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

Duan, Yonghong; Hu, Chunlei; Chai, Xiaojuan

doi:https://doi.org/10.1155/2018/8186247

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2018 | Article ID 8186247 | https://doi.org/10.1155/2018/8186247

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

Yonghong Duan,¹Chunlei Hu,²and Xiaojuan Chai²

Academic Editor: Nikos I. Karachalios

Received13 Mar 2018

Revised22 Jun 2018

Accepted02 Jul 2018

Published13 Aug 2018

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 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 in , in , in , we obtain thatTherefore 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 haveand taking as a test function in (15), we obtain thatSubtracting (70) from (69), it yieldsNote thatLetting , we haveOn the other hand, since , , we obtainThanks to the assumption on , we haveSinceOmitting them and taking in (71), we obtainThen 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) withThen 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 thatwe obtain thatwhich implies thatLet We getTherefore, for large enough we obtain thatFor any and large enough, we know that there exists such thatSinceIn view of (79), for any and any fixed , we can choose large enough such thatTherefore, from (87) we deduce thatwhich 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 thatSimilar to the proof of (26), we getPassing to the limit, we deduce thatThus, 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, withThanks 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 haveand 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 thatThanks 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 getThus, 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).

References

M. E. Taylor, Partial differential equations I. Basic theory, vol. 115 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 2011.
View at: Publisher Site | MathSciNet
J. Escher, “Global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions,” Mathematische Annalen, vol. 284, no. 2, pp. 285–305, 1989.
View at: Publisher Site | Google Scholar | MathSciNet
A. Miranville and S. Zelik, “Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions,” Mathematical Methods in the Applied Sciences, vol. 28, no. 6, pp. 709–735, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
G. R. Goldstein, “Derivation and physical interpretation of general boundary conditions,” Advances in Differential Equations, vol. 11, no. 4, pp. 457–480, 2006.
View at: Google Scholar | MathSciNet
A. Miranville, E. Rocca, G. Schimperna, and A. Segatti, “The Penrose-Fife phase-field model with coupled dynamic boundary conditions,” Discrete and Continuous Dynamical Systems - Series A, vol. 34, no. 10, pp. 4259–4290, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
M. Fila and P. Quittner, “Large time behavior of solutions of a semilinear parabolic equation with a nonlinear dynamical boundary condition,” in Topics in nonlinear analysis, vol. 35 of Progress in Nonlinear Differential Equations and Their Applications, pp. 251–272, Birkhäuser, Basel, Switzerland, 1999.
View at: Google Scholar | MathSciNet
J. M. Arrieta, P. Quittner, and A. b. Rodrguez-Bernal, “Parabolic problems with nonlinear dynamical boundary conditions and singular initial data,” Differential and Integral Equations, vol. 14, no. 12, pp. 1487–1510, 2001.
View at: Google Scholar | MathSciNet
J. L. Vázquez and E. Vitillaro, “Heat equation with dynamical boundary conditions of reactive-diffusive type,” Journal of Differential Equations, vol. 250, no. 4, pp. 2143–2161, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
C. G. Gal, “On a class of degenerate parabolic equations with dynamic boundary conditions,” Journal of Differential Equations, vol. 253, no. 1, pp. 126–166, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
Z.-H. Fan and C.-K. Zhong, “Attractors for parabolic equations with dynamic boundary conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 6, pp. 1723–1732, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
C. G. Gal and M. Grasselli, “The non-isothermal Allen-Cahn equation with dynamic boundary conditions,” Discrete and Continuous Dynamical Systems - Series A, vol. 22, no. 4, pp. 1009–1040, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
L. Yang and M. Yang, “Long-time behavior of reaction-diffusion equations with dynamical boundary condition,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 12, pp. 3876–3883, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
C. G. Gal, “The role of surface diffusion in dynamic boundary conditions: Where do we stand?” Milan Journal of Mathematics, vol. 83, no. 2, pp. 237–278, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
C. G. Gal and M. Warma, “Well posedness and the global attractor of some quasi-linear parabolic equations with nonlinear dynamic boundary conditions,” Differential and Integral Equations, vol. 23, no. 3-4, pp. 327–358, 2010.
View at: Google Scholar | MathSciNet
B. You and C. Zhong, “Global attractors for p-Laplacian equations with dynamic flux boundary conditions,” Advanced Nonlinear Studies, vol. 13, no. 2, pp. 391–410, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
M. a. Anguiano, P. Marín-Rubio, and J. Real, “Pullback attractors for non-autonomous reaction-diffusion equations with dynamical boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 383, no. 2, pp. 608–618, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
L. Yang, M. Yang, and P. E. Kloeden, “Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions,” Discrete and Continuous Dynamical Systems - Series B, vol. 17, no. 7, pp. 2635–2651, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
M. a. Anguiano, P. Marín-Rubio, and J. Real, “Regularity results and exponential growth for pullback attractors of a non-autonomous reaction-diffusion model with dynamical boundary conditions,” Nonlinear Analysis: Real World Applications, vol. 20, pp. 112–125, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
C. Sun and W. Tan, “Non-autonomous reaction-diffusion model with dynamic boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 443, no. 2, pp. 1007–1032, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
L. Yang, M. Yang, and J. Wu, “On uniform attractors for non-autonomous p-Laplacian equation with a dynamic boundary condition,” Topological Methods in Nonlinear Analysis, vol. 42, no. 1, pp. 169–180, 2013.
View at: Google Scholar | MathSciNet
F. Andreu, N. Igbida, J. M. Mazón, and J. Toledo, “Renormalized solutions for degenerate elliptic-parabolic problems with nonlinear dynamical boundary conditions and -data,” Journal of Differential Equations, vol. 244, no. 11, pp. 2764–2803, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, and J. L. Vázquez, “An -theory of existence and uniqueness of solutions of nonlinear elliptic equations,” Annali della Scuola Normale Superiore di Pisa, vol. 22, no. 2, pp. 241–273, 1995.
View at: Google Scholar | MathSciNet
A. Prignet, “Existence and uniqueness of ‘entropy’ solutions of parabolic problems with data,” Nonlinear Analysis. Theory, Methods & Applications, vol. 28, no. 12, pp. 1943–1954, 1997.
View at: Publisher Site | Google Scholar | MathSciNet
L. Boccardo and T. Gallouët, “Nonlinear elliptic and parabolic equations involving measure data,” Journal of Functional Analysis, vol. 87, no. 1, pp. 149–169, 1989.
View at: Publisher Site | Google Scholar | MathSciNet
X. Chai and W. Niu, “Existence and non-existence results for nonlinear elliptic equations with nonstandard growth,” Journal of Mathematical Analysis and Applications, vol. 412, no. 2, pp. 1045–1057, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
L. Boccardo and L. Orsina, “Some borderline cases of nonlinear parabolic equations with irregular data,” Journal of Evolution Equations (JEE), vol. 16, no. 1, pp. 51–64, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
C. Zhong and W. Niu, “Long-time behavior of solutions to nonlinear reaction diffusion equations involving data,” Communications in Contemporary Mathematics, vol. 14, no. 1, Article ID 1250007, 19 pages, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
W. Niu and C. Zhong, “Global attractors for the p-Laplacian equations with nonregular data,” Journal of Mathematical Analysis and Applications, vol. 392, no. 2, pp. 123–135, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
W. Niu, “Global attractors for degenerate semilinear parabolic equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 77, pp. 158–170, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative parabolic PDEs and the Theory Of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, UK, 2001.
View at: MathSciNet
J. Simon, “Compact sets in the space ,” Annali di Matematica Pura ed Applicata, vol. 146, pp. 65–96, 1987.
View at: Google Scholar
W. Niu and X. Chai, “Global attractors for nonlinear parabolic equations with nonstandard growth and irregular data,” Journal of Mathematical Analysis and Applications, vol. 451, no. 1, pp. 34–63, 2017.
View at: Publisher Site | Google Scholar | MathSciNet

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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.

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