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 L2(Ω) × L2(Γ) by proving that the corresponding semigroup satisfies the enhanced flattering property. Second, we apply asymptotic a priori estimate and obtain the exponential attractor in Lp(Ω) × L(Γ). Finally, we show the exponential attractor in (H1(Ω) ∩ Lp(Ω)) × L(Γ).


Introduction
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][2][3][4][5][6][7][8][9][10][11][12][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 [1], the authors showed existence of pullback attractors.In [8,9], the authors gave well posedness and global attractors in   (Ω) ×   (Γ).In [12,13]; the authors obtained uniform attractors and some asymptotic regularity of global attractors in ( 1 (Ω) ∩   (Ω)) ×   (Γ).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 where Ω ⊂ R  ,  ≥ 1, is a bounded domain with a smooth boundary Γ.Here ] is the outer unit normal on Γ.
In [14], 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][15][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  2 (Ω) ×  2 (Γ) for weak solutions, then combine asymptotical a prior estimate and show the exponential attractor in   (Ω) ×   (Γ).Finally, we derive the existence of the exponential attractor in the space ( 1 (Ω) ∩   (Ω)) ×   (Γ).

The Basic Results and Theorems.
Let  be a complete metric space and a one-parameter family of mappings (): let  → ( ≥ 0) be a semigroup.Here we omit the definitions of continuous or norm-to-weak semigroups, dynamical systems, global attractors, and exponential attractors (see [15][16][17][18][19][20]). Definition 1.Let  be a metric space and  be a bounded subset of .The Kuratowski measure of noncompactness () of  is defined as  () = inf { > 0 |  admits a finite cover by sets of diameter ≤ } .
(2) Theorem 2 (see [14]).Assume that  is a bounded absorbing set for discrete dynamical system () in ; then the following are equivalent.
Theorem 3 (see [14]).Assume that B is a bounded absorbing set for () in ; then the following are equivalent.
Definition 4 (see [14]  Inspired by [14,21], we easily obtain the following. Theorem 5. Let  be a uniformly convex Banach space and {()} ≥0 be a continuous or norm-to-weak continuous semigroup in .Then the following conditions are equivalent.
By Theorem 5, we can deduce the following.
Theorem 6.Let  be a uniformly convex Banach space and {()} ≥0 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.
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  > 0, the bounded subset B of   (Ω) has a finite -net in   (Ω) if there exists a positive constant  = (), such that Inspired by [16], we give the subsequent theorem which describes our new technique to construct an exponential attractor in a stronger topological space.Theorem 9. Assume that  >  > 0 and Ω ⊂ R  .Let S(t) be a continuous or norm-to-weak continuous semigroup on   (Ω) and   (Ω) and  0 be a positively invariant bounded absorbing set in   (Ω).If the following conditions hold true: (1) () has an exponential attractor   (Ω); (2) for any  > 0, there exist positive constant  = () such that then () has an exponential attractor in   (Ω).
Proof.Take  > 0, and let   = (); obviously   is a discrete dynamical system.On account of Condition (1), for   there exists an exponential attractor M in   (Ω).

The Solution Semigroup.
We can write Problem (1) as an evolution for unknown (, ) in Ω and V(, ) on Γ   +  +  = 0, (15) with the compatibility V = (()) ( is trace operator) for  > 0 and  > 0, where In the case Ω 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  2 (Ω) ×  2 (Γ), {  }  , 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]).
Proof.Let  2 = ( −   ), where   is denoted by the orthonormal projector as mentioned before.Multiplying (1) 1 by  2 and integrating by parts, we get Using (19), we can deduce that for any ,  ≥ 2. Therefore, ||  ≤ , ∀ ≥ , where the constant  is independent of  and .
If  > min{, }, the reasoning process mentioned earlier is not available.We note the following result in [13] after authors obtained regularity of global attractor.where the constant  is independent of  and .
where  1 is a positive constant which depends on .