GLOBAL EXPONENTIAL κ − DISSIPATIVE SEMIGROUPS AND EXPONENTIAL ATTRACTION

. Globally exponential κ − dissipativity, a new concept of dissipativity for semigroups, is introduced. It provides a more general criterion for the exponential attraction of some evolutionary systems. Assuming that a semigroup { S ( t ) } t ≥ 0 has a bounded absorbing set, then { S ( t ) } t ≥ 0 is globally exponentially κ − dissipative if and only if there exists a compact set A ∗ that is positive invariant and attracts any bounded subset exponentially. The set A ∗ need not be ﬁnite dimensional. This result is illustrated with an application to a damped semilinear wave equation on a bounded domain.


1.
Introduction. It has long been considered that the global attractor is the appropriate concept to describe the long-time behavior of a dissipative infinite dimensional dynamical system. A set A is called a global attractor of a semigroup {S(t)} t≥0 on a complete metric space (M, d) (the definition of the semigroup will be given later in Definition 2.6), if A is compact, invariant, i.e., S(t)A = A, ∀ t ≥ 0, and attracts any bounded subset B in M , i.e., ∀B ⊂ M bounded, By the definition, the global attractor contains all of the nontrivial dynamics of a dissipative system and is much smaller than the phase space. It was thought that the global attractor should have finite fractal dimension and that the dynamics restricted to the global attractor should also be finite fractal dimensional. The existence of the finite fractal dimensional global attractor, in fact, has been established for many classes of dissipative PDEs, see [2,16,24,26,29].
However, the global attractor may not be the ideal object to describe the long term behavior of some dissipative systems. Since the reduced dynamics given by the Hölder-Mañé projection [13,17,20,25] is only Hölder continuous, the projectors need to be Lipschitz to guarantee that the the reduced dynamics on the attractor is finite dimensional. Moreover, the global attractor may attract the trajectories at a rather slow rate. In addition, it may be difficult to express the convergence rate in terms of the physical parameters of the system. This can be seen in the following example of a real Ginzburg-Landau equation in one dimension [2,21]: Finally, the global attractor may be too sensitive to perturbations, which could prevent a clear understanding of the real system modelled by the given system, since a system is only an approximation of reality. Therefore, it is desirable to embed the global attractor in to a proper smooth manifold containing a subset, which is robust under small perturbations, attracts trajectories at a fast rate, and moreover, has finite fractal dimension.
In [14], Foias, Sell and Temam proposed the notion of an inertial manifold, which is positive invariant, is respresented by a finite dimensional Lipschitz graph, contains the global attractor and attracts the trajectories fast, i.e., at an exponential rate. The existence of inertial manifolds has since then been verified for many systems, see [4,24,26,29]. However, for many systems, the existence of an inertial manifold still remains open. Moreover, the construction of an inertial manifold is based on the spectral gap condition, which is very restrictive and often difficult to verify or does not even hold for some systems, see [14,27]. In addition, the inertial manifold has a smooth structure, but it is not always possible to embed the global attractor into a proper smooth finite dimensional manifold. All of these facts led Eden, Foias, Nicolaenko and Temam to propose the concept of exponential attractor (also called an inertial set) in [7]. Definition 1.1. Let X be a Banach space. A compact set M ⊂ X is an exponential attractor for S(t) if (i) it has finite fractal dimension, dim F M < +∞, (ii) it is positively invariant, i.e. for arbitrary t ≥ 0, S(t)M ⊂ M, (iii) it attracts exponentially the bounded subsets of X in the following sense: Several methods have been proposed to construct the exponential attractor for a semigroup, see [1,8,9,10,12,15,22]. Generally speaking, exponential attractors can be constructed for dissipative systems which possesses a certain kind of smoothing property. Actually, not only does the smoothing property provide us with an exponential attractive compact set M (i.e. the exponential attractivity of the semigroup), but it also ensures the finite dimensionality of the set. However, for those systems that do not possess smoothing properties, the situation could be much more complicated, on the other hand, many semigroups generated by evolution equations have infinite dimensional global attractors, see, e.g., [31,30].
In the present paper, we discuss the exponential attractivity of the semigroups which do not possess smoothing properties. We propose a new notion of global exponentially κ−dissipativity, which mainly focuses on the behavior of semigroups and pays no attention to the dimension of the attracting set. This concept involves on the Kuratowski measure of noncompactness (the definition of which will be given later in Definition 2.1).
where κ is the Kuratowski measure of noncompactness.
We recall that the Kuratowski measure of noncompactness has been used to describe the asymptotic compactness of a semigroup (such as the ω-limit compactness property [18]), and then used to to prove the existence of the global attractor in [28,32]. However, the notion was only used to obtain an attracting compact set without consideration about the rate of the attraction. Here, we shall prove that the dissipative rate of the measure of non-compactness does provide information about this rate of attraction. We shall provide several sufficient conditions that can be used to determine if a semigroup is globally exponentially κ−dissipative. In addition, we shall prove that if a globally exponentially κ−dissipative semigroup admits a bounded absorbing set, then there exists a positively invariant compact subset which attracts any bounded subset exponentially.
The paper is organized as follows. In Section 2, we recall some definitions and useful results that will be used later. In Section 3 and Section 4, we provide several sufficient conditions and also a property of global exponential κ−dissipative semigroup. In Section 5, we give a simple application of our abstract result. In particular, we show that the solution semigroup {S(t)} t≥0 of a damped semilinear wave equation is globally exponentially κ−dissipative.
2. The measure of noncompactness and some preliminary results. We recall the concept of measure of noncompactness and its basic properties and refer the reader to [5,18] for more details.
Lemma 2.3. Let M be an infinite dimensional Banach space and let B(ε) be a ball of radius ε, then κ(B(ε)) = 2ε.
Lemma 2.4. Let X be an infinite dimensional Banach space with the following decomposition: Let P : X → X 1 , Q : X → X 2 be the canonical projectors and let A be a bounded subset of X. If the diameter of QA is less than ε, then κ(A) < ε.
The next lemma will be used later in the proof of Theorem 3.2.
Proof. It is sufficient to prove that κ(B) ≤ 2γ + ε for any ε > 0. In fact, for each and thus Due to the compactness of A, there exist y 1 , y 2 , . . ., y m ∈ A such that Combining this with (2.1), we have Bearing in mind that diam N γ+ ε 2 (y i ) = 2γ + ε, the proof is completed.
The concept of ω−limit compactness of a semigroup, which is an important necessary and sufficient condition for the existence of the global attractor (see [18]), is now recalled along with the definition of a continuous semigroup.
3. Sufficient conditions for global exponential κ−dissipative semigroups. In this section, we consider several different sufficient conditions for a semigroup to be globally exponential κ−dissipative are considered here.
Proof. Since S(t) has a bounded absorbing set B 0 , then for each bounded subset The proof is finished by noticing that is valid for every s ≥ s 0 .
Corollary 1. Any uniformly compact semigroup {S(t)} t≥0 on a complete metric space is a global exponential κ−dissipative semigroup.
Remark 1. This conclusion shows that κ-contracting and uniformly compact semigroups are global exponentially κ−dissipative. Intuitively, by the contraction property κ (S(t 0 )B) ≤ βκ(B), the measure of noncompactness κ (S(t 0 )B) should decay to zero exponentially fast (related to β t ). The κ-contracting property (or uniformly compact) is actually a relatively strong condition and can be replaced by other more general conditions. Now it will be shown that the existence of an exponentially attracting compact set is a sufficient condition for the global exponential κ− dissipativity. In Theorem 4.1 of the following section it will be proved that this condition is also a necessary condition for global exponential κ−dissipativity when the semigroup has a bounded absorbing set.
Proof. By the assumption, for each bounded subset B, t≥0 S(t)B is bounded in M , so there exist positive constants C and α such that Recalling Lemma 2.5, it follows that Combining with Lemma 2.2(v), we complete the proof.
Applications to PDEs usually involve semigroups acting on Banach spaces. In order to prove that these semigroups are global exponential κ−dissipative semigroups, we present some useful methods.
In particular, let X be a Banach space with the decomposition and denote projections by P : X → X 1 and (I − P ) : X → X 2 . In addition, let {S(t)} t≥0 be a continuous semigroup on X.
Condition (C * ): For any bounded set B ⊂ X there exist positive constants t 0 , C and α such that for any ε > 0 there exists a finite dimensional subspace X 1 ⊂ X satisfying { P S(t)B } t≥0 is bounded, and where P : X → X 1 is a bounded projection.
In [29] the semigroup decomposition method was used to prove the asymptotic compactness, this motivates the following decomposition condition for a semigroup to be global exponentially κ−dissipative.
Theorem 3.4. Let {S(t)} t≥0 be a continuous semigroup on a Banach space X. Suppose, for each t > 0, the semigroup S(t) admits a decomposition S(t) = S 1 (t) + S 2 (t), where the operators S 1 (t) are uniformly compact for t large and S 2 (t) is a continuous mapping which satisfies: for every bounded set B ⊂ X there exist constants C, α such that S 2 (t)B ≤ Ce −αt for every t > 0.
In addition, suppose that {S(t)} t≥0 has a bounded absorbing set B 0 ⊂ X, then {S(t)} t≥0 is a global exponential κ−dissipative semigroup.
Proof. We omit the proof which similar to Theorem 3.3.

4.
Existence of exponentially attracting set for global exponential κ− dissipative semigroups. An important property for the long-time behavior of global exponential κ−dissipative semigroups will now be established.
Proceeding inductively, set s m = m. Then there exist a (4.1) Next, we will show that the set defined To prove the compactness of set A * , consider an arbitrary sequence {x n } in A * , it will be shown that {x n } has a convergent subsequence in A * . In fact, if {x n } has a subsequence in A, which is a compact subset, then {x n } has a convergent subsequence in A. Hence, without loss of generality, the proof reduces to the case that Then there exist m n , t n and 1 ≤ i n ≤ k mn for n = 1, 2, . . . such that Finally, the combination of estimates (4.1) and (4.2) gives The proof is completed.

Remark 2.
It follows from the proof of Theorem 4.1 that, the sequence {a (m) i }, i = 1, 2, · · · , k m , m = 1, 2, · · · can be chosen in any subset B 1 which is dense in B 0 . In particular, we can choose regular points {a (m) i } such that the orbits S(t)a (m) i are smooth. In fact, for any dense set B 1 of the absorbing set B 0 , we know that there exist points {b Similarly to what we did in the proof of Theorem 4.1, the conclusion of Theorem 4.1 is also valid for the set where A 1 is the set of positive trajectories of some points in the dense set B 1 . Therefore, the attracting set A * can be constructed as the union of global attractor and some smooth forward trajectories.
Next, we consider the following question: if the global attractor A is finite dimensional, whether the dimension of set A * is also finite?
Proof. It follows from the definition of Hausdorff dimension, for any ε > 0, δ ∈ (0, 1) and d > 0, there exist balls Due to the compactness of A, the number of the covering balls is finite, we will denote them by {B(x i , r i )} n0 i=1 . In addition, it follows from the definition of a global attractor that, for any u 0 ∈ B, there exists a time t 0 such that By Lipschitz (or Hölder) continuity, the length of the trajectory S(t)u 0 is finite in time t ∈ [0, t 0 ], so it can be covered by finite balls Finally, the property of Hausdorff dimension gives which completes the proof.
Combining the above Lemma with the countable stable property of Hausdorff dimension, we can obtain the following theorem. Since the fractal dimension does not have countable stability, so the above theorem does not hold for the fractal dimension. Indeed, let S(t) : 2 → 2 is the semigroup associated with the system in the Banach space if |x j n | ≥ 1 2 n ·n , 2 n · n · |x j n | if |x j n | < 1 2 n ·n . Obviously, the global attractor A = {0} is a single point set. On the other hand, the set {x ∈ 2 | |x j n | ≤ 1 2 n ·n } is compact and exponentially attracts any bounded subset of 2 , so the the semigroup S(t) is global exponentially κ−dissipative. However, this can be seen that the fractal dimension of any exponentially attracting set A * constructed as in Theorem 4.1 is infinite. Therefore, the finite fractal dimension of global attractor and κ−exponential dissipative are not insufficient to give the finite fractal dimensionality of exponential attracting set. More conditions are needed.
Note that the above semigroup S(t) is not Fréchet differentiable at the point set 0. Motivated by the method in [34], we assume that a semigroup is Fréchet differentiable in a neighborhood of global attractor A.
Following the process in [34], we provide the following theorem. Sketch of proof. Since {S(t)} t≥0 is a global exponentially κ−dissipative semigroup, thus by Theorem 4.1, the compact set A * exponentially attracts the set N (A). Therefore, for any λ > 0, there exists a time T * > 0, such that for all x ∈ N (A), the linear map D x S(x) D x S(T * )(x) admits a decomposition D x S(x) = K + C, where K is compact and C < λ.
Applying a process similar to that in [34], we have On the other hand, it is obvious that d F (A * \ N (A)) < ∞, since the set is the union of finite trajectories with finite length. Thus, we have Remark 3. When the fractal dimension of attractor is finite, the exponential attractor is powerful concept for studying the rate of attracting (see [21] page 133). Exponential attractors are as general as global attractors. To the best of our knowledge, exponential attractors exist for all equations of mathematical physics for which we can prove the existence of the finite fractal dimensional global attractors. When the fractal dimension of the attractor is infinite, our concept of κ−exponential dissipativity allows us to study the existence of a compact positive invariant exponential attracting set.