Time-Dependent Global Attractor for a Class of Nonclassical Parabolic Equations
Based on the recent theory of time-dependent global attractors in the works of Conti et al. (2013) and di Plinio et al. (2011), we prove the existence of time-dependent global attractors as well as the regularity of the time-dependent global attractor for a class of nonclassical parabolic equations.
Let be a bounded set of with smooth boundary . For any , we consider the following nonclassical equation: where are assigned data and is a decreasing bounded function satisfying and is such that
The nonlinearity , with , is assumed to satisfy the inequality along with the dissipation condition where is the first eigenvalue of in and .
The classical reaction diffusion equation has a long history in mathematical physics and appears in many mathematical models. It arises in several bead mark problems of hydrodynamics and heat transfer theory, such as heat transfer, as well as in solid-fluid in Hradionconfigurations and, of course, in standard situations mass diffusion and flow through porous media [1, 2]. In 1980, Aifantis in  pointed out that the classical diffusion equation does not suffice to describe transport in media with two temperatures or two diffusions as well as in cases where the diffusions substances behave as a viscous fluid. It turns out that new terms appear in the classical diffusion equation when such effects are considered. In particular, the mixed spatiotemporal derivative consistently appears in several generalized reaction-diffusion models [1, 3–6] and this is our motivation for studying (1) which contains, in addition, the nonlinear term and inhomogeneous term , along with the time-dependent parameter . The presence of the has some important consequences on the character of the solution of the partial differential equation under consideration. For example, the classical reaction diffusion equation has smoothing effect; for example, although the initial data only belongs to a weaker topology space, the solution will belong to a stronger topology space with higher regularity. However, for (1), if the initial data belongs to , then the solution with is always in and has no higher regularity because of the term .
In the case when is a positive constant, the asymptotic behavior of solutions to (1) has been extensively studied by several authors in [2, 7–16] and the references therein. In the general case of time dependence, that is, , the longtime behavior of the nonclassical equation has not been considered so far. In this paper, we borrow some ideas from the following previous contributors: Conti et al. in  who introduced the theory of time-dependent global attractors and apply the theory to the wave equations; Caraballo et al. who introduced a one-parameter family of Banach spaces in the context of cocycles for nonautonomous and random dynamical systems in  as well as time-dependent spaces  in the context of stochastic partial differential equations; and Flandoli and Schmalfuss in  who introduced a family of metric spaces depending on a parameter and applied it to the stochastic form of Navier-Stokes equations. In this paper, based on the recent theory of time-dependent global attractors of Conti et al.  and di Plinio et al. , we prove the existence of time-dependent global attractors as well as the regularity of the time-dependent global attractor for a class of nonclassical parabolic equations as described by (1).
The paper is organized as follows. In Section 2, we present some preliminaries, establish the necessary notation and functions spaces to be used in the subsequent analysis, and give some useful lemmas. In Section 3, we prove the existence of time-dependent global attractors for the nonclassical parabolic equation and its regularity. Our main results are Theorems 14 and 16.
In this section, we introduce some notations and definitions, along with a lemma.
We set , with inner product and norm . For , we define the hierarchy of (compactly) nested Hilbert spaces: Then, for and , we introduce the time-dependent spaces endowed with the time-dependent product norms: The symbol is always omitted whenever zero. In particular, the time-dependent phase space where we settle the problem is Then, we have the compact embeddings: with injection constants independent of .
Note that the spaces are all the same as linear spaces, and the norms and are equivalent for any fixed , .
According to (5), we have the following lemma.
Lemma 1. The following inequalities hold for some and :
3. Existence of the Time-Dependent Global Attractor
For any , we rewrite the problem (1) as
Using the Galerkin approximation method, we can obtain the following result concerning the existence and uniqueness of solutions; see, for example, [2, 7, 8, 15, 16].
Lemma 2. Under the assumptions of (2)–(5), for any , there is a unique solution of (1), on any interval with , Furthermore, for , let be two initial conditions such that and denote by the corresponding solutions to the problem of (11). Then the following estimates hold as follows: for some constant .
According to Lemma 2 above, the family of maps with acting as where is the unique solution of (11) with initial time and initial condition , defines a strongly continuous process on the family .
3.2. Time-Dependent Absorbing Set
Definition 3. A time-dependent absorbing set for the process is a uniformly bounded family with the following property: for every there exists such that
Lemma 4. Under the assumptions of (2)–(5), for , , let be the solution of (1); then, there exist positive constants , and an increasing positive function such that
Proof. Multiplying (11) by , we obtain Noting that and using (3) and Young and Poincaré inequalities, for small, we infer that By the Gronwall lemma, we have This completes the proof.
Lemma 5 (time-dependent absorbing set). Under the assumptions of (2)–(5), there exists a constant , such that the family is a time-dependent absorbing set for .
Proof. From the proof of Lemma 4, for , there exists , provided that ,
This concludes the proof of the existence of the time-dependent absorbing set.
We can assume that the time-dependent absorbing set is positively invariant (namely, for all ). Indeed, calling the entering time of such that we can substitute with the invariant absorbing family:
3.3. Time-Dependent Global Attractor
As introduced in , for , let be a family of normed spaces; we consider the collection When we say that the process is asymptotically compact.
Definition 6. One calls a time-dependent global attractor the smallest element of ; that is, the family such that for any element .
Theorem 7 (see ). If is asymptotically compact, then the time-dependent attractor exists and coincides with the set . In particular, it is unique.
According to Definition 6, the existence of the time-dependent global attractor will be proved by a direct application of the abstract Theorem 7. Precisely, in order to show that the process is asymptotically compact, we will exhibit a pullback attracting family of compact sets. To this aim, the strategy classically consists in finding a suitable decomposition of the process in the sum of a decaying part and of a compact one.
3.3.1. The First Decomposition of the System Equations
For the nonlinearity , following [11, 15, 17], we decompose as follows: where satisfy, for some ,
Noting that is a time-dependent absorbing set for , then for each initial data , we decompose as where and solve the following equations, respectively:
Lemma 8. Under assumptions of (2)–(5), (27)–(30), there exists such that
Proof. Multiplying (32) by , we obtain Using (28) and noting that , by Young and Poincaré inequalities, for small, we infer By the Gronwall lemma, we complete the proof.
Remark 9. From Lemmas 4 and 8, we have the uniform bound
Lemma 10. Under the assumptions of (2)–(5), (27)–(30), there exists such that
Proof. Multiplying (33) by , we obtain In view of Remark 9 and the growth of and , using the embedding , we have Noting that , by Young and Poincaré inequalities, for small, we infer By the Gronwall lemma, we complete the proof.
Remark 11. From Lemma 10, we immediately have the following regularity result: is bounded in (with a bound independent of ).
Theorem 12 (see ). If is a -closed process for some , which possesses a time-dependent global attractor , then is invariant.
Remark 13 (see ). If the process is closed, it is -closed, for any . Note that if the process is a continuous (or even norm-to-weak continuous) map for all , then the process is closed.
Theorem 14 (existence of the time-dependent global attractor). Under the assumptions of (2)–(5), the process generated by problem (1) admits an invariant time-dependent global attractor .
According to Lemma 10, we consider the family , where where is compact by the compact embedding ; besides, since the injection constants are independent of , is uniformly bounded. Hence, according to Lemmas 5, 8, and 10, is pullback attracting, and the process is asymptotically compact, which proves the existence of the unique time-dependent global attractor. In order to state the invariance of the time-dependent global attractor, due to the strong continuity of the process stated in Lemma 2, according to Remark 13, the process is closed, and it is -closed, for some ; then by Theorem 12, we know that the time-dependent global attractor is invariant.
3.4. Regularity of the Time-Dependent Global Attractor
3.4.1. The Second Decomposition of the System Equations
We fix and each initial data , decomposing as where and solve the following equations, respectively: As a particular case of Lemma 8, we learn that
Lemma 15. Under assumptions of (2)–(5), for some , one has the uniform bound
Proof. Multiplying (46) by , we obtain
Noting that (2) and using Young and Poincaré inequalities, for small, we infer
Denoting by a generic constant depending on the size of in , using the invariance of the attractor, we find Exploiting the embeddings , , we get this yields noting that , we infer By the Gronwall lemma, we can get (48) immediately.
This completes the proof.
Therefore, we have the following regularity result.
Theorem 16 (regularity of the time-dependent global attractor). Under the assumptions of (2)–(5), the time-dependent global attractor , is bounded in , with a bound independent of .
In fact, we define according to inequalities (47) and (48), for all , we have where denotes the Hausdorff semidistance in ; that is,
From Theorem 14, we know that the time-dependent global attractor is invariant; this means that
Hence, ; that is, is bounded in , with a bound independent of .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author expresses her sincere thanks to the anonymous reviewer for his/her careful reading of the paper, giving valuable comments and suggestions. She also thanks the editors for their kind help.
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