ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO NONCLASSICAL DIFFUSION EQUATIONS WITH DEGENERATE MEMORY AND A TIME-DEPENDENT PERTURBED PARAMETER

. This article concerns the asymptotic behavior of solutions for a class of nonclassical diffusion equation with time-dependent perturbation coefficient and degenerate memory. We prove the existence and uniqueness of time-dependent global attractors in the family of time-dependent product spaces, by applying the operator decomposition technique and the contractive function method. Then we study the asymptotic structure of time-dependent global attractors as t → ∞ . It is worth noting that the memory kernel function satisfies general assumption, and the nonlinearity f satisfies a polynomial growth of arbitrary order.

For simplicity, we let (H2) The nonlinearity f satisfies f ∈ C 1 , f (0) = 0, and the arbitrary order polynomial growth restriction and the dissipative condition where α i , β i (i = 1, 2) and l are positive constants.
When ε(t) is zero (or a positive constant) and memory term is non-degenerate, it is easy to show that the equation (1.1) becomes the usual reaction-diffusion equation (or nonclassical diffusion equation) with memory, under these circumstances, the asymptotic behavior of solutions has been researched by many scholars in recent years (see [9,17,18,19,31,35,36,37]). Especially to deserve to be mentioned, more recently, the authors considered the existence, regularity and upper semicontinuity of global and uniform attractors for autonomous and non-autonomous nonclassical diffusion equation lacking instantaneous damping −∆u in bounded and unbounded domain when the nonlinearity satisfies critical exponential growth and polynomial growth of arbitrary order respectively, see [10,29,32,33,36,38,39].
Nevertheless, for equation (1.1) with time-dependent parameter, the current studies focus on the nonclassical diffusion equation without memory (i.e., k(s) = 0 in (1.1)), see [23,24,30,40] and the references therein.In [23,24,40], the authors proved the existence of time-dependent global attractors in H t when the nonlinearity f satisfies the subcritical exponential growth, critical exponential growth, and polynomial growth of arbitrary p − 1 (p ≥ 2) order respectively.Particularly, when f meets polynomial growth of arbitrary order, the authors of [30] proved the existence, regularity and the asymptotic structure of the time-dependent global attractors for the equation (1.17) To sum up, we try to consider the long-term behavior of equation (1.17) with memory (i.e., the case of non-degenerate) and without linear damping.At this point, if the memory term satisfies classical assumption µ(s) + δµ ′ (s) ≤ 0 (see [31,35]), then the results we obtain are perfectly predictable.This allows us to think of the asymptotic behavior of problem (1.1) under the premises that the (degenerate) memory term satisfies the weaker assumption (1.7) and the nonlinearity fulfills polynomial growth of arbitrary order?So why would we consider the system (1.1) with the degenerated memory?In [4], the authors completed a pioneering work, namely, the uniform decay of solutions was obtained for degenerate problem where a(x) ∈ C 1 ([0, 1]) satisfies a(x) > 0 and a(x)| x=0,1 = 0, and b(x) ∈ C([0, 1]) meets b(x) ≥ 0. Whereafter, some authors investigated the asymptotic behavior and stability of solutions for above problem under suitable assumptions, see [16,1] and the references therein.In addition, many scholars considered wave equation with degenerate memory, see [5,6,7,27] and the references therein.In should be emphasized that the authors in [27] obtained regularity of global attractors for the following wave equation with degenerate memory Furthermore, Faria et al. showed the existence of global attractors for the following heat equation with degenerate memory in recent a study [15] when the nonlinear term f (θ) fulfills critical exponential growth and the memory kernel function satisfies the weaker conditions (see (H1)).
The ideas in [15,39] inspires us to take into account the existence and uniqueness of time-dependent global attractors for equation (1.1) when the assumptions (H1) and (H2) hold, which answers the question we posed earlier.Moreover, in this article, we also incidentally consider asymptotic structure of time-dependent global attractors based on existing studies in [11,12,30].Thus, it is a comprehensive and innovative problem for us to think about the existence, uniqueness and the asymptotic structure of time-dependent global attractors for the equation (1.1), and this article improves the existing work in [15,32,33,37].
Of course, we need to overcome the following two difficulties for solving foregoing problem: (i) On the one hand, because the nonlinearity f has polynomial growth of arbitrary order and equation (1.1) includes the memory term, we cannot use Sobolev compact embedding to verify asymptotic compactness of the solution process generated by equation (1.1) as [15].(ii) On the other hand, since the memory term is degenerate and the memory kernel function k(s) satisfies the weaker condition (H1), which does not allow us to the classical estimation methods from [18,31,35,37] in our problem.To solve the above difficulties, we use some ingenious analytical techniques, and the operator decomposition method is adopted to obtain constructive function.Then we can verify that the process {U (t, τ )} t≥τ generated by equation (1.1) is pullback asymptotically compact.Meanwhile, the asymptotic regularity of solutions for equation (1.1) is also obtained, and it follows that we can construct the contractive function and further show pullback asymptotical compactness of the corresponding process {U (t, τ )} t≥τ associated with the equation (4.1).In addition, the study of asymptotic structures shows that the limit relation between the time-dependent attractors for the problem (1.1) and the global attractor for the reaction-diffusion equation with degenerate memory of [15] with the same conditions by using the method in [11].
This article is organized as follows: In Section 2, we recall some basic concepts with respect to the time-dependent global attractors and other some useful results that will be used later.In Section 3, we first prove pullback asymptotic compactness of the process {U (t, τ )} t≥τ generated by problem (1.15) by establishing contractive function, and then the existence and uniqueness of time-dependent global attractors are attained for system (1.15)- (1.16).In Section 4, we obtain the limit relation between the time-dependent attractors for the equation (1.1) and the global attractor for the equation (1.1) in [15] satisfying the same conditions.

Preliminaries
In this section, we first give some notation used later, and then describe some basic concepts and theories of the existence of time-dependent global attractors, for details see [12,11].
Basic concepts and notation.Hereafter let |u| be the modular (or absolute value) of u and | • | p be the norm of L p (Ω)(p ≥ 1), and (•, •) be the inner product of As in [7,6], we assume that there is a Hilbert space We define the weight space a ds.As in [27] we define the regular Hilbert space We define the weight space ) and its inner product and norm Then phase space of equation (1.1) is , where {M t } t∈R represents a family of timedependent normed spaces, and it should be noted that the superscript is omitted when r = 0. Remark 2.1.As stated in [7,27,5], there exists reasonable continuous embedding a (Ω) does not hold.Next, we introduce some common notation based on processes of time-dependent space (see [23,12,11]).
Let {M t } t∈R be a family of normed spaces.Note that the ball of radius R in For any given ε > 0, we define the ε neighborhood of set B ⊂ M t as The process {U (t, τ )} t≥τ is called dissipative whenever it enters a pullback absorbing family B0 = {B 0 t } t∈R .Definition 2.5.A time-dependent absorbing set for the process U (t, τ ) is a uniformly bounded family B = {B t } t∈R with the following characteristic: for any R > 0, there exists t 0 = t 0 (t, R) ≥ 0, such that (ii) Ã is pullback attracting, namely, Ã is uniformly bounded and holds for all uniformly bounded family C = {C τ } τ ∈R and every fixed t ∈ R and τ ≤ t.
Remark 2.8.The pullback attracting essence can be equivalently described in the light of pullback absorbing: a (uniformly bounded) family K = {K t } t∈R is said to be pullback attracting if for all ε > 0 the family {O ε t (K t )} t∈R is pullback absorbing.Theorem 2.9.A time-dependent global attractor Ã exists and it is unique if and only if the process U (t, τ ) is asymptotically compact, i.e., the set It can be seen from Definition 2.7 that the time-dependent global attractor does not have to be invariant, which is because the process does not require to meet some continuity.If the process U (t, τ ) satisfies appropriate continuity, then the invariance of time-dependent global attractor Ã can be obtained.

Definition 2.10. The time-dependent global attractor
Lemma 2.11.If the time-dependent global attractor Ã exists and the process U (t, τ ) is a strongly continuous process, then Ã is invariant.
Next, we will state the definitions of contractive function and M t -contractive process, which will be utilized to prove asymptotic compactness of a family of process {U (t, τ )} t≥τ (see [22,25,28,34,35]).Definition 2.12.Let {M t } t∈R be a family of Banach spaces and B = {B t ⊂ M t } t∈R be a family of uniformly bounded subset.We call function φ(•, •), defined on M τ ×M τ , to be a contractive function on We use E(B τ ) to denote the set all contractive function on . where φ t T depends on T .Next, we give the method to prove the existence of time-dependent global attractors for evolution equations, which will be used in the later discussion.
Theorem 2.14 ([25]).Let {M t } t∈R be a family of Banach spaces, then U (t, τ ) has a time-dependent global attractor in {M t } t∈R , if the following conditions hold In what follows, we give two Lemmas, which shall be used to prove the compactness of solution process.Lemma 2.15 ([26]).Let X ⊂⊂ H ⊂ Y be Banach spaces, with X reflexive and T, τ be two constants with τ ≤ T .Suppose that u n is a sequence that is uniformly bounded in L 2 (τ, T ; X) and du n /dt is uniformly bounded in L p (τ, T ; Y ), for some p > 1.Then there is a subsequence of u n that converges strongly in L 2 (τ, T ; H).

Lemma 2.16 ([3]
). Suppose that the nonnegative function µ ∈ L 1 (R + ) is decreasing piecewise absolutely continuous, and it satisfies that if there exists

Existence and uniqueness of a time-dependent global attractor
It is easy to know that the key for existence and uniqueness is to verify the pullback asymptotic compactness of process generated by (1.1).To do this, we first prove the asymptotic regularity of solutions, then the contractive function can be constructed by it, which can ensure the pullback asymptotic compactness of corresponding process.
3.1.Well-posedness.We now describe the well-posedness for the equation (1.1), which can be obtained by using standard Faedo-Galerkin method (see e.g., [30,26,29]).For simplicity, we only give the final conclusion.Lemma 3.1.Let Ω be a bounded domain of R n (n ≥ 3) with smooth boundary ∂Ω, z τ = (u τ , η τ ) ∈ M τ , and (H1)-(H3) be satisfied.Then for any T > τ , the system (1.15)-(1.16)possesses a unique weak solution z(t) = (u(t), η t ) satisfying (3.1) are two weak solutions of (1.15)-(1.16),then for any t ≥ τ , it is easy to obtain a positive constant C ℜ independent of t, such that Remark 3.2.By Lemma 3.1, the following solution process can be defined on the family of time-dependent spaces In particular, from (3.2), it is easy to find the process U (t, τ ) is Lipschitz continuous.That is to say, U (t, τ ) is a strongly continuous process over the family of timedependent phase space {M t } t∈R .
3.2.Time-dependent absorbing sets.In this subsection, we shall study the dissipative feature for the process {U (t, τ )} t≥τ .To this end, we need a series of prior estimates.Throughout this subsection and subsequent sections, we always assume that Ω ⊂ R n (n ≥ 3) is a bounded domain with smooth boundary, the initial value (Ω) and the assumptions (H1)-(H3) hold.
then L(t) satisfies the differential inequality and where Proof.Combining with Hölder inequality, Young inequality and assumption (H3), we have In addition, (3.7)By (3.6) and (3.7), one can obtain (3.4).Then it is easy to obtain The proof is complete.
Proof.By (3.18) by letting the conclusion holds.□ The following Lemma can be obtained from Lemma 3.3.we omit its proof.
Lemma 3.6.Under the assumption of Lemma 3.3.Let Then B(t) satisfies the differential inequality and the estimate where To obtain the bounded absorbing set of M t , we need the following result.Lemma 3.7.Under the assumption of Lemma 3.4, for each t − τ ≥ T 1 = T (R), there exists Proof.Taking the inner product of η t and the second equation of (1.15), we have 1 2 For the right-hand side we have by (3.17), we have where γ 1 is appropriately small to ensure 1 − 2γ 1 δ > 0. By combining with (3.19) and (3.23), we obtain Then by Corollary 3.5 and (3.24)-(3.25),we have where c 3 = γ1 2(1+2γ1δ) .Applying Gronwall's Lemma to (3.26), we have From this, (3.24), and (3.27), there exists where Next, we show the existence of a bounded absorbing set.
Then, for any given R ∈ R + , there exists ρ 0 > 0 such that the process U (t, τ ) generated by (1.15) possesses a time-dependent bounded absorbing set B0 = {B 0 t } t∈R (: this is, there exists T 0 = max{T 0 , T 1 } ≥ 0, such that Proof.In Lemmas 3.4 and 3.7, just take ρ 0 = κ 0 + κ 1 and T 0 = max{T 0 , T 1 } ≥ 0. Then the above conclusion of the theorem follows.□ 3.3.Time-dependent global attractors.Now we prove the existence of timedependent global attractors in {M t } t∈R for the process defined by (3.3).This will be Theorem 3.17, but we first give the following lemmas.
Lemma 3.9.Under the assumption of Lemma 3.4, there exists positive constant Proof.According to (3.12), by letting c 4 = min{ Proof.Using ε(t)u t to make inner product with the first equation of (1.16) in L 2 (Ω), and combining with (1.4)-(1.5),we have Estimating the first term at the right-hand side hand of (3.34), we obtain Integrating (3.36) about t from s to t + 1, (t ≤ s ≤ t + 1); then by Corollary 3.5, we have from (1.11), we know that the above inequality can be turned into where (3.39) Thus, by Corollary 3.5 and (3.38), we have where To demonstrate asymptotic properties of solutions corresponding to the process {U (t, τ )} t≥τ , we decompose the solution z = (u, η t ) of problem (1.15)-(1.16)with initial data z τ = (u τ , η τ ) ∈ M τ into the sum where z 1 = U 1 (t, τ )z τ = (v(t), ζ t ) and z 2 = U 2 (t, τ )z τ = (w(t), θ t ) are two solutions of the following systems respectively: with initial-boundary conditions where μ > l (from (1.10)) is a constant, and (3.44) with initial-boundary conditions Next, we show that z 1 has M t -decay, but not necessarily exponential decay.
Proof.We divide the proof into two steps.
Step 1. Multiplying the first equation of (3.42) by v, then integrating over Ω, we obtain where then as in the proof of Lemma 3.3, we obtain that N 1 (t) satisfies the differential inequality Applying Gronwall's Lemma for (3.51), and taking limit about τ , we have lim Thus, from (3.50) it follows that 0 ≤ lim Step 2. As in Lemma 3.6, suppose that Then we have that N 2 (t) satisfies the inequality 2 ds.Taking the inner product of v and the second equation of (3.42) on V 0 , we obtain where γ2 is a constant to be defined later.Let Then we have for all t ≥ τ .
The proof of the above lemma is similar to the proof of Corollaries 3.5 and 3.10, Lemma 3.11, and (3.38) word by word.So we omit it here.Next we show that, for all time, the component z 2 belongs to a subset of M 1 t , uniformly as the initial data z τ belongs to the absorbing set B0 , given by (3.28).Lemma 3.14.For each R > 0, let ∥z τ ∥ Mτ ≤ R. Then there exists a constant Proof.Firstly, for degenerate memory term, we have (3.62) Then using −∆ω to make inner product over L 2 (Ω) for (3.62).At this time, we only need to deal with the right-hand side of (3.62); that is In addition, by combining with the second equation of (3.44), we obtain Next, applying −∆ω on the first equation of (3.44) on L 2 (Ω), then combining with Hölder inequality, Young inequality, and (3.62)-(3.63),we have 1 2 (3.64) Furthermore, using −∆θ t to make inner product on V 0 , we obtain 1 2 Then according to Corollary 3.5, we have where γ4 = 1 + m 0 (|∇a| 2 ∞ + 1 μ−l ).Applying Gronwall's lemma to (3.66), we have Letting K 3 = e γ4(t−τ ) (4μ 2 K 0 + 4|g| 2 2 ), the conclusion follows.□ The following result shall be used in the proof of asymptotic structure of timedependent global attractors.Lemma 3.15.For each t > τ , let C t := PU 2 (t, τ )B 0 τ , where P : ).Also we know that θ t (s) = t t−s ω(y)dy, so ∂ s θ t (s) = ω(t − s).Thus, from and Lemma 3.7 and Lemma 3.13, it follows that θ t and ω(t) are bounded in . Additionally, by Lemma 3.7 and Lemma 3.14, we obtain where From the above arguments, Lemma 2.16, and ( After the above preparations, we can prove the existence and uniqueness of time-dependent global attractors.The key to achieve this goal is to demonstrate the pullback asymptotic compactness of process U (t, τ ).We know from [20] that the method of the standard Kuratowski measure of non-compactness may be useful for verifying the asymptotic compactness of solution process generated by equation (1.15).But the contractive function method seems to be more concise for our problem, which is mainly based on our previous research [32,33,36].Thus, we only need to prove that U (t, τ ) is a M t -contractive process by Theorem 2.14.
Theorem 3.16.The family of process {U (t, τ )} t≥τ generated by (1.15) with initialboundary value coditions (1.16) is a family of M t -contractive process on B 0 )) respectively.By (3.41), we can establish the decomposition and by Lemma 3.12, we have lim Hence, for each ε > 0, there exists δ holds for any t ≥ T = T (ε) fixed.Moreover, it is easy to check that (ψ(t), is the solution of the system with initial-boundary value conditions Taking the inner product of ψ and ξ t on L 2 (Ω) and V 0 for the first and second equations of (3.69) respectively, by Hölder inequality, we have Taking ε ∈ (0, 1/δ), such that where C = max{ 2μ 2 μ−l , γ5 1−δε }.From (3.76) and (3.77), we have (3.78) Then we let (3.79) Combining this, Corollary 3.5, and Lemma 3.11, and applying Lemma 2.15, there exists a subsequence of {u n (s)} ∞ n=1 that converges strongly in L 2 (T, t; L 2 (Ω)).In other words, for any sequences Similarly, through Lemmas 3.13 and 3.14, it is easy to obtain that the set  [15] can be weakened to polynomial growth of arbitrary order (such as (1.9)), then the existence of global attractors for the reaction-diffusion equation with degenerate memory (i.e., the equation (4.1)) can still be obtained.
(ii) Similar to the study of [12], for sufficiently small δ, we can also obtain the regularity of time-dependent attractor A by Lemmas 3.12 and 3.14, i.e., A ⊂ {M 1 t } t∈R .This is so because we can obtain the uniform boundedness of ∥u(t, τ )z τ ∥ 2 M 1 t with respect to τ in Lemma 3.14 by introducing the functional of Lemma 3.3 when δ is sufficiently small.

Asymptotic structure of the attractor
Following the idea in [11], we study the relationship between the time-dependent attractor for problem (1.1)-(1.3)and the global attractor for the following limit system (4.1) with initial-boundary (1. where s)}ds.For this purpose, we introduce the following conclusions about the completely bounded trajectories (CBT); for more details see [11].
where Π is the projection from X × Y to X.The process U (t, τ ) defined by(3.3)possesses a time-dependent global attractor A = {A(t)} t∈R in {M t } t∈R , and A is non-empty, compact, invariant in {M t } t∈R and pullback attracting every bounded set in {M t } t∈R .Proof.By Theorems 3.8 and 3.16, the existence and uniqueness of time-dependent global attractor A for the process U (t, τ ) generated by equation (1.1) in timedependent product spaces {M t } t∈R .In addition, from Lemma 2.11 and (3.2) of Lemma 3.1, we can obtain the invariance of time-dependent global attractor A .□ Remark 3.18.(i) By fully utilizing the method in this paper, if the condition of the nonlinearity f (u) in Proof.By Lemma 3.17, we can obtain (4.2).For the conclusion (4.3), as in the proof of Lemma 3.11, we only need to use the inner product of u t and (1.15) on L 2 (Ω).Lemma 4.6.Under the assumptions of Lemma 4.4, for every sequence z n = (u n , η t n ) of CBT for the process U (t, τ ) generated by (1.1) and any t n → ∞ as n → ∞, there exists a CBT z = (ū, ηt ) of the semigroup S(t) generated by (4.1) such that, for each T > 0, up to a subsequence, A = {z : t → z(t) = (u(t), η t ) ∈ M t with z CBT of U (t, τ )}, 22 ≤ K 5 .(4.5)On the other hand, for any z ∈ A(t), by Theorem 3.17 we can obtain that By Lemma 3.15 and (4.6), we obtain that the sequence η •+tn t (s) is bounded in L ∞ ([−T , T ]; L 2 µ (R + ; H 2 a (Ω)) ∩ H 1 µ (R + ; L 2 (Ω))), which indicates that Theorem 3.8, Lemma 3.12, Lemma 3.14, Theorem 3.17 and Lemma 2.16 implies that θ t n (s) is relatively compact in L 2 µ (R + ; H 1 0 (Ω)) (the proof is similar to Lemma 3.15).Thus, we have − ∇η t (s)|22 ds → 0, as n → ∞, this implies that (4.11) holds.Meanwhile, in the distributional sense, there exists a subsequence such that∂ t υ n → ūt .whichimplies that z = (ū, ηt ) is solution of (4.1).Combining (1.12) and (4.5), it is clear that z is a CBT for the semigroup {S(t)} t≥0 .□ By Lemma 4.6 and Theorem 4.3, the following conclusion can be obtained at once.Theorem 4.7.If A := {A t } t∈R = {A(t)} t∈R and A ∞ is time-dependent global attractors and global attractors of {U (t, τ )} t≥τ and {S(t)} t≥0 generated by (1.1) and (4.1) respectively.Then there is lim t→∞ dist M0 (Π t A t , A ∞ ) = 0.