Pullback Attractors for Nonclassical Diffusion Equations in Noncylindrical Domains Cung

The existence and uniqueness of a variational solution are proved for the following nonautonomous nonclassical diffusion equation ut − εΔut − Δu f u g x, t , ε ∈ 0, 1 , in a noncylindrical domain with homogeneous Dirichlet boundary conditions, under the assumption that the spatial domains are bounded and increase with time. Moreover, the nonautonomous dynamical system generated by this class of solutions is shown to have a pullback attractor ̂ Aε, which is upper semicontinuous at ε 0.


Introduction
In recent years, the evolution equations on noncylindrical domains, that is, spatial domains which vary in time so their Cartesian products with the time variable are noncylindrical sets, have been investigated extensively see, e.g., 1-3 .Much of the progress has been made for nested spatial domains which expand in time.However, the results focus mainly on formulation of the problems and existence and uniqueness theory, while the existence of attractors of such systems has been less considered, except some recent works for the reaction-diffusion equation or the heat equation 4, 5 .This is not really surprising since such systems are intrinsically nonautonomous even if the equations themselves contain no time-dependent terms and require the concept of a nonautonomous attractor, which has only been introduced in recent years.
In this paper, we consider a class of nonautonomous nonclassical diffusion equations on bounded spatial domains which are expanding in time.First, we show how the first initial boundary value problem for these equations can be formulated as a variational problem International Journal of Mathematics and Mathematical Sciences with appropriate function spaces, and then we establish the existence and uniqueness over a finite time interval of variational solutions.Next, we show that the process of two parameter generated by such solutions has a nonautonomous pullback attractor.Finally, we study the upper semicontinuity of the obtained pullback attractor.
Let {Ω t } t∈R be a family of nonempty bounded open subsets of R N such that s < t ⇒ Ω s ⊂ Ω t . 1.1 From now on, we will frequently use the following notations: ∂Ω t × {t}.

1.2
In this paper we consider the following nonautonomous equation: where ε ∈ 0, 1 , the nonlinear term f and the external force g satisfy some conditions specified later on.This equation is called the nonclassical diffusion equation when ε > 0, and when ε 0, it turns to be the classical reaction-diffusion equation.Nonclassical diffusion equation arises as a model to describe physical phenomena, such as non-Newtonian flows, soil mechanics, and heat conduction see, e.g., 6-9 .In the last few years, the existence and long-time behavior of solutions to nonclassical diffusion equations has attracted the attention of many mathematicians.However, to the best of our knowledge, all existing results are devoted to the study of the equation in cylindrical domains.For example, under a Sobolev growth rate of the nonlinearity f, problem 1.3 in cylindrical domains has been studied 10-13 for the autonomous case, that is the case g not depending on time t and in 14, 15 for the nonautonomous case.In this paper, we will study the existence and long-time behavior of solutions to problem 1.3 in the case of noncylindrical domains, the nonlinearity f of polynomial type satisfying some dissipativity condition, and the external force g depending on time t.It is noticed that this question for problem 1.3 in the case ε 0, that is, for the reaction-diffusion equation, has only been studied recently in 4, 5 .
In order to study problem 1.3 , we make the following assumptions.

H1 The function f
for some p ≥ 2, where α 1 , α 2 , β, are nonnegative constants.By 1.4 , there exist nonnegative constants α 1 , α 2 , β such that where F u u 0 f s ds is the primitive of f.H2 The external force Since the open set Ω t changes with time t, problem 1.3 is nonautonomous even when the external force g is independent of time.Thus, in order to study the long-time behavior of solutions to 1.3 , we use the theory of pullback attractors.This theory has been developed for both the nonautonomous and random dynamical systems and has shown to be very useful in the understanding of the dynamics of these dynamical systems see 16 and references therein .The existence of a pullback attractor for problem 1.3 in the case ε 0, that is, for the classical reaction-diffusion equation, has been derived recently in 4 .In the case ε > 0, since 1.3 contains the term −εΔu t , this is essentially different from the classical reaction-diffusion equation.For example, the reaction-diffusion equation has some kind of "regularity"; for example, although the initial datum only belongs to a weaker topology space, the solution will belong to a stronger topology space with higher regularity, and hence we can use the compact Sobolev embeddings to obtain the existence of attractors easily.However, for problem 1.3 when ε > 0, because of −Δu t , if the initial datum u τ belongs to H 1 0 Ω τ ∩ L p Ω τ , the solution u t with the initial condition u τ u τ is always in H 1 0 Ω t ∩ L p Ω t and has no higher regularity, which is similar to hyperbolic equations.This brings some difficulty in establishing the existence of attractors for the nonclassical diffusion equations.Other difficulty arises since the considered domain is not cylindrical, so the standard techniques used for studying evolution equations in cylindrical domains cannot be used directly.Therefore, up to now, although there are many results on attractors for evolution equations in cylindrical domains see, e.g., 17, 18 , little seems to be known for the equations in noncylindrical domains.
In this paper, we first exploit the penalty method to prove the existence and uniqueness of a variational solution satisfying the energy equality to problem 1.3 .Next, we prove the existence of a pullback attractor A ε for the process associated to problem 1.3 .Finally, we study the continuous dependence on ε of the solutions to problem 1.3 , in particular we show that the solutions of the nonclassical diffusion equations converge to the solution of the classical reaction-diffusion equation as ε → 0. Hence using an abstract result derived recently by Carvalho et al. 19 and techniques similar to the ones used in 14 , we prove the upper semicontinuity of pullback attractors A ε in L 2 Ω t at ε 0. The last result means that the pullback attractors A ε of the nonclassical diffusion equations converge to the pullback International Journal of Mathematics and Mathematical Sciences attractor A 0 of the classical reaction-diffusion equations as ε → 0, in the sense of the Hausdorff semidistance.
The paper is organized as follows.In Section 2, for the convenience of readers, we recall some results on the penalty method and the theory of pullback attractors.After some preliminary results in Section 2, we proceed by a penalty method to solve approximated problem, and then we also prove the existence and uniqueness of the solution to problem 1.3 in Section 3. In Section 4, a uniform estimate for the solutions is then obtained under an additional assumption of the external force g, and this will lead to the proof of existence of a pullback attractor A ε in an appropriate framework.The upper semicontinuity of pullback attractors A ε at ε 0 is investigated in Section 5.In the last section, we give some discussions and related open problems.
Notations.In what follows, we will introduce some notations which are frequently used in the paper.Denote H r : L 2 Ω r and V r : H 1 0 Ω r for each r ∈ R, and denote by •, • r and | • | r the usual inner product and associated norm in H r and by •, • and • r the usual gradient inner product and associated norm in V r .For each s < t, consider V s as a closed subspace of V t with the functions belonging to V s being trivially extended by zero.It follows from 1.1 that {V t } t∈ τ,T can be considered as a family of closed subspaces of V T for each T > τ with 1.7 In addition, H r will be identified with its topological dual H * r by means of the Riesz theorem and V r will be considered as a subspace of H * r with v ∈ V r identified with the element f v ∈ H * r defined by The duality product between V * r and V r will be denoted by •, • .

Penalty Method
To study problem 1.3 , for each T > τ, we consider the following auxiliary problem: where τ ∈ R, u τ : Ω τ → R and g : Q τ → R are given functions.
The method of penalization due to Lions see 20 will be used to prove the existence and uniqueness of a solution to problem 2.1 satisfying an energy equality a.e. in τ, T and, as a consequence, the existence and uniqueness of a solution to problem 1.3 satisfying the energy equality a.e. in τ, ∞ .To begin, fix T > τ and for each t ∈ τ, T denote by the orthogonal subspace of V t with respect the inner product in V T and by P t ∈ L V T the orthogonal projection operator from V T onto V ⊥ T , which is defined as for each v ∈ V T .Finally, define P t P T for all t > T and observe that P T is the zero of L V T .
We will now approximate P t by operators which are more regular in time.Consider the family p t; •, • of symmetric bilinear forms on V T defined by p t; v, ω : It can be proved that the mapping τ, ∞ t → p t; v, ω ∈ R is measurable for all v, ω ∈ V T .Moreover, |p t; v, ω | ≤ v T ω T .For each integer k ≥ 1 and each t ≥ τ, we define and denote by P k t ∈ L V T the associated operator defined by Lemma 2.1 see 2, 4 .For any integer 1 ≤ h ≤ k, any t ≥ τ and every v, ω ∈ V T ,

2.7
Moreover, for every sequence Let J : V T → V * T be the Riesz isomorphism defined by International Journal of Mathematics and Mathematical Sciences and for each integer k ≥ 1 and each t ∈ τ, T , we denote

2.10
Obviously, A k t ∈ L V T , V * T , t ∈ τ, T , is a family of symmetric linear operators such that the mapping t ∈ τ, T → A k t ∈ L V T , V * T is measurable, bounded, and satisfies

2.11
Let u τ ∈ V T be given and for each k ≥ 1 consider the following problem:

2.12
The idea of the penalty method is as follows: for each k ≥ 1 we first prove the existence of a solution u k to problem 2.12 a problem in a cylindrical domain using standard methods such as the Galerkin method, and then show that u k converges to a solution to problem 2.1 a problem in a noncylindrical domain in some suitable sense, and as a consequence, the existence of a solution to problem 1.3 see Section 3 for details .

Pullback Attractors
Since the open set Ω t changes with time t, problem 1.3 is nonautonomous even when the external force g is independent of time.Thus, in order to study the long-time behavior of solutions to 1.3 , we use the theory of pullback D-attractors which is a modification of the theory in 16 .
Consider a process U •, • on a family of metric spaces For each t ∈ R, let dist t D 1 , D 2 be the Hausdorff semi-distance between nonempty subsets D 1 and D 2 of X t , which is defined as

2.19
Furthermore, A is minimal in the sense that if

The Upper Semicontinuity of Pullback Attractors
We now state some results on upper semicontinuity of pullback attractors, which are slight modifications of those in 19 .Because the proof is very similar to the one in 19 , so we omit it here.

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Definition 2.7.Let {U ε •, • : ε ∈ 0, 1 } be a family of evolution processes in a family of Banach spaces {X t } with corresponding pullback D-attractors {A ε t : ε ∈ 0, 1 , t ∈ R}.For any bounded interval I ∈ R, we say that for each T > 0, and for each compact subset K of X t−τ , the following conditions hold:

Existence and Uniqueness of Variational Solutions
For each T > τ, denote To prove the uniqueness of variational solutions to problem 2.1 , we need the following lemmas.

3.4
Then Using the open sets Ω t : Ω t T −T , τ ≤ t ≤ T , it is easy to see that under the conditions of 3.5 , one also has for every function Φ ∈ U τ,T .Then, for every pair τ ≤ s < t ≤ T of Lebesgue points of the inner product function v 1 , v 2 T it holds International Journal of Mathematics and Mathematical Sciences

3.8
If u is a variational solution of problem 2.1 , then τ is the Lebesgue point of |u| 2 T since the condition C4 is satisfied.The next corollary gives an obvious consequence of 3.8 .

3.9
Proof.If u is a variational solution of 2.1 , then we have

3.10
Applying Lemma 3.6 with v 1 v 2 u, we get

3.11
Hence, it implies the desired result.

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The aim of this section is to obtain a variational solution of 2.1 such that g r , u r T dr.

3.12
We will say that u satisfies the energy equality in τ, T if 3.12 is satisfied a.e. in τ, T .Analogously, if u is a variational solution of 1.3 , we will say that u satisfies the energy equality a.e. in τ, ∞ if for each T > τ the restriction of u to Q τ,T satisfies the energy equality 3.12 a.e. in τ, T .For any function v ∈ L 2 τ, T; H T and any t ∈ τ, T , we put

3.13
Then η v,T is a nondecreasing function.By Corollary 3.7, a variational solution u of 1.3 satisfies the energy equality a.e. in τ, T if and only if η u,T t 0 for a.e.t ∈ τ, T .In fact, using the continuity of the following mapping:

3.15
Then, u satisfies the energy equality a.e. in τ, T .

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Proof.It is sufficient to prove that η u,T T 0. Since t n → T and η u,T is nondecreasing, by Corollary 3.7, we have This completes the proof.

3.17
Hence, it implies the uniqueness of variational solutions to 2.1 satisfying the energy equality in τ, T .
Proof.Since u, u satisfy the energy equation, η u,T t η u,T t 0 for all t ∈ τ, T and

3.18
International Journal of Mathematics and Mathematical Sciences 13 On the other hand, Using this and 1.5 in 3.12 , one can conclude

3.21
By an application of Gronwall's inequality, we get 3.17 .
The method of penalization will now be used to prove the existence and uniqueness of a variational solution to problem 2.1 satisfying an energy equality a.e. in τ, T and, as a consequence, the existence and uniqueness of a variational solution to problem 1.3 satisfying the energy equality a.e. in τ, ∞ .Theorem 3.10.Let g ∈ L 2 τ, T; H T , u τ ∈ V τ ∩ L p Ω τ be given.Then problem 2.1 has a unique variational solution satisfying the energy equality a.e. in τ, T .
Proof.We divide the proof into two steps.
Step 1. Existence of a weak solution to problem 2.12 .We will use the Galerkin method see 20 .Take an orthonormal Hilbert basis {e j } of H T formed by elements of V T ∩ L p Ω T such that the vector space generated by {e j } is dense in V T and L p Ω T .Then, one takes a sequence {u τ m } converging to u τ in V T , with {u τ m } in the vector space spanned by the m first {e j }.For each integer m ≥ 1, one considers the approximation u k m t m j 1 γ k m ,j t e j , defined as a solution of

3.30
From 3.30 , we deduce that , one can check that {f u k m } is bounded in L q τ, T; L q Ω T with q p/ p − 1 , hence f u k m η in L q τ, T; L q Ω T .We now prove that η f u k .
Indeed, we have

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By the Aubin-Lions lemma 20, Chapter 1 , {u k m } is relatively compact in L 2 τ, T; H T .Therefore, one can assume that u k m → u k strongly in L 2 τ, T; H T , so 3 in 20 , we have f u k m f u k weakly in L q τ, T; L q Ω T .

3.31
This implies that u k is a weak solution of problem 2.12 .
Step 2. Existence of a variational solution to 2.1 satisfying the energy equality.
From 3.30 , we have

3.32
Consider the function Φ : L 2 τ, T; V T → R defined by

3.33
It is easy to see that Φ is a continuous and convex function.It follows that

3.34
Since {u k m } u k weakly in L 2 τ, T; V T , then, reasoning as above, 2εk

3.35
From the facts that u k m u k weakly in L ∞ τ, T; V T , u k m u k weakly in L 2 τ, T; V T and the weak lower semicontinuity of the norm, we deduce that

3.36
Since u τ ∈ V τ ∩ L p Ω τ , P k τ u τ , u τ T 0 for all k ≥ 1, 3.36 gives From Lemma 2.1, we have that is, P t u t 0 a.e. in τ, T or u t ∈ V t a.e. in τ, T , that is, P t u t 0 a.e. in τ, T .

3.37
Moreover, 3.36 and the equality

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give

3.39
It follows from 3.36 that u k t T ≤ C for all t ∈ τ, T and each k ≥ 1.Since the injection of V T into H T is compact, the set {v ∈ V T : v 2 T ≤ C} is compact in H T .By 3.39 and the Arzela-Ascoli theorem, there exists a subsequence, that will be still denoted by {u k }, such that

3.40
So, the condition C4 is satisfied.
On the other hand, {u k } is bounded in L ∞ τ, T; V T and {u k }is bounded in L 2 τ, T; V T , applying the Aubin-Lions lemma and Lemma 1.3 in 20, Chapter 1 , one has f u k f u weakly in L q τ, T, L q Ω T .

3.41
Since u k is the weak solution of the problem 3.42 taking to the limit as k → ∞ and using the fact that P t u t 0, P t u t 0 a.e. in τ, T , we can conclude that u is the solution of 2.1 .Now, we will show that u satisfies the energy equality in τ, T .Multiplying 3.22 by γ k m ,j and summing from j 1 to m, we obtain

3.43
Hence, we get

3.44
International Journal of Mathematics and Mathematical Sciences 19 or

3.47
Now

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This inequality and 3.47 give

3.49
Since u k u weakly in L 2 τ, T; V T , we get g r , u r T dr.

3.50
Applying Lemma 3.8 with t n T for all n, one concludes that u satisfies the energy equality on τ, T , and the desired uniqueness follows from Proposition 3.9.

Existence of Pullback D-Attractors
The aim of this section is to establish the existence of a pullback attractor for problem 1.3 .Suppose that g ∈ L 2 loc R N 1 .Then, according to Theorem 3.10, for each τ ∈ R and each u τ ∈ V τ ∩ L p Ω τ given, there exists a unique variational solution u •; τ, u τ of problem 1.3 satisfying the energy equality a.e. in τ, T for all T > τ. Define It is easy to check that the family of mappings {U t, τ A uniform estimate in V T ∩ L p Ω T will be obtained now for the variational solutions of 1.3 satisfying the energy equality, and since the compactness of the embedding V T ∩ L p Ω T → H T , this will immediately imply the existence of a pullback attractor for the process U •, • .The proof requires the following lemma.Lemma 4.1 see 21 .Let X ⊂ Y be Banach spaces such that X is reflexive and the injection of X in Y is compact.Suppose that {v n } is a bounded sequence in L ∞ t 0 , T; X such that v n v weakly in L p t 0 , T; X for some p ∈ 1, ∞ and v ∈ C 0 t 0 , T ; Y .Then, v t ∈ X for all t ∈ t 0 , T and , ∀η 1 , η 2 > 0.

4.10
International Journal of Mathematics and Mathematical Sciences 23 Choose η 2 < 1 and η 1 small enough such that σ T < min{1/ ε 1 , α 1 / α 2 , λ 1,T − 2η 1 }, we have International Journal of Mathematics and Mathematical Sciences 2 so combining with 4.13 and 4.14 , we have where C is independent of t, τ, ε, and k.Now, it is known that u k m u * k -weakly in L ∞ τ, T; V T as m → ∞.Hence, by 4.16 and Lemma 4.1, we can conclude that

4.18
International Journal of Mathematics and Mathematical Sciences 25 where we have used the fact that 4

The Upper Semicontinuity of Pullback D-Attractors at ε 0
It is proved in 4 , when ε 0, the existence of a pullback D-attractor A 0 {A 0 t : t ∈ R} in a family of spaces {H t } for problem P 0 .The aim of this section is to prove the upper semicontinuity of pullback attractors where I is an arbitrary bounded interval in R.
The following lemma is the key of this section.
Lemma 5.1.For each t ∈ R, each T > 0, and each compact subset K of V t−T , we have where the constant C is independent of τ and u τ .We have

Conclusion
In this paper we have proved the existence and uniqueness of variational solutions satisfying the energy equality to a class of nonautonomous nonclassical diffusion equations in noncylindrical domains.We have also proved the existence of pullback attractors A ε of the process generated by this class of solutions and the upper semicontinuity of { A ε : ε ∈ 0, 1 } at ε 0, which means that the pullback attractors A ε of the nonclassical diffusion equations converge to the pullback attractor A 0 of the reaction-diffusion equation in the sense of the Hausdorff semidistance.As far as we know, this is the first result on the existence and longtime behavior of solutions to the nonclassical diffusion equations in noncylindrical domains.
The result is obtained under the assumption 1.1 of spatial domains which are expanding in time.This assumption may be replaced by the assumption that the spatial domains Ω t in R N are obtained from a bounded base domain Ω by a C 2 -diffeomorphism, which is continuously differentiable in the time variable and are contained, in the past, in a common bounded domain see 5 for the related results in the case ε 0 .It is noticed that the obtained results seem to be not very satisfying because although the process U •, • associated to problem 1.3 is constructed in the family of spaces H 1 0 Ω t ∩ L p Ω t , we are only able to prove the existence and upper semicontinuity of the pullback attractor in L 2 Ω t .It would be very interesting if one can show the existence and upper semicontinuity of the pullback attractor in the space H 1 0 Ω t ∩ L p Ω t .For nonclassical diffusion equations in cylindrical domains, this question has been solved very recently in 15, 22 by using the so-called asymptotic a priori estimate method.However, in the case of non-cylindrical domains, this question seems to be more difficult and is still completely open.
The next lemma provides a sufficient condition for u to satisfy the energy equality a.e. in τ, T .Let u be a variational solution of 2.1 and suppose that there exists a sequence {t n } ⊂ τ, T of Lebesgue points of |u| 2 T such that t n → T and r T dr, 3.14 one can see that a variational solution u of 1.3 satisfies the energy equality a.e. in τ, T if and only if η u,T T 0.
∀t ∈ t 0 , T .Suppose that the assumptions in Theorem 3.10 hold and g ∈ L 2 loc R N 1 satisfies where σ T min{λ 1,T /2, 1/ ε 1 , α 1 / α 2 }, λ 1,T > 0 is the first eigenvalue of the operator −Δ in Ω T with the homogeneous Dirichlet condition, α 1 , α 2 are the constants in H1 , and the constant C is independent of t, τ, ε.Proof.Assume that u k m are the Galerkin approximations of u k defined by 3.22 .
Denote by D the class of all families D {D t ; D t ∈ V t ∩ L p Ω t , D t / ∅, t ∈ R} such that D t ⊂ B r t for some r t ∈ R.
and consider the family of closed balls B {B t ; t ∈ R}, whereB t {v ∈ V t : v t ≤ r 0 t }, t ∈ R.4.22 Then using 4.4 , it is not difficult to check that B is pullback D-absorbing for the process U •, • .Moreover, by the compactness of the injection of V t into H t , it is clear that B t is a compact set of H t for any t ∈ R.Then, it follows from Theorem 2.6 that the process U •, • has a pullback D-attractor A ε {A ε t : t ∈ R} in a family of spaces {H t }.
International Journal of Mathematics and Mathematical SciencesProof.Denote U ε t, τ u τ by u t , and U 0 t, τ u τ by v t .Let w t u t − v t , we have

ε
Δu t , w t −ε ∇u t , ∇w t ≤ ε u t t • w t .Now, we estimate the term on the right-hand side of 5.7 .Multiplying the first equation in 1.3 by u t and integrating over Ω t , we obtain Because lim t → −∞ sup r 0 t < ∞ due to 4.4 , from 5.22 we have ε∈ 0,1 t≤t 0 A ε t is bounded in L 2 Ω t 0 for given t 0 , 5.23 that is, condition ii is satisfied.From 5.21 , we can find that, for each t ∈ R, Ω t ⊂ L 2 Ω t compactly.Then condition iii holds.