.7 and (iii), it follows that lim n→∞ lim m→∞ ∫ t τ ∫ Ω (gn(x, s)− gm(x, s))(un(s)− um(s))d$ds = 0, (3.46) from Proposition 2.8, we have lim n→∞ lim m→∞ ∫ t τ ∫ Ω (gn(x, s)− gm(x, s))(unt (s)− umt (s))d$ds = 0. (3.47) On the other hand, since f(um) → f(u) weak star in L2(τ, T ;H), and exploiting (ii)-(iii), it follows that lim n→∞ lim m→∞ ∫ t τ ∫ Ω (f(un(r))− f(um(r)))(unt (r)− umt (r))d$dr = 0, (3.48) lim n→∞ lim m→∞ ∫ t τ ∫ Ω (f(un(r))− f(um(r)))(un(r)− um(r))d$dr = 0. (3.49) Therefore, from (3.46) and (3.49), we deduce that φT ∈ C(B,B; Σ,Σ), which com- pletes the proo. � Finally, by Theorems 3.6 and 3.7, we conclude the main result of this article. Theorem 3.8. Assume that (H1) and (H2) hold and g ∈ H(g0). Then a family of processes {Ug(t, τ)}, g ∈ H(g0) corresponding to (1.8)-(1.11) has a compactly uniform (w.r.t.g ∈ H(g0)) attractor AΣ in H. EJDE-2024/16 SUSPENSION BRIDGE EQUATIONS WITH MEMORY 15 Acknowledgments. The authors express their gratitude to the anonymous refer- ees for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (No. 11961059). References [1] N. U. Ahmed, H. 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Lulu Wang College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China Email address: wangll0526@126.com Qiaozhen Ma (corresponding author) College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China Email address: maqzh@nwnu.edu.cn 1. Introduction 2. Preliminaries 3. Well-posedness and uniformly bounded absorbing set Acknowledgments References