Exponential Attractors for Parabolic Equations with Dynamic Boundary Conditions
We study exponential attractors for semilinear parabolic equations with dynamic boundary conditions in bounded domains. First, we give the existence of the exponential attractor in by proving that the corresponding semigroup satisfies the enhanced flattering property. Second, we apply asymptotic a priori estimate and obtain the exponential attractor in . Finally, we show the exponential attractor in .
Parabolic equations with dynamical boundary conditions have strong backgrounds in mathematical physics. They arise in the heat transfer theory in a solid in contact with moving fluid, thermoelasticity, diffusion phenomena, heat transfer in two medium, problems in fluid dynamic, and so forth. At present, there are many monographs in the whole world (see [1–13]). Several approaches have been used for these equations, like the theory of semigroup, with Bessel potential and Besov space, and the variational setting. In particular, we are devoted to the long-time behavior of the solutions. For instance, In , the authors showed existence of pullback attractors. In [8, 9], the authors gave well posedness and global attractors in . In [12, 13]; the authors obtained uniform attractors and some asymptotic regularity of global attractors in . An exponential attractor, in contrast to a global attractor (or a uniform attractor), enjoys a uniform exponential rate of convergence of its solution. Because of this, exponential attractors possess more practical property. But to our knowledge, it does not seem to be in the literature any study of the existence of exponential attractors for this kind of equations.
This paper is concerned with existence of exponential attractors for the following reaction-diffusion equation with dynamic boundary condition where , , is a bounded domain with a smooth boundary . Here is the outer unit normal on .
In , the authors established some necessary and sufficient conditions for the existence of exponential attractors for continuous and norm-to-weak continuous semigroup and provided a new method for proving the existence of exponential attractors by combining with the flattering property. Motivated by some ideas in [14–16], we combine asymptotic a prior estimate with the enhanced flattening property and show sufficient and necessary existence of exponential attractors in uniformly convex Banach spaces. As an application, we prove the existence of exponential attractors for the reaction-diffusion equation with dynamic boundary condition.
This paper is organized as follows. In Section 2, we recall some basic results and then give our theorems, that is, Theorems 5, 6, and 9 and the solution semigroup corresponding to (1). In Section 3, we obtain the exponential attractor in for weak solutions, then combine asymptotical a prior estimate and show the exponential attractor in . Finally, we derive the existence of the exponential attractor in the space .
In the following, the constants , will always denote generic constants different in various occurrences. The symbol will denote a positive constant dependent of . We will write for for for the inner product in and for the inner product in . For convenience, we denote the norm of by in the space and in the space .
2.1. The Basic Results and Theorems
Let be a complete metric space and a one-parameter family of mappings : let be a semigroup. Here we omit the definitions of continuous or norm-to-weak semigroups, dynamical systems, global attractors, and exponential attractors (see [15–20]).
Definition 1. Let be a metric space and be a bounded subset of . The Kuratowski measure of noncompactness of is defined as
Theorem 2 (see ). Assume that is a bounded absorbing set for discrete dynamical system in ; then the following are equivalent.(1)The measure of noncompactness is exponentially decaying for dynamical system , that is, there exist such that .(2)For , there exist exponential attractors.
Theorem 3 (see ). Assume that B is a bounded absorbing set for in ; then the following are equivalent.(1)The measure of noncompactness is exponentially decaying for dynamical system , that is, there exist such that .(2)For , there exist exponential attractors.
Definition 4 (see  (Enhanced Flattening Property)). Let be a uniformly convex Banach space; for any bounded set of , there exist and , and a finite dimension subspace of , such that(1) is bounded and(2), for all , for all . Here is a bounded projector, is the dimension of , denotes the norm in , and is a real-valued function satisfying .
Theorem 5. Let be a uniformly convex Banach space and be a continuous or norm-to-weak continuous semigroup in . Then the following conditions are equivalent.(1)The measure of noncompactness is exponentially decaying for dynamical system , that is, there exist such that .(2) satisfies the enhanced flattening property.
Proof. On account of condition (1), for any bounded subset of and for any , there exist such that Namely, there exist a finite number of subset with diameter less than , such that Let , then Let , since is uniformly convex, there exist a projection , such that for any , . Hence, have been shown in Theorem 4.3 of , so we omitted it here.
By Theorem 5, we can deduce the following.
Theorem 6. Let be a uniformly convex Banach space and be a continuous or norm-to-weak continuous semigroup in X. Then, for dynamical system , there exist exponential attractors in if and only if(1)there is a bounded absorbing set , and(2)S(t) satisfies the enhanced flattening property.
In addition, we use later the following theorem about global attractors.
Theorem 7 (see [15, Corollary 5.7]). Let be a semigroup on , , be a continuous or weak continuous semigroup on for some , and have a global attractor in . Then, has a global attractor in if and only if(1) has a bounded absorbing set in , and(2)for any and any bounded subset , there exist positive constants and , such that where .
We give the following lemma concerning the covering of the set in two different topologies used later in the proof of Theorem 9.
Lemma 8 (see [15, Lemma 5.3]). For any , the bounded subset B of has a finite -net in if there exists a positive constant , such that(1) has a finite -net in ;(2), for any .
Inspired by , we give the subsequent theorem which describes our new technique to construct an exponential attractor in a stronger topological space.
Theorem 9. Assume that and . Let S(t) be a continuous or norm-to-weak continuous semigroup on and and be a positively invariant bounded absorbing set in . If the following conditions hold true:(1) has an exponential attractor in ;(2)for any , there exist positive constant such that then has an exponential attractor in .
Proof. Take , and let ; obviously is a discrete dynamical system. On account of Condition (1), for there exists an exponential attractor in . By Theorem 2, we find that there exist such that . By the definition of the measure of noncompactness, for all , there exist finite points such that . Then, there exist and such that
From Lemma 8 and Condition (2), it follows that has a finite -net in .
Let be replaced by , there and such that From Lemma 8 and Condition (2), it follows that has a finite -net in .
By induction, let be replaced by , there and such that From Lemma 8 and Condition (2), it follows that has a finite -net in . Note that So, the ratio of and is a constant for given , , , and . In fact, Substituting (13) into (11), we deduce easily that Choose that and , we know that has a finite -net in .
Combining (9), (10) (11), and (13), we can choose , where is the integer part of . Denoted by , we have has a finite -net in . By Theorem 2, an exponential attractor exists in for discrete semigroup . Using the same argument as in , it is easy to deduce that an exponential attractor exists in for discrete semigroup . Let . As an repetition of the general method developed by Li et al. , we can show that is the exponential attractor for dynamical system .
2.2. The Solution Semigroup
We can write Problem (1) as an evolution for unknown in and on with the compatibility is trace operator) for and , where In the case bounded, the operator has a compact resolvent and its spectrum, denoted by , forms an increasing sequence converging to infinity (see [11, Theorem 1.4]). Moreover, there exists an orthonormal basis in , , which are solutions of the eigenvalue problem where forms an increasing sequence converging to infinity. We denote by the orthonormal projector So, we can perform the Galerkin truncation by using orthonormal basis mentioned above and guarantee the following existence and uniqueness (see [1, 8, 9, 12]).
Theorem 10. Assume that the functions satisfying Then, Problem (1) has a unique weak solution, for any , given and , there exists a solution with and is continuous on .
By the last theorem, we can define the operator semigroup in as follows: which is continuous in .
Theorem 11. Under the assumptions of Theorem 10, the semigroup have -, -bounded absorbing sets, that is, for any bounded subset , there exists a positive constant , which is only dependent on the -norm of , such that where is a positive constant independent of , .
3. The Main Results
3.1. Exponential Attractors in
Theorem 12. Under the hypothesis of Theorem 11, has a -exponential attractor.
Proof. Let , where is denoted by the orthonormal projector as mentioned before. Multiplying (1)1 by and integrating by parts, we get Using (19), we can deduce that for any . Therefore, Thus, we know Applying the Gronwall-inequality, we have Obviously, the enhanced flattening property holds true. By Theorem 6, we obtain the exponential attractor in .
3.2. -Exponential Attractor
Theorem 13. Under the hypothesis of Theorem 11, then for any and any bounded subset , there exist two positive constants and such that where the constant is independent of and .
After Theorem 13, we obtain the subsequent result.
Theorem 14. Under the assumption of Theorem 11, then the semigroup generated by Problem (1) with initial data and has a -exponential attractor , that is, is compact, invariant in , and attracts every bounded in in the topology of .
If , the reasoning process mentioned earlier is not available. We note the following result in  after authors obtained regularity of global attractor.
Theorem 16. Under the hypothesis of Theorem 11, then for any and any bounded subset , there exist two positive constants and such that where the constant is independent of and .
Remark 18. In fact, for any integer , we can obtain Theorem 16 and Corollary 17 by Lemma 15. In other words, if , we can obtain Theorem 16 and Corollary 17 by Lemma 15 and need not use asymptotic a prior estimate. Here, we point this result obtained by the different procedure.
Corollary 19. There exists some such that
Now, we give some a prior estimates about .
Lemma 20 (see [8, 12]). Under the assumption of Theorem 11, for any bounded subset , there exists a positive constant which depends only on the -norm of B such that where is a positive constant which depends on .
We can easily obtain the following theorem.
Proof. We denote , where , , and has been introduced in Section 3.
Multiplying Problem (1) by , we can get By (25) and (26), we can deduce By Theorems 11 and 12, Corollary 19, and Lemma 20, we know that satisfies the enhanced flattering property in . On account of Theorem 6 and Corollary 19, we can deduce that Theorem 21 is valid.
The research was supported by the NSFC Grant (no. 11271339).
M. 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 | Zentralblatt MATH | MathSciNet
Y. Li, H. Wu, and T. Zhao, “Necessary and sufficient conditions for the existence of exponential attractors for semigroups, and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 17, pp. 6297–6305, 2012.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. K. Zhong, M. H. Yang, and C. Y. Sun, “The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations,” Journal of Differential Equations, vol. 223, no. 2, pp. 367–399, 2006.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Eden, C. Foias, B. Nicolaenko, and R. Temam, Exponential Attractors for Dissipative Evolution Equations, vol. 37 of Research in Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1994.View at: MathSciNet
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, 1997.View at: MathSciNet