Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 05, pp. 1–20. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu PULLBACK ATTRACTORS FOR NON-AUTONOMOUS BRESSE SYSTEMS RICARDO DE SÁ TELES Abstract. This article concerns the asymptotic behavior of solutions of non- autonomous Bresse systems. We establish the existence of pullback attractor and upper semicontinuity of attractors as a non-autonomous perturbations tend to zero. In addition we study the continuity of attractors with respect to a parameter in a residual dense set. 1. Introduction An important problem in dynamical systems is the study of the asymptotic behavior of evolution processes associated with mathematical models that appear in many applications to natural sciences. Our attention will be in evolution processes associated with non-autonomous problems, where the pullback attractor theory has been applied quite successfully. This theory is an extension of the autonomous concept of global attractor. A good presentation about this theory can be found in the book by Carvalho, Langa and Robinson [6]. Let L > 0 be given. This article concerns the long-time dynamics of the non- autonomous Bresse system for vibrations of curved beams, ρ1ϕtt − k(ϕx + ψ + lw)x − k0l(wx − lϕ) + g1(ϕt) + f1(ϕ,ψ,w) = h1(x, t), (1.1) ρ2ψtt − bψxx + k(ϕx + ψ + lw) + g2(ψt) + f2(ϕ,ψ,w) = h2(x, t), (1.2) ρ1wtt − k0(wx − lϕ)x + kl(ϕx + ψ + lw) + g3(wt) + f3(ϕ,ψ,w) = h3(x, t), (1.3) defined in (0, L) × R, where ϕ, ψ, w represent, vertical displacement, shear angle, and longitudinal displacement, respectively. The coefficients ρ1, ρ2, b, k, k0, l are positive constants, g1(ϕt), g2(ψt), g3(wt) are nonlinear damping terms, fi(ϕ,ψ,w), i = 1, 2, 3, are nonlinear forces and hi(x, t), i = 1, 2, 3, are time-dependent pertur- bations. To this system we add the Dirichlet boundary condition ϕ(0, t) = ϕ(L, t) = ψ(0, t) = ψ(L, t) = w(0, t) = w(L, t) = 0, t > τ, (1.4) and the initial condition (for t = τ), ϕ(·, τ) = ϕτ0 , ϕt(·, τ) = ϕτ1 , ψ(·, τ) = ψτ0 , ψt(·, τ) = ϕτ1 , w(·, τ) = wτ0 , wt(·, τ) = wτ1 . (1.5) 2010 Mathematics Subject Classification. 35B40, 35B41, 74K10. Key words and phrases. Asymptotic behavior of solutions; pullback attractor; Bresse system; upper semicontinuity; continuity of attractors. ©2022. This work is licensed under a CC BY 4.0 license. Submitted October 19, 2020. Published January 14, 2022. 1 2 R. S. TELES EJDE-2022/05 In recent years, the Bresse system has been studied by many authors. There are several works concerned with decay rates of solutions. For the linear case, see for instance [1, 2, 8, 9], and for the nonlinar case see [7, 12]. On the other hand, the long-time dynamics in autonomous case (hi(x, t) = 0) was discussed in [13], where the authors prove the existence of a global attractor and compare the Bresse system with the Timoshenko system, in the sense of upper-semicontinuity of their attractors as l→ 0. The non-autonomous case was not discussed there. Motivated by this, we study the long-time dynamics of the non-autonomous Bresse system (1.1)-(1.3) characterized by pullback attractors. To the best of our knowledge the first work concerned with Bresse systems with non-autonomous forces was [3], where the authors establish uniform decay rates of the energy. Since our problem has damping terms in all of the equations we shall not assume any equal wave speeds assumption which is a remarkable stability criteria to the Bresse systems. For more details see [3, 13]. The content of the paper is as follows: (i) Under appropriate assumptions on the forcing terms and on hi, i = 1, 2, 3, we establish the existence of a pullback attractor. Our main result is pre- sented in Theorem 3.1. (ii) We show the upper-semicontinuity of a family of pullback attractors as the non-autonomous perturbations tend to zero. To this end, we replace hi by εhi and let ε→ 0. See Theorem 4.1. (iii) We apply a recent result by Hoang, Olson and Robinson [11] to study the continuity of a family of pullback attractors Aε when ε ∈ J , a residual dense set of [0, 1]. See Theorem 5.1. 2. Preliminaries Let the Lp norm be denoted by ‖u‖p when p 6= 2, and by ‖u‖ when p = 2. In the Sobolev space H1(0, L) we have ‖u‖ 6 L π ‖ux‖ (2.1) and ‖u‖H1 0 = ‖ux‖. We consider weak solutions in the phase space H = H1 0 (0, L)3× L2(0, L)3, equipped with the norm ‖y‖2H = ‖ϕx‖2 + ‖ψx‖2 + ‖wx‖2 + ‖ϕ̃‖+ ‖ψ̃‖2 + ‖w̃‖2, where y = (ϕ,ψ,w, ϕ̃, ψ̃, w̃). For convenience, we use the equivalent norm ‖y‖2H = ρ1‖ϕ̃‖2 + ρ2‖ψ̃‖2 + ρ1‖w̃‖2 + b‖ϕx‖+ k‖ϕx + ψ + lw‖2 + k0‖wx − lϕ‖2. In fact, as proved in [13], there exists constants γ1 > 0, γ2 > 0 and γ3 > 0 such that ‖y‖2H 6 γ1‖y‖2H , (2.2) ‖y‖2H 6 γ2‖y‖2H, (2.3) ‖ϕx‖2 + ‖ψx‖2 + ‖wx‖2 6 γ3 ( b‖ψx‖2 + k‖ϕx + ψ + lw‖2 + k0‖wx − lϕ‖2 ) . (2.4) Next we present some definitions and results about pullback attractors, taken from [6] and [15]. Let (X, d) a metric space and {U(t, τ) : t > τ ∈ R} a evolution process in X. EJDE-2022/05 NON-AUTONOMOUS BRESSE SYSTEMS 3 Definition 2.1. Let A and B be non-empty subsets of X. We define the Hausdorff semidistance distX(A,B) = sup a∈A inf b∈B d(a, b). The Hausdorff metric in X is dX(A,B) = max{distX(A,B),distX(B,A)}. Definition 2.2. A process {U(t, τ) : t > τ ∈ R} in X is a two-parameter family of maps U(t, τ) : X → X such that (i) U(t, t) = I, where I is the identity operator in X, for all t ∈ R, (ii) U(t, s) ◦ U(s, τ) = U(t, τ), for all t > s > τ in R, (iii) (t, τ, x) 7→ U(t, τ)x is continuous for all t > τ and x ∈ X. Definition 2.3. A family of compact sets A = {A(t) : t ∈ R} in X is the pullback attractor for the process U(·, ·) if (i) A is invariant: U(t, τ)A(τ) = A(t), for all t > τ , (ii) A is pullback atracting: for any bounded set D ⊂ X and t ∈ R, distX(U(t, τ)D,A(t))→ 0 as τ → −∞, (iii) A is minimal, that is, if {C(t)} is any other family of compact sets that satisfies (i) and (ii), then A(t) ⊂ C(t), for all t ∈ R. The following definitions are useful to guarantee the existence of pullback at- tractor for a process. Definition 2.4. A family {B(t)} of non-empty subset of X is called pullback absorbing for the process U(·, ·) if given t ∈ R, ε > 0 and bounded subset D of X, there exists τε 6 t such that U(t, τ)D ⊂ B(t), for all τ 6 τε. Definition 2.5. An evolution process U(·, ·) is called pullback asymptotically com- pact if, for each t ∈ R, each sequence {τk} ⊂ (−∞, t], and each bounded sequence {xk} ⊂ X such that (i) τk → −∞ as k →∞, and (ii) {U(t, τk)xk} is bounded, it follows that the sequence {U(t, τk)xk} has a convergent subsequence. The following result gives conditions for obtaining the existence of pullback at- tractors for an evolution processes. Theorem 2.6. Let {U(t, τ); t > τ ∈ R} be an evolution process in a metric space X. Assume that the process {U(t, τ)}t>τ is pullback asymptotically compact and pos- sesses a pullback absorbing family B̂ = {B(t)}t∈R. Then the family  = {A(t)}t∈R given by A(t) = ∪ D⊂X, bounded ω(D, t) is a minimal pullback attractor for the process U(·, ·), where ω(D, t) = ∩s6t∪τ6sU(t, τ)D(τ). Next, we present a sufficient condition for pullback asymptotic compactness of an evolution process. 4 R. S. TELES EJDE-2022/05 Definition 2.7. Let X be a Banach space. Then, one says that the function Ψ : X×X → R is contractive on a bounded subset B of X if given a sequence {xn} ⊂ B there exists a subsequence {xnk} such that limk→∞ liml→∞Ψ(xnk , xnl) = 0. Theorem 2.8. Let X be a Banach space and U(·, ·) be a process which admits a pullback absorbing family of bounded subsets of X, B̂ = {B(t)}t∈R. Suppose that for all t ∈ R and ε > 0 there exist τε 6 t and a contractive function Ψε on B(τε), such that ‖U(t, τε)x− U(t, τε)y‖X 6 ε+ Ψε(x, y), for all , x, y ∈ B(τε). Then {U(t, τ) : t > τ ∈ R} is pullback asymptotically compact. Let Aε a family of pullback attractors for a process Uε(·, ·), with ε in a metric space Λ. We establish conditions ensuring the continuity of pullback attractor with respect to a parameter. Definition 2.9. We say that a family of pullback attractors {Aε}ε∈Λ is upper semicontinuous at ε0 ∈ Λ if lim ε→ε0 distX(Aε(t),A(t)) = 0, for all t ∈ R. Analogously, if we interchange Aε and Aε0 in the above limit, then we say that Aε is lower semicontinuous at ε0 ∈ Λ. Then, Aε is continous at ε ∈ Λ if lim ε→ε0 dX(Aε,Aε0) = 0. Proposition 2.10. Let Uε : X → X be a family of parametrized processes with ε ∈ [0, 1). Suppose that (i) Uε has a pullback attractor Aε for all ε ∈ [0, 1), (ii) For every t ∈ R, T > 0 and bounded subsets D ⊂ X, sup τ∈[0,T ],u0∈D d(Uε(t+ τ, t)u0, U0(t+ τ, t)u0)→ 0, as ε→ 0, (iii) There exists δ and t0 ∈ R such that ∪ε∈(0,δ) ∪s6t0 Aε(s) is bounded. Then the family of processes Uε(t, τ) : X → X is upper semicontinuous at ε = 0. We shall use a recent result in [11] for the continuity of pullback attractors with respect to a parameter. Theorem 2.11. Let Uε(·, ·) a family of processes on X with ε in a metric space Λ. Assume that Uε(·, ·) has a pullback attractor for every ε and (i) There exists a bounded set B ⊂ X such that Aε(t) ⊂ D, for every ε ∈ Λ and t ∈ R, (ii) For every τ ∈ R and t > τ , Uε(t, τ)x is continuous in ε, uniformly for x in bounded subsets of X, (iii) For every ε0 ∈ Λ and t ∈ R, there exists δ > 0 such that ∪BΛ(ε0,δ)Aε(t) (2.5) is compact. Then there exists a residual set J ⊂ Λ such that for every t ∈ R the function ε 7→ Aε(t) is continuous at each ε ∈ J . EJDE-2022/05 NON-AUTONOMOUS BRESSE SYSTEMS 5 2.1. Assumptions. Our hypothesis are similar to those in [13]. We consider f1, f2, f3 are locally Lipschitz and gradient type. Let us assume there exists a C2 function F : R3 → R such that ∇F = (f1, f2, f3), (2.6) and satisfies the following conditions: There exist β, mF > 0 such that F (u, v, w) > −β(|u|2 + |v|2 + |w|2)−mF , ∀u, v, w ∈ R, (2.7) where 0 6 β 6 π2 2γ3L2 , (2.8) and there exist p > 1 and Cf > 0 such that, for i = 1, 2, 3, |∇fi(u, v, w)| 6 Cf (1 + |u|p−1 + |v|p−1 + |w|p−1), ∀u, v, w ∈ R. (2.9) In particular, there exists CF > 0 such that F (u, v, w) 6 CF (1 + |u|p+1 + |v|p+1 + |w|p+1), ∀u, v, w ∈ R. Furthermore, we assume that for all u, v, w ∈ R, ∇F (u, v, w) · (u, v, w)− F (u, v, w) > −β(|u|2 + |v|2 + |w|2)−mF . (2.10) With respect to the damping functions gi ∈ C1(R), i = 1, 2, 3, we assume that gi is increasing, gi(0) = 0 and there exist constants mi, Mi > 0 such that mi 6 g ′ i(s) 6Mi,∀s ∈ R. (2.11) Furthermore, we assume h1, h2, h3 ∈ L2 loc(R;L2(0, L)) and satisfy, for some con- stant Ch > 0,∫ t −∞ e−σ(t−s)(‖h1(s)‖22 + ‖h2(s)‖22 + ‖h3(s)‖22 ) ds < Ch, ∀t ∈ R, (2.12) with σ ∈ [0, σ0], where σ0 > 0 is a constant dependent only on the parameters ρ1, ρ2, b, k, l, β, and will be defined later. 2.2. Energy of the system. The energy of the system along a solution y(t) = (ϕ(t), ψ(t), w(t), ϕt(t), ψt(t), wt(t)), t > τ , is E(t) = 1 2 ‖(ϕ(t), ψ(t), w(t), ϕt(t), ψt(t), wt(t))‖2H (2.13) and E(t) = E(t) + ∫ L 0 F (ϕ,ψ,w)dx. (2.14) Then, multiplying (1.1)-(1.3) by ϕt, ψt, wt, respectively, we obtain by integration over [0, L], d dt E(t) = − ∫ L 0 ( g1(ϕt)ϕt + g2(ψt)ψt + g3(wt)wt ) dx + ∫ L 0 ( h1(t)ϕt(t) + h2(t)ψt(t) + h3(t)wt(t) ) dx, ∀t > τ. (2.15) The following inequalities will be useful in next sections. 6 R. S. TELES EJDE-2022/05 Lemma 2.12. There exists β0 > 0, kF > 0 such that E(t) > β0‖U(t, τ)z‖2H − LmF , ∀t > τ, z; (2.16) E(t) 6 kF (1 + ‖U(t, τ)z‖p+1 H ), ∀t > τ (2.17) Proof. From (2.7) and (2.14), E(t) > E(t)− β(‖ϕ‖2 + ‖ψ‖2 + ‖w‖2)− LmF > ( 1− 2βγ3L 2 π2 ) El(t)− LmF . Then, we take β0 = 1 − 2βγ3L 2 π2 . Now, using (2.9) and the embedding of H1(0, L) in Lp(0, L) we obtain (2.17). � 2.3. Well-possedness. We shall write the system (1.1)-(1.5) as an abstract Cauchy problem, d dt y(t)− (B1 +B2)y(t) = F(y(t), t), y(τ) = yτ , (2.18) where y(t) = (ϕ(t), ψ(t), w(t), ϕ′(t), ψ′(t), w′(t)) ∈ H, ϕ′ = ϕt, ψ′ = ψt, w′ = wt, and yτ = (ϕτ , ψτ , wτ , ϕ ′ τ , ψ ′ τ , w ′ τ ) ∈ H is the initial condition. We let B1 : D(B1) ⊂ H → H be defined by B1y =  ϕ′ ψ′ w′ k ρ1 (ϕx + ψ + lw)x + k0l ρ1 (wx − lϕ) b ρ2 ψxx − k ρ2 (ϕx + ψ + lw) k0 ρ1 (wx − lϕ)x − kl ρ1 (ϕx + ψ + lw)  , with domain D(B1) = (H2(0, L) ∩ H1 0 (0, L))3 × H1 0 (0, L)3, and B2 : H → H, is given by B2y =  0 0 0 − g1(ϕ′) ρ1 − g2(ϕ′) ρ2 − g3(w′) ρ1  , with domain D(B2) = H. Also, we have the nonlinear function F : H → H defined by F(y, t) =  0 0 0 − f1(ϕ,ψ,w) ρ1 + h1 ρ1 − f2(ϕ,ψ,w) ρ2 + h2 ρ2 − f3(ϕ,ψ,w) ρ1 + h3 ρ1  . We establish the following result on the existence and uniqueness of a solution. EJDE-2022/05 NON-AUTONOMOUS BRESSE SYSTEMS 7 Theorem 2.13. Assume that (2.6)-(2.12) hold and h1, h2 h3 ∈ L2 loc(R;L2(0, L)). Then for each initial time τ ∈ R and data yτ ∈ H, problem (1.1)-(1.5) has a unique weak solution y = (ϕ,ψ,w, ϕt, ψt, wt) satisfying y ∈ C([τ,∞);H), y(τ) = yτ . In addition, if yτ ∈ D(B1), then the solution is strong. Also the weak solutions depend continuously on the initial data. Proof. It follows from [7, Theorem 2.2] that B1 + B2 is maximal monotone in H. Then the Cauchy problem d dt y(t)− (B1 +B2)y(t) = 0, y(τ) = yτ (2.19) has a unique solution. We will show that (2.18) is a locally Lipschitz perturbation of (2.19). Then from classical results in [5], we obtain a local solution defined in an interval [τ, tmax), and if tmax <∞, then lim t→t−max ‖y(t)‖H = +∞. (2.20) To show that F (y, t) is locally Lipschitz in y, for each t, let y1, y2 ∈ H, y1 = (ϕ1, ψ1, w1, ϕ1 t , ψ 1 t , w 1 t ), y2 = (ϕ2, ψ2, w2, ϕ2 t , ψ 2 t , w 2 t ). From (2.10), there exists a constant Cr > 0 depending of r = max{‖y1‖H , ‖y2‖H}, such that ∫ L 0 |fj(y1)− fj(y2)|2dx 6 Cr‖y1 − y2‖2H . Then ‖F(y1)−F(y2)‖2 6 3Cr‖y1 − y2‖2H , which shows that F is locally Lipschitz on H. To verify that the solution is global, that is, tmax = ∞, we consider the energy of system defined by (2.14). By density argument, we can assume that y is a strong solution. Then, of (2.15), we obtain d dt E(t) 6 −m1‖ϕt(t)‖2 −m2‖ψt(t)‖2 −m3‖wt(t)‖2 + ‖h1(t)‖2 m1 + ‖h2(t)‖2 m2 + ‖h3(t)‖2 m3 + m1 2 ‖ϕt(t)‖2 + m2 2 ‖ψt(t)‖2 + m3 2 ‖wt(t)‖2 6 ‖h1(t)‖2 m1 + ‖h2(t)‖2 m2 + ‖h3(t)‖2 m3 . (2.21) It follows that E(t) 6 E(τ) + 1 min{m1,m2,m3} ∫ t τ ( ‖h1(s)‖2 + ‖h2(s)‖2 + ‖h3(s)‖2 ) ds, ∀t > τ. By Lemma 2.12 we have that there exists a constant C > 0, independent of t, such that ‖y(t)‖2H 6 CE(t) + C, ∀t > τ. Consequently, ‖y(t)‖H <∞, t > τ , which shows that tmax =∞. Finally, using (2.9) and (2.11) we can check that for any solution of (1.1)-(1.5), y1 = (ϕ1, ψ1, w1, ϕ1 t , ψ 1 t , w 1 t ), y2 = (ϕ2, ψ2, w2, ϕ2 t , ψ 2 t , w 2 t ), 8 R. S. TELES EJDE-2022/05 with corresponding initial data y1 τ , y2 τ ∈ H, respectively, there exists constant C > 0, such that ‖y1(t)− y2(t)‖2H 6 C‖y1 τ − y2 τ‖2H , ∀t ∈ [τ, T ], for all T > τ . This shows the continuous dependence of solutions on the initial data. � Theorem 2.13 shows that the solution operator U(t, τ) : H → H, given by U(t, τ)y = (ϕ(t), ψ(t), w(t), ϕt(t), ψt(t), wt(t)), t > τ defines a continuous evolution process. 3. Pullback attractor Our main result reads as follows. Theorem 3.1. If (2.6)-(2.12) hold, then the evolution process generated by the problem (1.1)-(1.5) possesses a pullback attractor A = {A(t)}t∈R in the phase space H. The proof of the above theorem will be completed after the following two lemmas. Lemma 3.2. The process generated by problem (1.1)-(1.5) possesses a pullback absorbing family. Proof. From (2.15), we have d dt E(t) = − ∫ L 0 ( g1(ϕt)ϕt + g2(ψt)ψt + g3(wt)wt ) dx + ∫ L 0 ( h1(t)ϕt(t) + h2(t)ψt(t) + h3(t)wt(t) ) dx. (3.1) We define the perturbed energy Eα(t) = E(t) + αΦ(t) (α > 0), (3.2) where Φ(t) = ρ1 ∫ L 0 ϕ(t)ϕt(t)dx+ ρ2 ∫ L 0 ψ(t)ψt(t)dx+ ρ1 ∫ L 0 w(t)wt(t)dx. (3.3) Using Holder’s and Young’s inequalities, (2.1) and (2.3), we can estimate (3.3): |Φ(t)| 6 1 2 max { ρ1, ρ2, ρ1 L π , ρ2 L π } γ2‖y‖2H. (3.4) Using Lemma 2.12 and choosing α0 = (max{ρ1, ρ2, ρ1 L π , ρ2 L π }) −1 β0 2 γ −1 2 , we obtain α|Φ(t)| 6 α0|Φ(t)| 6 1 2 (E(t) + LMF ), since α 6 α0. So that Eα(t) = E + αΦ(t) 6 3 2 E(t) + L 2 MF , (3.5) Eα(t) = E + αΦ(t) > 1 2 E(t)− L 2 MF , (3.6) for each α ∈ [0, α0]. EJDE-2022/05 NON-AUTONOMOUS BRESSE SYSTEMS 9 Now we estimating Φ′(t), Φ′(t) = ρ1‖ϕt‖2 + ρ2‖ψt‖2 + ρ1‖wt‖2 + ∫ L 0 ϕ [ h1 + k(ϕx + ψ + lw)x + k0l(wx − lϕ)− g1(ϕt)− f1 ] dx + ∫ L 0 ψ [ h2 + bψxx − k(ϕx + ψ + lw)− g2(ψt)− f2 ] dx + ∫ L 0 w [ h3 + k0(wx − lϕ)x − kl(ϕx + ψ + lw)− g3(wt)− f3 ] dx. (3.7) From Green’s Formula and using (2.6) in (3.7) we obtain Φ′(t) = ρ1‖ϕt‖2 + ρ2‖ψt‖2 + ρ1‖wt‖2 − k‖ϕx + ψ + w‖2 − k0‖wx − lϕ‖2 − b‖ψx‖2 − ∫ L 0 ϕg1(ϕt)dx− ∫ L 0 ψg2(ψt)dx− ∫ L 0 wg3(wt)dx − ∫ L 0 ∇F (ϕ,ψ,w) · (ϕ,ψ,w)dx+ ∫ L 0 ϕh1dx + ∫ L 0 ψh2dx+ ∫ L 0 wh3dx (3.8) Using E(t) in (3.8), we have Φ′(t) = −E(t)− 1 2 ( k‖ϕx + ψ + w‖2 + k0‖wx − lϕ‖2 + b‖ψx‖2 ) + 3 2 ρ1‖ϕt‖2 + 3 2 ρ2‖ψt‖2 + 3 2 ρ1‖wt‖2 + ∫ L 0 [ F (ϕ,ψ,w)−∇F (ϕ,ψ,w) · (ϕ,ψ,w) ] dx − ∫ L 0 ϕg1(ϕt)dx− ∫ L 0 ψg2(ψt)dx− ∫ L 0 wg3(wt)dx+ ∫ L 0 ϕh1dx + ∫ L 0 ψh2dx+ ∫ L 0 wh3dx (3.9) From (2.1), (2.4), (2.10), (2.11) and Holder’s and Young’s inequalities, we can write∫ L 0 [ F (ϕ,ψ,w)−∇F (ϕ,ψ,w) · (ϕ,ψ,w) ] dx 6 βL2γ3 π2 ( b‖ψx‖2 + k‖ϕx + ψ + lw‖2 + k0‖wx − lϕ‖2 ) +MF , (3.10) ∫ L 0 |ϕg1(ϕt) + ψg2(ψt) + wg3(wt)|dx 6 L2γ3 β0π2 ( max{M1,M2,M3} )2(‖ϕt‖2 + ‖ψt‖2 + ‖wt‖2 ) + β0 4 ( b‖ψx‖2 + k‖ϕx + ψ + lw‖2 + k0‖wx − lϕ‖2 ) , (3.11) ∫ L 0 |ϕh1 + ψh2 + wh3|dx 6 L2γ3 β0π2 ( ‖h1‖2 + ‖h2‖2 + ‖h3‖3 ) + β0 4 ( b‖ψx‖2 + k‖ϕx + ψ + lw‖2 + k0‖wx − lϕ‖2 ) . (3.12) 10 R. S. TELES EJDE-2022/05 Using (3.10)-(3.12) in (3.9), we find that Φ′(t) 6 −E(t) + [3ρ1 2 + L2γ3 β0π2 (max{M1,M2,M3})2 ] ‖ϕt‖2 + [3ρ2 2 + L2γ3 β0π2 (max{M1,M2,M3})2 ] ‖ψt‖2 + [3ρ1 2 + L2γ3 β0π2 (max{M1,M2,M3})2 ] ‖wt‖2 + L2γ3 β0π2 ( ‖h1‖2 + ‖h2‖2 + ‖h3‖2 ) +MF + [1 2 + βL2γ3 π2 + β0 2 ]( b‖ψx‖2 + k‖ϕx + ψ + lw‖2 + k0‖wx − lϕ‖2 ) . (3.13) As [ 1 2 + βL2γ3 π2 + β0 2 ] = 0, by (3.13) we obtain Φ′(t) 6 −E(t) + c1‖ϕt‖2 + c2‖ψt‖2 + c1‖wt‖2 + c3(‖h1‖2 + ‖h2‖2 + ‖h3‖2) +MF , (3.14) where c1 = [3ρ1 2 + L2γ3 β0π2 ( max{M1,M2,M3} )2] c2 = [3ρ2 2 + L2γ3 β0π2 ( max{M1,M2,M3} )2] c3 = L2γ3 β0π2 . From (2.15), (3.2), and (3.14), taking c4 = max{c1, c2}, we obtain d dt Eα(t) 6 −m1 2 ‖ϕt‖2 − m2 2 ‖ψt‖2 − m3 2 ‖wt‖2 + ( 2 min{m1,m2,m3} + αc3 )( ‖h1‖2 + ‖h2‖2 + ‖h3‖2 ) + αc4(‖ϕt‖2 + ‖ψt‖2 + ‖wt‖2) + αMF − αE(t). (3.15) Setting α = min{α0, 1, 1 c4 min{m1,m2,m3}} and using (3.5), we obtain d dt Eα(t) 6 −2 3 αEα(t) + c5 ( ‖h1‖2 + ‖h2‖2 + ‖h3‖2 ) + α (L+ 3 3 ) MF , (3.16) where c5 = 2 min{m1,m2,m3} + αc3. Using Gronwall’s inequality and (3.5), (3.6) in (3.16), E(t) 6 3E(τ)e−σ0(t−τ) + 2c5 ∫ t τ e−σ0(t−τ) ( ‖h1‖2 + ‖h2‖2 + ‖h3‖2 ) ds + (3L+ 3)MF , (3.17) with σ0 = 2α 3 . (3.18) From (2.12) we can rewrite (3.17) as E(t) 6 3E(τ)e−σ0(t−τ) + 2c5ch + (3L+ 3)MF . (3.19) EJDE-2022/05 NON-AUTONOMOUS BRESSE SYSTEMS 11 Using (2.17) and (3.19) we can write, for y ∈ H, that ‖U(t, τ)y‖2H 6 3 β0 [ kF ( 1 + ‖y‖p+1 H )] e−σ0(t−τ) + c β0 + LMF β0 , (3.20) where c = 2c5ch + (3L+ 3)MF . So, there exists a uniformly bounded pullback absorbing family B(t) = BH(0, R), with R2 > c β0 + LMF β0 , for all t ∈ R. � Lemma 3.3. The process generated by system (1.1)-(1.5) is pullback asymptotically compact. Proof. Let y = y1 − y2 = (ϕ,ψ,w, ϕt, ψt, wt) be a solution of the problem ρ1ϕtt − k(ϕx + ψ + lw)x − k0l(wx − lϕ) = − [ g1(ϕ1 t )− g1(ϕ2 t ) ] + [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ1, w1) ] , ρ2ψtt − bψxx + k(ϕx + ψ + lw) = − [ g2(ψ1 t )− g2(ψ2 t ) ] + [ f2(ϕ2, ψ2, w2)− f2(ϕ1, ψ1, w1) ] , ρ1wtt − k0(wx − lϕ)x + kl(ϕx + ψ + lw) = − [ g3(ϕ1 t )− g3(ϕ2 t ) ] + [ f3(ϕ2, ψ2, w2)− f3(ϕ1, ψ1, w1) ] , with initial condition (ϕ(τ), ψ(τ), w(τ), ϕt(τ), ψt(τ), wt(τ)) = y1 τ − y2 τ ( y1 τ , y2 τ ∈ B(t) ) . Multiplying the above equations by ϕt, ψt and wt, respectively, and integrating in [0, L], we obtain d dt G(t) = − ∫ L 0 [ g1(ϕ1 t )− g1(ϕ2 t ) ] ϕt − ∫ L 0 [ g2(ψ1 t )− g2(ψ2 t ) ] ψt − ∫ L 0 [ g3(w1 t )− g3(w2 t ) ] wt + ∫ L 0 [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ1, w1) ] ϕt + ∫ L 0 [ f2(ϕ2, ψ2, w2)− f2(ϕ1, ψ1, w1) ] ψt + ∫ L 0 [ f3(ϕ2, ψ2, w2)− f3(ϕ1, ψ1, w1) ] wt, (3.21) where G(t) = ρ1 2 ‖ϕt‖2 + ρ2 2 ‖ψt‖2 + ρ1 2 ‖wt‖2 + b 2 ‖ψx‖2 + k 2 ‖ϕx + ψ + lw‖2 + k0 2 ‖wx − lϕ‖2. We define Gη(t) = G(t) + ηΨ(t), η > 0, (3.22) with Ψ(t) = ρ1 ∫ L 0 ϕϕtdx+ ρ2 ∫ L 0 ψψtdx+ ρ1 ∫ L 0 wwtdx. We observe that |Ψ(t)| 6 γ2 max { ρ1, ρ2, ρ1 L π , ρ2 L π } G(t). 12 R. S. TELES EJDE-2022/05 Choosing η0 = 1 2γ −1 2 ( max{ρ1, ρ2, ρ1 L π , ρ2 L π } )−1 , since η ∈ [0, η0], we have 1 2 G(t) 6 Gη(t) 6 3 2 G(t) ∀t > τ. (3.23) Now we estimate Ψ′(t), Ψ′(t) = ρ1‖ϕt‖2 + ρ2‖ψt‖2 + ρ1‖wt‖2 − b‖ψx‖2 − k‖ϕx + ψ + lw‖2 − k0‖wx − lϕ‖2 − ∫ L 0 [ g1(ϕ1 t )− g1(ϕ2 t ) ] ϕ− ∫ L 0 [ g2(ψ1 t )− g2(ψ2 t ) ] ψ − ∫ L 0 [ g3(w1 t )− g3(w2 t ) ] w + ∫ L 0 [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ1, w1) ] ϕ + ∫ L 0 [ f2(ϕ2, ψ2, w2)− f2(ϕ1, ψ1, w1) ] ψ + ∫ L 0 [ f3(ϕ2, ψ2, w2)− f3(ϕ1, ψ1, w1) ] w = −G(t)− 1 2 ( k‖ϕx + ψ + lw‖2 + k0‖wx − lϕ‖2 + b‖ψx‖2 ) + 3ρ1 2 ‖ϕt‖2 + 3ρ2 2 ‖ψt‖2 + 3ρ1 2 ‖wt‖2 + ∫ L 0 [ g1(ϕ1 t )− g1(ϕ2 t ) ] ϕ − ∫ L 0 [ g2(ψ1 t )− g2(ψ2 t ) ] ψ − ∫ L 0 [ g3(w1 t )− g3(w2 t ) ] w + ∫ L 0 [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ1, w1) ] ϕ + ∫ L 0 [ f2(ϕ2, ψ2, w2)− f2(ϕ1, ψ1, w1) ] ψ + ∫ L 0 [ f3(ϕ2, ψ2, w2)− f3(ϕ1, ψ1, w1) ] w (3.24) We observe that − ∫ L 0 [ g1(ϕ1 t )− g1(ϕ2 t ) ] ϕt 6 −m1‖ϕt‖2, (3.25) − ∫ L 0 [ g2(ψ1 t )− g2(ψ2 t ) ] ψt 6 −m2‖ψt‖2, (3.26) − ∫ L 0 [ g3(ϕ1 t )− g3(w2 t ) ] wt 6 −m3‖wt‖2, (3.27) − ∫ L 0 [ g1(ϕ1 t )− g1(ϕ2 t ) ] ϕ 6 Lγ3M 2 1 2π ‖ϕt‖2 + π 2Lγ3 ‖ϕ‖2, (3.28) − ∫ L 0 [ g2(ψ1 t )− g1(ψ2 t ) ] ψ 6 Lγ3M 2 2 2π ‖ψt‖2 + π 2Lγ3 ‖ψ‖2, (3.29) − ∫ L 0 [ g1(w1 t )− g1(w2 t ) ] w 6 Lγ3M 2 3 2π ‖wt‖2 + π 2Lγ3 ‖w‖2. (3.30) EJDE-2022/05 NON-AUTONOMOUS BRESSE SYSTEMS 13 Replacing (3.24)-(3.30) in (3.22), we see that d dt Gη(t) 6 −ηG(t) + ( −m1 + 3ρ1 2 η + Lγ3M 2 1 2π η ) ‖ϕt‖2 + ( −m2 + 3ρ2 2 η + Lγ3M 2 2 2π η ) ‖ψt‖2 + ( −m3 + 3ρ1 2 η + Lγ3M 2 3 2π η ) ‖wt‖2 + ∫ L 0 [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ1, w1) ] ϕt + ∫ L 0 [ f2(ϕ2, ψ2, w2)− f2(ϕ1, ψ1, w1) ] ψt + ∫ L 0 [ f3(ϕ2, ψ2, w2)− f3(ϕ1, ψ1, w1) ] wt + ∫ L 0 [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ1, w1) ] ϕ + ∫ L 0 [ f2(ϕ2, ψ2, w2)− f2(ϕ1, ψ1, w1) ] ψ + ∫ L 0 [ f3(ϕ2, ψ2, w2)− f3(ϕ1, ψ1, w1) ] w (3.31) Taking η = min { η0, 1, 2πm1 3ρ1π + Lγ3M2 1 , 2πm2 3ρ2π + Lγ3M2 2 , 2πm3 3ρ1π + Lγ3M2 3 } , from (3.31) we have d dt Gη(t) 6 −ηG(t) + ∫ L 0 [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ1, w1) ] ϕt + ∫ L 0 [ f2(ϕ2, ψ2, w2)− f2(ϕ1, ψ1, w1) ] ψt + ∫ L 0 [ f3(ϕ2, ψ2, w2)− f3(ϕ1, ψ1, w1) ] wt + ∫ L 0 [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ1, w1) ] ϕ + ∫ L 0 [ f2(ϕ2, ψ2, w2)− f2(ϕ1, ψ1, w1) ] ψ + ∫ L 0 [ f3(ϕ2, ψ2, w2)− f3(ϕ1, ψ1, w1) ] w (3.32) 14 R. S. TELES EJDE-2022/05 Using (2.9), and Holder’s and Young’s inequalities, we can estimate the terms on the right-hand side of (3.32),∫ L 0 [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ1, w1) ] ϕ + ∫ L 0 [ f2(ϕ2, ψ2, w2)− f2(ϕ1, ψ1, w1) ] ψ + ∫ L 0 [ f3(ϕ2, ψ2, w2)− f3(ϕ1, ψ1, w1) ] w 6 C1 ( 1 + ‖y1‖p−1 H + ‖y2‖p−1 H )( ‖ϕ‖22p + ‖ψ‖22p + ‖w‖22p ) =: k1(τ, t) ( ‖ϕ‖22p + ‖ψ‖22p + ‖w‖22p ) , (3.33) for some constant C1 > 0; and∫ L 0 [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ1, w1) ] ϕt + ∫ L 0 [ f2(ϕ2, ψ2, w2)− f2(ϕ1, ψ1, w1) ] ψt + ∫ L 0 [ f3(ϕ2, ψ2, w2)− f3(ϕ1, ψ1, w1) ] wt 6 C2 ε ( 1 + ‖y1‖2(p−1) H + ‖y2‖2(p−1) H )( ‖ϕ‖22p + ‖ψ‖22p + ‖w‖22p ) + ε (ρ1 2 ‖ϕt‖2 + ρ2 2 ‖ψt‖2 + ρ1 2 ‖wt‖2 ) =: k2(τ, t) ( ‖ϕ‖22p + ‖ψ‖22p + ‖w‖22p ) + ε (ρ1 2 ‖ϕt‖2 + ρ2 2 ‖ψt‖2 + ρ1 2 ‖wt‖2 ) , (3.34) for some constants C2 > 0 and ε > 0. Replacing (3.23), (3.33) and (3.34) in (3.32), we obtain d dt Gη(t) 6 ( − 2 3 η + 2ε ) Gη(t) + k(τ, t) ( ‖ϕ‖22p + ‖ψ‖22p + ‖w‖22p ) , (3.35) with k(τ, t) = max{k1(τ, t), k2(τ, t)}. Now, we take ε > 0 sufficiently small such that σ1 = − 2 3η+ 2ε < 0 and σ0 6 σ1, where σ0 was given in (3.18). Using Gronwall’s inequality in (3.35)andfrom(3.23), we infer that G(t) 6 3G(τ)e−σ1(t−τ) + 2 sup s∈[τ,t] k(τ, s) ∫ t τ e−σ1(t−s)(‖ϕ‖2p + ‖ψ‖2p + ‖w‖2p ) ds As G(t) = 1 2‖U(t, τ)y1 τ − U(t, τ)y2 τ‖2H and e−σ(t−s) < 1, for each s ∈ [τ, t], since y1 τ , y2 τ ∈ B = B(t) ⊂ BH(0, R) (defined in the Lemma 3.2), we have ‖U(t, τ)y1 τ − U(t, τ)y2 τ‖2H 6 3‖y1 τ − y2 τ‖2He−σ1(t−τ) + 4 sup s∈[τ,t] k(τ, s) ∫ t τ ( ‖ϕ‖2p + ‖ψ‖2p + ‖w‖2p ) ds 6 3R2e−σ1(t−τ) + 4 sup s∈[τ,t] k(τ, s) ∫ t τ ( ‖ϕ‖2p + ‖ψ‖2p + ‖w‖2p ) ds. (3.36) EJDE-2022/05 NON-AUTONOMOUS BRESSE SYSTEMS 15 Let t ∈ R and ε > 0. Then there exists τε = τ(t, B, ε) such that 3R2e−σ1(t−τε) < ε2. We defining Ψε on B(τε) by Ψε(z1, z2) = 2 ( Ct,τε ∫ t τε ( ‖ϕ‖22p + ‖ψ‖22p + ‖w‖22p ))1/2 . So, we can rewrite (3.36) as ‖U(t, τε)y − U(t, τε)z‖2H 6 ε+ Ψε(y, z) for any y, z ∈ B(τε). Let us prove that Ψε is a contractive function. In fact, let {zn} be a sequence in B(τε). We have ‖U(s, τε)zn‖H 6 R <∞ for any s ∈ [τε, t]. So, since U(s, τε)zn = (ϕn, ψn, wn, ϕnt , ψ n t , w n t ), we obtain (ϕn, ψn, wn) is bounded in L2(τε, t;H 1 0 (Ω)3), (ϕnt , ψ n t , w n t ) is bounded in L2(τε, t;L 2(Ω)3) From the embedding H1 0 (0, L) c ↪→ L2p(0, L) ↪→ L2(0, L) there exist a function z and a subsequence {znk} such that znk → z strongly in L2(τε, t;L 2p(Ω)3). Then, Ψε is contractive on B(τε). Consequently, the process generated by (1.1)- (1.5) is pullback asymptotically compact. � 4. Upper semicontinuity We consider (1.1)-(1.5), with hi(x, t) replaced by εhi(x, t), with i = 1, 2, 3; that is, ρ1ϕtt − k(ϕx + ψ + lw)x − k0l(wx − lϕ) + g1(ϕt) + f1(ϕ,ψ,w) = εh1(x, t), ρ2ψtt − bψxx + k(ϕx + ψ + lw) + g2(ψt) + f2(ϕ,ψ,w) = εh2(x, t), ρ1wtt − k0(wx − lϕ)x + kl(ϕx + ψ + lw) + g3(wt) + f3(ϕ,ψ,w) = εh3(x, t), As we will take ε→ 0, we can assume without loss of generality, that ε ∈ [0, 1]. The same procedure used before yields an evolution process Uε(t, τ) : H → H, for each ε ∈ [0, 1]. Noting that for ε = 0, the above problem becomes autonomous and has a C0 semigroup associated S(t) on H. As proved in [13], S(t) admits a global attractor A0. Furthermore, we can see that S(t) is a evolution process defined by U0(t, τ) = S(t − τ). Then the constant family A0 = {A0}, for any t ∈ R, is the pullback attractor by the process U0(t, τ). As the R from the absorbing ball, given in the Lemma 3.2, is independent of ε and t, we have {Bε(t)} with Bε(t) = B(0, R) is a absorbing family by the process Uε(t, τ) that absorbs all bounded sets in H. Theorem 4.1. The pullback attractors family Aε is upper-semicontinuous as ε→ 0, that is, lim ε→0 dist(Aε(t), A0) = 0, ∀t ∈ R. 16 R. S. TELES EJDE-2022/05 Proof. From Lemma 3.2 there exists a uniformly bounded family B0(t) = B(0, R). From the invariance of the attractor, we have that Aε(t) ⊂ B(0, R), for all t ∈ R and ε ∈ [0, 1]. And this prove (ii) from the Preposition 2.10. Given z ∈ D (D ⊂ H bounded) and τ 6 t, let Uε(t, τ)z = (ϕ1, ψ1, w1, ϕ1 t , ψ 1 t , w 1 t ) = u1, U0(t, τ)z = (ϕ2, ψ2, w2, ϕ2 t , ψ 2 t , w 2 t ) = u2. Then, we can see that u = u1 − u2 solves the problem ρ1ϕtt − k(ϕx + ψ + lw)x − k0l(wx − lϕ) = − [ g1(ϕ1 t )− g1(ϕ2 t ) ] + [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ2, w1) ] + εh1, ρ2ψtt − bψxx + k(ϕx + ψ + lw) = − [ g2(ψ1 t )− g2(ψ2 t ) ] + [ f2(ϕ2, ψ2, w2)− f2(ϕ1, ψ2, w1) ] + εh2, ρ1wtt − k0(wx − lϕ)x + kl(ϕx + ψ + lw) = − [ g3(w1 t )− g3(w2 t ) ] + [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ2, w1) ] + εh3, with null initial conditions. Multiplying the first equation by ϕt, the second equa- tion by ψt, and the third equation by wt, we obtain d dt [ ρ1‖ϕt‖2 + ρ2‖ψ‖2 + ρ1‖wt‖2 + b‖ψx‖2 + k‖ϕx + ψ + lw‖2 + k0‖wx − lϕ‖2 ] = −2 ∫ L 0 [ g1(ϕ1 t )− g1(ϕ2 t ) ] ϕtdx− 2 ∫ L 0 [ g2(ψ1 t )− g1(ψ2 t ) ] ψtdx − 2 ∫ L 0 [ g3(w1 t )− g3(w2 t ) ] wtdx+ 2 ∫ L 0 [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ1, w1) ] ϕt + 2 ∫ L 0 [ f2(ϕ2, ψ2, w2)− f2(ϕ1, ψ1, w1) ] ψt + 2 ∫ L 0 [ f3(ϕ2, ψ2, w2)− f3(ϕ1, ψ1, w1) ] wt + 2ε ∫ L 0 [ h1ϕt + h2ψt + h2wt ] dx. (4.1) Estimating the terms on the right-hand side of (4.1), we obtain −2 ∫ L 0 [ g1(ϕ1 t )− g1(ϕ2 t ) ] ϕtdx 6 −2m1‖ϕt‖2 6 0, −2 ∫ L 0 [ g2(ψ1 t )− g1(ψ2 t ) ] ψtdx 6 −2m2‖ψt‖2 6 0, −2 ∫ L 0 [ g3(w1 t )− g3(w2 t ) ] wtdx 6 −2m3‖wt‖2 6 0, EJDE-2022/05 NON-AUTONOMOUS BRESSE SYSTEMS 17 2 ∫ L 0 [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ1, w1) ] ϕt + 2 ∫ L 0 [ f2(ϕ2, ψ2, w2)− f2(ϕ1, ψ1, w1) ] ψt + 2 ∫ L 0 [ f3(ϕ2, ψ2, w2)− f3(ϕ1, ψ1, w1) ] wt 6 C ( 1 + ‖u1‖2(p−1) H + ‖u2‖2(p−1) H )( ‖ϕx‖2 + ‖ψx‖2 + ‖wx‖2 ) + 1 2 (‖ϕt‖2 + ‖ψt‖2 + ‖wt‖2) for some constant C > 0. Also 2ε ∫ L 0 [ h1ϕt + h2ψt + h2wt ] dx 6 2ε2 ( ‖h1‖2 + ‖h2‖2 + ‖h3‖2 ) + 1 2 ( ‖ϕt‖2 + ‖ψt‖2 + ‖wt‖2 ) . Replacing this estimates in (4.1), we can write d dt [ ρ1‖ϕt‖2 + ρ2‖ψ‖2 + ρ1‖wt‖2 + b‖ψx‖2 + k‖ϕx + ψ + lw‖2 + k0‖wx − lϕ‖2 ] 6 C(t, T,D) [ ρ1‖ϕt‖2 + ρ2‖ψ‖2 + ρ1‖wt‖2 + b‖ψx‖2 + k‖ϕx + ψ + lw‖2 + k0‖wx − lϕ‖2 ] + 2ε2 [ ‖h1‖2 + ‖h2‖2 + ‖h3‖2 ] , (4.2) where C(t, T,D) depends on t, T and D. Applying Gronwall’s inequality in (4.2), we obtain ‖u1 − u2‖2H 6 2ε2 ∫ t τ eC(t,T,D)(t−s)[‖h1(s)‖2 + ‖h2(s)‖2 + ‖h3(s)‖2 ] ds and, thus ‖Uε(t, τ)z − U0(t, τ)z‖2H 6 2ε2 ∫ t t−T eC(t,T,D)(t−s)[‖h1(s)‖2 + ‖h2(s)‖2 + ‖h3(s)‖2 ] ds, for all τ ∈ [t− T, t] and z ∈ D. As h1, h2, and h3 are locally integrable, we verify (iii) of Preposition 2.10. Consequently, we prove the upper semicontinuity of Aε when ε→ 0. � 5. Continuity of attractors Our last result establishes the continuity of pullback attractors with respect to some parameter. Theorem 5.1. Let t ∈ R. In the context of Theorem 3.1 there exists a set J dense in [0, 1] such that Aε is continuous with respect to any parameter ε0 ∈ J , that is, lim ε→ε0 dH(Aε,Aε0) = 0 ∀ε0 ∈ J. 18 R. S. TELES EJDE-2022/05 Proof. We can apply Theorem 2.11. Assumptions (i) and (ii) of Theorem 2.11 hold because of Theorem 3.1 and Lemma 3.2. Let D ⊂ H bounded. Given ε1, ε2 ∈ [0, 1], z ∈ D and τ 6 t, let us denote Uε1(t, τ)z = (ϕ1, ψ1, w1, ϕ1 t , ψ 1 t , w 1 t ) = u1, Uε2(t, τ)z = (ϕ2, ψ2, w2, ϕ2 t , ψ 2 t , w 2 t ) = u2. Then, u = u1 − u2 is solution of ρ1ϕtt − k(ϕx + ψ + lw)x − k0l(wx − lϕ) = − [ g1(ϕ1 t )− g1(ϕ2 t ) ] + [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ2, w1) ] + (ε1 − ε2)h1 ρ2ψtt − bψxx + k(ϕx + ψ + lw) = − [ g2(ψ1 t )− g2(ψ2 t ) ] + [ f2(ϕ2, ψ2, w2)− f2(ϕ1, ψ2, w1) ] + (ε1 − ε2)h2 ρ1wtt − k0(wx − lϕ)x + kl(ϕx + ψ + lw) = − [ g3(w1 t )− g3(w2 t ) ] + [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ2, w1) ] + (ε1 − ε2)h3 Multiplying the first equation by ϕt, the second equation by ψt, and the third equation by wt, we can write d dt [ ρ1‖ϕt‖2 + ρ2‖ψ‖2 + ρ1‖wt‖2 + b‖ψx‖2 + k‖ϕx + ψ + lw‖2 + k0‖wx − lϕ‖2 ] = −2 ∫ L 0 [ g1(ϕ1 t )− g1(ϕ2 t ) ] ϕtdx− 2 ∫ L 0 [ g2(ψ1 t )− g1(ψ2 t ) ] ψtdx − 2 ∫ L 0 [ g3(w1 t )− g3(w2 t ) ] wtdx+ 2 ∫ L 0 [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ1, w1) ] ϕt + 2 ∫ L 0 [ f2(ϕ2, ψ2, w2)− f2(ϕ1, ψ1, w1) ] ψt + 2 ∫ L 0 [ f3(ϕ2, ψ2, w2)− f3(ϕ1, ψ1, w1) ] wt + 2(ε1 − 2ε2) ∫ L 0 [ h1ϕt + h2ψt + h2wt ] dx. (5.1) Estimating the terms on the right-hand side of (5.1) we have −2 ∫ L 0 [ g1(ϕ1 t )− g1(ϕ2 t ) ] ϕtdx 6 −2m1‖ϕt‖2 6 0, −2 ∫ L 0 [ g2(ψ1 t )− g1(ψ2 t ) ] ψtdx 6 −2m2‖ψt‖2 6 0, −2 ∫ L 0 [ g3(w1 t )− g3(w2 t ) ] wtdx 6 −2m3‖wt‖2 6 0, EJDE-2022/05 NON-AUTONOMOUS BRESSE SYSTEMS 19 2 ∫ L 0 [ f1(ϕ2, ψ2, w2)− f1(ϕ1, ψ1, w1) ] ϕt + 2 ∫ L 0 [ f2(ϕ2, ψ2, w2)− f2(ϕ1, ψ1, w1) ] ψt + 2 ∫ L 0 [ f3(ϕ2, ψ2, w2)− f3(ϕ1, ψ1, w1) ] wt 6 C ( 1 + ‖u1‖2(p−1) H + ‖u2‖2(p−1) H )( ‖ϕx‖2 + ‖ψx‖2 + ‖wx‖2 ) + 1 2 ( ‖ϕt‖2 + ‖ψt‖2 + ‖wt‖2 ) for some constant C > 0. Also 2(ε1 − ε2) ∫ L 0 [ h1ϕt + h2ψt + h2wt ] dx 6 2|ε1 − ε2|2 ( ‖h1‖2 + ‖h2‖2 + ‖h3‖2 ) + 1 2 ( ‖ϕt‖2 + ‖ψt‖2 + ‖wt‖2 ) . Replacing these estimates in (5.1), we obtain d dt [ ρ1‖ϕt‖2 + ρ2‖ψ‖2 + ρ1‖wt‖2 + b‖ψx‖2 + k‖ϕx + ψ + lw‖2 + k0‖wx − lϕ‖2 ] 6 C(t, T,D) [ ρ1‖ϕt‖2 + ρ2‖ψ‖2 + ρ1‖wt‖2 + b‖ψx‖2 + k‖ϕx + ψ + lw‖2 + k0‖wx − lϕ‖2 ] + 2|ε1 − ε2|2 [ ‖h1‖2 + ‖h2‖2 + ‖h3‖2 ] , (5.2) where C(t, T,D) depends on t, T and D. Applying Gronwall’s inequality in (5.2), we obtain ‖u1 − u2‖2H 6 2|ε1 − ε2|2 ∫ t τ eC(t,T,D)(t−s)[‖h1(s)‖2 + ‖h2(s)‖2 + ‖h3(s)‖2 ] ds and, therefore ‖Uε(t, τ)z − U0(t, τ)z‖2H 6 2|ε1 − ε2|2 ∫ t t−T eC(t,T,D)(t−s)[‖h1(s)‖2 + ‖h2(s)‖2 + ‖h3(s)‖2 ] ds, for all τ ∈ [t − T, t] and z ∈ D. As h1, h2, and h3 are locally integrable, from (ii) of Theorem 2.11. This completes the proof. � References [1] Alabau Boussouira, F.; Muñoz Rivera, J. E.; Almeida Júnior, D. S.; Stability to weak dissi- pative Bresse system, J. Math. Anal. Appl., 374 (2011), 481-498. [2] Alves, M. O.; Fatori, L. H.; Jorge Silva, M. A.; Monteiro, R. N.; Stability and optimality of decay rate for a weakly dissipative Bresse system, Math. Methods Appl. Sci., 38 (2015), 898-908. [3] Araújo, R. O.; Marinho, S. S.; Prates Filho, J. S.; Uniform stability of a non-autonomous semilinear Bresse system, Applied Mathematics and Computation, 387 (2020), 124418. [4] Babin, A. V.; Vishik, M. 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[16] Temam, R.; Infinite dimensional dynamical systems in mechanics and physics, Springer- Verlag, New York, 1997. Ricardo de S’a Teles Department of Mathematics, Federal University of Technology (UTFPR), Londrina- PR, Brazil Email address: ricardo.sa.teles@gmail.com 1. Introduction 2. Preliminaries 2.1. Assumptions 2.2. Energy of the system 2.3. Well-possedness 3. Pullback attractor 4. Upper semicontinuity 5. Continuity of attractors References