Considered here is the first initial boundary value problem for a semilinear degenerate parabolic equation involving the Grushin operator in a bounded domain . We prove the regularity and exponential growth of a pullback attractor in the space for the nonautonomous dynamical system associated to the problem. The obtained results seem to be optimal and, in particular, improve and extend some recent results on pullback attractors for reaction-diffusion equations in bounded domains.
Let be a bounded domain in , with smooth boundary . In this paper, we consider the following problem:
is the Grushin operator, is given, the nonlinearity and the external force satisfy the following conditions.(H1) The nonlinearity satisfies
where , , () are positive constants. Relation (1.3) and (1.4) imply that
where , and are positive constants.(H2) satisfies
where is the first eigenvalue of the operator in with the homogeneous Dirichlet boundary condition.
The Grushin operator was first introduced in . Noting that if , then is not elliptic in domains of which intersect the hyperplane . 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 for the process associated to problem (1.1) is considered in .
In this paper we continue the study in the paper . First, we will prove the existence of pullback attractors in (see Section 2 for its definition) and . As we know, if the external force is only in , then solutions of problem (1.1) are at most in 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 . Noting that, to prove the existence of pullback attractors in , we only need assumption (H2) of the external force ; however, to prove the existence of pullback attractors in and , we need an additional assumption of , namely, (3.18) in Section 3. Next, following the general lines of the approach in , we give exponential growth conditions in 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 of the pullback attractors [11, 12]. Therefore, the obtained results seem to be optimal and, in particular when , 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 and by using the asymptotic a priori estimate method. In Section 4, under additional assumptions of , an exponential growth in for the pullback attractors is deduced.
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 defined as the completion of in the following norm:
and the scalar product
The following lemma comes from .
Lemma 2.1. Assume that is a bounded domain in . Then the following embeddings hold:(i) continuously;(ii) compactly if , where, .
Now, we introduce the space defined as the closure of with the norm
The following lemma comes directly from the definitions of and .
Lemma 2.2. Assume that is a bounded domain in , with smooth boundary . Then continuously.
It is known that (see, e.g., ) for the operator , there exist
and is a complete orthonormal system in .
2.2. Pullback Attractors
Let be a Banach space with the norm . denotes all bounded sets of . The Hausdorff semidistance between and is defined by
Let be a process in , that is, such that and for all , . The process is said to be norm-to-weak continuous if , as in , for all , . The following result is useful for proving the norm-to-weak continuity of a process.
Proposition 2.3 (see ). Let be two Banach spaces, and let be, respectively, their dual spaces. Suppose that is dense in , the injection is continuous, and its adjoint is dense, and is a continuous or weak continuous process on . Then is norm-to-weak continuous on if and only if for , , maps compact sets of into bounded sets of .
Definition 2.4. The process is said to be pullback asymptotically compact if for any , any , any sequence , and any sequence , the sequence is relatively compact in .
Definition 2.5. A family of bounded sets is called a pullback absorbing set for the process if for any and any , there exist and such that
Definition 2.6. The family is said to be a pullback attractor for if (1) is compact for all , (2) is invariant, that is,
(3) is pullback attracting, that is,
(4)if is another family of closed pullback attracting sets, then , for all .
Theorem 2.7 (see ). Let be a norm-to-weak continuous process which is pullback asymptotically compact. If there exists a pullback absorbing set , then has a unique pullback attractor and
In the rest of the paper, we denote by , the norm and inner product in , respectively, and by the norm in . By we denote the norm in . For a Banach space , will be the norm. We also denote by an arbitrary constant, which is different from line to line, and even in the same line.
3. Existence of Pullback Attractors in
It is well known (see, e.g.,  or ) that under conditions , problem (1.1) defines a processwhere is the unique weak solution of (1.1) with initial datum at time . The process has a pullback attractor in .
In this section, we will prove that the pullback attractor is in fact in .
Lemma 3.1. Assuming that and satisfy (H 1)-(H 2), is a weak solution of (1.1). Then the following inequality holds for :
where is a positive constant.
Proof. Multiplying (1.1) by and then integrating over , we get
Using hypothesis (H1) and the inequality , we have
Letting , by (H1), we have
Now multiplying (3.4) by and using (3.5), we get
Integrating (3.6) from to and to , respectively, we obtain
Multiplying (1.1) by and integrating over , we have
Combining (3.8) and (3.10), and using the uniform Gronwall inequality, we have
Using (H1) once again and thanks to as , we get the desired result from (3.11).
Lemma 3.2. Assume that (H 1), (H 2) hold. Then for any and any that is bounded, there exists such that
for any and any , where .
Proof. Integrating (3.10) from to and using (3.8) and (3.11), in particular we find
On the other hand, differentiating (1.1) and denoting , we have
Taking the inner product of (3.14) with in , we get
Using (1.5) and Young’s inequality, after a few computations, we see that
Combining (3.16) and (3.13) and using the uniform Gronwall inequality, we obtain
The proof is now complete because as .
3.1. Existence of a Pullback Attractor in
In this section, following the general lines of the method introduced in , we prove the existence of a pullback attractor in . In order to do this, we need an additional condition of
where , are defined as in (3.30).
Lemma 3.3. The process associated to problem (1.1) has a pullback absorbing set in .
Proof. Multiplying (1.1) by and integrating over , we get
From (1.3) and the fact that continuously, we have
On the other hand, by Cauchy’s inequality, we see that
Combining (3.19)–(3.22) imply that
Applying (3.2) and Lemma 3.2, we conclude the existence of a pullback absorbing set in for the process .
Lemma 3.4. For any , any , and any bounded set , there exists such that
where depends on , but not on , and .
Proof. We will prove the lemma by induction argument. Letting and denoting we prove that for , there exist and such that
where depends on and and depends only on . For , we have from (3.17). Integrating (3.16) and using continuously, we get . Assuming that (), () hold, we prove so are and . Multiplying (3.14) by and integrating over , we obtain
Using the imbedding once again, we get
Combining Holder’s and Young’s inequalities, we see that
where . Choose , such that
Then from (3.27), we infer that
Applying (3.26) and (3.31) in (3.25), we find that
Combining () and (3.32), using the uniform Gronwall inequality and taking into account assumption (3.18), we get . On the other hand, integrating (3.32) from to , we find . Now since , and taking , we get the desired estimate.
We will use the following lemma.
Lemma 3.5 (see ). If there exists such that , for all , then
Let in , and let be the orthogonal projection, where are the eigenvectors of operator . For any , we write
Lemma 3.6. For any , any and any , there exist and such that
Proof. Multiplying (3.14) by and then integrating over , using and Cauchy’s inequality we get
We multiply (3.36) by and use assumption (1.4). We get
Integrating (3.37) from to ,
Now integrating (3.38) with respect to from to , we infer that
By Lemma 3.5 and since as , there exist and such that
for all and . For the second term of the right-hand side of (3.40), using Holder’s inequality we have
From Lemmas (3.5)–(3.7), we see that there exist and such that
Let and , from (3.40), taking into account (3.41) and (3.43), we obtain (3.35).
Lemma 3.7 (see ). Let be a bounded subset in . If has a finite -net in , then there exists an , such that for any , the following estimate is valid:
Using Lemma 3.7 and taking into account Lemmas 3.2 and 3.6 we conclude that the set has a finite -net in . Therefore, we get the following result.
Lemma 3.8. For any , any that is bounded, and any , there exists and such that
Lemma 3.9 (see ). For any , any bounded set , and any , there exist and such that
where is the Lebesgue measure in and .
Lemma 3.10 (see ). Let be a norm-to-weak continuous process in and . Then is pullback asymptotically compact in if (i) is pullback asymptotically compact in ; (ii)for any , any bounded set , and any , there exist and
where is independent of , , , and .
We are now ready to prove the existence of a pullback attractor in .
Theorem 3.11. Assume that assumptions (1.3)–(1.7) and (3.18) hold. Then the process associated to problem (1.1) possesses a pullback attractor in .
Proof. Because of Lemma 3.10, since has a pullback absorbing set in , we only have to prove that for any , any , and any , there exist and such that
Taking the inner product of (1.1) with in , where
Some standard computations give us
Combining (3.50)–(3.53), we find
Applying Lemmas 3.7 and 3.8 to (3.54) we find there exist and such that
Repeating the above arguments with in place of , we have
for some and , where
Letting and we have
This completes the proof.
3.2. Existence of a Pullback Attractor in
In this section, we prove the existence of a pullback attractor in .
Lemma 3.12. The process associated to (1.1) has a pullback absorbing set in .
Proof. We multiply (1.1) by ; then, using , we have
Using , Cauchy’s inequality, and argument as in Lemma 3.3, from (3.59) we have
Taking into account (3.11), the proof is complete.
In order to prove the existence of the pullback attractor in , we will verify so-called “(PDC) condition”, which is defined as follow
Definition 3.13. A process is said to satisfy (PDC) condition in if for any , any bounded set and any , there exists and a finite dimensional subspace of such that (i) is bounded in ; and(ii), for all and , where is a canonical projection and is the identity.
Lemma 3.14 (see ). If a process satisfies (PDC) condition in then it is pullback asymptotically compact in . Moreover, if is convex then the converse is true.
Lemma 3.15 (see ). Assume that satisfies (1.3) and (1.5). Then for any subset , if in , then we have
where the Kuratowski noncompactness measure in a Banach space defined as
Theorem 3.16. Assume that satisfies (1.3)–(1.5), satisfies (1.7) and (3.18). Then the process generated by (1.1) has a pullback attractor in .
Proof. We consider a complete trajectory lies on pullback attractor in for , that is, and , for all . Denoting and multiplying (1.1) by we have
Using Holder’s inequality we get
Thanks to Lemmas 3.6 and 3.15 and the fact that , we see that satisfies condition (PDC) in . Now from Lemmas 3.3 and 3.14 we get the desired result.
4. Exponential Growth in of Pullback Attractors
In this section, we will give an exponential growth condition in for the pullback attractor .
First, we recall a result in  which is necessary for the proof of our results.
Lemma 4.1. Let , be Banach spaces such that is reflexive, and the inclusion is continuous. Assume that is bounded sequence in such that weakly in for some and . Then, for all and
In the following theorem, instead of evaluating the functions which are differentiable enough and then using Lemma 4.1, we will formally evaluate the function .
Theorem 4.2. Assume that satisfies (1.3)-(1.5), satisfies (H 2), (3.18) and the following conditions
Proof. We differentiate with respect to time in (1.1), then multiply by , we get
Integrating in the last inequality, in particular, we get
for all . Now, integrating with respect to , between and
for all , in paricular,
for all . Multiplying (1.1) by and then integrating on , we get
Using hypothesis (H1) and the fact that , we have
Integrating (4.9) from to , we have
Multiplying (1.1) by then integrating over , we have
Integrating now between and , we obtain
From (4.13) and using hypothesis (H1), we get
Integrating inequality (4.15) with respect to from to , we obtain
for all , . From (4.16) and (4.11), we obtain that
for all . From (4.7), taking and we have
for all . Analogously, and if we take and in inequlity (4.15), then