Continuity and pullback attractors for a semilinear heat equation on time-varying domains

We consider dynamics of a semilinear heat equation on time-varying domains with lower regular forcing term. Instead of requiring the forcing term f ( · ) to satisfy (cid:2) t – ∞ e λ s (cid:3) f ( s ) (cid:3) 2 L 2 ds < ∞ for all t ∈ R , we show that the solutions of a semilinear heat equation on time-varying domains are continuous with respect to initial data in H 1 topology and the usual ( L 2 , L 2 ) pullback D λ -attractor indeed can attract in the H 1 -norm, provided that (cid:2) t – ∞ e λ s (cid:3) f ( s ) (cid:3) 2 H –1 ( O s ) ds < ∞ and f ∈ L 2 loc ( R , L 2 ( O s )). Mathematics Subject Classiﬁcation: 35K57; 35L05; 35B40; 35B41


Introduction
Let O be a nonempty bounded open subset of R N with C 2 boundary ∂O, and let r = r(y, t) be a vector function We consider the following initial boundary value problem for a semilinear parabolic equation: where τ ∈ R, u τ : O τ → R, Q τ ,T := t∈(τ ,T) O t × {t} for all T > τ , Q τ := t∈(τ ,+∞) O t × {t}, that g ∈ C 1 (R, R) is a given function for which there exist nonnegative constants α 1 , α 2 , β, l, and p ≥ 2 such that -β + α 1 |s| p ≤ g(s)s ≤ β + α 2 |s| p , g (s) ≥ -l ∀s ∈ R, (1.4) and, moreover, g satisfies the Lipschitz condition: there exists a positive constant c 0 such that About the diffeomorphism r(•, •), as in Límacoet al. [6] and Kloedenet al. [5], we assume that the function r = r(x, t), where r(•, t) = r -1 (•, t) denotes the inverse of r(•, t), satisfies r ∈ C 2,1 Qτ,T ; R N for all τ < T. (1.6) The reaction-diffusion equation with nonlinear term g(•) satisfying assumptions (1.4) is one of the classical example models in the theory of infinite-dimensional dynamical systems, especially regarding to the theory of attractors; e.g., see the classical monographs in this field like [1,8,11].
About the dynamics of reaction-diffusion equation (1.3)-(1.4), the known results mainly concentrate in the L 2 phase space; e.g., see [8,11] for the fixed domain case (i.e., r(•, t) ≡ Id) and Kloeden et al. [5,13] for time-varying domain case; and the corresponding mathematical analysis is standard to some extent.When we try to improve the corresponding results to a more regular phase space, say H 1 , some essential difficulties arise, for example, the continuity with respect to the initial data and asymptotical compactness in H 1 topology.Indeed, even in the autonomous case, for any space dimension N and any growth power p ≥ 2 (comes from (1.4)), the question about the continuity of solution with respect to initial data in H 1 remained open until 2008; see Robinson [8].In 2008, for the autonomous case of (1.3) and with the same assumption (1.4) about the nonlinearity, Trujillo and Wang [12] used the method of differentiating the equation with respect to t to get the bounded estimate for tu t L 2 for t ∈ [0, T] and then obtained the uniform boundedness of tu(t) in L ∞ (0, T; H 2 ) and, finally, obtained the continuity in H 1 for any space dimension N and any growth power p ≥ 2 (to our knowledge, this is the first result).Later, Cao et al. [2] obtained such continuity for nonautonomous case by establishing some new a priori estimates for the difference of solutions near the initial time; see also [3,13] for further discussion in this direction.
Note that to obtain the continuity with respect to the initial data and existence of attractors in the H 1 topology, to our knowledge, the known results always required the force term to belong to L 2 ; e.g., see [2,3,12] for autonomous and stochastic case; and in [13], to obtain similar results as in [2] in the nonautonomous case, they required f (•) to satisfy for some proper positive constant λ.On the other hand, it is well-known that when we consider system The main aim of this paper is to establish the same continuity with respect to the initial data in the H 1 topology and H 1 -attraction as that in [2,3,12,13] and relax the assumption on the forcing term.To include the nonautonomous case, we consider systems (1.3)-(1.4)defined on a time-varying domain.Note that a semilinear heat equation on a time-varying domain is intrinsically nonautonomous even if the terms in the equation do not depend explicitly on time.
Our main result is the following theorem.
Moreover, the (L 2 , L 2 ) pullback attractor Â = {A (t) : t ∈ R} obtained in [5] can pullback attract in the topology of H 1 , i.e., for all t ∈ R and D = {D(t As mentioned previously, after the work [2], although (1.3) is defined on a time-varying domain, the continuity in (1.10) and attraction (1.11) is more or less expectable, in this paper, we give rigorous proofs about how to justify the approximation that is necessary due to relaxing the assumption on the forcing term.Note also that here we only additionally assume that f ∈ L 2 loc (R, L 2 (O t )), but not (1.7), which was required in [3,7,9,13] etc. for obtaining the boundedness in L p and H 1 .However, in the nonautonomous case, the question whether we can remove further the additional condition f ∈ L 2 loc (R, L 2 (O t )) remains open.

Functional spaces
We first recall some functional spaces and notations.
For a fixed finite time interval [τ , T], let (X t , • X t ) (t ∈ [τ , T]) be a family of Banach spaces such that X t ⊂ L 1 loc (O t ) for all t ∈ [τ , T].For any 1 ≤ q ≤ ∞, we denote by L q (τ , T; X t ) the vector space of all functions u ∈ L 1 loc (Q τ ,T ) such that u(t) = u(•, t) ∈ X t for a.e.t ∈ (τ , T) and the function u(•) X • defined by t → u(t) X t belongs to L q (τ , T).

Definitions of solutions
For the readers' convenience, in this subsection, we recall the definition of different solutions of equation (1.3); see Límaco et al. [6] and Kloeden et al. [5] for more detail.
For each T > τ , consider the auxiliary problem where τ ∈ R and u τ : O τ → R.

Definition 2.1 (Strong solution)
A function u = u(x, t) defined in Q τ ,T is said to be a strong solution of problem (2.1) if and the three equations in (2.1) are satisfied almost everywhere in their corresponding domains. Denote , and -∞ < τ ≤ T < ∞.We say that a function u is a weak solution of (2.1) if where u m is the unique strong solution of (2.1) corresponding to (u τ m , f m );

Preliminary lemmas
For later application, in the following, we collect some results for obtaining higher-order integrability, which can be proved by the standard methods; see [5,10] for the detailed proofs.
and the following energy equality is satisfied: = (k + 2) for some s ∈ R, we the following equality: where • stands for the usual inner product in R N .
and for any t In order the test function |u| k • u to make sense, we also recall the following L ∞ -estimate on the nice initial data, which can be obtained by applying the standard Stampacchia's truncation method; see [10] for a detailed proof.

Higher-order integrability
Along the ideas in [2], as the preliminaries, in this section, we obtain some higher-order integrability of the difference of two weak solutions near the initial time, which was firstly established in [2] for the (autonomous and fixed domain) stochastic case of (1.3), and later, similar results were obtained in [13] for (1.3)-(1.4) in the stochastic case (in time-varying case, but the forcing term was required to satisfy (1.7)).

A priori estimates for approximation solutions
To make our proof rigorous, we will use the approximation techniques.
For any (fixed) T ∈ R, throughout this section, we choose (we can do this by Lemma 2.6) and fix a family Then, for any τ < T and any u τ , v τ ∈ L 2 (O τ ), according to the definition of a weak solution, we know that there are two sequences such that and where u m and v m are the unique strong solution of (1.3) corresponding to the regular data (u τ m , f m ) and (v τ m , f m ), respectively.Without loss of generality, by (3.3) we can require that where and hereafter, Then w m (t) (m = 1, 2, . ..) is the unique strong solution of the following equation: , and the three equations in (3.7) are satisfied almost everywhere in their corresponding domains.
The main purpose of this subsection is to prove the following uniform (with respect to m) a priori estimates of w m defined in (3.6).
Proof By Lemma 2.7 we know that u m , v m ∈ L ∞ (Q τ ,T ) for each m = 1, 2, . . ., and so and for any 0 Consequently, we can multiply (3.7) by In the following, we will separate our proof into two steps.
Step 1 k = 1 At first, multiplying (3.7) by w m , from the definition of a strong solution and (1.4), applying Lemmas 2.4 and 2.5, we obtain that 1 2 Consequently, combining with the embedding we can deduce that Note that here the embedding constant c N,τ ,T in (3.12) depends only on the domain for a.e.t ∈ (τ , T).
To simplify the calculations, we denote by c, c i (i = 1, 2, . ..) the constants that depend only on N , Tτ , k, and l and may vary 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 . (3.15) Recall that b 1 = 1 + 1 2 was defined in (3.2).
Integrating this inequality over [τ , T] with respect to t, we obtain that where we have used (3.17).Consequently, applying embedding (3.12) again, we can deduce that Therefore, noticing (3.3) and (3.5), from (3.17) and (3.19) 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 Assuming that (A k ) and (B k ) hold for k ≥ 1, we will show that (A k+1 ) and (B k+1 ) hold.

Higher-order integrability near the initial time
Based on the a priori estimate in Theorem 3.1 for the approximation solutions, we can obtain the following higher-order integrability near the initial time: where w(t) = U(t, τ )u τ -U(t, τ )v τ , and Then from Theorem 3.1 we have that for any k = 1, 2, . . ., there exists a positive constant where u m and v m are the unique strong solutions of (1.3) corresponding to the regular data (u τ m , f m ) and (v τ m , f m ) on the interval [τ , T], respectively.From (3.4) we know that for each t ∈ [τ , T], there are two subsequences {u m j (t)} ⊂ {u m (t)} and {v m j (t)} ⊂ {v m (t)} satisfying where the subindex m j may depend on t.
Hence, since estimate (3.29) is independent of m, we can finish our proof by applying the Fatou lemma:

Proof of Theorem 1.1
We start with the following a priori estimates.

Lemma 4.1 Let Assumption I hold, and let f
Then for all τ ∈ R and u τ ∈ L 2 (O τ ), the corresponding weak solution u(t) = U(t, τ )u τ (t ≥ τ ) of equation (1.3) satisfies the following estimates: for any T > τ , Note that since we only assume that f ∈ L 2 loc (R, L 2 (O t )), we cannot obtain the uniform boundedness of the solutions in the L p sense as that in [3,9,13], i.e., our constant M above depends on the time tτ .However, we will show further that such boundedness is sufficient for Theorem 1.1.
Proof Since the results of the lemma are or less standard, we restrict ourselves by only formal derivation of estimate (4.1), which can be easily justified using, e.g., the methods as in Sect.3: first, deduce the a priori estimates for approximation solutions and then obtain (4.1) by Fatou's lemma.
First, multiplying (1.3) by u and integrating with respect to x ∈ O t , we have that 1 2 recall that • t denotes the L 2 (O t )-norm; Then using (1.4) and Cauchy's inequality, we obtain that Secondly, multiplying in (1.3) by |u| p-2 • u and integrating with respect to x ∈ O t , we have that where we have used Lemmas 2.4 and 2.5 and (1.4).Consequently, using Cauchy's inequality, we have that for a.e.t ∈ (τ , T), ( where the constants c 1 , c 2 , c 3 depend only on β, α 1 , and p.Now from (4.3) we know that there is and Therefore, for any t ∈ [ 2 , T], integrating (4.4) with respect to time from t 0 to t, we deduce that which, combined with (4.6) and (4.3), immediately implies (4.1).Now we are ready to prove our main results.
Consequently, we obtain the H 1 -pullback attraction (1.11) by the arbitrariness of ε and D.
In the following, we give the proof of the above claim.To make our proof rigorous, as in Sect.3, we will prove the claim firstly for approximation solutions and then take the limit.
This finishes the proof of the claim and thus the proof of the theorem.
{t} and the family {f m } may depend on T.