Pullback attractors for the nonclassical di ff usion equations with memory in time-dependent spaces

: In this paper, we consider the asymptotic behavior of nonclassical di ff usion equations with hereditary memory and time-dependent perturbed parameter on whole space R n . Under a general assumption on the memory kernel k , the existence and regularity of time-dependent global attractors are proven using a new analytical technique. It is remarkable that the nonlinearity f has no restriction on the upper growth.

Thus the system (1.1) with (1.2) can be rewritten as follows: and the initial data u(x, τ) = u τ (x, τ), η τ (x, s) = s 0 u τ (x, τ − r)dr. (1.15) From (1.13), it is easy to obtain the following estimate: For Eq (1.1), it is often used to describe some physical phenomena, for example, non-Newtonian flows, soil mechanics and heat conduction theory.Specifically, when we study heat conduction problems in fluid mechanics or solid mechanics, if the influence of viscosity is emphasized, the classic heat-conduction equation is often extended to the following form (see e.g., [8][9][10]): However, when we consider polymer and high viscous liquid, etc., some important factors such as the historical influence of u and the disturbance coefficient of viscosity must be included [11], that is, the following evolution equation: The asymptotic behavior of solutions of Eq (1.16) has been studied by many scholars (see e.g., [3,[12][13][14][15][16][17][18][19] and the references therein).However, the research focused on the nonclassical diffusion equation with constant coefficient and bounded smooth domain early on, see e.g., [20][21][22][23][24][25][26].In [11], the diffusion equation with memory was proposed in the study of heat conduction and relaxation of high viscosity liquids.The convolution term represents the influence of past history on its future evolution and describes more accurately the diffusive process in certain materials, such as high viscosity liquids at low temperatures and polymers.Hence, it is necessary and scientifically significant to study the nonclassical diffusion equation with the time-dependent coefficient (i.e., variable coefficient) and memory.Furthermore, for Eq (1.16), we focus on the nonclassical diffusion equation with memory and a time-dependent perturbed parameter ε(t) on a bounded domain.For example, for the case of k(s) = 0 in (1.1), in [27,28], the authors proved the existence and regularity of the time-dependent global attractors on time-dependent spaces when the nonlinearity satisfies | f ′′ (u)| ≤ c(1 + |u|) and critically exponential growth, respectively.In addition, Wang and Ma studied the existence, regularity and asymptotic structure of the time-dependent global attractors in [29] for this equation when f meets polynomial growth of arbitrary order.In particular, Wang et al. [6] proved the existence of the timedependent global attractors for the problem with the nonlinearity of critical exponential growth.
More recently, the fourth author of the article and other co-authors considered the autonomous and nonautonomous nonclassical diffusion equation with memory or without memory on bounded domains, see e.g., [4,5,19,[30][31][32].In these papers, the operator decomposition and contractive functional methods are used to obtain the asymptotic regularity of the solutions and verify asymptotic compactness.It is worth mentioning that Xie et al. [4] have obtained the existence of the timedependent global attractors on bounded domain for the nonclassical diffusion equation (1.16), lacking instantaneous damping with the nonlinearity that satisfies the polynomial growth of arbitrary order.However, we focus on the unbounded domain.Therefore, we can see that there is few relevant studies for the asymptotic behavior of solutions of Eq (1.1) in time-dependent whole spaces under assumptions (H1)-(H3).This is because there are two major difficulties to obtain the existence of time-dependent global attractors.
(i) First, because of the nonlinearity with no restriction on the upper growth, the higher asymptotic regularity of the solutions of Eq (1.1) can not be obtained using the method of [33,34].
(ii) Second, due to the influence of the time-dependent perturbed parameter ε(t) and the lacking of compact embedding theorem on unbounded domains R n , it is impossible to directly construct the contractive function to prove the asymptotic compactness for the corresponding process {U(t, τ)} t≥τ of Eq (1.1) (see e.g., [29,30,35]).
For solving these problems, a new analytical technique combined with the operator decomposition method is used to obtain contractive function, and then the pullback asymptotic compactness for the process {U(t, τ)} t≥τ of Eq (1.14) is proved.Furthermore, using this operator decomposition method, the asymptotic regularity of the solutions for Eq (1.14) is also proved.Then, the regularity of time-dependent global attractors for this equation on The time-dependent space H 1 t = H 1 (R n ) and H 2 t = H 2 (R n ) are equipped with the norms: It is necessary to point out here that ∥ • ∥ 2 In fact the following inequality is obvious: Denote the weight spaces , and their inner products and norms are defined as follows: With the above notation, the phase spaces of Eq (1.14) can be denoted as equipped the following norms: respectively.Particularly, we use {M t } t∈R to denote a family of normed time-dependent spaces.Moreover, we introduce some common notations based on processes of time-dependent space(see e.g., [35][36][37][38]).
Let {M t } t∈R be a family of normed time-dependent space.Note that the ball with radius of R in M t is For any given ε > 0, we define the ε neighborhood of set B ⊂ M t as follows: Hausdorff semidistance of between two nonempty sets A, B ⊂ X t is defined as The plan of this paper is as follows.In Section 2, we recall some basic concepts as the timedependent global attractors and useful results that will be used later.In Section 3, we first prove pullback asymptotic compactness of the process corresponding to problem (1.14) with (1.15) by constructing contractive function, and then we obtain the existence and the regularity of time-dependent global attractors to problem (1.14) with (1.15) in whole space R n .

Preliminaries
In this section, we will recall some basic concepts of time-dependent global attractors and theories of the existence of time-dependent global attractors (see e.g., [35][36][37]).Definition 2.1.Let {M t } t∈R be a family of normed spaces.A two-parameter family of operators U(t, τ) : M τ → M t is called a process if it satisfies the following properties: bounded and for all R > 0, there exists a constant t 0 = t 0 (t, R) ≤ t such that U(t, τ)B τ (R) ⊂ B t for any τ ≤ t 0 .
The process {U(t, τ)} t≥τ is called dissipative whenever it enters a pullback absorbing family B0 = {B 0 t } t∈R .Definition 2.4.A time-dependent absorbing set for the process U(t, τ) is a uniformly bounded family B = {B t } t∈R with the following characteristics: For any R > 0, there exists t 0 = t 0 (t, R) ≥ 0, such that holds for any uniformly bounded family C = {C τ } τ∈R and every fixed t ∈ R and τ ≤ t.
Remark 2.1.The pullback attracting nature 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 any ε > 0 the family {O ε t (K t )} t∈R is pullback absorbing.Theorem 2.1.A time-dependent global attractor Ã exists and it is unique if and only if the process U(t, τ) is asymptotic compact, i.e., the set It can be seen from Definition 2.6 that the time-dependent global attractor is not necessarily invariant.This is mainly because that the process is not required to meet some continuity.If the process U(t, τ) satisfies the appropriate continuity, then the invariance of time-dependent global attractor A can be obtained.Definition 2.7.We say that Ã = {A t } t∈R is invariant if Lemma 2.1.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 contractive process, which will be used to obtain asymptotic compactness of a family of process {U(t, τ)} t≥τ (see e.g., [23,35,[39][40][41][42][43]).Definition 2.8.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 B τ × B τ .
Definition 2.9.Let U(t, τ) be a process on {M t } t∈R and have a pullback bounded absorbing set B = {B t } t∈R .U(t, τ) is called M t -contractive process if for any given ε > 0, there exist T = T (ε) and , where φ t T depends on T .Next, we will give the method to prove the existence of time-dependent global attractors for evolution equations, which will be used in the later discussion.Theorem 2.2.[35] 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: (i) U(t, τ) has a pullback absorbing set Lemma 2.2.[44] Let X ⊂⊂ H ⊂ Y be Banach spaces, with X reflexive.Suppose that {u n } ∞ n=0 is a sequence, 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 } ∞ n=0 that converges strongly in L 2 (τ, T ; H).

Time-dependent global attractors in {M t } t∈R
In this section, we shall consider the existence of time-dependent global attractors in {M t } t∈R .For this purpose, we have to first discuss the well-posedness for Eq (1.14) with (1.15).

The well-posedness of equation
The well-posedness for Eq (1.14) with (1.15) can be obtained by using Faedo-Galerkin method (see e.g., [38,45]).To this end, we first give the definition of weak solution.
).Furthermore, the following identity hold: ), which continuously depends on the initial data in M τ , i.e., there exists a constant κ > 0 not related to t such that the process U(t, τ) is Lipschitz continuous By Lemma 3.1, we may define the process of solutions on time-dependent space {M t } t∈R : In addition, it's easy to obtain that the process U(t, τ) is a strongly continuous process on the timedependent phase space {M t } t∈R .

Time-dependent absorbing sets
In the following discussion, let C mean any positive constant and Q(•) be a monotonically increasing function on [0, ∞) which may be different from line to line even in the same line.
Then we get and where Then from (3.9) and (3.12) we get Applying Grönwall lemma, we have By (3.11), it follows that Let Then the proof is complete.
Proof.We now multiply the first equation of (1.14) by u t in L 2 (R n ), then we get Setting From Lemma 3.2 and Corollary 3.2, we get that for any t > τ, there exists t ⋆ ∈ (τ, t] such that Furthermore, associating with (1.10), it is easy to obtain the following inequality: Let Therefore, we get  and Combining with (3.4) and (3.15), it follows that Then, Then, there is constant δ 2 > 0, such that Thus, we can rewrite (3.17) as follows: Applying Grönwall lemma, we have holds for any τ < t.Then the proof is complete.

Time-dependent global attractors
In subsection, we will prove the existence of time-dependent global attractors in {M t } t∈R through the process U(t, τ) defined by (3.3).In order to prove Theorem 3.1, we first give the following lemmas.Lemma 3.4.For any R > 0 and z τ = (u τ , η τ ) ∈ B τ (R) ⊂ M τ , then there exists a positive constant holds for any t ≥ τ.
Proof.From (3.19) and the value of β, one gets Integrating t from t to t + 1 on both sides of (3.22), and organizing it to obtain , then the proof is complete.Lemma 3.5.Let B be any bounded subset M τ and z τ ∈ B. Then for any ε > 0, there exist positive constants K 1 large enough and t 1 ≤ t, such that where b > 0 is a constant.Setting θ k = θ( 2|x| 2 k 2 ) and multiplying the first equation of (1.14) by θ 2 k u in L 2 (R n ), then we obtain where . Similar to (3.7), we define the function N k (t) as follows: Then it is follows that We can define a functional with a undetermined coefficient κ as follows: Let 0 < κ < 1 2θ 2 be sufficiently small, then it follows that and Combining with (3.23), (3.25) and (3.26), we obtain From (1.9), there is Next, we will estimate each item on the right side of (3.30).According to the definition of θ, it can be seen that Therefore, Similarly, we also can obtain the following estimates: Combining with (3.28), then (3.30) can be rewritten as follows: By Grönwall lemma, it follows that Combining with Lemma 3.3, we have and for any ε > 0, there exists K 1 ≥ 0 large enough, such that for every k ≥ K 1 , By (3.27), then, In order to obtain the asymptotic regularity estimates later, we decompose the solution U(t, τ)z τ = (u(t), η t ) into the following sum: where U 1 (t, τ)z τ = (v(t), ξ t ) and K(t, τ)z τ = (ω(t), ζ t ) solve the following equations respectively: where l is a constant from (1.5).It has an initial data with the initial value conditions Furthermore, we define the energy-like functional Then we get and where a 3 = 1+γθ 2 and ã3 = γ 2(1+γ) > 0. Therefore, it follows that By (3.41), we have (3.50)By Corollary 3.2 and the Grönwall lemma, we have Next, we will verify the existence and regularity of the pullback global attractors Ã for Eq (1.1).
As the end of this article, we will deduce the main conclusion as the following theorem: Theorem 3.2.The process U(t, τ) defined by (3.3) possesses a time-dependent global attractor Ã in {M t } t∈R , and Ã is non-empty, compact, invariant in {M t } t∈R and pullback attracting in {M t } t∈R .Furthermore, Ã ⊂ {M 1 t } t∈R .Proof.Thanks to Lemma 3.2 and Theorem 3.1, it's easy to get the existence of time-dependent global attractor Ã for the process U(t, τ) defined by (3.3) in time-dependent spaces {M t } t∈R .According to Lemmas 3.7 and 3.8, the pullback asymptotic regularity of the solutions of Eq (1.1) is proved, and the regularity of the time-dependent global attractor Ã is obtained.By Lemma 2.1 and (3.2), it follows that the invariance of time-dependent global attractor Ã .

Conclusions
We conclude the existence, uniqueness and regularity of time-dependent global attractors on whole space.The findings of this study can be considered as a supplement to our previous works, such as [4,5].We overcome some essential difficulties for studying this kind of problem, including that the compact embedding is no longer valid under the case of unbounded domain and the nonlinear term fulfills the supcritical growth as well as the memory kernel satisfying more general assumptions.However, our results show that the method of operator decomposition that was proposed in [46] is available for dealing with the case of unbounded domain like (1.1).
Unfortunately, we fail to consider the existence of time-dependent global attractors for Eq (1.1) which lacks instantaneous damping on whole space, and further study the upper-semicontinuity of time-dependent global attractors between two kinds of equations.Future studies shall consider such issues using the ideas of the paper and [4,5], i.e., the asymptotic behavior of solutions for nonautonomous and autonomous equations (1.1) lacking instantaneous damping on whole space.

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Definition 3 . 1 .
For any R > 0 and T > τ, let I = [τ, T ], then the function z = (u, η t ) defined on R n × I is called a weak solution of the problem (1.14) with initial value
Acting on the first equation of (3.35) by −∆ω(t) on L 2 (R n ), we have holds for any τ ≤ t ∈ R.Proof.