Morse decomposition of global attractors with infinite components

Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume xx, Number 0x, xxxxx xxxx pp. 1–xx MORSE DECOMPOSITION OF GLOBAL ATTRACTORS WITH INFINITE COMPONENTS Tom´ as Caraballo, Juan C. Jara, Jos´ e A. Langa Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain Jos´ e Valero Centro de Investigaci´on Operativa, Universidad Miguel Hern´andez, Avda. de la Universidad, s/n, 03202-Elche, Spain (Communicated by Associate Editor) Abstract. In this paper we describe some dynamical properties of a Morse decomposition with a countable number of sets. In particular, we are able to prove that the gradient dynamics on Morse sets together with a separation assumption is equivalent to the existence of an ordered Lyapunov function associated to the Morse sets and also to the existence of a Morse decomposition -that is, the global attractor can be described as an increasing family of local attractors and their associated repellers. 1. Introduction. The asymptotic behaviour of a system of (ordinary or partial) differential equations modeling real phenomena from different areas of Science is usually described by the analysis of their global attractors, a compact invariant set for the associated semigroups attracting (uniformly) bounded sets forwards in time. This subject has received much attention throughout the last decades (see, for instance, [4], [9], [12], [16], [19], [18] or [20]). We recall now the definition of global attractor associated to a semigroup. First, 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) := {x∈X:d(x, a)<  for some a∈ A}. Definition 1.1. A family {T(t) : t≥0} ⊂ C(X)is a semigroup in a complete metric space Xif: •T(0) = IX, with IXbeing the identity map in X, •T(t+s) = T(t)T(s), for all t, s ∈R+, •R+×X3(t, x)7→ T(t)x∈Xis continuous. 2000 Mathematics Subject Classification. 37B25, 37L99, 35B40, 35B41. Key words and phrases. Morse decomposition; infinite components; gradient dynamics; Lyapunov function; gradient-like semigroup. Partially supported by FEDER and Ministerio de Econom´ıa y Competitividad (Spain) under grants MTM2011-22411 and MTM2012-31698, and by Junta de Andaluc´ıa under Proyecto de Excelencia P12-FQM-1492. 1 2 T. CARABALLO, J.C. JARA, J.A. LANGA, J. VALERO The notion of invariance plays a fundamental role in the study of the asymptotic behavior of semigroups. Definition 1.2. A subset Aof Xis said invariant under the semigroup {T(t) : t≥ 0}if T(t)A=Afor all t≥0. Given A, B ⊂X, the Hausdorff semidistance from Ato Bis given by d(A, B) := sup a∈A inf b∈Bd(a, b). Definition 1.3. Given two subsets A, B of Xwe say that Aattracts Bunder the action of the semigroup {T(t) : t≥0}if d(T(t)B, A)t→∞ −→ 0. We are now in a position to define global attractors. Definition 1.4. 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}. Definition 1.5. The semigroup {T(t) : t≥0}is eventually dissipative if for any bounded set Bthere exists t∗=t∗(B)≥0such that ∪t≥t∗T(t)Bis bounded. Remark 1.6. It is obvious that if T(t)possesses a global attractor, then it is eventually dissipative. One of the main properties in the study of attractors is referred to the description of their geometrical internal structure. Generically, a global attractor is characterized by a (finite or infinite) number of isolated invariant sets and the connecting orbits among them. This fact leads to a Morse decomposition of the global attractor in terms of a family of attracting-repeller pairs (see [8,17,11,14,15]). We now introduce this concept. Definition 1.7. Let {T(t) : t≥0}be a semigroup on X. We say that an invariant set E⊂Xfor the semigroup {T(t) : t≥0}is an isolated invariant set if there is an  > 0such that Eis the maximal invariant subset of O(E). Definition 1.8. A disjoint family of isolated invariant sets is a family {M1,· · · , Mn} of isolated invariant sets with the property that O(Mi)∩ O(Mj) = ∅,1≤i<j≤n, for some  > 0. Definition 1.9. 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 with ξ(0) = x. It is also well known that the global attractor is the union of all bounded complete global solutions of the semigroup T. Definition 1.10. Let {T(t) : t≥0}be a semigroup which possesses a disjoint family of isolated invariant sets M={M1,· · · , Mn}. A homoclinic structure associated to Mis a subset {Mk1,· · · , Mkp}of M(p≤n) together with a set of global solutions {ξ1,· · · , ξp}such that Mkj t→−∞ ←− ξj(t)t→∞ −→ Mkj+1 ,1≤j≤p, where Mkp+1 := Mk1. MORSE DECOMPOSITION WITH INFINITE COMPONENTS 3 Remark 1.11. Here, ξ(t)t→±∞ −→ Mmeans that d(ξ(t), M)→0as t→ ±∞. We will study the dynamics of the semigroup inside the global attractor A. We now define generalized dynamically gradient semigroups (see [6,5]). Definition 1.12. Let {T(t) : t≥0}be a semigroup with a global attractor Aand a disjoint family of isolated invariant sets M={M1,· · · , Mn}in A. We say that {T(t) : t≥0}is a generalized dynamically gradient semigroup relative to Mif: a) For any global solution ξ:R→ A there are 1≤i, j ≤nsuch that Mi t→−∞ ←− ξ(t)t→∞ −→ Mj. b) There is no homoclinic structure associated to M. Remark 1.13. The concept of generalized dynamically gradient semigroup is the same as the concept of gradient-like semigroup as given in [1],[5]. To introduce the notion of a Morse decomposition for the attractor Aof a semigroup {T(t) : t≥0}(see [8], [17] or [18]) we previously need the notion of attractorrepeller pair. We recall that the omega-limit set of B⊂Xis defined by ω(B) = ∩t≥0∪s≥tT(s)B. Definition 1.14. 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 an attractor-repeller pair for {T(t) : t≥0}. Note that if Ais a local attractor, then A∗is closed and invariant. Definition 1.15. Given an increasing family ∅=A0⊂A1⊂ · · · ⊂ An=A, of n+1 local attractors, for j= 1,· · · , n, define Mj:= Aj∩A∗ j−1.The ordered n-tuple M:= {M1, M2,· · · , Mn}is called a Morse decomposition for A. Definition 1.16. We will say that a semigroup {T(t) : t≥0}with a global attractor Aand a disjoint family of isolated invariant sets M={M1,· · · , Mn}in Ais a gradient semigroup with respect to M, if there exists a continuous function V: X→Rsuch that [0,∞)3t7→ V(T(t)x)∈Ris non-increasing for each x∈X\M, Vis constant in Mifor each 1≤i≤n, and V(T(t)x) = V(x)for all t≥0if and only if x∈ n S i=1 Mi. Vis called a Lyapunov function related to M. It has been proved in [1] that given a disjoint family of isolated invariant sets on the global attractor M={M1,· · · , Mn}for a semigroup T(t),the dynamical property of being generalized dynamically gradient, the existence of an associated ordered family of local attractor-repellers, and the existence of a Lyapunov functional related to M, are equivalent properties. Many of the arguments in [1] make a precise use of the fact that the number of Morse sets is finite. The aim of this paper is to generalize this result to the case of a countable number of Morse sets. Indeed, the general theory of Morse decomposition of invariant sets is generically adapted to the existence of a finite number of isolated Morse sets. However, it is not 4 T. CARABALLO, J.C. JARA, J.A. LANGA, J. VALERO unusual to have an infinite number of invariants in a global attractor. For instance, consider the scalar differential equation dy dt =f(y) with f(y) =      −y, if y≤0, (1 −e−y)sin π y,if 0 < y ≤1, 1−y, if y≥1. Note that the equation possesses the following fixed points: y1= 1, y2=1 2, y3=1 3, ..., yk=1 k, ..., y∞= 0, with their respective associated unstable manifolds (see Definition 4.2) Wu(1) = 1, Wu1 2= [1 2,1), ..., Wu1 k= [1 k,1 k−1), ..., Wu(0) = 0, and as global attractor A= [0,1].In [3] the authors study a multivalued version of the well-known Chafee-Infante equation, also leading to a global attractor with an infinite number of equilibria, which actually has motivated the necessity of developing the theory in this paper. We will consider this application in a subsequent work. In Section 2we recall some results on the dynamics related to an attractor-repeller pair. In Section 3we will generalize Definitions 1.12,1.15 and 1.16 to the case of an infinite number of disjoint isolated invariant sets M∞={Mi}∞ i=1 ∪M∞inside the global attractor. In Sections 4,5and 6we prove the main result of this paper, the equivalence between a generalized dynamically gradient semigroup referred to M∞with a suitable separation assumption, the existence of an ordered Lyapunov function associated to M∞, and the existence of a Morse decomposition on the global attractor. This is done in several steps: first, we prove that the property of the semigroup of being generalized dynamically gradient together with a separation assumption implies that a Morse decomposition can be constructed; then we prove that from a Morse decomposition related to M∞an ordered Lyapunov function can be defined; finally, we check that the existence of an ordered Lyapunov function implies that the semigroup is generalized dynamically gradient semigroup referred to M∞and that the separation assumption holds. 2. Preliminary results on attractor-repeller pairs. The following results on the dynamics on attractor-repeller pairs are taken from [1]. We recall that local attraction of Ain Ais equivalent to local attraction in X, for which we firstly need the following result. Lemma 2.1. Let {T(t) : t≥0}be a semigroup in Xwith a global attractor A. If A⊂ A is a compact invariant set for {T(t) : t≥0}and there is an  > 0such that Aattracts O(A)∩ A, then given δ∈(0, ε)there is a δ0∈(0, δ)such that γ+(Oδ0(A)) ⊂ Oδ(A), where γ+(Oδ0(A)) = S x∈Oδ0(A)S t≥0 {T(t)x}. The next result generalizes for semigroups a known result for groups given in [8] and shows that our definition of local attractor is equivalent to that one in [8,17]. MORSE DECOMPOSITION WITH INFINITE COMPONENTS 5 Lemma 2.2. If {T(t) : t≥0}is a semigroup in Xwith a global attractor Aand S(t) := T(t)|A, clearly {S(t) : t≥0}is a semigroup in the metric space A. If Ais a local attractor for {S(t) : t≥0}in the metric space A(that is, there exists ε > 0 with ω(O(A)∩ A) = A), and Kis a compact subset of Asuch that K∩A∗=∅, then Aattracts K. Furthermore Ais a local attractor for {T(t) : t≥0}in X. We now describe the dynamics on an attractor-repeller pair. Lemma 2.3. Let {T(t) : t≥0}be a semigroup in Xwith a global attractor Aand (A, A∗)an attractor-repeller for {T(t) : t≥0}. Then: (i)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∗. (ii)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. (iii)If x∈X\A, then T(t)xt→∞ −→ A∪A∗. Part (i) of the previous lemma is proved in Theorem 1.4 in [17]. Parts (ii) and (iii) can be found in [1]. 3. Generalized dynamically gradient semigroups. In this section we will introduce the concepts of generalized dynamically gradient semigroups and Morse decomposition for a countable set of isolated invariant sets. Definition 3.1. A disjoint (countable) family of invariant sets is a family M∞= {Mi}∞ i=1 ∪M∞of invariant sets with the property that, given j∈N, there exists δj such that Oδj(Mj)∩ Oδj(Mi) = ∅,for all i6=j,i∈N∪ {∞}.(3.1) Definition 3.2. Let {T(t) : t≥0}be a semigroup which possesses a disjoint family of invariant sets M∞={Mi}∞ i=1 ∪M∞with Mjisolated for each j∈N. A homoclinic structure associated to M∞is a finite subset {Mk1,· · · , Mkp}of M∞ together with a set of global solutions {ξ1,· · · , ξp}such that Mkj t→−∞ ←− ξj(t)t→∞ −→ Mkj+1 ,1≤j≤p, where Mkp+1 := Mk1. Remark 3.3. The set M∞is not assumed to be isolated. The reason is that typically in applications M∞is an accumulation set of the sequence Mnas n→ ∞. Hence, it is not isolated. This is the case in the example given in the introduction, and also, for instance, in the application for multivalued semiflows in [3]. Definition 3.4. Let {T(t) : t≥0}be a semigroup with a global attractor Aand a disjoint family of invariant sets M∞={Mi}∞ i=1 ∪M∞in Awith Mjisolated for each j∈N. We say that {T(t) : t≥0}is a generalized dynamically gradient semigroup relative to M∞if for any global solution ξ:R→ A such that ξ(t0)6∈ Mk, for some t0∈Rand any k∈N∪ ∞, it holds that Mj t→−∞ ←− ξ(t)t→∞ −→ Mi,for 1≤i<j≤ ∞.(3.2) Remark 3.5. It is obvious that condition (3.2) implies the following properties: •For any global solution ξ:R→ A there are 1≤i, j ≤ ∞ such that Mj t→−∞ ←− ξ(t)t→∞ −→ Mi. 6 T. CARABALLO, J.C. JARA, J.A. LANGA, J. VALERO •There is no homoclinic structure associated to M∞. When the number of sets Miis finite, then it is proved in [2]that these last two properties imply (3.2) for a suitable rearrangement of the sets. In fact, Definition 3.4 is the way in which it is defined a Morse decomposition of a global attractor in [18]. Note that, in particular, (3.2) implies that there is no global solution ξ(t) : R→ A with ξ(t0)6∈ M1for some t0∈Rsuch that lim t→−∞ d(ξ(t), M1)=0. The following lemma implies that an isolated invariant set inside a global attractor is compact. Lemma 3.6. Let Mbe an isolated invariant set which is relatively compact. Then Mis compact. Proof. We need to prove that Mis closed. Let yn→y, where yn∈M. By the continuity of Twe have that T(t)yn→T(t)yfor any t > 0. Hence, T(t)y∈M. Thus, T(t)M⊂Mfor all t≥0. On the other hand, as Mis invariant, for any t > 0 there exists zn∈Msuch that T(t)zn=yn. Since Mis relatively compact, passing to a subsequence we have zn→z∈M, and then T(t)z=y. Therefore, M⊂T(t)Mfor all t > 0. It follows that Mis invariant. As Mis an isolated invariant set, we get M=M. As a consequence of the first statement in Lemma 2.3 we obtain the following. Corollary 3.7. If {T(t) : t≥0}is a semigroup in Xwith a global attractor A and (A, A∗)is an attractor-repeller pair for {T(t) : t≥0}, then {T(t) : t≥0} is a generalized dynamically gradient semigroup associated to the disjoint family of isolated invariant sets {A, A∗}. Note that (3.1) implies Mi∩M∞=∅, for each i∈N.(3.3) Lemma 3.8. Condition (3.2) implies that there is no global solution ξ:R→ A with ξ(t0)∈ A \ M∞for some t0∈Rsuch that lim t→+∞d(ξ(t), M∞)=0.(3.4) Proof. It is obvious, as in (3.2), that the index icannot be ∞. Lemma 3.9. Let M∞={Mi}∞ i=1 ∪M∞be compact invariant sets such that Mj∩ Mi=∅for i6=j,i, j ∈N∪ ∞, and also suppose that the invariant compact set M∞⊂ A is such that lim i→∞ d(Mi, M∞)=0.(3.5) Then M∞is a disjoint family of invariant sets. Proof. Take j∈Narbitrary. We have to check (3.1). There exists δ1>0 such that Oδ1(Mj)∩ Oδ1(M∞) = ∅. In view of (3.5) there is N > j such that Mi⊂ Oδ1 2 (M∞) if i > N. MORSE DECOMPOSITION WITH INFINITE COMPONENTS 7 Hence, Oδ1(Mj)∩ Oδ1 2 (Mi) = ∅if i > N. Obviously, there exists δ2>0 for which Oδ2(Mj)∩ Oδ2(Mi) = ∅for 1 ≤i≤N,i6=j. Then the result follows for δj= min{δ1/2, δ2}. We can now introduce the concept of a Morse decomposition referred to M∞. Definition 3.10. Given an increasing family ∅=A0⊂A1⊂ · · · ⊂ An⊂ · · · A∞= Aof local attractors, for j∈Ndefine Mj:= Aj∩A∗ j−1,M∞=∩∞ j=0A∗ j.The ordered countable set M∞:= {Mi}∞ i=1 ∪M∞is called a Morse decomposition of A. The following properties of the sets Mjfollow. Lemma 3.11. M∞∩Aj=∅for any j∈N.Hence, M∞⊂A∞\ ∪∞ j=1 Ajand M∞∩Mj=∅for all j∈N. Proof. Let y∈M∞. Then y∈A∗ j,for any j∈N, implies y6∈ Ajfor all j∈N. Lemma 3.12. The sets Mj,j∈N∪ ∞, are compact. Proof. Since Mj⊂ A, they are relatively compact. Also, as Mjare the intersection of closed sets, they are closed. We can also give the following characterization. Proposition 3.13. Let {T(t) : t≥0}be a semigroup with the global attractor A and M∞={Mi}∞ i=1 ∪M∞a Morse decomposition for Awith the family ∅=A0⊂ A1⊂ · · · ⊂ A∞=Aof local attractors. Then, ∞ \ j=0 (Aj∪A∗ j)=( ∞ [ j=1 Mj)∪M∞. Proof. If z∈ ∞ S j=1 Mj, let k∈Nbe such that z∈Mk=Ak∩A∗ k−1. Hence z∈Ak⊂Ak+1 ⊂ · · · ⊂ A∞and z∈A∗ k−1⊂A∗ k−2⊂ · · · ⊂ A∗ 0. Thus z∈( ∞ \ j=k Aj)∩( k−1 \ j=0 A∗ j)⊂  ∞ \ j=k (Aj∪A∗ j) ∩  k−1 \ j=0 (Aj∪A∗ j) = ∞ \ j=0 (Aj∪A∗ j), proving that ∞ S j=1 Mj⊂ ∞ T j=0 (Aj∪A∗ j).If z∈M∞, then z∈ ∞ T j=0 A∗ j⊂ ∞ T j=0 (Aj∪A∗ j). Conversely, we take z∈ ∞ T j=0 (Aj∪A∗ j).If z∈ ∞ T j=0 A∗ j, then z∈M∞. Otherwise, z∈Ajfor some j∈N. Denote I:= {i1, i2,· · · , ik, . . . }and J:= {j1, j2,· · · , jl, .. . } such that I∪J=Z+with I∩J=∅and z∈Aifor all i∈Iand z∈A∗ jfor all j∈J. Clearly, if i:= min I, necessarily I={j≥i}and J={0,1,· · · , i−1}, consequently z∈Aiand z∈A∗ i−1. So, z∈Ai∩A∗ i−1=Mi, from which ∞ T j=0 (Aj∪A∗ j)⊂ ∞ S j=1 Mj. 8 T. CARABALLO, J.C. JARA, J.A. LANGA, J. VALERO 4. Construction of a Morse decomposition from the dynamics on M∞. In this section we describe the construction of a Morse decomposition of the global attractor Arelative to the disjoint family of invariant sets M∞={Mi}∞ i=1 ∪M∞ in Asuch that Mjis isolated if j∈Nand satisfying (3.2). By Lemma 3.8 we have that (3.4) does not hold. The following lemma will play an important role in what follows. Lemma 4.1. Let {T(t) : t≥0}be a semigroup with a global attractor Aand the disjoint family of invariant sets M∞={Mi}∞ i=1 ∪M∞={M1, . . . , Mn, . . . ;M∞}in Abe such that Mjare isolated for j∈N. Assume that Tis generalized dynamically gradient relative to M∞. Then, M1is a local attractor for {T(t) : t≥0}. Proof. We firstly prove that for all δ∈(0, δ1) there exists δ0∈(0, δ) such that γ+(Oδ0(M1)) ⊂ Oδ(M1), where δ1satisfies Oδ1(M1)∩ Oδ1(Mi) = ∅for i > 1 or i=∞. If not, there exist 0 < δ < δ1and sequences {tk}k∈Nof positive times and {xk}k∈N of points in Xsuch that for all k d(xk, M1)<1 k, d(T(tk)xk, M1) = δ and d(T(t)xk, M1)< δ for t∈[0, tk). Thus, if we define, for each k, ξk(t) := T(t+tk)xkfor t∈[−tk,∞),as tk→ k→∞ ∞, we conclude that there exists a global solution ξ:R→Xfor T(·) such that ξk→ k→∞ ξ uniformly in compact sets of times (see [7, Lemma 3.1]). Then, d(ξk(t), M1)≤δfor t∈[−tk,0] implies d(ξ(t), M1)≤δ < δ1for t≤0. But by (3.2) we have ξ(t)→Mj,with j > 1, as t→ −∞, a contradiction. M1is the maximal invariant set in Oε(M1) for some ε > 0. Thus, for δ < min{ε, δ1}take δ0∈(0, δ) such that γ+(Oδ0(M1)) ⊂ Oδ(M1), so that ω(Oδ0(M1)) ⊂ Oδ(M1)⊂ Oε(M1), and then, as ω(Oδ0(M1)) is invariant, ω(Oδ0(M1)) ⊂M1. The other inclusion is trivial, so that M1is a local attractor. For M1a local attractor, let M∗ 1={x∈ A :ω(x)∩M1=∅}be its associated repeller, so each Mi, with i≥2,is contained in M∗ 1and more generally the orbit ξ(R) of any global solution ξ:R→ A that converges to Mi,i≥2, when t→+∞,is contained in M∗ 1. Considering the restriction {T1(t) : t≥0}of {T(t) : t≥0}to M∗ 1 we have that {T1(t) : t≥0}satisfies (3.2) in the space M∗ 1with the invariant sets {Mi}∞ i=2 ∪M∞and we may assume, by the last lemma, that M2is a local attractor for the semigroup {T1(t) : t≥0}in M∗ 1. If M∗ 2,1is the repeller associated to the local attractor M2for {T1(t) : t≥0}in M∗ 1we may proceed and consider the restriction MORSE DECOMPOSITION WITH INFINITE COMPONENTS 9 {T2(t) : t≥0}of the semigroup {T1(t) : t≥0}to M∗ 2,1and then {T2(t) : t≥0} satisfies (3.2) in M∗ 2,1with the associated invariant sets {Mi}∞ i=3 ∪M∞. Setting A=: M∗ 0,−1and M∗ 1,0:= M∗ 1, for j≥1 we have that Mjis a local attractor for the restriction of {T(t) : t≥0}to M∗ j−1,j−2whose repeller will be indicated by M∗ j,j−1. Definition 4.2. Let {T(t) : t≥0}be a semigroup. The unstable set of an invariant set Mis defined by Wu(M) := {z∈X:there is a global solution ξ:R→X such that ξ(0) = zand lim t→−∞ d(ξ(t), M)=0}. Define A0:= ∅,A1:= M1and for j= 2,3,· · · , Aj:= Aj−1∪Wu(Mj) = j [ i=1 Wu(Mi).(4.1) Also, A∞=A. It is clear that A=∪∞ i=1Wu(Mi)∪Wu(M∞). Lemma 4.3. Assume the conditions of Lemma 4.1. Then M∞=∩∞ j=0A∗ j. Proof. Let z∈M∞. Then as M∞is invariant, ω(z)⊂M∞. Then zcannot be in Wu(Mj) for j∈N, as in such a case by (3.2) we would have ω(z)∩Mi6=∅for some i≤j, a contradiction. Thus, by (4.1) we have that z6∈ Ajfor j∈N. Hence, ω(z)∩Aj=∅, so that z∈ ∩∞ j=0A∗ j. Conversely, let z∈ ∩∞ j=0A∗ j. Then ω(z)∩Aj=∅for all j∈N. If z6∈ M∞, we take a global solution ξ(·) such that ξ(0) = z. Then by condition (3.2) we have that ξ(t)→Mias t→+∞for some i∈N. But then ω(z)∩Ai6=∅, a contradiction. Lemma 4.4. Assume the conditions of Lemma 4.1. Then the sets Mj,j∈N∪ ∞, are compact. Proof. We note that Mj⊂ A implies by Lemma 3.6 that the sets Mjare compact if j∈N. Also, Lemma 4.3 implies that M∞is closed, and then M∞⊂ A implies that it is compact. Lemma 4.5. Assume the conditions of Lemma 4.1. Suppose that, given j∈N, there exists δjsuch that Wu(Mj)∩ Oδj( ∞ [ i=j+1 Mi∪M∞)) = ∅.(4.2) Then, Aj∩ Oδj(( ∞ [ i=j+1 Mi)∪M∞) = ∅.(4.3) Proof. For j= 1 the result follows since A1=M1=Wu(M1). Suppose (4.3) is true for j−1 and we will show it for j. If not, there exists a sequence {xk}k∈Nin Ajsuch that for all k d(xk,( ∞ [ i=j+1 Mi)∪M∞)<1 k. As Aj:= Aj−1∪Wu(Mj) and we have (4.3) for j−1,then xk∈Wu(Mj), from which, by hypothesis, we get a contradiction. 16 T. CARABALLO, J.C. JARA, J.A. LANGA, J. VALERO We can now conclude our main theorem. Theorem 6.7. Let {T(t) : t≥0}be a semigroup with global attractor Aand consider a disjoint family of invariant sets M∞={Mi}∞ i=1 ∪M∞in Asuch that Mjare isolated for j∈N. Then, the following conditions are equivalent: 1. {T(t) : t≥0}is a generalized gradient semigroup with respect to M∞in the sense of the Definition 5.1 and M∞is ordered with respect to the respective Lyapunov function. 2. {T(t) : t≥0}is a generalized dynamically gradient semigroup with respect to M∞(as in Definition 3.4) satisfying (4.2). 3. M∞is a Morse decomposition of A. Proof. It is a straightforward consequence of Theorem 4.8 and Propositions 5.3,6.3 and 6.6. Corollary 6.8. Let {T(t) : t≥0}be a semigroup with global attractor Aand consider a disjoint family of invariant sets M∞={Mi}∞ i=1 ∪M∞in Asuch that Mjare isolated for j∈N. Assume that M∞is a Morse decomposition of A. Then A=∪∞ j=1Wu(Mj)∪Wu(M∞). Proof. In view of Theorem 6.7,{T(t) : t≥0}is a generalized gradient semigroup with respect to M∞in the sense of the Definition 5.1 and M∞is ordered with respect to the Lyapunov function. Hence, the result follows from Proposition 6.5. Acknowledgement. We would like to thank the referees for their helpful comments and suggestions which allowed us to improve the presentation of the paper. REFERENCES [1] Arag˜ao-Costa, E.R., Caraballo, T., Carvalho, A.N., & Langa, J.A., Stability of gradient semigroups under perturbation, Nonlinearity, 24 (2011), 2099–2117 . [2] Arag˜ao-Costa, E.R., Caraballo, T., Carvalho, A.N., & Langa, J.A., Continuity of Lyapunov functions and of energy level for a generalized gradient system, Topological Methods Nonl. 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