Pullback attraction in H 10 for semilinear heat equation in expanding domains

*Correspondence: hardxiao@126.com 1College of Mathematics and Computer Science, Northwest Minzu University, Lanzhou, P.R. China Abstract In this article, we consider the pullback attraction in H1 0 of pullback attractor for semilinear heat equation with domains expanding in time. Firstly, we establish higher-order integrability of difference about variational solutions; then, we prove the continuity of variational solution in H1 0(Ot). As application of continuity, we obtain the pullback Dλ1 attraction in H 1 0-norm. MSC: 35K57; 35L05; 35B40; 35B41


Introduction
Let {O t } t∈R be a family of nonempty bounded open subsets of R N such that ∂O t × {t}, ∀τ < T.
We consider the following initial boundary value problem with homogeneous Dirichlet boundary condition: © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
For each T > τ , consider the auxiliary problem where u τ : O τ → R for τ ∈ R, g satisfies (4)- (5) and f ∈ L 2 loc (R; L 2 (O t )). The issue of non-cylindrical region usually refers to the problem that spatial region changes with time, also known as the problem of variable region. Variable region problems are applied widely in physics, chemistry, and cybernetics, so have been focused on relevant experts. Compared with the invariant regional system, the study of variable regional problem can vividly describe the actual phenomena. In addition, the problem defined in the variable region is essentially non-autonomous, so the discussion of variable regional problem adds vitality to the development and perfection of theory of non-autonomous system.
Based on the actual requirement, many mathematical researchers began to focus on the variable region problems, for example, see [1, 4-8, 15, 16] and so on. Recently, the existence and uniqueness of variational solution of system (3) have been considered in [6] with monotonic increase region, and then (L 2 , L 2 ) pullback D λ 1 attractor has been established. In 2009, by means of differ-morphism, a similar conclusion of system (3) was obtained in [7]. Later, in [11], by the solution orbit being shifted via a fixed complete orbit, the authors obtained the pullback D λ 1 attraction of L 2 pullback attractor in higher-order integrable spaces.
The continuity of solution plays an important role in the study of dynamic systems, especially in pullback attraction, fractal dimension, and so on. For the invariant region, the continuity of strong solution with respect to the initial data in H 1 0 (O) was considered for the space dimension N ≤ 2, and the nonlinear term exponent p ≥ 2, but p ≤ 4 for N = 3 was required. For an autonomous system, in order to obtain continuity in H 1 0 (O) and L p (O), the concept of norm-to-weak continuity was given in [12], and then the existence of global attractor was established. Then, the norm-to-weak continuity concept to the case of a non-autonomous system was studied in [10]. However, for a long time, the continuity of solution in H 1 0 (O) with respect to initial data has still been an open problem. Until 2008, when the nonlinear term f of autonomous system satisfying (4) and (5) was introduced, the author obtained the uniform boundedness of tu(t) by differentiating equation about time t, then considered the continuity of solution about initial data, see details in [14]. However, for a non-autonomous system, we cannot differentiate equation, so the method in [14] cannot be shifted to solve non-autonomous problems. In order to overcome the difficulties deriving from the non-autonomous character, in 2015, for the case of random equation, [2] discussed the continuity in H 1 0 (O) by studying the higher-order integrability of solutions difference near the initial time. Then, a natural problem arose: Does it still hold for variable domains? As far as the author knows, the continuity of solution in H 1 0 (O t ) about initial data is still unknown.
Enlightened by the above, we consider the continuity of variational solution in H 1 0 (O t ) with respect to initial data when the region of system (3) is monotonically increasing. As an application of continuity, we establish the pullback D λ 1 This paper is organized as follows. In Sect. 2, we recall some concepts and related results about variational solution. In Sect. 3, we prove higher-order integrability of difference of variational solutions near initial data (Theorem 3.3) and the continuity in H 1 0 (O t ) (Theorem 3.4), then establish the pullback D λ 1 attraction in H 1 0 (O t ) (Theorem 3.5).

Variational solutions
For each t ∈ R, denoted by (·, ·) t and | · | t the usual inner product and related norm in L 2 (O t ) and by ((·, ·)) t and · t the usual gradient inner product and associated norm in The usual duality product between H 1 0 (O t ) and H -1 (O t ) is denoted by ·, · t . And (·, ·) t and · L p (O t ) represent the duality product between L p (O t ) and L q (O t ) with 1 p + 1 q = 1 and the associated norm.
We consider a process U on a Banach space X, i.e., a family {U(t, τ ); -∞ < τ ≤ t < +∞} of continuous mappings U(t, τ ) : X → X such that Suppose that D is a nonempty class of parameterized setsD = {D(t) : t ∈ R} ⊂ P(X), where P(X) denotes the family of all nonempty subsets of X.
(2)Â is pullback D -attracting, i.e., lim τ →-∞ dist X U(t, τ )D(τ ), A (t) = 0 for allD ∈ D and all t ∈ R; (3)Â is invariant, i.e., is the orthogonal subspace of H 1 0 (O t ) with respect to the inner product in H 1 0 (O T ). We may identify w with its null-expansion and by P(t) ∈ L(H 1 0 (O T )) the orthogonal projection operator from H 1 0 (O T ) to H 1 0 (O t ) ⊥ , which is defined as . Consider the family p(t; ·, ·) of symmetric bilinear forms on H 1 0 (O T ) defined by It can be proved that the mapping [τ , Then we know from the above that, for any integers 1 ≤ h ≤ k, any t ≥ τ , and any v, w ∈ For each T > τ , denote (7) is a function u such that
We can also obtain the following result.

Theorem 2.5 Under the assumptions of Theorem
Proof One can take an orthonormal Hilbert basis {w j } of L 2 and H 1 0 formed by the ele- with u τ n in the vector space spanned by the n first w j .
Consider the equality for a.e. t ∈ [τ , T]. Multiplying (10) by λ j r kn,j (t) and summing from j = 1 to n, we know that Combining (11) and Hölder's inequality, we have a.e. t ∈ (τ , T), integrate the inequality above from τ to t, so, we have and Note (8), we have By (12)- (14), we obtain , there exists a subsequence, denoted still by {u kn }, such that as n → ∞ ). There exists a subsequence, denoted still by {u k }, such that it is convergent weakly, convergent weakly star to the uniqueness variational solution u of (7) in L 2 (τ , T; hold and g satisfies (4). Then there exists a positive constant K which depends on u τ L ∞ (O τ ) , f L ∞ (Q τ ,T ) , β and α 1 such that the variational solution u of (7) satisfies By Theorem 2.3 and Remark 2.4, we know that, for any τ ∈ R and any u τ ∈ L 2 (O τ ), there exists a unique variational solution u(·; τ , u τ ) satisfying energy equality for a.e. t ∈ (τ , T) and any T > τ . Moreover, u ∈ C([τ , T]; L 2 (O T )) satisfying energy equality for all t ∈ (τ , T) with any T > τ . Define By the uniqueness of variational solution for (7) and u ∈ C([τ , T]; L 2 (O T )) satisfying energy equality for all t ∈ (τ , T) with any T > τ , we know U(·, ·) defined by (15) is a process for the family of Hilbert spaces To obtain main results, the following lemma is necessary.

Lemma 3.1 ([11]) For any k > 0 and any
, the following equality holds: where · stands for the usual inner product in R N .
In the following, suppose Due to the density of and it can be done that, for each m = 1, 2, . . . , Based on the above, applying the interpolation inequality to estimate the L 2p-2 -norm of approximation solution, we can establish the higher-order integrability near initial time τ for approximation solution as follows.
Proof For any fixed τ ∈ (-∞, T], denote where u m (t), v m (t) are the variational solutions of equation (7) corresponding to the data (u τ m , f m ), (v τ m , f m ) satisfying (18) respectively. By Theorem 2.3 and Theorem 2.6, we know for any φ ∈ U τ ,T . For any θ > 0, we have Hence, we can choose η|w m | θ w m as a test function to have for any η ∈ C 1 c (τ , T) holds. Therefore, for a.e. t ∈ (τ , T), By (5), for a.e. t ∈ (τ , T), we have In the following, we separate our proof into two steps.
Step 1. k = 1 Firstly, taking φ = w m in (22), from the definition of variational solution and (5), we obtain that and then, Consequently, combining with the embedding we can deduce that note that here the embedding constant c N,τ ,T in (24) depends only on the domain O T . Secondly, take θ = 2N N-2 -2 in (23), by Lemma 3.1, we have that a.e. t ∈ (τ , T).
In the following we denote by c, c i (i = 1, 2, . . .) the constants which depend only on N, T -τ , and l, which may differ from line to line. Then the above inequality can be written as and by multiplying both sides with (tτ ) 3N N-2 , we obtain that here b 1 = 1 + 1 2 . One direct result of (27) is that , and so Consequently, for any t ∈ [τ , T], integrating (28) over [τ , t], we obtain that Then, multiplying (27) by (tτ ) 2N N-2 , we obtain that: for a.e. t ∈ (τ , T), Integrating the above inequality over [τ , T] with respect to t, we obtain that Consequently, applying embedding (24) again, we can deduce that Therefor, noticing (18) and (19), from (29) and (30) we know that there is a positive constant M 1 , which depends only on N, τ , T, l, |u τ | τ , |v τ | τ such that (A 1 ) and (B 1 ) hold.
Step 2. Assume (A k ) and (B k ) hold for k ≥ 1, in the following, we will show that (A k+1 ) and (B k+1 ) hold.
Next, we start to establish the higher-order integrability near the initial time τ for the variational solution of equation (7). This result shows some decay rate of variational solution in L 2( N N-2 ) k+1 -norm near the initial time τ .
In [15], the existence of pullback D λ 1 attractor defined in time varying domains has been considered. Then we can establish the regularity attraction of (L 2 , L 2 ) pullback D λ 1 attractor.
Noticing the arbitrariness of ε andD, the conclusion is proved.