Non-autonomous morse-decomposition and Lyapunov functions for gradient-like processes

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 365, Number 10, October 2013, Pages 5277–5312 S 0002-9947(2013)05810-2 Article electronically published on April 1, 2013 NON-AUTONOMOUS MORSE-DECOMPOSITION AND LYAPUNOV FUNCTIONS FOR GRADIENT-LIKE PROCESSES E. R. ARAG˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA Abstract. We define (time dependent) Morse-decompositions for non-auton- omous evolution processes (non-autonomous dynamical systems) and prove that a non-autonomous gradient-like evolution process possesses a Morsedecomposition on the associated pullback attractor. We also prove the existence of an associated Lyapunov function which describes the gradient behavior of the system. Finally, we apply these abstract results to non-autonomous perturbations of autonomous gradient-like evolution processes (semigroups or autonomous dynamical systems). 1. Introduction The Fundamental Theorem of Dynamical Systems (see [8]) describes the flow of autonomous reversible dynamical systems (groups) in compact metric spaces as a decomposition of an ordered family of isolated invariant sets and connections between them which respects their ordering. In the terminology of [8], this is called a Morse-decomposition of a compact invariant set. The extension of this result to general autonomous dynamical systems (or semigroups) is done in [1] (see also [20] for the Morse-decomposition for semigroups in compact metric spaces). For autonomous dynamical systems, the Morse-decomposition plays a fundamental role in the process of understanding the structure of invariant sets. It allows us to decompose the dynamics into a gradient part and the dynamics in smaller isolated invariant sets, reducing the study of the structure of the attractors to the study of the structure of these isolated invariant sets. The aim of this paper is to extend the notion of Morse-decomposition to nonautonomous dynamical systems (or non-autonomous evolution processes)andto show the pullback attractors for some of these non-autonomous dynamical systems Received by the editors April 8, 2011 and, in revised form, September 21, 2011 and February 3, 2012. 2010 Mathematics Subject Classification. Primary 37B25, 37B35; Secondary 37B55, 35B40, 35B41. Key words and phrases. Non-autonomous Morse decomposition, Lyapunov functions for nonautonomous dynamical systems, robustness of dynamical structures. The first author was partially supported by CAPES/DGU 267/2008 and FAPESP 2008/50248- 0, Brazil. The second author was partially supported by FEDER and Ministerio de Ciencia e Innovaci´on grant # MTM2008-0088, # MTM2011-22411, PBH2006-0003-PC, and Junta de Andaluc´ıa grants # P07-FQM-02468, # FQM314 and HF2008-0039, Spain. The third author was partially supported by CNPq 302022/2008-2, CAPES/DGU 267/2008 and FAPESP 2008/55516-3, Brazil and Junta de Andaluc´ıa grant # P07-FQM-02468. The fourth author was partially supported by FEDER and Ministerio de Ciencia e Innovaci´on grants # MTM2008-0088, # MTM2011-22411, # PBH2006-0003-PC, and Junta de Andaluc´ıa grants # P07-FQM-02468, # FQM314 and HF2008-0039, Spain. c 2013 American Mathematical Society Reverts to public domain 28 years from publication 5277 Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5278 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A LANGA can also be decomposed as an ordered family of isolated invariant families and connections between them respecting their ordering. In particular, under some mild additional assumptions, we construct a non-autonomous Lyapunov function for the non-autonomous evolution process with a Morse-decomposition. We show that this decomposition is observed for non-autonomous perturbations of gradient autonomous dynamical systems (or gradient semigroups). Some practical examples are also presented. Our results contrast with those of [19] because, in our case, the definition of local attractors takes into account only the usual notion of attraction instead of using past and forwards notions. We also do not impose the reversibility required in [19]. Our results can be applied (under mild assumptions) to perturbations of autonomous dynamical systems (which cannot be said for the results in [19]). Furthermore, in the case of perturbations of autonomous dynamical systems and under mild assumptions, we prove some sort of continuity of the Morse-decomposition. To better describe the results in the paper we will need to introduce some terminology and a few definitions. Let Xbe a metric space with metric d:X×X→R+, where R+:= [0,∞),and denote by C(X) the set of continuous maps from Xinto X. Given a subset A⊂X,the-neighborhood of Ais the set O(A):={z∈X: d(z,a)<for some a∈A}. An evolution process in a metric space Xis a two-parameter family {T(t, s): t≥s}in C(X) such that 1) T(t, t)=I, for all t∈R, 2) T(t, σ)T(σ, s)=T(t, s),for t≥σ≥s, and 3) P×X((t, s),x)→T(t, s)x∈Xis continuous, where P:= {(t, s)∈R2:t≥s}. An evolution process {T(t, s):t≥s}is autonomous when T(t, s)=T(t−s, 0) for all t≥s, otherwise it is non-autonomous. A semigroup is a family {S(t):t≥0}⊂ C(X) such that 1) S(0) = I, for all t∈R, 2) S(t+s)=S(t)S(s),t,s≥0,and 3) R+×X(t, x)→ S(t)x∈Xis continuous. Note that {T(t, s):t≥s}is an autonomous evolution process iff {T(t, 0) : t≥0}is a semigroup. A continuous function ξ:R→Xis a global solution for the evolution process {T(t, s):t≥s}when it satisfies T(t, s)ξ(s)=ξ(t),for all (t, s)∈P. Next we recall the definition of Hausdorff semidistance and Hausdorff distance. Given A, B ⊂X,theHausdorff semidistance from Ato Bis given by dist(A, B):=sup a∈A inf b∈Bd(a, b)=sup a∈A d(a, B), where d(a, B):= inf b∈Bd(a, b) is the usual distance from the point ato the set B, and the Hausdorff distance between Aand Bis defined by dH(A, B):=dist(A, B)+dist(B,A). Definition 1.1. A family Ξ := {Ξ(t):t∈R}of subsets of Xis an invariant family for the evolution process {T(t, s):t≥s}when T(t, s)Ξ(s)=Ξ(t) for all t≥s.An Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use GRADIENT PROCESSES UNDER PERTURBATIONS 5279 invariant family {A(t):t∈R}is a pullback attractor when, for each t∈R,A(t) is compact, pullback attracts bounded subsets of X;thatis, lim s→−∞ dist(T(t, s)B,A(t)) = 0,for each t∈Rand bounded subset Bof X, and {A(t):t∈R}is minimal among all closed invariant families {C(t):t∈R} with the property that C(t) pullback attracts bounded subsets of Xfor each t∈R. Recall that, if {S(t):t≥0}is a semigroup in a metric space X,itissaidthat A⊂Xis invariant for {S(t):t≥0}if S(t)A=A, for all t≥0. It is important to notice that, in the autonomous context, the non-autonomous notion of invariance also allows for non-constant families. The notion of an unstable set of an invariant family plays a key role in the definition of attractor-repeller pairs. On their turn, the attractor-repeller pairs enter in an essential way in the definition of Morse-decomposition for the pullback attractor of a non-autonomous evolution process. Next we define these concepts beginning with the definition of an unstable set of an invariant family. Definition 1.2. Let {T(t, s):t≥s}be an evolution process. The unstable set of an invariant family Ξ = {Ξ(t):t∈R}is the set Wu(Ξ) := {(t, z)∈R×X:there is a global solution ξ:R→X such that ξ(t)=zand lim s→−∞ dist(ξ(s),Ξ(s)) = 0}. Also, Wu(Ξ)(t):={z∈X:(t, z)∈Wu(Ξ)}for each t∈R. Now we introduce the notions of isolated invariant sets and disjoint families of isolated invariant sets. Definition 1.3. Let Ξ := {Ξ(t):t∈R}be an invariant family for the evolution process {T(t, s):t≥s}.Ξisanisolated invariant family if there exists a δ>0 with the property that any global solution ξ:R→Xwith ξ(t)∈O δ(Ξ(t)) must satisfy that ξ(t)∈Ξ(t) for all t∈R.AsetΞ={Ξ1,··· ,Ξn}is said to be a disjoint set of isolated invariant families if each Ξi,1≤i≤n, is an isolated invariant family and there exists δ>0 such that Oδ(Ξi(t)) ∩O δ(Ξj(t)) = ∅, for all t∈R and for 1 ≤i<j≤n. Remark 1.4.Let {S(t):t≥0}be a semigroup in a metric space X.Itisimportant to notice that: (1) A subset Aof Xis an isolated invariant set for {S(t):t≥0}if it is invariant and there is a δ>0 such that Ais the maximal invariant set for {S(t):t≥0}in Oδ(A). (2) If ξ:R→Xis a uniformly continuous global solution for {S(t):t≥0}, then {{ξ(t)}:t∈R},t∈R, is an invariant family which is not isolated. Any isolated invariant family {Ξ(t):t∈R}such that ξ(t)∈Ξ(t) for all t∈Rmust satisfy that ξ(R)⊂Ξ(t) for all t∈R. In the autonomous context this remark says that, in some sense, the two notions of isolated invariant sets (autonomous and non-autonomous) are similar. (3) It is easy to see that if A⊂Xand δ>0 is such that Ais the maximal invariant set for {S(t):t≥0}in Oδ(A), then •If δ<δand x∈Xis such that γ+(x)⊂Oδ(A), then d(T(t)x, A)t→∞ −→ 0. •If δ<δand there is a global solution ξ:R→Xsuch that ξ(R−)⊂ Oδ(A), then d(ξ(t),A)t→−∞ −→ 0. Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5280 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A LANGA (4) The preceding remark ensures that a center in a two dimensional dynamical system or an equilibrium which is part of a continuum of equilibria will not be an isolated invariant set. The notions of a local attractor and of an atractor-repeller pair are defined next. Definition 1.5. Let {T(t, s):t≥s}be an evolution process in a metric space X with a pullback attractor {A(t):t∈R}. We say that an isolated invariant family A:= {A(t):t∈R}, with A(t)⊂A(t) for all t∈R, is a (non-autonomous) local attractor if Wu(A)(t)=A(t) for all t∈R. If Ais a local attractor, we define its associated repeller A∗:= {A∗(t):t∈R} by (1.1) A∗(t):={z∈A(t):dist(T(r+t, t)z,A(r+t)) −→ 0asr→∞},for each t∈R. The pair (A, A∗) is called an attractor-repeller pair. The following result ensures that the above definition extends its autonomous counterpart. We sketch its proof for completeness (see [1, Lemma 2.15] for a complete proof). Theorem 1.6. Let {S(t):t≥0}be a semigroup in a metric space Xwith a global attractor A. An isolated invariant set A⊂Ais a local attractor (autonomous) iff the constant family {A(t):A(t)=A, t ∈R}is a local attractor for {T(t, s): T(t, s):=S(t−s)for t≥s}, in the sense of Definition 1.5. Proof. Note that Wu(A)=Wu({A(t):t∈R})(t),∀t∈R, where Wu(A):={z∈A: there is a global solution ξ:R→Xwith ξ(0) = zand lim t→−∞d(ξ(t),A)=0}and that if Ais an isolated invariant set for S(·) such that Wu(A)=A, then for every δ>0thereexistsδ∈(0,δ) with γ+(Oδ(A)) ⊂O δ(A). The proof now follows easily from the fact that ω(Oδ(A)) attracts Oδ(A)and is invariant (hence it must be contained in A).  For autonomous evolution processes (semigroups) a repeller is automatically closed, but that is not the case for non-autonomous evolution processes. However, if there exists ε>0 such that Oε(A(t)) ∩O ε(A∗(t)) = ∅, for all t∈R,then A∗(t)isclosedforeacht∈R. Now, we define the notion of Morse-decomposition for non-autonomous evolution processes. Definition 1.7. Let {T(t, s):t≥s}be an evolution process in Xwith a pullback attractor {A(t):t∈R}and let A0={A0(t):t∈R},A1={A1(t):t∈R},···, An={An(t):t∈R}be n+ 1 local attractors with ∅=A0(t)⊂A1(t)⊂ ··· ⊂ An(t)=A(t)foreacht∈R. Define Ξj(t):=Aj(t)∩A∗ j−1(t)foreacht∈Rand j=1,2,··· ,n. The ordered set of invariant families Ξ:= {Ξ1,Ξ2,··· ,Ξn}is called a Morse-decomposition for the pullback attractor {A(t):t∈R}. Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use GRADIENT PROCESSES UNDER PERTURBATIONS 5281 In Section 2 we give sufficient conditions to obtain a Morse-decomposition for the pullback attractor of a gradient-like evolution process. Section 3 is devoted to obtain, under some mild assumptions, a Lyapunov function for a gradient-like (non-autonomous) evolution process restricted to the pullback attractor. We also give conditions so that the Lyapunov function is defined and is continuous in R×X. We prove in Section 4 that the results of Section 2 can be applied (under natural assumptions) to non-autonomous perturbations of autonomous gradient-like evolution processes. In this situation, thanks to the results from Section 3, the Lyapunov function can be defined and is continuous in R×X. The continuity of the Lyapunov function under perturbation is also considered. Section 5 exhibits some concrete examples where the theory developed in the previous sections can be applied. 2. Morse-decomposition of pullback attractors for generalized gradient-like evolution processes We first introduce the notion of a gradient-like process (see [5]). Let {T(t, s):t≥ s}be an evolution process with a pullback attractor {A(t):t∈R}which contains a disjoint set of isolated invariant families Ξ={Ξ1,··· ,Ξn},that is, Ξi(t)⊂A(t) for each iand t. We can now define the concept of pinned-chain recurrence (see [1, 5, 8, 15] for similar concepts in the autonomous case): Definition 2.1. Let δbe as in Definition 1.3 and fix 0∈(0,δ). For Ξ ∈Ξ and ∈(0, 0), an −pinned-chain from Ξ to Ξ is a sequence i,1≤i≤k,in {1,··· ,n}, a sequence of real numbers ti,σ i,τ iwith τi<σ i<t i, and a sequence of points zi∈X,1≤i≤k, such that zi∈O (Ξi(τi)), T(σi,τ i)zi/∈O 0(n i=1 Ξi(σi)) and T(ti,τ i)zi∈O (Ξi+1 (ti)), 1 ≤i≤k,withΞ=Ξ k+1 =Ξ 1.Wesaythat Ξ∈Ξis pinned-chain recurrent if there is an 0∈(0,δ)and−pinned-chain from Ξ to Ξ for each ∈(0, 0). Remark 2.2.We note that the introduction of 0in the above definition is only needed to account for the case k=1. Whenk>1, it is automatically true that the solution must leave O0(n i=1 Ξi(t)),for some t∈R,while going from one isolated invariant family to another. Before we proceed let us point out the distinctions between the notion of pinnedchain recurrence introduced in the above definition and the notion of chain recurrence defined in [8]. We start with the definition of chain recurrence given in [8]. Definition 2.3 (Conley).Let {S(t):t∈R}be a group in a metric space Xand Ξ⊂Xbe a compact invariant set. Given >0, t>0andξ,ξ∈Ξ, an (, t)−chain from ξto ξis a sequence ξ=ξ0,ξ 1,··· ,ξ n=ξ;t1,··· ,t nsuch that ti≥tfor all i and d(S(ti)ξi−1,ξ i)<,i=1,··· ,n. A subset R(Ξ) of an invariant set Ξ is called chain recurrent if for any ξ∈R(Ξ), >0andt>0thereisa(, t)-chain from ξto itself. Remark 2.4.The notions of pinned-chain recurrence and chain recurrence are different, even in the autonomous context. However, the first is strongly inspired by the second. In the pinned-chain recurrence notion the jumping points are pinned in the isolated invariant sets, and in chain recurrence notion the jumping points occur around points which are not pinned. Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5282 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A LANGA Another important difference is that an equilibrium is always chain recurrent (Conley’s concept), whereas they may not be pinned-chain recurrent (our concept). In fact, for gradient semigroups having a global attractor and with a finite set of equilibria, no equilibria is pinned-chain recurrent. To be pinned-chain recurrent there should exist solutions starting arbitrarily close to the isolated invariant set that leave a fixed neighborhood of it and come back arbitrarily close (possibly jumping only near other isolated invariant sets). The notion of pinned-chain recurrence, in the autonomous case, is closely related to the existence of homoclinic structures, as shown in Lemma 2.2 in [5]. It is tailored to describe the dynamical properties of a gradient semigroup, without a priori requiring the existence of a Lyapunov function. Definition 2.5. Let Xbe a metric space and {T(t, s):t≥s}be an evolution process in Xwith a pullback attractor {A(t):t∈R}and a disjoint set of isolated invariant families Ξ={Ξ1,··· ,Ξn}in {A(t):t∈R}.Wesaythat{T(t, s):t≥s} is a gradient-like evolution process with respect to Ξ if the following two conditions are satisfied: (H1) Any global solution ξ:R→Xin {A(t):t∈R}satisfies lim t→−∞ dist(ξ(t),Ξi(t)) = 0 and lim t→∞ dist(ξ(t),Ξj(t)) = 0, for some 1 ≤i, j ≤n. (H2) Ξ={Ξ1,··· ,Ξn}does not contain any pinned-chain recurrent isolated invariant family. Let us now define the notion of a homoclinic structure. Definition 2.6. Let {T(t, s):t≥s}be an evolution process which possesses a disjoint set of isolated invariant families Ξ={Ξ1,··· ,Ξn}.Ahomoclinic structure associated to Ξis a finite subset {Ξi1,··· ,Ξip}of Ξtogether with a set of global solutions {ξ1,··· ,ξ p}such that lim t→−∞ dist(ξk(t),Ξik(t)) = 0 and lim t→∞ dist(ξk(t),Ξik+1 (t)) = 0,1≤k≤p, where Ξip+1 := Ξi1. Next we prove the invariance of the repeller A∗of a local attractor A. Proposition 2.7. Let {T(t, s):t≥s}be an evolution process with a pullback attractor {A(t):t∈R}and let A={A(t):t∈R}be a local attractor. Then, the repeller A∗of Ais invariant. Proof. If A∗(t0)isemptyforsomet0∈R,thenitisemptyforallt∈Rand the proof is obvious. Assume that A∗(t)isnon-emptyforallt∈R, and let s∈R,t≥s and w∈A∗(s). If T(t, s)w∈ A∗(t)wehavethat lim τ→∞ dist(T(τ,t)T(t, s)w,A(τ)) = 0, that is, lim τ→∞ dist(T(τ,s)w,A(τ)) = 0, which contradicts the fact that w∈A∗(s) and proves that T(t, s)A∗(s)⊂A∗(t). Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use GRADIENT PROCESSES UNDER PERTURBATIONS 5283 Conversely, if z∈A∗(t)⊂A(t)=T(t, s)A(s), let w∈A(s) be such that z=T(t, s)w. It follows that w∈A∗(s), for otherwise 0 = lim τ→∞ dist (T(τ,s)w,A(τ)) = lim τ→∞ dist(T(τ,t)T(t, s)w,A(τ)) = lim τ→∞ dist(T(τ,t)z,A(τ)), contradicting the fact that z∈A∗(t) and proving that T(t, s)A∗(s)⊃A∗(t). We note that if A0={A0(t):t∈R},A1={A1(t):t∈R},···,An={An(t): t∈R}are n+ 1 local attractors with ∅=A0(t)⊂A1(t)⊂ ··· ⊂ An(t)=A(t), for each t∈R,then∅=A∗ n(t)⊂A∗ n−1(t)⊂···⊂A∗ 0(t)=A(t),for each t∈R. Next we describe the construction of a Morse-decomposition for the pullback attractor of a generalized gradient-like evolution process relative to the disjoint set of isolated invariant families {Ξ1,··· ,Ξn}, and of the associated collection of increasing local attractors starting from the collection of isolated invariant sets. The following result plays a fundamental role on that. Lemma 2.8. Let {T(t, s):t≥s}be a gradient-like evolution process with an associated disjoint set of isolated invariant families Ξ={Ξ1,··· ,Ξn}.Then,there exists i∈{1,··· ,n}such that Ξiis a local attractor. Proof. We first note that each Ξiis an isolated invariant family which is not pinnedchain recurrent, and we need to show that there is some i∈{1,··· ,n}such that Wu(Ξi)(t)=Ξ i(t)foreacht∈R. If that is not the case, for each 1 ≤i≤nthere is a global solution ξi(t)∈A(t) (with ξi(s)∈ Ξi(s)forsomes∈R) such that lim t→−∞dist(ξi(t),Ξi(t)) = 0. Since ξi(t) converges to some element of Ξas t→∞, this necessarily produces a homoclinic structure and provides a contradiction with (H2).  Let {T(t, s):t≥s}be a gradient-like evolution process with the associated disjoint set of isolated invariant families Ξ={Ξ1,··· ,Ξn}. If (after possible reordering) Ξ1is a local attractor and Ξ∗ 1as in Definition 1.5, then we have that each Ξi(s), for i≥2ands∈R,is contained in Ξ∗ 1(s)(alsoifξ:R→Xis a global bounded solution and dist(ξ(t),Ξi(t)) t→∞ −→ 0 with i>1, we have that ξ(s)∈Ξ∗ 1(s) for all s∈R)andthenforanyz∈A(t)\(Ξ1(t)∪Ξ∗ 1(t)) and global solution ξ:R→Xwith ξ(s)∈A(s), for each s∈R,andξ(t)=zwe have that lim s→−∞ dist(ξ(s),Ξ∗ 1(s)) = 0 and lim s→∞ dist(ξ(s),Ξ1(s)) = 0. We can repeat the reasoning in Lemma 2.8 to conclude that there is i≥2 such that Wu(Ξi)(t)∩Ξ∗ 1(t)=Ξ i(t) for all t. We relabel this isolated invariant family as Ξ2and define (2.1) Ξ∗ 2,1(t):={z∈Ξ∗ 1(t):dist(T(r+t, t)z,Ξ2(r+t)) −→ 0asr→∞}. Then we have that, for each t∈Rand i=3,··· ,n,Ξ i(t)⊂Ξ∗ 2,1(t)(alsoif ξ:R→Xis a global bounded solution and dist(ξ(t),Ξi(t)) t→∞ −→ 0 with i>2, we have that ξ(s)∈Ξ∗ 1(s) for all s∈R, and therefore ξ(s)∈Ξ∗ 2,1(s) for all s∈R). As before, we can assume that Wu(Ξ3)(t)∩Ξ∗ 2,1(t)=Ξ 3(t)foreachtand define Ξ∗ 3,2 in analogy to (2.1). Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5284 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A LANGA Proceeding in this way until all isolated invariant families are exhausted, we obtain a reordering of Ξ={Ξ1,··· ,Ξn}such that Ξ1is a local attractor for {T(t, s):t≥s},thesetΞ ∗ 1,0:= Ξ∗ 1,and Wu(Ξi)(t)∩Ξ∗ i−1,i−2(t)=Ξ i(t) for all tand i=2,··· ,n, where, for i=2,··· ,n, Ξ∗ i,i−1(t):={z∈Ξ∗ i−1,i−2(t):dist(T(r+t, t)z,Ξi(r+t)) −→ 0asr→∞}. Lemma 2.9. Let {T(t, s):t≥s}be a gradient-like evolution process with associated (reordered) disjoint set of isolated invariant families Ξ={Ξ1,··· ,Ξn}.Then, any global bounded solution ξ:R→Xsatisfies (2.2) lim t→−∞ dist(ξ(t),Ξi(t)) = 0 and lim t→∞ dist(ξ(t),Ξj(t)), with i≥j. Proof. Indeed, if j= 1 in (2.2) there is nothing to prove. If j= 2 and (2.2) holds, we have that ξ(t)∈Ξ∗ 1(t) for all t∈R,soif1=i<j=2,wehavethatξ(t)∈ Wu(Ξ1)(t)=Ξ 1(t)foreacht, which contradicts the fact that Ξ∗ 1(t)∩Ξ1(t)=∅for each t∈R. It follows that i≥j. For the general case, we suppose that j≥3 and that (2.2) holds. Then ξ(t)∈ Ξ∗ j−1,j−2(t) for all t,soifi<j,wehavethatΞ ∗ j−1,j−2(t)⊂Ξ∗ i−1,i−2(t) for all t and ξ(t)∈Wu(Ξi)(t)∩Ξ∗ i−1,i−2(t)=Ξ i(t) for every t. Thanks to the fact that the invariant families in Ξare isolated, we must have i=j, which is a contradiction and proves the lemma.  We will prove that this reordering of {Ξ1,··· ,Ξn}(which we denote the same) is a Morse-decomposition for {A(t):t∈R}with a suitably chosen sequence A0(t)⊂A1(t)⊂A2(t)⊂···⊂An(t) of local attractors: For each t∈R, define A0(t):=∅,A1(t):=Ξ 1(t), and for j=2,3,··· ,n, (2.3) Aj(t):=Aj−1(t)∪Wu(Ξj)(t)= j  i=1 Wu(Ξi)(t). It is clear that An(t)=A(t). Theorem 2.10. Let {T(t, s):t≥s}be a gradient-like process with a pullback attractor {A(t):t∈R}and an associated disjoint set of isolated invariant families Ξ={Ξ1,··· ,Ξn}reordered as explained above. Assume that there exists δ>0such that, for j=1,2,··· ,n−1and t∈R,it holds that (2.4) Oδ(Aj(t)) ∩⎛ ⎝ n  i=j+1 Ξi(t)⎞ ⎠=∅. Then, for each j=0,1,··· ,n, the invariant family Ajdefined in (2.3) is a local attractor for {T(t, s):t≥s}and Ξj(t)=Aj(t)∩A∗ j−1(t),for all t∈Rand 1≤j≤n. Consequently, Ξdefines a Morse-decomposition for {A(t):t∈R}. Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use GRADIENT PROCESSES UNDER PERTURBATIONS 5285 Proof. Clearly {Aj(t):t∈R}is invariant and Aj(t)⊂Wu(Aj)(t)foreacht∈ R. On the other hand, if z∈Wu(Aj)(t), there is a global solution ξ:R→ Xwith ξ(t)=zand lims→−∞ dist(ξ(s),A j(s)) = 0. Since {T(t, s):t≥s}is gradient-like and from (2.4) we have that there exists k∈{1,2,··· ,j}such that lims→−∞ dist(ξ(s),Ξk(s)) = 0. Therefore, z∈Wu(Ξk)(t) with k≤j. This implies that z∈Ajand Wu(Aj)(t)⊂Aj(t). Now we prove that {Aj(t):t∈R}is an isolated invariant family. Indeed, let δ>0 be as in (2.4) and ξ:R→Xbe a global solution with ξ(t)∈O δ(Aj(t)), for all t∈R.Since{T(t, s):t≥s}is gradientlike, there is a k∈{1,··· ,j}such that dist(ξ(t),Ξk(t)) t→−∞ −→ 0. It follows that ξ(t)∈Wu(Ξk)(t)⊂Aj(t) for all t∈R. This proves that {Aj(t):t∈R}is a local attractor. It is easy to see that Ξj(t)⊂Aj(t)∩A∗ j−1(t). For the reverse inclusion, note that if z∈Aj(t)∩A∗ j−1(t), there exists a global solution ξ:R→Xwith ξ(t)=z.Since z∈Aj(t) we must have that lims→−∞ dist(ξ(s),Ξk(s)) for some k≤j.As{T(t, s): t≥s}is gradient-like, there is i∈{1,2,··· ,n}such that lims→∞ dist(ξ(s),Ξi(s)) and, due to the fact that z∈A∗ j−1(t), it follows that i≥j. Since from Lemma 2.9 we must have that i≤k, it follows that k=i=j, and again taking into account that {T(t, s):t≥s}is gradient-like, we have that ξ(s)∈Ξj(s)foreachs∈R and, in particular, z=ξ(t)∈Ξj(t). This proves that Aj(t)∩A∗ j−1(t)⊂Ξj(t)and completes the proof.  Remark 2.11.Hypothesis (2.4) is not necessary in the autonomous case (see [1, Theorem 2.18]). The following result plays a key role in the proof of the main results in this paper concerning the continuity of the Lyapunov function of a gradient-like evolution process. It extends Lemma 2.11 in [1] to the non-autonomous case. Lemma 2.12. Let {T(t, s):t≥s}be an evolution process with a pullback attractor {A(t):t∈R}and let A={A(t):t∈R}be a local attractor which is not pinnedchain recurrent. Suppose that there exists ε>0with (2.5) A(t)∩O ε(A(t)) ∩O ε(A∗(t)) = ∅,for all t∈R. Then, for each δ∈(0,ε)there is δ∈(0,δ)such that T(t, s)(A(s)∩O δ(A(s))) ⊂A(t)∩O δ(A(t)) whenever t≥s. Proof. We argue by contradiction. Assume that there are δ>0, sequences (sj)j∈N, (tj)j∈Nin R,and(xj)j∈Nin Xwith xj∈A(sj)foreachj, such that sj≤tj, dist(xj,A(sj)) <1 j,but dist(T(tj,s j)xj,A(tj)) ≥δ, for all j∈N.Choosej0∈Nwith dist(xj,A(sj)) <1 j<εfor all j≥j0.Bythe definition of a repeller, for each j≥j0there is a τj≥tjsuch that dist(T(τj,s j)xj,A(τj)) <1 j. It follows that Ais pinned-chain recurrent. This is a contradiction, and the lemma is proved.  Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5292 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A LANGA Also, let (A, A∗)be an attractor-repeller pair for {T(t, s):t≥s}with Oε(A(t)) ∩O ε(A∗(t)) = ∅,for all t∈R, for some ε>0and, furthermore, satisfying the following conditions: (a)The conclusion of Lemma 2.12 holds, considering neighborhoods of Ain X. That is, for each δ>0there is δ∈(0,δ)such that T(r+t, t)(Oδ(A(t))) ⊂O δ(A(r+t)),forallr≥0and t ∈R. (b)For each t∈Rand z∈X\(A(t)∪A∗(t)) we have lim r→∞ dist(T(r+t, t)z,A(r+t)) = 0 or lim r→∞ dist(T(r+t, t)z,A∗(r+t)) = 0. Then, there exists a function f:R×X→Rsatisfying the following four properties: i)For each (t, z)∈R×X, the function [0,∞)r→ f(r+t, T(r+t, t)z)∈R is non-increasing. ii)If t∈Rand ft:X→Ris defined by ft(z):=f(t, z),foreachz∈X,then f−1 t(0) = A(t)and f−1 t(1) ∩A(t)=A∗(t). iii)Given t∈Rand z∈X,iff(r+t, T(r+t, t)z)=f(t, z)for all r≥0,then z∈A(t)∪A∗(t). iv)f:R×X→Ris continuous (in both variables). Proof. In fact, let k:R×X→Rbe the function given in Theorem 3.3 and h:R×X→Rbe the function given in Lemma 3.6. We define f:R×X→Rby f(t, z):=k(t, z)+h(t, z),(t, z)∈R×X. It follows immediately from the properties of hand kthat fis continuous (in both variables) and that, for each (t, z)∈R×X, the function [0,∞)r→ f(r+t, T(r+t, t)z)∈Ris non-increasing. Proof of ii). First, let z∈A(t)⊂A(t). Then h(t, z) = 0 and, by Theorem 3.3, we have that k(t, z)=0. Hencef(t, z)=0andA(t)⊂f−1 t(0).Conversely, if z∈Xis such that ft(z) = 0, we have that h(t, z)=0andk(t, z)=0,therefore z∈A(t), and by ii) in Theorem 3.3, z∈A(t). This proves that f−1 t(0) ⊂A(t). On the other hand, if z∈A∗(t)⊂A(t), we have that h(t, z) = 0, and thus, f(t, z)= k(t, z)+h(t, z)=k(t, z)=1andz∈A∗(t). It follows that A∗(t)⊂f−1(1) ∩A(t). Now, let z∈A(t) with f(t, z) = 1. It follows that h(t, z)=0andk(t, z)=1. Therefore z∈A∗(t) and the proof of ii)iscomplete. Proof of iii). Let (t, z)∈R×Xwith f(r+t, T (r+t, t)z)=f(t, z) for all r≥0. If z∈A(t)wehaveh(r+t, T (r+t, t)z) = 0 for all r≥0, and k(r+t, T(r+t, t)z)= k(t, z) for all r≥0, and, from iii) in Theorem 3.3, the conclusion follows. On the other hand, if z∈X\A(t), then lim r→∞dist(T(r+t, t)z,A∗(r+t)) = 0. Indeed, if that was not the case, by assumption lim r→∞dist(T(r+t, t)z,A(r+t)) = 0 it is easy to see that f(t, z) = lim r→∞ f(r+t, T(r+t, t)z) = lim r→∞ k(r+t, T(r+t, t)z) + lim r→∞ h(r+t, T(r+t, t)z)=0. (3.5) Thus, k(t, z)=0andz∈A(t), which is in contradiction with the fact that z∈ X\A(t). It follows that lim r→∞dist(T(r+t, t)z,A∗(r+t)) = 0. Now, using the same Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use GRADIENT PROCESSES UNDER PERTURBATIONS 5293 reasoning as in (3.5), we obtain f(t, z) = lim r→∞ f(r+t, T(r+t, t)z) = lim r→∞ k(r+t, T(r+t, t)z) + lim r→∞ h(r+t, T(r+t, t)z)=1+0=1, but this cannot be true since the fact that k(t, z)≥k(r+t, T (r+t, t)z) for all r≥0 implies 1 ≥k(t, z)≥lim r→∞k(r+t, T(r+t, t)z)=1,that is, k(t, z) = 1. Therefore z∈A∗(t),contradicting z∈X\A(t). This completes the proof.  Remark 3.8.We note that the function hplays a fundamental role in the proof of iii). Finally, analogously to Theorem 3.4 in [1], joining all of the results in this section, the following result holds. Theorem 3.9. Let {T(t, s):t≥s}be a gradient-like evolution process with respect to the isolated invariant families Ξ={Ξ1,··· ,Ξn}, reordered in the way described in Theorem 2.10, and with a pullback attractor {A(t):t∈R}. Let Aj(t)= j  i=1 Wu(Ξi)(t),for1≤j≤nand t∈R. Assume that the following conditions hold: (S1) There exists δ>0such that Oε(Aj(t)) ∩O ε(A∗ j(t)) = ∅, for all 1≤j≤n and t∈R. (S2) Given δ>0,thereisδ∈(0,δ)such that T(r+t, t)(A(t)∩O δ(Aj(t))) ⊂A(t+r)∩O δ(Aj(r+t)), for all r≥0,1≤j≤nand t∈R. Then, there exists a Lyapunov function V:R×X→Rwith properties i)-iii)of Definition 3.1, and such that V(t, Ξk(t)) = {k−1}, for all t∈Rand k=1,··· ,n. Moreover, if (S2) holds for neighborhoods of the Aj’s in Xand given (t, z)∈ R×Xand 1≤j≤n,lim r→∞dist(T(r+t, t)z,Aj(r+t)∪A∗ j(r+t)) = 0,then the Lyapunov function V:R×X→Rcan be chosen continuous in both variables. 4. Non–autonomous perturbations of autonomous evolution processes Now, we will exhibit a class of examples to which our previous abstract theory can be applied. This class consists of the non-autonomous perturbations of gradient-like semigroups. To begin with, we recall Theorem 3.9 in [5]: Theorem 4.1. Let Xbe a metric space and, for each η∈[0,1],let {Tη(t, s):t≥s} be an evolution process on Xwith a pullback attractor {Aη(t):t∈R}.Assume that the following conditions hold: (a) η∈[0,1]  t∈R Aη(t)is compact. (b){T0(t, s):t≥s}is an autonomous evolution process and {T0(t, 0) = S(t): t≥0}is a gradient-like semigroup relative to the set of equilibria E0= {y∗ 1,0,··· ,y∗ n,0}. Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5294 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A LANGA (c)For each η∈(0,1],{Tη(t, s):t≥s}possesses nisolated global solutions ξ∗ i,η :R→X,i=1,2,··· ,n,η∈(0,1],and lim η→0+ sup 1≤i≤n sup t∈R d(ξ∗ i,η(t),y∗ i,0)= 0. (d)For each compact set K⊂R+×X, lim η→0+sup s∈R sup (t,x)∈K d(Tη(t+s, s)x, T0(t+s, s)x)=0. (e)There exist μ>0and η1∈(0,1] such that, if ξη:R→Xis a bounded solution of {Tη(t, s):t≥s}with η≤η1so that there exists t0∈Rand some i∈ {1,··· ,n}with sup t≤t0 dist(ξη(t),ξ∗ i,η(R)) <μ(resp. sup t≥t0 dist(ξη(t),ξ∗ i,η(R)) <μ),then lim t→−∞d(ξη(t),ξ∗ i,η(t)) = 0 (resp. lim t→∞d(ξη(t),ξ∗ i,η(t)) = 0). Then, there exists η0∈(0,η 1]such that, for each η∈(0,η 0],{Tη(t, s):t≥s}is a non-autonomous gradient-like evolution process with respect to the disjoint set of isolated invariant families Eη={ξ∗ 1,η,··· ,ξ∗ n,η}. The following lemma will be crucial to establish our main results in this section (see Lemma 3.1 in [7] for the proof). Lemma 4.2. Let Xbe a metric space and {Tη(t, s):t≥s}η∈[0,1] be a family of evolution processes on Xwith pullback attractors {Aη(t):t∈R}η∈[0,1]. Assume that conditions (a),(b)and (d)of Theorem 4.1 are satisfied. Let {ηk}k∈Nbe a sequence in (0,∞)with ηk k→∞ −→ 0,{tk}k∈Nbe a sequence in R and {ξk}k∈Nbe a sequence in C(R,X)such that, for each k∈N,ξk:R→Xis a bounded global solution for {Tηk(t, s):t≥s}. Then, there is a subsequence of {ξk}k∈N, which we denote the same, and a global solution ξ:R→A0for {T0(t, 0) = S(t):t≥0}such that lim k→∞ sup |t|≤R d(ξk(t+tk),ξ(t)) = 0,for each R>0. Let Xbe a metric space and, for each η∈[0,1], {Tη(t, s):t≥s}an evolution process on Xwith a pullback attractor {Aη(t):t∈R}which satisfies conditions (a)-(e) in Theorem 4.1. We suppose that the set of equilibria, E0={y∗ 1,0,··· ,y∗ n,0}, of {T0(t, 0) = S(t):t≥0}is reordered so that it is a Morse-decomposition of the global attractor A0,as in Theorem 2.10 (see also Theorem 2.17 in [1] for the autonomous framework). For η∈[0,1], reordering the Eη={ξ∗ 1,η,··· ,ξ∗ n,η}so that sup 1≤i≤n sup t∈R d(ξ∗ i,η(t),y∗ i,0) η→0+ −→ 0, let A0,η(t):=∅,A1,η(t):={ξ∗ 1,η(t)}and (4.1) Aj,η(t):=Aj−1,η(t)∪Wu(ξ∗ j,η)(t),j=2,··· ,n, for each t∈R. Also, for each η∈[0,1], 1 ≤j≤nand t∈R,let (4.2) A∗ j,η(t):={z∈Aη(t):dist(Tη(r+t, t)z,Aj,η(r+t)) −→ 0asr→∞}. If we assume these constructions, we can prove the following results: Lemma 4.3. Consider a metric space Xand, for each η∈[0,1],{Tη(t, s):t≥s} an evolution process on Xwith a pullback attractor {Aη(t):t∈R}which satisfies conditions (a)-(e)in Theorem 4.1. Assume that E0={y∗ 1,0,··· ,y∗ n,0}is reordered Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use GRADIENT PROCESSES UNDER PERTURBATIONS 5295 so that it is a Morse-decomposition for {T0(t, 0) = S(t):t≥0}and that Eηis reordered in such a way that lim η→0+ sup 1≤i≤n sup t∈R d(ξ∗ i,η(t),y∗ i,0)=0. Then, the families of Aj,η and A∗ j,η defined by (4.1) and (4.2) behave upper semicontinuously as η→0+, that is, (4.3) lim η→0+sup t∈R dist(Aj,η(t),A j,0)=0 and (4.4) lim η→0+sup t∈R dist(A∗ j,η(t),A ∗ j,0)=0 for each j=1,2,··· ,n. Proof. Indeed, thanks to the fact that Aj,0and A∗ j,0are disjoint compact sets (see [1]) for all j,wecanpickε>0 such that, for all j=1,2,··· ,n, (4.5) Oε(Aj,0)∩O εA∗ j,0=∅. First, we will prove that lim η→0+max 1≤j≤nsup t∈R dist(Aj,η(t),A j,0)=0.If this is not true, there are δ∈(0,ε), j∈{1,2,··· ,n}, a sequence {ηk}k∈Nin (0,1] with ηk k→∞ −→ 0+, a sequence {τk}k∈Nin Rand a sequence {xk}k∈Nin Xsuch that, for each k∈N, xk∈Aj,ηk(τk)and (4.6) dist(xk,A j,0)≥δ. Now, by the definition of Aj,η(t), we can assume that there is i≤j, fixed, such that for each natural k, there is a global solution ξk:R→Xfor {Tηk(t, s):t≥s} with ξk(τk)=xkand (4.7) lim t→−∞ d(ξk(t),ξ∗ i,ηk(t)) = 0. On the other hand, by hypothesis (c),it follows the existence of ηδ>0such that, for all η≤ηδ, sup t∈R dist(ξ∗ i,η(t),A j,0)<δ 2. Consequently, by (4.6) and (4.7), for each kthere exists tk∈Rsuch that (4.8) dist(ξk(t),A j,0)<δfor all t<t k and (4.9) dist(ξk(tk),A j,0)=δ. Thus, we define  ξk:R→Xby  ξk(t):=ξk(t+tk),t∈R,and, by Lemma 4.2, we can assume that there is a global solution ξ:R→Xfor the semigroup {T0(t, 0) = S(t):t≥0}such that, for each R>0, lim k→∞ sup |t|≤R d( ξk(t),ξ(t)) = 0. By (4.8), we have that dist(ξ(t),A j,0)≤δfor all t≤0, so, by (4.5), we must have ξ(0) ∈Aj,0,but (4.9) means that dist(ξ(0),A j,0)=δ. This contradiction proves (4.3). Now, we prove that lim η→0+max 1≤j≤nsup t∈R dist(A∗ j,η(t),A ∗ j,0)=0.If it does not hold, there exist δ∈(0,ε), j=1,2,··· ,n, a sequence {ηk}k∈Nin (0,1] with ηk k→∞ −→ 0, Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5296 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A LANGA a sequence {τk}k∈Nin R, and a sequence {xk}k∈Nin Xsuch that, for each k∈N, xk∈A∗ j,ηk(τk)and (4.10) dist(xk,A ∗ j,0)≥δ. Now, by the definition of A∗ j,η(t) (and Theorem 4.1), we can assume that there is i≥j+1,fixed, such that for each k∈N, there is a global solution ξk:R→X for {Tηk(t, s):t≥s}with ξk(τk)=xkand (4.11) lim t→∞ d(ξk(t),ξ∗ i,ηk(t)) = 0. On the other hand, hyphotesis (c) implies the existence of η∗ δ>0 such that, for all η≤η∗ δ, sup t∈R dist(ξ∗ i,η(t),A ∗ j,0)<δ 2, whence, by (4.10) and (4.11), for each k∈N,thereexiststk∈Rsuch that (4.12) dist(ξk(t),A ∗ j,0)<δfor all t>t k and (4.13) dist(ξk(tk),A ∗ j,0)=δ. Thus, we define  ξk:R→Xby  ξk(t):=ξk(t+tk),t∈R,and, by Lemma 4.2, we can assume that there is a global solution ξ:R→Xfor the semigroup {T0(t, 0) = S(t):t≥0}such that, for each R>0, lim k→∞ sup |t|≤R d( ξk(t),ξ(t)) = 0. From (4.12) we have that dist(ξ(t),A ∗ j,0)≤δfor all t>0 and, as a consequence of (4.5), we must have ξ(0) ∈A∗ j,0. On the other hand, (4.13) implies that dist(ξ(0),A ∗ j,0)=δ. This is a contradiction and (4.4) is proved, completing the proofofthelemma.  Lemma 4.4. Let Xbe a metric space and, for each η∈[0,1],{Tη(t, s):t≥s} an evolution process on Xwith a pullback attractor {Aη(t):t∈R}which satisfies conditions (a)-(e)in Theorem 4.1. Assume that E0={y∗ 1,0,··· ,y∗ n,0}is reordered in such a way that it is a Morse-decomposition for {T0(t, 0) = S(t):t≥0}and that Eηis reordered in such a way that lim η→0+ sup 1≤i≤n sup t∈R d(ξ∗ i,η(t),y∗ i,0)=0. If the families of local unstable sets {Wu η,ρ(ξ∗ j)(t)}η∈[0,1] behave lower semicontinuously, then the families of Aj,η behave lower semicontinuously as η→0+; that is, lim η→0+sup t∈R dist(Aj,0,A j,η(t)) = 0. If the families of local stable sets {Ws η,ρ(ξ∗ j)(t)∩Aη(t)}η∈[0,1] behave lower semicontinuously and Tη(t, s):A(s)→A(t)is injective for each t≥s,η∈[0,1], then lim η→0+sup t∈R dist(A∗ j,0,A j,η(t)∗)=0. Proof. It follows analogously to the proof of Theorem 2.13 in [6].  Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use GRADIENT PROCESSES UNDER PERTURBATIONS 5297 Next we exhibit an example (pictorial) for which we have lower semicontinuity of repellers but for which we do not have structural stability. The purpose of this example is to show that the systems which satisfy the hypothesis imposed in the last part of Lemma 4.4 are a larger class than that of the structurally stable systems. Below, the figure labelled 01.a corresponds to the perturbed attractor-repeller pair (Ai,η,A ∗ i,η) with Ai,η pictured in black and A∗ i,η pictured in red, 0 ≤i≤5. The figure labelled 01.b corresponds to the limiting attractor-repeller (Ai,A ∗ i)pair with Aipictured in black and A∗ ipictured in red, 1 ≤i≤5. Of course A5,η (A∗ 0,η) and A5(A∗ 0) correspond to the global attractor, whereas A∗ 5,η (A0,η)andA∗ 5(A0) correspond to the empty set. Color can be seen online. q e5 qe2 q e1q e3 q e4 -  i  N  k Y Y   R ) K  KKI)   R R - Figure 00.a q e5 qe2 q e1q e3 q e4 -  i  N   k Y Y   R ) K  K KI)   R R Figure 00.b e1 q q e5 q e4 q e3qe2 - i  N  k Y Y   R ) K  KKI) ^   R R Figure 01.a e1 q q e5 q e4 q e3qe2 - i  N   k Y Y   R ) K  K KI) ^   R R Figure 01.b qe2 q e1 q e5 q e4 q e3 i  N  k Y Y   R ) K  KKI)   R R Figure 02.a qe2 q e1 q e5 q e4 q e3 Y  N   k Y Y   R ) K  K KI)   R R Figure 02.b qe2 q e1q e3 - q e5 q e4 Figure 03.a qe2 q e1q e3 -  q e5 q e4 Figure 03.b Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5298 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A LANGA qe2 q e1q e3 q e4 - R ) K  KKI)   R R - q e5 Figure 04.a qe2 q e1q e3 q e4 - R ) K  K KI)   R R q e5 Figure 04.b q e5 qe2 q e1q e3 q e4 -  i  N  k Y Y   R ) K  KKI)   R R - Figure 05.a q e5 qe2 q e1q e3 q e4 -  i  N   k Y Y   R ) K  K KI)   R R Figure 05.b Proposition 4.5. Consider a metric space Xand, for each η∈[0,1],{Tη(t, s): t≥s}an evolution process on Xwith a pullback attractor {Aη(t):t∈R}which satisfies conditions (a)-(e)in Theorem 4.1. Assume that E0={y∗ 1,0,··· ,y∗ n,0}is reordered so that it is a Morse-decomposition for {T0(t, 0) = S(t):t≥0}and that Eηis reordered in such a way that lim η→0+ sup 1≤i≤n sup t∈R d(ξ∗ i,η(t),y∗ i,0)=0. Then, there are ε>0and η0>0such that Oε(Aj,η(t)) ∩O ε⎛ ⎝ n  i=j+1 {ξ∗ i,η(t)}⎞ ⎠=∅ for each j=1,2,··· ,n−1,t∈Rand η∈[0,η 0]. Proof. Indeed, by Theorem 2.17 in [1], we can take ε>0 such that, for each j=1,2,··· ,n−1andt∈R, (4.14) O2ε(Aj,0)∩O 2ε⎛ ⎝ n  i=j+1 {y∗ i,0}⎞ ⎠=∅. We note that the proposition will be accomplished if we show the following stronger conditions: For each δ∈(0,ε]thereisηδ>0 such that (4.15) Aj,η(t)⊂O δ(Aj,0) and (4.16) n  i=j+1 {ξ∗ i,η(t)}⊂O δ⎛ ⎝ n  i=j+1 {y∗ i,0}⎞ ⎠ for all t∈R,j=1,2,··· ,n−1andη∈[0,η δ]. Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use GRADIENT PROCESSES UNDER PERTURBATIONS 5299 Clearly (4.16) follows from hyphotesis (c), and (4.15) is a direct consequence of (4.3) in Lemma 4.3. This completes the proof of the proposition.  Using Proposition 4.5 we can show that the sets Aj,η, defined above, are all local attractors for all suitable η. This is proved in the following proposition. Proposition 4.6. Let Xbe a metric space and, for each η∈[0,1],{Tη(t, s):t≥s} an evolution process on Xwith a pullback attractor {Aη(t):t∈R}which satisfies conditions (a)-(e)from Theorem 4.1. Assume that E0={y∗ 1,0,··· ,y∗ n,0}is reordered so that it is a Morse-decomposition for {T0(t, 0) = S(t):t≥0}and that Eηis reordered in such a way that lim η→0+ sup 1≤i≤n sup t∈R d(ξ∗ i,η(t),y∗ i,0)=0. Then, there exists η0>0such that the invariant family Aj,η := {Aj,η(t): t∈R}is a local attractor for the evolution process {Tη(t, s):t≥s},foreach j=1,2,··· ,n and η∈(0,η 0]. Proof. Indeed, by Lemma 4.3 and the proof of Proposition 4.5, we can choose ε>0, δ∈(0,ε]andη0>0 such that Aj,η(t)⊂O δ(Aj,0) and n  i=j+1 {ξ∗ i,η(t)}⊂O δ⎛ ⎝ n  i=j+1 {y∗ i,0}⎞ ⎠ for each j=1,2,··· ,n−1, t∈Rand η∈[0,η 0],where ε>0 satisfies O2ε(Aj,0)∩O 2ε⎛ ⎝ n  i=j+1 {y∗ i,0}⎞ ⎠=∅. Thus, if Γj,η is the trace of the family Aj,η, it follows that Γj,η ⊂O ε(Aj,0)for j=1,2,··· ,n−1andη∈[0,η 0], and therefore (4.17) Oε(Γj,η)∩O ε⎛ ⎝ n  i=j+1 {ξ∗ i,η(t)}⎞ ⎠=∅ for each j=1,2,··· ,n−1, t∈Rand η∈[0,η 0]. Now, for a fixed η∈[0,η 0], let ξη:R→Xbe a global solution for {Tη(t, s): t≥s}with ξη(t)∈O ε(Γj,η) for all t∈Rand, recalling that {Tη(t, s):t≥s}is a gradient-like evolution process with respect to Eη={ξ∗ 1,η,··· ,ξ∗ n,η}for η∈[0,η 0] (by Theorem 4.1),let i∈{1,2,··· ,n}such that lim t→−∞ d(ξη(t),ξ∗ i,η(t)) = 0. By (4.17), we must have i≤j,soξη(t)∈Wu(ξ∗ i,η)(t)⊂Aj,η(t) for all t∈R,which tells us that the family Aj,η ={Aj,η(t):t∈R}is invariant and isolated for each j=1,2,··· ,n and η∈[0,η 0]. On the other hand, let ξη:R→Xbe a global solution for {Tη(t, s):t≥s}with η∈[0,η 0] and lim t→−∞dist(ξη(t),A j,η(t)) = 0.If t0is such that dist(ξη(t),A j,η(t)) < εfor t≤t0,as Oε(Aj,η(t)) ∩O ε⎛ ⎝ n  i=j+1 {ξ∗ i,η(t)}⎞ ⎠=∅for all t∈R,using the above reasoning, we must have ξη(t)∈Aj,η(t) for all t∈R, which shows that Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5300 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A LANGA Wu(Aj,η)(t)⊂Aj,η(t)foreacht∈R. Now, the inclusion Aj,η(t)⊂Wu(Aj,η)(t), for each t∈R, holds thanks to the invariance of the family Aj,η,showing that Wu(Aj,η)(t)=Aj,η(t) for all t∈R, and completing the proof of the proposition.  The separation property between the local attractor and its repeller is also satisfied for the class of the non-autonomous perturbation of a gradient-like semigroup. This is what we will show in the next proposition. Proposition 4.7. Let Xbe a metric space and, for each η∈[0,1],{Tη(t, s):t≥s} an evolution process on Xwith a pullback attractor {Aη(t):t∈R}which satisfies conditions (a)-(e)in Theorem 4.1. Assume that E0={y∗ 1,0,··· ,y∗ n,0}is reordered so that it is a Morse-decomposition for {T0(t, 0) : t≥0}and that Eηis reordered in such a way that lim η→0+ sup 1≤i≤n sup t∈R d(ξ∗ i,η(t),y∗ i,0)=0. Consider the families of local attractors and corresponding repellers defined in (4.1) and (4.2). Then, there exist ε>0and η0>0such that, for all t∈R,j=1,2,··· ,n and η∈[0,η 0],we have Oε(Aj,η(t)) ∩O εA∗ j,η(t)=∅. Proof. Indeed, thanks to the fact that Aj,0and A∗ j,0are disjoint compact sets (see [1]) for all j,wecanchooseε>0 such that, for all j=1,2,··· ,n, (4.18) O2ε(Aj,0)∩O 2εA∗ j,0=∅. By Lemma 4.3, for each δ∈(0,ε], there is ηδ>0 such that Aj,η(t)⊂O δ(Aj,0) and (4.19) A∗ j,η(t)⊂O δ(A∗ j,0), for every η∈[0,η∗ δ], t∈Rand j=1,2,··· ,n. The result now follows.  Theorem 4.8. Let Xbe a metric space and, for each η∈[0,1],{Tη(t, s):t≥s} a nonlinear evolution process on Xwith a pullback attractor {Aη(t):t∈R}which satisfies conditions (a)-(e)of Theorem 4.1. Assume that E0={y∗ 1,0,··· ,y∗ n,0}is reordered so that it is a Morse-decomposition for {T0(t, 0) : t≥0}and that Eη is reordered in such a way that lim η→0+ sup 1≤i≤n sup t∈R d(ξ∗ i,η(t),y∗ i,0)=0. Also consider the families of local attractors Aj,η ={Aj,η(t):t∈R}defined in (4.1) and their associated repellers A∗ j,η ={A∗ j,η(t):t∈R}given by (4.2). Then, there exists η0>0such that for all η∈(0,η 0],t∈Rand j=1,2,··· ,n, Aj,η(t)∩A∗ j−1,η(t)={ξ∗ j,η(t)}. That is, the set Eη={ξ∗ 1,η,··· ,ξ∗ n,η}determines a Morse decomposition of the pullback attractor {Aη(t):t∈R}for {Tη(t, s):t≥s}for each η∈(0,η 0].In particular, such Morse-decomposition behaves continuously. Proof. Indeed,itisclearthatξ∗ j,η(t)∈Wu(ξ∗ j,η)(t)⊂Aj,η(t)foreach1≤j≤n, t∈Rand η∈[0,1]. Now, using Proposition 4.5, let ε>0andη0>0 be such that, for all t∈R,2≤j≤nand η∈[0,η 0],we have (4.20) O2ε(Aj−1,η(t)) ∩O 2ε⎛ ⎝ n  i=j {ξ∗ i,η(t)}⎞ ⎠=∅. Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use GRADIENT PROCESSES UNDER PERTURBATIONS 5301 From (4.20), dist(Tη(r+t, t)ξ∗ j,η(t),A j−1,η(t+r)) = dist(ξ∗ j,η(t+r),A j−1,η(t+r)) ≥ εfor all r≥0. Hence ξ∗ j,η(t)∈A∗ j−1,η(t) for all t∈R. Therefore, {ξ∗ j,η(t)}⊂ Aj,η(t)∩A∗ j−1,η(t) for every η∈(0,η 0], t∈Rand 1 ≤j≤n. Conversely, we note that, by Lemma 4.3, given δ∈(0,ε]thereexistsηδ>0such that for all η∈(0,η δ],t∈Rand 1 ≤j≤nwe have that Aj,η(t)∩A∗ j−1,η(t)⊂Oδ 2(Aj,0)∩Oδ 2A∗ j−1,0. Now, if the statement Aj,η(t)∩A∗ j−1,η(t)⊂{ξ∗ j,η(t)}for all t∈R,1≤j≤nand for all η>0 suitably small does not hold, there are j∈{1,··· ,n}, a sequence {ηk}k∈Nin (0,1] with ηk→0+, a sequence {tk}k∈Nin R, and a sequence {zk}k∈Nin Xwith zk∈Aj,ηk(tk)∩ A∗ j−1,ηk(tk)\{ξ∗ j,ηk(tk)}for all k∈N. Hence, for each k∈N, there exists a global solution ξk:R→Xfor {Tηk(t, s):t≥s}with ξk(tk)=zk, and we can assume that there are i≤jand l≥j, fixed, with lim t→−∞d(ξk(t),ξ∗ i,ηk(t)) = 0 and lim t→∞ d(ξk(t),ξ∗ l,ηk(t)) = 0. Moreover, ξk(t)∈Oδ 2(Aj,0)∩Oδ 2A∗ j−1,0for each k∈Nand t∈R.Therefore, if we choose, for suitably large k,τkand σkwith d(ξk(t),y∗ i,0)<δ 2for t≤τkand d(ξk(t),y∗ l,0)<δ 2for t≥σk, we must have y∗ i,0,y ∗ l,0∈O δ(Aj,0)∩O δA∗ j−1,0. Then, by (4.20), i=l=j; that is, y∗ i,0=y∗ l,0=y∗ j,0.Thusξ∗ i,ηk=ξ∗ l,ηk=ξ∗ j,ηk, which means that the solution ξk:R→Xis a homoclinic solution (because zk=ξ∗ j,ηk(tk) for every k), contradicting the fact that {Tηk(t, s):t≥s}is gradient-like with respect to Eηk={ξ∗ 1,ηk,··· ,ξ∗ n,ηk},completing the proof.  Definition 4.9. We say that the family {Tη(t, s):t≥s}η∈[0,1] of evolution processes in a metric space Xis collectively asymptotically compact if {Tηk(tk+ τk,τ k)xk}k∈Nis relatively compact whenever {ηk}k∈Nis a sequence in (0,1], {xk}k∈N is a bounded sequence in X,{tk}k∈Nis a sequence in R+and {τk}k∈Nis a sequence in Rwith ηk k→∞ −→ 0andtk k→∞ −→ ∞ . Lemma 4.10. Let {Tη(t, s):t≥s}η∈[0,1] be a family of evolution processes in a metric space Xwith a corresponding family of pullback attractors {Aη}η∈[0,1] := {Aη(t):t∈R}η∈[0,1], which satisfies conditions (a)and (d)in Theorem 4.1.Assume that the family {Tη(t, s):t≥s}η∈[0,1] is collectively asymptotically compact and that T0(t, s)=S(t−s)for all t≥s,where{S(t):t≥0}is a semigroup. Let {Aη}η∈[0,1] := {Aη(t):t∈R}η∈[0,1] be such that Aη(t)⊂Aη(t)for every t∈ Rwith A0a local attractor for the semigroup {S(t):t≥0}(that is, ω(Oε(A0)) = A0for some ε>0). If {Aη}η∈[0,1] is continuous at η=0,that is, lim η→0+sup t∈R dH(Aη(t),A 0)=0, then, given δ∈(0,ε),thereexistδ∈(0,δ)and ηδ>0such that, for all η∈[0,η δ], it holds that Tη(t, s)(Oδ(Aη(s))) ⊂O δ(Aη(t)) whenever t≥s. Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5308 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A LANGA for all η∈[0,η 3], s∈Rand w∈O σ3(z). So, hη(s, w)= sup 0≤t≤t0 dist(Tη(t+s, s)w,Aη(t+s)) for η∈[0,η 3], s∈Rand w∈O σ3(z). Note that, for all w∈X,η∈[0,1], s∈Rand t≥0, we have that |dist(Tη(t+s, s)w,Aη(t+s))−dist(S(t)w,A0)|≤dH(Aη(t+s),A0) +d(Tη(t+s, s)w,S(t)w), so that, for all η∈[0,η 3], sup s∈R sup w∈Oσ3(z) |hη(s, w)−h0(w)|≤sup s∈R dH(Aη(s),A0) +sup s∈R sup w∈Oσ3(z) sup 0≤t≤t0 d(Tη(t+s, s)w,S(t)w), and so, it is easy to see that, given ε>0, there exist σ∈(0,σ 3]andη4∈(0,η 3] such that sup s∈R sup w∈Oσ(z) |hη(s, w)−h0(w)|≤εfor all η∈[0,η 4]. Case 2: dist(z,A0)=0; that is, z∈A0.From Lemma 4.10, given ε>0, there are ε∈(0,ε 2)andη0∈(0,1] such that (4.38) Tη(t, s)(Oε(Aη(s))) ⊂O ε 2(Aη(t)) for each t≥sand η∈[0,η 0]. Also, since {Aη}η∈[0,1] is lower semicontinuous at η=0,wecanchooseη1∈ (0,η 0] such that A0⊂O ε 2 (Aη(s)) whenever η∈[0,η 1]ands∈R. It follows that Oε 2 (A0)⊂O ε(Aη(s)) for all η∈[0,η 1]ands∈R. From this and (4.38) we have that Tη(t, s)(Oε 2 (A0)) ⊂O ε 2(Aη(t)) if η∈[0,η 1]andt≥s. Consequently, hη(s, w)=sup t≥0 dist(Tη(t+s, s)w,Aη(t+s)) ≤ε 2for all η∈[0,η 1] and w∈O ε 2 (A0), so that sup s∈R sup w∈O ε 2 (A0) |hη(s, w)−h0(w)|≤εfor all η∈[0,η 1]. In these conditions, given ε>0andz∈X,thereareσ=σ(ε, z)>0and η=η(ε, z)>0 such that sup s∈R sup w∈Oσ(z) |hη(s, w)−h0(w)|≤εfor all η∈[0,η]. We now obtain the convergence lim η→0+sup t∈R sup z∈K |hη(t, z)−h0(z)|= 0 by an argument similar to that of Step 2. This completes the proof.  5. Applications In this section we consider some applications of the abstract results in the previous sections and conclude that some non-autonomous evolutions processes possess a Morse-decomposition and, therefore, a Lyapunov function. Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use GRADIENT PROCESSES UNDER PERTURBATIONS 5309 Example 5.1. Consider the initial boundary value problem (5.1) ⎧ ⎪ ⎨ ⎪ ⎩ ut=uxx +λfη(t, u),x∈(0,π),t>τ, ux(0,t)=ux(π,t)=0,t>τ, u(·,τ)=φ0∈H1 0(0,π), where λ∈[0,∞),η∈[0,1] and fη∈C2(R2,R) satisfying: 1) There exists M>0 such that fη:R2→Ris such that fη(t, u)u<0 for all (t, u)∈R2with |u|>M. Suppose also that f0(t, u)=f0(u) for all t∈R. 2) For each r>0, lim η→0+ sup t∈R sup |u|≤r {|fη(t, u)−f0(u)|+|(fη)u(t, u)−f 0(u)|+|(fη)uu(t, u)−f 0(u)|} =0. It is not difficult to see that, under these assumptions, problem (5.1) possesses a pullback attractor {Aη(t):t∈R}. If (5.1) with η= 0 has a finite number of equilibria, all of them hyperbolic, the hypotheses of Theorem 4.12 are clearly satisfied and there exists η0>0 such that {Aη(t):t∈R}has a Morse-decomposition and a continuous (uniformly in R×K for each compact subset Kof H1 0(0,π)) Lyapunov function Vη:R×H1 0(0,π)→R (in the sense of Definition 3.1). Example 5.2. Consider a general cascade system of the form (5.2) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ˙u1=f1(u1), ˙u2=f2(u1,u 2), . . .. . .. . . ˙un=fn(u1,··· ,u n), where fj:Rj→R,1≤j≤n,areC1functions. Assume that there is ξ>0 such that fj(u1,··· ,u j)uj<0 for all (u1,··· ,u j)∈Rjsuch that |uj|≥ξand for all 1 ≤j≤n. Assume that all equilibria of (5.2) are hyperbolic. It follows from the results in [1] that the semigroup associated to (5.2) is gradient-like (therefore gradient). Now we consider a small non-autonomous perturbation of the above problem; that is, (5.3) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ˙u1=f1(u1)+g1(t, u1,··· ,u n), ˙u2=f2(u1,u 2)+g2(t, u1,··· ,u n), . . .. . .. . . ˙un=fn(u1,··· ,u n)+gn(t, u1,··· ,u n). If gj(t, ·):Rn→Ris a C1(Rn,R) function with the property that sup t∈R sup (u1,··· ,un)∈Rn|gj(t, u1,··· ,u n)|+ n  i=1  ∂gj ∂ui (t, u1,··· ,u n) is suitably small, it follows from the results in [5] that the evolution process {Tg(t, s) :t≥s}associated to (5.3) is gradient-like and, as a consequence of the results in Section 4, the pullback attractor of {Tg(t, s):t≥s}has a Morse-decomposition and a Lyapunov function. One can easily see that cascade systems of semilinear parabolic problems can also be considered. Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5310 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A LANGA 6. Conclusive remarks In this paper we have presented a theory for the Morse-decomposition of pullback attractors of evolution processes suitable for the class of non-autonomous perturbations of semigroups. Our approach extends the notions and main results of the associated autonomous theory in [8, 20, 1]. In particular, it is shown that a nonautonomous perturbation of a gradient semigroup is a gradient non-autonomous evolution process, in the sense that it has a (time-dependent) continuous (in the whole phase space) Lyapunov function. Moreover, this Lyapunov function behaves continuously with respect to the underlying perturbation. Concerning the autonomous theory, two main remarks are well worth being emphasized: a) According to Definition 1.5, a local attractor must be an isolated invariant family. This property allows us prove that our definition of local attractor extends the notion of local attractor of the autonomous case (see Theorem 1.6). Again, a hypothesis of uniform separation on the attractor-repeller pair (automatic in the autonomous case) ensures that the associated repeller is closed. In Remark 1.4 we discuss some properties of isolated invariant sets in the autonomous framework, among them, a dichotomous behavior of solutions around them (topological hyperbolicity). b) In [1], restricted to the autonomous case, it is shown that gradient systems are stable under perturbation by showing that a gradient semigroup (with a Lyapunov functional) is gradient-like in the sense of Definition 2.5 (or Definition 2.8 in [1]) and that a gradient-like semigroup is gradient (has a Lyapunov function). Note that, in the non-autonomous framework, we do not have that the existence of a Lyapunov functional implies that the evolution process is gradient-like. This fact deserves further attention. In a non-autonomous framework, Rasmussen [19] (see also Kloeden and Rasmussen [12]) adopts a different approach towards a Morse-decomposition for evolution processes, specially adapted for finite dimensional phase spaces. In particular, if {T(t, s): t⩾s}is an invertible non-linear evolution process with pullback attractor {A(t): t∈R}, a compact invariant family {A(t): t∈R}with A(t)⊆A(t) for every t∈Ris defined as a local pullback attractor if there exists η>0 such that lim s→−∞ distH(T(t, s)Oη(A(s)),A(t)) = 0,for all t∈R, where Oη(A(s)) . ={x∈A(s): dist H(x, A(s)) <η}, for all s∈R. Observe that a local pullback attractor is a pullback attractor with respect to the attraction universe Dcontaining all the families {Oζ(A(t)) : t∈R}where ζ∈(0,η]. On the other hand, a compact invariant family A∗={A∗(t): t∈R} with A∗(t)⊂A(t) for every t∈Ris called a local repeller if there exists η>0such that lim s→−∞ distH(T(s, t)Oη(A∗(t)),R(s)) = 0,for all t∈R. This is not clear, and we think that it would need deeper research if similar results to ours in this paper could be developed from this approach of pullback attraction for the local attractors. The same kind of situation is found in the case of random dynamical systems (see Liu [14]) related to some stochastic differential equations. Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use GRADIENT PROCESSES UNDER PERTURBATIONS 5311 Acknowledgement The authors wish to express their gratitude to the anonymous referees whose comments and suggestions have greatly improved the readability of the article. References [1]E.R.Arag˜ao-Costa, T. Caraballo, A. N. Carvalho and J.A. Langa, Stability of gradient semigroups under perturbation, Nonlinearity 24 (2011), 2099-2117. MR2805595 [2] E.R.Arag˜ao-Costa, T. Caraballo, A. N. 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A LANGA Instituto de Ciˆ encias Matem´ aticas e de Computac¸˜ ao, Universidade de S˜ ao Paulo- Campus de S˜ ao Carlos, Caixa Postal 668, 13560-970 S˜ ao Carlos SP, Brazil E-mail address:[email protected] Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla Spain E-mail address:[email protected] Instituto de Ciˆ encias Matem´ aticas e de Computac¸˜ ao, Universidade de S˜ ao Paulo- Campus de S˜ ao Carlos, Caixa Postal 668, 13560-970 S˜ ao Carlos SP, Brazil E-mail address:[email protected] Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla Spain E-mail address:[email protected] Licensed to Universidad de Sevilla. Prepared on Wed Feb 25 06:38:26 EST 2015 for download from IP 150.214.182.201. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use