Continuity of Lyapunov Functions and of Energy Level for a Generalized Gradient Semigroup

CONTINUITY OF LYAPUNOV FUNCTIONS AND OF ENERGY LEVEL FOR A GENERALIZED GRADIENT SEMIGROUP E. R. ARAG˜ AO-COSTA1, T. CARABALLO2, A. N. CARVALHO3, AND J. A. LANGA4 Abstract. The global attractor of a gradient-like semigroup has a Morse decomposition. Associated to this Morse decomposition there is a Lyapunov function (differentiable along solutions)-defined on the whole phase spacewhich proves relevant information on the structure of the attractor. In this paper we prove the continuity of these Lyapunov functions under perturbation. On the other hand, the attractor of a gradient-like semigroup also has an energy level decomposition which is again a Morse decomposition but with a total order between any two components. We claim that, from a dynamical point of view, this is the optimal decomposition of a global attractor; that is, if we start from the finest Morse decomposition, the energy level decomposition is the coarsest Morse decomposition that still produces a Lyapunov function which gives the same information about the structure of the attractor. We also establish sufficient conditions which ensure the stability of this kind of decomposition under perturbation. In particular, if connections between different isolated invariant sets inside the attractor remain under perturbation, we show the continuity of the energy level Morse decomposition. The class of Morse-Smale systems illustrates our results. 1. Introduction Qualitative properties of infinite-dimensional dynamical systems has been receiving very much attention throughout the last four decades (see, for instance, [5], [9], [14] or [2]). The analysis of compact attracting invariant sets has developed a profound area of research, providing crucial information for an increasing number of models for phenomena from Physics, Biology, Economics, Engineering and others. The asymptotic behaviour of a dissipative system can be described by a study of its associated global attractor. Moreover, a careful study of the geometrical structure -and its stability under perturbations- of the global attractor leads to the understanding of its internal dynamics, which, essentially, describes the long time behaviour of the whole system. 1Partially supported by CAPES/DGU 267/2008 and FAPESP 2008/50248-0, Brazil. 2Partially supported by Ministerio de Ciencia e Innovaci´on grant # MTM2008-00088, PBH2006-0003-PC, and Junta de Andaluc´ıa grants # P07-FQM-02468, # FQM314 and HF2008-0039, Spain. 3Partially supported by CNPq 305447/2005-0 and 451761/2008-1, CAPES/DGU 267/2008 and FAPESP 2008/53094-4, Brazil, Ministerio de Ciencia e Innovaci´on grant # MTM2008-00088, Spain, and Junta de Andaluc´ıa grant # P07-FQM-02468. 4Partially supported by Ministerio de Ciencia e Innovaci´on grants # MTM2008-00088, # PBH2006-0003- PC, and Junta de Andaluc´ıa grants # P07-FQM-02468, # FQM314 and HF2008-0039, Spain. 1 2 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA The most general result in this line follows from [4], which describes any flow on a compact metric space as a decomposition of chain recurrent isolated invariant sets and connections between them. In the terminology of [4], this is called a Morse decomposition of a compact invariant set (see Definition 2.10 below), and has been considered in different frameworks, as in the case of flows ([4]) and semiflows on compact spaces ([13]), or even compact and non-compact topological spaces ([8, 11, 12]). Recently, it has been introduced in [3] the so-called gradient-like semigroups with respect to a disjoint family of isolated invariant sets Ξ= (Ξ1,· · · ,Ξn) on the global attractor (see Definition 2.8 below) in Banach spaces, as an intermediate concept between gradient semigroups (i.e., those possessing a Lyapunov function) and semigroups possessing a gradient-like attractor (that is, an attractor that is characterized as the union of the unstable sets of associated isolated invariant sets). In [1], given a gradient-like semigroup in a general metric space, we construct a differentiable (along solutions) generalized Lyapunov function proving that gradient-like semigroups are in fact gradient semigroups. This function is not only constant on each isolated invariant set as in the classical theory of [4], but it also detects the points in the phase space with orbits having a single value of this function, a crucial property of Lyapunov functions (see, for instance, [5]). Indeed, we will say that a semigroup {T(t) : t≥0}with a global attractor Aand a disjoint family of isolated invariant sets Ξ={Ξ1,· · · ,Ξn}is a generalized gradient semigroup with respect to Ξif there exists a continuous function V:X→Rsuch that, V is constant in each in each Ξi, 1 ≤i≤n, [0,∞)3t7→ V(T(t)x)∈Ris decreasing for each x∈X, and V(T(t)x) = V(x) for all t≥0 if and only if x∈Sn i=1 Ξi. For the construction of the Lyapunov function, it is proved in [1] that the disjoint family of isolated invariant sets of a gradient-like semigroup on a general metric space can be reordered in such a way that it becomes a Morse decomposition for the global attractor. A refinement of the results from [4] leads to define a generalized Lyapunov function, not only on the attractor but on the whole phase space. In addition, the Lyapunov function V:X→Rof a generalized gradient-like semigroup can be chosen in such a way that V(Ξj) = j. Moreover, as gradient-like semigroups are stable under perturbation (see [3]), we conclude that gradient semigroups are stable under perturbation. In other words, the existence of a continuous Lyapunov function is robust under perturbation. In this paper, we are able to go further in this direction, i.e., we provide conditions for which not only a perturbation of a gradient semigroup is still gradient, but also the associated Lyapunov functions move continuously under the perturbation. A careful study of the upper and lower semicontinuity of local attractors and repellers will be crucial in our argument. On the other hand, observe that any Morse decomposition Ξ= (Ξ1,· · · ,Ξn) of a compact invariant set Aleads to a partial order among the isolated invariant sets Ξi; that is, we can define an order between two isolated invariant sets Ξiand Ξjif there is a chain of global solutions {ξ`,1≤`≤j−i}, with limt→∞ ξ`(t) = Ξi+`−1and limt→−∞ ξ`(t) = Ξi+`, ENERGY LEVEL DECOMPOSITION 3 1≤`≤j−i. This defines a partial order and some of the isolated invariant sets in Ξmay not be comparable. In Section 4 we rewrite and expand the construction in [1] of a new Morse decomposition of the attractor for a generalized gradient-like semigroup which improves the construction and dynamical properties of its associated Lyapunov function. Indeed, we show that, given any generalized gradient-like semigroup with respect to the disjoint family of isolated invariant sets Ξ= (Ξ1,· · · ,Ξn), there exists another Morse decomposition given by the so-called energy levels N= (N1,N2,· · · ,Np), p≤n, which can be totally ordered by the flow. Each of the levels Ni, 1 ≤i≤pis made of a finite union of the isolated invariant sets in Ξand Nis totally ordered. The associated Lyapunov function takes different values in any two different sets of Nand any two elements of Ξwhich are contained in the same element of N(same energy level) are not connected. Because of this energy level decomposition can be made from any gradient-like semigroup (i.e., for any Morse decomposition with a finite number of components), when we start form the finest Morse decomposition of an invariant set in the sense of [12], we claim that our new dynamical decomposition is optimal, since its associated Lyapunov function is the simplest one in order to describe connected isolated invariant sets inside the global attractor. We recall that, given a Morse decomposition of an attractor, it can be continuous under perturbation even if the connections between sets are destroyed (see figures in Section 3). This is saying that when we describe the geometric structure of the attractor using the associated isolated invariant subsets it may, under perturbation, change drastically the way these isolated invariant subsets are connected. In Section 5 we prove that, if connections are kept under perturbation, then the energy level decomposition is stable under perturbation. There exists a general class of semigroups satisfying this last property, being Morse-Smale systems ([7], [6]) the prototype of them. 2. Morse decomposition of global attractors for generalized gradient-like semigroups Let Xbe a metric space with metric d:X×X→R+, where R+= [0,∞). Given a subset A⊂X, the −neighborhood of Ais the set O(A) = {x∈X:d(x, a)<  for some a∈A} Definition 2.1. A family of mappings {T(t) : t≥0}is a semigroup in Xif •T(0) = IX, with IXbeing the identity map in X, •T(t+s) = T(t)T(s), for all t, s ∈R+and •R+×X3(t, x)7→ T(t)x∈Xis continuous. The notion of invariance plays a fundamental role in the study of the asymptotic behavior of semigroups Definition 2.2. A subset Aof Xis said invariant under the action semigroup {T(t) : t≥0} if T(t)A=Afor all t≥0. 4 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA Given A, B ⊂X, the Hausdorff semidistance between Aand Bis given by dist(A, B) = sup a∈A inf b∈Bd(a, b), and the Hausdorff distance by dH(A, B) := dist (A, B) + dist (B, A). For any subsets A, B and Cin Xit holds dist (A, C)≤dist (A, B) + dist (B, C). Definition 2.3. Given two subsets A, B of Xwe say that Aattracts Bunder the action of the semigroup {T(t) : t≥0}if dist(T(t)B, A)t→∞ −→ 0and we say that Aabsorbs Bunder the action of {T(t) : t≥0}if there is a tB>0such that T(t)B⊂Afor all t≥tB. With this we are in condition to define global attractors. Definition 2.4. A subset Aof Xis a global attractor for a semigroup {T(t) : t≥0}if it is compact, invariant under the action of {T(t) : t≥0}and for every bounded subset Bof Xwe have that Aattracts Bunder the action of {T(t) : t≥0}. 2.1. Gradient-like semigroups and Morse decomposition of attractors. Next we seek to introduce the notion of generalized gradient-like semigroups (see [3]). To that end we first need the definition of isolated invariant set. Definition 2.5. Let {T(t) : t≥0}be a semigroup. We say that an invariant set Ξ⊂Xfor the semigroup {T(t) : t≥0}is an isolated invariant set if there is an  > 0such that Ξis the maximal invariant subset of O(Ξ). A disjoint family of isolated invariant sets is a family {Ξ1,· · · ,Ξn}of isolated invariant sets with the property that, for some  > 0, O(Ξi)∩ O(Ξj) = ∅,1≤i<j≤n. Definition 2.6. A global solution for a semigroup {T(t) : t≥0}is a continuous function ξ:R→Xwith the property that T(t)ξ(s) = ξ(t+s)for all s∈Rand for all t∈R+. We say that ξ:R→Xis a global solution through x∈Xif it is a global solution and ξ(0) = x. Definition 2.7. Let {T(t) : t≥0}be a semigroup which has a disjoint family of isolated invariant sets Ξ={Ξ1,· · · ,Ξn}. A homoclinic structure associated to Ξis a subset {Ξk1,· · · ,Ξkp}of Ξ(p≤n) together with a set of global solutions {ξ1,· · · , ξp}such that Ξkj t→−∞ ←− ξj(t)t→∞ −→ Ξkj+1 ,1≤j≤p where Ξkp+1 := Ξk1. We are now ready to define generalized gradient-like semigroups ([3]) ENERGY LEVEL DECOMPOSITION 5 Definition 2.8. Let {T(t) : t≥0}be a semigroup with a global attractor Aand a disjoint family of isolated invariant sets Ξ={Ξ1,· · · ,Ξn}. We say that {T(t) : t≥0}is a generalized gradient-like semigroup associated to Ξif •For any global solution ξ:R→ A there are 1≤i, j ≤nsuch that Ξi t→−∞ ←− ξ(t)t→∞ −→ Ξj. •There is no homoclinic structure associated to Ξ. Now we will introduce the notion of a Morse decomposition for an attractor Aof a gradientlike semigroup {T(t) : t≥0}. We start with the notion of attractor-repeller pairs. Definition 2.9. Let {T(t) : t≥0}be a semigroup with a global attractor A. We say that a non-empty subset Aof Ais a local attractor if there is an  > 0such that ω(O(A)) = A. The repeller A∗associated to a local attractor Ais the set defined by A∗={x∈ A :ω(x)∩A=∅}. The pair (A, A∗)is called attractor-repeller pair for {T(t) : t≥0}. Note that if Ais a local attractor, then A∗is closed and invariant. Definition 2.10. Given an increasing family ∅=A0⊂A1⊂ · · · ⊂ An=A, of local attractors, define Ξj:= Aj∩A∗ j−1,1≤j≤n. The ordered n-upla Ξ:= (Ξ1,Ξ2,· · · ,Ξn)is called a Morse decomposition on A. Remark 2.11. Observe that Ξis a local attractor if and only if it is compact, invariant and attracts O(Ξ) for some  > 0. We observe that the above definition differs slightly from the usual definition since the local attractor is required to attract a neighborhood of Ξin Xand not in Aas in [4, 13]. The following results are proved in Arag˜ao-Costa et al. [1]: Lemma 2.12. Let {T(t) : t≥0}be a semigroup in Xwith a global attractor Aand an attractor-repeller (A, A∗). A global solution ξ:R→Xof {T(t) : t≥0}with the property that ξ(t)∈ Oδ(A∗)for all t≤0for some δ > 0such that Oδ(A∗)∩A=∅must satisfy d(ξ(t), A∗)t→−∞ −→ 0. Lemma 2.13. Let {T(t) : t≥0}be a semigroup in Xwith a global attractor Aand (A, A∗) an attractor-repeller for {T(t) : t≥0}. If ξ:R→Xis a global bounded solution for {T(t) : t≥0}through x /∈A∪A∗, then ξ(t)t→∞ −→ Aand ξ(t)t→−∞ −→ A∗. Furthermore, if x∈X\A then, T(t)xt→∞ −→ A∪A∗. Corollary 2.14. If {T(t) : t≥0}is a semigroup in Xwith a global attractor Aand (A, A∗)is an attractor-repeller pair for {T(t) : t≥0}, then {T(t) : t≥0}is a generalized gradient-like semigroup associated to the disjoint family of isolated invariant sets {A, A∗}. 6 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA In [1] we describe the construction of a Morse decomposition of the attractor of a gradientlike semigroup associated to the disjoint family of isolated invariant sets {Ξ1,· · · ,Ξn}and of the associated collection of increasing local attractors starting from the collection of isolated invariant sets {Ξ1,· · · ,Ξn}. For the sake of completeness, we recall such a construction here. Let {T(t) : t≥0}be a generalized gradient-like semigroup with associated family of isolated invariant sets Ξ={Ξ1,· · · ,Ξn}. If (after possible reordering) Ξ1is a local attractor for {T(t) : t≥0}and Ξ∗ 1={a∈ A :ω(a)∩Ξ1=∅} each Ξi,i > 1 is contained in Ξ∗ 1and that for any a /∈ A\{Ξ1∪Ξ∗ 1}and global solution φ:R→ A with φ(0) = awe have that Ξ∗ 1 t→−∞ ←− φj(t)t→∞ −→ Ξ1. Considering the restriction T1(t) of T(t) to Ξ∗ 1we have that T1(t) is a generalized gradientlike semigroup in Ξ∗ 1=: Ξ1,0with isolated invariant sets {Ξ2,· · · ,Ξn}and we may assume without loss of generality that Ξ2is a local attractor for the semigroup {T1(t) : t≥0}in Ξ∗ 1. If Ξ∗ 2,1is the repeller associated to the isolated invariant set Ξ2for {T1(t) : t≥0}in Ξ∗ 1 we may proceed and consider the restriction {T2(t) : t≥0}of the semigroup {T1(t) : t≥0} to Ξ∗ 2,1and {T2(t) : t≥0}is a generalized gradient-like semigroup in Ξ∗ 2,1with associated isolated invariant sets {Ξ3,· · · ,Ξn}. Proceeding with this until all isolated invariant sets are exhausted we obtain a reordering of {Ξ1,· · · ,Ξn}in such a way that Ξjis a local attractor for the restriction of {T(t) : t≥0} to Ξ∗ j−1,j−2(Ξ∗ 0,−1:= A). Definition 2.15. Let {T(t) : t≥0}be a semigroup. The unstable set of an invariant set Ξ is defined by Wu(Ξ) = {z∈X:there is a global solution ξ:R→X such that ξ(0) = zand lim t→−∞ dist(ξ(t),Ξ) = 0}. Define A0=∅,A1= Ξ1and for j= 2,3,· · · , n Aj=Aj−1∪Wu(Ξj) = j [ i=1 Wu(Ξi).(2.1) It is clear that An=A. Theorem 2.16. (Arag˜ao-Costa et al. [1]) Let {T(t) : t≥0}be a generalized gradient-like semigroup with associated family of isolated invariant sets Ξ={Ξ1,· · · ,Ξn}reordered in such a way that Ξjis an attractor for the restriction of {T(t) : t≥0}to Ξ∗ j−1,j−2. Then Aj defined in (2.1) is a local attractor for {T(t) : t≥0}in X, and Ξj=Aj∩A∗ j−1,1≤j≤n. As a consequence, Ξdefines a Morse decomposition on A. ENERGY LEVEL DECOMPOSITION 7 2.2. A Lyapunov function for a generalized gradient-like semigroup. Let us now recall some definitions and results from [1]. Definition 2.17. We say that a semigroup {T(t) : t≥0}with a global attractor Aand a disjoint family of isolated invariant sets Ξ={Ξ1,· · · ,Ξn}is a generalized gradient semigroup with respect to Ξif there is a continuous function V:X→Rsuch that, Vis constant in Ξi, for each 1≤i≤n,[0,∞)3t7→ V(T(t)x)∈Ris decreasing for each x∈Xand V(T(t)x) = V(x)for all t≥0if and only if x∈Sn i=1 Ξi. A function Vwith the properties above is called a Lyapunov function for the generalized gradient semigroup {T(t) : t≥0} with respect to Ξ. Proposition 2.18. Let {T(t) : t≥0}be a semigroup in a metric space (X, d)with global attractor A, and let (A, A∗)be an attractor-repeller pair in A.Then, there exists a function f:X→Rsatisfying the following: (i)f:X→Ris continuous in X. (ii)f:X→Ris non-increasing along solutions. (iii)f−1(0) = Aand f−1(1) ∩ A =A∗. (iv)Given z∈X, if f(T(t)z) = f(z)for all t≥0, then z∈(A∪A∗). Theorem 2.19. (Aragao-Costa et al. [1]) Let {T(t) : t≥0}be a semigroup with global attractor Aand a disjoint family of isolated invariant sets Ξ={Ξ1,· · · ,Ξn}. Then, {T(t) : t≥0}is a generalized gradient semigroup with respect to Ξif and only if it is a generalized gradient-like semigroup with respect to Ξ. Moreover, [0,∞)3t7→ V(T(t)z)is differentiable for all z∈X. Finally, the Lyapunov function V:X→Rof a generalized gradient-like semigroup may be chosen in such a way that V(Ξk) = k,k= 1,· · · , n. 2.3. Stability under perturbations of generalized gradient semigroups. We introduce the notions of continuity and asymptotic compactness for a parameter dependent family. We start with the notion of continuity for a family of semigroups. Definition 2.20. A family of semigroups {Tη(t) : t≥0}η∈[0,1] is said to be continuous at η= 0 if Tη(t)xη→0 −→ T0(t)xuniformly for (t, x)in compact subsets of R+×X. Definition 2.21. A family of semigroups {Tη(t) : t≥0}η∈[0,1] is said to be collectively asymptotically compact at η= 0 if, given a sequence (ηk)k∈Nwith ηk k→∞ −→ 0, a bounded sequence (xk)k∈Nin Xand a sequence (tk)k∈Nin R+with tk k→∞ −→ ∞, then (Tηk(tk)xk)is relatively compact. We are now ready to state the following result from [3]. Theorem 2.22 (Carvalho-Langa).Let {Tη(t) : t≥0}η∈[0,1] be a collectively compact family of semigroups which is continuous at η= 0. Assume that 8 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA a) {Tη(t) : t≥0}possesses a global attractor Aηfor each η∈[0,1] and ∪η∈[0,1]Aηis bounded. b) There exists n∈Nsuch that Aηhas nisolated invariant sets Ξη={Ξ1,η,· · · ,Ξn,η} for all η∈[0,1], and sup1⩽i⩽ndH(Ξi,η,Ξi,0)η→0 −→ 0. c) {T0(t) : t≥0}is a generalized gradient-like semigroup. Then, there exists η0>0such that, for all η⩽η0,{Tη(t) : t≥0}is a generalized gradient-like semigroup associated to Ξηand consequently Aη=∪n i=1Wu(Ξi,η),∀η∈[0, η0]. As an immediate consequence of this result and the ones in Section 2.2 we have the following result. Corollary 2.23. Under the assumption of Theorem 2.22, there exists η0>0such that, for all η⩽η0,{Tη(t) : t≥0}is a generalized gradient semigroup. Corollary 2.24. Under the assumption of Theorem 2.22, suppose there exists n∈Nsuch that Aηhas nstationary solutions Sη={ξ1,η,· · ·, ξn,η}for all η∈[0,1] and sup1⩽i⩽nd(ξi,η, ξi,0) η→0 −→ 0. Then, there exists η0>0such that, for all η⩽η0,{Tη(t) : t≥0}is a gradient semigroup in the sense of [5]. Remark 2.25. The previous theorem supposes the continuity of the isolated invariant sets in order to prove the stability of the generalized gradient-like semigroups under perturbation. Note (cf. [3]) that from a perturbation of a gradient-like semigroup it could emerge a gradientlike semigroup with a different collection of isolated invariant sets. On the other hand, for a generalized gradient-like semigroup, even when the isolated sets behave continuously under perturbation, some of the connections between them may change. So the dynamics under perturbation could suffer drastic changes. This fact allows that the Lyapunov functions that we have constructed behave discontinuously under perturbation. 3. Continuity of the Lyapunov function under perturbation Now, we will analyze the continuity of the Lyapunov function under suitable perturbations. Definition 3.1. Let (Aη)η∈[0,1] be a family of sets in a metric space Xwith distance d: X×X→R+.We say that this family is upper semicontinuous (u.s.c.) at η= 0 if lim η→0+dist (Aη, A0) = 0. We say that this family is lower semicontinuous (l.s.c.) at η= 0 if lim η→0+dist (A0, Aη) = 0. ENERGY LEVEL DECOMPOSITION 9 Finally, the family is said to be continuous at η= 0 if it is upper and lower semicontinuous, i.e., when it holds lim η→0+dH(Aη, A0)=0. Lemma 3.2. Let {Tη(t) : t≥0}η∈[0,1] be a family of collectively asymptotically compact semigroups in a metric space Xwhich is continuous at η= 0 (see Definition 2.20). Assume that each {Tη(t) : t≥0}has a global attractor Aηand that S η∈[0,1] Aηis bounded in X. Let (Aη)η∈[0,1] be a family of subsets in Xsuch that Aη⊂ Aηand A0is a local attractor for {T0(t) : t≥0}with ω(Oε(A0)) = A0,for some ε > 0. Then, if (Aη)η∈[0,1] is continuous at η= 0,given δ∈(0, ε)there exist δ0∈(0, δ)and η0>0 such that for all η∈[0, η0]it holds γ+ η(Oδ0(Aη)) ⊂ Oδ(Aη), where γ+ η(Oδ0(Aη)) denotes the positive orbit of the set Oδ0(Aη)associated to {Tη(t) : t≥0}. Proof. Suppose not, then there exist δ∈(0, ) and sequences (zj)j∈Nin X, (ηj)j∈Nin [0,1] and (tj)j∈Nin Rsuch that ηj j→∞ →0+, tj j→∞ → ∞, dist zj, Aηj<1 jfor all j, dist Tηj(t)zj, Aηj< δ for all t∈[0, tj) and all j∈N and dist Tηj(tj)zj, Aηj=δfor all j∈N. If, for each j, we now define ξj: [−tj,∞)→Xby ξj(t) := Tηj(t+tj)zj,then, by the collective asymptotic compactness and the uniform convergence in compact sets, it is not difficult to see that there exist a bounded global solution ξ0:R→Xfor {T0(t) : t≥0}and a subsequence for (ξj)j∈N,denoted the same, such that for all t, ξ0(t) = lim j→∞ξj(t). On the other hand, given t < 0,for all jbig enough it holds dist (ξj(t), A0)≤dist ξj(t), Aηj+ dist Aηj, A0, from where, by the u.s.c. of (Aη)η∈[0,1] ,we obtain that for all t < 0 dist (ξ0(t), A0)≤δ, and from δ= dist ξj(0) , Aηj≤dist (ξj(0) , A0) + dist A0, Aηj,by the l.s.c. of (Aη)η∈[0,1], it follows that dist (ξ0(0) , A0) = δ. But, as δ < ε, then A0attracts K={ξ0(t) : t≤0},which contradicts the fact that dist (ξ0(0) , A0) = δ.  We also have the following lemma: 16 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA On the other hand, by the lower semicontinuity of (Aη)η∈[0,1] at η= 0,let η1∈(0, η0] such that A0⊂ Oε0 2(Aη) if η∈[0, η1],and then, Oε0 2(A0)⊂ Oε0(Aη) if η∈[0, η1],and thus (3.17) implies that γ+ ηOε0 2(A0)⊂ Oε 2(Aη) if η∈[0, η1], from which hη(w) = sup t≥0 dist (Tη(t)w, Aη)≤ε 2for all η∈[0, η1] and w∈ Oε0 2(A0), so that we conclude sup w∈O ε0 2 (A0) |hη(w)−h0(w)| ≤ εfor all η∈[0, η1]. In these conditions, given ε > 0,each z∈Xpossesses a neighbourhood Oσ(z),with σ=σ(ε, z)>0,and there exists an index η0=η0(ε, z)>0 such that sup w∈Oσ(z) |hη(w)−h0(w)| ≤ εif η∈[0, η0], so that we conclude the convergence of hη η→0+ →h0uniformly in compact sets of Xby a similar argument to the one in Step 2. This completes the proof.  Remark 3.5. Let {Tη(t) : t≥0}η∈[0,1] be a family of semigroups in a metric space X satisfying hypotheses of Theorem 2.22 with isolated invariant sets Ξη:= {Ξ1,η,· · · ,Ξn,η} reordered in such a way that Ξjis a local attractor for the restriction of {Tη(t) : t≥0}to (Ξη)∗ j−1,j−2. For suitably small η,{Tη(t) : t≥0}is a gradient-like semigroup associated to Ξη,and such that Ξηis a Morse decomposition with associated local attractors A0,η := ∅and for each j= 1,2,· · · , n Aj,η := j [ i=1 Wu η(Ξi,η). Then, the repellers are given by: A∗ n,η := ∅and for each j= 0,1,· · · , n −1 A∗ j,η := n [ i=j+1 Ws η(Ξi,η), where, for each η∈[0,1] and all i= 1,2,· · · , n Ws η(Ξi,η) := nz∈ Aη:Tη(t)zt→∞ →Ξi,ηo and Ws loc,η (Ξi,η)or Wu loc,η (Ξi,η)(in the context of generalized gradient-like semigroups) is the intersection of Ws η(Ξi,η)or Wu η(Ξi,η)with a neighborhood of Ξi,η. Thus, we must look for sufficient conditions to obtain continuity of stable (restricted to the attractors) and unstable manifolds in order to obtain the continuity of Lyapunov functions. This last remark leads us to the following result: ENERGY LEVEL DECOMPOSITION 17 Corollary 3.6. Let {Tη(t) : t≥0}η∈[0,1] be a family of semigroups in a metric space (X, d) satisfying the hypotheses of Theorem 2.22 with isolated invariant sets Ξη:= {Ξ1,η,· · · ,Ξn,η}. If the stable Ws η(Ξj,η)η∈[0,1] and unstable Wu η(Ξj,η)η∈[0,1] manifolds are continuous at η= 0,for all j= 1,2,· · · , n, then the Lyapunov functions associated to {Tη(t) : t≥0}, given with the aid of Proposition 3.4, for ηsmall enough behave continuously at η= 0. 4. Energy level decomposition of a generalized gradient-like semigroup We now give a dynamical description of a generalized gradient-like semigroup by reordering and regrouping the corresponding isolated invariant subsets to obtain a totally ordered family of isolated invariant sets that we will refer to as energy levels. This new family of isolated invariant sets is a Morse decomposition of Awith fewer invariant sets but in such a way that it still gives us a Lyapunov function that is constant only in the solutions lying in the original isolated invariant sets. In a certain sense, this decomposition is the coarsest decomposition which still gives us a Lyapunov function which is constant only in the solutions lying in the original isolated invariant sets. Assume that {T(t) : t≥0}is a generalized gradient-like semigroup with respect to the disjoint family of isolated invariant sets Ξ={Ξ1,Ξ2,· · · ,Ξn}. (a) Given Ξl1and Ξl2∈Ξ, we say that Ξl1precedes Ξl2(we write Ξl1≺Ξl2), if there exists a global solution ξ:R→Xof {T(t) : t≥0}such that ξ(R)*Ξl1∪Ξl2and lim t→−∞ d(ξ(t),Ξl2) = 0 and lim t→∞ d(ξ(t),Ξl1) = 0. (b) Let us consider M1:= {Ξ`∈Ξ: there is no element Ξ ∈Ξthat preceeds Ξ`} and, for any integer k≥2 Mk:= {Ξ`∈Ξ: if Ξ ∈Ξand Ξ ≺Ξ`then Ξ ∈ Mk−1}. Note that, by definition, Mk⊂ Mk+1. (c) We now define the sets N1:= [ Ξ∈M1 Ξ,and Nk:= [ Ξ∈Mk\Mk−1 Ξ, for all k≥2. Since Ξis finite, there exists a positive integer qsuch that Mk=Mqfor each k > q, so that, Nk=∅for all k > q. Thus, let N1,N2,· · · ,Np,the level sets with p:= min{q∈N:Mk=Mqfor each k > q}. We have the following first result related to this family of sets: 18 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA Lemma 4.1. (see [1]) Let {T(t) : t≥0}be a semigroup with global attractor A. Assume that {T(t) : t≥0}is a generalized gradient-like semigroup with respect to the disjoint family of isolated invariant sets Ξ={Ξ1,Ξ2,· · · ,Ξn}. Then each element of Ξis contained in Nk, for some k≤p. The following result will show that N= (N1,N2,· · · ,Np) is a Morse decomposition for A. Theorem 4.2. Let {T(t) : t≥0}be a semigroup with global attractor A.If {T(t) : t≥0}is a generalized gradient-like semigroup with respect to Ξ={Ξ1,Ξ2,· · · ,Ξn}, then (N1,N2,· · · ,Np)is a Morse decomposition for A. Proof. Clearly {T(t) : t≥0}is a generalized gradient-like semigroup with respect to N. The proof of the result now follows from Theorem 2.16 (see [1]).  In order to see that the continuity of local unstable manifolds is not sufficient to obtain the continuity of Lyapunov functions one may consider the example in the following picture -  ? 6 q d1 q d2 q c q b q a- i  ^ i Y Y   Figure 01 ^ ) K  KKK)   R ^ - ? 6 q c1 q c2 q b2q b1 q a- - i  ^ i Y Y   Figure 02 ^ ) K  KKK)   R ^ Note that, in both cases the semigroup associated are gradient-like. Also, the semigroup associated to Figure 01 has energy levels N1={a},N2={b},N3={c},N4={d1, d2}, while the semigroup associated to Figure 02 has energy levels N1={a},N2={b1, b2}, N3={c1, c2}. This clearly shows that, even if all equilibria are hyperbolic, if the connections between them are not stable under perturbations, the level sets may be discontinuous. ENERGY LEVEL DECOMPOSITION 19 Remark 4.3. All the concepts and results in the previous section can be written in the particular case in which we have a finite set of equilibria. Indeed, let {T(t) : t≥0}be a gradient-like semigroup in Xwith global attractor Awith equilibrium points E={ζ1,· · · , ζn}. Then, there exists an energy level decomposition in Amade of equilibrium points. 5. Energy levels for a generalized gradient-like semigroup under perturbation Again, for each η∈[0,1], let {Tη(t) : t≥0}a semigroup on a metric space X, with global attractor Aη,and a finite family of isolated bounded sets Ξη={Ξ1,η,Ξ2,η,· · · ,Ξp,η},such that each {Tη(t) : t≥0}is a generalized gradient-like semigroup with respect to Ξη.We suppose sup 1≤i≤p dH(Ξi,η,Ξi,0)η→0+ →0. Under these conditions, we give sufficient conditions so that the energy levels are continuous under perturbation. Lemma 5.1. For each η∈[0,1], let {Tη(t) : t≥0}be a semigroup on a metric space X, with global attractor Aηand a finite family of isolated bounded sets Ξη={Ξ1,η,Ξ2,η,· · · ,Ξn,η}, such that each {Tη(t) : t≥0}is a generalized gradient-like semigroup with respect to Ξη,and let Nη= (N1,η, ..., Np(η),η)be the corresponding Morse decomposition formed by the energy levels. Assume the hypotheses of Theorem 2.22. Let Ξ0∈ N1,0and (Ξη)η∈(0,1], with Ξη∈Ξη for each η∈(0,1],the unique family such that dH(Ξη,Ξ0)η→0+ →0. Then there exist δ > 0and η1∈(0,1] such that, for any η∈(0, η1],if z∈Xis such that dist (z, Ξη)< δ then dist (Tη(t)z, Ξη)t→∞ →0.Moreover, for i≥2if Ξ0∈ Ni,0and (Ξη)η∈(0,1], with Ξη∈Ξηfor each η∈(0,1],is the unique family such that dH(Ξη,Ξ0)η→0+ →0,then there exist δ > 0and ηi∈(0,1] such that for any η∈(0, ηi],if z∈Xsatisfies dist (z, Ξη)< δ, then, either dist (Tη(t)z, Ξη)t→∞ →0,or dist Tη(t)z, M(i−1),ηt→∞ →, where M(i−1),η := i−1 S j=1 Nj,η, in particular dist(Tη(t)z, Mi,η)t→0 −→ 0. Proof. For the case Ξ0∈ N1,0, suppose not. Then, there exist ηk→0+in (0,1],(zk)k∈Nin Xand (Ξηk)k∈Nsubfamily of (Ξη)η∈(0,1] with d(zk,Ξηk)<1 ksuch that Tηk(t)zkdoes not converge to Ξηkwhen t→ ∞,for all k. Fix δ0>0 such that Oδ0(Ξi,η)∩ Oδ0(Ξj,η) = ∅ for i6=jand ηsmall enough. Then, as each {Tηk(t) : t≥0}is a generalized gradient-like semigroup with respect to Ξηk, so that, for each k, dist Tηk(t)zk,Ξ(k) ηkt→∞ →0 for some Ξ(k) ηk∈Ξηk\ {Ξηk}and so for kbig enough, we can find τk>0 such that dist (Tηk(t)zk,Ξηk)< δ0for t∈[0, τk) and (5.1) dist (Tηk(τk)zk,Ξηk) = δ0.(5.2) 20 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA By the uniform convergence Tη η→0+ →T0on compacts of [0,∞)×X, from dH(Ξηk,Ξ0)k→∞ →0 and by (5.2) we have that τk k→∞ → ∞.Thus, consider, for each kbig enough, the map ξk: [−τk,∞)→Xgiven by ξk(t) := Tηk(t+τk)zkt∈[−τk,∞).By the collective compactness and from (5.1) there exists a global bounded solution ξ0:R→Xfor {T0(t) : t≥0}such that lim k→∞ξk(t) = ξ0(t) for all t∈Rand lim t→−∞dist (ξ0(t),Ξ0) = 0.But, as Ξ0in N1,0with {T0(t) : t≥0}generalized gradient-like, we have ξ0(t)∈Ξ0for all t∈R,which contradicts d(ξ0(0) ,Ξ0) = δ0,which comes from (5.2) as k→ ∞. For i= 2, we also argue by contradiction. Then we obtain ηk→0+in (0,1],(zk)k∈Nin Xand (Ξηk)k∈Nsubfamily of (Ξη)η∈(0,1] with dist (zk,Ξηk)<1 ksuch that Tηk(t)zkdoes not converge to Ξηkwhen t→ ∞ and Tηk(t)zkdoes not converge to N1,ηkwhen t→ ∞.Now, let δ > 0 such that the conclusion of the previous case is satisfied in Oδ(N1,ηk) for all kbig enough and with Oδ(Ξi,η)∩ Oδ(Ξj,η) = ∅for i6=jand ηsmall enough. Thus, for all t≥0 and kwe have dist (Tηk(t)zk,N1,ηk)≥δ. (5.3) On the other hand, as each {Tηk(t) : t≥0}is a generalized gradient-like semigroup, for each kwe have that dist Tηk(t)z, Ξ(k) ηkt→∞ →0 for some Ξ(k) ηk∈Ξηk\ {Ξηk}and, consequently, for each klarge enough, there exists τk>0 satisfying dist (Tηk(t)zk,Ξηk)< δ for t∈[0, τk) and (5.4) dist (Tηk(τk)zk,Ξηk) = δ. (5.5) Again, by (5.5) ,it holds τk k→∞ → ∞ and then, if we define ξk: [−τk,∞)→Xgiven by ξk(t) := Tηk(t+τk)zkt∈[−τk,∞),from the collective compactness and (5.4) ,we obtain the existence of a global bounded solution for {T0(t) : t≥0}, ξ0:R→X, such that lim k→∞ξk(t) = ξ0(t) for all t∈Rwith lim t→−∞d(ξ0(t),Ξ0)=0.Since {T0(t) : t≥0} is a generalized gradient-like semigroup with respect to Ξ0,there exists Ξl1∈Ξ0with lim t→∞dist (ξ0(t),Ξl1) = 0.As Ξ0∈ N2,0,it holds that Ξl1∈ N1,0,from where, for τ > 0 with dist (ξ0(τ),Ξl1)<δ 2,we deduce, for kbig enough, dist (Tηk(τ+τk)zk,N1,ηk)<δ 2,which contradicts (5.3). A similar argument for the remaining cases finishes the proof.  Theorem 5.2. Suppose the hypotheses of the previous lemma. Let N1,η,· · · ,Np(η),η the energy levels associated to the family Ξη={Ξ1,η,· · · ,Ξn,η}for η∈(0,1],and suppose that, (H)if (ηk)k∈Nis a sequence in (0,1] with ηk k→∞ →0+and (Ξηk)k∈Nsatisfy that, for some i∈T k∈N {1,2,· · · , p(ηk)},Ξηk∈ Ni,ηkand dH(Ξηk,Ξ0)k→∞ →0then Ξ0∈ Ni,0. Then, if pdenotes the number of energy levels for {T0(t) : t≥0},written as N1,0,· · ·,Np,0, there exists η∗∈(0,1] such that for all η∈(0, η∗]the semigroup {Tη(t) : t≥0}possesses ENERGY LEVEL DECOMPOSITION 21 also penergy levels, N1,η,· · · ,Np,η (i.e., p(η) = pfor all η∈(0, η∗]),and dH(Ni,η,Ni,0)η→0+ →0for all i= 1,2,· · · , p. Proof. Let us write the energy levels for the limit case. For i= 1,2,· · · , p,Ni,0=nΞ(i) l1,0,· · · ,Ξ(i) lk(i),0o. If we define, for each η∈(0,1] and i= 1,2,· · · , p the sets Hi,η := nΞ(i) l1,η,· · · ,Ξ(i) lk(i),ηoand H0 i,η := k(i) S j=1 Ξ(i) lj,η,then, Theorem 2.22 implies dHH0 i,η,Ni,0η→0+ →0,for all i= 1,2,· · · , p. The sets Hi,η’s are the natural candidates to be the energy levels for Tη(·).Indeed, let us prove that it holds that Hi,η =Ni,η,for i= 1,2,· · · , p and ηsmall enough, i.e., H1,η,H2,η,· · · ,Hp,η are the energy levels of {Tη(t) : t≥0}for ηsmall enough. For i= 1 let Ξ(1) l1,ηη∈(0,1] be the family within the set H1,η.Then, there exists θ1∈(0,1] such that Ξ(1) l1,η ∈ N1,η for all η∈(0, θ1].Indeed, if not, we can find a sequence ηk→0+and global solutions ξk:R→Xfor {Tηk(t) : t≥0}such that lim t→−∞dist ξk(t),Ξ(1) lj,ηk= 0 and lim t→∞dist (ξk(t),Ξηk) = 0,for some isolated invariant set Ξηk∈Ξηk, but with Ξηk6= Ξ(1) l1,ηkfor all k. Choose now, for each kbig enough, τksuch that dist ξk(t),Ξ(1) lj,ηk< δ0for all t<τkand dist ξk(τk),Ξ(1) l1,ηk=δ0,(5.6) where δ0>0 satisfies Oδ0(Ξi,η)∩ Oδ0(Ξj,η) = ∅for i6=jand ηsmall enough. If we define for kbig enough, ζk:R→Xby ζk(t) := ξk(t+τk)t∈R,we get ζ0:R→X a global solution of {T0(t) : t≥0}and a subsequence of (ζk)k∈N,written the same, satisfying lim k→∞dist (ζk(t), ζ0(t)) = 0 for all t∈R,with lim t→−∞dist ζ0(t),Ξ(1) l1,0= 0.Since Ξ(1) l1,0∈ N1,0, from the definition of N1,0it follows that ζ0(t)∈Ξ(1) l1,0for all t∈R. But this fact contradicts that dist ζ0(0) ,Ξ(1) l1,0=δ0,which comes from (5.6) as k→ ∞. The same argument for j= 2,· · · , k (1) leads to η1∈(0,1] such that Ξ(1) lj,η ∈ N1,η for j= 1,· · · , k (1) y η∈(0, η1], that is, H1,η ⊂ N1,η for all η∈(0, η1]. On the other hand, there exists η0 1∈(0, η1] such that N1,η ⊂ H1,η when η∈(0, η0 1].If not, there exist a sequence ηk→0+and, for each k, an isolated invariant set Ξηk∈ N1,ηk\H1,ηk such that the sequence (Ξηk)k∈Nsatisfies dH(Ξηk,Ξ0)k→∞ →0.However, from (H), we have Ξ0∈ N1,0,which contradicts that Ξηk/∈ H1,ηkfor any k. Thus, we conclude that H1,η =N1,η for η∈(0, η0 1]. We now show that there exists η2∈(0, η0 1] such that H2,η ⊂ M2,η if η∈(0, η2].Note that, if this claim holds, from the proof of the above case we have that H2,η ⊂ M2,η\N1,η =N2,η, once H2,η is disjoint of H1,η =N1,η for all η∈(0, η2]. 22 E. R. ARAG ˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA To get the existence of η2, take the family Ξ(2) l1,ηη∈(0,η1]of the elements in H2,η’s. Then there exists θ2∈(0, η1] such that Ξ(2) l1,η ∈ M2,η for all η∈(0, θ2].If not, by the same argument above, we get a subsequence ηk→0+and corresponding global solutions ξk:R→Xfor {Tηk(t) : t≥0}such that lim t→−∞dist ξk(t),Ξ(2) l1,ηk= 0 and lim t→∞d(ξk(t),Ξηk) = 0,for some isolated invariant sets Ξηk∈Ξηkwith Ξηk/∈ M1,ηk=N1,ηkand Ξηk6= Ξ(2) l1,ηkfor all k. As above and for the same δ0let, for each k, τk∈Rsuch that dist ξk(t),Ξ(2) l1,ηk< δ0for all t < τkand dist ξk(τk),Ξ(2) l1,ηk=δ0. From lemma 5.1, let δ > 0 and ¯η1∈(0, η0 1] such that the asymptotic stability of the elements in N1,η are satisfied in Oδ(N1,η) if η∈[0,¯η1].Then, dist (ξk(t),N1,ηk)≥δ, for all t∈Rand all k∈N.(5.7) If we define the solutions of ζk:R→Xby ζk(t) = ξk(t+τk)t∈R,we get again a global solution ζ0:R→Xde {T0(t) : t≥0}such that lim k→∞ζk(t) = ζ0(t) for all t∈Rwith lim dist t→−∞ ζ0(t),Ξ(2) l1,0= 0. As {T0(t) : t≥0}is a generalized gradient-like semigroup, there exists Ξ0∈Ξ0such that lim dist t→∞ (ζ0(t),Ξ0) = 0 and since Ξ(2) l1,0∈ N2,0we get Ξ0∈ N1,0. Thus, let τ > 0 such that dist (ζ0(τ),Ξ0)<δ 2,from which it follows the existence of k0∈N such that dist (ξk(τ+τk),N1,ηk)< δ for all k≥k0,which contradicts (5.7) . The same argument can be used for all j= 2,· · · , k (2) and so we obtain η2∈(0,1] such that H2,η ⊂ M2,η if η∈(0, η2]. Again, we get η0 2∈(0, η2] such that N2,η ⊂ H2,η if η∈(0, η0 2],from which we conclude that H2,η =N2,η for all η∈(0, η0 2]. Finally, repeating the reasoning for i= 3,· · · , p and recalling that Ξη=H1,η ∪ · · · ∪ Hp,η for each η, the proof is completed.  In the following theorem we state a sufficient condition for the hypotheses in the previous result. In particular, we prove that the stability of connecting orbits under perturbation gives the desired result on the continuity of the energy level sets. Consider {Tη(t) : t≥0}η∈[0,1] a family of generalized gradient-like semigroups with respect to Ξη={Ξ1,η,· · · ,Ξn,η}for each η∈[0,1].Suppose that: (HG) For each Ξl1,0,Ξl2,0∈Ξ0such that Ξl1,0≺Ξl2,0,if Ξl1,η,Ξl2,η are in Ξηfor η∈(0,1] and satisfy dH(Ξl1,η,Ξl1,0)η→0+ →0 and dH(Ξl2,η,Ξl2,0)η→0+ →0,then Ξl1,η ≺Ξl2,η for all ηsmall enough. Theorem 5.3. Suppose hypotheses in Theorem 2.22, and that (HG)is satisfied for {Tη(t) : t≥0}η∈[0,1]. ENERGY LEVEL DECOMPOSITION 23 Then, if (ηk)k∈Nis a sequence in (0,1] with ηk k→∞ →0+and for some i∈T k∈N {1,2,· · · , p(ηk)} (Ξηk)k∈Nis a sequence with Ξηk∈ Ni,ηkfor all kand dH(Ξηk,Ξ0)η→0+ →0for some Ξ0∈Ξ0, then Ξ0∈ Ni,0. Proof. Indeed, if for i= 1 and Ξ0does not belong to N1,0there exits Ξl1∈Ξ0with Ξl1≺Ξ0 but with Ξl16= Ξ0.Then, let Ξl1,ηkk∈Nthe sequence with Ξl1,ηk∈Ξηk,for all k, such that dH(Ξl1,η,Ξl1)η→0+ →0.By (HG) we have Ξl1,ηk≺Ξηkfor all kbig enough, which contradicts that Ξηk∈ N1,ηk. Thus the result is true for i= 1 and from it and the first part of the proof in Theorem 5.2, we get η1∈(0,1] such that H1,η,are N1,η for η∈(0, η1]. For i= 2, if Ξ0does not belong to N2,0=M2,0\N1,0we have, on the one hand, that Ξ0is not in N1,0,since if Ξ0∈ N1,0,as we have seen above N1,η =H1,η and so dH(N1,η,N1,0)η→0+ →0, so that Ξηk∈ N1,ηkfor all kbig enough, which contradicts that Ξηk∈ N2,ηkfor all k. Thus, Ξ0∈ N3,0∪ N4,0∪ · · · ∪ Nn,0and so we can find Ξl1∈Ξ0with Ξl1≺Ξ0such that Ξl1is not in N1,0.Let (Ξl1,ηk)k∈Nthe sequence with Ξl1,ηk∈Ξηk,for all k, such that dH(Ξl1,ηk,Ξl1)k→∞ →0.By (HG) we have Ξl1,ηk≺Ξηkfor all kbig enough, but, as Ξηk∈ N2,ηk for each k, then Ξl1,ηk∈ N1,ηkfor each k, but then we get that Ξl1∈ N1,0,which is a contradiction, so that the case i= 2 is also proved. Thus, by the second part in the proof of Theorem 5.2 we get η2∈(0, η1] such that the sets H2,η,defined as in the previous theorem, are the sets N2,η for η∈(0, η1],from which, in particular, dH(N2,η,N2,0)η→0+ →0. For i= 3,suppose Ξ0/∈ N3,0.Again, we then have that Ξ0/∈ N1,0∪ N2,0=M2,0,since as dH(Ni,η,Ni,0)η→0+ →0 for i= 1 and 2,if Ξ0∈ N1,0∪N2,0we would have that Ξηk∈ N1,ηk∪N2,ηk for all kbig enough, which contradicts that Ξηk∈ N3,ηkfor all k. Thus, Ξ0∈ N4,0∪ · · · ∪ Nn,0and then we can find Ξl1∈Ξ0with Ξl1≺Ξ0such that Ξl1/∈ M2,0.As in the above cases, let (Ξl1,ηk)k∈Nthe sequence with Ξl1,ηk∈Ξ0,for all k, such that dH(Ξl1,ηk,Ξl1)k→∞ →0.From (HG) we have that Ξl1,ηk≺Ξηkfor all kbig enough, but since Ξηk∈ N3,ηkfor each k, then Ξl1,ηk∈ M2,ηkfor each k, but then Ξl1must be in N1,0∪ N2,0=M2,0,which contradicts the way it was chosen. The argument must stop in a finite number of steps and so the proof is finished.  References [1] E.R. Arag˜ao-Costa, T. Caraballo, A.N. Carvalho and J.A. 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Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Berlin (1988; second edition 1996). (T. Caraballo and J. A. Langa) Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla Spain E-mail address, T. Caraballo: [email protected] E-mail address, J. A. Langa: [email protected] (A. N. Carvalho and E. R. Arag˜ao-Costa) 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, E. R. Arag˜ao–Costa: [email protected] E-mail address, A. N. Carvalho: [email protected]