Regularity and Exponential Growth of Pullback Attractors for Semilinear Parabolic Equations Involving the Grushin Operator

and Applied Analysis 3 The content of the paper is as follows. In Section 2, for the convenience of the reader, we recall some concepts and results on function spaces and pullback attractors which we will use. In Section 3, we prove the existence of pullback attractors in the spaces S0 Ω and L2p−2 Ω by using the asymptotic a priori estimate method. In Section 4, under additional assumptions of g, an exponential growth in S0 Ω ∩ L2p−2 Ω for the pullback attractors is deduced. 2. Preliminaries 2.1. Operator and Function Spaces In order to study the boundary value problem for equations involving the Grushin operator, we have usually used the natural energy space S0 Ω defined as the completion of C ∞ 0 Ω in the following norm:


Introduction
Let Ω be a bounded domain in R N 1 × R N 2 N 1 , N 2 ≥ 1 , with smooth boundary ∂Ω.In this paper, we consider the following problem: where is the Grushin operator, u τ ∈ L 2 Ω is given, the nonlinearity f and the external force g satisfy the following conditions.
where λ 1 is the first eigenvalue of the operator −G s in Ω with the homogeneous Dirichlet boundary condition.
The Grushin operator G s was first introduced in 1 .Noting that if s > 0, then G s is not elliptic in domains of R N 1 × R N 2 which intersect the hyperplane {x 1 0}.In the last few years, the existence and long-time behavior of solutions to parabolic equations involving the Grushin operator have been studied widely in both autonomous and nonautonomous cases see, e.g., 2-7 .In particular, the existence of a pullback attractor in S 1 0 Ω ∩ L p Ω for the process associated to problem 1.1 is considered in 2 .
In this paper we continue the study in the paper 2 .First, we will prove the existence of pullback attractors in S 2 0 Ω see Section 2 for its definition and L 2p−2 Ω .As we know, if the external force g is only in L 2 Ω , then solutions of problem 1.1 are at most in L 2p−2 Ω ∩ S 2 0 Ω and have no higher regularity.Therefore, there are no compact embedding results that hold for this case.To overcome the difficulty caused by the lack of embedding results, we exploit the asymptotic a priori estimate method which was initiated in 8, 9 for autonomous equations and developed recently for nonautonomous equations in the case of pullback attractors in 10 .Noting that, to prove the existence of pullback attractors in S 1 0 Ω ∩ L p Ω , we only need assumption H2 of the external force g; however, to prove the existence of pullback attractors in S 2 0 Ω and L 2p−2 Ω , we need an additional assumption of g, namely, 3.18 in Section 3. Next, following the general lines of the approach in 11 , we give exponential growth conditions in S 2 0 Ω ∩ L 2p−2 Ω for the pullback attractors.It is noticed that, as far as we know, the best known results on the pullback attractors for nonautonomous reaction-diffusion equations are the boundedness and exponential growth in H 2 Ω of the pullback attractors 11, 12 .Therefore, the obtained results seem to be optimal and, in particular when s 0, improve the recent results on pullback attractors for the nonautonomous reaction-diffusion equations in 11-15 .
The content of the paper is as follows.In Section 2, for the convenience of the reader, we recall some concepts and results on function spaces and pullback attractors which we will use.In Section 3, we prove the existence of pullback attractors in the spaces S 2 0 Ω and L 2p−2 Ω by using the asymptotic a priori estimate method.In Section 4, under additional assumptions of g, an exponential growth in S 2 0 Ω ∩ L 2p−2 Ω for the pullback attractors is deduced.

Operator and Function Spaces
In order to study the boundary value problem for equations involving the Grushin operator, we have usually used the natural energy space S 1 0 Ω defined as the completion of C ∞ 0 Ω in the following norm: and the scalar product

2.2
The following lemma comes from 16 .
Then the following embeddings hold: Now, we introduce the space S 2 0 Ω defined as the closure of C ∞ 0 Ω with the norm

2.3
The following lemma comes directly from the definitions of S 1 0 Ω and S 2 0 Ω .
It is known that see, e.g., 3 for the operator A −G s , there exist {e j } j≥1 such that e j , e k δ jk , Ae j λ j e j , j,k 1, 2, . . ., and {e j } j≥1 is a complete orthonormal system in L 2 Ω .

Pullback Attractors
Let X be a Banach space with the norm 0 Ω .For a Banach space E, • E will be the norm.We also denote by C an arbitrary constant, which is different from line to line, and even in the same line.

Existence of Pullback Attractors in
It is well known see, e.g., 2 or 14 that under conditions H1 − H2 , problem 1.1 defines a process where U t, τ u τ is the unique weak solution of 1.1 with initial datum u τ at time τ.The process {U t, τ } has a pullback attractor in S 1 0 Ω ∩ L p Ω .In this section, we will prove that the pullback attractor is in fact in S 2 0 Ω ∩ L 2p−2 Ω .
Lemma 3.1.Assuming that f and g satisfy (H 1)-(H 2), u t is a weak solution of 1.1 .Then the following inequality holds for t > τ: where C is a positive constant.
Proof.Multiplying 1.1 by u and then integrating over Ω, we get 1 2

3.4
Letting F s s 0 f τ dτ, by H1 , we have Now multiplying 3.4 by e λ 1 t and using 3.5 , we get 3.6 Integrating 3.6 from τ to s ∈ τ, t − 1 and s to s 1, respectively, we obtain t τ e λ 1 r g r 2 2 dr .

3.8
Multiplying 1.1 by u t and integrating over Ω, we have 3.9 Thus

3.10
Abstract and Applied Analysis 7 Combining 3.8 and 3.10 , and using the uniform Gronwall inequality, we have Using H1 once again and thanks to e λ 1 τ |u τ | 2 2 → 0 as τ → −∞, we get the desired result from 3.11 .

Lemma 3.2. Assume that (H 1), (H 2) hold. Then for any t ∈ R and any
g s Proof.Integrating 3.10 from r to r 1, r ∈ τ, t − 1 and using 3.8 and 3.11 , in particular we find ds .

3.13
On the other hand, differentiating 1.1 and denoting v u t , we have Taking the inner product of 3.14 with v in L 2 Ω , we get 1 2 Using 1.5 and Young's inequality, after a few computations, we see that

3.16
Combining 3.16 and 3.13 and using the uniform Gronwall inequality, we obtain g s 2 2 ds .

8
Abstract and Applied Analysis

Existence of a Pullback Attractor in
In this section, following the general lines of the method introduced in 9 , we prove the existence of a pullback attractor in L 2p−2 Ω .In order to do this, we need an additional condition of where m, m are defined as in 3.30 .

3.19
From 1.3 and the fact that

3.20
On the other hand, by Cauchy's inequality, we see that Proof.We will prove the lemma by induction argument.Letting β N s / N s − 2 > 1 and denoting v u t we prove that for k 0, 1, 2, . .., there exist τ k and M k s such that where τ k depends on k and B and M k depends only on k.
For k 0, we have P 0 from 3.17 .Integrating 3.16 and using Assuming that P k , Q k hold, we prove so are P k 1 and Q k 1 .Multiplying 3.14 by |v| 2β k 1 −2 v and integrating over Ω, we obtain

3.26
Combining Holder's and Young's inequalities, we see that

3.30
Then from 3.27 , we infer that

3.31
Applying 3.26 and 3.31 in 3.25 , we find that

3.32
Combining Q k and 3.32 , using the uniform Gronwall inequality and taking into account assumption 3.18 , we get P k 1 .On the other hand, integrating 3.32 from t to t 1, we find Q k 1 .Now since β > 1, and taking k ≥ log β p/2, we get the desired estimate.
We will use the following lemma.

3.40
By Lemma 3.5 and since λ m → ∞ as m → ∞, there exist τ 1 and m 1 such that

3.41
for all τ ≤ τ 1 and m ≥ m 1 .For the second term of the right-hand side of 3.40 , using Holder's inequality we have

3.42
From Lemmas 3.5 -3.7 , we see that there exist τ 2 and m 2 ∈ N such that
Lemma 3.7 see 9 .Let B be a bounded subset in L q Ω q ≥ 1 .If B has a finite ε-net in L q Ω , then there exists an M M B, ε , such that for any u ∈ B, the following estimate is valid: |u| q dx < ε.

3.44
Using Lemma 3.7 and taking into account Lemmas 3.2 and 3.6 we conclude that the set {u t s : s ≤ t, u τ ∈ B} has a finite -net in L 2 Ω .Therefore, we get the following result.Lemma 3.8.For any t ∈ R, any B ⊂ L 2 Ω that is bounded, and any ε > 0, there exists τ 0 ≤ t and 3.45 Lemma 3.9 see 9 .For any t ∈ R, any bounded set B ⊂ L 2 Ω , and any ε > 0, there exist τ 0 and where mes is the Lebesgue measure in R N and Ω u t ≥ M {x ∈ Ω : u t, x ≥ M}.
ii for any t ∈ R, any bounded set D ⊂ L 2 Ω , and any > 0, there exist M > 0 and τ 0 ≤ t where C is independent of M, τ, u τ , and .
We are now ready to prove the existence of a pullback attractor in L 2p−2 Ω .
Theorem 3.11.Assume that assumptions 1.3 -1.7 and 3.18 hold.Then the process {U t, τ } associated to problem 1.1 possesses a pullback attractor Proof.Because of Lemma 3.10, since {U t, τ } has a pullback absorbing set in L 2p−2 Ω , we only have to prove that for any t ∈ R, any B ⊂ L 2 Ω , and any ε > 0, there exist τ 2 ≤ t and

3.48
Taking the inner product of 1.

3.50
Some standard computations give us Abstract and Applied Analysis Combining 3.50 -3.53 , we find

3.54
Applying Lemmas 3.7 and 3.8 to 3.54 we find there exist τ 0 and M 0 such that Repeating the above arguments with for some τ 1 ≤ t and M 1 > 0, where

3.58
This completes the proof.

Existence of a Pullback Attractor in S 2 0 Ω
In this section, we prove the existence of a pullback attractor in S 2 0 Ω .Lemma 3.12.The process {U t, τ } associated to 1.1 has a pullback absorbing set in S 2 0 Ω .
Proof.We multiply 1.1 by −G s u; then, using f 0 0, we have

3.59
Using f u ≥ − , Cauchy's inequality, and argument as in Lemma 3.3, from 3.59 we have

3.60
Taking into account 3.11 , the proof is complete.

Abstract and Applied Analysis 15
In order to prove the existence of the pullback attractor in S 2 0 Ω , we will verify socalled " PDC condition", which is defined as follow Definition 3.13.A process {U t, τ } is said to satisfy PDC condition in X if for any t ∈ R, any bounded set B ⊂ L 2 Ω and any ε > 0, there exists τ 0 ≤ t and a finite dimensional subspace X 1 of X such that i P τ≤τ 0 U t, τ B is bounded in X; and ii I X − P U t, τ u τ X < ε, for all τ ≤ τ 0 and u τ ∈ B, where P : X → X 1 is a canonical projection and I X is the identity.Lemma 3.14 see 13 .If a process {U t, τ } satisfies (PDC) condition in X then it is pullback asymptotically compact in X.Moreover, if X is convex then the converse is true.Lemma 3.15 see 9 .Assume that f satisfies 1.3 and 1.5 .Then for any subset

of Pullback Attractors
In this section, we will give an exponential growth condition in S 2 0 Ω ∩ L 2p−2 Ω for the pullback attractor A τ .
First, we recall a result in 17 which is necessary for the proof of our results.
Lemma 4.1.Let X, Y be Banach spaces such that X is reflexive, and the inclusion X ⊂ Y is continuous.Assume that {u n } is bounded sequence in L ∞ t 0 , T; X such that u n → u weakly in L q t 0 , T; X for some q ∈ 1, ∞ and u ∈ C 0 t 0 , T ; Y .Then, u t ∈ X for all t ∈ t 0 , T and In the following theorem, instead of evaluating the functions u n which are differentiable enough and then using Lemma 4.1, we will formally evaluate the function u.Theorem 4.2.Assume that f satisfies 1.3 -1.5 , g satisfies (H 2), 3.18 and the following conditions Proof.We differentiate with respect to time in 1.1 , then multiply by u t , we get 1 2

4.4
Integrating in the last inequality, in particular, we get u r

Proposition 2.3 see
• .B X denotes all bounded sets of X.The following result is useful for proving the norm-to-weak continuity of a process.9 .Let X, Y be two Banach spaces, and let X * , Y * be, respectively, their dual spaces.Suppose that X is dense in Y , the injection i : X → Y is continuous, and its adjoint i The process {U t, τ } is said to be pullback asymptotically compact if for any t ∈ R, any D ∈ B X , any sequence τ n → −∞, and any sequence {x n } ⊂ D, n → x in X, for all t ≥ τ, τ ∈ R. * : Y *

the sequence {U t, τ n x n } is relatively compact in X. Definition 2.5. A family of bounded sets B {B t : t ∈ R} ⊂ X is called a pullback absorbing set for the process {U t, τ } if for any t ∈ R and any D ∈ B X , there exist τ 0 τ 0 D, t ≤ t and B t ∈ B such that
In the rest of the paper, we denote by | • | 2 , •, • the norm and inner product in L 2 Ω , respectively, and by |•| p the norm in L p Ω .By • we denote the norm in S 1 Lemma 3.3.The process {U t, τ } associated to problem 1.1 has a pullback absorbing set in L 2p−2 Ω .
Proof.Multiplying 1.1 by |u| p−2 u and integrating over Ω, we get For any s ∈ R, any 2 ≤ p < ∞, and any bounded set B ⊂ L 2 Ω , there exists τ 0 such that Applying 3.2 and Lemma 3.2, we conclude the existence of a pullback absorbing set in L 2p−2 Ω for the process U t, τ .Ω |u t s | p dx ≤ M, ∀τ ≤ τ 0 , u τ ∈ B, 3.24where M depends on s, p but not on B, and u t s d/dt U t, τ u τ | t s .
2 e 2 , ..., e m } in L 2 Ω , and let P m : L 2 Ω → H m be the orthogonal projection, where {e i } ∞ i 1 are the eigenvectors of operator A −G s .For any u ∈ L 2 Ω , we write u P m u I − P m u u 1 u 2 .For any t ∈ R, any B ⊂ L 2 Ω and any ε, there exist τ 0 t, B, ε and m 0 ∈ N such that | I − P m v|22 < ε, ∀τ ≤ τ 0 , ∀u τ ∈ B, m ≥ m 0 .
Lemma 3.5 see 15 .If there exists σ > 0 such that t −∞ e σs |ϕ s | 2 ds < ∞, for all t ∈ R, then lim Assume that f satisfies 1.3 -1.5 , g satisfies 1.7 and 3.18 .Then the process {U t, τ } generated by 1.1 has a pullback attractor A {A t : t ∈ R} in S 2 0 Ω .Thanks to Lemmas 3.6 and 3.15 and the fact that g ∈ C loc R; L 2 Ω , we see that {U t, τ } satisfies condition PDC in S 2 0 Ω .Now from Lemmas 3.3 and 3.14 we get the desired result.