FRACTIONAL APPROXIMATIONS OF ABSTRACT SEMILINEAR PARABOLIC PROBLEMS

. In this paper we study the abstract semilinear parabolic problem of the form du dt + Au = f ( u ) , as the limit of the corresponding fractional approximations du , in a Banach space X , where the operator A : D ( A ) ⊂ X → X is a sectorial op- erator in the sense of Henry [22]. Under suitable assumptions on nonlinearities f : X α → X ( X α := D ( A α )), we prove the continuity with rate (with respect to the parameter α ) for the global attractors (as seen in Babin and Vishik [4] Chapter 8, Theorem 2.1). As an application of our analysis we consider a fractional approximation of the strongly damped wave equations and we study the convergence with rate of solutions of such approximations.


(Communicated by José A. Langa)
Abstract. In this paper we study the abstract semilinear parabolic problem of the form du dt + Au = f (u), as the limit of the corresponding fractional approximations du dt in a Banach space X, where the operator A : D(A) ⊂ X → X is a sectorial operator in the sense of Henry [22]. Under suitable assumptions on nonlinearities f : X α → X (X α := D(A α )), we prove the continuity with rate (with respect to the parameter α) for the global attractors (as seen in Babin and Vishik [4] Chapter 8, Theorem 2.1). As an application of our analysis we consider a fractional approximation of the strongly damped wave equations and we study the convergence with rate of solutions of such approximations.
1. Introduction. In this paper we study the abstract semilinear parabolic problem of the form du dt as the limit of the corresponding fractional approximations du dt where α ∈ [0, 1], X is a Banach space and A : D(A) ⊂ X → X is such that −A is a generator of an analytic strongly continuous semigroup on X with 0 ∈ ρ(A). Here A α : D(A α ) ⊂ X → X is the fractional power associated with operator A.
Our aim is to obtain the convergence with rate of the dynamics of (2) with 0 < α < 1 to the dynamics of (1), as α 1, in a variety of applied problems (several different possibilities for A). These problems arise in many models from mathematical physics, acoustics, elastodynamics, optics, geophysics and biomechanics, to name but a few, and to analyze the asymptotic behavior of solutions of such problems.
The theory of fractional powers of densely defined closed operators has been extensively used in the geometric theory of semilinear equations, let us mention [1,6,7,10,15,18,22,23] and references therein. See Carvalho and Cholewa [10] (see also Chen and Triggiani [18]) for the usage of fractional powers in the "damping" of a semilinear wave equation that gives to these wave problems a parabolic structure; see Amann [1] and Henry [22] for the definition and basic properties of fractional power of sectorial operators and examples.
In this paper, we investigate the behavior of the asymptotic sets of states (global attractors) for the evolution differential problems in a rather abstract sense so that it may be applied to many different models.
A key point of our analysis is the proof of convergence of the operators A −α ∈ L(X) in the uniform operator topology to A −1 ∈ L(X), as α 1, with rate 1 − α. Namely, we prove that there exists 0 < α 0 < 1 such that the following estimate is holds A −α − A −1 L(X) ≤ C(1 − α), for all α ∈ (α 0 , 1), where C > 0 is independent of α .
We will study the dependence of solutions u α of the problem (3) on the parameter α and obtain the convergence with rate of the nonlinear semigroup, with respect to the parameter α, and we will use this to prove upper and lower semicontinuity of the family of attractors {A α : 1 2 ≤ α ≤ 1}, at α = 1. This paper is organized as follows. In Section 3, we will prove that the family {A −α ∈ L(X) : α ∈ [0, 1)} converges in the uniform operator topology to A −1 ∈ L(X) with rate 1 − α, this is one of the key points of our work. In the Section 4 we will study spectral properties of the operators A α , α ∈ (0, 1]. In the Section 5 we will analyze the continuity properties of the linear and nonlinear semigroups. In the Section 6, we will study the linearization around a hyperbolic equilibrium and the existence of unstable manifolds as a graph, and at the end of this section we prove the continuity with rate 1 − α of local unstable manifolds. In the Section 7, we will prove the continuity with rate of attractors using the continuity properties of the local unstable manifolds of equilibrium solutions of the Section 5. Finally, Section 8 is dedicated to applications. 2. Abstract theory. In order to better explain the results in the paper we will introduce some concepts and terminology. Let X be a Banach space with norm · , the underlying dynamical system is described by a semigroup of nonlinear operators (or 'nonlinear semigroup'), that is, a family {T (t) : t ≥ 0} (or T (·) for short) consisting of continuous operators from X into itself such that (1) T (0) = I, (2) T (t)T (s) = T (t + s) for each t, s ≥ 0, (3) [0, +∞) × X (t, x 0 ) → T (t)x 0 is continuous.
If each T (t) is linear then we call {T (t) : t ≥ 0} a semigroup of linear operators ('linear semigroup' for short).
In the next we recall same definitions and results about theory of sectorial operator in the sense of Henry [22,Definition 1.3.1], see also [13, Chapter 6, Section 6.4]. 1 2 and a constant L U > 0 such that Assume also that f is bounded and continuously Frechét differentiable function.
For each α ∈ [0, 1], consider the Cauchy problem    du dt Under the above assumptions the problem (3) (see Henry [22] and Pazy [26]), is globally well posed (for each α ∈ [0, 1]) in X For each α ∈ [0, 1] we consider the nonlinear semigroup {T α (t) : t ≥ 0} defined by T α (t)u 0 = u α (t, u 0 ). We assume that {T α (t) : t ≥ 0} has a compact global attractor, that we will denote by A α . Recall the definition of a global attractor for a nonlinear semigroup T (·), for more details see [13] and [21]. For this purpose we start remembering the definition of Hausdorff semi-distance between two subsets A and B of a metric space (X, d): Definition 2.3. If B and C are subsets of X, we say that the set B attracts the We remark that a system may have a number of compact invariant sets; for example, compact invariant sets can be made of equilibria (or a continuum of them), periodic orbits, etc.
A globally-defined solution (or simply a global solution) through x is a function ξ : R → X such that ξ(0) = x and for all s ∈ R and t ≥ 0 we have T (t)ξ(s) = ξ(t+s). The global attractor is exactly the union of all orbits (ξ(R)) of global bounded solutions ξ : R → X, that is, there is a bounded global solution through x}.
By upper and lower semicontinuity of a family of sets we understand the following: Definition 2.6 ( [13]). Let η be a parameter varying in the interval [0, 1] and X be a Banach space. For each η ∈ [0, 1] let A η be a subset of X. We say that the family • Continuous at η = 0 if it is upper and lower semicontinuity at η = 0.
We will assume, throughout the text, unless otherwise stated, that the following holds. Hypothesis (S): Let X be a Banach space and A : D(A) ⊂ X → X be a positive sectorial operator. Suppose that there exists K ≥ 1 independent of α ∈ (0, 1) such that for each λ ∈ −Σā ,θ := {w ∈ C : |arg (w +ā)| < π − θ},ā > 0 and θ ∈ (0, π 2 ). Remark 1. It is clear that positive self-adjoint operators satisfy Hypothesis (S). For some matrix type operators we will prove, in Section 8, that it is also satisfied.
Remark 2. Suppose that A : D(A) ⊂ X → X be a positive sectorial operator. It is clear that A α is a positive sectorial operator and −A α is the generator of a strongly continuous analytic semigroup on X for all α ∈ (0, 1], that we will denote by {e −tA α : t ≥ 0}, (see Kreǐn [25] and Tanabe [27]). Remark 3. Let A be a positive sectorial operator in X, α close to 1 and let {e −A α t : t ≥ 0} be the analytic semigroup generated by −A α . It follows from [23, where Γ α is a closed curve in ρ(−A α ) and that runs from ∞e −iαφ to ∞e iαφ , π 2 < φ < π, and Γ α is oriented so that Im(λ) increases along Γ α .
For each fixed α, from Cauchy Theorem the convergence of the integral in (5) independent of the choose of the closed curve Γ α in ρ(−A α ). Moreover, since α is close to 1 the curve Γ α can be chosen (uniformly) independent of α. We will denote this curve by Γ.
Theorem 2.7. Let A be a positive sectorial operator and suppose that Hypothesis (S) is holds. Then there are positive constants C and δ, independents of α (α ∈ (0, 1)), such that e −tA α L(X) ≤ Ce −δt , for all t > 0. Proof. The proof is essentially the same as in [22,Theorem 1.3.4]. We include here for completeness. Recall that where Γ is a contour in ρ(−A α ). Putting µ = λt (t > 0) in the integral (6), we get and the contour is still Γ since the integrand is analytic. From this and from (4) and the result is proved.
Theorem 2.8. Suppose that A satisfies the same conditions as in Theorem 2.7. Then there are positive constants C and δ (independents of α (α ∈ (0, 1])), such that A α e −tA α L(X) ≤ Ct −1 e −δt , for all t > 0. Proof. In the same way of the proof of Theorem 2.7, we have and the result follow.
Remark 4. If the same hypothesis of the Theorem 2.8 hold and {e −A α t : t ≥ 0} is the analytic semigroup generated by −A α in X, then Proof. This result is a immediate consequence of moment inequality (see [22,Theorem 1.4.4]), Theorems 2.7 and 2.8.
The next result is inspired by [29, (2.127)] and we will include the proof for completeness. Proposition 1. Let A be a positive sectorial operator in a Banach space X. Then, for all β ∈ (0, ∞) and α ∈ (0, 1] we have where Γ is any piece-wise smooth simple curve running C \ R + from ∞e −iφ to ∞e iφ for some φ ∈ ( π 2 , π) such that σ(−A α ) lies strictly to the left of Γ. Proof. First note that for such curve Γ, for all β > 0 and See [1, p. 154] and [23], respectively.
The result is complete if we can proof that From this, consider Γ with the same restrictions of the Γ given in the theorem and such that Γ is to the right of Γ. Then From Cauchy theorem, it is not difficult to see that Then we have for all α ∈ (0, 1] and β > 0.
Proof. The proof is a immediate consequence of Proposition 1.
3. Convergence with rate of the resolvents. Let X be a Banach space and suppose that A is a positive sectorial operator in X. Denote by e −At the semigroup generated by −A. This semigroup satisfies e −At L(X) ≤ M e −ωt , for some M ≥ 1 and ω > 0 (to express this fact we say that the operator A is of type G(M, −ω)).
We begin this section showing that the family of linear operators {A −α ∈ L(X) : α ∈ ( 1 2 , 1)} converges in the uniform operator topology to A −1 ∈ L(X), as α 1, with rate 1 − α. Theorem 3.1. Suppose that A is a positive sectorial operator in X. Then for all α ∈ ( 1 2 , 1), the following estimate holds where C > 0 is independent of α; that is, the linear operators A −α ∈ L(X) converges in the uniform operator topology to A −1 ∈ L(X), as α 1, with rate 1 − α. If σ p (A) contains an isolated eigenvalue µ = 1 then, this rate is optimal.
Proof. For the last statement, if µ is an isolated eigenvalue of A, just note that A has an invariant subspace (the kernel of µI − A) where it is a multiple of the identity.
From this and using (9) it follows that Note that and where we have used that For the estimation of the other term we compute the maximum of Since α ∈ ( 1 2 , 1), it is not difficult to see that the maximum of f in the variable x is attained at With this we obtain for all r ≥ r 0 , Observe that lim α→1 − (2α − 1) 2α−1 2−2α = e −1 . Using (11), (12), (13) and (14) we where C can be chosen independent of α.
In a similar way, we can show that the rate of convergence of 1 2πi Γ3 (λ −α − λ −1 )(λI − A) −1 dλ to zero as α 1 in L(X) is equal to 1 − α. Finally, we study the integral over .
With this, we can conclude that the rate of convergence of 1 2πi Γ2 (λ −α −λ −1 )(λI− A) −1 dλ to zero as α 1 in L(X) is also equal to 1 − α. It follows from (10) that the rate of convergence of A −α − A −1 to zero as α 1 in L(X) is equal to 1 − α.
Remark 5. Note that, Hypothesis (S) is not required for the proof of the above result.
The next result shows the convergence of the spectrum of −A α at α = 1.
The identities in the following lemma are easy to check.
is a bounded subset then, for α ∈ (0, 1) suitably close to 1, B ⊂ ρ(−A α ) and the following identities hold and Proof. It is not difficult to see that the identities are hold. We only show that Since the inverse of A α exists, if the inverse of (λA −α + I) exists, we conclude that the inverse of λI + A α there exists, that is, λ ∈ ρ(−A α ). In fact, and we have for all α ∈ (α 0 , 1) and λ ∈ B. From the fact that I + λA −1 has bounded inverse we conclude that B ⊂ ρ(−A α ), for all α ∈ (α 0 , 1).
. There exists 0 < α 0 < 1 such that the following estimate is holds Proof. Using (8) we conclude that (17) is a direct consequence of (16).
Proof. First note that using Proposition 2, it follows that From the moment inequality for 0 < 1 2 < α, we have The proof of the next result is a immediate consequence of the Lemma 3.2 and Proposition 6.
4. Spectral properties. Let X be a Banach space and suppose that A be a positive sectorial operator in X. We begin this section studying the properties of convergence of spectral projections associated with −A α , after we will study the eigenvalues of −A α .
4.1. Spectral projection. Let C be a positively oriented (counterclockwise) contour (closed, rectifiable and simple curve) in ρ(−A). From Lemma 3.2, we have Proposition 7. There exists 0 < α 0 < 1 such that the following estimate is holds Proposition 8. There exists 0 < α 0 < 1 such that the following estimate is holds From (16) we obtain whereM is defined in the Proposition 5, and by compactness of the contour C, it follows that where C = C |λ| |dλ| > 0 is independent of α.

4.2.
Characterization and convergence of eigenvalues. In this section we characterize the eigenvalues of A α , in terms of the eigenvalues of A, and we show that they also converge with rate 1 − α, we will use the fact that if A is an operator with compact resolvent, all points in the spectrum σ(A) of A are eigenvalues, since A is a closed operator (see Kato [24, Theorem 6.29, p. 187]).
Theorem 4.1. Suppose that A is a positive sectorial operator with compact resolvent. On eigenvalues of the operator A α , α ∈ (α 0 , 1), we have (a) The eigenvalues {µ α,n } n∈N of the operator A α are given by µ α,n = µ α n , n ∈ N, where {µ n } n∈N denotes the eigenvalues of the operator A.
for some constant C n,C > 0 independent of α.
Proof. Part (a). Since A −1 is compact and σ(A) ∩ R − = ∅ it follows, from the Spectral Mapping Theorem (see [28,Section V.9]), that the isolated eigenvalues of A −α are given by µ −α n , n ∈ N, where {µ n } n∈N denotes the sequence of eigenvalues of A. The result now follows immediately. Part (b) is trivial from the expression of µ α,n .

5.
Convergence of the linear and nonlinear semigroups. In this section we study the linear and nonlinear semigroups generated by operator −A and by −A α , α ∈ [0, 1]. Let X be a Banach space and suppose that A is a positive sectorial operator in X.
As in previous section we note that A is of type G(M, −ω) ⊂ G(M, 0). From this, the semigroup generated by −A can be view as a semigroup of contractions, otherwise we renorming the Banach space X (see Pazy [26, p. 19]).
If A is of type G(1, 0), then −A is a dissipative operator, and it follows that On the other hand, we can write for 0 ≤ α < 1, that [1,23,27]). We can show that −A α is a dissipative operator, that is that is, −A α is a dissipative operator. Additionally, suppose that there is λ 0 > 0 such that the range R(λ 0 I + A α ) = X. From Lumer-Phillips theorem it follows that −A α is the infinitesimal generator of a semigroup of contractions, that is, A α is of type G(1, 0). Remark 6. Let X = H be a Hilbert space. Suppose that A is a positive selfadjoint operator. From [22, p. 27] it follows that A α is self-adjoint for all α > 0. In a similar way we can show that A α is a dissipative operator, then so is (A α ) * for all α > 0. It is not difficult to see that R(I + A α ) = H, and then from Lumer-Phillips theorem (for λ 0 = 1 > 0) it follows that −A α is the infinitesimal generator of a semigroup of contractions in H.
In the next result, we obtain a version of the Trotter-Kato theorem for convergence of the linear semigroups.
Lemma 5.1. If A be as before then, A and A α are in G(1, 0) and for all x ∈ X, and t ≥ 0, we have as α 1, in X. This convergence is uniform for t in bounded intervals.
Proof. Fix x ∈ X and T > 0. For t ∈ [0, T ], we have Observe that e −A α t L(X) ≤ 1 for all t ∈ [0, T ], then it follows from Theorem 3.1 that sup Integrating the above identity from 0 to t, we have Then It follows from (18) that for every x ∈ D(A), is a dense subset of X, it follows that the above expression holds for all x ∈ X.
Remark 7. Note that for x ∈ D(A 2 ), the convergence of linear semigroup obtained in the Lemma 5.1, has rate 1 − α. In fact, from proof of the Lemma 5.1, it follows that for all x ∈ D(A 2 ), there exists C > 0 independent of α such that for each α sufficiently close to 1 and we obtain the rate of convergence from Theorem 3.1.
The above results guarantee uniform convergence in compact sets of X. Proof. Given > 0 and T > 0, choosing δ = /4, n ∈ N and {x 1 , denotes the ball of radius δ centered in x i , i = 1, . . . , n. From the Lemma 5.1, the exists α = α( ) > 0 such that Now we consider the nonlinear semigroups: if t → T (t)u denotes the solution of (3) with α = 1 , then Our goal is to investigate, following the ideas of [3], the rate of convergence of the linear semigroup generated by −A α to the linear semigroup generated by −A. For this, we will use the uniform convergence of A −α to A −1 as α 1.
From this, we get From this, we get Finally, it follows from (19) and (20) that for some θ ∈ (0, 1 2 ) and all t > 0.
Lemma 5.4. There exist a constant C > 0 such that Now, in a similar way as in the proof of Lemma 5.3, we can show that From this, and from (21) we obtain for all θ ∈ (0, 1 2 ). Lemma 5.5. Let a > 0, α ∈ ( 1 2 , 1] and θ ∈ (0, 1 2 ). Then, there exists a positive constant C(a, α, θ) < ∞ such that Proof. The result follows by a simple change of variable.
It follows from Corollary 1 that and using Lemma 5.4 we get for any t > 0. Additionally, we know that As f : X 1 2 → X is globally Lipschitz and uniformly bounded, it follows from Corollary 1 that for some L > 0.
In the next, we will establish the upper semicontinuity of attractors using the continuity properties of the corresponding nonlinear semigroups. which is continuous at α = 1 and possesses a global attractors A α (in X 1 2 ). Furthermore, for each α 0 ∈ ( 1 2 , 1), α∈[α0,1) A α is precompact in X Proof. The existence of attractors it follows from as in [15, Theorems 1.6, 1.16].
The precompactness of α∈[α0,1) A α follows directly from the Variation of Constants Formula, from the exponential decay of the semigroup generated by A α (Hypothesis (S)), from the fact that A has compact resolvent and from the fact that f is bounded.
Proof. We will show that lim where A α is the global attractor to T α (t) (in X 1 2 ) and A 1 is the global attractor to T (t) (in X 1 2 ). Since A 1 is the global attractor to T (t), then A 1 attracts each bounded subsets B of X 1 2 and in particular attracts A α (A α is compact and then is a bounded subset of X 1 2 ). From this, given > 0, there exists t 0 > 0 such that Moreover, it follows from Theorem 5.6 that for the given above, there exists α 0 sufficiently close to 1 such that , for all α ∈ (α 0 , 1).
From this and using the invariance property of A α and A 1 we obtain 6. Linearization around a hyperbolic equilibrium. Let X be a Banach space. Suppose that A is a positive sectorial operator with compact resolvent in X, and that f : X 1 2 → X is a bounded locally Lipschitz continuous function and continuously Frechét differentiable.
The main theme of this section is the control of behavior of the equilibrium of the problem (3) (see definition below), in terms of the difference A −1 − A −α L(X) , and therefore with rate 1 − α (see Theorem 3.1), according to, analysis carried out in the previous sections, for this we will use similar arguments which were used in [2,8,9,12] and [14]. Denote by E the set of equilibrium solutions for (3) with α = 1 .
We say that an equilibrium u * of (3) with α = 1 is hyperbolic if the spectrum σ(A − f (u * )) of A − f (u * ) is disjoint from the imaginary axis and the set is compact in X The unstable manifold of an equilibrium u * , W u (u * ) is defined as follows  We will assume that each element of E is hyperbolic. Then E = {u * 1 , u * 2 , . . . , u * n } is a finite set, and additionally we will assume that the problem (3) with α = 1 is gradient. Consequently, the semigroup generated by −A has a gradient-like attractor A, that is To prove the continuity of local unstable manifolds and that are given as graphs, we will use similar arguments that were used in [8,9,15] and [22].
Let u * i be an equilibrium solution for (3) with α = 1 ; that is, If u(t) is a solution to (3) with α = 1 by the change of variable z = u − u * i , we translate all the dynamics around u * i to the zero solution, so that we can consider where Note that 0 is an equilibrium solution for (27), h(0) = 0, and h (0) = 0 ∈ L(X).
From Definition 6.1, we can choose a smooth closed simple curve γ in ρ(B) ∩ {λ ∈ C : Re λ > 0} that is positively oriented and encloses σ + , where ρ(B) denotes the resolvent set of B. We can then define the spectral projection in X If X ≤M e βt for all t ≤ 0, The existence of local stable and unstable manifolds as graphs near a hyperbolic equilibrium is well known (see [22]). The following result is well known, and therefore we will omit the proof. Theorem 6.2. If u * i is a hyperbolic equilibrium then for suitably small > 0 there are Lipschitz functions Next, we give the definition of equilibrium solution for the problem (3). Our goal is to obtain the continuity of the set of equilibria and then study the continuity of the linearization around each equilibrium. Denote by E α the set of equilibrium solutions to (3). Lemma 6.4. Let u * be a hyperbolic equilibrium solutions of (3). There are constantsᾱ ∈ ( 1 2 , 1] and C > 0, independent of α, such that Proof. The proof is essentially the same as in [9, Lemma 2.3] and then we omit the proof here (see also [13,Lemma 14.16]. Now, we will prove the continuity of the set of equilibria in α = 1. We start with the proof of upper semicontinuity at α = 1. To prove lower semicontinuity of the set of equilibria at α = 1 we use the following auxiliary lemma. Lemma 6.5. Let X be a Banach space, Λ be a topological space and A λ in X, λ ∈ Λ. If A λ0 is compact and for any u ∈ A λ0 , there exists a sequence {x λn }, with x λn ∈ A λn , n ∈ N, λ n → λ 0 such that x λn → x, then {A λ ; λ ∈ Λ} is lower semicontinuous at λ = λ 0 . Proof. If A λ0 is compact and for any x ∈ A λ0 there is a sequence {x λn } with x λn ∈ A λn , λ n → λ 0 , which converges to x and {A λ } is not lower semicontinuous at λ 0 then, there are > 0 and sequence {λ n } with λ n → λ 0 such that sup x∈A λ 0 dist(x, A λn ) ≥ 3 , n ∈ N. Thus, for some x λn ∈ A λ0 , we have that dist(x λn , A λn ) ≥ 2 , n ∈ N. Since A λ0 is compact we may assume that {x λn } converges to some x ∈ A and that dist(x, A λn ) ≥ , n ∈ N. From our assumptions, there is a sequence y λn ∈ A λn such that y λn → x and which is a contradiction.
Hereafter we assume that f : X 1 2 → X is a continuously Fréchet differentiable function with f : X 1 2 → L(X 1 2 , X) being a globally Lipschitz continuous function. This condition is natural in the applications to semilinear partial differential equations.
We turn now to prove the lower semicontinuity of the set of equilibria at α = 1.
From Proposition 3, there is 1 2 < α γ < 1 such that γ is in ρ(A α ), where ρ(A α ) denotes the resolvent set of A α , for all α γ < α ≤ 1, and consider the spectral projection in X α generates an analytic semigroups on X ≤M e βt for all t ≤ 0, ≤M e −βt for all t ≥ 0.
Remark 8. Since the Hypothesis (S) holds, using Proposition 2 and that f (u * α ) is uniformly bounded in α it is not difficult to see that there are ω > 0, θ < π 2 , a sector Σ µ,θ , andM (independent of α) such that This is sufficient to ensure that the constants in two above estimates are independents of α.
Definition 6.6. The unstable manifold of an equilibrium u * α , W u (u * α ) is defined as follows W u (u * α ) = u ∈ X 1 2 : there is a backwards solution u of (3) with u(0) = u and such that lim The local unstable manifolds of u * α is defined as follows. Given a neighborhood Let us consider the following setting for the potentials. 1) and V := f (u * ) with u * ∈ E. There exist 0 < α 0 < 1 and C > 0 independent of α such that Proof. Let u * be an equilibrium solution to the problem (3) with α = 1 and fix α 0 ∈ ( 1 2 , 1). Let u * α ∈ E α such that u * α − u * for some constant C > 0 independent of α, α ∈ (α 0 , 1) (see Proposition 12). Moreover, if 1]. In order to prove (30) we note by Lipschitz property of f and f that , and L f and L f are the Lipschitz constants of f , and f , respectively. Thus for all α ∈ (α 0 , 1) we have for some positive constant C independent of α. Therefore, it follows from Theorems 3.1 and 6 that there exists C > 0 independent of α such that for all α ∈ (ᾱ, 1).
It is not difficult to see that because of the above analysis, all the convergences obtained to follow they occur at a rate of 1 − α.
Proof. This follows easily using the decomposition and applying Theorem 6 and Theorem 6.7.
Recall that from our notations, B α = A α + V α and B = A + V . It is easy to see that the following holds: Proof. The following identity holds and therefore, , applying Theorem 6 and Lemma 6.8 we obtain the result.
Proposition 14. The operators Q α converges in the uniform operator topology L(X, X 1 2 ) to Q as α 1.
Proof. Let us prove first, the inequality .
Note that and using the convergence in the uniform operator topology of : α 1 < α < 1} is uniformly bounded in α, and therefore 1 2πi is uniformly bounded for all α 1 < α < 1, and it follows that the operators Q α for all α 1 < α < 1 converge in the uniform operator topology to Q as α 1.
We decompose the space X 1 2 through the projection Q α , that is, X ). This induces a decomposition of (3) in the following sense: if z is a solution of (3), we write z + = Q α z and z − = (I − Q α )z, and then where Since W u (u * α ) is invariant, for an initial data u * α + (Q α w, Σ * ,u α (Q α w))) ∈ W u (u * α ) the solution of (31) stays in the graphic of Σ * ,u α for all t ∈ R. This ensures that z − (t) = Σ * ,u α (z + (t))) and we can be rewritten as Furthermore, the solution (z + (t), z − (t)) must go to zero as t → −∞ and in particular it must stay bounded. Since , Σ * ,u α (z + (s)))ds and in particular Thus, we should have Σ * ,u α as a fixed point of a map defined in a space of functions suitable. Once this is accomplished, the convergence of linearized semigroups give us the continuity of unstable manifolds, with a similar argument as it is done in [2] and [22]. Theorem 6.9. Assume that y * is a hyperbolic equilibrium of (3) with α = 1 Then, by Proposition 12 the problem (3) has a unique equilibrium u * α in a small neighborhood of y * . Then there exist δ > 0 andᾱ ∈ [α 1 , 1) such that u * α has an unstable local manifold W u loc (u * α ) ⊂ X α for α ∈ [ᾱ, 1], and if we denote where C > 0 is independent of α.

7.
Continuity of attractors. We conclude this part of the work proving the continuity of the attractors at α = 1, following the ideas of [11]. Let X be a Banach space. Suppose that A is a positive sectorial operator with compact resolvent in X.
7.1. Lower semicontinuity of the attractors. In this subsection we analyze the lower semicontinuity of the attractors at α = 1.
Proof. Let z ∈ A 0 . As T (·) is a gradient systems, we have A 0 = u * ∈E W u loc (u * ) and then z ∈ W loc (u * ) for some u * ∈ E. Let τ ∈ R and v ∈ W u δ (u * ) such that T (τ )v = u. Let z * α such that z * α → z * as α 1. From convergence of unstable manifolds there is a sequence {v α }, v α ∈ W u δ (u * α ) such that v α → v as α 1. Finally, from continuity of nonlinear semigroup, we obtain T α (t)v α → T (τ )v = z. To conclude we use Lemma 6.5 and observe that if z α = T α (t)v α , then z α ∈ A α , and z α converges to z.
It follows from Theorem 5.6, Proposition 12, Theorem 6.9, together with Theorem 7.2 are hold, the convergence with rate of the global attractors A α given by (32) as α 1.
8. Strongly damped wave equations. In this section, we apply the results of the analysis in the previous sections to study the convergence of the dynamical of a strongly damped wave equation. Consider a parabolic strongly damped wave equation of the form where Ω is a bounded smooth domain in R N , N ≥ 3, and f ∈ C 1 (R) satisfies for some and lim sup |s|→∞ f (s) s < µ 1 , with µ 1 being the first eigenvalue of the negative Dirichlet Laplacian −∆ D in L 2 (Ω).
Recall that if X = L 2 (Ω) and A : D(A) ⊂ X → X is defined by then A is a positive self-adjoint sectorial operator and −A generates a compact analytic C 0 -semigroup in X.
Denote by X α the fractional power spaces associated to operator A; that is, X α = D(A α ) with the norm A α · X : X α → R + . For α > 0 define also X −α as the completion of X with the norm A −α · X . Observe that with this notation X 1 2 = H 1 0 (Ω) and X 1 = H 2 (Ω) ∩ H 1 0 (Ω). Observe also from [1, Chapter V] that X −α = (X α ) .
With the above set-up we can define the operator Lemma 8.1. If A and Λ are as in (34) and in (35) respectively then we have all the following.
Now, we will show that the following estimates are hold: where M is a positive constant independent of α and δ is some positive constant. Suppose that where M is independent of α. It follows from (42) that there exists a positive constantM (independent of α) such that (λI + Λ α ) −1 This proves that Hypothesis (S) is satisfied whenever (42) is.
Remark 10 (Proof of (42)). Remember that X = L 2 (Ω) and A : D(A) ⊂ X → X is defined by Au = −∆ D u for u ∈ D(A) = H 2 (Ω) ∩ H 1 0 (Ω), then A is a positive self-adjoint operator in the Hilbert space X.
From ( [22, p. 27]) since A is a self-adjoint positive definite operator, then so is A α 2 , for all α > 0. From this, there exist a positive constant a such that A α 2 u, u X ≥ a u 2 .
Combining the growing conditions and sign on the f , spectral properties of the operator Λ and its fractional powers (37), and the properties of embedding of the fractional powers spaces X α = D(A α ) we can conclude that the initial value problems (45) and (46) are globally well posedness and they possesses a global attractor in some space in the scale of fractional powers spaces of the operator Λ.
From Proposition 9 we obtain the upper semicontinuity of the global attractors at α = 1. To show the lower semicontinuity of the global attractors, we assume that the equilibrium of (33) are hyperbolic, and consequently, they are finite in number and the property of continuity of the set of equilibria of the problem (46) at α = 1 follows from Proposition 11 and Proposition 6.5. The continuity with rate of the local unstable manifolds it follows from Theorem 6.9, and the property of lower semicontinuity of the family of global attractors {A α : 1 2 ≤ α ≤ 1} at α = 1 follows from Theorem 7.1.

Remark 11.
Other examples where this analysis can be carried out are damped wave equations considered in [6], the nonlinear Schrödinger equation in [16,17] and [26]. In each example a careful study of the fractional power of the operator that governs the problem it is necessary.