Attractors for the nonclassical reaction–diffusion equations on time-dependent spaces

In this paper, based on the notation of time-dependent attractors introduced by Conti, Pata and Temam in (J. Differ. Equ. 255:1254–1277, 2013), we prove the existence of time-dependent global attractors in Ht\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{H}_{t}$\end{document} for a class of nonclassical reaction–diffusion equations with the forcing term g(x)∈H−1(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g(x)\in H^{-1}(\varOmega )$\end{document} and the nonlinearity f satisfying the polynomial growth of arbitrary 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}) order, which generalizes the results obtained in (Appl. Anal. 94:1439–1449, 2015) and (Bound. Value Probl. 2016: 10, 2016).

In this paper, we consider Eq. (1.1) with the nonlinearity f satisfying polynomial growth of arbitrary p -1 (p ≥ 2) order, which makes that the Sobolev compact embedding is no longer valid and brings more difficulty for verifying the corresponding asymptotic compactness of the solutions process {U(t, τ )} t≥τ . In order to overcome the difficulty mentioned above, we verify the existence of the time-dependent global attractorsÂ in H t for the process {U(t, τ )} t≥τ by applying the contractive function methods as in [6,13,14,19,22,27] (see Theorem 3.8).

Preliminaries
In this section, we firstly review briefly some notations, basic definitions and results about processes on time-dependent spaces (see [7][8][9]19] for details).

Notations
Let {X t } t∈R be a family of normed spaces, we introduce the R-ball of X t as For any given > 0, the -neighborhood of a set B ⊂ X t is defined as We denote the Hausdorff semidistance of two (nonempty) sets B, C ⊂ X t by Moreover, we introduce the time-dependent space H t endowed with the norms where · 2 denotes the usual norm in L 2 (Ω).

Some concepts
In this subsection, we give some concepts about the time-dependent global attractors.

Definition 2.1
Let {X t } t∈R be a family of normed spaces. A process is a two-parameter family of mappings U(t, τ ) : The process {U(t, τ )} t≥τ is called dissipative whenever it admits a pullback absorbing family.

Definition 2.4
A time-dependent absorbing set for the process {U(t, τ )} t≥τ is a uniformly bounded familyB = {B t } t∈R with the following property: for every R ≥ 0 there exists a t 0 = t 0 (R) ≥ 0 such that

Definition 2.5
The process {U(t, τ )} t≥τ is said to be pullback asymptotically compact if for any t ∈ R, any bounded sequence {x n } ∞ n=1 ⊂ X τ n and any sequence

Definition 2.6
The time-dependent global attractor for the process (iii)Â is pullback attracting, i.e., it is uniformly bounded and the limit holds for every uniformly bounded familyĈ = {C t } t∈R and every fixed t ∈ R.
Remark 2.7 The attracting property can be equivalently stated in terms of pullback absorbing: a (uniformly bounded) family K = {K t } t∈R is called pullback attracting if for any > 0 the family {O t (K t )} t∈R is pullback absorbing.
Similarly to Theorem 4.2 in [8], we have the following theorem.

Theorem 2.8 The time-dependent global attractorÂ exists and it is unique if and only if the process {U(t, τ )} t≥τ is asymptotically compact, namely, the set
is not empty.

Some results
In order to obtain the time-dependent global attractors of Eq. (1.1), we need the following definitions and conclusions, which are similar to those in [6,13,14,19,22,27].

Definition 2.9
Let {X t } t∈R be a family of Banach spaces andĈ = {C t } t∈R be a family of uni- We denote the set of all contractive functions on C τ × C τ by Contr(C τ ).
By induction, we can obtain that, for each m ≥ 1, there exists a subsequence {U(t, Similarly to Theorem 3.3 in [19], we have the following conclusion, which will be used to verify the existence of the time-dependent global attractor.

Theorem 2.11
Let {U(t, τ )} t≥τ be a process on Banach space {X t } t∈R , then {U(t, τ )} t≥τ has a time-dependent global attractor in {X t } t∈R if the following conditions hold:

Time-dependent global attractors
In this section, we will establish the existence of the time-dependent global attractors.

Existence and uniqueness of solutions
In this subsection, we consider the well-posedness of the solutions for Eq. (1.1) with (1.4)-(1.5). At first, we define the weak solutions as follows.

Definition 3.1 A weak solution of Eq. (1.1) is a function u ∈ C([τ , T];
H t ) ∩ L 2 (τ , T; H 1 0 (Ω)) ∩ L p (τ , T; L p (Ω)) for all T > τ , with u(τ ) = u τ and such that, for all ϕ ∈ H 1 0 (Ω), it satisfies   The following theorem gives the existence of the weak solutions, which is similar to that in [10] and can be obtained by the Faedo-Galerkin methods.
Proof Let {w j } j≥1 ⊂ H 1 0 (Ω) ∩ L p (Ω) be a Hilbert basis of L 2 (Ω) such that span{w j } j≥1 is dense in H 1 0 (Ω) ∩ L p (Ω). In order to establish the existence of the weak solutions, we need the approximate system for any m ≥ n seekingũ m (t, We will provide a priori estimates that show that these solutions are well-defined in the interval [τ , t] for any t > τ . Step 1: First a priori estimates. Multiplying each equation in the above system by γ mj (t), respectively, and summing from j = 1 to m, we obtain where we have used the Hölder and Young inequalities. Furthermore, by (1.5), we know that Integrating it in [τ , t], we have Hence, where q = p/(p -1). Then there exist functionsũ ∈ L ∞ (τ , t; H t ) ∩ L 2 (τ , t; H 1 0 (Ω)) ∩ L p (τ , t; L p (Ω)) andχ ∈ L q (τ , t; L q (Ω)) for all t > τ , and a subsequence such that u m →ũ weakly in L 2 (τ , t; H 1 0 (Ω)), u m →ũ weakly in L p (τ , t; L p (Ω)), f (ũ m ) →χ weakly in L q (τ , t; L q (Ω)).

(3.3)
Step 2: Uniform estimate for the time derivatives. Multiplying each equation of the approximate system by γ mj (t) and summing from j = 1 to m, we arrive at By the Hölder and Young inequalities, we have Integrating it from τ to t, and from (1.6) we can get for all t ≥ τ and any m ≥ n. and for all t > τ . Then there exist functionsũ ∈ L ∞ (τ , t; H 1 0 (Ω) ∩ L p (Ω)) andũ t ∈ L 2 (τ , t; H t ) for all t > τ , which improve the regularity ofũ obtained in Step 1.
Thus, together with (3.3) and (3.8), by taking the limit in the equations satisfied by {ũ m } and, thanks to the fact that span{ω j } j≥1 is dense in H 1 0 (Ω) ∩ L p (Ω), we conclude thatũ is a weak solution of Eq. (1.1).
Step 3: Proof of the general statement by density. For each n ∈ N, we define u n τ = Σ n j=1 (u τ , ω j )ω j . (Due to the fact that {ω j } j≥1 is a Hilbert basis of L 2 (Ω), it is easy to check that u n τ → u τ in H τ .) Let also consider a sequence {g n } ∞ n=1 ⊂ L 2 (Ω) converging to g ∈ H -1 (Ω). Denote by u n the corresponding solution to Eq. (1.1) with g replaced by g n and initial data u n τ . Then, by the energy equality for each u n , we have u n (t) Similar to the reasoning process in Step 1, we get for all t > τ . Now, combining with (1.5) and (3.9), we see that {f (u n )} is bounded in L q (τ , t; L q (Ω)) for all t > τ .
Moreover, we may improve some of the above convergence. Taking into account the energy equality for u nu m , we have By (3.11), we know that Thus, we have u n → u a.e. in Ω × (τ , ∞). Therefore, as before, combining with (3.10) and [15] (Lemma 1.3, p. 12) we obtain χ = f (u); and from (3.10) we may take the limit in the equations satisfied by u n and conclude that u is a weak solution of Eq. (1.1).
For the solutions of Eq. (1.1), the following theorem shows the uniqueness and continuity with respect to initial data.  1) is unique. Moreover, for every two solutions u 1 (t) and u 2 (t) (with different initial data), the following Lipschitz continuity holds: where ω(t) = u 1 (t)u 2 (t).