Pullback Attractors for a Nonautonomous Retarded Degenerate Parabolic Equation

Abstract

This paper is devoted to a nonautonomous retarded degenerate parabolic equation. We first show the existence and uniqueness of a weak solution for the equation by using the standard Galerkin method. Then we establish the existence of pullback attractors for the equation by proving the existence of compact pullback absorbing sets and the pullback asymptotic compactness.

1. Introduction

In this paper, we study the following nonautonomous retarded degenerate parabolic equation defined on an arbitrary domain (bounded or unbounded) :

with the initial value and boundary condition

where is a fixed positive constant, and we suppose that satisfies the following assumptions:

: when is bounded, we suppose that and for some and each ;

: when is unbounded, we suppose that satisfies condition and for some .

For , we denote by the Banach space which is consisted of all continuous functions endowed with norm . For all real number , , and any continuous function , we denote by the element of , and for .

is a nonlinear function satisfying the following conditions:(i);(ii)there exists a positive continuous function with

for some positive integers such that if , and , then we have(iii)there exists a positive constant such that, for any , and ,

(iv).

is a given function satisfying

It is well known that problem (1) can be seen as a simple model for neutron diffusion (see [1]). For this case, and represent the neutron flux and neutron diffusion, respectively.

As we know, the most interesting problem about the partial differential equations is to investigate the asymptotic behavior of the solutions when time tends to infinity. In addition, we can get the useful information about the future of the model from it. Besides, when we want to model some phenomena arising in physics, chemistry, biology and other fields, some hereditary variables such as aftereffect, time-lag, and time delay can be added in the variables. For example, the stochastic retarded reaction-diffusion equation on unbounded domains was considered in [2]. For this kind of equation with the delay term, the reader is referred to [2–4] and the references therein.

The theory of pullback attractors is a good choice to investigate the long time behavior of evolution systems. Anh and Bao [5] proved the existence of a pullback attractor for a nonautonomous semi-linear degenerate parabolic equation. Cui and Li [6] investigated the existence and upper semicontinuity of random attractors for stochastic degenerate parabolic equations with multiplicative noises. For nonautonomous stochastic PDEs, the existence of attractors and well-posedness of the problem have been studied in [7–15]. In addition, the existence of a random attractor is generally ensured by the well-posedness (i.e., existence and uniqueness) of the problem combined with a compact absorbing set. When we want to get the compactness of the absorbing sets and get the compactness of the system, Sobolev compact embedding is a good choice, see, e.g., [5, 16].

Motivated by [5] and [2], in this article, we devote to study a nonautonomous retarded degenerate parabolic equation. In fact, as we can see, the term of delay makes the phase space not reflexive and the uniform estimates of solutions will be difficult. The main result of the paper is Theorem 2.12.

The rest of this paper consists of the following contents. For convenience, in Section 2, we first introduce some preliminaries about function spaces and operators as well as dynamical systems and pullback attractors. Then, in Section 3, we get the existence and uniqueness of a weak solution for the problem (1) by using the standard Galerkin method. Section 4 is devoted to prove the existence of the pullback attractors.

2. Preliminaries

2.1. Function Spaces and Operators

We recall some basic concepts related to the function spaces and operators which we will use, the reader is referred to [5, 7, 16] for more details.

In order to study problem (1), we define the space as the closure of with respect to the norm

This space is a Hilbert space with respect to the inner product

We denote by , , , the norms and inner products in and , respectively, and the norm in . And also, define

Let be a normal positive number which might vary from line to line. We have the following lemma which is a generalized version of the Poincaré inequality in [5].

Lemma 2.1. Suppose that is a bounded (unbounded) domain of and suppose that the condition () is satisfied. Then there is a positive constant such that

The following results can be found in [16].

Remark 2.2. Let , andThe given exponent has the role of the critical exponent in the classical Sobolev embedding.

Lemma 2.3. Suppose that is a bounded domain in and satisfies (). We get the following embedding:(1) continuously,(2) compactly for every .

Lemma 2.4. Suppose that is an unbounded domain in and satisfies (). We get the following embedding:(1) continuously, if ,(2) compactly, if .

We now investigate the case where is a bounded domain (the unbounded case is considered similarly).

We define a linear operator determined by the leading term in (1)

Under the conditions or , the operator is a positive and self-adjoint linear operator with the domain of definition

The eigenvectors of the operator construct the complete orthonormal family in and the relevant spectrum is discrete and represented by such that

and

Furthermore

Noting that

Define as the conjugate space of the space . Since , we get an evolution triple

2.2. Dynamical Systems and Pullback Attractors

Next, we will show some theories about the dynamical systems and pullback attractors, the following contents can be found in [2, 5, 7].

Let be a nonempty set and be a metric space endowed with metric . For any , define the Hausdorff semidistance between and

Definition 2.5. A family of mapping from to itself is said to be a family of shift operates on if for , the following two group properties are satisfied:(i), for any ,(ii), for any and .

Definition 2.6. Suppose that is a family of shift operators on . Then a mapping on is called a continuous -cocycle if for any and , it satisfies:(i) is the identity on ,(ii),(iii) is continuous.In what follows, we will always suppose that is a continuous -cocycle on and is a collection of families of subsets of :

Definition 2.7. Suppose that is a collection of families of subsets of . Then is said to be inclusion closed if and with for any imply that .

Definition 2.8. Suppose that is a collection of families of subsets of . Then is said to be a -pullback absorbing set for if for any and , there is such that

Definition 2.9. Suppose that is a collection of families of subsets of . Then is called -pullback asymptotically compact in if for all , possesses a convergent sub-sequence in whenever and with .

Definition 2.10. Suppose that is a collection of families of subsets of and . Then is said to be a -pullback attractor for if for any :

(i) is compact,(ii) is invariant, that is,

(iii) attracts every set in , that is, for any ,

Proposition 2.11. Suppose that is an inclusion-closed collection of families of subsets of and is a continuous -cocycle in . Assume that is a closed absorbing set for in and is -pullback asymptotically compact in . Then possesses a unique -pullback attractor such that

In this article, let be the collection of families of subsets of . Then, we will give the main theorem of this article.

Theorem 2.12. (Main Theorem) Assume that (i)–(iv) hold and (6) satisfies. Then the problem (1) and (2) has a -pullback attractor .

3. Existence and Uniqueness of the Solution

In this section, we will prove the existence and uniqueness of the solution for problem (1) and (2).

Theorem 3.1. Suppose that satisfies (i)–(iv) and , then for any given , , the problem (1) and (2) has a unique weak solutionand the equation is satisfied in the sense of distribution.

Proof. Consider the approximating solution in the formwhere are eigenvectors of the linear operator . We have the following equations for Hence, we get the local existence of . We now give some priori estimates for . We haveBy the Young inequality, we getwhere is given by (5), is a positive constant. By the condition (iv) of , we can choose small enough such that , thenIntegrating (32) on , , then from the conditions (iii) and (iv) of , we can getNoting that , we obtainBy (34), we also havefor fixed , we obtain that, for ,and for ,Hence, from (36) and (38), we get for all The (34) and (38) imply that for each ,Noting that from the condition of , we conclude is bounded in then we getHence we getWe also have the following equation which has another formwe obtain that is bounded in . We also see the following tripleapplying the compactness lemma we can suppose that strongly in . Thus in and in . Since is continuous, in , we getTherefore by (46) we getBy a standard argument, and using the strong convergence in and the Doubinskii’s theorem [4], we get that any weak-∗limit is a solution of problem (1) subject to the initial conditions. Hence we obtain the existence of the solution.
Next we investigate the uniqueness of the solution. Let be the two solutions of problem (1) with the initial value , respectively. Thus from (1) we getBy the condition (ii) of we conclude thatIntegrating on , ,Therefore, for fixed , we have that, for ,for ,Thus, from (53) and (54) we get that, for all ,By using the Gronwall inequality we obtain that, for all ,Hence we obtain the uniqueness (if ) of the solution.

4. Existence of Pullback Attractors

In this section, we will prove the existence of -pullback attractors of this nonautonomous retarded degenerate parabolic equation by proving the existence of the -pullback absorbing sets and the -pullback asymptotic compactness.

We will first construct a -cocycle for the nonautonomous retarded degenerate parabolic equation defined on . As in the case of bounded domains, the Theorem 3.1 implies that the problem (1) and (2) is well defined. In order to construct a cocycle for the problem (1) and (2), let and define a shift operator on , for any ,

Then we can define as a mapping from to such that

where , , , and is the solution of the problem (1) and (2). It is not difficult to see that is a continuous -cocycle on .

Then, we will derive uniform estimates on the weak solutions of problem (1) and (2) when , which is essential to prove the existence of a bounded pullback absorbing set and the pullback asymptotic compactness for .

Lemma 4.1. Suppose that (i)–(iv) hold and (6) satisfies. Then there exists such that for any ,which implies the existence of -pullback absorbing set in .

Proof. Taking the inner product of (1) with over , we haveBy using the Young inequality and replacing with in (35) and (38), we can get for all ,Since , let bethen for any , is bounded. Hence we get the lemma immediately.

Lemma 4.2. Suppose that (i)–(iv) hold and (6) satisfies. Then for any ,

Proof. Noting that from (32) we obtain thatIntegrating (65) over the interval , we get thatIt follows from the Young inequality and (iii) of thatBy (66) and (67), we find thatThus, thanks to , we have thatTherefore,where is given by (59). It follows from (6) and Lemma 4.1 we find that is bounded. Thus we finish the proof of the Lemma 4.2.

Lemma 4.3. Suppose that (i)–(iv) hold and (6) satisfies. Then for any and ,where , are two positive constants.

Proof. Taking the inner product of (1) with we getApplying the Young inequality, we find thatIt follows from (73)–(75) thatThus,Let be the constant in Lemma 4.1, then take and . Integrating (77) over the interval leads toIntegrating the above inequality with respect to over the integral , we obtain thatThus we get, for all ,From Lemma 4.1, Lemma 4.2, and (6) we know is bounded. Let . Integrating (77) over the interval we getThus, for all and ,where is given by (59), is given by (71). Then we let beIt is not difficult to find that is bounded. Thus we finish the proof.

Lemma 4.4. is -pullback asymptotically compact in ; that is, for any , the sequence possesses a convergence subsequence in provided , , and .

Proof. The proof is quite similar to the Lemma 18 in [2], more details can be found in [7, 9, 17, 18] and thus we omit the proof here.

Proof of Theorem 2.12. Noting that from Lemma 4.1, we get that has a -pullback absorbing set. In addition, is -pullback asymptotically compact in by Lemma 4.4. Therefore, the existence of a unique -pullback attractor for follows from Proposition 2.11.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The second author is supported by the Natural Science Foundation of Shandong under Grant No. ZR2018QA002, the National Natural Science Foundation of China No. 11901342 and China Postdoctoral Science Foundation No. 2019M652350. The third author is supported by the NSF of China (No. 11371183) and the NSF of Shandong Province (No. ZR2013AM004).