Quasi-stability and continuity of attractors for nonlinear system of wave equations


 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.

In [28], Guo et al. studied the local and global well-posedness of the coupled nonlinear wave equations in Ω × ( , T), (1.2) in a bounded domain Ω ⊂ R n 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. Moreover, 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 (−∆) α u t , (−∆) α v t and considering small perturbations of autonomous external forces ϵh , ϵh . 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 useful 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 [ , ] such that the family of global attractors {Aϵ} ϵ∈ [ , ] associated to problem (1.1) converges upper and lower-semicontinuously to the corresponding global attractor associated with the limit problem when the parameter ϵ → ϵ for all ϵ ∈ I * . Moreover, the upper semicontinuity for all ϵ ∈ [ , ] 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, we assume the nonlinear damping terms with polynomial growth to include functions of type g i (u) = |u| p− 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. Indeed, 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 → ϵ on a residual dense set I * ⊂ [ , ] in the same sense proposed in Hoang et al. [31]. The upper semicontinuity or all ϵ ∈ [ , ] 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.

Preliminaries
In this section, we present preliminaries including notations and assumptions.

. Notations and assumptions
The following notations will be used for the rest of the paper.
In this work, we consider the Hilbert space with the following inner product and norm and for any U = (u, v, ϕ, φ) andŨ = (ũ,ṽ,φ,φ) in H.
and there exist p ≥ and C > such that where ≤ β < λ and λ > denotes the Poincaré's constant. Moreover (iv) Concerning to the nonlinear damping g i , we assume that

10)
and there exists constants m, M , q ≥ , such that

11)
and, if q ≥ , there exist l > q − and M > such that Remark 2.1. Observe that assumption (2.11) implies the monotonicity property, that is, In this case, we have the assumptions (2.7)-(2.8) hold with m F = and p = . Noting that then (2.9) also holds.

. Energy identities
The total energy of system (1.1) is de ned by as and we also de ne the total energy by Then we have the following lemma.
Lemma 2.1. The total energy given in (2.15) satis es Moreover, there exists a positive constant C such that Proof. Multiplying (formally) the rst equation in (1.1) by u t , and second by v t , respectively and using integration by parts, we can easily get (2.16). It follows from (2.8) and Poincaré's inequality that and thus and using the estimate we obtain the rst inequality in (2.17) with Using (2.7) we can see that the second inequality in (2.17) holds. The proof is complete.

. Well-posedness
Let us write the problem (1.1) as an equivalent Cauchy problem where and H de ned in (2.3) and A : D(A) ⊂ H → H is the nonlinear operator de ned by (2.21) The domain of A is given by The forcing terms are represented by a nonlinear function F : H → H de ned by We have the following global existence result for the problem (1.1).
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 di erence of solutions.

Long-time dynamics
On account of Theorem 2.2, we can de ne the one-parameter family of operators Sϵ(t) : H → H by where (u, v, u t , v t ) 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.
(iii) Every trajectory stabilizes to the set Nϵ, namely, for any U ∈ H one has In particular, there exists a global minimal attractor A min

Moreover, every trajectory
. The proof of this theorem will be achieved in the end of this section.

. 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
Proof. Using the notations with boundary Dirichlet conditions and initial data Multiplying the equations in (3.26) by u and v, respectively, and integrating over (3.28) We shall estimate the right-hand side of (3.28).
Step 1. From Hölder's and Poincarés inequalities, we deduce for some constant C > . Using the assumption (2.13), we obtain Step 2. By using the Young's inequality, continuous embedding H (Ω) → D((−∆) s ) for < s < , we nd Analogously, (3.30) Step 3. From Young's inequality and (2.13), we obtain Using the assumption (2.11), we deduce To estimate the second term in the right side of (3.31), we consider three cases separately: Case 1. q = . In this case, it is easy see that Case 2. q ≥ . For this situation we use (2.12) and Holder's inequality to get where we have used the fact that q− l < . Case 3. < q < . As in case 2 we obtain Combining the three last estimates and (3.31) we conclude that there exist C > and θ ≥ such that (3.32) Analogously, Then, since U , U ∈ B, we deduce that by (2.16) and (2.17) that there exists C B > such that Combining the last estimate with (3.32) and (2.12) and using the embedding L θ (Ω) → L θ (Ω) we obtain Step 4. Using (2.7), the Hölder's inequality with conjugated exponentsθ = θ θ− , θ and , and the embedding (in 2D) H (Ω) → L s (Ω), ≤ s < ∞, we obtain Combining the two last estimates, there exists C B,T > such that Inserting the estimates (3.29)-(3.35) into (3.28), we conclude that there exist C B > and C B,T > such that (3.36)

Step 5. Multiplying the equation in (3.26) by u t and v t , respectively, and then integrating over [s, T]× Ω, yields
we have Similarly to (3.34), we deduce that Inserting the last estimate into (3.38) with ϵ = T , we nd that Integrating the last estimate over [ , T] with respect to s, we conclude that there exist a constant C B,T > such that Step 6. The estimates (3.37) and (3.41) with ϵ = imply (3.43) Substituting the above estimate in (3.36), we nd that This estimate and (3.42) yields Choosing T > C B , we nd that For any t ≥ , there exists m ∈ N and r ∈ [ , T) such that t = mT + r. Then, by (2.22) we obtain Therefore, The proof is complete.

. 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, Consequently, u t = v t = , for all t ≥ , a.e. in Ω. Therefore u t (t) = u and v t (t) = v for all t ≥ . Then we can obtain that Sϵ(t)U = U(t) = (u , v , , )

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, , ) and U satis es the equations Multiplying (3.49) and (3.50) by u and v, respectively, and then integrating over Ω, we get By (2.8) and (2.9) we have and therefore, in light of (2.18), Hence, using the estimate (2.19), we deduce The proof is complete.

. Proof of Theorem 3.1 (i) We consider a bounded positively invariant set
with a(t) = e C T . We denote by X = H (Ω) × H (Ω) and de ne the semi-norm Since the embedding (in 2D) H (Ω) → L θ (Ω) is compact, we know that n X is a compact semi-norm on X. By Lemma 3.2, we can get Since B ⊂ H is bounded, we know that c(t) is locally bounded on [ , ∞). From [21, De nition 7.9.2], we get that the dynamics system (H, Sϵ(t)) is quasi-stable on any bounded positively invariant set B ⊂ H.
Moreover, there exists R > such that From (3.54), (1.1), the embedding L ∞ (R; H (Ω)) → L ∞ (R; D((−∆) α i )) for < α i < / and the fact that < ϵ < , we obtain For solution U(t) with initial data z = U( ) ∈ B, we can conclude from the positive invariance of B that there exists C B > such that for any ≤ t ≤ T, which gives us for any ≤ t < t ≤ T, (3.56) From (3.56), we conclude that for any z ∈ B, the map t → Sϵ(t)z is Hölder continuous in the extended space H − with exponent δ = . Then we can get the existence of a generalized exponential attractor whose fractal dimension is nite in H − .
Following the same arguments in [21], we can obtain the existence of exponential attractors in H −δ with δ ∈ ( , ).
Thus, the proof of Theorem 3.1 is complete.

Upper and lower semicontinuity of global attractors
In this section, we study the continuity of the attractors Aϵ as ϵ → ϵ . Firstly, we prove that this family of attractors converges upper and lower semicontinuously to the global compact attractor Aϵ of the limiting semi-ow Sϵ (t) on a residual subset of [ , ], 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 ϵ ∈ [ , ].
Let Λ be a complete metric space and S λ (t) a parametrised family of semigroups on X. Suppose that (L1) S λ (t) has a global attractor A λ for every λ ∈ Λ, (L2) There is a bounded subset D of X such that A λ ⊂ D for every λ ∈ Λ, (L3)For t > , S λ (t)x is continuous in λ, uniformly for x in bounded subsets of X. Then A λ 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] Then U = (u, v, u t , v t ) satis es the following system Multiplying the rst equation in (4.58) by u t , the second by v t , respectively, and using integration by parts, we obtain d dt (4.59) Using (2.7), Hölder's inequality and the embedding (in 2D) H (Ω) → L s (Ω) for ≤ s < ∞, we deduce (4.60) Using the fact that Eϵ(t) is a non-increasing function and (2.17), we nd that for i = , , Inserting the above estimate into (4.60) and using Young's inequality, we see that Ω F (u, v)u t dx ≤ C B ( ∇u + ∇v ) u t ≤ C B ( ∇u + ∇v ) + u t .

Analogously
, Adding the last two estimates, we conclude that By the monotonicity property (2.13), we get In addition, Applying Gronwall's inequality to (4.64) and using that U( ) H = , we conclude that This implies  Proof. The argument is inspired by (see, e.g. [25,30]). We proceed by contradiction as in [35]. Suppose that (4.65) does not hold. Then there exist δ > and a sequences ϵn → ϵ and U n ∈ Aϵ n such that dist H (U n , Aϵ ) ≥ δ > , ∀n. Let U n (t) = (u n (t), v n (t), u n t (t), v n t (t)) be a full trajectory from the attractor Aϵ n such that U n ( ) = U n . From uniform estimate (3.24), we know {U n } is bounded in L ∞ (R; H ). (4.67) Since H is compactly embedded into H, using Simon's Compactness Theorem (see [37]), we obtain a subsequence {U n k } and U ∈ C ([−T Using the same argument as in the proof of property (L3) above, we can see that solves the limiting equations (ϵ = ϵ ) Therefore U(t) is a bounded full trajectory for the limiting semi-ow Sϵ (t). Consequently, which is contradict (4.66). The proof is complete.

Con ict of interest:
On behalf of all authors, the corresponding author states that there is no con ict of interest.