Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 8186247, 13 pages
https://doi.org/10.1155/2018/8186247

Research Article

Asymptotic Behavior of Solutions to Reaction-Diffusion Equations with Dynamic Boundary Conditions and Irregular Data

Yonghong Duan,¹ Chunlei Hu,² and Xiaojuan Chai²

¹Department of Mathematics, Taiyuan University, Taiyuan 030032, China
²School of Mathematical Science, Anhui University, Hefei 230601, China

Correspondence should be addressed to Chunlei Hu; chunleihu_ahu@163.com

Received 13 March 2018; Revised 22 June 2018; Accepted 2 July 2018; Published 13 August 2018

Academic Editor: Nikos I. Karachalios

Copyright © 2018 Yonghong Duan et al. 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.

Abstract.

This paper is concerned with the asymptotic behavior of solutions to reaction-diffusion equations with dynamic boundary conditions as well as -initial data and forcing terms. We first prove the existence and uniqueness of an entropy solution by smoothing approximations. Then we consider the large-time behavior of the solution. The existence of a global attractor for the solution semigroup is obtained in . This extends the corresponding results in the literatures.

1. Introduction

We consider the asymptotic behavior of solutions to the following parabolic equations with dynamic boundary conditions and irregular data,

where is a bounded domain in with smooth boundary . The first equation is the standard reaction-diffusion equation, and the second equation is the boundary equation, in which the value of is assumed to be the trace of the function defined for ,   is the Laplace-Beltrami operator on [1], plays the role of a surface diffusion coefficient, and is the outward normal to . By irregular data, we mean that , and . We also assume that is and there exist positive constants and such that

Partial differential equations with dynamic boundary conditions like (1) have applications in various fields such as hydrodynamics, heat transfer theory, and thermoelasticity [2–5]. The existence and uniqueness of solutions for problem (1) have been studied extensively in various contexts; see, e.g., [2, 6–9]. The long time behavior of solutions to (1) and related models have also aroused much interests in recent years. For autonomous equations, in [10] the existence of global attractors was derived under the assumption . Then in [11–13] the existence of global attractors and their fractal dimensions were further studied for certain semilinear reaction-diffusion equations with dynamic boundary conditions, while in [9, 14, 15], the existence of global attractors was obtained for more general quasilinear parabolic equations with dynamic boundary conditions. For the nonautonomous case, the existence of pullback attractors for parabolic equations with dynamic boundary conditions was first obtained in [16], and then in [17–19], while the existence of uniform attractors for linear and quasilinear parabolic equations with dynamic boundary conditions was investigated in [12, 20].

In the references aforementioned, the initial data and forcing terms involved are mostly assumed to be regular (belonging to or even ), and few results were known when the initial data and forcing terms are not regular, such as functions [21]. This motivates us to investigate the existence and uniqueness results as well as the large-time behavior of solutions to problem (1) with data. Due to irregular initial data and forcing terms, the usual framework for the existence and uniqueness of solutions does not work here. Also the large-time behavior for parabolic equations with data is much more involved. The less regularity of the data influences the regularity of the solutions greatly, which in turn causes some crucial difficulties in investigating the asymptotic behaviors of solutions.

In this paper, to derive the existence and uniqueness result we shall work in the framework of entropy solutions, which was first introduced in [22] for elliptic equations involving measure data and was then adapted to parabolic equations with data in [23]. We will borrow some ideas in [23, 24] and use smooth approximations to derive the existence of the entropy solution, whereas, to cope with the dynamic boundary conditions, some delicate analysis must be addressed. For the large-time behavior of the entropy solution, we prove the existence of global attractor in . It is well known that to obtain global attractors, the most essential step is to derive the compactness of the semigroup, which more or less relies on certain uniform estimates in higher order spaces. Here to overcome the difficulties brought by the irregular initial data and forcing terms, we will perform some delicate Marcinkiewicz estimates on the solution and use the Aubin-Simon type compactness results to derive the compactness of the solution semigroup.

We mention that the existence and uniqueness results for elliptic or parabolic equations with Dirichlet boundary condition and or measure data have been studied extensively in the past years; see [24–26] and large amount of references therein. The large-time behaviors for parabolic equations with Dirichlet boundary conditions involving irregular data have also been studied by many authors; see, e.g., [27–29]. The results obtained here might be viewed as an extension of the results therein to problems with more general boundary conditions.

The rest of this paper is organized as follows. In Section 2, we provide some preliminaries and the main results of this paper. Then in Sections 3 and 4, we provide the proof for the main results. For convenience, in the following we use , , to denote generic constants in various occasions, and we will denote ,   as and , respectively. For a function on and , we set .

2. Preliminaries and the Main Result

To deal with the dynamic boundary conditions, we introduce the Lebesgue spaces as follows (see [14] for more details). Let be a bounded domain with smooth boundary . For , define the Lebesgue space as

with norm

where on is defined by for any measurable set . Define the Sobolev space as

with norm

It is easy to see that we can identify with under this norm. Hereafter, we denote by the dual space of .

Let be the truncation at height ,

and denote as its primitive function, i.e.,

It is obvious that and .

We work within the framework of entropy solutions defined as follows.

Definition 1. A function is called an entropy solution of problem (1), if for any and , ; moreover,

for all and such that

Our first result concerns on the existence and uniqueness of entropy solutions.

Theorem 2. Under assumptions (2)–(3) and suppose that and , there exists a unique entropy solution for problem (1).

To give the second result on the long time behavior of solutions, we recall the definition of global attractors.

Definition 3 (see [30]). Let be a semigroup on a Banach space . A subset is called a global attractor for the semigroup if is compact in and enjoy the following properties:

(i) is invariant, i.e., for any ;(ii) attracts every bounded subset of , i.e., for any bounded subset of and any neighborhood of the set , there exists a such that

Theorem 4. Assume that ,   and satisfies assumptions (2)–(3); then the semigroup generated by problem (1) admits a global attractor in ; i.e., is compact, invariant in and attracts every bounded subset of in the norm topology of

To prove the theorems above, let us first provide some preliminaries.

For , define the Marcinkiewicz space as the set of measurable functions such that

for some positive constant and all . We have the following.

Lemma 5. Let be positive constants such that and let be a function defined on . If , then . In particular, for all such that .

The following is the well-known Aubin-Simon compactness result.

Lemma 6 (see [31]). Assume with compact imbedding ( and are Banach spaces). Let be bounded in , where , and be bounded in . Then is relatively compact in .

3. Existence and Uniqueness of Entropy Solutions

In this section, we provide the proof for Theorem 2. We begin with the existence and uniqueness results for the problem with regular data.

Definition 7. Assume that , and satisfies (2)-(3). A function is called a weak solution of problem (1), if for any , and , and moreover

for all .

Theorem 8. Assume that ,  ,   satisfies (2)-(3). Then problem (1) admits a weak solution .

Proof. The proof of this theorem is based on the standard Galerkin approximation method as in [10]; we thus omit the details for concision.

Proof of Theorem 2. Now provide the proof of the existence and uniqueness of entropy solutions for problem (1). For simplicity, we assume that Let ,  , and be three sequences of functions strongly convergent, respectively, to in , to g in and to in such that

Let us consider the approximation problem of (1),

By virtue of Theorem 8, there is an unique weak solution to (15) for each , with

Next, we shall follow the ideas of [24] to prove that, up to a subsequence, converges to a measurable function , which is the entropy solution of problem (1). Let us divide the proof into several steps. Hereafter, without indication all the convergence should be understood in the sense of subsequences.

Step 1. converges to in
Taking as a test function in (15), we deduce that

Since

Note that From the definition of we obtain

If we choose and taking the above inequality in consideration, we deduce from (19) that

By the standard Gronwall’s inequality, we obtain that

Note that

By the definition of , we have

Therefore, we get

for any
Furthermore, integrating (19) between and , it is easy to obtain

Setting and using (27), we deduce that

Similarly we can obtain

Combining (28) and (29), it yields

which implies that is bounded in Hence, we conclude from Lemma 5 that is bounded in for , which implies that is bounded in for . Therefore, is bounded in
Furthermore, we obtain that is bounded in . Let ,  ,  ; using Lemma 6, we know that is relatively compact in . Thus, up to a subsequence convergence to in .

Step 2. converges to in for any given .
By Vitali’s convergence theorem, it is enough to prove

Given , we define

Let be a sequence of real smooth increasing functions with and as Taking as a test function in (15), we deduce that for any

where is the primitive function of Note that the first integral is nonnegative; thus discarding it and passing to the limit in , we obtain that

From (2), we know that when we have

and when , we have Combining this with (35), we obtain that for any

where . This implies that

for some positive constant Thus, we deduce from (34) that

Since for any , converges to in ; there exists a positive constant independent of , such that

Then we have

For any , we can always find a positive constant such that

Now we analyze each term of the right hand side of (38); note that converges to in , and

For any given , the first term on the right hand side can be strictly less then whenever Thanks to (41), for large enough (), the second term can be strictly less than for all (absolute continuity of the Lebesgue integral). Also, we can always find a positive constant , such that

So setting , we get

Similarly, setting , we can get

Then for any there exists a , such that

Taking (41)–(46) into (38), we obtain that for any there exists , such that

uniformly in On the other hand, from (36) we have

uniformly in , whenever is small enough.
Note that for any

Thus, for any , is equi-integrable in and due to Vitali’s convergence theorem, converges to in .

Step 3. converges to a function in .
For , taking as a test function in (15) one can deduce that

Note that , , , and are convergent in , and , for we obtain that

where

Using Hölder inequality, we get

where . Now let us bound meas (); we have

in ; hence, we have . Then we have

In the same way, we have . So we can obtain that

That is is a Cauchy sequence in . Discarding the nonnegative term in the left hand side of (50) we can deduce that .

Step 4. Pass to the limits.
Similar to [29], taking as a test function in (15), we have

where . Let us study the limit for of each term.
We have seen that in ; hence, . Since is k-Lipschitz continuous one has, when

Since

and in , one has similarly

We now pass to the limit in . Using the hypothesis ,   Since weakly in , we have

Moreover,

Since weakly in , we have

It follows from Fatou’s lemma that

Combining (62)–(64) we have

Similarly, we can obtain

Finally since converges to      and in ,   in ,   in , we obtain that

Therefore passing to the limit in (57), it yields

Step 5. Uniqueness of entropy solutions
Assume that there is another entropy solution to the problem. For any , taking as a test function we have

and taking as a test function in (15), we obtain that

Subtracting (70) from (69), it yields

Note that

Letting , we have

On the other hand, since , , we obtain

Thanks to the assumption on , we have

Since

Omitting them and taking in (71), we obtain

Then Gronwall's inequality implies that for all , and hence .

4. Existence of Global Attractors

This section is devoted to the proof on the existence of the global attractor for the solution semigroup .

Proof of Theorem 4. We first prove continuity of the semigroup in . Let be a family of operators corresponding to the entropy solution of problem (1) with

Then for any , the map is continuous map from to

The proof is similar to [32]. However, for the sack of completeness, we provide the details here. Let and be two solutions of problem (1) with initial data , respectively. Assume that and are the solution of the approximate problem with initial data , and , respectively, such that for any and , there exists a such that for any

Here we may suppose that we have already extracted from a proper subsequence (the one converges to in , which is still denoted by

Note that satisfies the following equation:

Taking as a test function, and using the fact that

we obtain that

which implies that

Let We get

Therefore, for large enough we obtain that

For any and large enough, we know that there exists such that

Since

In view of (79), for any and any fixed , we can choose large enough such that

Therefore, from (87) we deduce that

which means that is continuous in .

Now let us prove that has an absorbing set in ; i.e., for any bounded set , there exist a , such that, for all ,   Repeating the proof of Section 3 Step 1, taking as a test function in (97), we obtain that

Similar to the proof of (26), we get

Passing to the limit, we deduce that

Thus, the semigroup possesses an absorbing set in , which can actually be chosen as the ball centered at zero with radius in .

Finally, we prove that is compact in for any ; i.e., for any bounded sequence , the sequence has convergent subsequences in . Here for every positive integer ,   denotes the unique entropy solution to the following problem:

Note, for any , the entropy solution of the above problem can be obtained by approximations as in the last section; indeed, consider the following approximate problem for (93):

where ,  ,   converges, respectively, to , , in -norm, with

Thanks to the analysis in Section 3, we know that satisfies the estimates similar to (26) and (27), i.e.,

Moreover, up to subsequence converges to the unique entropy solution in , converges to in , and converges to in Passing to the limit in n, we have

and for any fixed

Here the positive constant may depend on ,  ,  ,  ,   and -norm of the sequence , and they are independent of . Thanks to the estimates (97)–(99), we can obtain that is bounded in for any , and for any satisfying in , we can deduce that is bounded in . Then we deduce from the equation that

Thanks to the Aubin-Simon type compactness result, as is shown in Lemma 6, there exists a subsequence of , denoted by which converges to a function in , i.e.,

Thus, up to a subsequence, converges to in for almost all . Especially for any , there exist such that, in . Because the map is continuous map from we can get

Thus, is compact in From the standard results on the existence of global attractors [30] we conclude that the semigroup possesses a global attractor in , which is compact, invariant in and attracts every bounded subset of in -norm.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by NSF of China (11701002) and NSF of Anhui Province (1708085MA02).

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