Stability of Gradient Semigroups Under Perturbations

STABILITY OF GRADIENT SEMIGROUPS UNDER PERTURBATIONS E. R. ARAG˜ AO-COSTA1, T. CARABALLO2, A. N. CARVALHO3, AND J. A. LANGA4 Abstract. In this paper we prove that gradient-like semigroups (in the sense of [4]) are gradient semigroups (possess a Lyapunov function). This is primarily done to provide conditions under which gradient semigroups, in a general metric space, are stable under perturbation exploiting the known fact (see [4]) that gradient-like semigroups are stable under perturbation. The results presented here were motivated by the work carried out in [6] for groups in compact metric spaces (see also [14] for the Morse Decomposition of an invariant set for a semigroup on a compact metric space). 1. Introduction The analysis of qualitative properties of semigroups in general phase spaces (infinitedimensional Banach spaces or general metric spaces) has received much attention throughout the last four decades (see, for instance, [7], [10], [15] or [3]). In particular, the study of compact attracting invariant sets has developed into a large and deep research area, providing vital information for an increasing number of models for phenomena from different areas of Science such as Physics, Biology, Economics, Engineering and others. When a system is shown to possess a global attractor, all its asymptotic behavior can be described by a detailed analysis of the internal dynamics on this compact invariant set. To this aim, a careful study of the geometrical structure -and its stability under perturbations- of the global attractor arises as a crucial fact. Probably the most general result in this line is what is now known as the Fundamental Theorem of Dynamical Systems, suggested in [11] from the results of [6], which describes any flow on a compact metric space as a decomposition of an ordered family of chain recurrent isolated invariant sets and order compatible connections between them. In the terminology of [6], this is called a Morse decomposition 1Partially supported by CAPES/DGU 267/2008 and FAPESP 2008/50248-0, Brazil. 2Partially supported by Ministerio de Ciencia e Innovaci´on grant # MTM2008-0088, PBH2006-0003-PC, and Junta de Andaluc´ıa grants # P07-FQM-02468, # FQM314 and HF2008-0039, Spain. 3Research partially supported by grants CNPq 302022/2008-2, CAPES/DGU 267/2008 and FAPESP 2008/55516-3, Brazil, Ministerio de Ciencia e Innovaci´on grants # MTM2008-0088, and Junta de Andaluc´ıa grant # P07-FQM-02468, Spain. 4Partially supported by Ministerio de Ciencia e Innovaci´on grants # MTM2008-0088, # 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 of a compact invariant set (see Definition 2.10), and has been considered in different frameworks, as in the case of flows ([6]) and semiflows on compact spaces ([14]), or even compact and non-compact topological spaces ( [9, 12, 13]). In [12, 13] the authors obtain (as a corollary of their general theory and using Conley’s proof) the existence of a Lyapunov function from a Morse Decomposition for an open semiflow on a compact Hausdorff space or for a completely regular space. We do not assume that the semiflow is open or any additional property of the metric space nor we use Conley’s proof. On the other hand, very recently, it has been introduced in [4] the so-called gradient-like semigroups (see Definition 2.8) in Banach spaces (which does not requires the existence of a Lyapunov function) as an intermediate concept between gradient semigroups (i.e., those possessing a Lyapunov function) and semigroup possessing a gradient-like attractor; that is, an attractor that is characterized as the union of unstable sets of associated isolated invariant sets. In this paper, given a gradient-like nonlinear semigroup in a general metric space, we construct a differentiable Lyapunov function for it proving that gradient-like nonlinear semigroups are in fact gradient semigroups. This is done without any compactness assumption on the associated group or semigroup and any additional assumption on the phase space in which it is defined. Our proofs, in comparison with the classical works as [6, 14], are quite different and considerably extends the results there. We adopt an approach that enables us to use the results on stability of gradient-like (see [4]) semigroups to obtain the stability of gradient semigroups under perturbations. For the construction of the Lyapunov function we will firstly prove that a 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. Again, the proofs are intuitive, focused on the dynamics of the semigroup and, for instance, not based on chain recurrence and related more classical notions in this theory. A refinement of the results from [6] would lead us to define a Lyapunov function, not only on the attractor but on the whole phase space. 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 gradient semigroup with respect to Ξif there exists a continuous function V:X→Rsuch that [0,∞)∋t7→ V(T(t)x)∈Ris decreasing for each x∈X,Vis constant in Ξifor each 1≤i≤nand V(T(t)x) = V(x) for all t≥0 if and only if x∈ n S i=1 Ξi. Our main result can now be stated as follows Theorem 1.1. {T(t) : t≥0}is a gradient semigroup with respect to Ξif and only if it is a gradient-like semigroup with respect to Ξ. In addition the Lyapunov function V:X→Rof a gradient-like semigroup may be chosen in such a way that V(Ξk) = kfor each k= 1,2,···, n. GRADIENT SEMIGROUPS UNDER PERTURBATIONS 3 One of the straighforward consequences of the previous result is that, if the disjoint family of isolated invariant sets E={z1,···, zn}is made of stationary points, then a nonlinear semigroup is gradient in the sense of [7] (see Definition 3.8.1) if and only if it is gradient-like (see Definition 1.3 in [4]). Then, as gradient-like nonlinear semigroups are stable under perturbation (see [4]), we conclude that gradient semigroups are stable under perturbation; that is, the existence of a continuous Lyapunov function is robust under perturbation. Observe that any Morse decomposition Ξ= (Ξ1,···, Ξn) of a compact invariant set A leads 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 {ξℓ,1≤ℓ≤r}, with limt→−∞ ξℓ(t) = Ξℓand limt→∞ ξℓ(t) = Ξℓ+1, 1 ≤ℓ≤ℓ−1. This defines a partial order and some of the isolated invariant sets in Ξmay not be comparable. In Section 5 we introduce a new Morse decomposition of the attractor of a gradient-like semigroup which improves the construction and dynamical properties of its associated Lyapunov function. Indeed, we prove that, given any 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) which guarantees a total order in A. Each of the levels Ni, 1 ≤i≤pis made of a finite union of the isolated invariant sets in Ξ and Nis totally ordered. The associated Lyapunov function has 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. Before we proceed, let us exhibit several classes of examples that, so far, were not known to be gradient semigroups and that the results in this paper show that they indeed are. Example 1.2. Let A= (ai,j)n i,j=1 ∈Mn×n(R)be a n×nmatrix which satisfies ai,j = 0 if j > i and f∈C1(R,R)be such that there is a ξ > 0with ai,is2+f(s)s < 0whenever |s| ≥ ξ and 1≤i≤n. It is easy to see that the semigroup {T(t) : t≥0}associated to the problem ˙u=Au +F(u) u(0) = u0∈Rn(1.1) where u= (u1,···, un)and F(u) = (f(u1),···, f(un))⊤, has a global attractor Ain Rn. Assume that all equilibria of (1.1) are hyperbolic. Following [4] it is not difficult to see that all global bounded solutions of (1.1) are forwards and backwards asymptotic to equilibria and that homoclinic structures are not allowed. Hence the semigroup associated to (1.1) is gradient-like. We remark that it is not known whether there is a Lyapunov function for (1.1) and that the results proved next ensure that there is a continuous function V:Rn→Rwhich is continuous, decreasing along solutions and with the property that V(T(t)u) = V(u)for all t≥0if and only if uis an equilibrium point of (1.1), that is the semigroup {T(t) : t≥0} (1.1) is gradient (in Rn). 4 E. R. ARAG˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA The problem (1.1) can be changed to a general cascade system of the form ˙u=G(u) u(0) = u0∈Rn(1.2) with G∈C1(Rn,Rn),G(u) = (G1(u),···, Gn(u)) such that (1.2) has a global attractor, the equilibria of (1.2) are all hyperbolic and Gi(u) = G1(u1,···, ui),1≤i≤n. Under these assumptions, the semigroup associated to (1.2) is gradient in Rn. Similar cascade systems for partial differential equations can easily be constructed and we remark that a C1small perturbation of (1.2) will also be gradient-like (according to the results in [4]). Consequently, a C1small perturbation of (1.2) will also give us a gradient semigroup. We present next another example of coupled parabolic partial differential equations with a different nature to show the variety of applicability of our results. Example 1.3. If f∈C2(R),lim sup|u|→∞ f(u) u<0,Ω⊂Rnis a bounded smooth domain, P0∈¯ Ωand p > n, then      ut= ∆u+f(u), x ∈Ω, t > 0, ∂nu(x, t) = 0, x ∈∂Ω, t > 0, u(·,0) = u0∈W1,p(Ω)      vt=vxx +f(v), x ∈(0,1), t > 0 v(0, t) = u(P0, t), vx(1, t) = 0, v(·,0) = v0∈W1,p(0,1) and for a∈C(¯ Ω,Rn)      utt +βut= ∆u+ǫa(x)· ∇u+f(u), x ∈Ω, t > 0 u(x, t) = 0, x ∈∂Ω, t > 0 u(·,0) = u0∈H1 0(Ω), ut(·,0) = v0∈L2(Ω) correspond to gradient-like semigroups in W1,p(Ω)×W1,p(0,1) and H1 0(Ω)×L2(Ω) respectively if all the associated equilibria are hyperbolic and ǫ > 0is suitable small (see [1, 2] and [4]). Hence the associated semigroups are gradient. With this, it becomes evident that the class of semigroups known to be gradient increases considerably after our result is proved. In a general metric space X, consider a gradient-like semigroup {T(t) : t≥0}with respect to the isolated invariant sets Ξ={Ξ1,··· , Ξn}and with a global attractor A. In Section 2 we prove that Ξcan be reordered in such a way that it becomes a Morse decomposition for A. In Section 3 we construct a Lyapunov function in Xfor {T(t) : t≥0}, showing that a gradient like semigroup is a gradient semigroup. Section 4 is dedicated to a recollection GRADIENT SEMIGROUPS UNDER PERTURBATIONS 5 of the results in [4] to conclude that the gradient semigroups are stable under perturbation. In Section 5 we introduce the level grouping isolated invariant sets with same dynamical characteristics. Finally, in Section 6 we make some final comments and present some of our future research on the subject. Acknowledgement. We are very grateful to the anonymous referees for their careful reading and comments that led to a considerable improvement of the presentation of the paper and, at some points, have made the statements and proofs clearer. 2. Morse decomposition of global attractors for gradient-like semigroups In this section we present the notions of gradient-like semigroups and of Morse decomposition for a global attractor as well as the relationship between them. In order to do that we will need to introduce some basic notions and results (see [7] for example). 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 ǫ-neighborhood of Ais the set Oǫ(A) := {x∈X:d(x, a)< ǫ for some a∈A} Next we introduce the notion of semigroups in the metric space X. Definition 2.1. A family {T(t) : t≥0} ⊂ C(X)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+×X∋(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. Now we will introduce the notions of attraction and absorption. For that we recall the definition of Hausdorff semi-distance. Given A, B ⊂X, the Hausdorff semidistance from A to Bis given by dH(A, B) := sup a∈A inf b∈Bd(a, b). Definition 2.3. Given two subsets A, B of Xwe say that Aattracts Bunder the action of the semigroup {T(t) : t≥0}if dH(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 the position 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}. 6 E. R. ARAG˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA Next we seek to introduce the notion of gradient-like semigroups (see [4]). 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⊂X for the semigroup {T(t) : t≥0}is an isolated invariant set if there is an ǫ > 0such that Ξ is the maximal invariant subset of Oǫ(Ξ). Adisjoint family of isolated invariant sets is a family {Ξ1,···, Ξn}of isolated invariant sets with the property that, for some ǫ > 0, Oǫ(Ξi)∩ Oǫ(Ξj) = ∅,1≤i < j ≤n. Definition 2.6. Aglobal 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. 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 := Ξk1and, if p= 1,ξ1(R)(Ξk1. We are now ready to define gradient-like semigroups. 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 gradientlike semigroup relative to Ξif: •For any global solution ξ:R→Athere are 1≤i, j ≤nsuch that Ξi t→−∞ ←− ξ(t)t→∞ −→ Ξj. •There is no homoclinic structure associated to Ξ. It is important to notice that Definition 2.8 only uses dynamical properties of the semigroup (that is, only the inner structure of the attractor) and does not assume a priori the existence of any kind of Lyapunov function (see Definition 3.1 below). Differently from the concept, with the same name, in [6] and [14] (see Section I.6 in [6] and Definition 5.2 in [14]) that assumes the existence of a Lyapunov function. Now we will introduce the notion of a Morse decomposition for an attractor Aof a semigroup {T(t) : t≥0}(see [6] and [14]). We start with the notion of attractor-repeller pairs. GRADIENT SEMIGROUPS UNDER PERTURBATIONS 7 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 ǫ > 0such that ω(Oǫ(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 n+ 1local attractors, for j= 1,···, n, define Ξj:= Aj∩A∗ j−1.The ordered n-upla Ξ:= (Ξ1, Ξ2,···, Ξn)is called a Morse decomposition for A. Observe that Ais a local attractor if and only if it is compact invariant and attracts Oǫ(A) for some ǫ > 0. We observe that the above definition differs slightly from the usual definition since the local attractor is required to attract a neighborhood of Ain Xand not only in Aas in [6, 14]. That is needed because, in this way, we can to show the continuity of the Lyapunov function on whole space X(see Proposition 3.3 below) and not only on the attractor, as is done in [6]. We will prove next that local attraction inside Ais equivalent to local attraction in whole space X. To that end, the next result plays a crucial role, actually it is the key result to almost everything that we do in this work. In some sense, it helps us to deal with the absence of reversibility of the semigroup in the proof of Proposition 3.3 and, furthermore, it improves part a) of Proposition 1.3 in [14]. Lemma 2.11. 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 ǫ > 0such that Aattracts Oǫ(A)∩Athen, given δ∈(0, ε)there is a δ′∈(0, δ)such that γ+(Oδ′(A)) ⊂ Oδ(A), where γ+(Oδ′(A)) = S x∈Oδ′(A)S t≥0 {T(t)x}. Proof. If that conclusion is false, there is an 0 < δ < ǫ such γ+(Oδ′(A)) 6⊂ Oδ(A) for each δ′∈(0, δ). So there are x∈A,X∋xn n→∞ −→ xand R∋tn n→∞ −→ ∞ such that d(T(tn)xn, A) = δand T(t)xn∈ Oδ(A), t∈[0, tn). Since the {T(t) : t≥0}has a global attractor it is not difficult to see that there is a global solution ξ:R→Xsuch that ξn: [−tn,∞)→X, given by ξn(t) := T(tn+t)xn,satisfies ξn(t)n→∞ −→ ξ(t) for each t∈R. Clearly ξ(t)∈Oδ(A)∩A⊂ Oǫ(A)∩Afor all t≤0, and d(ξ(0), A) = δ, and consequently Acannot attract Oǫ(A)∩A. We remark that if Ais a local attractor for a semigroup {T(t) : t≥0}with a global attractor A, then Ais in the conditions of the above lemma. 8 E. R. ARAG˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA The next result generalizes to semigroups a known result for groups given in [6], and shows that our definition of local attractor is equivalent to that found in [6, 14]. Lemma 2.12. 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 is a ε > 0with ω(Oǫ(A)∩A) = A) and Kis a compact subset of Asuch that K∩A∗=∅, then Aattracts K. Furthermore A is a local attractor for {T(t) : t≥0}in X. Proof. Let Kbe a compact subset of Asuch that K∩A∗=∅. From Lemma 2.11, given 0< δ < ǫ, there is a 0 < δ′< δ such that γ+(Oδ′(A)) ⊂ Oδ(A). If A=ω(Oǫ(A)∩A) does not attracts K, there are tn n→∞ −→ ∞,x∈Kand K∋xn n→∞ −→ xsuch that d(T(t)xn, A)⩾δ′, 0⩽t⩽tn. Hence d(T(t)x, A)⩾δ′for all t⩾0 proving that ω(x)∩A=∅. It follows that x∈A∗and that is a contradiction. For the rest of the proof note that ω(Oδ′(A)) ⊂ Oǫ(A)∩Aand, consequently, ω(Oδ′(A))∩ A∗=∅. From the invariance of ω(Oδ′(A)) and from the property that Aattracts Oǫ(A)∩A, we must have that ω(Oδ′(A)) ⊂A. Since ω(Oδ′(A)) attracts Oδ′(A), the result follows.  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}. 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\Athen, T(t)xt→∞ −→ A∪A∗. Proof. Part (i) follows from the Theorem 1.4 in [14]. Part (ii) is divided into two cases (ξ(R)∩A∗=∅and ξ(R)∩A∗6=∅). If ξ(R)∩A∗=∅, from Lemma 2.12 we have that ξ(R)⊂Awhich gives a contradiction. On the other hand, if ξ(R)∩A∗6=∅, it follows from (i) that d(ξ(t), A∗)t→−∞ −→ 0. This completes the proof of (ii). Part (iii) is proved as follows. Let δ > 0 be such that ω(Oδ(A)) = Aand we chose δ′∈(0, δ) with γ+(Oδ′(A)) ⊂ Oδ(A).Then, if there is a t0>0 such that T(t0)x∈ Oδ′(A), we have that lim t→∞d(T(t)x, A) = 0. On the other hand, if d(T(t)x, A)≥δ′for all t≥0 then ω(x)∩A=∅and we must have that ω(x)⊂A∗. Since ω(x) atracts xwe must have that T(t)xt→∞ −→ A∗, completing the proof of (iii).  Part (i) of the previous lemma is done in Theorem 1.4 in [14], the conclusion (iii) (not found in [14, 6]) is needed, in our case, to obtain the continuity of the Lyapunov function in points that do not belong to A. GRADIENT SEMIGROUPS UNDER PERTURBATIONS 9 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 gradient-like semigroup with respect to to the disjoint family of isolated invariant sets {A, A∗}. Next we describe the construction of a Morse decomposition for the attractor of a gradientlike semigroup (relative to the disjoint family of isolated invariant sets {Ξ1,···, Ξn}). This is done by explaining how we can obtain the increasing collection of local attractors starting from the collection of isolated invariant sets {Ξ1,···, Ξn}. The following lemmas play a fundamental role on that. Lemma 2.15. Let {T(t) : t≥0}be a semigroup with a global attractor Aand let Ξbe a compact isolated invariant set such that Wu(Ξ) = Ξ. Then Ξis a local attractor; that is, there is a δ > 0such that ω(Oδ(Ξ)) = Ξ. Proof. Let δ0>0 be such that Ξis the maximal invariant set in Oδ0(Ξ). Let us prove that, given δ∈(0, δ0), there exists δ′∈(0, δ) such that γ+(Oδ′(Ξ)) ⊂ Oδ(Ξ). In fact, if the result was not true, there would exist a δ∈(0, δ0), a sequence {xℓ}in Xwith d(xℓ, Ξ)ℓ→∞ −→ 0 and a sequence {tℓ}in (0,∞) with tℓ ℓ→∞ −→ ∞ (this follows from the invariance of Ξand from the continuity of the semigroup) such that d(T(tℓ)xℓ, Ξ) = δand d(T(t)xℓ, Ξ)≤δfor all 0≤t≤tℓ. Since {T(t) : t≥0}has a global attractor, there is a global solution ξ:R→A such that ξℓ: [−tℓ,∞)→X,ξℓ(t) := T(t+tℓ)xℓfor t≥ −tℓ, satisfies ξℓ(t)→ξ(t) for all t∈R (see Lemma 3.1 in [5] for more details). Clearly ξ(t)∈ Oδ(Ξ) for all t≤0 and d(ξ(0), Ξ) = δ. Now, if z=ξ(0), αξ(z) := {x∈X: limn→∞ ξ(−tn) = x, for some sequence tn n→∞ −→ ∞} is invariant, attracts zand, since Ξis maximal invariant in Oδ0(Ξ), is contained in Ξ. Hence we must have that ξ(t)t→−∞ −→ αξ(z)⊂Ξand that is a contradiction with the fact that Wu(Ξ) = Ξ. It follows from the above that, for any δ∈(0, δ0), there is a δ′∈(0, δ) such that ω(Oδ′(Ξ)) ⊂γ+(Oδ′(Ξ)) ⊂Oδ(Ξ)⊂ Oδ0(Ξ). Now, since Ξis an isolated invariant set, we must have that ω(Oδ′(Ξ)) ⊂Ξ. The other inclusion is trivial and the result follows.  Lemma 2.16. Let {T(t) : t≥0}be a gradient-like semigroup with respect to the disjoint family of isolated invariant sets Ξ={Ξ1,···, Ξn}and let Abe its global attractor. Then, there is a k∈ {1,···, n}such that Ξkis a local attractor for {T(t) : t≥0}. Proof. Assume, by contradiction, that Wu(Ξj)6=Ξjfor all 1 ≤j≤n. Then, for each 1≤j≤n, there is a global solution ξj:R→Asuch that ξj(t)t→−∞ −→ Ξj. From the fact that {T(t) : t≥0}is gradient-like, ξj(t) converges (as t→ ∞) to some element of Ξ. This produces a homoclinic structure and gives a contradiction. Hence, there exists k∈ {1,···, n}such that Wu(Ξk) = Ξkand, from Lemma 2.15, Ξkis a local attractor.  Let {T(t) : t≥0}be a gradient-like semigroup with respect to the disjoint family of isolated invariant sets Ξ={Ξ1,···, Ξn}. If (after possible reordering) Ξ1is a local attractor 16 E. R. ARAG˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA This implies that zis in A⊂Aand that contradicts the fact that z∈X\A. This proves that lim t→∞d(T(t)z, A∗) = 0. Using the same reasoning as in (3.1) we conclude that f(z) = lim t→∞ f(T(t)z) = lim t→∞ k(T(t)z) + lim t→∞ h(T(t)z) = 1 + 0 = 1. This is a contradiction, since k(z)≥k(T(t)z) for all t≥0, and 1 = lim t→∞k(T(t)z)≤k(z)≤ 1, that is, k(z) = 1, but f(z) = k(z) + h(z) and we must have that h(z) = 0 which is a contradiction with the fact that z /∈A. Thus, f(T(t)z) = f(z) for all t≥0 if and only if z∈A∪A∗and this completes the proof.  We observe that the proof of continuity of the Lyapunov function for an attractor-repeller pair found in [6] strongly uses that {T(t) : t∈R}is a group and that Uis a neighborhood for Athen, for all t∈R,T(t)Uis also a neighborhood for A. Since we do not assume that the maps T(t)’s are homeomorphisms, we use Lemma 2.11 to overcome this difficulty, essentially in the proof of the Case 2 above. Also, the role of the function h:X→R, given in Lemma 3.2, is to ensure that, given z∈Xthe only way to have f(T(t)z) = f(z) for all t≥0 is that z∈A∪A∗. We must observe that this can not be true, in the general case, for the function kalone. Finally, the proof of the Case 3) above does not appear in [6], since there the author does not consider points that do not belong to A. That made necessary to prove (iii) in the Lemma 2.13. Theorem 3.4. 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 gradient semigroup with respect to Ξ,in the sense of the Definition 3.1, if and only if it is a gradientlike semigroup with respect to Ξ,in the sense of Definition 2.8. In addition, the corresponding Lyapunov function V:X→Rmay be chosen in such a way that V(Ξk) = k−1,k= 1,···, n. Proof. It is clear that a gradient semigroup with respect to Ξis a gradient-like semigroup with respect to Ξ(See, for example, [7]). Suppose that {T(t) : t≥0}is a gradient-like semigroup relatively to Ξreordered in such a way that it is a Morse decomposition for A. Let ∅=A0⊂A1⊂ · · · ⊂ An=Abe the sequence of local attractors defined in (2.2) and ∅=A∗ n⊂A∗ n−1⊂ · · · ⊂ A∗ 0=Atheir corresponding repellers such that for each j= 1,2,···, n, we have Ξj=Aj∩A∗ j−1. Let fj:X→Rbe the function constructed in Proposition 3.3 for the attractor-repeller pair (Aj, A∗ j), j= 1,···, n and f0=hwith hgiven by Lemma 3.2. Define the continuous function V:X→Rby V(z) := n X j=0 fj(z), z ∈X. GRADIENT SEMIGROUPS UNDER PERTURBATIONS 17 Then V:X→Ris a Lyapunov function for the gradient semigroup {T(t) : t≥0}with respect to Ξ. Indeed, since each fj:X→R, 0 ≤j≤n, are decreasing along solutions of {T(t) : t≥0}, Vis also decreasing along solutions of {T(t) : t≥0}. Now, if z∈Xis such that V(T(t)z) = V(z) for all t≥0, then, using that each fj, 0 ≤ j≤n, are decreasing along solutions of {T(t) : t≥0}, we conclude that fj(T(t)z) = fj(z) for all t≥0 , and for each j= 0,···, n. It follows that f0(z) = 0 and consequently z∈A. From part (iv) of Proposition 3.3, we have that z∈(Aj∪A∗ j), for each j= 0,1,···, n; that is, z∈ n T j=0 (Aj∪A∗ j). From Lemma 2.19 we have that n \ j=0 (Aj∪A∗ j) = n [ j=1 Ξj, and so z∈ n S j=1 Ξj. If k∈ {1,···, n}and z∈Ξk=Ak∩A∗ k−1, it follows that z∈Ak⊂Ak+1 ⊂ · · · ⊂ An=A and z∈A∗ k−1⊂A∗ k−2⊂ · · · ⊂ A∗ 0=A. Hence fj(z) = 0 if k⩽j⩽n,f0(z) = 0 and fj(z) = 1 if 1 ⩽j⩽k−1. Hence, V(z) = n X j=0 fj(z) = k−1 X j=0 fj(z) + n X j=k fj(z) = k−1 X j=0 1 + n X j=k 0 = k−1.  Now, we show that it is possible to improve the last result building a Lyapunov function which is strictly decreasing outside the isolated invariant sets and that is differentiable along solutions. Next we present such a construction. Proposition 3.5. Let {T(t) : t≥0}be a semigroup which possesses a global attractor A. Assume that {T(t) : t≥0}is a gradient-like semigroup with respect to the disjoint family of isolated invariant sets Ξ={Ξ1,···, Ξn}. Then, there is a function W:X→Rwhich is a Lyapunov function for the gradient-like semigroup {T(t) : t≥0}with respect to Ξand such that (i) [0,∞)∋t7→ W(T(t)z)is differentiable for all z∈Xand (ii) [0,∞)∋t7→ W(T(t)z)is strictly decreasing whenever z /∈ n S j=1 Ξj. Proof. Let V:X→Rbe the function defined in Theorem 3.4 and W:X→Rbe the function defined by W(z) := Z∞ 0 e−tV(T(t)z)dt, z ∈X. 18 E. R. ARAG˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA It is easy to see that the function is well defined. Next we prove that this function has the desired properties. We start with the continuity of W. First note that W(z)≤V(z) for all z∈X. Now, from the fact that the semigroup {T(t) : t≥0}is eventually bounded and the definition of Vit is easy to see that for each z∈Xthere is an ǫz>0 and tz>0 such that V(γ+(T(tz)Oǫz(z))) is bounded. Hence, given ε > 0 and z∈Xwe choose ¯ t≥tzand a neighborhood Bof zsuch that, Z∞ ¯ t e−tdt < ε 4(MB+ 1),(3.2) where MB:= sup{V(T(t)w) : w∈B, t ≥tz} ≥ 0. Now, from the continuity of Vand of the mapping [0,∞)×X∋(t, x)7→ T(t)x∈X, it is easy to see that there exists δ > 0 such that, if z∈Xsatisfies d(z, z′)< δ then, Z¯ t 0 e−s|V(T(t)z)−V(T(t)z)|dt ≤ε 2. This and (3.2) show that, for z∈Xwith d(z, z)< δ , |W(z)−W(z)| ≤ Z¯ t 0 e−s|V(T(t)z)−V(T(t)z)|dt + 2MBZ∞ ¯ t e−tdt ≤ε 2+ε 2=ε. Clearly, W:X→Ris decreasing along solutions of {T(t) : t≥0}. Now, if z∈ n S j=1 Ξj, we have that T(t)z∈ n S j=1 Ξjfor all t≥0, and V(T(t)z) is constant for all t≥0, yielding that W(T(t)z) is constant. Conversely, if z∈Xis such that W(T(t)z) = R∞ 0e−sV(T(t+s)z)dt is constant, then V(T(t)z) is constant for all t≥0 and, consequently, z∈ n S j=1 Ξjfrom the properties of V. Next, given z∈X\ n S j=1 Ξjlet us prove that [0,∞)∋t7→ W(T(t)z) is strictly decreasing. Indeed, given t > 0, we have that W(T(t)z)−W(z) = Z∞ 0 e−s[V(T(s+t)z)−V(T(s)z)]ds. From this we see that, if for some t > 0 we have that W(T(t)z)−W(z) = 0, then V(T(s+ t)z)−V(T(s)z) = 0 for all s≥0. In particular, V(T(t)z) = V(z) and, as a consequence of that, V(T(t)z) = V(T(s)z) = V(z) for all s∈[0, t]. Repeating this reasoning we conclude that V(T(s)z) = V(z) for all s≥0, which is in contradiction with the choice of z. Now, given z∈X,t≥0 and h∈Rwe have that W(T(t+h)z)−W(T(t)z) h=et h(eh−1) Z∞ t+h e−sV(T(s)z)ds −Zt+h t e−sV(T(s)z)ds, GRADIENT SEMIGROUPS UNDER PERTURBATIONS 19 which converges to etZ∞ t e−sV(T(s)z)ds −V(T(t)z)≤0, proving the differentiability of [0,∞)∋t7→ W(T(t)z)∈R. The idea of the proof for the last result comes from [6]. Note however that our Lyapunov function V:X→Ris not necessarily bounded as in [6] (the fact that the semigroup is eventually bounded helps to overcome the difficulty that arises). Besides our Lyapunov function is defined on whole (not necessarily compact) space X. 4. Stability under perturbations of gradient semigroups The equivalence between gradient semigroups and gradient-like semigroups, proved in the previous section, together with the results in [4] prove that gradient semigroups are stable under perturbations. In this section we briefly describe this fact. To that end, we first need to introduce a parameter dependent family of semigroups, as well as 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 4.1. We say that a family of semigroups {Tη(t) : t≥0}η∈[0,1], is continuous at η= 0 if Tη(t)xη→0 −→ T0(t)xuniformly for (t, x)in compact subsets of R+×Xas η→0. The notion of collectively asymptotic compactness which is given next plays a fundamental role in the proof of the main result in [4]. Definition 4.2. We say that a family of semigroups {Tη(t) : t≥0}η∈[0,1] is 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 [4]. Theorem 4.3 (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 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⩽n[distH(Ξ∗ i,η, Ξ∗ i,0) + distH(Ξ∗ i,0, Ξ∗ i,η)] η→0 −→ 0. c) {T0(t) : t≥0}is a gradient-like semigroup with respect to Ξ0. Then, there exists η0>0such that, for all η⩽η0,{Tη(t) : t≥0}is a gradient-like semigroup and consequently Aη=∪n i=1Wu(Ξ∗ i,η),∀η∈[0, η0]. 20 E. R. ARAG˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA As an immediate consequence of this result and the ones in Section 3 we have the following result. Corollary 4.4. Under the assumption of Theorem 4.3, there exists η0>0such that, for all η⩽η0,{Tη(t) : t≥0}is a gradient semigroup with respect to Ξη. Corollary 4.5. Under the assumption of Theorem 4.3, suppose there exists n∈Nsuch that Aηhas nstationary solutions Ξη={ξ1,η,···, ξn,η}for all η∈[0,1] and sup1⩽i⩽ndist(ξ∗ i,η, ξ∗ i,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 [7]. 5. Energy level decomposition of a gradient-like system Let {T(t) : t≥0}be a semigroup in Xwith global attractor A. Next we will give a dynamical description of a gradient-like system by reordering and regrouping the corresponding disjoint 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 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 original isolated invariant sets. Assume that {T(t) : t≥0}is a 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: = {Ξℓ∈Ξ: there is no element Ξ∈Ξthat precedes Ξℓ} and, for any integer k≥2 Mk:= {Ξℓ∈Ξ: if Ξ∈Ξand Ξ≺Ξℓthen 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, we consider N1,N2,···,Np,with p:= min{q∈N:Mk=Mqfor each k > q}. GRADIENT SEMIGROUPS UNDER PERTURBATIONS 21 We have the following first result related to this family of sets: Lemma 5.1. Let {T(t) : t≥0}be a semigroup with global attractor A. Assume that {T(t) : t≥0}is a gradient-like semigroup with respect to the disjoint family of isolated invariant sets Ξ={Ξ1, Ξ2,···, Ξn}. Then each element of Ξbelongs to Nk, for some k≤p. Proof. Suppose that N1is empty and fix Ξ∈Ξ. Then, since Ξ /∈N1=∅, by the definition of N1,there exists Ξ1∈Ξwith Ξ1≺Ξand Ξ16=Ξ. Analogously, Ξ1/∈N1, from which we find Ξ2∈Ξwith Ξ2≺Ξ2and Ξ26=Ξ1. Also Ξ26=Ξsince, otherwise, there would be a homoclinic structure Ξ≺Ξ1≺Ξ2=Ξ. Following in this way, as Ξis finite we arrive at a contradiction in a finite number of steps. If Ξ=N1, we are done. If not, we must have that N2is non-empty. Otherwise, given Ξ∈Ξ\N1with Ξ /∈ N2. Then, there exists Ξ1∈Ξ\N1with Ξ≺Ξ1. As before, the fact that {T(t) : t≥0}is a gradient-like semigroup with respect to Ξimplies that Ξ16=Ξ. Since Ξ1/∈N2, there is a Ξ2∈Ξ\N1with Ξ1≺Ξ2and, again, Ξ26=Ξ1and Ξ26=Ξ. As before, we arrive at a contradiction in a finite number of steps, so that N26=∅. If Ξ=N1∪N2=M2,we finish. If not, again N36=∅.As this argument has to finish in a finite number of steps completing the proof.  The following result will show that N= (N1,N2,···,Np) is a Morse decomposition for A. Theorem 5.2. Let {T(t) : t≥0}be a nonlinear semigroup with global attractor A.If {T(t) : t≥0}is a 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 gradient-like semigroup with respect to N. The proof of the result now follows as the proof of Theorem 2.18.  Remark 5.3. Given a compact invariant set A,we define the finest Morse decomposition on A(cf. [13]) as (Ξ1, . . . , Ξn)in which, for each isolated compact invariant set Ξithe unique Morse decomposition on it is the trivial one. Then, the energy level decomposition of the finest Morse decomposition can be considered as the optimal decomposition of A, as it gives a total description of Aby a Lyapunov (energy) function which is constant on each level and strictly decreasing on connecting global solutions among all the different levels. 5.0.1. Energy level decomposition for a gradient-like system. 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: Definition 5.4. Let {T(t) : t≥0}be a gradient-like semigroup in Xwith global attractor Awith equilibrium points E={ζ1,···, ζn}. 22 E. R. ARAG˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA (a)Given ζrand ζs∈E, we say that ζrprecedes ζs(we write ζr≺ζs), if there exists a non-constant global solution ξ:R→Xof {T(t) : t≥0}such that lim t→−∞ d(ξ(t), ζs) = 0 and lim t→∞ d(ξ(t), ζr) = 0. (b)Let us consider M1: = {ζℓ∈E:there is no element ζ∈Ethat preceeds ζℓ} and, for any integer k≥2 Mk:= {ζℓ∈E:if ζ∈ E eζ≺ζℓthen ζ∈ Mk−1}. Note that, by definition, Mk⊂ Mk+1. (c)We now define the sets N1:= M1and Nk:= Mk\Mk−1, for all k≥2. Since Eis finite, there exists a positive integer qsuch that Mk=Mqfor each k > q, so that, Nk=∅for all k > q. Thus, we consider N1,N2,···,Np,with p:= min{q∈N:Mk=Mqfor each k > q}. Lemma 5.5. If {T(t) : t≥0}is a gradient-like semigroup, then every ζ∈Eis in Ni for some i= 1,· · ·, p, where p≥1is the maximum number of non-void energy levels in {T(t) : t≥0}. Lemma 5.6. Let {T(t) : t≥0}be a gradient-like semigroup with global attractor Aand a finite set of equilibria E={ζ1,···, ζn}, and (N1,··· ,Np)the ordered n-upla of energy levels {T(t) : t≥0}. Then, (N1,···,Np)is a Morse decomposition in A. 6. Final comments and further research We have proved that a gradient-like semigroup with respect to a disjoint family of isolated invariant sets Ξis gradient (has a Lyapunov function) with respect to Ξ. In particular, a gradient-like semigroup with respect to a finite set of equilibria is gradient (in the sense of [7]), concluding that gradient semigroups (in the sense of [7]) are stable under perturbation. Note that any global attractor for a semigroup admits at least a Morse decomposition -the trivial one-, being common to have a better description of the internal dynamics of it in several non-trivial examples. Our results apply to any Morse decomposition of a global attractor (which behave continuously with respect to the parameter). The applications considered in the introduction show that the class of semigroups known to be gradient (in the sense of [7]) increases considerably after the results in this paper have been proved. 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Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Berlin (1988; second edition 1996). 24 E. R. ARAG˜ AO-COSTA, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA (E. R. Arag˜ao-Costa and A. N. Carvalho) 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] (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]