On the regularity of global attractors

This note is focused on a novel technique in order to establish the boundedness in more regular spaces for global attractors of dissipative dynamical systems, without appealing to uniform-in-time estimates. As an application of the abstract result, the semigroup generated by the strongly damped wave equation $$u_{tt}-\Delta u_t-\Delta u+\phi(u)=f$$ with critical nonlinearity is considered, whose attractor is shown to possess the optimal regularity.


Introduction
The evolution of many physical phenomena is ruled by a differential equation generating a semigroup of operators {S(t)} t≥0 , otherwise called a dynamical system, acting on a suitable infinite-dimensional Banach space H. In mathematical terms, the presence of some dissipation mechanism in the model often reflects into the existence of an absorbing set for the semigroup. This is, by definition, a bounded set B 0 ⊂ H enjoying the following property: for any R ≥ 0, there exists an entering time t R ≥ 0 such that Clearly, an absorbing set is attracting as well, whereas the existence of an attracting set implies the existence of an absorbing one. On the other hand, an attracting set is more likely to possess nice additional properties, such as compactness and finite fractal dimension. A relevant object providing the ultimate description of the asymptotic dynamics is the global attractor : the unique compact set A ⊂ H which is at the same time attracting and fully invariant under the action of S(t), that is, Roughly speaking, A is the smallest possible set where the evolution is eventually confined. Accordingly, any possible further regularity of the attractor is extremely important for a better understanding of the longterm behavior of the semigroup. For more details on the theory of dynamical systems and their attractors, we address the reader to the classical textbooks [2,13,14,20] (see also the more recent [3,4,15]).
A standard way to prove the existence of the global attractor for a (strongly continuous) semigroup is to exhibit a compact attracting set. In that case, S(t) is called asymptotically compact, and the attractor A turns out to be the ω-limit set of any absorbing set B 0 : This is usually attained through the limit where, for every fixed t, C(t) is a bounded subset (say, a closed ball about the origin) of another Banach space V compactly embedded into H. Even though (1.1) yields the global attractor, no conclusion can be drawn at this stage on the regularity of A. However, if V is reflexive, so that closed balls of V are closed in H, and one is able to produce the uniform-in-time estimate it is immediate to see that A is norm-bounded in V by the very constant ̺. As a rule, verifying (1.1) in concrete situations requires a reasonable effort; on the contrary, showing the uniform bound (1.2), and in turn the V-boundedness of A, is generally a much harder task, if not out of reach. A paradigmatic example is the semigroup of the damped wave equation with a critical nonlinearity lacking monotonicity properties, whose global attractor in the weak-energy space was found in [1], but its optimal regularity has been obtained only several years later [12,22].
The aim of this note is to present a new and easy to handle technique apt to establish the boundedness of A in the higher space V without making use of the uniform estimate (1.2), proving that, to some extent, (1.1) alone suffices (see Section 3). Actually, we say more: we find a ball C of V attracting exponentially fast all bounded subsets of H; precisely, for some ω > 0 and some increasing function J. Remark 1.1. A sufficiently regular exponentially attracting sets is crucial in order to demonstrate the existence of an exponential attractor : a compact set E ⊂ H of finite fractal dimension and positively invariant (S(t)E ⊂ E for all t ≥ 0), which attracts bounded subsets of H at an exponential rate, contrary to the global attractor, whose attraction rate can be arbitrarily slow and not measurable in terms of the structural parameters of the problem [6,7,8,15]. In this respect, an exponential attractor happens to be more helpful than the global one for practical purposes, e.g. numerical simulations.
As an application, in the final Section 4, we consider the dynamical system generated by the strongly damped wave equation with a nonlinearity of critical growth, providing a simple proof of the optimal regularity of the related attractor.

A Basic Inequality
We begin with some notation. Given a Banach space V and R ≥ 0, we set We denote by I the space of continuous increasing functions J : R + → R + , and by D the space of continuous decreasing functions β : R + → R + such that β(∞) < 1.
The family U(t) is called a semigroup if it fulfills the further property Remark 2.2. The above maps are closely related to the study of differential equations in Banach spaces: suppose that, for all initial data z ∈ V, there is a unique solution (in some weak sense) ζ : where A(·, t) is a family of operators densely defined on V. Then, we can write When the system is autonomous, i.e. A does not depend explicitly on time, U(t) is a semigroup.
The inequality of the following lemma, although not more than a trivial observation, is really the key idea of the paper.

Lemma 2.3. Let U(t) be a solution operator on a Banach space V. Assume that
for some β ∈ D, J ∈ I. Then, for any t ⋆ > 0 large enough so that β ⋆ := β(t ⋆ ) < 1, Here is a first remarkable consequence of (2.1).

Corollary 2.4. If U(t) is also a semigroup, then it possesses an absorbing set.
Proof. If z V ≤ R ⋆ , from (2.1) and the semigroup properties we readily get For an arbitrary t ≥ 0, we write t = nt ⋆ + τ , with n ∈ N and τ ∈ [0, t ⋆ ). This yields with an entering time In other words, the ball B V (κR ⋆ ) is an absorbing set for U(t).

The Abstract Theorem
Throughout the section, let S(t) be a semigroup of closed operators acting on a Banach space H (cf. [18]). This is a semigroup for which the implication holds at any fixed time t ≥ 0, whenever x n , x, ξ ∈ H. The semigroup S(t) is also required to possess an absorbing set B 0 ⊂ H. Without loss of generality, Within the above assumptions, our main result reads on H with the following properties: (i) For any two vectors y, z ∈ H satisfying y + z = x, (iii) There are β ∈ D and J ∈ I such that Then, B 0 is exponentially attracted by a closed ball of V; namely, there exist (strictly) positive constants ̺, K, ω such that As a byproduct, we have Indeed, as mentioned in the introduction, if S(t) is strongly continuous it is well known that (3.1) implies the existence of the global attractor A subject to the bound A V ≤ ̺ (see e.g. [20]). Same thing if S(t) is only a semigroup of closed operators (see [18]).
Proof of Theorem 3.2. Let x ∈ B 0 be arbitrarily fixed, and select t ⋆ > 0 large enough such that For every n ∈ N, we claim that the vector admits the decomposition x n = y n + z n , for some y n , z n satisfying the bounds . We proceed by induction on n ∈ N. The case n = 0 is verified by y 0 = x, z 0 = 0. Assume the claim true for all n ≤ m ∈ N. Choosing we obtain the equality Observing that y m ∈ B 0 , and using (2.1), we derive the estimates This proves the claim. Let then t ≥ 0. Writing t = nt ⋆ + τ , with n ∈ N and τ ∈ [0, t ⋆ ), with κ > 1 as in (2.2). Thus, setting , the required exponential attraction property (3.1) follows.
Incidentally, Corollary 3.3 is still true under weaker hypotheses.
For every x ∈ B 0 , let there exist two operators V x and U x on H with the following properties: (i) For any two vectors y, z ∈ H satisfying y + z = x, (iii) There are β ⋆ < 1 and J ⋆ ≥ 0 such that

Then, S(t) possesses the global attractor A bounded in V.
Proof. Let x ∈ B 0 be fixed. Arguing exactly as in the proof of Theorem 3.2, S(nt ⋆ )x = y n + z n , ∀n ∈ N, which is enough to establish the existence of A (cf. [18]). Since the attractor is fully invariant and contained in the absorbing set B 0 , Hence, letting n → ∞, we conclude that In concrete cases, a commonly adopted strategy leading to the global attractor A is finding a decomposition (3.2) S(t)x = η(t; x) + ζ(t; x), ∀x ∈ B 0 , such that, for some function µ vanishing at infinity and some J ∈ I, However, in order to deduce the V-boundedness of A, estimate (3.4) need be uniform in time, same as requiring that Let us first dwell on a simple, albeit quite interesting, situation.
Example 3.5. For two (linear and nonlinear, respectively) operators A 0 , A 1 , assume that the differential equation generates a semigroup S(t) on H. Besides, let the linear semigroup L(t), generated by the equation with A 1 ≡ 0, be exponentially stable on both spaces H and V, i.e.
where ξ(t) = S(t)x (actually, (3.3) follows directly from exponential stability). This kind of decomposition has been successfully employed several times (e.g. [10,11]), and typically works for subcritical problems. Then, setting V x (t)y = L(t)y and U x (t)z = L(t)z + ζ(t; x), hypotheses (i)-(iii) of Theorem 3.2 are easily verified. Hence, in contrast to the standard procedure, our approach gives at once the V-boundedness of A, with no need of (3.5).
In general, a semigroup decomposition of the form (3.2), complying with (3.3)-(3.4), can be much more complicated (cf. [1,16]). Nonetheless, whenever (3.2)-(3.4) occur, we have a strong evidence that the conclusions of Theorem 3.2 hold true, as in the quite challenging case of the strongly damped wave equation with critical nonlinearity, discussed below.

A Concrete Application
Consider the semilinear strongly damped wave equation in a smooth bounded domain Ω ⊂ R 3 subject to Dirichlet boundary conditions where f ∈ L 2 (Ω) is independent of time, and the nonlinear term ϕ ∈ C(R) satisfies the critical growth condition Notation. For r ∈ R, we introduce the scale of Hilbert spaces (we will always omit the index r when r = 0) and we define the product spaces H r = H r+1 × H r .
Equation (4.1) generates a strongly continuous semigroup S(t) on H possessing the global attractor A (see [5,16]). In fact, for a nonlinearity of (critical) growth of polynomial order 5, the existence itself of the attractor remained an open question for a long time. Replacing (4.3) with the more restrictive assumption the boundedness of A in H 1 has been demonstrated in [17], by means of a "parabolic" approach. The same paper indicates the way to obtain the result also within (4.3), by means of a rather complicated scheme borrowed from [22]. This has been recently carried out in detail by some other authors [19,21].
Our goal is a simpler proof, which does not actually require anything more than what already contained in [16]. To this end, let us first recall some results therein. ⋄ S(t) has an absorbing set B 0 ⊂ H. From now on, c 0 > 0, ν 0 > 0 and J 0 ∈ I will denote generic constants and a generic function, respectively, depending only on B 0 . ⋄ The following uniform estimate holds: ⋄ For every x ∈ B 0 , the solution S(t)x = (u(t), u t (t)) splits into the sum Remark 4.1. In the same spirit of [1], the proofs of [16] lean on the decomposition where the continuous functions ϕ 0 and ϕ 1 satisfy (4.2) and (4.3), respectively, along with ϕ 0 (u)u ≥ 0 and |ϕ 1 (u)| ≤ c(1 + |u|).
A standard Gronwall-type lemma will also be needed.
As a matter of fact, Theorem 3.2 in its full strength, together with the transitivity property of exponential attraction devised in [9], yield a stronger result, whose proof is left to the interested reader.   3) is easily seen to hold replacing H 1 with the more regular space H 2 × H 2 , provided that ϕ ∈ C 1 (R).