TIME-DEPENDENT ASYMPTOTIC BEHAVIOR OF THE SOLUTION FOR PLATE EQUATIONS WITH LINEAR MEMORY

. In this article, we consider the long-time behavior of solutions for the plate equation with linear memory. Within the theory of process on time- dependent spaces, we investigate the existence of the time-dependent attractor by using the operator decomposition technique and compactness of translation theorem and more detailed estimates. Furthermore, the asymptotic structure of time-dependent attractor, which converges to the attractor of fourth order parabolic equation with memory, is proved. Besides, we obtain a further regular result.

With respect to the memory component, we assume that where ρ is a positive constant. The problem (1.1) stems from the elastic equation established by Woinowsky-Krieger( [21]). However, the strictly mathematical analysis and the survey of the global solutions as well as the asymptotic behavior for the linear plate equations should start with Ball who studied the stable property of the linear elastic beam equation in 1973 ( [2,3]).
When ε(t) is a positive constant independent of time t, the problem (1.1) has been extensively studied in many literatures. For instance, when µ(s) ≡ 0, Yang and Zhong in [24,25] achieved the existence of global attractors for the plate equation on a bounded domain, including not only the autonomous system with nonlinear damping but also the non-autonomous system with localized damping and critical nonlinearity. Asymptotic dynamics of plate equations on the unbounded domain were investigated by several authors such as Khanmamedov [9,10] and Xiao [22]. Ma and Yang obtained the existence of exponential attractors for the plate equation with strong damping in [13]. Recently, the random attractors of stochastic plate equation with strong damping and additive noise were considered in [14].
In the case when ε(t) is a positive decreasing function which vanish at infinity, the problem (1.1) becomes more complex and interesting; the reason is that the corresponding dynamical system is still understood under the non-autonomous framework even the forcing term in the equation is independent of time t. In order to deal with these problems, in [5], Conti, Pata and Temam presented a notion of time-dependent attractor exploiting the minimality with respect to the pullback attraction property, and gave a sufficient condition proving the existence of time-dependent attractor based on the theory established by Plinio, Duane and Temam ( [8]). Besides, they applied the new methods into the following weak damped wave equations with time-dependent speed of propagation ε(t)u tt + αu t − ∆u + f (u) = g(x). (1.8) Also, they proved that the time-dependent global attractor of (1.8) converged in a suitable sense to the attractor of the parabolic equation αu t − ∆u + f (u) = g(x) when ε(t) → 0 as t → +∞ ( [6]). Successively, in [7], they continued to show the asymptotic structure of time-dependent global attractor to the following specific one-dimensional wave equation sufficient conditions of the existence of time-dependent global attractor borrowed from the ideas in [15].
To the best of our knowledge, in [11,12] Liu and Ma have studied the existence of time-dependent attractors for the plate equations without linear memory. Just for the problem (1.1), the presence of the memory make it impossible to utilize (I − P m )u as the test function to capture the asymptotic compactness of the solution process, so the methods in [17] is out of action to our problem. For our purpose, we firstly construct a relatively complicated triple solution space by introducing a new variable. Secondly, we capitalize the decomposition techniques and compactness transitivity theorem to conquer the barriers induced by the critical nonlinearity and the history memory.
It is worth mentioning that the dissipative condition (1.5) is weaker than one in [16], because the authors made use of the dissipative condition, i.e., lim inf For convenience, hereafter, C (or c) denotes an arbitrary positive constant which may be different from line to line even in the same line.
The rest of this article consists of five Sections. In the next Section, we define some functions sets and iterate some useful lemmas. In Section 3, the existence of the time-dependent global attractor is obtained. In Section 4, the existence of the attractor is proved, and then we prove the upper-semicontinuous convergence of non-autonomous attractors of (1.1) to the global attractor of (4.1). Finally, in Section 5, we consider the further regularity of attractors.

Preliminaries.
2.1. Mathematical setting. As in ( [4,18]), we introduce a new variable η as follows then we can rewritten (1.1) as with initial and boundary conditions where Without loss of generality, set H = L 2 (Ω), endowed with the inner product ·, · and norm · , respectively. For s ∈ R + , we define the hierarchy of (compactly) nested Hilbert spaces especially, we have the embeddings H s+1 → H s .

TINGTING LIU AND QIAOZHEN MA
For s ∈ R + , let L 2 µ (R + ; H s ) be the family of Hilbert spaces of functions ϕ : R + → H s , equipped with the inner product and norm, respectively, Now, for t ∈ R and s ∈ R + , write the following symbols, . The letter s is always omitted whenever zero. Especially, we consider the timedependent phase space We will use sometimes also the space Notations and concepts. In the following, we review briefly the notations, some definitions and abstract results about process on time-dependent spaces, see ( [5,6]) for more details.
Let X t be a family of normed spaces, we introduce the R−ball of X t B t (R) = {z ∈ X t : z Xt ≤ R}. We denote the Hausdorff semidistance of two sets B, C ⊂ X t by: Definition 2.1. Let X t be a family of normed spaces. A process is a two-parameter family of mappings {U (t, τ ) : Definition 2.3. A time-dependent absorbing set for the process U (t, τ ) is a uniformly bounded family B = {B t } t∈R with the following property: for every R > 0 there exists a t 0 such that Definition 2.4. The time-dependent global attractor for U (t, τ ) is the smallest family A = {A t } t∈R , with the following properties: (ii) A is pullback attracting, namely, it is uniformly bounded and the limit holds for every uniformly bounded family C = {C t } t∈R and every t ∈ R.
Theorem 2.5. The time-dependent attractor A exists and it is unique if and only if the process U (t, τ ) is asymptotically compact, namely, the set Theorem 2.8. When the attractor A = {A t } t∈R is invariant, it coincides with the sets of all CBT of the process U (t, τ ), namely [24,25]) From (1.5), we can get for 0 < ν < 1 and c i > 0 (i = 1, 2), there hold Lemma 2.10. ( [4,18]) Assume that µ ∈ C 1 (R + ) ∩ L 1 (R + ) is a nonnegative function, and satisfies the following: if there exists s 0 ∈ R + such that µ(s 0 ) = 0, then µ(s) = 0, for all s ≥ s 0 holds. Moreover, let B 0 , B 1 , B 2 be Banach spaces, where B 0 , B 2 are reflexive and satisfies with m ≥ 0. Then, 3. Existence of the time-dependent global attractor.

3.1.
A priori estimates. Global existence of solution u to (2.2)-(2.3) is classical, by using the standard Galerkin approximation method, see ( [4,1]); that is, if (1.2)-(1.7) hold, then the problem (2.2)-(2.3) has a unique weak solution z(t) = (u(t), u t (t), η t (·)). Thus combining with Lemma 3.1, we can define: To prove Lemma 3.1, we first need the following estimate: Proof. Denote where u 2 2 ≥ λ 1 u 2 , ∀u ∈ H 2 (Ω). Now taking δ small enough, such that 1 − δL 2 λ1α > 0, and set so we conclude that d dt Together with (2.4) and Hölder, Young, Poincaré inequalities, it follows that where, we use ν − 2δ 2 L λ1 − δ 2 > ν 4 for δ < ν small enough. In addition, by Hölder, Young inequalities, and together with (1.6)-(1.7) we obtain (3.5) Combining with the above estimates and (2.5), Hölder, Young inequalities, it leads to where m 1 = c 1 + 2 λ1ν g 2 , m 2 = 2 λ1ν g 2 + 2c 2 . So, for any K 0 > m2 C1 , there exists t 0 > τ such that On the other hand, from the above discussion, there exist a positive constant R 0 such that and denote by C a generic positive constant depending on R but independent of z i . We first observe that the energy estimate in Lemma 3.2 above ensures: Multiplying the above equation by 2ū t and integrating over Ω, we obtain By exploiting (1.4), (3.7), and Hölder, Young inequality, and the embedding H 2 → L 2n n−4 (n ≥ 5), it yields Combining with the above estimates, we have then applying the Gronwall lemma on [τ, t], we obtain From Lemma 3.1 and Lemma 3.2, we can get the following conclusion: where γ is defined in (3.9). According to Lemma 3.3 we know that B = {B t (R 0 )} t∈R is a time-dependent absorbing set. Then, for any z(τ ) ∈ B τ (R 0 ), we split U (t, τ )z(τ ) into the sum solve the systems, respectively, and In what follows, the generic constant C > 0 depends only on B.

TINGTING LIU AND QIAOZHEN MA
As a consequence d dt thus, according to (3.20) and taking δ < ρ small enough, such that d dt Combining with (3.20), we can get the boundedness of K(t, τ )z(τ ) in H σ t . For proving this theorem, we need first to verify the compactness of the memory term.
Thus, combining with Lemma 2.10, it is easy to know that K T is relatively compact in L 2 µ (R + , H 2 ). Furthermore, using the compact embedding H σ+2 × H σ → H 2 × H we get: Lemma 3.8. Under the conditions of Lemma 3.7, for any T > τ, K(T, τ )B τ is relatively compact in H T .
Proof of Theorem 3.6. According to Lemma 3.5, 3.7, we consider the family R = {K t } t∈R , where K t = {z ∈ H σ t : z H σ t ≤ M }, Applying the compact embedding H σ t → H t and together with Lemma 3.8, we know that K t is compact; since the injection constant M is independent of t, the set R is uniformly bounded.
Besides, from Lemma 3.2, 3.3, 3.4, it's easy to know that R is pullback attracting, indeed, Hence the process {U (t, τ )} is asymptotically compact, which allows the application of Theorem 2.5 and achieve the existence of the unique time-dependent global attractor A = {A t } t∈R . Due to Lemma 3.1, we know the process {U (t, τ )} is strongly continuous, so A is invariant([11, Theorem 5.6]).

TINGTING LIU AND QIAOZHEN MA
For δ > 0 small and some C > 0 (depending on g ) large enough such that Taking δ < ρ small enough, we get Then, exploiting σ = 1 4 in Lemma 3.5, (3.19), and the embeddings H where C depends on δ, L. We finally end up with and an application of the Gronwall lemma, recalling (3.28) we can get the uniform boundedness of K 1 (t, τ )z(τ ) H 1 t . Proof of Theorem 3.9. Let (3.27) and Lemma 3.10, for all t ∈ R, it yields Since A is invariant, this means Hence, A t ⊂K 1 t = K 1 t , proving that A t is bounded in H 1 t with a bound independent of t ∈ R. Now from Lemma 3.3 and Theorem 3.9, the following conclusion is obtained immediately: Lemma 3.11. For any τ ∈ R, u ∈ A t , there exists a positive constant C such that 4. Asymptotic structure of the time-dependent attractor. We now investigate the relationship between the time-dependent global attractor of U (t, τ ) and the global attractor of the limit equation (4.1) formally corresponding to (1.1) when t → +∞.

Attractor of fourth order parabolic equation with linear memory.
If ε(t) ≡ 0 in (2.2) we obtain the following system where Ω is an open bounded set of R n (n ≥ 5) with smooth boundary ∂Ω, and with boundary and initial conditions Then under the conditions (1.4)-(1.7), and applying the standard semigroup theory [19,23] and a priori estimates (4.2), we can obtain that the problem (4.1) has a unique solution (u, η t ). Therefore, we can define the semigroup {S(t)} t≥0 acting on the space H 2 ×L 2 µ (R + , H 2 ) associated with the problem (4.1), such that (u(t), η t (·)) = S(t)(u 0 , η 0 ), where (u 0 , η 0 ) is the initial data of (4.1).
Proof of Theorem 4.1. Similar to Lemma 3.7, 3.8, the compactness of the memory term in (4.1) is checked, then combining with Lemma 4.3, we know that S(t) is asymptotically compact ( [9,23]), so the semigroup S(t) possesses a global attractor A ∞ in H 2 × L 2 µ (R + , H 2 ). 4.2. Asymptotic structure. From Theorem 4.1, we know that the semigroup S(t) associated with (4.1) has a unique attractor A ∞ in H 2 × L 2 µ (R + , H 2 ), moreover, according to Theorem 2.8, for all s ∈ R, we have The main result of this part establishes the asymptotic closeness of the timedependent global attractor A = {A t } t∈R of the process generated by (2.2)-(2.3) and the global attractor A ∞ of the semigroup S(t) generated by (4.1).
Theorem 4.4. The following limit holds: where Π t A t denotes the projection of A t into it's first component, that is, To prove (4.18), we need to verify the following result: Lemma 4.5. For any sequence z n = (u n , ∂ t u n , η t n ) of CBT of the process U (t, τ ) and any t n → ∞, there exists a CBT y = (w, ξ t ) of the semigroup S(t) such that, for every T > 0, sup and sup as n → ∞, up to a subsequence.
Proof. Let z n = (u n , ∂ t u n , η t n ) be a CBT of U (t, τ ) and t n → +∞ be given, owing to (3.29), for every T > 0, Then a direct application of [22, corollary 4] shows that u n (· + t n ) is relatively compact in C([−T, T ], H 2 ), for every T > 0. By Lemma 3.7, the sequence η t n (· + t n ) in the space . Hence there exists a function H 2 ) such that for sequence u n , η t n , the convergence u n (· + t n ) → w(·), η t n (· + t n ) → ξ t (s) hold, and y ∈ H 2 × L 2 µ (R + , H 2 ). Besides, recalling (3.29), sup we are left to show that y solves (4.1). We define v n (t) = u n (t + t n ), ε n (t) = ε(t + t n ), ξ t n (t) = η t n (t + t n ), then we write Eq.(2.2) for u n in the form We first prove that the sequence ε n (t)∂ tt v n converges to zero in the distributional sense. Indeed, for every fixed T > 0 and every smooth H-valued function ϕ, supported on (−T, T ), we have Exploiting again (3.29), we find where the generic constant c > 0 depends also on ϕ. Since in the topology of L ∞ (−T, T ; H −2 ). At the same time, the convergence ∂ t v n (t) → w t (t), ∂ t ξ t n → ∂ t ξ t , hold in the distributional sense. Therefore, we end up with the equality αw t + ∆ 2 w + f (w) + ∞ 0 µ(s)∆ 2 ξ t (s)ds = g(x), which together with (4.21), proves that y(t) is a CBT of the the semigroup S(t).
The proof of Theorem 5.1 is based on the following lemma.