Time-Dependent Global Attractor for a Class of Nonclassical Parabolic Equations

Abstract

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.

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Academic Editor:Gilles Lubineau

Department of Mathematics, Longqiao College of Lanzhou Commercial College, Lanzhou, Gansu 730101, China

Received 14 May 2014; Revised 12 August 2014; Accepted 13 August 2014; 28 August 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let Ω be a bounded set of ^R3 with smooth boundary ∂Ω . For any τ∈R , we consider the following nonclassical equation: [figure omitted; refer to PDF] where a:Ω[arrow right]R are assigned data and [varepsilon](t) is a decreasing bounded function satisfying [figure omitted; refer to PDF] and L>0 is such that [figure omitted; refer to PDF]

The nonlinearity f∈^C2 (R) , with f(0)=0 , is assumed to satisfy the inequality [figure omitted; refer to PDF] along with the dissipation condition [figure omitted; refer to PDF] where _λ1 is the first eigenvalue of -Δ in ^H01 (Ω) and g(x)∈^L2 (Ω) .

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 [1] 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 Δ_ut 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 f(u) and inhomogeneous term g(x) , along with the time-dependent parameter [varepsilon](t) . The presence of the Δ_ut 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 _u0 belongs to ^H01 (Ω) , then the solution u(t,x) with u(0,x)=_u0 is always in ^H01 (Ω) and has no higher regularity because of the term -Δ_ut .

In the case when [varepsilon](t)=[varepsilon] 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, [varepsilon]=[varepsilon](t) , 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 [17] 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 [18] as well as time-dependent spaces [19] in the context of stochastic partial differential equations; and Flandoli and Schmalfuss in [20] 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. [17] and di Plinio et al. [21], 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.

2. Preliminaries

In this section, we introduce some notations and definitions, along with a lemma.

We set H=^L2 (Ω) , with inner product Y9;·,·YA; and norm ||·|| . For 0...4;σ...4;2 , we define the hierarchy of (compactly) nested Hilbert spaces: [figure omitted; refer to PDF] Then, for t∈R and 0...4;σ...4;2 , we introduce the time-dependent spaces ^Htσ =_Hσ+1 endowed with the time-dependent product norms: [figure omitted; refer to PDF] The symbol σ is always omitted whenever zero. In particular, the time-dependent phase space where we settle the problem is [figure omitted; refer to PDF] Then, we have the compact embeddings: [figure omitted; refer to PDF] with injection constants independent of t∈R .

Note that the spaces _Ht are all the same as linear spaces, and the norms ^{||z||_Ht 2} and ^{||·||_Hτ 2} are equivalent for any fixed t , τ∈R .

According to (5), we have the following lemma.

Lemma 1.

The following inequalities hold for some 0<ν<1 and _c1 ...5;0 : [figure omitted; refer to PDF]

3. Existence of the Time-Dependent Global Attractor

3.1. Well-Posedness

For any τ∈R , we rewrite the problem (1) as [figure omitted; refer to PDF]

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 a∈_Hτ , there is a unique solution u of (1), on any interval [τ,t] with t...5;τ , [figure omitted; refer to PDF] Furthermore, for i=1,2 , let ^uτi ∈_Hτ be two initial conditions such that _{||^uτi ||Hτ} ...4;R and denote by _ui the corresponding solutions to the problem of (11). Then the following estimates hold as follows: [figure omitted; refer to PDF] for some constant K=K(R)...5;0 .

According to Lemma 2 above, the family of maps with t...5;τ∈R [figure omitted; refer to PDF] acting as [figure omitted; refer to PDF] where u is the unique solution of (11) with initial time τ and initial condition _uτ =_Hτ , defines a strongly continuous process on the family {_Ht}t∈R .

3.2. Time-Dependent Absorbing Set

Definition 3.

A time-dependent absorbing set for the process U(t,τ) is a uniformly bounded family B={_Bt}t∈R with the following property: for every R...5;0 there exists _θe =_θe (R)...5;0 such that [figure omitted; refer to PDF]

Lemma 4.

Under the assumptions of (2)-(5), for _uτ ∈_Hτ , t...5;τ , let U(t,τ)_uτ be the solution of (1); then, there exist positive constants δ , _Cν,||g|| and an increasing positive function Q such that [figure omitted; refer to PDF]

Proof.

Multiplying (11) by u , we obtain [figure omitted; refer to PDF] Noting that ^{[varepsilon][variant prime]} <0 and using (3) and Young and Poincaré inequalities, for δ>0 small, we infer that [figure omitted; refer to PDF] By the Gronwall lemma, we have [figure omitted; refer to PDF] This completes the proof.

Lemma 5 (time-dependent absorbing set).

Under the assumptions of (2)-(5), there exists a constant _R1 >0 , such that the family B={_Bt (_R1 )_}t∈R is a time-dependent absorbing set for U(t,τ) .

Proof.

From the proof of Lemma 4, for u∈_Bτ (R) , there exists _θe ...5;0 , provided that t-τ...5;_θe , [figure omitted; refer to PDF] This concludes the proof of the existence of the time-dependent absorbing set.

We can assume that the time-dependent absorbing set _Bt =_Bt (_R1 ) is positively invariant (namely, U(t,τ)_Bτ ⊂_Bt for all t...5;τ ). Indeed, calling _θe the entering time of _Bt such that [figure omitted; refer to PDF] we can substitute _Bt with the invariant absorbing family: [figure omitted; refer to PDF]

3.3. Time-Dependent Global Attractor

As introduced in [17], for t∈R , let _Xt be a family of normed spaces; we consider the collection [figure omitted; refer to PDF] When K...0;Ø we say that the process is asymptotically compact.

Definition 6.

One calls a time-dependent global attractor the smallest element of K ; that is, the family A={_At}t∈R ∈K such that [figure omitted; refer to PDF] for any element R={_Kt}t∈R ∈K .

Theorem 7 (see [17]).

If U(t,τ) is asymptotically compact, then the time-dependent attractor A exists and coincides with the set A={_At}t∈R . 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 f , following [11, 15, 17], we decompose f as follows: [figure omitted; refer to PDF] where _f0 ,_f1 ∈^C2 (R) satisfy, for some c...5;0 , [figure omitted; refer to PDF]

Noting that B={_Bt (_R1 )_}t∈R is a time-dependent absorbing set for U(t,τ)_uτ , then for each initial data _uτ ∈_Bτ (_R1 ) , we decompose U(t,τ) as [figure omitted; refer to PDF] where v=_U1 (t,τ)_uτ and w=_U2 (t,τ)_uτ solve the following equations, respectively: [figure omitted; refer to PDF]

Lemma 8.

Under assumptions of (2)-(5), (27)-(30), there exists δ=δ(B)>0 such that [figure omitted; refer to PDF]

Proof.

Multiplying (32) by v , we obtain [figure omitted; refer to PDF] Using (28) and noting that ^{[varepsilon][variant prime]} <0 , by Young and Poincaré inequalities, for δ>0 small, we infer [figure omitted; refer to PDF] By the Gronwall lemma, we complete the proof.

Remark 9.

From Lemmas 4 and 8, we have the uniform bound [figure omitted; refer to PDF]

Lemma 10.

Under the assumptions of (2)-(5), (27)-(30), there exists M=M(B)>0 such that [figure omitted; refer to PDF]

Proof.

Multiplying (33) by ^A(1/3) w , we obtain [figure omitted; refer to PDF] In view of Remark 9 and the growth of f and _f0 , using the embedding _H1 ⊂^L6 (Ω) , we have [figure omitted; refer to PDF] Noting that ^{[straight epsilon][variant prime]} <0 , by Young and Poincaré inequalities, for δ>0 small, we infer [figure omitted; refer to PDF] By the Gronwall lemma, we complete the proof.

Remark 11.

From Lemma 10, we immediately have the following regularity result: _At is bounded in ^Ht1/3 (with a bound independent of t ).

Theorem 12 (see [17]).

If U(t,τ) is a T -closed process for some T>0 , which possesses a time-dependent global attractor A , then A is invariant.

Remark 13 (see [17]).

If the process U(t,τ) is closed, it is T -closed, for any T>0 . Note that if the process U(t,τ) is a continuous (or even norm-to-weak continuous) map for all t...5;τ , then the process is closed.

Theorem 14 (existence of the time-dependent global attractor).

Under the assumptions of (2)-(5), the process U(t,τ):_Hτ [arrow right]_Ht generated by problem (1) admits an invariant time-dependent global attractor A={_At}t∈R .

According to Lemma 10, we consider the family K={_Kt}t∈R , where [figure omitted; refer to PDF] where _Kt is compact by the compact embedding ^Ht1/3 ⊂_Ht ; besides, since the injection constants are independent of t , K is uniformly bounded. Hence, according to Lemmas 5, 8, and 10, K is pullback attracting, and the process U(t,τ) 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 U(t,τ) stated in Lemma 2, according to Remark 13, the process U(t,τ) is closed, and it is T -closed, for some T>0 ; then by Theorem 12, we know that the time-dependent global attractor A is invariant.

3.4. Regularity of the Time-Dependent Global Attractor

3.4.1. The Second Decomposition of the System Equations

We fix τ∈R and each initial data _uτ ∈_Aτ , decomposing U(t,τ) as [figure omitted; refer to PDF] where v=_U3 (t,τ)_uτ and w=_U4 (t,τ)_uτ solve the following equations, respectively: [figure omitted; refer to PDF] As a particular case of Lemma 8, we learn that [figure omitted; refer to PDF]

Lemma 15.

Under assumptions of (2)-(5), for some M=M(A)>0 , one has the uniform bound [figure omitted; refer to PDF]

Proof.

Multiplying (46) by A_wt +Aw , we obtain [figure omitted; refer to PDF] We denote [figure omitted; refer to PDF] noting that [figure omitted; refer to PDF] Noting that (2) and using Young and Poincaré inequalities, for δ>0 small, we infer [figure omitted; refer to PDF]

Denoting by C>0 a generic constant depending on the size of _At in ^Ht1/3 , using the invariance of the attractor, we find [figure omitted; refer to PDF] Exploiting the embeddings _H1/3 ⊂^L18/7 , _H4/3 ⊂^L18 , we get [figure omitted; refer to PDF] this yields [figure omitted; refer to PDF] noting that ^{[varepsilon][variant prime]} <0 , we infer [figure omitted; refer to PDF] 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 A={_At}t∈R , _At is bounded in ^Ht1 , with a bound independent of t .

In fact, we define [figure omitted; refer to PDF] according to inequalities (47) and (48), for all t∈R , we have [figure omitted; refer to PDF] where dist... denotes the Hausdorff semidistance in _Ht ; that is, [figure omitted; refer to PDF]

From Theorem 14, we know that the time-dependent global attractor A={_At}t∈R is invariant; this means that [figure omitted; refer to PDF]

Hence, _At ⊂_Et ¯=_Et ; that is, _At is bounded in ^Ht1 , with a bound independent of t .

Acknowledgments

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.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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