Skew Product Semiflows and Morse Decomposition

SKEW PRODUCT SEMIFLOWS AND MORSE DECOMPOSITION M. C. BORTOLAN 1, T. CARABALLO2, A. N. CARVALHO3, AND J. A. LANGA4 Abstract. This paper is devoted to the investigation of the dynamics of non-autonomous differential equations. The description of the asymptotic dynamics of non-autonomous equations lies on dynamical structures of some associated limiting non-autonomous - and autonomous - differential equations (one for each global solution in the attractor of the driving semigroup of the associated skew product semi-flow). In some cases, we have infinitely many limiting problems (in contrast with the autonomous - or asymptotically autonomous - case for which we have only one limiting problem; that is, the semigroup itself). We concentrate our attention in the study of the Morse decomposition of attractors for these non-autonomous limiting problems as a mean to understand some of the asymptotics of our non-autonomous differential equations. In particular, we derive a Morse decomposition for the global attractors of skew product semiflows (and thus for pullback attractors of non-autonomous differential equations) from a Morse decomposition of the attractor for the associated driving semigroup. Our theory is well suited to describe the asymptotic dynamics of non-autonomous differential equations defined on the whole line or just for positive times, or for differential equations driven by a general semigroup. 1. Introduction Recently the analysis of qualitative properties of evolution processes and non-autonomous dynamical systems in general phase spaces - infinite-dimensional Banach spaces or general metric spaces - has received much attention (see, for instance, [4, 5, 6, 7, 10]). In particular, the study of pullback attractors has started to develop into a wide and deep research area, providing qualitative information for the asymptotic dynamics of an increasing number of non-autonomous models of 1Partially supported by FAPESP 2010/50690-5, Brazil. 2Partially supported by FEDER and Ministerio de Econom´ıa y Competitividad grant # MTM2011-22411. 3Partially supported by CNPq 305230/2011-5, CAPES/DGU 238/2011 and FAPESP 2008/55516-3, Brazil, and FEDER and Ministerio de Econom´ıa y Competitividad grant # MTM2011-22411. 4Partially supported by FEDER and Ministerio de Econom´ıa y Competitividad grant # MTM2011-22411. 1 2 M. C. BORTOLAN, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA phenomena from different areas of science, such as physics, biology, economics, engineering and others. The pullback attractors are a natural extension for the notion of global attractors of semigroups, in the sense that the global attractors and the pullback attractors are dynamical objects that contain all global bounded solutions. They help us to glance at the inner structure of the asymptotics of an evolution process {T(t, s) : t⩾s}, and enjoy pullback attraction property (attraction at a fixed time t∈Rwhen s→ −∞). Other kinds of attraction, such as pullback backwards attraction (attraction at time s∈Rwhen t→ −∞, for the case when T(s, t) is invertible and defining T(t, s) = T(s, t)−1,s > t), backwards attraction (attraction at −∞ when time t→ −∞) and forwards attraction (attraction at +∞when t→+∞) are harder to study and, in general, cannot be obtained from pullback attraction. Also, as we will see in Theorem 2.7, the uniform attractor, that is, the projection on the first coordinate of the global attractor for the associated skew product semiflow, is given as a union of (possibly) infinite pullback attractors. So, in order to provide a non-autonomous structure for the uniform attractor, which by itself contains no significant information about the asymptotics of our original non-autonomous system, we need to understand the evolution processes enclosed in the non-autonomous dynamical system; more specifically, understand the pullback attractors of each one of these evolution processes. One of the drawbacks of the theory of pullback attractors is that it requires the vector field to be defined for all times in R, and many models only consider the phenomenon after a given initial time. Of course one can artificially define the vector field for times preceding the given initial one and study the behavior of such system. But then, the pullback attractor would change for each extension and the object “pullback attractor” would lose its importance in the study of the dynamics. The crucial point here is that, to understand the forwards dynamics of a non-autonomous evolution process (as a general rule), we must understand the dynamics of many (possibly infinite) non-autonomous evolution processes, one for each global solution in the attractor for the driving semigroup of the skew product semiflow. This is in contrast with the autonomous case for which we have only one limiting problem (the semigroup itself) or with the asymptotically autonomous case for which we also have only one limiting problem (the limiting semigroup). SKEW PRODUCT SEMIFLOWS AND MORSE DECOMPOSITION 3 To illustrate the many possible limiting evolution processes that will play a role in the understanding of the dynamics of a given non-autonomous differential equation, consider the initial value problem    ˙x=r(t, x), t ≥0 x(0) = x0∈Rn, (1.1) where r: [0,∞)×Rn→Rnis given in Section 3.1. In this case, r(·+tn,·) with tn→ ∞ may converge to f:Rn→Rn,g:Rn→Rn(both independent of t) or to a map ξ:R×Rn→Rn (ψ:R×Rn→Rn) which is backwards asymptotic to g(to f) and forwards asymptotic to f(to g). We note that there are infinitely many ξ’s and ψ’s. To a limiting vector field η(say η=f, g, ξ or ψ) we associate the limiting evolution process Tη(·,·) given by Tη(t, s)x0=x(t, s, x0, η) where x(t, s, x0, η) is the solution of    ˙x=η(t, x), t ≥s x(s) = x0∈Rn, (1.2) at time t. The limiting evolution process Tη(·,·) will play an important role in the understanding of the limiting states for the solutions of (1.1) (as seen in Theorem 2.7). Once one realizes this feature, it becomes clear how rich and difficult is the subject “dynamics of non-autonomous dynamical systems”. We already have some insights of this difficulty when we look at a simple concept like hyperbolicity. Indeed, in the autonomous case we choose an equilibrium, linearize it and we can compute the spectrum of the linearized operator to decide whether we have hyperbolicity or not. In the non-autonomous context, we have no way to single out which solutions will be hyperbolic and, if we were able to single out these solutions, how to verify that they actually are hyperbolic is not a trivial task. If we consider a non-autonomous dynamical system given as a non-autonomous perturbation of an autonomous dynamical system which possesses an hyperbolic equilibria, then the simple matter of obtaining that the perturbed non-autonomous dynamical system has a global hyperbolic solution, near the hyperbolic equilibria of the autonomous dynamical system, involves highly non-trivial results on the roughness of exponential dichotomies (see [3],[4]). Before we proceed, let us recall the notion of global attractor for semigroups. Let Zbe a metric space with metric d:Z×Z→R+, and C(Z) be the space of continuous maps from Zinto Z. Definition 1.1. A family of mappings {T(t) : t≥0} ⊂ C(Z)is a semigroup in Zif 4 M. C. BORTOLAN, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA •T(0) = IZ, with IZbeing the identity map in Z, •T(t+s) = T(t)T(s), for all t, s ∈R+, and •R+×Z∋(t, x)7→ T(t)x∈Zis continuous. Definition 1.2. A subset Aof Zis said to be invariant under the action of the semigroup {T(t) : t≥0}if T(t)A=Afor all t≥0. Now we will introduce the notions of attraction and absorption. Given A, B ⊂Z, the Hausdorff semidistance from Ato Bis given by dH(A, B) := sup a∈A inf b∈Bd(a, b). Definition 1.3. Given two subsets A, B of Zwe say that Aattracts Bunder the action of the semigroup {T(t) : t≥0}if dH(T(t)B, A)t→∞ −→ 0, and 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. Thanks to these previous definitions, we can now define global attractors. Definition 1.4. A subset Aof Zis said to be 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 Zwe have that Aattracts Bunder the action of {T(t) : t≥0}. Let us now consider a general non-autonomous differential equation to illustrate the approach we will carry out in this paper. Consider the initial value problem    ˙x=f(t, x), t > 0 x(0) = x0∈Rn, (1.3) where f:R+×Rn→Rnis a “nice” function which belongs to a metric space C. Assume that, for each f∈ C,x0∈Rn, the solution of (1.3) is defined for all t≥0; that is, for each x0∈Rn, there is a unique continuous function R+∋t7→ x(t, f, x0)∈Rnsatisfying (1.3). Now, following [13] we define the skew product semiflow associated to (1.3) in the following way Π(t) : Rn×C→Rn×C Π(t)(x0, f) = (x(t, f, x0), ft), SKEW PRODUCT SEMIFLOWS AND MORSE DECOMPOSITION 5 where ft(s, x) = f(t+s, x) for all t, s ≥0 and x∈Rn. Suppose that the map R+×C×Rn∋ (t, f, x0)7→ (x(t, f, x0), ft)∈Rn×Cis continuous. It is easy to see that x(t+s, f, x0) = x(t, fs, x(s, f, x0)). From this we have that Π(t+s)(x0, f) = (x(t+s, f, x0), ft+s) = (x(t, fs, x(s, f, x0)), ft(fs)) = Π(t)(x(s, f, x0), fs) = Π(t)Π(s)(x0, f). Assume that R+×C×Rn∋(t, f, x0)7→ Π(t)(x0, f)∈Rn×Cis continuous ({Π(t) : t⩾0}is a semigroup in Z=Rn×C) and that {Π(t) : t≥0}possesses a global attractor Ain Z. Then, it may seem that we have found a proper way to study the asymptotic dynamics of (1.3). In fact, the set Apossesses dynamics associated to {Π(t) : t≥0}but it does not have any dynamics immediately associated to (1.3). An element of Ais an element of Rn×C; that is, an initial condition y0∈Rnand a vector field g(which is not fin general) and Π(t)(y0, g) = (x(t, g, y0), gt) has no straightforward relation to (1.3). Let us try to unravel a little the connection of the points in Awith the dynamics of (1.3). The first step to study the dynamics of (1.3) is to understand the attraction property of {Π(t) : t≥0}as t→ ∞ relatively to the solution operator of (1.3). Given a bounded subset Bof Rn×C,Aattracts Bunder the action of {Π(t) : t≥0}if lim t→∞ dH(Π(t)B,A) = 0. If, for a given bounded subset B⊂Rn, we only consider a bounded subset Bof the form B×{f}, this attraction property can be written as lim t→∞ dH(x(t, f, B)× {ft},A)≥lim t→∞ dH(x(t, f, B), A), where A={x∈Rn: there exists g∈ C such that (x, g)∈ A}. This means that the compact set A⊂Rnattracts bounded subsets of Rn. Although the set Adoes not have any dynamical property relatively to (1.3), we will see that some families in it are crucial to understand the dynamics of (1.3). Given a non-autonomous differential equation such as (1.3), we can refer to three different but closely related dynamical systems: •The driving system {Θ(t) : t⩾0}associated to the dynamics of the time-dependent terms appearing in the equation, and which is defined by Θ(t)f(·, x) = ft(·, x) = f(t+·, x), 6 M. C. BORTOLAN, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA •the skew-product semiflow {Π(t) : t⩾0}defined on the product space Z=Rn×C, •and the associated non-autonomous dynamical system (ϕ, Θ)(Rn,C)with ϕ(t, Θsf)x0=x(t+ s, f, x0). Next, we give the precise definition of the skew product semiflow associated to a non-autonomous dynamical system. Definition 1.5. Consider two metric spaces (X, dX)and (P, dP). A non-autonomous dynamical system (NDS), denoted by (ϕ, Θ)(X,P), consists of two ingredients: (i) Adriving semigroup {Θ(t) : t⩾0}in P. (ii) Acocycle ϕ:R+×P×X→Xover Θ, that is, a continuous map such that the family of mappings ϕ(t, p) : X→Xsatisfies the cocycle property: 1) ϕ(0, p) = IXfor all p∈P, 2) ϕ(t+s, p) = ϕ(t, Θ(s)p)ϕ(s, p)for all t, s ⩾0and p∈P. The associated skew product semiflow (SPSF) {Π(t) : t≥0} ⊂ C(X×P)is given by Π(t)(x, p) = (ϕ(t, p)x, Θ(t)p). It is clear that {Π(t) : t⩾0}is a semigroup with phase space1X×P. The aim of this paper is to describe the internal structure and dynamics (in the sense of Morse decomposition and Lyapunov functions) of the global attractor for the skew product semiflow and its relation with the pullback attractors for the associated (limiting) non-autonomous dynamical systems. In Section 2 we recall some results, define some basic concepts and establish the notation we will use throughout the paper. Our first result (Theorem 2.7) says that for each global solution ηin the attractor of the driving system {Θ(t) : t⩾0}we have an associated non-autonomous evolution process {Tη(t, s) : t⩾s}with a pullback attractor {Aη(t) : t∈R}. There are some results in this direction in the literature (see [5] and [2]) which consider that the driving semigroup {Θ(t) : t⩾0} is in fact a group defined in a compact metric space. In Section 4 we construct a Morse decomposition (see Definitions 2.18 and 2.19) for the global attractor of the skew product semiflow {Π(t) : t⩾0}from a Morse decomposition for the global attractor of the driving system {Θ(t) : t⩾0}, using the lift of this Morse decomposition (Theorem 4.3); that is, if we have a Morse decomposition {M1,··· ,Mn}for the attractor of {Θ(t) : t⩾0} 1We denote the metric in X×Psimply by d. SKEW PRODUCT SEMIFLOWS AND MORSE DECOMPOSITION 7 then, the family of lifts {M1,··· ,Mn}(Mi={(x, p)∈ A :p∈Mi},i= 1,··· , n), is a Morse decomposition for the attractor Aof the skew product semiflow {Π(t) : t≥0}. We also verify under which conditions, a Morse decomposition in the global attractor of the skew product semiflow {Π(t) : t⩾0}generates a Morse decomposition in the global attractor of the driving system {Θ(t) : t⩾0}(Theorem 4.6); that is, if {M1,··· ,Mn}is a Morse decomposition of the global attractor Aof {Π(t) : t⩾0}and {M1,··· ,Mn}(with Mi={p∈P: there exists x∈ Xsuch that (x, p)∈ Mi}, 1 ≤i≤n) are disjoint, then they constitute a Morse decomposition for the global attractor Aof {Θ(t) : t⩾0}. In Section 5 we construct a Morse decomposition for the pullback attractor of the non-autonomous dynamical system (ϕ, Θ)(X,P)and we obtain some dynamical properties of this Morse decomposition. In Theorem 5.6 we prove that, under some hypotheses, each solution of the non-autonomous dynamical system converges forwards to ∪ p∈Mi Mi(p), for some i= 1,· · · , n; where {M1,··· ,Mn} is a Morse decomposition of {Θ(t) : t⩾0}and {M1,··· ,Mn}are their lifts. In Theorem 5.7 we prove that a Lyapunov function for {Θ(t) : t⩾0}generates a Lyapunov function for {Π(t) : t⩾0}, which in turn is a non-autonomous Lyapunov function for the non-autonomous dynamical system (ϕ, Θ)X×P. In Section 6 we describe how a Morse decomposition of a pullback attractor is stable under perturbation of the parameter of the associated driving system and particularly, in Theorem 6.6 (Theorem 6.8), under certain stability assumptions, we prove that if ξis a global solution for {S(t, s) : t⩾s}, where {S(t, s) : t⩾s}is a forwards (backward) asymptotically autonomous evolution process to a semigroup {S0(t) : t⩾0}with Morse decomposition {Γ1,0,··· ,Γn,0}, then ξ(t)→Γi,0, for some i= 1,··· , n, when t→ ∞ (t→ −∞). Finally, in Section 7 we present some applications of our theory that can help us understand a little more of its aspects. An asymptotically autonomous (backward and forwards) non-autonomous differential equation, i.e.    ˙x=f(t, x), t ∈R, x(0) = x0∈Rn, where sup x∈Rn kf(t, x)−f2(x)kRnt→−∞ −→ 0,and sup x∈Rn kf(t, x)−f1(x)kRnt→∞ −→ 0, 8 M. C. BORTOLAN, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA for suitable functions f1and f2, and also a master-slave example, i.e. a system of partially coupled equations            ˙v=f(u, v)t > 0 ˙u=g(u), t > 0 u(0) = u0∈Rn, v(0) = v0∈Rn in which the second equation for u(t) acts as a driving system for the unknown v(t). Finally we present a more concrete example to illustrate the use of the abstract theory, studying the behavior of a planar system of ODE’s given by d dt(x, y) = F(t, (x, y)), t ∈R, where F:R×R2→R2is forwards and backwards asymptotically autonomous to some special vector fields. 2. Preliminaries In this section we state the definitions and some known results which will be used throughout the following sections. In particular, we pay special attention to the concept of pullback attractor. 2.1. Non-autonomous dynamical systems and pullback attractors. With Definition 1.5 in mind we can see that given an NDS (ϕ, Θ)(X,P)and a set R⊂Pwhich is invariant for {Θ(t) : t⩾0}, we can consider the restriction Θ(t)|R:R→Rand the restriction ϕ|R+×R×X: R+×R×X→X, so that we have a new NDS. In this case, the associated skew product semiflow is {Π(t)|X×R:t⩾0}in the phase space X×R. Definition 2.1. A family of subsets {D(t)}t∈Rof Xis called a non-autonomous set. If each fiber D(t)is closed/compact/open, then {D(t)}t∈Ris called a non-autonomous closed/compact/open set. Definition 2.2. Aglobal solution for a semigroup {Θ(t) : t≥0}is a continuous function η:R→ Psuch that Θ(t)η(s) = η(t+s)for all s∈Rand all t∈R+. We say that η:R→Pis a global solution through p∈Pif it is a global solution with η(0) = p. Definition 2.3. Given a global bounded solution η:R→Pof the driving system {Θ(t) : t⩾0}, a non-autonomous set {D(t)}t∈Ris said to be η-forwards invariant under the NDS (ϕ, Θ)(X,P)if SKEW PRODUCT SEMIFLOWS AND MORSE DECOMPOSITION 9 ϕ(t, η(s))D(s)⊂D(t+s)for all s∈Rand t⩾0. It is said to be η-invariant if ϕ(t, η(s))D(s) = D(t+s)for all s∈Rand t⩾0. Definition 2.4. Given a global bounded solution η:R→Pof the driving system {Θ(t) : t⩾0} and two non-autonomous sets {D(t)}t∈Rand {A(t)}t∈R, we say that {A(t)}t∈Rη-pullback attracts {D(t)}t∈Rif lim t→∞ dH(ϕ(t, η(s−t))D(s−t), A(s)) = 0,for each s∈R. Definition 2.5. Auniverse Dis a collection of nonempty non-autonomous sets which is closed with respect to set inclusion, i.e. if {D1(t)}t∈R∈Dηand D2(t)⊂D1(t)for all t∈R, then {D2(t)}t∈R∈D. A non-autonomous compact set {A(t)}t∈R∈Dηis called a (D, η)-pullback attractor of (ϕ, Θ)(X,P)if (i) {A(t)}t∈Ris η-invariant; (ii) {A(t)}t∈Rη-pullback attracts all families {D(t)}t∈R∈D. Remark 2.6. The above definitions are a simple rewriting of the known definitions for the nonautonomous setting given in [4] for the case of a non-injective driving system {Θ(t) : t⩾0}, where there may be more than one global solution through a given point p∈P. Another important fact is the relationship between the global attractor of a skew product semiflow and the pullback attractors of the evolution processes it may contain. Such a relation is expressed in our next result. Theorem 2.7. Assume that the skew product semiflow {Π(t) : t≥0}possesses a global attractor A, consequently, the driving system {Θ(t) : t≥0}has a global attractor A. If η(·) : R→Pis a global bounded solution for {Θ(t) : t≥0}then, the evolution process {Tη(t, s) : t≥s}given by Tη(t, s)x=ϕ(t−s, η(s))x, x ∈X, possesses a (D, η)-pullback attractor {Aη(t) : t∈R}with the property that Aη(t) = {x∈X: (x, η(t)) ∈ A}, where Dis the collection of all non-autonomous sets {D(t)}t∈Rsuch that ∪ t∈RD(t) is bounded in X. Of course, A=([ t∈R Aη(t)× {η(t)}, η(·)is a global bounded solution for {Θ(t) : t≥0}) 16 M. C. BORTOLAN, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA We can also see that Θ(t2 n)r→ξ1(t, x) := h1(t)f(x) + (1 −h1(t))g(x) uniformly in compact subsets of R×Rnand Θ(t4 n)r→ξ2(t, x) := h2(t)f(x) + (1 −h2(t))g(x) uniformly in compact subsets of R×Rn. It is clear that ξ1and ξ2are global solutions for the driving system {Θ(t) : t⩾0},ξ1is a connection between fand gand ξ2is a connection between gand f. We note that Θ(τ)ξ1(Θ(τ)ξ2) is also a connection between fand g(gand f), for any τ∈R. In this way we have found some of the possible limiting vector fields that may arise in the closure of the positive orbit (in the attractor of the driving semigroup). As one can see, these limiting vector fields may have completely different structure from that of r. In particular in what was described above, there are four different vector fields involved; that is, f,g, a non-autonomous map that is backwards asymptotic to fand forwards asymptotic to gand a map that is backwards asymptotic to gand forwards asymptotic to f. After the analysis done in Section 2.2, it will become clear that fand gbelong to the same Morse set for any given Morse Decomposition of the global attractor of the driving system {Θ(t) : t≥0} in γ+(r). Remark 3.1. 1. This example gives us a first understanding of how rich (and difficult) becomes the behaviour of non-autonomous dynamical systems; that is, if the non-autonomous dynamical system associated to (1.1) has a uniform attractor and {tn}n∈Nis a sequence with the property that tn n→∞ −→ ∞, there is a subsequence {tnk}k∈Nand ψ:R×Rn→Rnsuch that r(·+tnk, x)k→∞ −→ ψ(·, x), uniformly for compact subsets of R×Rn. Therefore, the forwards behavior of the pullback attractor of this equation (when restricted to this sequence, as in Theorem 2.7) is not related to the initial equation. It is related to the equation    ˙x=ψ(t, x), t ∈R, x(0) = x0∈Rn, which is the limiting problem along this subsequence (that is, a point in the ω-limit of r(·, x)under the action of the driving semigroup of translations). In the same way, for r:R×Rn→Rnand sequences converging to −∞, we can see that the backward behavior of the non-autonomous equation is related to another non-autonomous equation (also a limiting problem associated with a point in the α-limit of the driving semigroup of translations). SKEW PRODUCT SEMIFLOWS AND MORSE DECOMPOSITION 17 But this analysis works only for this subsequence, so that we could have another subsequence of {tn}n∈Nconverging to a different non-autonomous function. This is in contrast to the autonomous or forwards (backwards) asymptotically autonomous cases for which the forwards attractor (backwards attractor) is a single point. Also, we can see that the behavior of a non-autonomous equation (both forwards and backwards) are related to points in the ω-limit and α-limit of points r(·, x)under the action of the semigroup of translations; which in general are a lot smaller than the whole hull γ(r) of the function r. Therefore, contrary to what we frequently see in the literature, to deal with the asymptotic dynamics of the non-autonomous differential equation we just have to take into account the global attractor of the driving semigroup of translations defined on the hull γ(r). 2. In Example 3.1, we could have that h1(t) = 1 for t≤ −1and h1(t) = 0 for t≥1and h2= 1 −h1(for instance) that would give rise to a semigroup for which the connections reach the equilibria in finite time (a feature that does not occur when we are working with semigroups generated by ODE’s). 3. Also in Example 3.1, if we construct the function hin a different way, different phenomena may arise, for instance, taking hcontinuous, h≡1in the intervals [a1 n, a2 n],h≡0in the intervals [a3 n, a4 n]and linear in the intervals [a2 n, a3 n]and [a4 n, a5 n],n∈N, the points in ω(r) will include the continuum of equilibria for {Θ(τ) : τ≥0}given by the segment connecting fand g; that is, {αf + (1 −α)g:α∈[0,1]}(see Figure 1(b)). Indeed, with the function h constructed in this manner, we would have h(s) = s−a3 n a2 n−a3 n ,if s∈[a2 n, a3 n], thus if α∈[0,1] and t2 n=a2 n+ (1 −α)(a3 n−a2 n), we have h(t+t2 n) = t a2 n−a3 n +α, for as long as t+t2 n⩽a3 n. Therefore, for tin a compact set of R+we have that h(t+t2 n)→αas n→ ∞, which in turn implies that Θ(t2 n)r(t, x)→αf(x) + (1 −α)g(x)as n→ ∞. 4. In Example 3.1, the semigroup {Θ(τ) : τ≥0}in Z=γ+(r)has two invariant sets, namely Ξ1={f}and Ξ2={g}, and both of them are chain recurrent (see Figure 1(a)). In the 18 M. C. BORTOLAN, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA context of part 3above, {f}and {g}are not isolated and there is a continuum of equilibria for the semigroup {Θ(τ) : τ≥0}in Z=γ+(r) (the segment whose endpoints are fand g, see Figure 1(b)). b b f g (b)Continuum of equilibria ξ2 ξ1 (a)Chain recurrent equilibria b b f g Figure 1 4. Morse decomposition for a skew product semiflow In this section we will describe the relationship between the Morse decompositions of the skew product semiflow, the driving semiflow and the pullback attractors associated to evolution processes related to the global solutions of the driving semiflow. Indeed, our primary interest is to obtain a Morse decomposition for the global attractor Aof the skew product semiflow {Π(t) : t⩾0}in terms of a Morse decomposition of the global attractor A of the driving system {Θ(t) : t⩾0}. 4.1. The lift of a Morse decomposition from Pto X×P. Definition 4.1. Given any R⊂Pand D ⊂ X×P, we define the subset LD R⊂X×Pby LD R={(x, p)∈ D :p∈R}. The set LD Ris called the lift of Rin D. Remark 4.2. If ψ2:X×P→Pis the projection on the second coordinate, that is, ψ2(x, p) = p for all (x, p)∈X×P, then we can see that LD R=ψ−1 2(R)∩ D. We can now prove the main theorem of this section: SKEW PRODUCT SEMIFLOWS AND MORSE DECOMPOSITION 19 Theorem 4.3. Let (ϕ, Θ)(X,P )be a non-autonomous dynamical system and {Π(t) : t⩾0}the associated skew product semiflow. Assume that {Π(t) : t⩾0}has a global attractor A(hence, {Θ(t) : t⩾0}has a global attractor)and that the global attractor Aof the driving semigroup {Θ(t) : t⩾0}possesses a Morse decomposition {M1,··· ,Mn}. Define, for each i= 1,··· , n, the set Mi. =LA Mi. Then, the family {M1,··· ,Mn}is a Morse decomposition for the global attractor Aof {Π(t) : t⩾0}. Moreover, the set Micoincides with the global attractor of the semigroup {Πi(t) : t⩾0}defined on X×Miby Πi(t) = Π(t)|X×Mifor each i= 1,··· , n. Proof: First we prove the non-emptiness of each Mi. Let p∈Mi(since it is non-empty, such p exists) and take x∈X. Since Aattracts points of X×Punder the action of {Π(t) : t⩾0}, we have that lim t→∞ dH(Π(t)(x, p),A) = 0. Since Ais compact and non-empty, we know that there exist a sequence {tn}n∈Nand a point (x0, p0)∈ A such that tn→ ∞ and Π(tn)(x, p)→(x0, p0). But Π(tn)(x, p) = (ϕ(tn, p)x, Θ(tn)p), and thus ϕ(tn, p)x→x0and Θ(tn)p→p0. Since Miis closed and invariant, p0∈Mi. Hence (x0, p0)∈ A and p0∈Mi, which shows that (x0, p0)∈ Mi. Now we prove the invariance of each Miunder the action of {Π(t) : t⩾0}. Let (x, p)∈ Mi. Then Π(t)(x, p) = (ϕ(t, p)x, Θ(t)p), and, since Ais invariant by Π, (ϕ(t, p)x, Θ(t)p)∈ A. From the invariance of Mi, Θ(t)p∈Miand thus Π(t)(x, p)∈ Mi. Now, if (y, q)∈ Mi, from the invariance of Miunder {Θ(t) : t≥0}, there exists p∈Misuch that Θ(t)p=q. Let η:R→Abe a global solution of {Θ(t) : t≥0}through pat t= 0. It follows that η(t) = q. Define the process {Tη(t, s) : t≥s}in Xas in Theorem 2.7 and let {Aη(t) : t∈R} be its pullback attractor. Since (y, q)∈ A we have that y∈ Aη(t). Choose x∈ Aη(0) such that Tη(t, 0)x=y, consequently, Π(t)(x, p) = (y, q). Since (x, p)∈ Aη(0) × {η(0)} ⊂ A and p∈Mi, we have that (x, p)∈ Mi. Since Mi⊂ A, to show that it is compact it remains to show that it is closed. Assume that {(xn, pn)}n∈Nis a sequence in Misuch that (xn, pn)→(x, p) as n→ ∞. Clearly (x, p)∈ A and since {pn}n∈N⊂Mi,p∈Miand thus (x, p)∈ Mi. 20 M. C. BORTOLAN, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA Now, if (x, p)∈ Mi∩ Mj, with i6=j, then it implies that p∈Mi∩Mj, which is a contradiction since they are disjoint. Therefore Miand Mjare disjoint if i6=j. Now, given a global solution ξ:R→ A for {Π(t) : t⩾0}we have that ξ(t) = (x(t), η(t)),for all t∈R, where η:R→Pis a global solution for {Θ(t) : t⩾0}and x:R→Xsatisfies ϕ(t, η(s))x(s) = x(t+s) for all t⩾0 and s∈R. Since {M1,··· ,Mn}is a Morse decomposition for P, there exist 1⩽i < j ⩽nsuch that Mj t→−∞ ←− η(t)t→∞ −→ Mi. We will show now that ξ(t)→ Mi. For this purpose, assume that this is not the case, that is, assume that there exist ǫ0>0 and a sequence {tn}n∈Nwith tn→ ∞ as n→ ∞ and d(ξ(tn),Mi)⩾ǫ0,for all n∈N. But {ξ(tn)}n∈N⊂ A and we can also assume that ξ(tn)→(x, p)∈ A. Since η(tn)→Mi, it follows that p∈Miand thus (x, p)∈ Mi, but d((x, p),Mi)⩾ǫ0, and since Miis compact, this is a contradiction. Analogously, ξ(t)→ Mjas t→ −∞. In a similar way, we can prove that there are no homoclinic structures in {M1,··· ,Mn},as there are no homoclinic structures in {M1,··· ,Mn}. To prove the second statement, fix i= 1,··· , n and let Aibe the global attractor of {Πi(t) : t⩾0}. It is easy to see that Ai⊂ A. Now if (x, p)∈ Ai, then p∈Miand thus (x, p)∈ Mi. Conversely, if (x, p)∈ Mi, then (x, p)∈ A, and thus there exists a global solution ξ:R→ A. But, by the invariance of Mi, we have that ξ(t)∈ Aifor all t∈R, and hence (x, p)∈ Ai. 4.2. The projection of a Morse decomposition from X×Pto P. We are now interested in the opposite problem. Indeed, we investigate when a given Morse decomposition on the global attractor Aof the skew product semiflow {Π(t) : t⩾0}generates a Morse decomposition in the global attractor Aof the driving system {Θ(t) : t⩾0}. Definition 4.4. Given any D ⊂ X×Pand R⊂P, we define the subset QR Dby QR D={p∈R: (x, p)∈ D for some x∈X}. The set QR Dis called the P-projection of Dover R. SKEW PRODUCT SEMIFLOWS AND MORSE DECOMPOSITION 21 Remark 4.5. Notice again that if ψ2:X×P→Pdenotes the projection on the second coordinate, then it is clear that QR D=ψ2(D)∩R. Theorem 4.6. Assume that {M1,··· ,Mn}is a Morse decomposition for the global attractor A of the skew product semiflow {Π(t) : t⩾0}. Let Abe the global attractor of the driving system {Θ(t) : t⩾0}. Define Mi. =QA Mi, for each i= 1,··· , n and assume that the family {M1,··· ,Mn}is disjoint. Then, {M1,··· ,Mn}is a Morse decomposition for A. Proof: We know that Miis compact and non-empty for each i= 1,· · · , n. To see that Miis invariant under the action of {Θ(t) : t⩾0}let p∈Miand x∈Xsuch that (x, p)∈ Mi. Since Miis invariant for {Π(t) : t⩾0}, we have that Π(t)(x, p) = (ϕ(t, p)x, Θ(t)p)∈ Mi, therefore Θ(t)p∈Mi. Now, let q∈Miand y∈Xsuch that (y, q)∈ Miand from the invariance of Mi, there exists (x, p)∈ Misuch that Π(t)(x, p) = (y, q). Thus p∈Miand Θ(t)p=qand concludes the invariance of Miunder the action of {Θ(t) : t⩾0}. Now, given a global solution η:R→Aof {Θ(t) : t⩾0}, consider the associated evolution process {Tη(t, s) : t⩾s}, given by Theorem 2.7, and {Aη(t) : t∈R}its pullback attractor. Let x:R→Xbe a global solution for this evolution process; that is, x(t)∈ Aη(t) for all t∈Rand Tη(t, s)x(s) = x(t) for all t⩾s. Define ξ:R→ A by ξ(t) = (x(t), η(t)),for all t∈R. In this way, ξis a global solution for the skew product semiflow {Π(t) : t⩾0}and since {M1,··· ,Mn}is a Morse decomposition for A, we have that, for some i < j, Mj t→−∞ ←− ξ(t)t→∞ −→ Mi, which means that Mj t→−∞ ←− η(t)t→∞ −→ Mi. Again, there are no homoclinic structures in {M1,··· ,Mn}since there are no homoclinic structures in {M1,··· ,Mn}. 22 M. C. BORTOLAN, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA 5. Morse decomposition for pullback attractors We begin this section stating some known results about the Morse decomposition for pullback attractors of NDS in [2], where the approach used there requires that the set Pbe compact and that {Θ(t) : t⩾0}be in fact a group over P, i.e. Θ(t) : P→Pis invertible, with its continuous inverse given by Θ(t)−1= Θ(−t) (in particular P=Aand there is a unique global solution through each point p∈P). Given a set D ⊂ X×P, we define the p-section of Das the set D(p)⊂Xas D(p) = {x∈X: (x, p)∈ D}. In the same way, given a non-autonomous set {D(p)}p∈P, with D(p)⊂Xfor all p∈P, we define D ⊂ X×Pas D=[ p∈P D(p)× {p}. Also, given D ⊂ X×Por a non-autonomous set {D(p)}p∈Pand R⊂Pwe define the X- projection of D(or the X-projection of {D(p)}p∈P) along Rby PR(D) = PR({D(p)}p∈P) = ∪ p∈RD(p). In this setting, we can define the pullback attractor for the NDS (ϕ, Θ)(X,P)in the following way: Definition 5.1. Let Bbe the collection of all bounded sets in X. A non-autonomous set {S(p)}p∈P is a pullback attractor of (ϕ, Θ)(X,P)if: (i) PP({S(p)}p∈P)∈ B and each S(p)is compact; (ii) {S(p)}p∈Pis invariant under the NDS (ϕ, Θ)(X,P), i.e. ϕ(t, p)S(p) = S(Θ(t)p),for all t⩾0and p∈P; (iii) {S(p)}p∈Ppullback attracts all non-autonomous sets {D(p)}p∈Psuch that PP({D(p)}p∈P)∈ B, i.e. lim t→∞ dH(ϕ(t, Θ(−t)p)D(Θ(−t)p),S(p)) = 0,for all p∈P. Theorem 5.2 (Theorem 3.3 in [2]; see also Proposition 3.31 in [10]).Assume that A=[ p∈P A(p)× {p}is the global attractor of the skew product semiflow {Π(t) : t⩾0}. Then, the associated non-autonomous set {A(p)}p∈Pis the pullback attractor of the NDS (ϕ, Θ)(X,P). SKEW PRODUCT SEMIFLOWS AND MORSE DECOMPOSITION 23 Definition 5.3. Let Abe the global attractor of the skew product semiflow {Π(t) : t⩾0}. A nonautonomous compact pair ({H(p)}p∈P,{H∗(p)}p∈P)is called a pullback attractor-repeller pair in {A(p)}p∈Pif the associated pair (H,H∗)is an attractor-repeller of the global attractor Aof Π(t). Definition 5.4. Let Abe the global attractor of the skew product semiflow {Π(t) : t⩾0}. Let ({Hi(p)}p∈P,{H∗ i(p)}p∈P),i= 0,··· , n, be pullback attractor-repeller pairs in {A(p)}p∈Pwith ∅=H0(p) H1(p) ··· Hn(p) = A(p) and A(p) = H∗ 0(p)!H∗ 1(p)!···!H∗ n(p) = ∅ for all p∈P. Then, the family {M1(p)}p∈P,··· ,{Mn(p)}p∈Pof invariant non-autonomous compact sets, defined by Mi(p) = Hi(p)∩ H∗ i−1(p),for all p∈Pand 1≤i≤n, is called a pullback Morse decomposition of {A(p)}p∈P, and each {Mi(p)}p∈Pis called pullback Morse set. By Theorem 4.3 and Theorem 5.1 in [2] we can now prove the following result describing the internal asymptotic dynamics of a pullback Morse decomposition for the pullback attractor {A(p)}p∈P: Theorem 5.5 (Theorem 5.1 in [2]).Assume that the driving system {Θ(t) : t⩾0}has a global attractor Awith a Morse decomposition {M1,··· ,Mn}. Define, for each i= 1,··· , n as in Theorem 4.3, the set Mi. =LA Mi. The family {M1,··· ,Mn},is then a Morse decomposition for the global attractor Aof {Π(t) : t⩾0}. Then, the NDS (ϕ, Θ)(X,A)has a pullback Morse decomposition {M1(p)}p∈A,··· ,{Mn(p)}p∈A, where Mi(p) = {x∈X: (x, p)∈ Mi}, is the p-section of Mi, for all p∈Aand i= 1,··· , n. Assume that {M1(p)}p∈A,··· ,{Mn(p)}p∈A is described by pullback attractor-repeller pairs ({Hi(p)}p∈A,{H∗ i(p)}p∈A),i= 1,...,n such that PA({Hi(p)}p∈A)∩PA({H∗ i(p)}p∈A) = ∅for i= 1,...,n. Then, the collection of pullback Morse sets determines the limiting behavior of NDS ϕon {A(p)}p∈A. More precisely, we have: 24 M. C. BORTOLAN, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA (i) For any singleton non-autonomous set {x(p)}p∈Ain {A(p)}p∈A, by writing x=Sp∈Ax(p)× {p},if dH(x, n [ i=1 H∗ i)>0or x∈ n [ i=1 H∗ i, we have lim t→+∞dH(ϕ(t, Θ(−t)p)x(Θ(−t)p), n [ i=1 Mi(p)) = 0, for each p∈A. (ii) If ϕis invertible on {A(p)}p∈Athen, for any singleton non-autonomous set {x(p)}p∈Ain {A(p)}p∈Awith dH(x, n [ i=1 Hi)>0or x∈ n [ i=1 Hi we have lim t→+∞dH(ϕ(−t, Θ(t)p)x(Θ(t)p), n [ i=1 Mi(p)) = 0, for each p∈Aand ϕ(−t, Θ(t)p). = (ϕ(t, p))−1. (iii) Moreover, under the hypotheses of Theorems 5.2 and 5.5, the family {Mi(p)}p∈Miis the pullback attractor of the NDS (ϕ, Θ)(X,Mi). In particular, we have that lim t→∞ sup p∈Mi dH(ϕ(t, p)B, PMi({Mi(p)}p∈Mi)) = 0, for all B⊂Xbounded. We can now state the main new result of this section, concerning the asymptotic behavior of the solutions for the NDS (ϕ, Θ)(X,P)as t→ ∞ and for any p∈P. Theorem 5.6. Let (ϕ, Θ)(X,P)be a non-autonomous dynamical system and {Π(t) : t⩾0}the associated skew product semiflow. Assume that {Π(t) : t⩾0}possesses a global attractor A(hence the driving system {Θ(t) : t⩾0}has a global attractor A) and let {M1,··· ,Mn}be a Morse decomposition for A. Let {M1,··· ,Mn}be the corresponding Morse decomposition for the global attractor Aof {Π(t) : t⩾0}, given by Theorem 4.3. Then, given (x, p)∈X×P, there exists i= 1,··· , n such that lim t→∞ dH(ϕ(t, p)x, PMi(Mi)) = 0. SKEW PRODUCT SEMIFLOWS AND MORSE DECOMPOSITION 25 Proof: Let (x, p)∈X×P, and consider the solution ξ:R+→X×Pfor the skew product semiflow {Π(t) : t⩾0}starting at (x, p). We know that ξ(t) = Π(x, p) = (ϕ(t, p)x, Θ(t)p),for all t⩾0. Since {M1,··· ,Mn}is a Morse decomposition for the global attractor Aof the driving system {Θ(t) : t⩾0}, there exists i= 1,··· , n such that Θ(t)p→Mias t→ ∞. We claim that ξ(t)→ Mi. Assume, by contradiction, that this is not the case, i.e. there exist ǫ0>0 and a sequence {tn}n∈Nsuch that tn→ ∞ as n→ ∞ and d(ξ(tn),Mi)⩾ǫ0,for all n∈N. But d(ξ(tn),A)n→∞ −→ 0 and we can assume that there exists (x0, p0)∈ A such that ξ(tn)→(x0, p0). But this implies that p0∈Mi, which in turn implies that (x0, p0)∈ Miand gives a contradiction, since 0 = d((x0, p0),Mi)n→∞ ←− d(ξ(tn),Mi)⩾ǫ0>0. Thus, ξ(t)→ Mi, and then ϕ(t, p)x→PMi(Mi). Our last result in this section provides the equivalence between the existence of Morse decomposition and Lyapunov function for the skew product semiflow {Π(t) : t≥0}from the existence of a Lyapunov function (and thus a Morse decomposition) for the global attractor of the driving system {Θ(t) : t≥0}. Theorem 5.7. Let (ϕ, Θ)(X,P)be an NDS and {Π(t) : t⩾0}the associated skewproduct semiflow. Assume that {Π(t) : t⩾0}has a global attractor Aand let Abe the global attractor for the driving semigroup {Θ(t) : t⩾0}. If there is a Lyapunov function for {Θ(t) : t⩾0}, then there is a nonautonomous Morse decomposition {M1(p)}p∈P,··· ,{Mn(p)}p∈Pand a continuous Lyapunov function L:X×P→R+with the following properties: (i) L(ϕ(t, p)x, Θ(t)p)⩽L(x, p)for any (x, p)∈X×Pand t⩾0. (ii) L(ϕ(t, p)x, Θ(t)p) = L(x, p)when x∈ ∪n i=1Mi(p)for all t⩾0, and Ltakes different constant values on different Morse sets. (iii) L(ϕ(t, p)x, Θ(t)p)< L(x, p)when x∈X\ ∪n i=1Mi(p)for all t > 0. Proof: The Lyapunov function for {Θ(t) : t⩾0}generates a Morse decomposition for its global attractor A. This Morse decomposition generates a Morse decomposition for the global attractor Ain the phase space X×P. So the result follows from [1, Theorem 3.4 and Proposition 3.5]. 32 M. C. BORTOLAN, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA Assume also that {Θ(t) : t⩾0}has a global attractor Awith a Morse decomposition {M1,··· ,Mn} and also that the generated skew product semiflow {Π(t) : t⩾0}has a global attractor A. Then we have that for every pair of points (u0, v0)∈Rn×Rn, the solution v(t, v0, u0)of the problem    ˙v(t) = f(Θ(t)u0, v(t)), t > 0 v(0) = v0, satisfies v(t, v0, u0)t→∞ −→ PMi0(Mi0), for some i0= 1,··· , n, where i0is such that Θ(t)u0 t→∞ −→ Mi0. Example 7.2. Consider a function f:R×Rn→Rnand the non-autonomous dynamical system    ˙x=f(t, x), t ∈R x(0) = x0. (7.2) Assume that there are functions f1, f2:Rn→Rnsuch that sup x∈Rn kf(t, x)−f2(x)kRnt→−∞ −→ 0,and sup x∈Rn kf(t, x)−f1(x)kRnt→∞ −→ 0. Now considering the space C=Cb(I×Rn,Rn), we can see that kΘ(t)f(s, x)−f2(x)kk= sup x∈Bk,s∈Ik kf(t+s, x)−f2(x)kRnt→−∞ −→ 0, and kΘ(t)f(s, x)−f1(x)kk= sup x∈Bk,s∈Ik kf(t+s, x)−f1(x)kRnt→∞ −→ 0, for all k∈N, which means that Θ(t)ft→−∞ −→ f2and Θ(t)ft→∞ −→ f1in the uniform convergence on bounded sets, thus K. =γ(f)C={Θ(t)f}t⩾0∪ {f1, f2}and this set has a Morse decomposition {M1,M2}given by M1={f1}and M2={f2}. If the skew product semiflow {Π(t) : t⩾0}given by Π(t)(x0, g) = (x(t, g, x0), gt)in the phase state Rn×Khas a global attractor Athen {M1,M2}is a Morse decomposition for A, where M1=A(f1)× {f1}and M1=A(f2)× {f2}. Therefore, the solution x(t, f, x0)of the problem (7.2)converges to A(f1)as t→ ∞ and A(f2)as t→ −∞. Moreover, we know that A(f1) (A(f2)) is the global attractor of the problem ˙x=f1(x) ( ˙x=f2(x)), and if A(f1) (A(f2)) has a Morse decomposition {M1 1,··· , M1 m}({M2 1,··· , M2 p}) then, under the hypothesis (b′)and (c′)of Theorem 6.8 (Theorem 6.6)we conclude that there exists i= 1,··· , m (j= 1,··· , p)such that x(t, f, x0)converges to M1 ias t→ ∞ (M2 jas t→ −∞). SKEW PRODUCT SEMIFLOWS AND MORSE DECOMPOSITION 33 Remark 7.3. Hypothesis (b′) of Theorem 6.6 (Theorem 6.8)can be verified, for example, using the roughness of hyperbolic points (exponential dichotomy of global solutions or normally hyperbolic periodic orbits)under small non-autonomous perturbations. Hypothesis (c’)is checked with hyperbolicity for hyperbolic points (exponential dichotomy for global solutions or normal hyperbolicity for periodic orbits)(see, for instance, [9, 13, 4]). Example 7.4. This last example illustrates how we can use the general theory in a more concrete case. Let us consider the planar system d dt(x, y) = F(t, (x, y)) , t ∈R.(7.3) Assume that F(t, (x, y)) →F1(x, y)as t→ −∞ and that F(t, (x, y)) →F2(x, y)as t→ ∞, where F1, F2:R2→R2satisfy 1. F1(x, y) = (f(x), g(x, y)), where f(x) = x−x3and g(x, y) = (1 −x2)y−y3. Clearly they satisfy the conditions of Example 7.1; 2. F2is given in polar coordinates by F2(rcos θ, r sin θ) = G(r, θ) = (−r(r−1)(r−2),1). From Example 7.1, we know the Morse decomposition {M0,M1,M2}for the global attractor Aof the planar system    ˙x=f(x) ˙y=g(x, y) , given a Morse decomposition {M0,M1,M2}for the global attractor A= [−1,1] ⊂Rof the equation ˙x=f(x), where M0={−1},M1={1},M2={0}. Indeed, it is not hard to see that the Morse decomposition is this case is given by M0={(−1,0)},M1={(1,0)}and M2={(0, y) : y∈ [−1,1]}. Furthermore, we already know that the system d dt(x, y) = F2(x, y), t > 0,(7.4) generates a generalized gradient-like system, with invariant sets given by Ξ0={0},Ξ1={(1, θ) : θ∈[0,2π]}and Ξ2={(2, θ) : θ∈[0,2π]}. Thus, by Example 7.2, we know that every solution ξ:R→R2of the system (7.3)satisfies (see Figure 2) (a) ξ(t)→ Mifor some i= 0,1,2as t→ −∞, 34 M. C. BORTOLAN, T. CARABALLO, A. N. CARVALHO, AND J. A. LANGA (b) ξ(t)→Ξjfor some j= 0,1,2as t→ ∞. −∞ ∞ b b b b b b b ξ(t) b ψ(t) Figure 2: Asymptotic behaviors for solutions ξ, ψ of (7.3). Acknowledgements. We would like to thank the referee for the helpful and interesting suggestions and comments which allowed us to significantly improve the presentation of this paper. References [1] Arag˜ao-Costa, E. R.; Caraballo, T.; Carvalho, A. N.; Langa, J. A. Stability of gradient semigroups under perturbations. Nonlinearity 24 (2011), no. 7, 2099–2117. [2] T. Caraballo, J. C. Jara, J. A. Langa and Z. 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P¨otzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathemathics 2002, Springer-Verlag, New York 2010. [8] C. Conley, Isolated invariant sets and the Morse index. CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence, R.I. (1978). [9] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, (1981). [10] P. Kloeden and M. Rasmussen, Nonautonomous dynamical systems, Mathematical Surveys and Monographs 176, AMS, Providence RI 2011. [11] K. Mischaikow, H. Smith and H. R. Thieme, Asymptotically autonomous semiflows: chain recurrent and Lyapunov functions, Trans. Amer. Math. Soc. 347 (5) (1995), 1669–1685. [12] K. P. Rybakowski, The homotopy index and partial differential equations, Universitext, Springer-Verlag (1987). [13] G. R. Sell and Y. You, Dynamics of evolutionary equations. Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002. (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 M. C. Bortolan) 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, M. C. Bortolan: [email protected].br E-mail address, A. N. Carvalho: [email protected]r