Quasi-Stability and Continuity of Attractors for Nonlinear System of Wave Equations

Nonauton. Dyn. Syst. 2021; 8:27–45

Research Article Open Access

M. M. Freitas*, M. J. Dos Santos, A. J. A. Ramos, M. S. Vinhote, and M. L. Santos Quasi-stability and continuity of attractors for nonlinear system of wave equations https://doi.org/10.1515/msds-2020-0125 Received November 22, 2020; accepted March 8, 2021 Abstract: In this paper, we study the long-time behavior of a nonlinear coupled system of wave equations with damping terms and subjected to small perturbations of autonomous external forces. Using the recent approach by Chueshov and Lasiecka in [21], we prove that this dynamical system is quasi-stable by establishing a quasistability estimate, as consequence, the existence of global and exponential attractors is proved. Finally, we investigate the upper and lower semicontinuity of global attractors under autonomous perturbations.

Keywords: Wave equations; quasi-stable systems; global attractor; exponential attractor; continuity of attractors

MSC: 35B40; 35B41; 35L05; 35L75

1 Introduction

In this paper, we are interested in the long-time behavior of the coupled system of wave equations given by

 α1 + utt − ∆u + (−∆) ut + g1(ut) = f1(u, v) + ϵh1, in Ω × R ,   α2 + vtt − ∆v + (−∆) vt + g2(vt) = f2(u, v) + ϵh2, in Ω × R ,  + u = v = 0, on ∂Ω × R , (1.1)  u(0) = u , u (0) = u , in Ω,  0 t 1  v(0) = v0, vt(0) = v1, in Ω.

2 Here Ω ⊂ R is a bounded domain with a smooth boundary ∂Ω, α1, α2 ∈ (0, 1) and ϵ ∈ [0, 1] . The nonlinear- α1 α2 ities f1(u, v), f2(u, v) are with supercritical exponents representing strong sources, (−∆) ut and (−∆) vt are the fractional damping, while g1(ut) and g2(vt) act as nonlinear damping. The terms ϵh1, ϵh2 are autonomous perturbations of external forces. In the literature there is a large number of work with damping and source terms see, e.g., [1, 2, 24, 32, 39– 42] for problems with subcritical or critical sources, and [9–12, 22, 23, 26–28, 43] for problems with supercritical sources and with fractional damping see, e.g., [3, 7, 13–19, 34, 38, 46] and references therein.

*Corresponding Author: M. M. Freitas: Faculty of Mathematics, Federal University of Pará, Raimundo Santana Cruz Street, S/N, 68721-000, Salinópolis, Pará, Brazil, E-mail: [email protected] M. J. Dos Santos: Faculty of Exact Sciences and Technology, Federal University of Pará, Manoel de Abreu St, Abaetetuba, PA, 68440-000, Brazil, E-mail: [email protected] A. J. A. Ramos: Faculty of Mathematics, Federal University of Pará, Raimundo Santana Cruz Street, S/N, 68721-000, Salinópolis, Pará, Brazil, E-mail: [email protected] M. S. Vinhote: PhD Program in Mathematics, Federal University of Pará, Augusto Corrêa Street, 01, 66075-110, Belém, Pará, Brazil, E-mail: [email protected] M. L. Santos: PhD Program in Mathematics, Federal University of Pará, Augusto Corrêa Street, 01, 66075-110, Belém, Pará, Brazil, E-mail: [email protected]

Open Access. © 2021 M. M. Freitas et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. 28 Ë M. M. Freitas, M. J. Dos Santos, A. J. A. Ramos, M. S. Vinhote, and M. L. Santos

In [28], Guo et al. studied the local and global well-posedness of the coupled nonlinear wave equations ( utt − ∆u + g (ut) = f (u, v), in Ω × (0, T), 1 1 (1.2) vtt − ∆v + g2(vt) = f2(u, v), in Ω × (0, T),

n in a bounded domain Ω ⊂ R with a nonlinear Robin boundary condition on u and a zero boundary conditions on v. By employing nonlinear semigroups and the theory of monotone operators, the authors obtained several results on the existence of local and global weak solutions, and uniqueness of weak solutions. More- over, they proved that such unique solutions depend continuously on the initial data. Under some restrictions on the parameters, they also proved that every weak solution to our system blows up in nite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. The present system (1.1) is obtained by the model (1.2) by adding the fractional dissipations α1 α2 (−∆) ut , (−∆) vt and considering small perturbations of autonomous external forces ϵh1, ϵh2. Our main interest in this paper is to study the long-time behavior of the autonomous dynamical system generated by the nonlinear coupled system of wave equations (1.1). In this context, the concept of global attractor is a use- ful objective to learn the dynamical behavior of a dynamical system [5, 8, 21, 29, 33, 44, 45]. By using the quasi-stability theory of Chueshov and Lasiecka [20, 21], we prove that the dynamical system generated by the problem (1.1) possesses a compact global attractor with nite fractal dimension. We also prove the regularity of solutions on the global attractor. Moreover, we obtain the existence of a generalized exponential attractor with fractal dimension nite in extended spaces. We also established the stability of global attractors on the perturbation of the parameter ϵ. More precisely, we use the recent theory in [31] to prove that there exists a set I* dense in [0, 1] such that the family of global attractors {Aϵ}ϵ∈[0,1] associated to problem (1.1) converges upper and lower-semicontinuously to the corresponding global attractor associated with the limit problem when the parameter ϵ → ϵ0 for all ϵ0 ∈ I*. Moreover, the upper semicontinuity for all ϵ ∈ [0, 1] is analyzed. The main contributions of this paper are: (i) We consider the system with fractional and nonlinear dissipation acting on the same equation. Here, p−1 we assume the nonlinear damping terms with polynomial growth to include functions of type gi(u) = |u| u. (ii) Instead of showing the existence of an absorbing set, we prove the system is gradient, and hence obtain the existence of a global attractor, which is characterized as unstable manifold of the set of stationary solutions. (iii) The quasi-stability of the system is obtained by establishing a quasistability estimate and therefore obtain the nite dimensionality and smoothness of the global attractor and exponential attractor. (iv) We investigate the continuity of the family of global attractors under autonomous perturbations. In- deed, we prove that the family of global attractors indexed by ϵ converges upper and lower-semicontinuously to the attractor associated with the limiting problem when ϵ → ϵ0 on a residual dense set I* ⊂ [0, 1] in the same sense proposed in Hoang et al. [31]. The upper semicontinuity or all ϵ ∈ [0, 1] is proved. This paper is organized as follows. In section 2, we introduce some notations and preliminary results. We also give the well-posedness of the system without proof in this section. Section 3 is devoted to proving the existence of global attractors and exponential attractors. The arguments are based on the methods developed by Chueshov and Lasiecka [20, 21]. The continuity of global attractors will be proved in section 4.

2 Preliminaries

In this section, we present preliminaries including notations and assumptions. Quasi-stability and continuity of attractors for nonlinear system of wave equations Ë 29

2.1 Notations and assumptions

The following notations will be used for the rest of the paper.

kukp = kukLp(Ω), p ≥ 1, (u, v) = (u, v)L2(Ω).

In this work, we consider the Hilbert space

1 1 2 2 H = H0(Ω) × H0(Ω) × L (Ω) × L (Ω), (2.3) with the following inner product and norm

˜ ˜ ∇ ∇ ∇ ∇ (2.4) U, U H = (ϕ, ϕ) + (φ, φ˜) + ( u, u˜) + ( v, ˜v) and 2 2 2 2 2 kUkH = kϕk2 + kφk2 + k∇uk2 + k∇vk2 (2.5) for any U = (u, v, ϕ, φ) and U˜ = (u˜, ˜v, ϕ˜ , φ˜) in H.

Assumption 2.1. We assume that 2 (i) The external forces h1, h2 ∈ L (Ω). 2 2 (ii) There is a function F ∈ C (R ) such that ∇F = (f1, f2) (2.6) and there exist p ≥ 1 and C > 0 such that

p−1 p−1 |∇fi(u, v)| ≤ C(1 + |u| + |v| ), ∀u, v ∈ R. (2.7)

(iii) There is β0 > 0 and mF > 0 so that

2 2 F(u, v) ≤ β0(|u| + |v| ) + mF , ∀u, v ∈ R, (2.8)

λ1 where 0 ≤ β0 < 2 and λ1 > 0 denotes the Poincaré’s constant. Moreover 2 2 ∇F(u, v) · (u, v) − F(u, v) ≤ β0(|u| + |v| ) + mF , ∀u, v ∈ R. (2.9)

(iv) Concerning to the nonlinear damping gi, we assume that

1 gi ∈ C (R), gi(0) = 0, (2.10)

and there exists constants m, M1, q ≥ 1, such that

′ q−1 m ≤ gi(u) ≤ M1(1 + |u| ), ∀u ∈ R, (2.11)

and, if q ≥ 3, there exist l > q − 1 and M2 > 0 such that

l gi(u)u ≥ M2|u| , |u| ≥ 1. (2.12)

Remark 2.1. Observe that assumption (2.11) implies the monotonicity property, that is,

2 (gi(u) − gi(v))(u − v) ≥ m|u − v| , ∀u, v ∈ R. (2.13)

Remark 2.2. An simple example of the function F in Assumption 2.1 can be

4 2 2 F(u, v) = −|u + v| + |u + v| − c1|uv| , c1 > 0.

In this case, we have ∂F f (u, v) = = −4(u + v)3 + 2(u + v) − 2c uv2, 1 ∂u 1 ∂F f (u, v) = = −4(u + v)3 + 2(u + v) − 2c u2v. 2 ∂v 1 30 Ë M. M. Freitas, M. J. Dos Santos, A. J. A. Ramos, M. S. Vinhote, and M. L. Santos

Since 1 F(u, v) ≤ max{−ξ 4 + ξ 2} = , ξ∈R 4 1 the assumptions (2.7)-(2.8) hold with mF = 4 and p = 3. Noting that 1 ∇F(u, v) · (u, v) − F(u, v) ≤ −3|u + v|4 + |u + v|2 ≤ ≤ m , 12 F then (2.9) also holds.

2.2 Energy identities

The total energy of system (1.1) is dened by as 1 1 E(t) = k(u, v, u , v )k2 = ku k2 + kv k2 + k∇uk2 + k∇vk2, (2.14) 2 t t H 2 t 2 t 2 2 2 and we also dene the total energy by Z Z Eϵ(t) = E(t) − F(u, v) dx − ϵ (h1u + h2v)dx. (2.15) Ω Ω Then we have the following lemma.

Lemma 2.1. The total energy given in (2.15) satises

d α1 2 α2 2 2 2 E (t) ≤ −k(−∆) 2 u k + k(−∆) 2 v k − m(ku k + kv k ). (2.16) dt ϵ t 2 t 2 t 2 t 2

Moreover, there exists a positive constant C0 such that

2 p+1 C0k(u, v, ut , vt)kH − CF ≤ Eϵ(t) ≤ CF(1 + k(u, v, ut , vt)kH ), ∀ϵ ∈ [0, 1]. (2.17)

Proof. Multiplying (formally) the rst equation in (1.1) by ut, and second by vt, respectively and using integration by parts, we can easily get (2.16). It follows from (2.8) and Poincaré’s inequality that Z 2 2 F(u, v)dx ≤ β0(kuk2 + kvk2) + mF|Ω| Ω β0 2 2 ≤ (k∇uk2 + k∇vk2) + mF|Ω| λ1 β0 2 ≤ k(u, v, ut , vt)kH + mF|Ω|, λ1 and thus Z 1 β0 2 Eϵ(t) ≥ − k(u, v, ut , vt)kH − mF|Ω| − ϵ (h1u + h2v)dx. 2 λ1 Ω Letting 1 2β0 C0 = 1 − > 0, (2.18) 4 λ1 and using the estimate Z 2 2 1 2 2 ϵ (h1u + h2v) dx ≤ C0λ1 kuk2 + kvk2 + kh1k2 + kh2k2 , 4C0λ1 (2.19) Ω we obtain the rst inequality in (2.17) with

1 2 2 CF = mF|Ω| + kh1k2 + kh2k2 . 4C0λ1 Using (2.7) we can see that the second inequality in (2.17) holds. The proof is complete. Quasi-stability and continuity of attractors for nonlinear system of wave equations Ë 31

2.3 Well-posedness

Let us write the problem (1.1) as an equivalent Cauchy problem  dU  + AU = F(U), dt (2.20) U(0) = U0 = (u0, v0, u1, v1) ∈ H, where

U(t) = (u(t), v(t), ϕ(t), φ(t)) ∈ H, ϕ = ut , φ = vt , and H dened in (2.3) and A : D(A) ⊂ H → H is the nonlinear operator dened by  −ϕ   −φ  U =   . (2.21) A  α1   −∆u + (−∆) ϕ + g1(ϕ)  α2 −∆v + (−∆) φ + g2(φ)

The domain of A is given by 1 1 D(A) = U = (u, v, ϕ, φ) ∈ H : ϕ ∈ H0(Ω), φ ∈ H0(Ω), α1 2 − ∆u + (−∆) ϕ + g1(ϕ) ∈ L (Ω),

α2 2 − ∆v + (−∆) φ + g2(φ) ∈ L (Ω) .

The forcing terms are represented by a nonlinear function F : H → H dened by  0     0  F(U) =   .  f1(u, v) + ϵh1  f2(u, v) + ϵh2

We have the following global existence result for the problem (1.1).

Theorem 2.2. Suppose that Assumption 2.1 holds, then we have: (i) If initial data U0 ∈ H, then problem (2.20) has a unique mild solution U(t) ∈ C([0, ∞), H) with U(0) = U0 given by t Z At (t−τ)A U(t) = e U0 + e F(U(τ))dτ. 0

(ii) If U0 ∈ D(A), then the above mild solution is a strong solution. 1 2 (iii) If U (t) and U (t) are two mild solutions of problem (2.20) then there exists a positive constant C0 = C(U1(0), U2(0)), such that

1 2 C0 T 1 2 kU (t) − U (t)kH ≤ e kU (0) − U (0)kH, 0 ≤ t ≤ T. (2.22)

Proof. It is easy to see that the operator A is a maximal monotone operator. In addition, by (2.7) we see that F is a locally Lipschitz on H. Therefore, applying the theory of maximal nonlinear monotone operators (see e.g. [6, 21]) items (i)-(ii) are concluded. The continuous dependence (iii) is also obtained by using standard computations in the dierence of solutions.

3 Long-time dynamics

On account of Theorem 2.2, we can dene the one-parameter family of operators Sϵ(t): H → H by

Sϵ(t)(u0, v0, u1, v1) = (u(t), v(t), ut(t), vt(t)), t ≥ 0, (3.23) 32 Ë M. M. Freitas, M. J. Dos Santos, A. J. A. Ramos, M. S. Vinhote, and M. L. Santos

where (u, v, ut , vt) is the unique solution of problem (1.1). Thus, the pair (H, Sϵ(t)) constitutes a dynamical system that will describe the long-time behavior of problem (1.1). The main result for long-time dynamics is given in the following theorem.

Theorem 3.1. Under the assumptions of Theorem 2.2, we have: (i) The dynamical system (H, Sϵ(t)) given in (3.23) is quasi-stable on any bounded positively invariant set B ⊂ H. (ii) The dynamical system (H, Sϵ(t)) possesses a unique compact global attractor Aϵ ⊂ H, which is characterized by the unstable manifold Aϵ = M+(Nϵ) of the set of stationary solutions ( )

−∆u = f1(u, v) + ϵh1 Nϵ = (u, v, 0, 0) ∈ H . −∆u = f2(u, v) + ϵh2

(iii) Every trajectory stabilizes to the set Nϵ, namely, for any U ∈ H one has

lim distH(S(t)U, Nϵ) = 0. t→+∞

min min In particular, there exists a global minimal attractor Aϵ given by Aϵ = Nϵ . f (iv) The attractor Aϵ has nite fractal and Hausdor dimension dimHAϵ . (v) If αi ∈ (0, 1/2), i = 1, 2, the global attractor Aϵ is bounded in

2 1 2 1 2 H1 = (H (Ω) ∩ H0(Ω)) × (H0(Ω)) .

Moreover, every trajectory U = (u, v, ut , vt) in Aϵ satises

2 2 2 2 k(u, v)k 2 1 2 + k(ut , vt)k 1 2 + k(utt , vtt)k 2 2 ≤ R1, (3.24) (H (Ω)∩H0 (Ω)) (H0 (Ω)) (L (Ω))

for some constant R1 > 0 independent of ϵ ∈ [0, 1]. (vi) The dynamical system (H, Sϵ(t)) possesses a generalized fractal exponential attractor. More precisely, for exp any δ ∈ (0, 1], there exists a generalized exponential attractor Aϵ,δ ⊂ H, with nite fractal dimension in extended space He −δ, dened as interpolation of

2 −1 2 He 0 := H, and He −1 := [L (Ω) × H (Ω)] .

The proof of this theorem will be achieved in the end of this section.

3.1 Quasistability estimate

The aim of this section is to derive quasistability estimate which is the main tool in proving nite dimensionality and smoothness of attractors.

Lemma 3.2. Suppose that Assumptions 2.1 holds. Let B ⊂ H be a positively invariant bounded subset and let i i i i i i Sϵ(t)U = (u (t), v (t), ut(t), vt(t)), i = 1, 2, be weak solution of (1.1) with initial conditions U ∈ B. Then there exist constants ωB , ϑB , CB > 0 independent of ϵ such that

−ωB t 2 2 E(t) ≤ ϑB E(0)e + CB sup (ku(σ)k2θ + kv(σ)k2θ), ∀t ≥ 0, (3.25) σ∈[0,t] for some θ ≥ 2, where u = u1 − u2 and v = v1 − v2.

Proof. Using the notations

1 1 2 2 1 2 1 2 Fi(u, v) = fi(u , v ) − fi(u , v ), G1(u) = g1(u ) − g1(u ), G2(v) = g2(v ) − g2(v ), Quasi-stability and continuity of attractors for nonlinear system of wave equations Ë 33 then, the dierences u = u1 − u2 and v = v1 − v2 satises

( α1 utt − ∆u + (−∆) ut + G (ut) = F (u, v), in Ω × (0, T), 1 1 (3.26) α2 vtt − ∆v + (−∆) vt + G2(vt) = F2(u, v), in Ω × (0, T), with boundary Dirichlet conditions and initial data 1 2 (u(0), v(0), ut(0), vt(0)) = U − U . (3.27) Multiplying the equations in (3.26) by u and v, respectively, and integrating over [0, T] × Ω, yields T T Z Z T Z 1 2 2 E(t)dt = − (uut + vvt)dx + (kutk2 + kvtk2)dt 2 0 0 Ω 0 T 1 Z Z α1 α1 α2 α2 − (−∆) 2 u (−∆) 2 u + (−∆) 2 v (−∆) 2 vdxdt 2 t t 0 Ω (3.28) T 1 Z Z − (G (u )u + G (v )v)dxdt 2 1 t 2 t 0 Ω T 1 Z Z + (F (u, v)u + F (u, v)v)dxdt. 2 1 2 0 Ω We shall estimate the right-hand side of (3.28). Step 1. From Hölder’s and Poincarés inequalities, we deduce Z T (uut + vvt)dx ≤ C(E(T) + E(0)) 0 Ω for some constant C > 0. Using the assumption (2.13), we obtain T T Z 1 Z Z (ku k2 + kv k2)dt ≤ (G (u )u + G (v )v )dxdt. (3.29) t 2 t 2 m 1 t t 2 t t 0 0 Ω 1 s Step 2. By using the Young’s inequality, continuous embedding H0(Ω) ,→ D((−∆) 2 ) for 0 < s < 1, we nd T T Z Z α1 α1 Z α1 α1 (−∆) 2 ut(−∆) 2 udxdt ≤ k(−∆) 2 utk2k(−∆) 2 uk2dt 0 Ω 0 T Z α1 ≤ C k∇uk2k(−∆) 2 utk2dt 0 T T Z Z 1 2 α1 2 ≤ k∇uk dt + C k(−∆) 2 u k dt. 4 2 t 2 0 0 Analogously, T T T Z Z Z Z α2 α2 1 2 α2 2 (−∆) 2 v (−∆) 2 vdxdt ≤ k∇vk dt + C k(−∆) 2 v k dt. t 4 2 t 2 0 Ω 0 0 Then T Z Z α1 α1 α2 α2 (−∆) 2 ut(−∆) 2 u + (−∆) 2 vt(−∆) 2 v dxdt

0 Ω (3.30) T T Z Z 1 α1 2 α2 2 ≤ E(t)dt + C (k(−∆) 2 u k + k(−∆) 2 v k )dt. 2 t 2 t 2 0 0 34 Ë M. M. Freitas, M. J. Dos Santos, A. J. A. Ramos, M. S. Vinhote, and M. L. Santos

Step 3. From Young’s inequality and (2.13), we obtain

T T T Z Z 1 Z Z 1 Z Z |u|2 G1(ut)udxdt ≤ G1(ut)ut dxdt + G1(ut) dxdt. 2 2 ut 0 Ω 0 Ω 0 Ω

Using the assumption (2.11), we deduce

Z 1 Z M Z G (u )udx ≤ G (u )u dx + 1 (1 + |u1|q−1 + |u2|q−1)|u|2dx. 1 t 2 1 t t 2 t t (3.31) Ω Ω Ω

To estimate the second term in the right side of (3.31), we consider three cases separately: Case 1. q = 1. In this case, it is easy see that   Z Z 1 q−1 2 q−1 2 2 1 1 2 2 (1 + |ut | + |ut | )|u| dx ≤ 3kuk2 1 + (g1(ut )ut + g1(ut )ut )dx . Ω Ω

Case 2. q ≥ 3. For this situation we use (2.12) and Holder’s inequality to get

q−1   l Z Z 1 q−1 2 q−1 2 2 1 l 2 l (1 + |ut | + |ut | )|u| dx ≤ Ckuk 2l  (1 + |ut | + |ut | )dx l−q+1 Ω Ω q−1   l Z 2 1 1 2 2 ≤ Ckuk 2l  (1 + g1(ut )ut + g1(ut )ut )dx l−q+1 Ω   Z 2 1 1 2 2 ≤ Ckuk 2l  (1 + g1(ut )ut + g1(ut )ut )dx , l−q+1 Ω

q−1 where we have used the fact that l < 1. Case 3. 1 < q < 3. As in case 2 we obtain

q−1   2 Z Z 1 q−1 2 q−1 2 2 1 2 2 2 (1 + |ut | + |ut | )|u| dx ≤ Ckuk 4  (1 + |ut | + |ut | )dx 3−q Ω Ω q−1   2 Z 2 1 1 2 2 ≤ Ckuk 4  (1 + g1(ut )ut + g1(ut )ut )dx 3−q Ω   Z 2 1 1 2 2 ≤ Ckuk 4  (1 + g1(ut )ut + g1(ut )ut )dx . 3−q Ω

Combining the three last estimates and (3.31) we conclude that there exist C > 0 and θ ≥ 2 such that   Z 1 Z Z G (u )udx ≤ G (u )u dx + Ckuk2 (1 + g (u1)u1 + g (u2)u2)dx . (3.32) 1 t 2 1 t t θ  1 t t 1 t t  Ω Ω Ω Analogously,   Z 1 Z Z G (v )vdx ≤ G (v )v dx + Ckvk2 (1 + g (v1)v1 + g (v2)v2)dx . (3.33) 2 t 2 2 t t θ  2 t t 1 t t  Ω Ω Ω Quasi-stability and continuity of attractors for nonlinear system of wave equations Ë 35

1 2 Then, since U , U ∈ B, we deduce that by (2.16) and (2.17) that there exists CB > 0 such that Z 1 1 2 2 (1 + g1(ut )ut + g1(ut )ut )dx ≤ CB , Ω Z 1 1 2 2 (1 + g2(vt )vt + g1(vt )vt )dx ≤ CB . Ω

Combining the last estimate with (3.32) and (2.12) and using the embedding L2θ(Ω) ,→ Lθ(Ω) we obtain

T T 1 Z Z 1 Z Z − (G (u )u + G (v )v)dxdt ≤ (G (u )u + G (v )v )dxdt 2 1 t 2 t 2 1 t t 2 t t 0 Ω 0 Ω 2 2 + CB,T sup ku(σ)k2θ + kv(σ)k2θ . σ∈[0,T]

˜ 2θ Step 4. Using (2.7), the Hölder’s inequality with conjugated exponents θ = θ−1 , 2θ and 2, and the embedding 1 s (in 2D) H0(Ω) ,→ L (Ω), 1 ≤ s < ∞, we obtain Z F1(u, v)udx ≤ C(∇f1)(kuk2θ + kvk2θ)kvk2 Ω (3.34) ≤ CB kuk2θ + kvk2θ kvk2 2 2 ≤ CB kuk2θ + kvk2θ where 1 p−1 2 p−1 1 p−1 2 p−1 C(∇f1) = C 1 + ku k + ku k + kv k + kv k . (p−1)θ˜ (p−1)θ˜ (p−1)θ˜ (p−1)θ˜ Analogously, Z 2 2 F2(u, v)vdx ≤ CB kuk2θ + kvk2θ . Ω

Combining the two last estimates, there exists CB,T > 0 such that

T Z Z 1 2 2 (F1(u, v)u + F2(u, v)v)dxdt ≤ CB,T sup ku(σ)k2θ + kv(σ)k2θ . (3.35) 2 σ∈[0,T] 0 Ω

Inserting the estimates (3.29)-(3.35) into (3.28), we conclude that there exist CB > 0 and CB,T > 0 such that

T Z E(t)dt ≤ C(E(T) + E(0)) 0 T Z α1 2 α2 2 + CB (k(−∆) 2 utk2 + k(−∆) 2 vtk2)dt 0 (3.36) T Z Z + CB (G1(ut)ut + G2(vt)vt)dxdt 0 Ω 2 2 + CB,T sup ku(σ)k2θ + kv(σ)k2θ . σ∈[0,T] 36 Ë M. M. Freitas, M. J. Dos Santos, A. J. A. Ramos, M. S. Vinhote, and M. L. Santos

Step 5. Multiplying the equation in (3.26) by ut and vt, respectively, and then integrating over [s, T]×Ω, yields

T Z α1 2 α2 2 E(T) = E(s) − (k(−∆) 2 utk2 + k(−∆) 2 vtk2)dt 0 T Z Z − (G1(ut)ut + G2(vt)vt)dxdt (3.37) 0 Ω T Z Z + (F1(u, v)ut + F2(u, v)vt)dxdt. 0 Ω Since T Z Z − (G1(ut)ut + G2(vt)vt)dxdt ≤ 0, 0 Ω we have T Z Z E(T) ≤ E(s) + (F1(u, v)ut + F2(u, v)vt)dxdt. (3.38) 0 Ω Similarly to (3.34), we deduce that Z F1(u, v)ut dx ≤ CB(kuk2θ + kvk2θ)kutk2 Ω (3.39) C ≤ ϵku k2 + B kuk2 + kvk2 . t 2 4ϵ 2θ 2θ Similarly, Z C F (u, v)v dx ≤ ϵkv k2 + B kuk2 + kvk2 . 2 t t 2 4ϵ 2θ 2θ (3.40) Ω Therefore, Z C (F (u, v)u + F (u, v)v )dx ≤ ϵE(t) + B kuk2 + kvk2 . 1 1 2 t 4ϵ 2θ 2θ (3.41) Ω 1 Inserting the last estimate into (3.38) with ϵ = T , we nd that

T T 1 Z Z E(T) ≤ E(s) + E(t)dt + TC kuk2 + kvk2 dt T B 2θ 2θ 0 0

Integrating the last estimate over [0, T] with respect to s, we conclude that there exist a constant CB,T > 0 such that T Z 2 2 TE(T) ≤ 2 E(t)dt + CB,T sup ku(σ)k2θ + kv(σ)k2θ . (3.42) σ∈[0,T] 0 Step 6. The estimates (3.37) and (3.41) with ϵ = 1 imply

T T Z Z Z α1 2 α2 2 (k(−∆) 2 utk2 + k(−∆) 2 vtk2)dt + (G1(ut)ut + G2(vt)vt)dxdt 0 0 Ω (3.43) T Z 2 2 ≤ E(T) + E(0) + E(t)dt + TCB sup ku(σ)k2θ + kv(σ)k2θ . σ∈[0,T] 0 Quasi-stability and continuity of attractors for nonlinear system of wave equations Ë 37

Substituting the above estimate in (3.36), we nd that

T Z 2 2 E(t)dt ≤ CB(E(T) + E(0)) + CB,T sup ku(σ)k2θ + kv(σ)k2θ . σ∈[0,T] 0

This estimate and (3.42) yields

2 2 TE(T) ≤ CB(E(T) + E(0)) + CB,T sup ku(σ)k2θ + kv(σ)k2θ . σ∈[0,T]

Choosing T > 2CB, we nd that

2 2 E(T) ≤ γT E(0) + CB,T sup ku(σ)k2θ + kv(σ)k2θ , (3.44) σ∈[0,T] where CB γT = < 1. T − CB Dening 2 2 χm = sup ku(σ)k2θ + kv(σ)k2θ , m = 0, 1, 2, ... , σ∈[m,(m+1)T] the estimate (3.44) we can rewrite as

E(T) ≤ γT E(0) + CB,T χ0. (3.45)

By iterating the above estimate on intervals [mT, (m + 1)T], m ∈ N, we deduce

m m X m+1−l E(mT) ≤ γT E(0) + CB,T γT χk−1 k=1 m CB,T 2 2 ≤ γT E(0) + sup ku(σ)k2θ + kv(σ)k2θ . 1 − γT σ∈[0,mT]

For any t ≥ 0, there exists m ∈ N and r ∈ [0, T) such that t = mT + r. Then, by (2.22) we obtain

t −1 T CB,T 2 2 E(t) ≤ E(mT) ≤ γT γT E(0) + sup ku(σ)k2θ + kv(σ)k2θ . 1 − γT σ∈[0,t]

Therefore, −ωB t 2 2 E(t) ≤ ϑB E(0)e + CB sup (ku(σ)k2θ + kv(σ)k2θ), ∀t ≥ 0, σ∈[0,t] with −1 ln(γT) CB,T ϑB = γT , ωB = − , CB = . T 1 − γT The proof is complete.

3.2 Gradient system and stationary solutions

We recall that a dynamical system (H, S(t)) is gradient if it possesses a strict Lyapunov functional. That is, a functional Φ : H → R is a strict Lyapunov function for a system (H, S(t)) if, (i) the map t → Φ(S(t)z) is non-increasing for each z ∈ H, (ii) if Φ(S(t)z) = Φ(z) for some z ∈ H and for all t, then z is a stationary point of S(t), that is, S(t)z = z.

Lemma 3.3. Suppose that Assumption 2.1 holds. Then the dynamical system (H, Sϵ(t)) is gradient, that is, there exists a strict Lyapunov function Φϵ dened in H. In addition,

Φϵ(U) → ∞ ⇐⇒ kUkH → ∞. (3.46) 38 Ë M. M. Freitas, M. J. Dos Santos, A. J. A. Ramos, M. S. Vinhote, and M. L. Santos

Proof. In order to prove this lemma we will show that the functional total energy Eϵ(t) dened in (2.15) is a Lyapunov function Φϵ. Indeed, let U0 = (u0, v0, u1, v1) ∈ H, then (2.16) shows that t 7→ Φϵ(S(t)U0) is a non-increasing function. Now suppose that Φϵ(S(t)U0) = Φϵ(U0), ∀t ≥ 0. Using again (2.16), we conclude that

α1 2 α2 2 k(−∆) 2 utk2 + k(−∆) 2 vtk2 = 0, ∀t ≥ 0.

Consequently,

ut = vt = 0, for all t ≥ 0, a.e. in Ω.

Therefore ut(t) = u0 and vt(t) = v0 for all t ≥ 0. Then we can obtain that Sϵ(t)U0 = U(t) = (u0, v0, 0, 0) is a stationary solution, i.e., Sϵ(t)U0 = U0, for all t ≥ 0. Now, from the second inequality in (2.17) we have

p+1 Φϵ(U) ≤ CF(1 + kUkH ), ∀t ≥ 0. (3.47)

Considering the last estimate and taking Φϵ(U) → ∞ we have kUkH → ∞. On the other hand, by the rst inequality 2.17 we get 2 Φϵ(U) + CF kUkH ≤ , (3.48) C0 from where we conclude that kUkH → ∞ implies Φϵ(U) → ∞, proving (3.46).

Lemma 3.4. Suppose that Assumption 2.1 holds. Then the set Nϵ of the stationary points of (H, Sϵ(t)) is bounded in H.

Proof. Let U ∈ Nϵ be arbitrary. We know that U = (u, v, 0, 0) and U satises the equations

−∆u = f1(u, v) + ϵh1, (3.49)

−∆v = f2(u, v) + ϵh2. (3.50) Multiplying (3.49) and (3.50) by u and v, respectively, and then integrating over Ω, we get Z Z 2 2 k∇uk + k∇vk = f1(u, v)u + f2(u, v)v dx + ϵ (h1u + h2v)dx, Ω Ω

By (2.8) and (2.9) we have Z 2 2 f1(u, v)u + f2(u, v)v dx ≤ 2β0 kuk2 + kvk2 + 2mF|Ω| Ω 2β0 2 2 ≤ k∇uk2 + k∇vk2 + 2mF|Ω|, λ1 and therefore, in light of (2.18), Z 2 4C0kUkH ≤ 2mF|Ω| + ϵ (h1u + h2v)dx. Ω

Hence, using the estimate (2.19), we deduce

2 1 2 2 3C0kUkH ≤ 2mF|Ω| + kh1k2 + kh2k2 . (3.51) 4C0λ1 The proof is complete. Quasi-stability and continuity of attractors for nonlinear system of wave equations Ë 39

3.3 Proof of Theorem 3.1

i (i) We consider a bounded positively invariant set B ⊂ H with respect to Sϵ(t), denote Sϵ(t)U = i i i i i 1 2 1 2 (u (t), v (t), ut(t), vt(t)) for U ∈ B, i = 1, 2, and set u = u − u , v = v − v , as before. It follows from (2.22) that

1 2 2 1 2 2 kSϵ(t)U − Sϵ(t)U kH ≤ a(t)kU − U kH, (3.52)

C0 T 1 1 with a(t) = e . We denote by X = H0(Ω) × H0(Ω) and dene the semi-norm

1 2 2 2 nX(u, v) := kuk2θ + kvk2θ .

1 2θ Since the embedding (in 2D) H0(Ω) ,→ L (Ω) is compact, we know that nX is a compact semi-norm on X. By Lemma 3.2, we can get

1 2 2 2 2 kSϵ(t)U − Sϵ(t)U kH ≤ b(t)kU1 − U2kH + c(t) sup nX(u(s), v(s)) , (3.53) s∈[0,t]

−ωB t where b(t) = ϑB e and c(t) = CB. It is easy to get that

1 + b(t) ∈ L (R ) and lim b(t) = 0. t→∞

Since B ⊂ H is bounded, we know that c(t) is locally bounded on [0, ∞). From [21, Denition 7.9.2], we get that the dynamics system (H, Sϵ(t)) is quasi-stable on any bounded positively invariant set B ⊂ H. (ii) Since the system (H, Sϵ(t)) is quasi-stable, applying Proposition 7.9.4 in [21], then (H, Sϵ(t)) is asymp- totically smooth. Thus, noting Lemmas 3.3 and 3.4 and using Corollary 7.5.7 in [21], we know that (H, Sϵ(t)) has a compact global attractor given by Aϵ = M+(Nϵ). (iii) Combining Theorem 3.1-(ii) and Theorem 7.5.10 in [21], we can get the result. (iv) From the above, (H, Sϵ(t)) is quasi-stable on the attractor Aϵ. Thus, using Theorem 7.9.6 in [21], we f know that the attractor Aϵ has nite fractal dimension dimHAϵ. (v) Since the system (H, Sϵ(t)) is quasi-stable on the attractor Aϵ, it follows from Theorem 7.9.8 in [21] that any complete trajectory U = (u, v, ut , vt) in Aϵ enjoys the following regularity properties

∞ 1 2 ∞ 1 2 ut ∈ L (R, H0(Ω)) ∩ C(R, L (Ω)), vt ∈ L (R, H0(Ω)) ∩ C(R, L (Ω)), and ∞ 2 ∞ 2 utt ∈ L (R, L (Ω)), vtt ∈ L (R, L (Ω)). Moreover, there exists R > 0 such that

2 2 2 k(ut , vt)k 1 2 + k(utt , vtt)k 2 2 ≤ R . (3.54) (H0 (Ω)) (L (Ω))

∞ 1 ∞ αi From (3.54), (1.1), the embedding L (R; H0(Ω)) ,→ L (R; D((−∆) )) for 0 < αi < 1/2 and the fact that 0 < ϵ < 1, we obtain α1 k∆uk2 ≤ kuttk2 + k(−∆) utk + kg1(ut)k2 + kf1(u, v)k2 + kh1k2 ≤ C, (3.55) α2 k∆vk2 ≤ kvttk2 + k(−∆) vtk + kg2(vt)k2 + kf2(u, v)k2 + kh2k2 ≤ C.

Therefore (3.24) is obtained by (3.54) and (3.55). Since the global attractors Aϵ are also characterized as

Aϵ = {U(0) : U is a bounded full trajectory of Sϵ(t)}, we conclude the Aϵ is bounded in H1. (vi) Now we take B = {U : Φϵ(U) ≤ R}, where Φϵ is the strict Lyapunov functional considered in Lemma 3.3. Then we know that the set B is a positively invariant absorbing set for R large enough. Hence the system (H, Sϵ(t)) is quasi-stable on B. 40 Ë M. M. Freitas, M. J. Dos Santos, A. J. A. Ramos, M. S. Vinhote, and M. L. Santos

For solution U(t) with initial data z = U(0) ∈ B, we can conclude from the positive invariance of B that there exists CB > 0 such that for any 0 ≤ t ≤ T,

kUt(t)k ≤ CB, He −1 which gives us for any 0 ≤ t1 < t2 ≤ T,

t Z 2 kSϵ(t1)z − Sϵ(t2)zk ≤ kUt(τ)k dτ ≤ CB|t1 − t2|. (3.56) He −1 He −1

From (3.56), we conclude that for any z ∈ B, the map t 7→ Sϵ(t)z is Hölder continuous in the extended space He −1 with exponent δ = 1. Then we can get the existence of a generalized exponential attractor whose fractal dimension is nite in He −1. Following the same arguments in [21], we can obtain the existence of exponential attractors in He −δ with δ ∈ (0, 1). Thus, the proof of Theorem 3.1 is complete.

4 Upper and lower semicontinuity of global attractors

In this section, we study the continuity of the attractors Aϵ as ϵ → ϵ0. Firstly, we prove that this family of attractors converges upper and lower semicontinuously to the global compact attractor Aϵ0 of the limiting semi-ow Sϵ0 (t) on a residual subset of [0, 1], by using the abstract result in [31, Theorem 5.2], where the results were obtained as a extension of the previous results in [4]. Finally, the upper semicontinuity is proved for all ϵ0 ∈ [0, 1]. Let Λ be a complete metric space and Sλ(t) a parametrised family of semigroups on X. Suppose that (L1) Sλ(t) has a global attractor Aλ for every λ ∈ Λ, (L2)There is a bounded subset D of X such that Aλ ⊂ D for every λ ∈ Λ, (L3)For t > 0, Sλ(t)x is continuous in λ, uniformly for x in bounded subsets of X. Then Aλ is continuous on all Λ* where Λ* is a residual set dense in Λ (see [31, Theorem 5.2]). The following lemma will be used to obtain a uniform (with respect to the parameter ϵ of the problem) bound for the attractor that will be used to verify the property (L2) above (see [21, Remark 7.5.8]).

Lemma 4.1. Under the assumptions of Lemma 3.3, the following relation is valid:

sup Φϵ(U) ≤ sup Φϵ(U). U∈Aϵ U∈Nϵ

Theorem 4.2 (Residual continuity). Under the assumptions of Theorem 3.1, there exists a set I* dense in [0, 1] such that Aϵ is continuous at ϵ0 ∈ I*, that is,

lim dH(Aϵ , Aϵ0 ) = 0, ∀ϵ0 ∈ I*, (4.57) ϵ→ϵ0 where dH denotes the Hausdor distance between bounded sets B, C ⊂ H, dened as

dH(B, C) = max{distH(B, C), distH(C, B)}.

Proof. We shall apply the abstract result in [31, Theorem 5.2] with Λ = [0, 1]. The argument follows the lines of the proof of Theorem 4.1 in [36]. Theorem 3.1-(ii) indicates that (L1) holds. Now, we obtain uniform (with Quasi-stability and continuity of attractors for nonlinear system of wave equations Ë 41 respect to the parameter ϵ of the problem) bounds for the attractor. By (2.17) and Lemma 4.1, we obtain

2 supU∈Aϵ Φϵ(U) + CF sup kUkH ≤ U∈Aϵ C0 sup Φϵ(U) + C ≤ U∈Nϵ F C0 C sup kUkp+1 + 2C ≤ F U∈Nϵ H F . C0

Hence, by (3.51), we can conclude that there exists a constant C0 > 0 independent of ϵ such that

2 sup kUkH ≤ C0, ∀ϵ ∈ [0, 1]. U∈Aϵ

2 Then, D = {U ∈ H : kUkH ≤ C0} is a bounded set independent of ϵ such that

Aϵ ⊂ D, ∀ϵ ∈ [0, 1], and thereby (L2) holds. Let B be a bounded set of H. Given ϵ1, ϵ2 ∈ [0, 1] and U0 ∈ B, let us denote

i i i i Sϵi (t)U0 = (u (t), v (t), ut(t), vt(t)), i = 1, 2, and u = u1 − u2, v = v1 − v2.

Then U = (u, v, ut , vt) satises the following system

( α1 utt − ∆u + (−∆) ut = F (u, v) − G (ut) + (ϵ − ϵ )h , 1 1 1 2 1 (4.58) α2 vtt − ∆v + (−∆) vt = F2(u, v) − G2(vt) + (ϵ1 − ϵ2)h2, where 1 1 2 2 Fi(u, v) = fi(u , v ) − fi(u , v ), i = 1, 2, and 1 2 1 2 G1(ut) = g1(ut ) − g1(ut ), G2(vt) = g2(vt ) − g2(vt ).

Multiplying the rst equation in (4.58) by ut, the second by vt, respectively, and using integration by parts, we obtain 1 d Z kUk2 = (F (u, v)u + F (u, v)v ) dx 2 dt H 1 t 2 t Ω α1 2 α2 2 − (k(−∆) 2 utk2 + k(−∆) 2 vtk2) Z (4.59) − (G1(ut)ut + G2(vt)vt) dx Ω Z − (ϵ1 − ϵ2) (h1ut + h2vt) dx. Ω 1 s Using (2.7), Hölder’s inequality and the embedding (in 2D) H0(Ω) ,→ L (Ω) for 1 ≤ s < ∞, we deduce Z p−1 p−1 F1(u, v)ut dx ≤ C(1 + kSϵ1 (t)U0kH + kSϵ2 (t)U0kH )(kuk2p + kvk2p)kutk2 Ω (4.60) p−1 p−1 ≤ C(1 + kSϵ1 (t)U0kH + kSϵ2 (t)U0kH )(k∇uk2 + k∇vk2)kutk2.

Using the fact that Eϵ(t) is a non-increasing function and (2.17), we nd that for i = 1, 2,

k kp+1 p−1 Eϵi (0) + CF CF(1 + U0 H ) + CF kSϵi (t)U0kH ≤ ≤ ≤ CB , ∀U0 ∈ B. C0 C0 42 Ë M. M. Freitas, M. J. Dos Santos, A. J. A. Ramos, M. S. Vinhote, and M. L. Santos

Inserting the above estimate into (4.60) and using Young’s inequality, we see that Z F1(u, v)ut dx ≤ CB(k∇uk2 + k∇vk2)kutk2 Ω 2 2 2 ≤ CB(k∇uk2 + k∇vk2) + kutk2.

Analogously, Z 2 2 2 F2(u, v)vt dx ≤ CB(k∇uk2 + k∇vk2) + kvtk2. Ω Adding the last two estimates, we conclude that Z 2 (F1(u, v)ut + F2(u, v)vt) dx ≤ CDkUkH. (4.61) Ω By the monotonicity property (2.13), we get Z − (G1(ut)ut + G2(vt)vt + G3(wt)wt) dx ≤ 0. (4.62) Ω In addition, Z 1 1 (ϵ − ϵ ) (h u + h v ) dx ≤ (ku k2 + kv k2) + |ϵ − ϵ |2 kh k2 + kh k2 1 2 1 t 2 t 2 t 2 t 2 2 1 2 1 2 2 2 Ω (4.63) 1 1 ≤ kUk2 + |ϵ − ϵ |2 kh k2 + kh k2 . 2 H 2 1 2 1 2 2 2 Substituting the estimates (4.61)-(4.63) into (4.59), we obtain d kUk2 ≤ C kUk2 + |ϵ − ϵ |2 kh k2 + kh k2 . (4.64) dt H B H 1 2 1 2 2 2

2 Applying Gronwall’s inequality to (4.64) and using that kU(0)kH = 0, we conclude that 2 1 CB t 2 2 2 kU(t)kH ≤ e − 1 kh1k2 + kh2k2 |ϵ1 − ϵ2| , t > 0. CB This implies

r 1 CD t 2 2 kSϵ1 (t)U0 − Sϵ2 (t)U0kH ≤ e − 1 kh1k2 + kh2k2 |ϵ1 − ϵ2|, t > 0, CD and thereby (L3) holds. Therefore all the assumptions of Theorem [31, Theorem 5.2] are fullled. Then there exists a dense set I* ⊂ [0, 1] such that (4.57) holds. The proof is complete.

Theorem 4.3 (Upper-semicontinuity). Under the assumptions of Theorem 3.1, the family of global attractors Aϵ is upper semicontinuous at ϵ ∈ [0, 1], that is,

lim distH(Aϵ , Aϵ0 ) = 0, ∀ϵ ∈ [0, 1]. (4.65) ϵ→ϵ0

Proof. The argument is inspired by (see, e.g. [25, 30]). We proceed by contradiction as in [35]. Suppose that n (4.65) does not hold. Then there exist δ > 0 and a sequences ϵn → ϵ0 and U0 ∈ Aϵn such that

n distH(U0 , Aϵ0 ) ≥ δ > 0, ∀n. (4.66)

n n n n n n n Let U (t) = (u (t), v (t), ut (t), vt (t)) be a full trajectory from the attractor Aϵn such that U (0) = U0 . From uniform estimate (3.24), we know

n ∞ {U } is bounded in L (R; H1). (4.67) Quasi-stability and continuity of attractors for nonlinear system of wave equations Ë 43

Since H1 is compactly embedded into H, using Simon’s Compactness Theorem (see [37]), we obtain a subse- quence {Unk } and U ∈ C([−T, T]; H) such that

nk lim max kU (t) − U(t)kH = 0. (4.68) k→∞ t∈[−T,T]

By (4.67) and (4.68), we conclude that sup kU(t)kH < ∞. t∈R Using the same argument as in the proof of property (L3) above, we can see that

U(t) = (u(t), v(t), ut(t), vt(t)) solves the limiting equations (ϵ = ϵ0)

( α1 utt − ∆u + (−∆) ut + g1(ut) = f1(u, v) + ϵ0h1,

α2 vtt − ∆v + (−∆) vt + g2(vt) = f2(u, v) + ϵ0h2.

Therefore U(t) is a bounded full trajectory for the limiting semi-ow Sϵ0 (t). Consequently,

nk U0 → U(0) ∈ A0, which is contradict (4.66). The proof is complete. Conict of interest: On behalf of all authors, the corresponding author states that there is no conict of interest.

Acknowledgments: A.J.A. Ramos thanks the CNPq for nancial support through the projects “Asymptotic stabilization and numerical treatment for carbon nanotubes” (CNPq Grant 310729/2019-0).

Data Availability Statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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