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Discrete Dynamics in Nature and Society
Volume 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

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

Correspondence should be addressed to Chunlei Hu; moc.361@uha_uhielnuhc

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 [25]. The existence and uniqueness of solutions for problem (1) have been studied extensively in various contexts; see, e.g., [2, 69]. 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 [1113] 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 [1719], 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 [2426] 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., [2729]. 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 aswith normwhere on is defined by for any measurable set . Define the Sobolev space aswith normIt 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 thatfor 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 moreoverfor 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 thatLet us consider the approximation problem of (1),By virtue of Theorem 8, there is an unique weak solution to (15) for each , withNext, 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 thatSinceNote that From the definition of we obtainIf we choose and taking the above inequality in consideration, we deduce from (19) thatBy the standard Gronwall’s inequality, we obtain thatNote thatBy the definition of , we haveTherefore, we getfor any
Furthermore, integrating (19) between and , it is easy to obtainSetting and using (27), we deduce thatSimilarly we can obtainCombining (28) and (29), it yieldswhich 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 proveGiven , we defineLet 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 thatFrom (2), we know that when we haveand when , we have Combining this with (35), we obtain that for any where . This implies thatfor some positive constant Thus, we deduce from (34) thatSince for any , converges to in ; there exists a positive constant independent of , such thatThen we haveFor any , we can always find a positive constant such thatNow we analyze each term of the right hand side of (38); note that converges to in , andFor 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 thatSo setting , we getSimilarly, setting , we can getThen for any there exists a , such thatTaking (41)–(46) into (38), we obtain that for any there exists , such thatuniformly in On the other hand, from (36) we haveuniformly 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 thatNote that , , , and are convergent in , and , for we obtain thatwhereUsing Hölder inequality, we getwhere . Now let us bound meas (); we have in ; hence, we have . Then we haveIn the same way, we have . So we can obtain thatThat 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 havewhere . Let us study the limit for of each term.
We have seen that in ; hence, . Since is k-Lipschitz continuous one has, when Sinceand in , one has similarlyWe now pass to the limit in . Using the hypothesis ,   Since weakly in , we haveMoreover,Since weakly in , we haveIt follows from Fatou’s lemma thatCombining (62)–(64) we haveSimilarly, we can obtainFinally since converges to      and <