Upper semicontinuity of attractors for nonclassical diffusion equations with arbitrary polynomial growth

In this paper, we mainly investigate upper semicontinuity and regularity of attractors for nonclassical diffusion equations with perturbed parameters ν and the nonlinear term f satisfying the polynomial growth of arbitrary order p−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p-1$\end{document} (p≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p \geq 2$\end{document}). We extend the asymptotic a priori estimate method (see (Wang et al. in Appl. Math. Comput. 240:51–61, 2014)) to verify asymptotic compactness and upper semicontinuity of a family of semigroups for autonomous dynamical systems (see Theorems 2.2 and 2.3). By using the new operator decomposition method, we construct asymptotic contractive function and obtain the upper semicontinuity for our problem, which generalizes the results obtained in (Wang et al. in Appl. Math. Comput. 240:51–61, 2014). In particular, the regularity of global attractors is obtained, which extends and improves some results in (Xie et al. in J. Funct. Spaces 2016:5340489, 2016; Xie et al. in Nonlinear Anal. 31:23–37, 2016).


Introduction
In this paper, we consider the following perturbed nonclassical diffusion equation: (1.1) The problem is supplemented with the boundary condition u(x, t)| ∂ = 0 for all t ≥ 0 (1.2) and the initial condition where is a bounded smooth domain in R n (n ≥ 3), ν ∈ [0, +∞) is a perturbed parameter, and g ∈ L 2 ( ) is a given external force term.
(1. 6) This equation appears as an extension of the usual diffusion equation in fluid mechanics, solid mechanics, and heat conduction theory (see, e.g., [4][5][6][7]). Equation (1.1) with a first-order time derivative appearing in the highest-order term is called pseudo-parabolic or Sobolev-Galpern equation [8][9][10]. In [4], Aifantis proposed a general frame for establishing this equation for certain classes of materials such as polymer and high-viscosity liquids.
In this paper, our main purpose is to consider upper semicontinuity of attractors for Eq. (1.1) with the nonlinearity satisfying arbitrary-order polynomial growth condition, which makes the Sobolev compact embedding no longer valid and brings more difficulties for verifying the corresponding asymptotic compactness of the family of solutions semigroup {S ν (t)} t≥0 , ν ∈ [0, +∞). In the existing literature, many methods are not applicable to overcome these difficulties (see, e.g., [26][27][28][29][30][31][32]). In order to overcome the diffi-culty mentioned above, we introduce the asymptotic contractive function method to verify asymptotic compactness of a family of semigroups for autonomous dynamical systems (see Theorem 2.2) by referring to the methods and ideas in [2,3]. Then, by using the new operator decomposition method, we obtain the corresponding asymptotic regularity of the solutions for Eq. (1.1), which ensures the existence of asymptotic contractive function for our problem, which generalizes the results obtained in [1].
For convenience, hereafter let |u| be the modular (or absolute value) of u and | · | p be the norm of L p ( ) (p ≥ 1). Let V = H 1 0 ( ) and · 0 = |∇ · | 2 be the norm of V. Denote A = -with domain D(A) = H 2 ( ) ∩ H 1 0 ( ). C means any positive constant and Q(·) is a monotonically increasing function on [0, ∞), which may be different from line to line, even in the same line. Let I ⊂ [0, +∞) be a bounded closed interval.
The main results of this paper are given in the following two theorems, which will be proved in Sect. 2 and Sect. 3 respectively.
It is worth noting that Theorem 1.2 is also interesting in the nonautonomous case (i.e., g is dependent on t) (see, e.g., [1,22,24]). Obviously the results obtained herein are also applicable to considering upper semicontinuity of global attractors for Eq. (1.1) with memory [3] or in unbounded domain [1,2,23]. Particularly, the nonlinearity f is assumed to satisfy the polynomial growth of arbitrary order instead of critical nonlinearity (see, e.g., [20,23]).
The plan of this paper is as follows. In Sect. 2, we recall some basic concepts and results that are used later. In Sect. 3, by using the ideas in [23], we first verify the asymptotic regularity of the solutions of Eq.
We denote the set of all asymptotic contractive functions on B 2 × I 2 by E(B, I).
In the following theorem, we present a new method(or technique) to verify the relative compactness of a two-parameter sequence for the family of semigroups generated by evolutionary equations, which will be used in our later discussion.

Theorem 2.2
Let (X, · X ) be a Banach space, B be a bounded subset of X, and I be a parameter interval. Assume further that {S ν (t)} t≥0 , ν ∈ I, is a family of semigroups on X which satisfy the following conditions: Then the sequence {S ν n (t n )x n } is precompact in X for any {x n } ⊂ B, {ν n } ⊂ I, and t n n→∞ +∞), and t n → +∞ as n → ∞. By (i), for every x ∈ B 0 , t n = t nt N > 0, and ν ∈ I. Then we just need to consider this case as The following work is to prove the existence of a Cauchy subsequence of {S ν n (t n )x n } by the diagonal method. Taking Since t n → +∞, for T 1 fixed, we assume that t n T 1 is so large that S ν n (t n -T 1 )x n ∈ B 0 for each n ≥ N and ν n ∈ I. Let Due to the definition of E(B 0 , I) and ϕ T 1 ∈ E(B 0 , I), we know that there exist a contractive function φ T 1 and a subsequence {(y (1) Therefore, there exists N 1 such that Then the proof is complete.
For more details of the standard theory of global attractors, we recommend the readers to refer to [33]. Now, we present the following theorem to verify upper semicontinuity of global attractors in autonomous dynamical systems.

Lemma 2.4 ([33]
) Let X ⊂⊂ H ⊂ Y be Banach spaces with X reflexive. Suppose that u n is a sequence which is uniformly bounded in L 2 (0, T; X) and du n /dt is uniformly bounded in L p (0, T; Y ) for some p > 1. Then there exists a subsequence of u n that converges strongly in L 2 (0, T; H).

Global attractors in H 1 0 ( ) 3.1 A priori estimates
We start with the following general existence and uniqueness of solutions for the nonclassical diffusion equations which can be obtained by the Galerkin approximation methods (see [33] for more details), here we only formulate the results.
where κ > 0 is a constant.
Combining with (3.1) and taking δ → 0 for (3.14), we know that, for any ν ∈ I, S ν maps the bounded set of V into a bounded set for all t ≥ 0, and there is the following corollary.    Set For any t ≥ 0, we integrate (3.17) to t from 0 to t, then we have According to Lemma 3.5, Corollary 3.4, and (3.9)-(3.10), we know that H(0) is bounded and H(t) ≥ -β 1 mes( ) -|g| 2 |u(t)| 2 , then there exists a positive constant K 0 independent of ν such that the conclusion is true.

Lemma 3.7
There is a positive constant K 1 ; for any t > 0, the following estimate u t (t) Proof Differentiating about t for Eq. (1.1), we obtain u ttu tν u tt + f (u)u t = 0. (3.19) Multiplying (3.19) by u t and then integrating over leads to (3.20) By Lemma 3.6, for any 0 < δ < 1, there is s ∈ (0, δ) such that Now, we integrate (3.20) about t from s to t + δ (t > 0), we obtain Combine with Lemma 3.6 and Corollary 3.4, and let δ → 0, then for any t > 0 it follows that Setting holds for any t > 0.

The asymptotic regularity
In the following, we prove the asymptotic regularity of solutions for system (1.1) with initial-boundary conditions(1.2)-(1.3) in V by using a new decomposition method (or technique). In order to obtain the asymptotic regularity estimates later, we decompose the solution S ν (t)u 0 = u(t) into the following sum: where S ν 1 (t)u 0 = v(t) and K ν (t)u 0 = ω(t) are solutions of the following equations respectively: and where the constant μ ≥ 2l and l are from (1.5).
Remark 3.8 It is easy to verify the existence and uniqueness of the decomposition (3.24) corresponding to (3.25) and (3.26).
In fact, u is the unique solution of Eq. (1.1) with (1.2)-(1.3), thus g +μu ∈ L 2 ( ) is known. The existence and uniqueness of solutions ω corresponding to Eq. (3.26) can be obtained by the Galerkin approximation method (see [33]). By the superposition principle of solutions of partial differential equations, the existence and uniqueness of solutions v for Eq. (3.25) can be proved.
We use a priori estimate to get the asymptotic regularity of solutions for Eq. (1.1) with (1.2)-(1.3), which are the basis of our analysis. The proof is similar to [2], but the estimate about v 0 is not mentioned, only ν v 0 .