CRITERIA ON THE EXISTENCE AND STABILITY OF PULLBACK EXPONENTIAL ATTRACTORS AND THEIR APPLICATION TO NON-AUTONOMOUS KIRCHHOFF WAVE MODELS

. In this paper, we are concerned with the existence and stability of pullback exponential attractors for a non-autonomous dynamical system. (i) We propose two new criteria for the discrete dynamical system and continuous one, respectively. (ii) By applying the criteria to the non-autonomous Kirchhoﬀ wave models with structural damping and supercritical nonlinearity we construct a family of pullback exponential attractors which are stable with respect to perturbations.


1.
Introduction. It is well known that pullback attractor and pullback exponential attractor are two basic concepts to study the longtime dynamics of infinite dimensional non-autonomous dynamical system. To be more precise, a process acting on the Banach space E is a two-parametrical family of operators {U (t, τ ) : E → E|t, τ ∈ R, t ≥ τ } (or t, τ ∈ Z for discrete time) satisfying U (t, s)U (s, τ ) = U (t, τ ), U (τ, τ ) = I (identity operator), t, s, τ ∈ R, t ≥ s ≥ τ.
A family of nonempty compact subsets {A(t)} t∈R in E is said to be a pullback attractor of the process {U (t, τ )} if it is invariant, i.e., U (t, s)A(s) = A(t), t ≥ s, and it pullback attracts all bounded subsets of E, i.e., for every bounded subset D ⊂ E and t ∈ R, lim Pullback attractor is usually used to describe the longtime behavior of a nonautonomous dynamical system (cf. [2,16]). However, the pullback attractor may have some drawbacks: (i) the rate of convergence in (1) may be slow, which leads to the fact that it is difficult to estimate the pullback attracting rate in term of the physical parameters of the system; (ii) in many situations, one cannot show the finiteness of the fractal dimension for the sections of pullback attractor, which results in that the pullback attractor may be unobservable in experiments or in numerical simulations. In order to overcome these drawbacks, Efendiev et al [13] proposed the concept of pullback exponential attractor, which contains pullback attractor, pullback attracts every bounded subset at an exponential rate and is of finite fractal dimension. ln (1/ ) and N (A, ) denotes the cardinality of the minimal covering of the set A by the closed subsets of diameter ≤ 2 ; (iii) it pullback attracts every bounded subset B in E at an exponential rate, i.e., for some β > 0, where t, s ∈ R, s ≥ 0, B E = sup x∈B x E .
Lately, Efendiev et al [14] once again gave more general definitions on nonautonomous dynamical system and pullback exponential attractor which will be used in the present paper. Definition 1.2. [14] Let E be a Banach space with the norm · E , M be a subset of E, which is a metric space equipped with the distance d(x, y) = x − y E , the family {U (t, τ )} be a process acting on M . Then the triple (U (t, τ ), M, E) is said to be a non-autonomous dynamical system, M and E are said to be the phase space and the universal space, respectively. Obviously, M coincides with E in Definition 1.1. In addition, the existence of pullback exponential attractor implies the existence of finite dimensional pullback attractor (cf. [5]).
But there is also a question: pullback exponential attractors lose their uniqueness and may be sensitive to perturbations, that is, they may change dramatically under small perturbations of the system. However, a given system is usually an approximation of reality and it is thus essential that the pullback exponential attractors are robust under small perturbations. Since the pullback exponential attractors of a process are not unique, the optimal choice of a pullback exponential attractor which is robust under small perturbations is very important. So does the pullback attractor.
Generally speaking, in contrast to pullback attractor, it is expected that pullback exponential attractor is more robust under perturbations and numerical approximations because of its exponentially pullback attractability for any bounded subset in phase space.
In order to discuss the stability of pullback exponential attractors on the perturbations, it is imperative to construct a family of pullback exponential attractors {M σ (t)} t∈R for the family of processes {U σ (t, τ )}, σ ∈ Σ (Σ denotes an index set or a symbol space) such that the map σ → M σ (t) is, in some sense, stable.
Efendiev et al [13] proposed first criterion to construct pullback exponential attractors which are robust for the discrete process (the criterion is a development of that for discrete semigroup (cf. [12])), and they gave an explicit algorithm for the discrete process and an application to non-autonomous reaction-diffusion systems.
Langa et al [18], Czaja and Efendiev [10] extended the existence results in [13] to the continuous process, but the assumptions in [10,18] require the strong regularity on time t of the process, which seems typical for parabolic problems but is hard to realize for hyperbolic problems. Carvalho and Sonner [3] and Efendiev et al [14] further gave some alternative criteria on the existence of pullback exponential attractors for the continuous process, which remove the requirement for strong regularity on time t as in [10,18] and give some more relaxed assumptions of asymptotical compactness. In [4], Carvalho and Sonner applied the abstract criterion in [3] to the non-autonomous damped wave equations to obtain the existence of pullback exponential attractors of the process related to Eq. (3). We mention that there have been extensive investigations on the existence of pullback exponential attractors (see for example [1,11,25] and references therein). But all those investigations do not give any results on the stability of pullback exponential attractors with respect to perturbations except [13].
Recently, based on the assumptions on the stability and quasi-stability estimate, Chueshov [9] has established an abstract criterion on the existence of exponential attractors for a discrete semigroup, which includes many of others before as its special case because of its weaker assumptions.
Motivated by the idea in [9,13], we establish two new abstract criteria on the existence and stability of a family of pullback exponential attractors for the discrete dynamical system and continuous one, respectively (see Theorem 2.3 and Theorem 3.1), which are of more relaxed assumptions and applicability and are the developments of construction in [13]. By applying these criteria to non-autonomous Kirchhoff wave models with structural damping and supercritical nonlinearity with α ∈ (1/2, 1), we construct a family of pullback exponential attractors {M g (t)} t∈R and show their stability on the perturbations g ∈ Σ (see Theorem 4.4).
For the physical background of Eq. (4) and related researches, one can refer to [7,8,15,22,23,24] in detail. Recently, for the autonomous Kirchhoff wave model (4), i.e., g(x, t) ≡ g(x), Yang et al [24] have found an optimal supercritical exponent p α ≡ N +4α (N −4α) + (rather thanp ≡ N +2α (N −2) + as known before in [7], where a + = max{a, 0}). By the way, here the growth exponent p * , with p * ≡ N +2 N −2 (< p α ), N ≥ 3, is said to be critical relative to the natural energy space X = H 1 0 (Ω) ∩ L p+1 (Ω) × L 2 (Ω) for H 1 (Ω) → L p+1 (Ω) as p ≤ p * . It is also shown in [24] that when the growth exponent p of the nonlinearity f (u) is up to the supercritical range 1 ≤ p < p α , (i) the well-posedness and longtime behavior of solutions of Eq. (4) are of characteristics of parabolic equations. In particular, the solutions are of higher global regularity (rather than higher partial one as usual) as t > 0; (ii) the related solution semigroup S(t) has in X a global and an exponential attractor, respectively.
A challenging question is that whether the similar results on pullback exponential attractors hold for more complicated non-autonomous Kirchhoff model (4)? Unfortunately, the previous theory developed by Carvalho and Sonner [3] cannot be applied to the existence of pullback exponential attractor of the particular example considered in the current paper because its quasi-linear structure and supercritical nonlinearity cause that the traditional decomposition method of the evolution process as used in [4] ceases to be effective.
In this paper, the treatment for non-autonomous Eq. (4) is a continuation of researches in [24], under the same assumptions as in [24] except g ∈ H 1 b (R; L 2 ), by virtue of the criterion established above, we prove the existence of pullback exponential attractor and show their stability with respect to perturbations. To the best of the authors' knowledge, it is the first result on the existence and stability of pullback exponential attractors for the non-autonomous Kirchhoff wave models with structural damping and supercritical nonlinearity.
The main contributions of the paper are that (i) We establish two new criteria on the existence and stability of pullback exponential attractors for the non-autonomous discrete dynamical system (see Theorem 2.3 and the simplified proof for Theorem 2.4) and continuous one (see Theorem 3.1), respectively. Compared with the already published literature on this topic, the importance of these criteria lies in that they are based on recently developed quasi-stability method rather than traditional decomposition of the evolution process, which makes that they are of greater applicability because of their more relaxed assumptions than before. A model example (see Section 4) shows that they can be used to deal with more complicated quasilinear hyperbolic problem with supercritical nonlinearity. The paper provides answers for examples not tractable with the existing theory.
(ii) By applying above criterion, we construct a family of pullback exponential attractors {M g (t)} t∈R for a family of processes {U g (t, τ )}, g ∈ Σ generated by the Kirchhoff wave model (4), which are stable with respect to perturbations g ∈ Σ (symbol space) (see Theorem 4.4).
The paper is arranged as follows. In Section 2 and Section 3, we discuss two abstract criteria on the existence and stability of pullback exponential attractors for the non-autonomous discrete dynamical system and continuous one, respectively. In Section 4, we apply the criterion to non-autonomous Kirchhoff wave model (4) to construct a family of pullback exponential attractors which are stable with respect to perturbations.
2. Criterion 1 (Discrete case). In this section, we give a criterion on the existence and stability of pullback exponential attractors {M σ (n)} n∈Z for a discrete non-autonomous dynamical system (U σ (m, n), M, E).
(ii) there exist a Banach space Z and a compact seminorm n Z (·) on Z, and there exists a mapping K σ n : M → Z for each σ ∈ Σ, n ∈ Z such that sup where η ∈ (0, 1), L > 0 are constants independent of σ and n. Then for each θ ∈ (η, 1), σ ∈ Σ there exists a family {M σ θ (n)} n∈Z of compact subsets of M possessing the following properties: (ii) (boundedness of the fractional dimension) where where C is a positive consant. That is, the family {M σ θ (n)} n∈Z is a pullback exponential attractor of the non- C > 0 and λ : 0 < λ < 1 are constants independent of σ.
Proof. Condition (6) implies that the operator U σ (n) is an α-contraction (cf. [9]), i.e., Without loss of generality we assume that α(M ) < 1 because one can easily deduce from above inequality that α(U σ (n, n − k)M ) ≤ η k α(M ) < 1 for k suitably large, which means N (M, 1/2) < ∞, where N (B, ) denotes the cardinality of the minimal covering of the set B(⊂ E) by the closed subsets of diameter ≤ 2 .
be the minimal covering of M by its closed For every n ∈ Z and σ ∈ Σ, we define the pseudometric on M : where m ρ σ n (B, ) is the maximal cardinality of a subset {z k } in B such that ρ σ n (z k , z l ) > and We claim that sup n∈Z,σ∈Σ Therefore, by virtue of the linearity of the seminorm, where ℵ{· · · } denotes the maximal number of elements with the given properties. By the arbitrariness of and B(⊂ M ), we get (12), i.e., the claim is valid. Obviously, Thus (see (6)), Therefore (see (11)- (12)), Replacing M in (13) which implies that there exists a family of finite sets {V σ k (n)} such that where For any σ 0 ∈ Σ, we split the set Σ into the union Σ = Σ 1 ∪ Σ 2 , where (1) When σ ∈ Σ 1 , let Obviously (see (15)- (16)), We show that the family of sets {M σ θ (n)} n∈Z is of properties (i)-(iii) of Theorem 2.3.

ZHIJIAN YANG AND YANAN LI
The combination of (24)- (26) gives By the arbitrariness of b ∈ M , we have Let Taking account of the fact we infer from (18) and (23) that where [k Γ ] denotes the integer part of k Γ , C > 0 is a constant independent of k, n and σ. Now, we show that the family of sets {M σ θ (n)} n∈Z is of properties (i)-(iv) of Theorem 2.3 for every σ ∈ Σ 2 . ( then by the continuity of U σ (n) on M , (ii) (Pullback exponential attractability) By (27) and (15), we have (iii) (Boundedness of the fractal dimension) Based on (28), repeating the same proof as in case (1): (iii) we obtain estimate (8) and the compactness of M σ θ (n) in E for σ ∈ Σ 2 . (iv) (Stability w.r.t. perturbations) For any σ ∈ Σ 2 , a ∈ k≥1 E σ k (n), there must be a ∈ E σ k (n) for some k. When 1 ≤ k ≤ k Γ , by (24), (24) and (19), By the arbitrariness of a ∈ k≥1 E σ k (n), we obtain Similarly, repeating the proof of (29) (changing the position of σ and σ 0 ) and making use of (25), we obtain The combination of (29) and (30) gives (10).
By virtue of Theorem 2.3 here, one can easily deduce the criterion in [13].

Theorem 2.4. [13]
Let H and H 1 be two Banach spaces, H 1 compactly embed into H, and B be a bounded subset of H 1 . For given positive constants δ and K, we define a class S δ,K (B) of nonlinear operators S : Then the non-autonomous dynamical system (U (m, n), B, H 1 ) possesses a pullback exponential attractor {M U (n)} n∈Z , where B is equipped with the distance d(x, y) = x − y H1 . Moreover, the map U → M U (n) is uniformly Hölder continuous in the following sense: for each process U 1 and U 2 satisfying U i (n) ∈ S δ,K (B), n ∈ Z, i = 1, 2, we have where the positive constants C i , i = 1, 2, 3, α, β and κ depend only on B, H, H 1 , δ and K, and are independent of n and of the specific choice of the U i , Proof. Under the assumptions of Theorem 2.4, taking K U n ≡ I(identity operator) : H 1 → H 1 , then, for every process U acting on H 1 and satisfying U (n) ∈ S δ,K (B) for every n ∈ Z, we have that where η ∈ (0, 1) and K U n : M → Z is Lipschitz continuous, i.e., (ii) One sees from the above proof that the conditions of Theorem 2.3 are greatly weaker than those of Theorem 2.4, which leads to the fact that Theorem 2.3 is of greater applicability.
3. Criterion 2 (Continuous case). In this section, on the basis of Theorem 2.3, we further establish a criterion on the existence of a class of robust pullback exponential attractors for the time-continuous non-autonomous dynamical system (U σ (t, τ ), M, E). It is more applicable for a large class of evolution problem.
Theorem 3.1. (Continuous case) Let Σ be an index set or a symbol space, M be a bounded closed subset of the Banach space E, which is equipped with the distance d(x, y) = x − y E , and (U σ (t, τ ), M, E) be a non-autonomous dynamical system for each σ ∈ Σ. And assume that (i) there exist constants T > 0, L T > 0 such that, for any τ ∈ R, (ii) there exist a Banach space Z and a compact seminorm n Z (·) on Z, and there exists a mapping K σ n : M → Z for each σ ∈ Σ, n ∈ Z such that for any where η ∈ (0, 1), L > 0 are constants independent of σ and n.
Proof. For any σ ∈ Σ, we define a discrete process {U σ (m, n)} acting on the phase space M : which means (36).

Corollary 1.
Under the assumptions of Theorem 3.1, if {U σ (t, τ )}, σ ∈ Σ is also a family of processes acting on the Banach space E, and the bounded closed set M in Theorem 3.1 is a uniformly (w.r.t. τ ∈ R) absorbing set of the process {U σ (t, τ )} for each σ ∈ Σ, i.e., for any bounded subset B in E, there exists a T = T (B) > 0 such that τ ∈R U σ (t + τ, τ )B ⊂ M for all t ≥ T , then the pullback exponential attractor {M σ θ (t)} t∈R of the non-autonomous dynamical system (U σ (t, τ ), M, E) as shown in Theorem 3.1 is also a pullback exponential attractor of the dynamical system (U σ (t, τ ), E).

4.
Application to non-autonomous Kirchhoff wave models. We consider the existence and stability of pullback exponential attractors of the following nonautonomous Kirchhoff model with structure damping where α ∈ (1/2, 1), Ω is a bounded domain in R N (N ≥ 1) with the smooth boundary ∂Ω, and the nonlinearity f (u) and external force term g will be specified later.
For brevity, we use the following abbreviations: , H k are the L 2 -based Sobolev spaces. The notation (·, ·) for the L 2 -inner product will also be used for the notation of duality pairing between dual spaces, the sign H 1 → H 2 denotes that the functional space H 1 continuously embeds into H 2 and H 1 → → H 2 denotes that H 1 compactly embeds into H 2 , and C(· · · ) stands for positive constants depending on the quantities appearing in the parenthesis.
Rewriting Eq. (41) at an abstract level, we obtain where A = −∆, with Dirichlet boundary condition. Obviously, the operator A is self-adjoint in L 2 and strictly positive on V 1 . Then we can define the power A s of A (s ∈ R), and the spaces V s = D(A s 2 ) are the Hilbert spaces with the scalar products and the norms respectively. We define the phase spaces with α ∈ (1/2, 1), which are equipped with the usual graph norms, for example, Obviously, they are the Banach spaces, and In particular, where λ 1 (> 0) is the first eigenvalue of the operator A, and or else, there exist constants C 0 > 0, C 1 ≥ 0 such that be a symbol space, where {T (h)} h∈R is a translation group acting on L 2 loc (R; L 2 ), Lemma 4.1. [21] Let X be a Banach space with dual X , u, g ∈ L 1 (a, b; X). Then the following three conditions are equivalent: (i) There exists a ξ ∈ X such that in the scalar distribution sense on (a, b). If one of the conditions (i)-(iii) holds, then g = ∂ t u is the (X-valued) distribution derivative of u, u ∈ C([a, b]; X) and where the constant C = C(b − a).
. Then for every g ∈ Σ (see (47)), the (L 2 -valued) distribution derivative ∂ t g ∈ L 2 b (R; L 2 ), and sup Proof. For every g ∈ Σ 0 , g = T (h)g 0 for some h ∈ R. For any [a, b] ⊂ R, due to which means (by the arbitrariness of [a, b] ⊂ R), For any a ∈ R, it follows from Lemma 4.1 that g ∈ H 1 (a, a + 1; By the arbitrariness of a ∈ R, we have g ∈ C b (R; L 2 ) and Therefore, Σ is a bounded set in C b (R; L 2 ).
Proof. Similar to the arguments to the autonomous case (see [24]), one can easily obtain the conclusions of Theorem 4.3 except property (i). Thus we only prove property (i) here.

ZHIJIAN YANG AND YANAN LI
For every τ ∈ R, g ∈ Σ, we define the operator where u is the weak solution of problem (43)-(44). Theorem 4.3 shows that U g (t, τ ) is well-defined and {U g (t, τ )}, g ∈ Σ constitutes a family of processes on X, and the following translation identity holds: Theorem 4.4. Let Assumption (H) be valid. Then (i) for each g ∈ Σ, the non-autonomous dynamical system (U g (t, τ ), X) has a pullback exponential attractor {M g (t)} t∈R , where the sections M g (t) are uniformly (w.r.t. g ∈ Σ and t ∈ R) bounded in X 1+α ; (ii) there exists a δ > 0 such that for any g ∈ Σ, with g − g 0 where C > 0 and ν ∈ (0, 1) are some constants independent of g.
In order to prove Theorem 4.4, we first give a lemma.
where T 1 > 0 is a constant. Moreover, for each g ∈ Σ, B is a uniformly (w.r.t. τ ∈ R) absorbing set of the non-autonomous dynamical system (U g (t, τ ), X).
Obviously, there exists a T 0 > 0 such that g∈Σ U g (t, 0)B 0 ⊂ B 0 for t ≥ T 0 . Let We claim that B is the desired absorbing set. Indeed, (i) for every bounded set B ⊂ X, there exists a time t 0 = t 0 (B) > 0 such that g∈Σ U g (t, 0)B ⊂ B 0 for t ≥ t 0 . We have known that for any τ ∈ R, g ∈ Σ and t ≥ t 0 , with t 0 = t 0 + 1 + T 0 , there exist two elements g 1 , g 2 ∈ Σ (cf. Lemma 2.1 in [19]) such that i.e., B is a uniformly (w.r.t. τ ∈ R) absorbing set of the non-autonomous dynamical system (U g (t, τ ), X).
Proof of Theorem 4.4. (i) We first show that the family of non-autonomous dynamical systems (U g (t, τ ), B, X α ) (B is equipped with the distance d(x, y) = x − y Xα and g ∈ Σ) has a robust family of pullback exponential attractors.