Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system

CHARACTERIZATION OF NON-AUTONOMOUS ATTRACTORS IN PERTURBED INFINITE-DIMENSIONAL GRADIENT SYSTEMS ALEXANDRE N. CARVALHO†, JOS´ E A. LANGA‡, JAMES C. ROBINSON∗, AND ANTONIO SU´ AREZ? Abstract. In this paper we determine the exact structure of the pullback attractors in non-autonomous problems that are perturbations of autonomous gradient systems with attractors that are the union of the unstable manifolds of a finite set of hyperbolic equilibria. We show that the pullback attractors of the perturbed systems inherit this structure, and are given as the union of the unstable manifolds of a set of hyperbolic global solutions which are the non-autonomous analogues of the hyperbolic equilibria. We also prove, again parallel to the autonomous case, that all solutions converge as t→+∞to one of these hyperbolic global solutions. We then show how to apply these results to systems that are asymptotically autonomous as t→ −∞ and as t→+∞, and use these relatively simple test cases to illustrate a discussion of possible definitions of a forwards attractor in the non-autonomous case. 1. Introduction 1.1. Overview. The study of the global attractors that arise in many infinite-dimensional dynamical systems has been developed extensively over the past thirty years, and for autonomous systems much of the theory is now classical (see, for example, the books by Hale [9], Ladyzhenskaya [13], or Temam [24]). However, given the underlying models that arise in various branches of the sciences it is very natural to try to extend the theory to treat non-autonomous equations. In the autonomous case the concept of a global attractor is settled and for gradient systems (those that possess a Liapunov function) the structure of the attractor is well understood: it is given as the union of the unstable manifolds of the equilibria. However, for nonautonomous dynamical systems the appropriate definition of ‘a global attractor’ is still not entirely settled, and there are few examples with attractors whose structure is known. In this paper we identify a class of non-autonomous systems in which the structure of the pullback attractor can be determined exactly: these are uniformly small non-autonomous perturbations of gradient systems with a finite number of hyperbolic equilibria. Loosely speaking, the main result proved in this paper is that the structure of the attractor is unchanged by such non-autonomous perturbations. More precisely, we show that the pullback attractor is the union of the unstable manifolds of hyperbolic global solutions. These ‘global hyperbolic solutions’ are the non-autonomous analogue of hyperbolic equilibria, being solutions defined for all t∈R, the linearizations around which enjoy exponential dichotomies. †Partially supported by CNPq grant # 305447/2005-0 and by FAPESP grant # 03/10042-0, Brazil. ‡Partially supported by FAPESP grant # 06/51612-2, Brazil and Projects MTM2005-01412 and PCI20061198, MEC-Spain. ∗Currently a Royal Society University Research Fellow. Many thanks to the Society for their support. ?Partially supported by Projects BFM2003-06446 and MTM2006-07932, MEC-Spain. 1 2 A. N. CARVALHO, J. A. LANGA, J. C. ROBINSON, AND A. SU ´ AREZ Even in the case of autonomous systems this provides new examples in which the structure of the attractor is known explicitly. (A proof of this result for a restricted class of finitedimensional systems is given by Langa et al. [16], but the argument there uses time reversal and so is not applicable in the infinite-dimensional setting.) We then show how to adapt our results to the case of asymptotically autonomous systems. As well as obtaining results that are interesting in their own right, we use these simple models as a basis for a discussion of the possible definitions of an attractor in non-autonomous systems. 1.2. Semigroups and processes. In order to describe the results of this paper in more detail we need to introduce some terminology. Although in the main body of the paper we choose to work with a particular model for which we are able to prove that some key properties hold (see Section 1.4, below) more generally we are interested in non-autonomous perturbations of an underlying autonomous process. Taking a Banach space Zas our phase space, the underlying autonomous system is naturally described by a semigroup of nonlinear operators (or ‘nonlinear semigroup’), i.e. a family {S(t) : t≥0}(or S(·) for short) consisting of continuous operators from Zinto itself such that 1) S(t) = I, 2) S(t)S(s) = S(t+s),for each t, s ≥0,and 3) t7→ S(t)z0is continuous for t≥0, z0∈ Z. If each S(t) is linear then we call {S(t) : t≥0}a linear semigroup. Upon addition of a non-autonomous perturbation the initial time becomes as important as the final time, and the dynamics is then described by a nonlinear process, i.e. a two parameter family {S(t, τ) : t≥τ∈R}(or S(·,·) for short) of continuous operators from Z into itself such that 1) S(τ, τ) = I, 2) S(t, σ)S(σ, τ) = S(t, τ),for each t≥σ≥τ, and 3) (t, τ)7→ S(t, τ)z0is continuous for t≥τ, z0∈ Z. Again, if each S(t, τ) is linear then we refer to S(·,·) as a linear process. For a nonlinear process S(·,·) with the property that S(t, τ) = S0(t−τ) for all t≥τ∈R, i.e. for a process that is really a nonlinear semigroup in disguise, the behaviour of solutions as t→ ∞, which we refer to as ‘the forwards dynamics’, is the same as the behaviour of solutions as τ→ −∞, ‘the pullback dynamics’. However, for general processes these ‘dynamical limits’ are totally unrelated and can produce entirely different qualitative properties (see [6, 17]). We believe that this point is made forcibly in Section 4, where we consider asymptotically autonomous gradient systems and are able to describe both dynamical limits completely. 1.3. Attractors. Our main tool for describing the long-term dynamics of both the autonomous and non-autonomous systems we consider is the theory of attractors. Here we first recall the definition of a global attractor for a nonlinear semigroup S(·) (see [9] or [24]), and then discuss how this concept can be generalised to the attractor of a non-autonomous process S(·,·). If Band Care subsets of Z, we say that the set Battracts the set Cunder S(t) if dist(S(t)C, B)→0 as t→ ∞,where dist(A, B) = supa∈Ainfb∈B|a−b|. CHARACTERIZATION OF NON-AUTONOMOUS ATTRACTORS 3 A set A⊂ Z is said to be invariant under {S(t) : t≥0}if, for any z∈A, there is a complete orbit γ(z) through zsuch that γ(z)⊂Aor equivalently if S(t)A=Afor any t⩾0. Definition 1.1. A set A ⊆ Z is said to be the global attractor for S(·)if it is (i) compact, (ii) invariant, i.e. S(t)A=Afor all t≥0, and (iii) it attracts bounded subsets Bof Z, dist(S(t)B, A)→0as t→ ∞. The notion of a global attractor for a nonlinear process S(·,·) requires much more care. Since any fixed set Awill not, in general, be invariant in the above sense for a nonautonomous process, it is natural to define invariance in this context as follows: •A family {A(t)⊂ Z :t∈[σ, ∞)}is invariant under S(·,·) if S(t, τ)A(τ) = A(t) for all t≥τ≥σ. With this in mind one might think that a non-autonomous attractor should be defined as follows: •A family {A(t)⊂ Z :t∈R}with A(t) compact for all t∈Ris a non-autonomous attractor if it is invariant under S(·,·) and attracts bounded sets; that is, for each bounded set B⊂ Z and τ∈R lim t→∞ dist(S(t, τ)B, A(t)) = 0. Unfortunately such a definition is likely to be satisfied only in some very specific and restrictive situations (e.g. if {T(t, τ) : t≥τ∈R}is uniformly asymptotically compact in the sense of [7]). Some very simple examples of systems that we would expect to possess a ‘global non-autonomous attractor’ will not have an attractor in the sense of this definition; this is essentially due to the fact that some of the forwards dynamics may be associated with solutions that blow up in finite backwards time (see Section 4.2). Central to much of what follows is the concept of a globally-defined solution. In the autonomous case, a globally-defined solution (or simply a global solution) through zis a function ξ:R→ Z such that ξ(0) = zand for all s∈Rand t≥0 we have S(t)ξ(s) = ξ(t+s). In the autonomous case the attractor is exactly the union of all such orbits [24], A={z: there is a bounded global solution through z}.(1.1) In the non-autonomous case, the definition of an ‘attractor’ that has the same characterization as the union of all globally-defined bounded orbits, {A(t) : t∈R}={ξ(t) : ξ(·) : R→ Z is bounded and S(t, τ)ξ(τ) = ξ(t)},(1.2) is the pullback attractor: Definition 1.2. A family of compact sets {A(t)⊂ Z :t∈R}with ∪t∈RA(t)compact is a pullback attractor for {S(t, τ) : t≥τ∈R}if it is invariant and attracts all bounded subsets of Z‘in the pullback sense’, i.e. lim τ→−∞ dist(S(t, τ)B, A(t)) = 0,∀t∈R. 4 A. N. CARVALHO, J. A. LANGA, J. C. ROBINSON, AND A. SU ´ AREZ (See [7], where the sets A(t) are referred to as kernel sections, and also [12, 22] who use the terminology ‘pullback attractor’). It is shown in [7] that when a pullback attractor exists it is given by (1.2). Note that the requirement that ∪t∈RA(t) be compact is not a standard one in the literature on pullback attractors, and without it the definition still reduces to the familiar one in the autonomous case. Indeed, there are examples in which allowing A(t) to be unbounded, particular as t→+∞, is a useful weakening of the definition. Nevertheless, the uniformity imposed here occurs in most interesting applications, and allows for stronger results, while ruling out some potentially pathological behaviour, e.g. unstable sets that do not belong to the attractor, see Theorem 5.2 in [14].) For autonomous problems, it is clear that the concept of a pullback attractor coincides with the standard definition of the attractor, while the characterization in (1.2) shows that this notion is in some sense a ‘natural’ generalization. However, as is well-known and demonstrated here by the example presented in Section 4.2, the pullback attractor will not necessarily enjoy any kind of forward attraction. Except in very specific situations, for example when the non-autonomous nonlinear process is asymptotically autonomous backwards and forwards to the same nonlinear semigroup, the pullback behaviour and the forwards behaviour will not be related (see Theorems 4.2 and 4.5, and [6, 21, 15] for other specific cases). Ideally, therefore, one would describe the pullback attractor of a non-autonomous system, and give some information on the limits of solutions as t→ ∞. We accomplish both aims in the particular class of systems that we consider here. 1.4. Gradient systems and ‘gradient-like’ attractors. Our result considers small nonautonomous perturbations of autonomous gradient systems. In order to make it clear where our work differs from previous results, we need to draw a distinction between gradient systems and systems with ‘gradient-like’ attractors. If T0(·) is a gradient nonlinear semigroup (i.e. T0(·) has a Liapunov function, see Definition 2.4) that has a global attractor A0and a finite number of stationary solutions y∗ i, 1 ≤i≤n, then every solution converges to one of the equilibria as t→+∞, and every solution defined for all t≤0 is also backwards asymptotic to one of the equilibria. This implies, in particular, that the attractor A0is the union of the unstable manifolds Wu 0(y∗ i) of the equilibria, i.e. A0= n [ j=1 Wu(y∗ j),(1.3) and so the structure of A0is completely understood. This is essentially the class of nonlinear semigroups in Banach spaces for which a detailed knowledge of the structure of the attractor is available. An attractor of the form (1.3) we term ‘gradient-like’. Clearly the class of systems with gradient-like attractors is larger than those that are strictly gradient. The argument that leads to our main result has two ingredients. We consider an underlying semigroup T0(·), and a parametrized family Tη(·,·) of non-autonomous processes that converge to T0(·) (in a sense which will of course be made precise) as η→0. First, we assume that the equilibria of T0(·) become hyperbolic global solutions for Tη(·,·) for ηsufficiently small, and that the corresponding stable and unstable manifolds change continuously (this is made precise in Section 2). In this case, if one only assumes that the attractor of T0(·) is ‘gradient-like’ and all the y∗ j are hyperbolic, then it is possible to show that the pullback attractors Aηof Tη(·,·) behave CHARACTERIZATION OF NON-AUTONOMOUS ATTRACTORS 5 continuously as η→0, i.e. sup t∈R dist(Aη(t), A0)→0 as η→0, where dist(X, Y ) = max[dist(X, Y ),dist(Y, X)], see [16, 5]. In fact this continuity result is proved by showing that the pullback attractors for the perturbed problem contain (possibly strictly) the union of the unstable manifolds of the global hyperbolic solutions, while the remaining part of the pullback attractor for the perturbed problem (if it exists) is small. Our main result here is that under the additional assumption that the unperturbed problem is truly gradient, i.e. has a Liapunov function, then there is no ‘remainder’, and the pullback attractor has the same structure as the autonomous attractor, i.e. Aη(t) = n [ j=1 Wu(ξj,η(·))(t), where the ξj,η(·) are the hyperbolic global solutions corresponding to the hyperbolic equilibria y∗ jin the original problem. We also show that every solution converges to one of the ξj,η(·) as t→+∞. To obtain these results we make continual use of the Liapunov function for the unperturbed problem: the structure of the attractor for Tη(·,·) cannot be deduced from the continuity of the attractor under perturbation. Our results provide new classes of systems in which the exact structure of the attractor is known, even in the autonomous case. For example, if we consider an autonomous dynamical system that is gradient and perturb it in such a way that the perturbed dynamical system is still autonomous but no longer has a Liapunov function, the results in [5, 9] prove that the attractors behave continuously but do not ensure that the perturbed attractor is exactly the union of unstable manifolds of hyperbolic equilibria, which is what we are able to prove here. (Section 5 gives the striking example of a damped hyperbolic equation which is not gradient but whose attractor is nevertheless gradient-like.) It is a natural question whether our results can be obtained for small perturbations of a larger class than autonomous gradient systems. One might hope to prove a similar result starting from a “generalized gradient dynamical system”, a reasonable definition of which is a dynamical system (autonomous or non-autonomous) that has a pullback attractor given as the union of the unstable manifolds of finitely many global hyperbolic solutions, and for which every solution is forwards asymptotic to one of the (finite) set of global hyperbolic solutions. However, our arguments are completely unable to treat this case, since we use the Liapunov function of the limiting system throughout our proof. We like to think of the characterization result of this paper as midway between full structural stability (which one would expect to involve assumptions such as the transversality of stable and unstable manifolds) and the weaker property of continuity of attractors (as in [5, 16]) valid under less stringent conditions. We suspect that extending our results to treat generalized gradient systems will require techniques more akin to those involved in considerations of structural stability. For example, in a system whose vector field is periodic in time, a hypothesis such as the transversality of stable and unstable manifolds should lead to a similar characterization of the attractor and also guarantee the preservation of the connections between hyperbolic orbits, since if the associated Poincar´e map is Morse-Smale one can apply the results due to Oliva (see Oliva [19] or Hale et al. [10]) to show that the system is topologically stable. 6 A. N. CARVALHO, J. A. LANGA, J. C. ROBINSON, AND A. SU ´ AREZ 1.5. Detailed summary of results. We now specify the particular model that we will consider in detail, and give a formal summary of our main results. Our choice of model is motivated by the need to guarantee that the stable and unstable manifold structure near a hyperbolic equilibrium perturbs continuously. Such results were shown in [5] for a class of semilinear problems on a Banach space Z, and it is these models that we consider in what follows. At the risk of labouring the point, our results could be stated and proved within a more abstract setting (an abstract process Tη(·,·) that is a perturbation of a semigroup T0(·) with the relevant additional properties), but here we choose to concentrate on this particular example for the sake of concreteness. We will consider the semilinear autonomous problem ˙y=By+f0(y) with y(τ) = y0∈ Z,(1.4) and the non-autonomous family for η∈(0,1] ˙y=By+fη(t, y) with y(τ) = y0∈ Z,(1.5) where B:D(B)⊂ Z → Z is the generator of a C0-semigroup of bounded linear operators and, for η∈[0,1], fη(t, ·) is a differentiable function which is Lipschitz continuous in bounded subsets of Zwith Lipschitz constant independent of ηand t. Assume that, for each τ∈R and y0∈ Z, unique solutions of (1.4) and (1.5) exist for all t≥τ. Then the solution t→T0(t−τ)y0of (1.4) defines a nonlinear semigroup on Z, and the solution t7→ Tη(t, τ)y0 of (1.5) gives rise to a family of nonlinear processes on Z. Some authors have considered models with additional properties, e.g. Shen & Yi [23] have considered coefficients that are almost periodic, but we prefer to consider more general non-autonomous terms and, indeed, it seems that the extra properties of almost periodic equations would not help us in the direction we are pursuing here. As remarked above, we choose this particular model because it is shown in [5] that if fηis aC1perturbation of f0then to each hyperbolic equilibrium point of T0(·) there corresponds a hyperbolic global solution ξ∗ i,η(·) of Tη(·,·) and the corresponding stable and unstable manifolds behave continuously as η→0; these results are recalled in Section 2. Using these results and the assumption that T0(·) is gradient with a finite number of equilibria y∗ i, all of which are hyperbolic, we show the following in our main theorem, Theorem 2.11. ◦Structure of the pullback attractors for the perturbed systems Aη(t) = n [ i=1 Wu η(ξ∗ i,η)(t), for all t∈R,where Wu η(ξ∗ i,η)(t) denotes the unstable manifold associated to the global hyperbolic solutions ξ∗ i,η (these are shown in [5] to be given as a graph near each of the hyperbolic equilibrium points y∗ i). ◦Dimension of the pullback attractors for the perturbed systems For each t∈R, dimH(Aη(t)) = dimH(A0) and give an explicit expression for this dimension in (2.21). ◦Backwards and forwards limits of global solutions For every bounded global solution of (1.5), ξη(t), there are j, k with 1 ≤j≤nand CHARACTERIZATION OF NON-AUTONOMOUS ATTRACTORS 7 1≤k≤nsuch that lim t→∞ kξη(t)−ξ∗ j,η(t)kZ= 0 and lim t→−∞ kξη(t)−ξ∗ k,η(t)kZ= 0. ◦Forwards limits of all solutions For each (τ, y0)∈R× Z there is a 1 ≤j≤nsuch that lim t→∞ kTη(t, τ)y0−ξ∗ j,η(t)kZ= 0. In the second part (Section 4) of this paper we consider asymptotically autonomous dynamical systems in the case that the limiting system is gradient. Because asymptotically autonomous systems can be analysed by considering non-autonomous perturbations of an autonomous equation, we are able to take advantage of the above results to describe the structure of the attractors in this case. Moreover, we show that every solution converges to one of the hyperbolic global solutions of the non-autonomous problem; these are the true time-dependent (and invariant) attracting structures, rather than their asymptotic limits (which are invariant only for the limit system), cf. [3, 4]. We highlight the fact that if the backwards and forwards limit systems are different then, although both the forwards and pullback dynamics can be described in detail, they can be entirely unrelated. Ideally, we would also like to characterize a forwards attractor for bounded sets (when possible), insisting on the requirement that this be invariant. However, there are non-trivial problems with defining such a forwards attractor, and these are also discussed in Section 4. In Section 5 we present a number of examples that illustrate the broad applicability of our results, and finally we make some general comments and conjectures in Section 6. 2. Background Results and Statement of the Main Theorem We start by describing some previous results that are central in the proof of our main theorem, namely results on the continuity of stable and unstable manifolds proved in [5], and classical results on the structure of attractors in gradient systems. If t7→ T0(t−τ)y0denotes the solution of ˙y=By+f0(y) with y(τ) = y0,(2.1) then T0(t−τ)y0=eB(t−τ)y0+Zt τ eB(t−s)f0(T0(s−τ)y0)ds, (2.2) while if we denote by t7→ Tη(t, τ)y0the solution of ˙y=By+fη(t, y) with y(τ) = y0,(2.3) we have Tη(t, τ)y0=eB(t−τ)y0+Zt τ eB(t−s)fη(s, Tη(s, τ)y0)ds. (2.4) The following result on the continuity of these solution operators as η→0 is easy to prove. Theorem 2.1. Assume that lim η→0sup t∈R sup z∈B(0,r) kfη(t, z)−f0(z)kZ= 0,for each r > 0.(2.5) Then, for each r > 0and T > 0, lim η→0sup{kTη(t+τ, τ)z−T0(t)zkZ, τ ∈R, t ∈[0, T]and kzkZ≤r} → 0.(2.6) 8 A. N. CARVALHO, J. A. LANGA, J. C. ROBINSON, AND A. SU ´ AREZ A solution of (2.1) is an equilibrium solution if it satisfies By+f0(y) = 0.(2.7) Suppose that y∗ 0is solution of (2.7). It follows that, if A=B+f0 0(y∗ 0), then Agenerates a strongly continuous semigroup {eAt:t⩾0}of bounded linear operators. Definition 2.2. An equilibrium solution y∗ 0to (2.1) is said to be hyperbolic if the following are satisfied: (1) The spectrum of Adoes not intersect the imaginary axis and the set σ+={λ∈ σ(A) : Reλ > 0}is compact. This allows us to choose a smooth closed simple curve γin ρ(A)∩{λ∈C: Reλ > 0} that is positively orientated and encloses σ+; we can then define the projection Q=Q(σ+) = 1 2πiZγ (λI − A)−1dλ. (2.8) If Z+=Q(Z),Z−= (I−Q)(Z), and A±=A|Z±, then Z=Z+⊕Z−,A−generates a strongly continuous semigroup on Z−and A+∈L(Z+). (2) There are constants ¯ M≥1and β > 0such that keA+tkL(Z+)⩽¯ Meβt, t ⩽0, keA−tkL(Z−)⩽¯ Me−βt, t ⩾0.(2.9) The stable and unstable manifolds of an equilibrium y∗ 0,Ws(y∗ 0) and Wu(y∗ 0) respectively, are defined as follows: Ws(y∗ 0) = {z∈ Z : lim t→+∞kT0(t)z−y∗ 0kZ= 0}. Wu(y∗ 0) = {z∈ Z :there is a backwards solution y(t) of (2.1) satisfying y(τ) = zand such that lim t→−∞ ky(t)−y∗ 0kZ= 0}. The intersection of the unstable manifold with a neighbourhood of y∗ 0is termed the local unstable manifold, which we write Wu loc(y∗ 0). The existence of local stable and unstable manifolds as graphs near a hyperbolic equilibrium is well-known: Theorem 2.3. If y∗ 0is a hyperbolic equilibrium then for suitably small ² > 0there are Lipschitz functions B(0, ²)3w7→ Σ∗,u(Qw)∈(I− Q)Z B(0, ²)3w7→ Σ∗,s((I− Q)w)∈ QZ such that Wu loc(y∗ 0)={y∗ 0+ (Qw, Σ∗,u(Qw)),kwkZ≤²} Ws loc(y∗ 0)={y∗ 0+ (Σ∗,s((I−Q)w),(I−Q)w),kwkZ≤²}, where Qis the projection from (2.8). Next we recall the definition of a gradient nonlinear semigroup. (Note that we have slightly strengthened the definition from that in Hale [9], since to ensure that ξ(·) is an equilibrium we only require V(ξ(t)) to be constant on a semi-infinite interval.) Definition 2.4. We say that a nonlinear semigroup {T0(t) : t≥0}is gradient if {T0(t)z: t≥0}is relatively compact for each z∈ Z and there exists a continuous function V:Z → R such that CHARACTERIZATION OF NON-AUTONOMOUS ATTRACTORS 9 •t7→ V(T0(t)z) : [0,∞)→ Z is non-increasing for each z∈ Z. •If z∈ Z is such that there is a global solution ξ(·) : R→ Z through ξ(0) = zand there exists a t∗∈Rsuch that V(ξ(t)) = V(z)for all t≥t∗or for all t≤t∗, then z is a solution of (2.7) (and so in fact V(ξ(t)) = V(z)for all t∈R). The function V:Z → Ris called a Liapunov function for {S(t) : t≥0}. In a gradient system with a finite number of hyperbolic equilibria the attractor has a particularly simple structure, and all global solutions in it are both forwards and backwards asymptotic to an equilibrium, which we state formally in the following theorem (see [9]). Note that the assumptions in the theorem are satisfied (at least generically) for many examples that have a gradient structure. Theorem 2.5. If T0(·)is a gradient system that has a global attractor A0,V:Z → Ris its Liapunov function and (2.7) has a finite number of solutions y∗ i,1≤i≤n, then A0is given by A0= n [ i=1 Wu 0(y∗ i),(2.10) i.e. the attractor is ‘gradient-like’. Furthermore if we denote by {n1,· · · ,np}the set of all distinct values of V(y∗ j), ordered so that ni<nj,1≤i < j ≤p≤n, and define Ek={y∗ i∈ E :V(y∗ i) = nk}, then if y(·) : R→ Z is a global solution for (2.1), there are k1, k2with 1≤k1< k2≤p,y∗ i∈ Ek1and y∗ j∈ Ek2, such that lim t→−∞ y(t) = y∗ jand lim t→+∞y(t) = y∗ i. While a characterization of the pullback attractor for small non-autonomous perturbations of finite-dimensional gradient systems is given by Langa et al. [16], such a characterization is not currently available in the literature for any class of infinite-dimensional problems. Our first task is to find a non-autonomous analogue of a hyperbolic equilibrium points. In [5] it is shown that near each of the hyperbolic equilibrium y∗ ithere is a unique global solution ξ∗ i,η which enjoys a hyperbolic structure. In order to be more precise we need the notion of an exponential dichotomy, which we now introduce. Definition 2.6. We say that a linear process {U(t, τ) : t≥τ∈R}has an exponential dichotomy with exponent ωand constant Mif there exists a family of projections {Q(t) : t∈R} ⊂ L(Z)such that (1) Q(t)U(t, s) = U(t, s)Q(s), for all t≥s. (2) The restriction U(t, s)|R(Q(s)) ,t≥sis an isomorphism from R(Q(s)) into R(Q(t)); we denote its inverse by U(s, t) : R(Q(t)) →R(Q(s)). (3) There are constants ω > 0and M≥1such that kU(t, s)(I−Q(s))kL(Z)≤Me−ω(t−s)t≥s kU(t, s)Q(s)kL(Z)≤Meω(t−s), t ≤s. (2.11) Now we will define the analog of a hyperbolic equilibrium for non-autonomous problems (2.3). But first we need to introduce some more terminology. Consider the problem ˙z=Az+Bη(t)z z(τ) = z0∈ Z,(2.12) 16 A. N. CARVALHO, J. A. LANGA, J. C. ROBINSON, AND A. SU ´ AREZ convergence of Tηto T0that there is a fixed T2>0 such that ξηk(t2 k+T2)∈ O` νfor some ` < j. So we can restart from (3.5) but with Oj νreplaced by O` νfor some ` < j. Case (b). There exists a kν∈Nsuch that for all k≥kν, nj+ν≤V(ξηk(t2 k)) <nj+ν0. Note first that we may assume that V(ξηk(t)) >nj−ν∀t≥t2 k, k ∈N(3.7) for otherwise, we must have V(ξηk(t3 k)) ≤nj−ν for some t3 k> t2 kand for infinitely many values k. In this case would can return to case (a), but with t2 kreplaced by t3 k. Now set B={ξηk(θ) : V(ξηk(θ)) ≤nj+ν0,for some θ∈Rand some k∈N}. Using Lemma 3.2, we can find t∗ ν>0 such that sup{V(T0(t)B)}<nj+ν 2∀t≥t∗ ν.(3.8) Now use the continuity of Vand the uniform convergence of Tηto T0to choose kνsufficiently large that sup{V(Tηk(t+s, t)B)}<nj+ν∀s∈[t∗ ν,2t∗ ν]∀k≥kν.(3.9) It follows by induction that if ξηk(t2 k+t)∈Bthen ξηk(t2 k+t+τ)∈Bfor all τ≥t∗ ν, and hence that for all k≥kνwe must have V(ξηk(t2 k+t)) <nj+ν∀t≥t∗ ν.(3.10) Now, (3.10) together with (3.7) contradicts our assumption in (3.4). It follows, therefore, that case (b) is impossible and that case (a) must always occur. But case (a) can only occur a finite number of times, and so we obtain a contradiction and (3.2) must hold. Proof of (3.3):The argument to show that every bounded global solution must end backwards in Oν(E) is similar to the forwards case. We start from the assumption that there exists a sequence ηk→0 and corresponding bounded solutions ξηk(·) of (2.3) (with η=ηk) such that for any t∈R,there is a τ < t such that ξηk(τ)/∈ Oν(E).(3.11) Case (a). For infinitely many kwe have V(ξηk(t2 k)) ≥nj+νis almost identical, except for the obvious changes and its proof will be omitted. Case (b). There exists a kνsuch that for all k≥kν, nj−ν0< V (ξηk(t2 k)) <nj−ν. We may assume that V(ξηk(t)) ≤nj+νfor all t≤t2 k, k ∈N,(3.12) CHARACTERIZATION OF NON-AUTONOMOUS ATTRACTORS 17 for otherwise, using the continuity of V, we must have V(ξηk(t3 k)) ≥nj+ν for some t3 k< t2 kand for infinitely many values of k. Hence, we can return to case (a) with t2 k replaced by t3 k. Since case (a) can only be repeated a finite number of times, we eventually find ourselves in case (b) with (3.12) valid. In this situation we choose ν00 with ν0> ν00 > ν0> ν and set B={ξηk(θ) : V(ξηk(θ)) ≤nj−ν, for some θ∈Rand some k∈N}. Using Lemma 3.2, we can find t∗ ν>0 such that sup{V(T0(t)B)}<nj−ν00 ∀t≥t∗ ν. Now use the continuity of Vand the uniform convergence of Tηto T0to choose kνsufficiently large that sup{V(Tηk(t, t −s)B)}<nj−ν0∀s∈[t∗ ν,2t∗ ν]∀k≥kν.(3.13) It follows by induction that if ξηk(t2 k−t)∈Bthen ξηk(t2 k−t+τ)∈Bfor all τ≥t∗ ν. We now claim that for all k≥kνwe must have V(ξηk(t2 k−t)) >nj−ν∀t≥t∗ ν. Indeed, suppose not. Then it follows that for some t≥t∗ νthat V(ξηk(t2 k−t)) ≤nj−ν, i.e. ξηk(t2 k−t)∈B. But then (3.13) shows that V(ξηk(t2 k)) <nj−ν0, a contradiction. It follows that, there exist kνand t∗ νsuch that nj−ν < V (ξηk(t2 k−t)) <nj+ν∀k≥kν, t ≥t∗ ν. As before, this contradicts our initial assumption (3.11), and the argument is concluded as before. We are now ready to prove the main result of this paper. Proof of Theorem 2.11: The proof of (1) follows from (2) and the proof of (3) follows in the same way as the forwards argument in (2). To prove (2), let ² > 0 and η0>0 be such that in Bj ²=B(y∗ j, ²) there is a unique global hyperbolic solution ξ∗ j,η with the stable and unstable manifolds given as graphs for all 0≤η≤η0. Hence, if a global solution ξη(·) is such that ξη(t)∈ Bj ²for all t≥t²(or for all t≤ −t²) for some t²>0, then ξη(t)∈Ws j(ξη(·)) (or ξη(t)∈Wu j(ξη(·))). Hence, to prove (2.22), it is enough to show that there is a η0>0 such that every globally defined bounded solution of (2.3) ends forwards and backwards in B². Suppose that ηk k→∞ −→ 0 and that ξηk(·) does not end forwards or backwards in B²=∪n j=1Bj ². Taking a subsequence if necessary, we may assume that given ν≤ν0there exists kνsuch that ξηkends forward (respectively backward) in Oj νfor some fixed jwith 1 ≤j≤pwhenever k≥kν. Hence we have a sequence tksuch that kξηk(tk)−y∗ ikZ≥²for all y∗ iwith V(y∗ i) = nj. Consequently, there is a subsequence (which we again denote by ξηk) such that lim k→∞ ξηk(t+tk)→y(t) uniformly for tin compact subsets of R, where y(·) is a solution of (2.1). It is clear that V(y(0)) = njand since y(0) is not an equilibrium of (2.1) and Vis non-increasing it follows 18 A. N. CARVALHO, J. A. LANGA, J. C. ROBINSON, AND A. SU ´ AREZ (using the convergence of Tηto T0) that for a suitable choice of T > 0 (respectively T < 0) we must have V(ξηk(tk+T)) /∈(nj−ν0,nj+ν0) which leads to a contradiction. Finally we prove (2.21). We will use the following two properties of the Hausdorff dimension (see Falconer [8], for example): it is non-increasing under Lipschitz mappings, Σ:Z → Z with kΣ(z1)−Σ(z2)kZ≤Lkz1−z2kZ⇒dimH(Σ(X)) ≤dimH(X), and it is stable under countably infinite unions, dimHÃ∞ [ j=1 Xj!= sup 1≤j≤∞ dimH(Xj).(3.14) First, observe that dH(Wu loc(ξ∗ i,η)(τ)) = rank(Qi η(τ)) = rank(Qi), since sufficiently close to ξ∗ i,η the unstable manifold is given as a Lipschitz graph over Qi η(τ)Zand from the continuity of the projections we have that rank(Qi η(τ)) = rank(Qi). Note that Wu η(ξ∗ i,η)(t) = ∞ [ n=0 Tη(t, t −n)Wu loc,η(t−n). Since each Tη(t, τ) : Z → Z is Lipschitz it follows that dimH(Tη(t, t −n)Wu loc(ξ∗ i,η)(t−n)) ≤dimH(Wu loc(ξ∗ i,η)(t−n)), and hence, using (3.14), that dimH(Wu(ξ∗ i,η)) = rank(Qi). The equality in (2.21) then follows using (3.14) once again. 4. Asymptotically autonomous problems As in Section 1 we consider a Banach space Zand the semilinear problem ˙y=By+f(t, y) y(τ) = y0,(4.1) where B:D(B)⊂ Z → Z is the generator of a C0-semigroup of bounded linear operators and f(t, ·) is a differentiable function that is Lipschitz continuous in bounded subsets of Z with Lipschitz constant independent of t. If we denote by t7→ T(t, τ)y0the solution for (2.3), then {T(t, τ) : t≥τ∈R}defines a nonlinear process. We will assume that the problem (4.1) has a pullback attractor {A(t) : t∈R}. 4.1. Asymptotically Autonomous Problems at −∞.Assume that lim t→−∞ sup z∈B(0,r) {kf(t, z)−f0(z)kZ+kfy(t, z)−f0 0(z)kL(Z)}= 0,for each r > 0,(4.2) and that (2.1) has an autonomous attractor A0. Suppose that (2.7) is gradient and has a finite number of solutions, all of them hyperbolic: then A0is given by (2.10) (Theorem 2.5). We now prove that the non-autonomous system possesses global solutions that are backwards asymptotic to the equilibria of the limiting autonomous problem. CHARACTERIZATION OF NON-AUTONOMOUS ATTRACTORS 19 Proposition 4.1. Assume that (4.2) holds and that all solutions of (2.7) are hyperbolic. Then, there are solutions ξ∗ i−:R→ Z,1≤i−≤n−, such that lim t→−∞ max 1≤i−≤n−kξ∗ i−(t)−y∗ i−kZ= 0.(4.3) Furthermore, there is a τ∈Rsuch that ˙y=Ai−y+Bi−(t)y(4.4) has an exponential dichotomy in (−∞, τ], where Ai−=B+f0 0(y∗ i−)and Bi−(t) = fy(t, ξ∗ i−(t))− f0 0(y∗ i−). Proof: The proof of this result reduces to the proof of (2) in Proposition 2.10, cutting the nonlinearities fand f0around y∗ i−in such a way that the fixed point argument works. To be more specific, we fix 1 ≤i−≤n−and consider the change of variables z=y−y∗ i−in (4.1). In this new variable (4.1) becomes ˙z=Ai−z+ ˜gi−(t, z) (4.5) where ˜gi−(t, z) = f(t, z +y∗ i−)−f0(y∗ i−)−f0 0(y∗ i−)z. Cut ˜gi−outside a small neighbourhood of z= 0 and suitably large negative times t≤τin such a way that it becomes globally Lipschitz and bounded with very small Lipschitz constant and bound. Denote by gi−the new nonlinearity and consider, for t≤τ, z(t) = eAi−(t−τ)z(τ) + Zt τ eAi−(t−s)gi−(s, z(s)) ds. Hence Qi−z(t) = Zt ∞ eAi−(t−s)Qgi−(s, z(s)) ds and (I− Qi−)z(t) = Zt −∞ eAi−(t−s)(I− Q)gi−(s, z(s)) ds. Consequently, there exists in a small neighbourhood of z= 0 a globally defined solution of (4.1) if and only if Ti−(z)(t) = Zt ∞ eAi−(t−s)Qi−gi−(s, z(s)) ds +Zt −∞ eAi−(t−s)(I− Qi−)gi−(s, z(s)) ds has a unique fixed point in the set {z:R→ Z : sup t∈R kz(t)kZ≤²} for ²sufficiently small. This follows assuming that, for z, z1, z2∈B(0, ²), kgi−(t, z)kZ≤δand that kgi−(t, z1)−gi−(t, z2)kZ≤δkz1−z2kZ, with δ > 0 sufficiently small. As a consequence of this it follows that ξ∗ i−(·) is uniformly close to y∗ i−. This solution is hyperbolic on R. Hence, ξ∗ i−is a hyperbolic solution of (4.5) for all tlarge and negative. Hence, y∗ i−+ξ∗ i−is a hyperbolic solution of (4.1) in (−∞, τ] with −τ > 0 suitably large. This also ensures that (1) below holds, and we can show that all globally defined bounded solutions are backwards asymptotic to one of the solutions from Proposition 4.1 (themselves asymptotic to the equilibria of the limiting autonomous system). 20 A. N. CARVALHO, J. A. LANGA, J. C. ROBINSON, AND A. SU ´ AREZ Theorem 4.2. Let f:R× Z → Z be a differentiable function that satisfies (4.2). Consider the initial value problem (4.1). Assume that (2.1) is gradient and that all solutions of (2.7) are hyperbolic equilibrium solutions for (2.1). (1) If we write Wu(ξ∗ i)(τ) = {(τ, ζ)∈R× Z :there is a backwards solution z(t, τ, ζ) of (4.1)satisfying z(τ, τ, ζ) = ζand such that lim t→−∞ kz(t, τ, ζ)−ξ∗ i(t)kZ= 0}, then the attractor {A(τ) : τ∈R}of (4.1) is given by A(τ) = ∪n i=1Wu(ξ∗ i)(τ). (2) For each globally defined bounded solution ξ(·)of (4.1) there is an i−with 1≤i−≤n− such that lim t→−∞ kξ(t)−ξ∗ i−(t)kZ= 0.(4.6) Proof: The proof of (1) is a consequence of Proposition 4.1 and Theorem 2.11 if we analyse (4.1) by considering the small non-autonomous perturbations of (2.1) obtained by replacing f(t, y) by fν(t, y) = ½f(t, y),if t≤ −ν f(ν, y),if t > −ν. ¿From Theorem 2.11, for suitably large ν, there exists a pullback attractor {Aν(s) : s∈R} for ˙y=By+fν(t, y) y(τ) = y0 (4.7) given by Aν(s) = ∪n i=1Wu ν(ξ∗ i,ν)(s). To obtain the pullback attractor for (4.1) we first note that (4.7) and (4.1) coincide for t≤τ≤ −ν. Hence A(t) = Aν(t) for t≤ −ν. To recover A(t) for t≥ −νwe only have take advantage of the invariance to see that A(t) = T(t, τ)A(ν), for all τ≤ −ν≤t. Now, (2) is also essentially proved since, by (4.3), every global solution approaches one of the equilibria y∗ i−as t→ −∞,so that, in particular, (2) holds. It is clear from the above proof that in order to characterize the pullback attractor {A(t) : t≥0}it is not necessary that A(t) remains bounded as t→ ∞. This accounts for many cases in the existing literature where the pullback attractors do not remain bounded as t→ ∞ (see [12, 22]) (cf. comments after Definition 1.2). 4.2. Time-dependent forwards attractors. Before considering problems that are asymptotically autonomous as t→+∞we consider in general the problem of defining forwards attractors in non-autonomous problems. It is relatively easy to give a definition of an attractor for individual solutions (‘the point attractor’) in the non-autonomous setting: Definition 4.3. For any fixed t0∈R, a family {A(t) : t≥t0}is the forwards point attractor of a process S(·,·)for t≥t0if •A(t)is non-empty and compact for each t≥t0; •A(t)is invariant, in the sense that S(t, s)A(s) = A(t)for all t≥s≥t0; CHARACTERIZATION OF NON-AUTONOMOUS ATTRACTORS 21 •A(t)attracts each individual solution, dist(S(t, s)z0, A(t)) →0as t→ ∞ for all s∈R,z0∈ Z; and •A(t)is the minimal set with this property, in that if C(t)is another such family, we have A(t)⊆C(t)for all t≥t0. A somewhat simpler definition would have A(t) defined for all t∈R, but this does not seem appropriate for the asymptotically autonomous problems that we are considering here. Allowing for attractors that are only defined on semi-infinite intervals makes them more widely applicable, and we can use the minimality to show that even given this freedom the forwards attractor is essentially unique. Indeed, suppose that t1> t0, and {A0(t) : t≥t0}is a point attractor for t≥t0and {A1(t) : t≥t1}is a point attractor for t≥t1, then due to the minimality property it is immediate that we have A1(t)⊆A0(t) for all t≥t1, while the reverse inclusion follows if we define {˜ A1(t) : t≥t0}by setting ˜ A1(t) = A1(t) for all t≥t1and ˜ A1(t) = {z∈ Z :S(t1, t)z∈A1(t1)},for t∈[t0, t1). That ˜ A1(·) so defined is compact and invariant follows since A1(t)⊆A0(t), and so solutions starting in A1(t1) can be extended back to t=t0. It follows that A0(t) = A1(t) for all t≥t1, showing that ‘asymptotic behaviour’ of the point attractor is uniquely specified by this definition. Identifying the correct concept of a forwards global attractor (i.e. a forwards attractor of bounded sets) for non-autonomous problems is still something that requires further reflection. One would certainly desire that any definition of such a global attractor would include all globally defined bounded solutions, and as discussed in the introduction this is sufficient to define the global attractor in autonomous problems, and gives rise to the pullback attractor in non-autonomous problems. So the pullback attractor should certainly be a subset of the ‘global attractor’ in a non-autonomous problem. However, to see that the pullback attractor will not in general describe all the interesting asymptotic behaviour of a truly non-autonomous problem, consider the equation ˙x=λ(t)x−x3(4.8) with λ:R→Rbeing a smooth function with the property that 0 ≤λ(t)≤1, λ(t) = 0 for all t≤0 and λ(t) = 1 for all t≥1. While the pullback attractor for (4.8) is A(t) = {0}for all t∈R, for t≥1 the equation is ˙x=x−x3, which has three stationary solutions x0= 0, x−=−1 and x+= 1 with the equilibrium x0= 0 being unstable. If we look at the solution with x(1) = 1 and solve the equation for t≤1 we see that x(0,1,1) = x1>0 and therefore x(t, 1,1) = 1 p2t+x2 1 for −x2 1 2< t ≤0, 22 A. N. CARVALHO, J. A. LANGA, J. C. ROBINSON, AND A. SU ´ AREZ which blows up as t→ −x2 1/2. We also see that the set {−1,0,1}attracts points of Rand that [−1,1] attracts bounded subsets of Rforwards in time. In this case it is natural to define A+(t) = [−1,1] for each t > 1. This set has the property that T(t, τ)A+(τ) = A+(t) for all t≥τ≥1 and one can easily see that dist(T(t, τ)B, A+(t)) → 0 as t→ ∞. We now give a general definition along these lines: Definition 4.4. We say that a family {A+(t)⊂ Z :t≥t0}is a time-dependent forwards attractor for (4.1) if: •A+(t)is compact for each t≥t0, •{A+(t) : t≥τ}is invariant in the sense that T(t, τ)A+(τ) = A+(t)for all t≥τ≥t0, and •dist(T(t, τ)B, A+(t)) →0as t→ ∞ for each bounded set B⊂ Z and for any τ≥t0. However, it is important to note that even in our simple example, the ‘natural’ choice A+(t) = [−1,1] for t > 1 is not the only possibility that satisfies our definition. Indeed, if K is any compact set whose interior contains {0}and t0∈Ris fixed then it is easy to see that A+(t) = T(t, t0)K has the properties required by our definition. This implies, in particular, that we cannot impose uniqueness by requiring either maximality or minimality of the forwards attractor. Whether there can be a definitive notion of a forwards attractor for bounded sets is an outstanding open problem. Equally important would be to determine conditions under which the pullback attractor also attracts solutions forwards in time (for examples where this does occur, see [15] and [17]). In the next section we discuss forwards attractors in the context of asymptotically autonomous problems. In this case we can identify the forwards point attractor, and also find a candidate set that satisfies our definition of a forwards global attractor. 4.3. Asymptotically Autonomous Problems at +∞. Assume that lim t→+∞sup z∈B(0,r) {kf(t, z)−f0(z)kZ+kfy(t, z)−f0 0(z)kL(Z)}= 0,for each r > 0,(4.9) and that (2.1) has an autonomous attractor A0. We note that the nonlinearity f0in this subsection may be different from that in the previous subsection and consequently the attractor A0in this subsection may be different from that in the previous one. We assume in addition (and crucially) that is gradient, and that (2.7) has a finite number of solutions, all of them hyperbolic: it follows from Theorem 2.5 that A0is given by (2.10). Assume that (4.1) gives rises to a nonlinear process {T(t, τ) : t≥τ∈R}for which there is an absorbing ball B(0, r0). Consider fk(t, z) the function which coincides with fin [k, ∞)× Z and which is equal to f(k, z) for all t < k and z∈ Z. Then lim k→+∞sup t∈R sup z∈B(0,r0) {kfk(t, z)−f0(z)kZ+k(fk)y(t, z)−f0 0(z)kL(Z)}= 0.(4.10) It has been proved in [5] that the family of attractors for ˙y=By+fk(t, y) y(τ) = y0 (4.11) CHARACTERIZATION OF NON-AUTONOMOUS ATTRACTORS 23 behaves upper and lower semicontinuously as k→ ∞ with the limit attractor being the attractor for (2.1), i.e. sup t∈R dist(Ak(t), A0)→0 as k→ ∞, where dist(A, B) is the symmetric Hausdorff distance defined in Section 1.4. If we denote by {Ak(t) : t∈R}the pullback attractor for (4.11) then Aj(t) = Ak(t),for all j > k and t≤k. Let k0be such that for k≥k0the pullback attractor of (4.11) coincides with the union of the unstable manifolds of all those {ξ∗ i,k}with supt∈Rkξ∗ i,k(t)−y∗ ikZ→0 as k→ ∞. Define A+(t) = Ak0(t) for t≥k0. Note that A+(t) is in fact the forwards image of the global attractor of the autonomous system ˙y=By+f(k0, y) under the non-autonomous process T(·,·). If we define T∞(t, τ) = T0(t−τ),it follows from the fact that sup t≥τ kTk(t, τ)B−T∞(t, τ)BkZ→0,as k→ ∞ and from the lower semicontinuity of attractors that, given ² > 0 there is a T²>0 such that, for all t≥T², T∞(t, τ)B⊂O²(A0) and an N∈Nsuch that Tk(t, τ)B⊂O²(T∞(t, τ)B)⊂ O2²(A0)⊂O3²(A+(t)) for all t≥k≥N. This proves the following result: Theorem 4.5. There is a t0∈Rand a time dependent forwards attractor {A+(t) : t≥t0} for (4.1). We now show that there is a finite number of hyperbolic solutions that attract all other solutions as t→ ∞. First we show that there are hyperbolic solutions asymptotic (as t→ ∞) to each of the equilibria of (2.1) Proposition 4.6. Assume that (4.9) holds. Then, there are solutions ξ∗ j+:R→ Z,1≤j≤ n+, such that lim t→+∞max 1≤j+≤n+kξ∗ j+(t)−y∗ j+kZ= 0.(4.12) Furthermore, there is a t0∈Rsuch that ˙y=Aj+y+Bj+(t)y(4.13) has an exponential dichotomy in [t0,+∞), where Aj+=B+f0 0(y∗ j+)and Bj+(t) = fy(t, ξ∗ j+(t))− f0 0(y∗ j+). Proof: Again, the proof of this result reduces to the proof of (2) in Proposition 2.10, cutting the nonlinearities fin the same way as before to make (4.10) hold. To be more specific, we fix 1 ≤j+≤n+and consider the change variables z=y−y∗ j+in (4.1). In this new variable (4.1) becomes ˙z=Aj+z+ ˜gj+(t, z) (4.14) where ˜gj+(t, z) = f(t, z +y∗ j+)−f0(y∗ j+)−f0 0(y∗ j+)z. Cut ˜gj+outside a small neighbourhood of z= 0 and sufficiently large times in such a way that it becomes globally Lipschitz and bounded with very small Lipschitz constant and bound. Let gj+be the new nonlinearity and proceed exactly as in the previous section (asymptotically autonomous in −∞) to obtain the existence of a global hyperbolic solution ξ∗ j+(·) for the modified equation which is uniformly close to y∗ j+. Now, ξ∗ j+is a solution of (4.14) for all tlarge enough. Hence, y∗ j++ξ∗ j+is a 24 A. N. CARVALHO, J. A. LANGA, J. C. ROBINSON, AND A. SU ´ AREZ solution of (4.1) in [τ, ∞) with τ > 0 suitably large. This solution is hyperbolic on R. Hence, ξ∗ j+is a hyperbolic solution of (4.5) for all tlarge enough. Hence, y∗ j++ξ∗ j+is a hyperbolic solution of (4.1) in [τ, +∞) with τ > 0 suitably large. Ball and Peletier [4] (see also [3]) prove that, further to (4.12), given each (τ, y0)∈R× Z, there exists a j+ with 1 ≤j+≤nsuch that lim t→∞ kT(t, τ)y0−y∗ j+kZ= 0.(4.15) For us, this is a corollary of the following: Corollary 4.7. Let f:R×Z → Z be a differentiable function which satisfies (4.9). Consider the initial value problem (4.1). Assume that (2.1) is gradient and has a global attractor A0, and that all solutions of (2.7) are hyperbolic equilibrium solutions for (2.1). Then, for each (τ, y0)∈R× Z,there exists a j+with 1≤j+≤nsuch that lim t→∞ kT(t, τ)y0−ξ∗ j+(t)kZ= 0.(4.16) In particular, for each globally defined bounded solution ξ(·)of (4.1) there is a j+with 1≤j+≤nsuch that lim t→∞ kξ(t)−ξ∗ j+(t)kZ= 0.(4.17) Note that results on asymptotically autonomous systems in the literature usually show that the forwards asymptotic behaviour of the equations tends to limiting structures within the limit attractor, for instance equilibria of the limit equations, which, in general, are not solutions of the non-autonomous system. (Although there are non-gradient examples showing that the limiting behaviour can differ from that of the limit system, e.g. [20, 25].) Corollary 4.7 goes a little further, since it describes the forwards long time dynamics by means of hyperbolic solutions of the non-autonomous equations. Observe that we also get (4.15) from (4.16) and (4.12). 5. Examples In this section we give three examples to illustrate the wide applicability of our results: an autonomous damped wave equation (a striking example of an autonomous system in which the attractor is still gradient-like even though the underlying system is not gradient), an asymptotically autonomous parabolic equation that illustrates some of the peculiarities of non-autonomous systems, and a simple non-autonomous scalar ordinary differential equation whose pullback attractor we can describe very fully. We hope that our results will further the understanding of non-autonomous attractors in an even wider array of examples. 5.1. A gradient-like attractor for a damped wave equation. Let Ω be a bounded smooth domain in R3. For η∈[0,1], assume that g:R→Ris a twice differentiable function that is bounded with bounded derivatives up to second order. For a∈C(¯ Ω,R3) and η≥0, consider the damped hyperbolic equation utt +βut−∆u=η a(x)· ∇u+g(u) in Ω (5.1) with the boundary condition u= 0 in ∂Ω.(5.2) CHARACTERIZATION OF NON-AUTONOMOUS ATTRACTORS 25 The initial data for (5.1), (5.2) will be taken in the space Z=H1 0(Ω) ×L2(Ω), where the norm in H1 0(Ω) is defined by kϕkH1 0(Ω) =k∇ϕkL2(Ω),ϕ∈H1 0(Ω). It is easy to (see [1, 2]) that (5.1), (5.2) defines a nonlinear semigroup {Tη(t), t ≥0}on Zwhere Tη(t)(ϕ, ψ)=(u(t), ut(t)) with (u(t), ut(t)) being the solution of (5.1), (5.2) such that u(0) = ϕand ut(0) = ψ. If we let A:D(A)⊂L2(Ω) →L2(Ω) be −∆ with homogeneous Dirichlet boundary conditions, then D(A) = H2(Ω)∩H1 0(Ω). We consider (5.1), (5.2) as an abstract evolutionary equation in Z: ˙z=Cz+fη(z), z(0) = z0∈ Z (5.3) where z=µz1 z2¶∈ Z, C=µ0I −A−β¶, and fη(z)(x) = µ0 η a(x)· ∇z1(x) + g(z1(x))¶,for x∈Ω. Under these assumptions fηis continuously differentiable (see [1]), and it is not difficult to see that (2.16) is satisfied. Using the energy V:Z → Rdefined by V(z) = 1 2ZΩ |∇z1|2+δZΩ z1z2+1 2ZΩ z2 2+ZΩ G(z1), where δ > 0 is chosen suitably and G(z1) = Zz1 0 g(s)ds, it follows in a similar way as in [1] that (5.3) has a global attractor in Z. We note that the equilibrium points of (5.1) with η= 0 are of the form z∗ 0= (u∗ 0,0) where u∗ 0is a solution of Au =g(u).(5.4) Furthermore, if u∗ 0is a solution of (5.4) such that 0 /∈σ(−∆−g0(u∗ 0)I) (which is true generically) then (u∗ 0,0) is a hyperbolic equilibrium point of (5.1) with η= 0. As a consequence of Theorem 2.11 the following result holds: Theorem 5.1. Assume that gis twice continuously differentiable with bounded derivatives up to second order and that 0/∈σ(A−g0(u∗ 0)I)whenever u∗ 0is a solution of (5.4). Then, the nonlinear semigroup associated to (5.3) has a global attractor Aη,η∈[0,1], and from the results in [5] this family of attractors is upper and lower semicontinuous at η= 0. Additionally, as a consequence of Theorem 2.11, for suitably small η > 0,Aηhas a gradientlike structure, i.e. it is exactly the union of the unstable manifolds of hyperbolic equilibria.