Recent developments in dynamical systems: three perspectives

RECENT DEVELOPMENTS IN DYNAMICAL SYSTEMS: THREE PERSPECTIVES F. BALIBREA Universidad de Murcia, Dpto. de Matem´aticas, 30100-Murcia, Spain E-mail: balibre[email protected] T. CARABALLO Universidad de Sevilla, Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Apdo. de Correos 1160, 41080-Sevilla (Spain) E-mail: carabal[email protected] P.E. KLOEDEN Institut f¨ur Mathematik, Goethe-Universit¨at, D-60054 Frankfurt am Main, Germany E-mail: [email protected] J. VALERO Univ. Miguel Hern´andez de Elche, Centro de Investigaci´on Operativa, Avda. Universidad s/n, 03202 Elche (Alicante), Spain E-mail: jvaler[email protected] Received September 15, 2009; Revised October 30, 2009 Dedicated to the memory of Valery S. Melnik The aim of this paper is to give an account of some problems considered in past years in the setting of Dynamical Systems, some new research directions and also state some open problems Keywords: Topological entropy, Li-Yorke chaos, Lyapunov exponent, continua, nonautonomous systems, difference equations, ordinary and partial differential equations, set-valued dynamical system, global attractor, non-autonomous and random pullback attractors. Mathematics Subject Classifications (2000): 35B40, 35B41, 35K40, 35K55, 37B25, 37B30, 37B40, 37B45, 37B55, 37E05, 37E15, 39A20 58C06, 60H15. 1. Introduction The theory of dynamical systems has a long prehistory, but essentially started in its present form with the work of Poincar´e and Birkhoff on problems connected with celestial mechanics. In particular, the theory of discrete dynamical systems, which mainly uses iteration theory, is one of the most relevant topics in the subject. The purpose of this tutorial is to give a par- 1 2Balibrea, Caraballo, Kloeden & Valero tial account of the progress obtained in discrete and continuous systems is recent years, to present some open and new problems. It consists of four sections. In Sections 2 and 3 we consider some results in the topological dynamics in non-autonomous discrete systems and in Section 4 we give a rather complete review of multi-valued dynamical systems arising from models involving partial differential equations. Finally, in Section 5 we provide an overview of many problems in non-autonomous and random dynamical systems, in particular on new concepts of nonautonomous and random attractors. 2. Autonomous discrete dynamical systems The development of the theory of topological dynamics began in the earlier part of the last century. It focused in particular on problems related to autonomous discrete dynamical systems given by the pair (X, f), where Xis a topological space and fa continuous map of Xinto itself. The crucial problem was the study of properties of all orbits of all points in the space state X. For x∈X, the orbit of xby fis the sequence (fn(x))∞ n=0, where fn= f(fn−1) for all n≥1 and f0=idX(identity on X). In most cases Xis a compact metric space and, in particular, extensive results were obtained when X =I= [a, b] because many phenomena from social, natural and economical sciences can be formulated as systems evolving with time in a discrete way in such spaces. Moreover, when X=Rd, or a subset thereof, we generally speak of problems on difference equations. Although the number of new results has been impressive, we will include here some of them from one of the subjects more active in the field in last years, dynamical systems on continua. 2.1. Autonomous dynamical systems on continua One line of research that has been very active in recent years is that of dynamical systems on continua, i.e., where Xis a continuum (a compact and connected topological space) and f∈C(X, X). For definitions and detailed account of results see [Nadler, 1995]. Problems such as periodic structure using methods from combinatorial dynamics have been studied on circles, trees and finite graphs (see [Alsed`a et al., 2000]), as well as on centers and depth of centers [Ye, 1993, Kato, 1995, Kato, 1998, Efremova & Makhrova, 2003] on trees, graphs and dendrites, and on the structure of ω-limit sets [Kocan et al., 2010, Balibrea & Garc´ıa Guirao, 2005], in particular, ω-limit sets on hereditarily locally connected continua [Spitalsk´y, 2008], etc. Within dynamical systems theory one of the most interesting topics is that of minimal systems, i.e., systems that do not contain non-trivial subsystems. A system (X, f) is minimal if there is no proper subset Y⊆Xwhich is non-empty, closed and f-invariant, i.e. satisfies f(Y)⊆Y. It is immediate that (X, f) is minimal if and only if the forward orbit of all points in Xare dense in X. We will say also that in this case fis also minimal. Here we will concentrate on the progress on minimal systems in the case that the phase space Xis one-dimensional. The topological characterization of minimal sets of one-dimensional Xhas been carried out for intervals, circles, trees, finite graphs and dendrites. In the interval case, these are finite and Cantor sets (see [Block & Coppel, 1992]), while for the circle case the circle itself can also be minimal. These results can be generalized to graphs, where minimal sets are characterized as finite sets, Cantor sets and also unions of finitely many pairwise disjoint circles [Balibrea et al., 2003, Mai, 2005]. Adendrite is defined as a locally connected continuum which contains no simple closed curve. In this case, besides partial results in [Balibrea et al., 2003], a complete characterization has been given recently in [Balibrea et al., 2009] as a consequence of a more general result based on the new notion of almost totally disconnected spaces. A space Xis almost totally disconnected if the set of its degenerate components is dense in X. The main results says that an almost totally disconnected space admits a minimal map if and only if it is either a finite set or has no isolated point. By a brain we mean a cantoroid whose degenerate components are dendrites and form a null family (for any ε > 0, only a finite number of its members have diameters greater than ε). A cantoroid is a compact metric and almost totally disconnected space without isolated points. With these ingredients we can now state the characterization in [Balibrea et al., 2009]. Three perspectives 3 Theorem 2.1. Let Dbe a dendrite and let Mbe a subset of D. Then Mis a minimal set for some dynamical system (D, f)if and only if Mis either a finite set or a brain. In addition, the following characterization was given in [Balibrea et al., 2009] for general almost totally disconnected spaces. Theorem 2.2. An almost totally disconnected compact metric space admits a minimal dynamical system if and only if it is either a finite set or a cantoroid. Question 2.3. Is it possible to give a topological characterization of minimal sets in other families of one-dimensional continua like arc-like, tree-like or circle-like ? (For definitions see [Nadler, 1995]). 3. Non-autonomous discrete systems We now consider the situation in which the map describing the evolution of the dynamics is itself allowed to change with time. This admits the following formulation. Given a compact topological space Xand a sequence of continuous self-maps (fn)∞ n=1 =f1,∞from Xinto itself, the pair (X, f1,∞) will be called a non-autonomous discrete system where the orbit of a point x∈Xis described by the sequence x, f1(x), f2(f1(x)), . . . , fn(fn−1)...(f2(f1(x)))...) We will use the notation fn 1=fn◦fn−1◦... ◦f2◦f1 which was introduced in [Kolyada & Snoha, 1996]. When all of the maps are the same, i.e., fn=f for all n∈N, then we have a autonomous discrete dynamical system or, simply, a discrete dynamical system, which we considered above. There is not a large literature on truly nonautonomous discrete systems and, as consequence, there are a few results. In the following subsections we will survey some interesting developments. Other results can be found in the expository article [Kloeden, 2000]. 3.1. Topological entropy The authors of [Kolyada & Snoha, 1996] defined the topological entropy of a non-autonomous discrete system, which they denoted by h(f1,∞), using the technique of open covers of Xas in the original paper [Adler et al., 1965] on autonomous systems. In the case that Xis metric or metrizable, they also used separated and spanning sets as in [Bowen, 1970]. It is easy to see that such definitions give similar results when the system is in fact autonomous. What is really interesting, is that it is now possible to define the entropy h(f1,∞, Y ), when Yis a subset of Xwhich is not necessarily compact or invariant under f(f1,∞). Such an extension was necessary to deal with other problems considered in [Kolyada & Snoha, 1996]. An interesting and surprising consequence of the paper [Kolyada & Snoha, 1996] was a proof of the commutativity of the entropy autonomous dynamical systems, i.e., the entropy of the composition of two continuous self-maps on a compact space does not depend on the order in which they are taken, i.e., h(f◦g) = h(g◦f). The first time that the commutativity of the entropy was mentioned seems to have been in [Dinaburg, 1970]. Inspired by [Kolyada & Snoha, 1996], the commutativity or non-commutativity of other types of entropy such sequence entropy were proved in [Balibrea et al., 1999a, Balibrea et al., 1999b]). It is well known (see [Misiurewicz,1989]) that for interval maps, the entropy is positive if and only if one of its iterates has a structure called horseshoe. An interval map fhas a horseshoe if there are two disjoint intervals Jand Ksuch that f(J)∩f(K)⊃J∪K. It is difficult to extend this definition to non-autonomous interval systems because it is associated to a unique map and f1,∞is a sequence. If such notion were possible while retaining properties similar to those of horseshoes for autonomous dynamical interval systems, then one can ask: Question 3.1. If a non-autonomous interval system f1,∞has positive entropy, does it possess a structure of horseshoe type? Another way to obtain non-autonomous dynamical systems is to allow not only maps to change, but also spaces. This leads to the following generalization: a non-autonomous dynamical system is the pair (X, f) given by X= (Xn)∞ n=1 and f= (fn)∞ n=1), where each Xnis a compact metric 4Balibrea, Caraballo, Kloeden & Valero space and fn:Xn→Xn+1 for n∈N. Such a notion is used for example in [Kolyada et al., 1999] to construct a class of smooth triangular maps on the unit square of type 2∞having positive topological entropy and thus extending a previous result from [Balibrea et al., 1995]. Another example is to consider skew-product maps which were introduced in [Bowen, 1970] and are often called triangular maps when defined on [0,1]2, i.e. with F(x, y) = (f(x), g(x, y)) = (f(x), gx(y)), where f∈C(I, I) and gx∈C(I, I) are continuous. Dynamical systems generated by continuous maps on hereditarily locally connected continua such as dendrites can also be interpreted as a sequence of spaces and maps between them and hence as as generalized non-autonomous systems. The sequence of spaces can be, for example, trees approaching dendrites (see [Nadler, 1995]). Results on entropy can be also obtained in such cases. The above generalization was used also to obtain a formula for entropy in terms of the number of pieces of monotonicity of the functions fn 1 when all the Xnare compact real intervals and maps are piecewise monotone. This is a generalization of the well-known Misiurewicz-Szlenk formula for a continuous piecewise interval map (see [Misiurewicz & Slenk,1980]). 3.2. Periodic non-autonomous systems We will restrict now our attention to the case Xn= Ior Rfor each n∈Nsince a lot of results have been obtained in this setting. Let us suppose that there exists a positive integer p(prime period) such that fn+p=fnfor each n≥0. Then the system xn+1 =fn(xn) is called a p-periodic non-autonomous system. In the setting of economical theory, the two periodic case has an interpretation in terms of Parrondo’s paradox [C´anovas et al., 2006]. This paradox says that two losing games can result, under random or periodic alternation of their dynamics, in a winning game. That is, it can happen that combination of losing + losing can result winning. For more information on this paradox see [Harmer & Abbott, 1999a, Harmer & Abbott, 1999b, Parrondo, Harmer & Abbott, 2000]. A partial analysis of when the paradox is impossible is given in [C´anovas, 2010]. It would be interesting to consider examples in two dimensional settings and analyse whether the paradox is possible or not. Motivated by Sharkovskii’s results on the coexistence of periodic orbits of certain periods ([Sharkovskii, 1994, Sharkovskii, 1964]) and forcing relationships, we wonder if the same type of such relationships can be obtained in the non-autonomous case. There are several papers on this for interval maps ([Alsharawi et al., 2006, Alves, 2009, C´anovas & Linero, 2006]). It is stated in [Alves, 2009] that if the condition Card{x∈I, fi(x) = fj(x)}<∞,∀i6=j(mod p) holds, then if the system has a periodic orbit of period rsuch that lcm(r,p) pis odd and larger than 1, then the system has periodic orbits of all periods forced by the Sharkovskii ordering. When p= 1 we recover the Sharkovskii’s original result. To clarify the situation, the sets Aq=n∈Z+: lcm(n, p) = pqfor q∈Z+ were introduced in [Alsharawi et al., 2006]. If ≺denotes Sharkovskii’s ordering, then we have Theorem 3.2. If Al∩P6=∅for some l∈Z+, then Aq∩P6=∅for all l≺q. It is necessary distinguish the periods of the pperiodic equation in Aq. To this end we introduce Q={n∈P:p.n}. Note, by the definition of Aq, we have Aq∩pZ+= {pq}for every q∈Z+. As a consequence Aq∩P= {pq}whenever Aq∩P6=∅and Aq∩Q=∅. Finally, if l∈Z+, then Sl={pq ∈Z+:q≺l, or Aq∩Q6=∅ With such notation we reformulate the former theorem. Theorem 3.3. If Al∩P6=∅for some l∈Z+, then S\Sl= (Q∪pZ+)\Sl Reasons for interest of knowing the elements of Qare given in [Alves, 2009] and the Three perspectives 5 case of two periodic equations is studied in [C´anovas & Linero, 2006], where it is shown that if such an equation has an odd period larger than 1, then P=Q∪2Z+. Such results underline the role played by the set Q. Question 3.4. Is it possible to obtain a general formula for all periods of a p-periodic difference equation similar to that above? 3.3. Lyapunov exponents in periodic nonautonomous systems Let us start with an example coming from using the Poincar´e map to understand the behavior of orbits in non-linear and non-autonomous differential equations of second order (see for example [Chac´on, 2001]), namely xn+1 = [α+ε(bn+βcn)]xn=anxn(1) with α > 1, 0 < β < 1, bn=√2 sin n,cn= √2[2K(m)(n+ Θ)/π;m]. That is, for the sake of clarity and possibility of comparing results we choose the main resonance regime Tsin =Tsn = 2π, where sn denotes the Jacobian elliptic function of parameter m, while K(m) is the complete elliptic integral of the first kin and Θ is the initial phase, i.e. with 0 ≤Θ≤2π. The model is a particular case of general nonautonomous systems xn+1 =fn(xn), where f1,f2, ... are continuous maps of Xinto itself, and X= Ror X=I= [0,1]. The form of the perturbation is designed to capture only the effect of weak non-autonomous excitations on the stability of a generic unstable limit cycle. The elliptic function sn is chosen to introduce in the control excitation in a simple way the effect of the excitation waveform on the control scenario. The behavior of the system is investigated by exploiting the connection between the sensitivity to initial conditions and Lyapunov exponents in autonomous systems: when a point x∈Xhas a positive Lyapunov exponent, then its orbit and that of a point nearby diverge at a positive exponential rate. This connection was explored in [Abraham et al., 2004], while the connection between Lyapunov exponents and positive metric entropy was clarified in [Barrio, 2007]. We can extend the notion of Lyapunov exponent for an autonomous dynamical system (X, f) to a non-autonomous systems (X, f1,∞) by the formula λ(x) = lim n→∞ 1 nlog |(fn◦...f2◦f1)′(x)| = lim n→∞ 1 n n−1 X j=1 log |f′ j(x(j))|(2) As a consequence we can give a notion of chaotic behavior for systems. We will say that a nonautonomous discrete system has chaotic behavior if there is a Lebesgue measurable set of points Lin the space state Xwith positive Lyapunov exponent (in the extended sense). In turn this is interpreted in the application considered in the above paper as the existence of homoclinic chaos, i.e., the existence in the phase space of the non-linear second order differential equation of a homoclinic orbit in the neighborhood of a separatrix. The system is non-chaotic if there is no such set L. Exercise 3.5. Propose new examples and applications of the extension of Lyapunov exponents and develop a complete theory in the setting of (X, f1,∞) with X=Ior X=R. 3.4. Li-Yorke chaos in non-autonomous systems Since the notion of chaos in the sense of Li and Yorke is given in terms of the behavior of orbits of points in X, we can extend it to the setting of nonautonomous systems given by sequences of maps f1,∞ Definition 3.6. The system (X, f1,∞) is chaotic in the sense of Li and Yorke (resp., δ-chaotic) if there is an uncountable set S⊂X(called a scrambled set) such that for all pairs x,y∈Xwith x6=y (i) lim sup n→∞ d(fn 1(x), fn 1(y)) >0 (> δ, resp.) (ii) lim inf n→∞ d(fn 1(x), fn 1(y)) = 0. As a consequence, we can state similar problems in this new setting similar to what has been proved in the case of autonomous discrete dynamical systems. 6Balibrea, Caraballo, Kloeden & Valero Exercise 3.7. Prove or disprove that, when X= I, if there is a Li-Yorke pair, then there is an uncountable number of pairs. For topological entropy we have Question 3.8. If h(f1,∞)>0, is it true that f1,∞ is Li-Yorke chaotic? There are many problem for autonomous systems with zero topological entropy which can also be considered in the setting of non-autonomous systems. 3.5. Li-Yorke chaos in Rnwith n > 1 The notion of snap-back repeller for maps f∈ C1(Rn,Rn) was introduced in [Marotto, 1978] as follows. Assume that x⋆is an expanding fixed point of fin the ball Br(x⋆), i.e. where f(x⋆) = x⋆, and all eigenvalues of Df(x) are greater than one in norm for every x∈Br(x⋆)). Then x⋆is a snapback repeller for fif there is a point x0∈Br(x⋆) with x06=x⋆such that fp(x0) = x⋆for some positive integer pand the determinant det Dfp(x06= 0. In the same paper, Marotto proved that the existence of a snap-back repeller is a sufficient condition to have Li-Yorke chaos (in a generalized sense of the usual notion). Marotto’s theorem was improved after by a theorem of Shi and Chen in [Shi & Chen, 2004]. Other extensions to the Banach space setting can be found in [Kloeden & Li, 2006] . Exercise 3.9. It would be interesting to know if such theory could be extended to non-autonomous systems. That would require also extending the notion of snap-back repeller to another notion available in the new setting. 3.6. Non-autonomous difference equations The field of difference equations has developed quickly over the last thirty years. In many cases, problems can be formulated in terms of autonomous systems given by the pair (X, f) where Xis either (0,∞) or Rand f:X→Xis a non necessarily continuous map. The main problem is the study of properties of solutions of the equation which are given by the sequences (xn)∞ n=0 where x0is the initial point and xn=fn(x0) with the same meaning that in previous paragraphs. In many recent applications we have to consider xn+1 =fn(xn), where (fn)∞ n=0 is a sequence of maps of Xinto itself, which need not be continuous. The traditional problems to be solved are: boundedness, periodicity, convergence, local and global stability of solutions. Equations with delays which are also important in applications are formulated by xn+1 =fn(xn, xn−1, ..., xn−k), where fn:Rk+1 →Ris a continuous map for each n= 0, 1,...,k≥1 and x−k,x−k+1,...,x0∈ R1. These can be reformulated as first order (i.e. without delay) vector valued difference equations in terms of Xn:= (xn−k, xn−k+1,...,xn)∈Rk+1. 3.6(A). Examples of non-autonomous rational equations Non-autonomous rational equations are nonautonomous differential equations involving rational functions. They are common in applications in population dynamics (see [Kulenovic & Ladas, 2002]). 1. The non-autonomous delay Pielou logistic equation: xn+1 =anxn 1 + xn−k , where (an)∞ n=0 is a positive periodic sequence with period p, i.e. with an+p=an, n = 0,1, .... When an=αthe equation is autonomous and the asymptotic behavior and oscillation of its positive solutions have been studied in [Kulenovic & Ladas, 2002]. When (an)∞ n=0 is bounded and persistent (below bounded), [Kocic & Ladas, 1993] gives sufficient conditions for boundedness and the global attractivity of solutions. In the periodic case, sufficient conditions in the coefficients which ensure that there are periodic solutions are also given. Three perspectives 7 2. The previous model is an extension of the periodically forced Beverton-Holt model xn+1 =rKnxn Kn+ (r−1)xn , which represents a population with inherent growth rate r > 1 and a carrying capacity represented by the periodic positive sequence (Kn)∞ n=1 which is assumed to have a prime period p≥2. 3. Non-autonomous Lyness equation: Let (bn)∞ n=1 be a sequence of positive numbers. The non-autonomous Lyness equation xn+1 =xn+bn xn−1 , x0>0, x1>0, is a non-autonomous version of the Lyness equation given by xn+1 =xn+b xn−1 , x0>0, x1>0, which is well-known in the literature. See for example [Camouzis & Ladas, 2007], where a complete report of such equation is given and stated some interesting open problems. In the non-autonomous case, it has been conjectured that solutions are bounded and persist (i.e. are bounded away form zero and infinity) if (bn)∞ n=1 has the same properties. One can ask which other more general sequences (bn)∞ n=1 ensure that the solution (xn)∞ n=0 is bounded and persistent. An interesting example in [Angelis, 2004] shows that (xn)∞ n=0 can even be non-bounded when (bn)∞ n=1 attains only two values and proves that is sufficient for (bn)∞ n=1 to be a monotone sequence to prove that the conjecture is true. 4. It is well-known [Camouzis & Ladas, 2007] that the solutions of the difference rational equation with two delays xn+1 =xn+xn−2 xn−1 ,(3) tends to a period-four solution. It was shown for extension of the model in [Papaschinopoulos & Schinas, 2008], xn+1 =pn+qnxn+qn xn−k , n= 0,1, . . . , k = 1,2,..., with positive initial values x−k,x−k+1,..., x0, that the solutions are bounded, persist and have certain periodicity properties under some conditions on the sequences (pn)∞ n=1 and (qn)∞ n=1, The following open problem was stated in [Papaschinopoulos & Schinas, 2008]. Question 3.10. Consider the difference equation (3) with k= 3,4,... where (pn)∞ n=1 and (qn)∞ n=1 are positive sequences such that (pn)∞ n=1 is bounded and either pn−pn−k≥0and qn=qn−k, n =k, k + 1,... or pn−pn−k≤0and qn=qn−k, n =k, k + 1,... Is every solution of (3) bounded and persistent? 4. Multi-valued dynamical systems 4.1. Equations without uniqueness The theory of semigroups of operators has been a powerful tool for studying the properties (and, in particular, the asymptotic behavior) of solutions of autonomous differential equations in partial derivatives with uniqueness of the Cauchy problem. In this situation the existence and properties of global attractors have been established for a wide type of equations such as reaction-diffusion systems, the two-dimensional Navier-Stokes equations, wave equations and many others (see, e.g., [Babin & Vishik, 1992, Hale, 1988, Ladyzhenskaya, 1991, Sell & You, 1995, Temam, 1988]). However, in many situations concerning systems of physical relevance either uniqueness fails or it is not known to hold. In such cases we cannot define a classical semigroup of operators, so that another theory involving multi-valued maps is necessary, namely, the theory of multi-valued dynamical systems. Often a system is really multi-valued (example of this cases will be given) but, in other cases, we are simple not able to prove the uniqueness of solutions. Hence, the theory of multi-valued dynamical systems allows us to continue when the proof of uniqueness fails due to technical problems, 8Balibrea, Caraballo, Kloeden & Valero and it is still possible to study the asymptotic behavior of solutions no matter we have uniqueness or not. There exists now in the literature a great number of equations of physical interest for which uniqueness of solutions fails. Among these we can cite, for example, the three-dimensional Navier- Stokes system, the Ginzburg-Landau equation, the Lotka-Volterra system with diffusion, the wave equation or differential inclusions (including some models from climatology). This is, of course, one important reason justifying the interest of multivalued dynamical systems. Another (not less important) reason is the fact that there is usually a gap between the conditions that we need to impose to obtain existence of solutions and the conditions necessary to prove uniqueness. Hence, we can weaken the conditions imposed on a differential equations and then consider more general situations. We shall give now in more detail some examples of equations in which either uniqueness is not true or is not known to hold. 4.1(A). Three-dimensional Navier-Stokes equations Let Ω ⊂R3be a bounded open subset with smooth boundary. For given ν > 0 we consider the Navier- Stokes system            ∂u ∂t −ν∆u+ (u·∇)u=−∇p+f, in (0, T)×Ω, div u = 0, u|∂Ω= 0, u (0, x) = u0(x), (4) where u(t, x) = (u1(t, x), u2(t, x), u3(t, x)) is the velocity of an incompressible fluid, pis the pressure and fis an external force. If we consider the usual function spaces V={u∈(C∞ 0(Ω))3:div u = 0}, H=cl(L2(Ω))3V, V =cl(H1(Ω))3V,(5) and assume that f∈H, then the following results are well known (see e.g. [Lions, 1969, Temam, 1979, Temam, 1988]): 1. For every u0∈Vthere exists a unique strong solution of problem (4) which exists in some interval [0, T (ku0kV)). 2. For every u0∈Hthere exists at least one weak solution of (4) which exists in the whole half-line [0,+∞). Therefore, strong solutions are unique, but they cannot be defined globally in time. On the other hand, weak solutions exist globally in time, but it is not known whether uniqueness holds or not. Thus, if we intend to study the asymptotic behavior of solutions as time goes to infinity, then we need to work with weak solutions and we have a problem for which uniqueness fails. In the two-dimensional case, i.e., Ω ⊂R2, uniqueness of weak solutions is true (see, e.g., [Temam, 1979]), so a semigroup of operators can be defined in the phase space H, and the existence of a finite-dimensional global attractor is a well known result (see [Ladyzhenskaya, 1982, Temam, 1988, Babin & Vishik, 1992]). 4.1(B). Reaction-diffusion systems Let d > 0 and N≥1 be integers and let Ω ⊂RN be a bounded open subset with smooth boundary. We denote by |·| the norm in the space Rd(or R), and by (·,·) the scalar product in Rd. Consider the problem            ∂u ∂t −a∆u+f(u) = h(x),(t, x)∈(0, T)×Ω, u|x∈∂Ω= 0 or ∂u ∂ν |x∈∂Ω= 0, u(0, x) = u0(x), (6) where T > 0, x∈Ω, νis the unit outward normal, u= (u1(t, x),...,ud(t, x)), ais a real d×d matrix with a positive symmetric part 1 2(a+at)≥ βI with β > 0 and h∈L2(Ω)d. Moreover, f= (f1,...,fd) is a continuous function satisfying the following conditions: d X i=1 |fi(u)| pi pi−1≤C1(1 + d X i=1 |ui|pi),(7) (f(u), u)≥α d X i=1 |ui|pi−C2,(8) where pi≥2, α, C1, C2>0. Let H=L2(Ω)d,V=H1 0(Ω)dfor Dirichlet boundary conditions and V=H1(Ω)dfor Neumann boundary conditions. Also, for p= (p1, ..., pd) Three perspectives 9 define the spaces Lp(Ω) def =Lp1(Ω) ×···×Lpd(Ω) , Lp(0, T;Lp(Ω)) def =Lp1(0, T;Lp1(Ω)) × ×···×Lpd(0, T ;Lpd(Ω)) . By a globally defined weak solution of (6) we mean a function u(·) which belongs to L∞(0, T;H)∩L2(0, T;V)∩Lp(0, T;Lp(Ω)) for all T > 0 and satisfies the equation in the sense of distributions. As the regularity of uimplies that u∈C([0,∞), H), the initial condition makes sense [Chepyzhov & Vishik, 1996]. Under these conditions there exists at least one globally defined weak solution for every initial condition u0∈L2(Ω)d [Chepyzhov & Vishik, 2002b; p.283]. Uniqueness is true, for example, if the following conditions holds (see e.g. [Chepyzhov & Vishik, 2002b; p.283]): the function fis continuously differentiable and (fu(u)w, w)≥ −C3|w|2,∀u, w ∈Rd,(9) where C3≥0 and fudenotes the Jacobian matrix of u7→ f(u). If we consider a bit more general situation where f=f(x, u) is a Carath´eodory function, then we can give an example for which at least two weak solutions corresponding to a given initial condition exist. Let λ1>0 be the first eigenvalue of −∆ in H1 0(Ω), and ψ1be the corresponding eigenfunction. Without loss of generality we can assume that ψ1(x) >0 for any x∈Ω. It is known that ψ1∈C(Ω), so that max x∈Ω|ψ1(x)| ≤ K. We put f(x, u) =    −λ1u−pψ1(x)√u, u ∈[0,1], −λ1u−pψ1(x)u+u2(u−1), u 6∈ [0,1], and h≡0. It is easy to check that f(x, u) satisfies conditions (7)-(8) with p= 4. Suppose that a= 1, u0= 0. Then u(t, x)≡0 is a trivial solution of the Cauchy problem (6). For fixed r≥0 we define ur(t, x) :=          0,0≤t≤r, 1 4(t−r)2ψ1(x), r ≤t≤r+2 √K, vr(t, x)tr≤t≤T, where tr=r+2 √Kand vr(t, x) is a solution on [tr, T] with vr(tr, x) = ψ1(x) K. It is easy to see that the function uτ(t, x) is a solution of (6) until 1 4(t− r)2ψ1(x)≤1, and it is clear that |ur(t, x)| ≤ 1 for all t∈[r, tτ]. Hence, ur(t, x) is another solution of problem (6). This example shows that conditions (7)-(8) are not enough to give uniqueness. Let us consider some models for which condition (9) fails. The complex Ginzburg-Landau equation The complex-valued Ginzburg-Landau equation is the following:    ∂u ∂t = (1 + iη) ∆u+Ru −(1 + iβ)|u|2u+g(x), u|∂Ω= 0, u (x, 0) = u0(x), (10) where u=u(t, x) = u1(t, x) + iu2(t, x) for (x, t)∈ Ω×[0, T] and g(x) = g1(x)+ig2(x)∈L2(Ω,C) for η,β∈Rand R > 0. We assume that gi∈L2(Ω). For v=u1, u2and u=u1+ıu2, equation (10) can be written as the real-valued system                ∂v ∂t =  1−η η1 ∆v+  g1(x) g2(x)  +  Ru1−u1 2+u2 2u1−βu2 Ru2−u1 2+u2 2βu1+u2  and conditions (7)-(8) hold with p= (4,4). Also, condition (9) holds if |β| ≤ √3 [Chepyzhov & Vishik, 2002b; p.42]. Hence, if |β| ≤ √3, there exists a unique solution for every initial data in L2(Ω)2. We note that, if N= 1,2, uniqueness is proved for every β[Temam, 1988; p.224]. However, if N≥3 and we do not assume the condition |β| ≤ √3, then it is not known whether this equation possesses the property of uniqueness of the Cauchy problem or not. 16 Balibrea, Caraballo, Kloeden & Valero initial condition x0if x(t+τ)∈G(t, x(τ)) for all t,τ∈R+and x(0) = x0. The m-semi-flow G: R+×X→P(X) is said to be time-continuous if it is the union of continuous trajectories for all x0∈ X, i.e., G(t, x0) = (x(t) : x(·) is a trajectory and x(·)∈C(R+,X)). Theorem 4.9. [Melnik & Valero, 1998; Theorem 5] Let us suppose that the conditions of Theorem 4.7 are satisfied. Assume also that Gis a strict time-continuous m-semi-flow with connected values (i.e., G(t, x)is a connected set for every (t, x)). If the space Xis connected, then the global attractor Ais connected. Remark 4.10. We can avoid the assumptions that Gis strict and time continuous, if we assume the existence of a bounded connected set in Xcontaining the global attractor A[Amig´o et al., 2009; Theorem 4.5]. We note that in order to prove the connectivity is essential to keep the property of upper semi-continuity of G(t, ·). Let us apply these results to the m-semi-flow (26). First note that the following estimate is true [Kapustyan & Valero, 2000; p.625]: ku(t)k2≤e−δt ku0k2+K, (30) for any weak solution of (6), where the constants δ and K > 0 are universal. Then a bounded absorbing set exists (for example, a ball in Xof radius √1 + Kcentered at 0) and γ+ 0(B)∈ B(X) for all B∈ B(X). Also, the following lemma holds: Lemma 4.11. [Kapustyan & Valero, 2006; Lemma 2] Let {un}be an arbitrary sequence of solutions of (6) with un(0) →u0weakly in H. Then for any tn→t0, where tn,t0∈(0, T], there exists a subsequence such that un(tn)→u(t0)in H, where u(·)is a weak solution of (6) and u(0) = u0. As a consequence of this lemma we have that the map G(t, ·) is compact for every t > 0.Hence, Lemma 4.4 implies that Gis asymptotically compact. Also, it follows that the graph of Gis closed and that Ghas compact values. Finally, by a contradiction argument (see the proof of Corollary 7 in [Kapustyan & Valero, 2006]) it can be shown that G(t, ·) is upper semi-continuous. On the other hand, it is proved in [Kapustyan & Valero, 2009a] that the map G has connected values, and the other conditions in Theorem 4.9 are obviously satisfied. Hence, by Theorems 4.7, 4.9 we have: Theorem 4.12. The m-semi-flow Ggenerated by (6) possesses a global compact minimal attractor, which is invariant and connected. Let us consider now the method of generalized semi-flows. One of the difficulties of the method of multivalued semi-flows is that we have to work with multi-valued maps. This is avoided in this other method. Also, the method of generalized semi-flows has the advantage of working directly with solutions. Following [Ball, 1997, Ball, 2000] we have: Definition 4.13. A generalized semi-flow Son X is a family of maps ϕ: [0,∞)→X(called solutions) which satisfy: (H1) For each x∈Xthere exists at least one ϕ∈S such that ϕ(0) = x; (H2) (Translation) If ϕ∈Sand τ≥0, then ϕτ∈ S, where ϕτ(t) = ϕ(τ+t). (H3) (Concatenation) If ϕ,ψ∈Swith ψ(0) = ϕ(t1),then φ∈S, where φ(t) =    ϕ(t),0≤t≤t1, ψ(t−t1), t ≥t1. (H4) If ϕj∈Swith ϕj(0) →z, then there exist ϕ ∈Swith ϕ(0) = zand a subsequence such that ϕj(t)→ϕ(t), for all t≥0. We note here that we can define a multi-valued semi-flow G:R+×X→P(X) by G(t, x) = [ ϕ∈S, ϕ(0)=x ϕ(t).(31) Three perspectives 17 Then conditions (H2)−(H3) imply that Gis a strict multi-valued semi-flow and condition (H4) implies that the map x7→ G(t, x) is upper semi-continuous and has compact values. Therefore, Definition 4.13 contains stronger conditions than Definition 4.1. If we consider the reaction-diffusion system (6), then we have seen that the weak solutions satisfy properties (H1) −(H3). Also, by Lemma 4.11 we have (H4). Hence, taking Sas the union of all weak solutions we obtain a generalized semi-flow. It is obvious that the associated multi-valued semi-flow defined in (31) coincides with the semiflow given in (26). We introduce in this case the concepts of ωlimit set and global attractor. For B∈B(X) we set ω(B) = z∈X: there exist ϕj∈S, with ϕj(0) ∈B, and tj→+∞such that ϕj(tj)→z.  It is clear that ω(B) coincides with the omega-limit set (27) for the associated multi-valued semi-flow given in (31). Definition 4.14. The set Ais said to be a global attractor for the generalized semi-flow Sif the associated multi-valued semi-flow Gsatisfies the attraction property (28) and A ⊂ G(t, A), that is, it is negatively semi-invariant. In other words, Ais a global attractor for S if it is a global attractor for the associated multivalued semi-flow G. In [Ball, 1997, Ball, 2000] it is assumed also that the global attractor has to be compact and invariant, but, as we have commented before, we prefer to give a more general definition and add these properties in the statement of the theorem. The reason is that, although usually in applications these additional properties hold (as is the case of the reaction-diffusion system (6)), sometimes they can fail. When the global attractor is compact, one can check that it is also minimal and unique. As before in order to obtain the existence of a global attractor we need some previous definitions. Definition 4.15. The generalized semi-flow Sis asymptotically compact if for any sequence ϕj∈ Swith ϕj(0) bounded, any sequence ϕj(tj), where tj→+∞, is precompact in X. It is clear that Sis asymptotically compact if and only if the associated multi-valued semi-flow is asymptotically compact. Definition 4.16. The generalized semi-flow Sis said to be point dissipative if there exists a bounded set B0such that for any ϕ∈Sthere exists T(ϕ) such that ϕ(t)∈B0for all t≥T. This definition of point dissipativity is weaker than the one given for multi-valued semi-flows in (29). The difference is that if (29) holds for the associated semi-flow G, this implies that all the solutions starting at ϕ(0) entry uniformly in a bounded set B0after some time, that is, there exists a time T(ϕ(0)) such that for any solution ϕ∈Swe have ϕ(t)∈B0for all t≥T. The following lemma implies that an asymptotically compact and point dissipative generalized semi-flow satisfies property (29). Lemma 4.17. [Ball, 2000; Lemma 3.5] Let Sbe an asymptotically compact and point dissipative generalized semi-flow. Then there exists a bounded set B1such that given any compact K⊂Xthere exist ε=ε(K)>0, t1=t1(K)>0such that G(t, Nε(K)) ⊂B1for all t≥t1. It is important to remark that property (H4) is crucial in the proof of this lemma. We now state the theorem concerning the existence of a global attractor. Theorem 4.18. [Ball, 2000; Theorem 3.3] Let S be an asymptotically compact and point dissipative generalized semi-flow. Then there exists a compact global attractor given by A=[ B∈B(x) ω(B), which, moreover, is invariant. With respect to the connectivity of the attractor we have: Theorem 4.19. [Ball, 2000; Corollary 4.3] Assume the conditions of Theorem 4.18. Let Xbe 18 Balibrea, Caraballo, Kloeden & Valero a connected space. Assume that the operator Ghas connected values and that every ϕ∈Sbelongs to C((0,∞) ; X). Then the global attractor Ais connected. We note that the definition of the generalized semi-group Simplies that the associated semi-flow Gis strict and that the map G(t, ·) is upper semicontinuous. Also, it follows from the fact that every ϕ∈Sbelongs to C((0,∞) ; X) that Gcan be represented as the union of trajectories which are continuous on t∈(0,∞). Thus the conditions imposed in this theorem are almost the same as in Theorem 4.9 (in fact, the only difference is that there it is assumed that the trajectories are continuous on t∈[0,∞). Let us apply these results to equation (6). As we have seen before it follows from (30) and Lemma 4.11 that Sis an asymptotically compact and point dissipative generalized semi-flow. Also, as the weak solutions of (6) are continuous, it is clear that every ϕ∈Sbelongs to C((0,∞) ; X). The space X= L2(Ω)dis connected and we have seen that G has connected values. Hence, by Theorems 4.18, 4.19 we have: Theorem 4.20. The generalized semi-flow Sgenerated by (6) possesses a global compact minimal attractor, which is invariant and connected. Remark 4.21. Since the associated multi-valued semi-flow Gcoincides with the semi-flow defined in (26) and the global attractor is unique, we see that the global attractors given in Theorems 4.12, 4.20 are the same. Remark 4.22. Under some conditions the global attractors given in Theorems 4.7, 4.18 are described as the union of all bounded complete trajectories of the semi-flow [Ball, 2000, Kapustyan et al., 2008, Simsen & Gentile, 2008]. In conclusion, we can say that these two methods are very close and allow rather similar results to be obtained. The main difference is that in the method of generalized semi-flows several extra conditions are included in the definition, whereas in the method of multi-valued semi-flows such conditions are added when they become necessary. On the other hand, the method of generalized semi-flows allows to use a weaker definition of point dissipativity. These approaches have been very useful and productive and have allowed results about existence and properties of attractors to be obtained for a wide class of dissipative systems without uniqueness. For example, these results have been applied fruitfully to reactiondiffusion equations [Iovane & Kapustyan, 2006, Kapustyan, 2002, Kapustyan & Shkundin, 2003, Kapustyan & Valero, 2006, Morillas & Valero, 2005], the three-dimensional Navier-Stokes equation [Ball, 1997, Ball, 2000, Cheskidov, 2006, Cheskidov & Foias, 2006, Kapustyan et al., 2007, Kapustyan & Valero, 2007, Kloeden & Valero, 2007, Rosa, 2006], the wave equation [Ball, 2004, Horban & Stanzhyts’kyi, 2008, Iovane & Kapustyan, 2005, Wang & Zhou, 2007], differential inclusions [Carvalho & Gentile, 2003, Cheban & Mammama, 2006, Kapustyan & Valero, 2000, Kenmochi & Yamazaki, 2001, Kloeden & Li ,2005, Li et al., 2008, Melnik & Valero, 1998, Segatti, 2006, Valero, 2000, Valero, 2001, Yamazaki, 2004], delay ordinary differential equations [Caraballo et al., 2005, Caraballo et al., 2007], degenerate parabolic equations [Elmounir & Simonolar, 2000], lattice systems [Morillas & Valero, 2009], the phase-field equation [Kapustyan, 1999, Kapustyan et al., 2003, Morillas & Valero, 2008, Rossi et al., 2008, Valero, 2005] or the Cahn-Hilliard equation [Segatti, 2007, Schimperna, 2007]. See also the monographs [Kapustyan et al., 2008, Cheban, 2004]. Of course, in the case of uniqueness of solutions all results in the two methods coincide with the classical ones. 4.2(B). The method of trajectory attractors In order to avoid the problem of nonuniqueness of the Cauchy problem in the papers [Chepyzhov & Vishik, 1996, Chepyzhov & Vishik, 1997, Chepyzhov & Vishik, 2002a, Malek & Necas, 1996, Sell, 1996] (see also [Chepyzhov & Vishik, 2002b, Three perspectives 19 Malek & Prazak, 2002]) a new method for studying the asymptotic behavior of solutions was proposed, which is now known as the method of trajectory attractors. The main idea of this method is the following. Let us consider some collection of solutions of an equation, denoted by K+={ϕ(s), s ≥0}. These solutions belong to some topological space W, which usually is a space with some local convergence topology on any interval [t1, t2]⊂R+. If we assume that for any ϕ∈ K+we have that ϕτ(·) := ϕ(·+τ)∈K+for all τ≥0, then the translation semi-group S(t, ·) : W→Wgiven by S(t)ϕ(·) = ϕt(·) = ϕ(·+t) is well defined. This method has the advantage that a global attractor for such problems as the threedimensional Navier-Stokes equations can be constructed, whereas this question remains still open using the theory of multi-valued semi-flows or generalized semi-flows. However, as pointed out in [Ball, 2000], the disadvantage is that the direct connection with the evolution of the system in the physical phase space is lost. In the cited papers different approaches are used in order to construct a trajectory attractor, i.e., a global attractor for the translation semigroup. We shall describe briefly the methods given in [Chepyzhov & Vishik, 2002a, Sell, 1996] and their application to the three-dimensional Navier-Stokes system (4). In [Sell, 1996] it is considered that Wis a complete metric space. Then the standard theory of attractors for classical semi-groups is used. On the other hand, in [Chepyzhov & Vishik, 2002a] a bit different approach is used. Let us take some Banach spaces Eand E0such that E⊆E0with continuous embedding (it is possible to have E=E0), and let W=L∞(R+;E)∩C(R+;E0). We note that if ϕ(·)∈W, then ϕ(·)∈Cw(R+;E) (i.e., it is continuous with respect to the weak topology of E) and kϕ(t)kE≤ kϕkL∞(R+;E),∀t≥0. The convergence in C(R+;E0) is given by fn(·)→f(·) in C(R+;E0)⇐⇒ ΠMfn(·)→ΠMf(·) in C([0, M]; E0),∀M > 0, where ΠMis the operator of restriction on [0, M]. We observe that the translation semigroup is continuous in the space C(R+;E0) [Chepyzhov & Vishik, 2002a; Proposition 1.1]. Definition 4.23. The set U⊂ K+is called a trajectory attractor (with respect to the space of trajectories K+in the topology C(R+;E0), if: 1. Uis compact in C(R+;E0) and bounded in L∞(R+;E); 2. Uis invariant, that is, T(t)U=Ufor all t≥ 0; 3. Uis an attracting set, that is, for every set B ⊂ K+, bounded in L∞(R+;E), we have that for any M > 0, distC([0,M];E0)(ΠMT(t)B, ΠMU)→0,(32) as t→ ∞. The function ϕ(·)∈L∞(R;E)∩C(R;E0) is called a complete trajectory for K+, if Π+ϕh(·)∈ K+,∀h∈R, where ϕh(s) = ϕ(s+h) and Π+is the operator of restriction on [0,∞).Let Kbe the union of all complete trajectories for K+. Theorem 4.24. [Chepyzhov & Vishik, 2002a] If there exists an attracting set P⊂ K+, which is compact in C(R+;E0)and bounded in L∞(R+;E), then there exists a trajectory attractor U⊆Pand U= Π+K.(33) In the case where uniqueness of the Cauchy problem holds (and then a semi-group Scan be given), in [Chepyzhov & Vishik, 2002a; Corollary 2.1] it is shown a relation between the trajectory attractor Uand the global attractor of the semigroup S. Namely, assume that for any z∈Ethere exists a unique trajectory ϕ(·)∈K+such that ϕ(0) =z. Then the semi-group given by S(t, z) = ϕ(t) for all t≥0 is well defined. Then if we assume also that the set ∪t≥0S(t, B) is bounded in Efor any set B, which is bounded in E, then under the assumptions of Theorem 4.24 the set A=U(0) is 20 Balibrea, Caraballo, Kloeden & Valero a global (E, E0)-attractor for S. This means that the set Ais bounded in E, compact in E0, it is invariant (i.e., S(t, A) = Afor all t≥0) and attracts every bounded set of Ein the metric of E0. This result can be generalized to the case of non-uniqueness, that is, when we assume that for any z∈Ethere exists at least one trajectory ϕ(·)∈ K+such that ϕ(0) = z. In such a case we define a multi-valued semi-flow Ggiven by G(t, z) = {ϕ(t) : ϕ(·)∈ K+such that ϕ(0) = z} and assuming that ∪t≥0G(t, B) is bounded in Efor any set B, bounded in E, we can obtain the same result [Kapustyan & Valero, 2009a]. Let us consider system (4). In the paper [Chepyzhov & Vishik, 2002a] the set K+is the union of all weak solutions of (4) satisfying a suitable energy inequality. We take E=Hand E0=H−δ, where His given in (5) and Hσstands for the scale of Hilbert spaces corresponding to the Hilbert space H. Then it is proved that an attracting set U, which is compact in C(R+;H−δ) and bounded in L∞(R+;H), exists. Therefore, Theorem 4.24 implies the existence of a trajectory attractor. Now let us consider Sell’s approach [Sell, 1996]. There the set K+is the union of all weak solutions of (4) satisfying some regular assumptions and suitable dissipation and energy inequalities, as well. The space Wis defined as the set K+endowed with the topology of L2 loc (0,∞;H), which is a complete metric space. Then, using the classical theory of attractors for semi-groups [Ladyzhenskaya, 1991, Sell & You, 1995], it is proved that the translation semi-group has a compact global attractor Uin W, that is, Uis compact in W, it attracts every bounded set of Wand is invariant (S(t)U=Ufor all t≥0). As the two previous methods, the method of trajectory attractors has been fruitfully applied to different partial differential equations. For example we can cite applications to the three-dimensional Navier-Stokes equation and related systems [Chepyzhov & Vishik, 1997, Chepyzhov & Vishik, 2002a, Cutland, 2005, Feireisl, 2000, Flandoli & Schmalfuß, 1999, Malek & Necas, 1996, Malek & Prazak, 2002, Norman, 1999, Sell, 1996, Vorotnikov & Zvyagin, 2007], reactiondiffusion systems [Chepyzhov & Vishik, 1996, Efendiev & Zelik, 2001], differential inclusions [Hetzer, 2001, Hetzer & Tello, 2002], the wave equations [Chepyzhov & Vishik, 1997] or equations with delay [Chepyzhov et al., 2006]. 4.3. Technical difficulties and open problems in multi-valued dynamical systems In the previous subsection we reviewed the main points of two different theories for multi-valued dynamical systems. It is natural to ask about additional technical problems which arise if we compare them with the classical case for which uniqueness holds. 4.3(A). Asymptotic compactness and the 3D Navier-Stokes system One of the difficulties which appears as a consequence of the lack of uniqueness is that, unlike the case with uniqueness, there are strong restrictions in the kind of estimates that we can obtain. This occurs because, usually, we do not have a regular approximation for each solution of the equation and, as a result, we cannot obtain formally the necessary estimates and justify them through suitable approximations. We are only able to obtain the estimates that the regularity properties of the solutions allow. In particular, this problem arises when we need to prove the property of asymptotic compactness of a multi-valued semi-flow or a generalized semigroup which, as we have seen, it is crucial in order to prove the existence of a global attractor. For example, let us consider the reactiondiffusion system (6). In this case the asymptotic compactness of the multi-valued semi-flow (or the single-valued semi-group if we have uniqueness) is proved by checking that the map G(t, ·) is compact for any t > 0. If the monotonicity assumption (9) holds (so that we have uniqueness), then for any weak solution we can obtain (formally) an estimate of the type ku(t)kV≤C(t, ku0kH),∀t > 0,(34) where V=H1(Ω)dor H1 0(Ω)dand H= L2(Ω)d, and C(t, ku0kH)→ ∞ as t→0+ [Chepyzhov & Vishik, 1996; p.67]. Then using the compact embedding V⊂Hwe obtain that the Three perspectives 21 semi-group S(t) is compact for any t > 0. Since the solution is unique, we can justify this result as follows: we take a suitable regular approximation sequence for which the estimate is correct (for example, a Galerkin approximative sequence) and then passing to the limit we obtain the estimate for the weak solutions of the equation. When condition (9) is not satisfied, this argument fails, as we cannot state that every weak solution can be approximated by a Galerkin sequence: we just can say that at least one weak solution can be approximated. Hence, we cannot obtain (34) for all weak solutions. The same problem appear in other equations as the phase-field system (21) or the wave equation (19), although it is worth to point out that in some multi-valued systems we have enough regularity in order to obtain estimates of the kind (34), as for example equation (12) [Valero, 2001]. To avoid this problem two appropriate tools have been used in several papers: the method of the energy equation and the monotonicity method. For example, the first one has been applied in [Ball, 2004] to the wave equation (19) and in [Morillas & Valero, 2005] to the reactiondiffusion system (6) in unbounded domains, and the second one in [Kapustyan et al., 2003] to the phase-field system (21) and to the reactiondiffusion system (6) in [Morillas & Valero, 2005, Kapustyan & Valero, 2006]. We note that the method of the energy equation has been also successfully used in many papers for equations with uniqueness of solutions (see e.g. [Ghidaglia, 1994, Lu & Wang, 2001, Moise et al., 1998, Rosa, 1998, Wang, 1999]). The key idea in these methods is the following. Let the phase space Xbe a Hilbert space (of course, more general situations are possible). In such spaces if xn→xweakly and kxnk → kxk, then xn→xstrongly. If we consider the property of asymptotic compactness, then we take a sequence yn∈G(tn, xn) with xn∈ B(X), where Gis the multi-valued semi-flow generated by the equation, and we need to prove that {yn}is precompact. Assuming that the a priori estimates of the equation give us that this sequence is bounded, we obtain passing to a subsequence that yn→yweakly in X. Then kyk ≤ lim inf n→∞ kynk and in order to obtain kynk → kyk(so yn→y strongly) it is enough to show that lim sup n→∞ kynk ≤ kyk. Hence, estimates in more regular spaces are avoided. The two methods diverge in the way of proving the last inequality, as one can check in the cited papers. We observe that these methods fail in the case of the three-dimensional Navier-Stokes system (4), so that the problem of obtaining the property of asymptotic compactness for this equation remains open so far. For example, the method of monotonicity does no work because the weak solutions are not known to be continuous (just weakly continuous) and this property is necessary in this method. Nowadays, if we consider an arbitrary external force f∈H, then the question about the asymptotic compactness for the solutions of (4) is open. In the papers [Ball, 2000, Cheskidov & Foias, 2006, Kapustyan & Valero, 2007, Rosa, 2006] some conditional results have been obtained in this direction (that is, under assumptions that are not known to hold). On the other hand, in the papers [Birnir Svanstedt, 2004, Bondarevski, 1997] under some quite restrictive conditions on the external force f, it is shown that for every bounded set B all the weak solutions starting at Bbecome regular after a time T(B), and in this way it is proved the existence of a global attractor. On the other hand, there can be also difficulties in obtaining an absorbing set. Although this problem does not appear in reaction-diffusion systems or in the phase-field system (as we have enough regularity to obtain similar estimates as in the case of uniqueness), again some extra effort is needed for the three-dimensional Navier-Stokes system. If we multiply equation (4) by uand make some standard operations, one can obtain (formally) an estimate in the norm of the space Hof the kind: ku(t)k2≤e−νλ1tku(0)k2−1 ν2λ2 1kfk2 +1 ν2λ2 1kfk2,∀t≥0.(35) However, we cannot state that this holds for every weak solution, and just for the one that was obtained via the Galerkin approximations. Neverthe- 22 Balibrea, Caraballo, Kloeden & Valero less, estimate (35) can be proved if we take a suitable subset of the set of weak solutions. Namely, in [Ball, 2000] it is taken only the subset of weak solutions satisfying the following energy inequality: V(u(t)) ≤V(u(s)) ,∀t≥s, a.a. s > 0 and s= 0, (36) where V(u(t)) := 1 2ku(t)k2 H +νZt 0ku(r)k2 Vdr −Zt 0 (f(r), u (r)) dr. For such solutions (35) holds [Ball, 2000] and then a bounded absorbing set exists. However, such a choice of solutions has another problem. If we define the translation uτ(t) = u(t+τ), then property (36) fails for u(·) at s= 0. Thus, property (H2) of the definition of generalized semi-flow is not satisifed. Also, for the associated multi-valued map G(t, ·) the key property G(t+s, x)⊂G(t, G (s, x)) fails. We could try to modify inequality (36) in the following way: V(u(t)) ≤V(u(s)) (37) holds for all t≥sand a.a. s > 0. Then the map uτ(t) also satisfies this property and a multi-valued semi-flow can be correctly defined. However, in such case we cannot prove (35). One can check that the following holds [Kapustyan et al., 2008; Lemma 5.29]: ku(t)k2≤e−νλ1(t−s)ku(s)k2−1 ν2λ2 1kfk2 +1 ν2λ2 1kfk2,∀t≥sand a.a. s > 0,(38) but this is not enough to obtain an absorbing set. Also, we note that in this case the concatenation of two solutions do not have to satisfy necessarily (37) and then we cannot state that the semi-flow is strict. Moreover, property (H3) fails and a generalized semi-flow cannot be defined in this case. We observe that, without defining a multivalued semiflow, the uniform convergence of the weak solutions starting in a bounded set to the universal attractor (which consists of points in bounded complete trajectories) is proved in [Foias et. al, 2001]. Finally, a multi-valued semi-flow is defined in [Kapustyan & Valero, 2007, Kapustyan et al., 2008] with solutions satisfying (37) and having a bounded absorbing set by restricting the phase space to a ball of radius R ≥R0, where R2 0=1 ν2λ2 1kfk2. Namely, put BR= {u∈H:kuk ≤ R}and for any R≥R0define the multi-valued map GR:R+×BR→P(BR) as GR(t, u0) :=    u(t) : u(·) is a globally defined weak solution with u(0) = u0such that (37) holds and ku(r)k ≤ R, for all r≥0.   This map is well defined since at least one such solution exists [Kapustyan & Valero, 2007, Kapustyan et al., 2008] and, as the phase space is bounded, GRhas a bounded absorbing set. Then the existence of a global attractor can be proved in the space BRendowed with the weak topology of H. We note that in [Kapustyan & Valero, 2007] inequality (38) is used also in the definition of GR, but this is not necessary as (37) implies (38). 4.3(B). The structure and fractal dimension of attractors of multi-valued dynamical systems We have seen so far that a lot of results have been proved about the existence of global attractors for different multi-valued dynamical systems. Moreover, some topological properties such as its connectedness has been studied. However, not much is known about the structure of these attractors and also about their fractal dimension. Comparing with the classical single-valued semi-groups these problems are much more difficult to solve in the multivalued case. Let us consider the simplest structure of a global attractor. If we consider a continuous classical semi-group having a global compact connected attractor and a Lyapunov (also called energy) function, then, when the set of stationary points is finite, it is well known that the attractor consists of the set of stationary points and all the bounded complete trajectories (heteroclinic connections) joining them [Ladyzhenskaya, 1990]. In particular applications we need to determine which connections exist, as this will give us a full description of the attractor. The Chafee-Infante equation is a very well known model in which this problem has been solved. In this equation the number of Three perspectives 23 stationary points is finite (this number depends on a parameter) and a global compact connected attractor, together with a Lyapunov function, exist. Moreover, the set of stationary points is ordered by the Lyapunov function V. In [Henry, 1985] a complete description of this attractor is given by showing that the two stationary points with less energy are stable, whereas the other ones are unstable, and that the necessary and sufficient condition for the existence of an heteroclinic connection from a point vto a point uis that V(v)> V (u), i.e., if the energy of vis greater than the energy of u. More general and complicated cases can be found, for example, in [Brunovsky & Fiedler, 1989, Fiedler & Rocha, 1996]. Can we obtain such results in multi-valued systems? It is possible, of course, but it is much more difficult than in the single-valued case. For example, if we consider the property of stability of stationary points, then one can see that the method of linearization is not applicable, as the non-linear terms of equation as (6), (12), (19) or (21) are not differentiable. Hence, one of the main tools for studying stability in non-linear differential equations cannot be used in multi-valued systems. Nevertheless, some partial results are known so far. In [Arrieta et al., 2006] it is studied the following differential inclusion          ∂u ∂t −∂2u ∂x2−H0(u)∋0,on (0,1) ×(0, T), u(0, t) = u(1, t) = 0, u(x, 0) = u0(x), (39) where H0(u) =      −1,if u < 0 [−1,1] ,if u= 0, 1,if u > 0, is the Heaviside function. This inclusion is a particular case of (12). First, it was proved that equation (39) has an infinite, but countable, number of equilibria v0= 0, v± 1, v± 2, . . .,v± k, . . . ,which can be ordered using a natural energy (or Lyapunov function) E(u): Ev+ 1=Ev− 1< E v+ 2=Ev2 1< ... ... < E v+ k=Ev− k< ... < E (v0). Then, it is shown that v± 1are asymptotically stable fixed points, and all the other ones are unstable. The fixed point v≡0 possesses the following remarkable property: for any fixed point vkdifferent from 0 there exists a solution, u(t), with initial value u(0) = 0, such that u(t) converges to vkas t→+∞. We note that the existence of a Lyapunov function implies in this case, as occurs for the Chafee-Infante problem, that the global attractor can be described completely by the equilibria and the heteroclinic connections between them. The natural question is then to establish which connections actually exist. The global attractor of (39) is approximated by a sequence of attractors Aεncorresponding to an approximating sequence of Chafee-Infante problems, for which, as we have seen, all existing connections are known. The natural conjecture is that the connections are the same when we pass to the limit case, that is, that a connection exists from the fixed point vto the fixed point v∗if and only if E(v) > E (v∗). Of course, since the energy is decreasing along the trajectories, no connections can exist if E(v)≤E(v∗). It is natural to expect that (39) is equivalent to a Chafee-Infante problem that has undergo all the typical bifurcation cascade of these type of problems, and thus all connections should be present. In [Arrieta et al., 2006] some of these connections are established, but the complete description of the attractor remains an open problem. On the other hand, in [D´ıaz et al., 1997, D´ıaz et al., 2002] the authors show that equation (16) also has an infinite, but countable, number of equilibria. Probably, the structure of the attractor of this equation is the same as for (39). For other problems like (6), (19) or (21) nothing is known in this direction. Finally, let us consider the problem of obtaining estimates of the fractal and Hausdorff dimensions of the global attractor. This question has been studied fruitfully and widely for nonlinear single-valued semi-groups and in this way deep results have been proved in a lot of papers. To mention some of them, for example, estimates of the fractal and Hausdorff dimensions depending on the physical parameter of the equation were proved for the twodimensional Navier-Stokes system, for reactiondiffusion systems and for the wave equation (see e.g. [Ladyzhenskaya, 1982, Temam, 1988]). 24 Balibrea, Caraballo, Kloeden & Valero In the multi-valued case almost nothing has been done in this direction. Some abstract theorems concerning the fractal dimension of multivalued semi-flows can be found in [Melnik, 1998] (see also [Kapustyan et al., 2008]), a result which was applied in [Valero, 2000] to a differential inclusion. However, the conditions imposed in [Melnik, 1998] are too strong and, in fact, in the application given in [Valero, 2000] the multi-valued semi-flow is single-valued when we restrict it to the global attractor. The main difficulty in the multi-valued case is that the technique used in the single-valued one is not suitable due to the fact that either a Lipschitz property of the semi-group [Ladyzhenskaya, 1982, Ladyzhenskaya, 1990] or a strong differentiability property [Temam, 1988] is required. In particular, such properties do not hold for multi-valued semi-flows. Nevertheless, from our point of view there is not an objective reason to think that the global attractors of problems like (6), (19) or (21), for example, do not satisfy similar estimates of the fractal dimension as in the case of uniqueness. Thus, we are convinced that we need just a new technique in which the semi-flow would not need to be either Lipschitz or differentiable. On the other hand, there are multi-valued problems in which we could expect to have an infinite dimensional global attractor. An example is given in [Valero, 2000] for a differential inclusion. Also, we think that, probably, this is true also for the attractor of equation (39). The heuristic reason is the following: we have seen that for any fixed point vk different from 0 there exists a solution, u(t), with initial value u(0) = 0, such that u(t) converges to vkas t→+∞, and in some sense this means (as the number of fixed points vkis infinite) that the unstable manifold of 0 (if it exists, of course) is infinite-dimensional. Also, we note that the global attractor of (39) is approximated by a sequence of attractors of Chafee-Infante problems with growing dimension. The same could be true for equation (16). In conclusion, we state the following relevant open problems in the theory of multi-valued dynamical systems: 1. Obtain the existence of a global compact attractor for the three-dimensional Navier- Stokes system (4) in the phase space Hfor an arbitrary external force f∈H. 2. Describe the structure of the global attractor in some applications. 3. Study the stability of stationary points in some applications. 4. Develop a new method for obtaining estimates of the fractal and Hausdorff dimensions. 5. Non-autonomous and random dynamical systems Undoubtedly, non-autonomous and random systems are also of great importance and interest as they appear in many applications to natural sciences. On some occasions, some phenomena are modeled by nonlinear evolutionary equations which do not take into account all the relevant information of the real systems. Instead some neglected quantities can be modeled as an external force which in general becomes time-dependent (sometimes periodic, quasi-periodic or almost periodic due to seasonal regimes) or even can contain random/stochastic features (also called noise). In the finite-dimensional framework (i.e. for non-autonomous ordinary differential equations in RN) the long-time behaviour of non-autonomous dynamical systems has been widely studied by means of the theory of skew-product flows (see the pioneering works [Miller, 1965] and [Sell, 1996]). However, most of the progress in the infinitedimensional context, i.e., for non-autonomous partial differential equations and specially for systems appearing in mathematical physics, has been done during the last two decades. The first attempts to extend the notion of global attractor to the non-autonomous case led to the concept of the so-called uniform attractor (see [Chepyzhov & Vishik, 1996]). It is remarkable that the conditions ensuring the existence of the uniform attractor parallel those for autonomous systems. To this end, non-autonomous systems are lifted in [Vishik, 1992] to autonomous ones by expanding the phase space. Then, the existence of uniform attractors relies on some compactness property of the solution operator associated to the system. However, one disadvantage of this uniform Three perspectives 25 attractor is that it needs not be “invariant” unlike the global attractor for autonomous systems. At the same time, the theory of pullback (or cocycle) attractors has been developed for both the nonautonomous and random dynamical systems (see [Crauel et al., 1995, Langa & Schmalfuß, 2004, Kloeden & Siegmund, 2005, Kloeden & Schmalfuß, 1998, Schmalfuß, 2000]), and has shown to be very useful in the understanding of the dynamics of non-autonomous dynamical systems. In this case, the concept of pullback (or cocycle or non-autonomous) attractor provides a time-dependent (or random in the stochastic case) family of compact sets which attracts families of sets in a certain universe (e.g. the bounded sets in the phase space) and satisfying an invariance property, what seems to be a natural set of conditions to be satisfied for an appropriate extension of the autonomous concept of attractor. Moreover, this cocycle formulation allows to handle more general time-dependent terms in the models not only the periodic, quasi periodic or almost periodic ones (see, for instance, [Caraballo & Real, 2004, Caraballo et al., 2004] for non-autonomous models containing hereditary characteristics). Readers can consult the monographs [Carvalho et al., 2010, Kloeden & Rasmussen, 2010] for more references, examples and information. 5.1. Motivation of the process and skewproduct flow formalisms The formulation of an autonomous dynamical system as a group or semi–group of mappings depends on the fact that such systems depend only on the elapsed time since starting t−t0and not directly on the current time tor starting time t0themselves. For a non-autonomous system, both the current time tand starting time t0are important. The most natural generalization of a semi–group formalism to non-autonomous dynamical systems includes both tand t0instead of only t−t0. This leads to the two-parameter semi–group or process formalism of a non-autonomous dynamical system. An alternative method includes an autonomous dynamical system as a driving mechanism which is responsible for, e.g., the temporal change of the vector field of a non-autonomous dynamical system. This leads to the skew product flow formalism of a nonautonomous dynamical system. 5.2. Processes Consider an initial value problem for a nonautonomous ordinary differential equation in Rd, ˙x=f(t, x), x(t0) = x0.(40) In contrast to the autonomous case, the solutions may now depend separately on the actual time t and the starting time t0rather than only on the elapsed time t−t0since starting. For example, the scalar initial value problem ˙x=−2tx , x(t0) = x0, has the explicit solution x(t) = x(t, t0, x0) = x0e−(t2−t2 0), where t2−t2 0= (t−t0)2+ 2(t−t0)t0cannot be expressed in terms of t−t0alone. Assuming global existence and uniqueness of solutions in forward time, the solutions form a continuous mapping (t, t0, x0)7→ x(t, t0, x0)∈Rdfor t≥t0with t, t0∈Rand x0∈Rd, with the initial value and evolution properties (i) x(t0, t0, x0) = x0for all t0∈Rand x0∈Rd, (ii) x(t2, t0, x0) = xt2, t1, x(t1, t0, x0)for all t0≤t1≤t2and x0∈Rd. The evolution property (ii) is a consequence of the causality principle that the solutions are determined uniquely by their initial values (for the given differential equation). 5.2(A). Process formulation of nonautonomous systems Solution mappings of non-autonomous ordinary differential equations are one of the main motivations for the process formulation [Dafermos, 1971] (see also [Hale, 1988]) of an abstract non-autonomous system on a complete metric state space (X, d) and time set T, where T=Rfor a continuous time process and T=Zfor a discrete time process. Definition 5.1. Aprocess is a continuous mapping (t, t0, x0)7→ φ(t, t0, x0)∈Xfor t≥t0with t, t0∈Tand x0∈X, with the initial value and evolution properties 32 Balibrea, Caraballo, Kloeden & Valero In general, a non-autonomous set Mis said to have a topological property (such as compactness or closedness) if each fiber of Mhas this property. 5.6(B). Attractors of processes There are basically two ways to define attraction of a compact and invariant non-autonomous set A for a process φon a metric space (X, d) with time set T. The first, and perhaps more obvious, corresponds to the attraction in Lyapunov asymptotic stability, is called forward attraction and involves a moving target, while the latter, called pullback attraction, involves a fixed target set with progressively earlier starting time. In general, these two types of attraction are independent concepts, while for the autonomous case, they are equivalent. Definition 5.18. Let φbe a process. A nonempty, compact and invariant non-autonomous set Ais said to be (i) forward attracting if lim t→∞distφ(t, t0, x0), At= 0 for all x0∈Xand t0∈T, (ii) and pullback attracting if lim t0→−∞distφ(t, t0, x0), At= 0 for all x0∈Xand t∈T. Moreover, if the forward attraction in (i) is uniform with respect to t0∈T, or equivalently, if the pullback attraction in (ii) is uniform with respect to t∈T, then Ais called uniformly attracting. Figures 5.6(B) and 5.6(B) illustrate forward and pullback attraction, respectively, of a nonautonomous set with singleton sets as fibers At= {ρ(t)}, i.e., an entire solution of the process. In an autonomous system, the solutions depend only on the elapsed time t−t0. Moreover, t−t0→ ∞when either t→ ∞ with t0fixed or as t0→ −∞ with tfixed, so pullback and forward convergence are equivalent for an autonomous system. Two types of non-autonomous attractors for processes are possible, depending which of the above types of attraction is used. It is required that the component subsets of such attractors are t x t0 x(·, t0, x0)¯ϕ Fig. 2. Forward attraction t x0 t0t0t0 x(·, t0, x0)¯ϕ Fig. 3. Pullback Attraction compact and that they attract bounded subsets D of initial values in X(rather than just individual points), in the sense that distφ(t, t0, D), At→0, either as t→ ∞ with t0fixed (forward case), or as t0→ −∞ with tfixed (pullback case). Definition 5.19. Let φbe a process. A nonempty and invariant non-autonomous set Ais called (i) a forward attractor if it forward attracts bounded subsets of X, (ii) a pullback attractor if it pullback attracts bounded subsets of X, and (iii) a uniform attractor if it uniformly attracts bounded subsets of X. Forward and pullback attractors are independent concepts and one can exist without the other. Example 5.20. The non-autonomous set T× {0}, i.e., the zero solution, is a forward attractor but Three perspectives 33 not a pullback attractor of the system ˙x=−2tx (50) with the general solution x(t, t0, x0) = x0e−(t2−t2 0), and a pullback attractor but not a forward attractor of the system ˙x= 2tx (51) with the general solution x(t, t0, x0) = x0et2−t2 0. This example demonstrates that forward attraction can be seen as an attraction concept for the future of the system, since the coefficient −2t of (50) is negative for t > 0. On the other hand, pullback attraction means basically attraction for the past of the system, see the negativity of 2tfor t < 0 in (51). The concept of a uniform attractor, however, is concerned with attractivity for the entire time. Example 5.21. Consider the non-autonomous scalar ordinary differential equation ˙x=−x+ 2 sin t . (52) If x1(t) and x2(t) are any two solutions, then their difference z(t) = x1(t)−x2(t) satisfies the homogeneous linear differential equation ˙z=−z with the explicit solution z(t) = z(t0)e−(t−t0), so x1(t)−x2(t)=x1(t0)−x2(t0)e−(t−t0)→0 as t→ ∞, from which it follows that all solutions converge to each other in time. What do they converge to? The explicit solution of the non-autonomous differential equation (52) with initial value x(t0) = x0is x(t, t0, x0) = x0e−(t−t0)+ 2e−tZt t0 essin sds =x0−(sin t0−cos t0)e−(t−t0) + (sin t−cos t), from which it is clear that the forward limit limt→∞ x(t, t0, x0) does not exist. On the other hand, the pullback limit does exist for all tand x0, i.e., lim t0→−∞x(t, t0, x0) = sin t−cos t=: ρ(t), and is independent of x0, i.e., lim t0→−∞x(t, t0, x0)−ρ(t)= 0 . Hence, the non-autonomous set Ahaving the singleton fibers At:= {ρ(t)}is pullback attracting for the solution process. Moreover, it is easily shown that ρ(t) is a solution of the non-autonomous differential equation (52) and since all solutions converge to each other forward in time, the forward convergence lim t→∞x(t, t0, x0)−ρ(t)= 0 also holds. Remark 5.22. The non-autonomous set Ain Example 5.21 is both a pullback and forward attractor of the non-autonomous differential equation (52). It is not difficult to find other forward attractors of (52) which are not pullback attractors. This fact demonstrates that forward attractors can be nonunique, and this is quite typical for forward attractors. For pullback attractors, however, the following uniqueness result can be proved. Proposition 5.23. Suppose that a process φhas two pullback attractors Aand ¯ Asuch that both St≤0Atand St≤0¯ Atare bounded. Then A=¯ A. Proof. The boundedness of St≤0Atimplies for all t∈Tthat distAt,¯ At= lim t0→−∞distφ(t, t0, At0),¯ At ≤lim t0→−∞distφ(t, t0,Sτ≤0Aτ),¯ At= 0 . Analogously, one shows that dist¯ At, At= 0, which finishes the proof, since both the sets Atand ¯ Atare compact. Exercise 5.24. Aφ-invariant family of sets such as a pullback attractor A= (At)t∈Tconsists of entire solutions, see Lemma 5.12. Give an example of a process φwhich has entire solutions that are not contained in the pullback attractor. 5.7. Attractors of skew product flows Let (θ, ϕ) be a skew product flow on a base space Pand a state space Xwith time set T=Ror Z, where (P, dP) and (X, dX) are metric spaces. 34 Balibrea, Caraballo, Kloeden & Valero Then π= (θ, ϕ) is an autonomous semi– dynamical system on the extended state space X:= P×X. A global attractor for an autonomous semi– dynamical system πis, specifically, a nonempty compact subset Aof Xwhich is π-invariant, i.e., which satisfies π(t, A) = Afor all t∈T+is called aglobal attractor of πif lim t→∞distX(π(t, D),A) = 0 for every nonempty bounded subset Dof X. Suppose that Pis compact and θ-invariant. Then the global attractor Aof πhas the form A=[ p∈P(p, x) : x∈Ap, where Apis a nonempty compact subset of Xfor each p∈P, and the π-invariance property π(t, A) = Afor t∈T+is equivalent to the ϕ-invariance property ϕ(t, p, Ap) = Aθt(p)for all t∈T+and p∈P. A global attractor of the autonomous system π is a possible candidate for an attractor for the nonautonomous dynamical system described by the skew product flow. A disadvantage of this definition is that the extended state space Xincludes the base space Pas a component, which often does not have the same physical significance as the state space X. Other types of attractors consisting of a family of nonempty compact subsets of the state space X have also been proposed for a skew product (θ, ϕ). These are analogues of the forward and pullback attractors of a process. Definition 5.25. Let (θ, ϕ) be a skew product flow. A nonempty, compact and invariant nonautonomous set Ais called a pullback attractor of (θ, ϕ) if the pullback convergence lim t→∞distX(ϕ(t, θ−t(p), D), Ap) = 0 holds for every nonempty bounded subset Dof X and p∈P, and is called a forward attractor if the forward convergence lim t→∞distXϕ(t, p, D), Aθt(p)= 0 holds for every nonempty bounded subset Dof X and p∈P. As for processes, the concepts of forward and pullback attractors for skew products are generally independent of each other, and one can exist without the other existing. If the above limit is replaced by limt→∞ supp∈PdistX(·,·), then the attractors are called uniform pullback and uniform forward attractors, respectively. If either of the limits is uniform in this sense, then so is the other and the attractor is both a uniform pullback and a uniform forward attractor, so will be called simply a uniform attractor. The relationship between these different kinds of non-autonomous attractors can be seen in [Cheban et al., 2002] and in the monograph [Kloeden & Rasmussen, 2010]. Example 5.26. Reconsider the non-autonomous scalar ordinary differential equation (52), now writing p(t), a periodic function with period 2π, instead of sin t, ˙x=−x+ 2p(t), with the initial condition x(0) = x0. In the spirit of skew product flows as introduced in Section 5.4, the general solution of this equation depends on both pand x0. The initial value problem has thus the explicit solution x(t) = x(t, p, x0) given by x(t) = x0e−t+ 2e−tZt 0 esp(s) ds . We introduce shift operators on the space P= p(t+·) : 0 ≤t≤2πdefined by θt(p(·)) = p(t+·) and consider the solution corresponding to the driving term θ−τ(p(·)) = p(−τ+·) at time τ, i.e., xτ, θ−τ(p(·)), x0=x0e−τ+ 2e−τZτ 0 esθ−τ(p(s)) ds =x0e−τ+ 2e−τZτ 0 esp(s−τ) ds =x0e−τ+ 2 Zτ 0 es−τp(s−τ) ds =x0e−τ+ 2 Z0 −τ etp(t) dt , where the substitution t:= s−τhas been used. The pullback limit as τ→ ∞ gives lim τ→∞xτ, θ−τ(p(·)), x0=α(p(·)) := 2 Z0 −∞ etp(t) dt Three perspectives 35 for x0in an arbitrary bounded subset D. It consists of singleton fibers Ap=α(p(·)),p∈P, and corresponds to the entire solution ρ(t) := sin t−cos t in the process version of this differential equation in Example 5.21, i.e., ρ(t) = αθt(p(·))for all t∈R. The pullback attraction in this example is uniform, and the pullback attractor is also a forward attractor, and hence a uniform attractor. Moreover, the autonomous semi–dynamical system π= (θ, ϕ) on the extended state space P×Rhas a global attractor given by A=[ p(·)∈Pp(·), α(p(·)). Remark 5.27. The analysis of this system as a skew product flow is somewhat more complicated and less transparent than its analysis as a process in Example 5.21. This is typical and is why the process formulation will often be used in subsequent examples (whenever it is possible). Example 5.28. The difference equation in Example 5.9 generates a discrete time skew product flow with cocycle mapping ϕ(n, p, x) = νnx+ n−1 X j=0 νn−1−jqjfor all n∈N, (53) on the state space X=R. The base space P= [−1,1]Zis the space of bi-infinite sequences p= (qn)n∈Ntaking values in [−1,1] and θis the left shift operator on this sequence space. Replacing pby θ−n(p) in (53) implies ϕn, θ−n(p), x=νnx+ n−1 X j=0 νn−1−jq−n+j∀n∈N, which can be re-indexed as ϕn, θ−n(p), x=νnx+−1 X k=−n ν−k−1qk,∀n∈N. Taking pullback convergence gives lim n→∞ϕn, θ−n(p), x=α(p) := −1 X k=−∞ ν−k−1qk. The pullback attractor Athus consists of singleton fibers Ap={α(p)}for p= (qn)n∈N∈P. Since the pullback convergence here is in fact uniform in p∈ P, the pullback attractor is also a uniform forward attractor, and hence a uniform attractor. Moreover, Ais also the global attractor of the autonomous semi–dynamical system π= (θ, ϕ) on the extended state space P×R. 5.7(A). Existence of pullback attractors In order to prove the existence of the attractor (in both the autonomous and non-autonomous cases) the simplest, and therefore the strongest, assumption is the compactness of the solution operator associated with the system, which is usually available for parabolic systems in bounded domain. However, this kind of compactness does not hold in general for parabolic equations in unbounded domains and hyperbolic equations on either bounded or unbounded domain. Instead we often have some kind of asymptotic compactness. In this situation, there are several approaches to prove the existence of the global (or non-autonomous) attractor. Roughly speaking, the first one ensures the existence of the global (resp. non-autonomous) attractor whenever a compact attracting set (resp. a family of compact attracting sets) exists. The second method consists in decomposing the solution operator (resp. the cocycle or two-parameter semi-group) into two parts: a compact part and another one which decays to zero as time goes to infinity. However, as it is not always easy to find this decomposition, one can use a third approach which is based on the use of the energy equations which are in direct connection with the concept of asymptotic compactness. This third method has been used fo example in [Moise et al., 1998] and [Lukaszewicz & Sadowski, 2004] to extend to the non-autonomous situation the corresponding one in the autonomous framework (see [Rosa, 1998]), but related to uniform asymptotic compactness. Our aim here is to consider the case without uniformity properties and show how the pullback theory works in this situation. In this sense, it is worth mentioning that the compact case has been treated in [Flandoli & Schmalfuß, 1996] in the random case, so our results can be considered as natural complements (as it happens in the deterministic autonomous case). 36 Balibrea, Caraballo, Kloeden & Valero There are generalizations of the theorems ensuring existence of an attractor for autonomous systems to pullback attractors for processes and skew product flows. These are also based on the supposed existence of an absorbing set, which is now absorbing in the pullback sense. Definition 5.29. Let φbe a process on a metric space (X, d). A nonempty compact subset Bof Xis called pullback absorbing if for each t∈T and every bounded subset Dof X, there exists a T=T(t, D)>0 such that φ(t, t0, D)⊂Bfor all t0∈Twith t0≤t−T . Definition 5.30. Let (θ, φ) be a skew product flow on a metric space (X, d). A nonempty compact subset Bof Xis called pullback absorbing if for each p∈Pand every bounded subset Dof X, there exists a T=T(p, D)>0 such that ϕt, θ−t(p), D⊂Bfor all t≥T . The existence theorems will be presented here under basic but restricted assumptions, which can then be relaxed and generalized. 5.7(B). Existence of pullback attractors for processes The following theorem is a simple generalization of the corresponding one for attractors of autonomous semi–dynamical systems. Theorem 5.31. Let φbe a process on a complete metric space Xwith a compact pullback absorbing set Bsuch that φ(t, t0, B)⊂Bfor all t≥t0. Then there exists a pullback attractor Awith fibers in Buniquely determined by At=\ τ≥0[ t0≤−τ φ(t, t0, B)for all t∈T.(54) The formula (54) is a kind of a non-autonomous ω-limit set of the set B. As seen in the introduction of this chapter, a naive definition of a non-autonomous ω-limit set leads to a set, which is not positively invariant. However, the pullback construction used in (54) gives an invariant set and can be regarded as a proper version of a non-autonomous ω-limit set (see [Crauel & Flandoli, 1994] for details). Example 5.32. Consider a non-autonomous dynamical system in Rdgiven by ˙x=f(t, x),(55) where fis continuously differentiable and satisfies the uniform dissipative condition x, f(t, x)≤K−Lkxk2,∀x∈Rd, t ∈R(56) with positive constants Kand L. These assumptions ensure that the differential equation (55) generates a process. Moreover, any solution x(t) of (55) satisfies d dtkx(t)k2= 2x(t),˙x(t) = 2x(t), f(t, x(t))≤K−Lkx(t)k2, from which, on integrating, it follows that kx(t)k2≤ kx(t0)k2e−L(t−t0)+K L1−e−L(t−t0). Suppose that for a bounded subset Dof Rdwith kDk:= supd∈Dkdk>1, we have x(t0)∈D, and define T:= 1 Lln (LkDk). Then kx(t)k2≤1 L+K L=K+ 1 L for x(t0)∈Dand t0≤t−T. Thus, the closed ball Bd0,p(K+ 1)/L:= x∈Rd:kxk2≤(K+1)/L is pullback absorbing and positively invariant. From Theorem 5.31, it follows that the process generated by the differential equation (55) has a pullback attractor in Rdwith components subsets At⊂B:= Bd0,p(K+ 1)/L. 5.7(C). Existence of pullback attractors for skew product flows The counterpart of Theorem 5.31 for skew product flows is the first part of the following theorem. The second part provides some information about a form of forwards convergence of the cocycle mapping, which is different from that in the definition of a forward attractor. Three perspectives 37 Theorem 5.33. Let (θ, ϕ)be skew product flow on a complete metric space Xwith a compact pullback absorbing set Bsuch that ϕ(t, p, B)⊂Bfor all t≥0and p∈P . (57) Then there exists a unique pullback attractor Awith fibers in Buniquely determined by Ap=\ τ≥0[ t≥τ ϕ(t, θ−t(p), B)for all p∈P . (58) If, in addition, (P, dP)is a compact metric space, then lim t→∞ sup p∈P distϕ(t, p, D), A(P)= 0 (59) for any bounded subset Dof Rd, where A(P) := Sp∈PAp⊂B. 5.7(D). A continuous time example Consider a non-autonomous dynamical system in Rdgiven by ˙x=f(p, x) (60) with the driving system θon a compact metric space (P, dP). Suppose that fis regular enough to ensure that the differential equation (60) generates a skew product flow. In addition, suppose that fsatisfies the uniform dissipative condition x, f(p, x)≤K−Lkxk2,∀p∈P, x ∈Rd(61) with positive constants Kand L. Then, similarly to Example 5.32, a solution x(t) satisfies the differential inequality d dtkx(t)k2≤K−Lkx(t)k2, from which it follows that the closed ball B:= Bd0,p(K+ 1)/L is pullback absorbing and positively invariant. From Theorem 5.33, it follows that the skew product flow has a pullback attractor in Rdwith components subsets Ap⊂B,p∈P. Suppose instead that the vector field fsatisfies the uniform one-sided dissipative Lipschitz conditions x1−x2, f(p, x1)−f(p, x2)≤ −Lkx1−x2k2(62) for all p∈Pand x1, x2∈Rdwith some constant L > 0. Then fsatisfies the uniform dissipative condition (61) with constants K′=2 Lsup p∈Pkf(0)kand L′=L 2, and the closed ball B′:= Bd0,p(K′+ 1)/L′is pullback absorbing and positively invariant. Thus, the skew product flow has a pullback attractor with component subsets Apin this ball. In fact, the fibers of the pullback attractor are singleton sets. The proof uses the fact that due to the uniform one-sided dissipative Lipschitz condition (62), the system satisfies kx1(t)−x2(t)k ≤ e−Ltkx0,1−x0,2k(63) for any pair of solutions with the same initial value p∈Pof the driving system. This follows from d dt x1(t)−x2(t)  2 =d dtx1(t)−x2(t), x1(t)−x2(t) = 2x1(t)−x2(t),˙x1(t)−˙x2(t) = 2x1(t)−x2(t), fθtp, x1(t)−fθtp, x2(t) ≤ −2L x1(t)−x2(t)  2, which is integrated to give kx1(t)−x2(t)k2≤e−2Ltkx0,1−x0,2k2. Taking square roots yields the desired result. Theorem 5.34. The pullback attractor Aof the skew product flow (θ, ϕ)generated by the differential equation (60) consists of singleton fibers Ap={ap} for each p∈Pwhen the vector field fsatisfies the uniform one-sided dissipative Lipschitz condition (62). Moreover, t7→ aθt(p),t∈R, is an entire solution of (60) for each p∈P. Proof. Since Ap⊂B′for all p∈P, it follows that kApk:= maxa∈Apkak ≤ R:= (K′+ 1)/L′for each p∈P. Now consider a fixed p∈P, and suppose that there exists an ε0>0 and points a1,a2∈Ap such that ka1−a2k=ε0. Moreover, choose T > 0 such that 2Re−LT =ε0. The ϕ-invariance of the pullback attractor gives ϕT, θ−T(p), Aθ−T(p)= 38 Balibrea, Caraballo, Kloeden & Valero Ap, which means that there exist a′ 1, a′ 2∈Aθ−Tp such that ϕT, θ−T(p), a′ 1=a1and ϕT, θ−T(p), a′ 2=a2. Then, from the inequality in (63), it follows that 0< ε0=ka1−a2k = ϕT, θ−T(p), a′ 1−ϕT, θ−T(p), a′ 2  ≤e−LT ka′ 1−a′ 2k ≤ Re−LT =1 2ε0, which is not possible. Hence, a1=a2. Finally, from the ϕ-invariance of the pullback attractor, ϕ(t, p, ap) = aθt(p)for all t∈Rand p∈P, so the singleton sets forming the pullback attractor define an entire solution of the system. It follows from inequality (63) that this entire solution also forward attracts all other solutions with the same initial value of the driving system, so the pullback attractor is also a forward attractor. (There will be more than one such entire solution when Pis not a minimal subset for the autonomous dynamical system θ). The above theorem generalizes the following autonomous result in [Stuart & Humphries, 1996]. Corollary 5.35. An autonomous differential equation with a vector field fwhich satisfies a one-sided dissipative Lipschitz condition such as (62) (i.e., without the pvariable) has a unique globally asymptotically stable equilibrium point. 5.7(E). Pullback attractors for absorbing families and attraction universes To take into account non-uniformities that are ubiquitous in non-autonomous dynamical systems, greater generality can be attained in the definition of a pullback attractor by considering arbitrary non-autonomous sets Band Dinstead just a single compact absorbing set Band single attracted bounded set D. This allows local as well as global attraction to be handled at the same time. The skew product flows (θ, ϕ) in this subsection are on a metric state space (X, dX) with a metric base space (P, dP) and a time set T. Definition 5.36. An attraction universe Dof a skew product flow (θ, ϕ) is a collection of bounded non-autonomous sets D, which is closed in the sense that if ∅(D′⊆ D for some D ∈ D, then D′∈D. The definitions of pullback convergence and pullback attractor need to be extended accordingly. Definition 5.37. Let (θ, ϕ) be a skew product flow on P×X. A nonempty, compact and invariant nonautonomous set Ais called pullback attractor with respect to an attraction universe Dif the pullback convergence lim t→∞distϕt, θ−t(p), Dθ−t(p), Ap= 0 holds for all p∈Pand D ∈ D. Exercise 5.38. Show that a pullback attractor is unique within a given attraction universe D. The pullback absorbing property now depends on the attraction universe Dunder consideration. Definition 5.39. Let Dbe an attraction universe of a skew product flow (θ, ϕ) on P×X. A nonempty and compact non-autonomous set B ∈ Dis called pullback absorbing with respect to Dif for each D ∈ Dand p∈P, there exists a T=T(p, D)>0 such that ϕt, θ−t(p), Dθ−t(p)⊂Bpfor all t≥T . Theorem 5.33 on the existence of a pullback attractor assuming that of a pullback absorbing set generalizes to attraction universes and pullback absorbing families. Theorem 5.40. Let (θ, ϕ)be a skew product flow on P×X, and suppose that the compact nonautonomous set Bis pullback absorbing with respect to an attraction universe D. Then (θ, ϕ)has a pullback attractor Awith respect to D, where the fibers Apare defined for each p∈Pby Ap=\ s>0[ t>s φ(t, θ−t(p), Bθ−t(p)).(64) The proof is a direct modification of that of Theorem 5.33. Similar theorems can be found in [Schmalfuß, 1992, Flandoli & Schmalfuß, 1996, Kloeden & Schmalfuß, 1996, Kloeden, 1997], see also [Crauel et al., 1995, Cheban & Fakhih, 1994]. Remark 5.41. The assumption that the absorbing sets in Theorems 5.33 and 5.40 are compact is no Three perspectives 39 restriction in a state space such as Rd, which is locally compact and thus closed and bounded subsets are equivalently compact. This is not true for a general state space. In particular, for infinite dimensional spaces, compact subsets are “thin” and it is much easier to determine an absorbing property for a closed and bounded subset, such as a unit ball, rather than a compact subset. Counterparts of Theorems 5.33 and 5.40 then hold, if the cocycle mapping is assumed to be compact, i.e., the mapping ϕ(t, p, ·) : X→Xmaps bounded subsets into precompact subsets for all t > 0 and p∈P, or more generally, asymptotically compact. These generalizations will be considered in the next section on infinite dimensional dynamical systems, i.e., with an infinite dimensional state space X, and dealing with possibly multi-valued non-autonomous and random dynamical systems. It is also remarkable that, when the problem contains some randomness or stochastic features, the parameter space possesses a measurable structure (i.e., is a probability space), and, consequently, the measurability must appear in the definitions. This will be highlighted in the next subsection. In addition, it is also worth mentioning that not all stochastic differential equation generate a random dynamical system (see [Arnold, 1998] for a more detailed exposition on this point). Specifically, in the case of stochastic partial differential equations, it is only known when the noise possesses a very particular form (e.g. either additive or linear multiplicative). 5.8. Existence of pullback and random attractors for multi-valued nonautonomous and random dynamical systems Although we could have developed our theory for the existence of pullback attractors for processes or cocycles directly in the multi-valued case, we have preferred to first introduce and establish the theory in the single-valued framework because, in this way, it may be clearer for a general audience. Now, we will establish the theory in the more general set-up of multi-valued dynamical systems generalizing the theory in Section 4 for non-autonomous/random dynamical systems (see [Caraballo et al., 2008] for more details). A pair (Ω, θ), where Ω is a set and θ= (θt)t∈R is a flow on Ω, i.e., with θ:R×Ω→Ω satisfying θ0= idΩ, θt+τ=θt◦θτ=: θtθτfor t, τ ∈R, is called a non-autonomous perturbation. As an example which describes typical non-autonomous perturbations we consider Ω = Rand θtτ=t+τfor τ=ω∈Ω, t ∈R. Let P:= (Ω,F,P) be a probability space. On this probability space we consider a measurable non-autonomous flow θ: θ: (R×Ω,B(R)⊗F)→(Ω,F). In addition, Pis supposed to be ergodic with respect to θ, which means that every θt-invariant set has measure zero or one, t∈R. Hence Pis invariant with respect to θt. The quadruple (Ω,F,P, θ) which is the model for a noise is called a metric dynamical system. If we replace in the definition of a metric dynamical system the probability space Pby its completion Pc:= (Ω,¯ F,¯ P) the above measurability property is not true in general, see Appendix A [Arnold, 1998; Appendix A]. But for fixed t∈R we have that the mapping θt: (Ω,¯ F)→(Ω,¯ F) is measurable. From now on, let X= (X, dX) be a Polish space. Let D:ω→D(ω)∈2Xbe a multi-valued mapping. The set of multi-functions D:ω→ D(ω)∈2Xwith closed and non-empty images is denoted by C(X).Let also denote by Pf(X) the set of all non-empty closed subsets of the space X. Thus, it is equivalent to write that Dis in C(X), or D: Ω →Pf(X). Let D:ω→D(ω) be a multi-valued mapping in Xover P. Such a mapping is called a random set if ω→inf y∈D(ω)dX(x, y) is a random variable for every x∈X. It is well known that a mapping is a random set if and only if for every open set Oin Xthe inverse image {ω:D(ω)∩O6=∅} is measurable, i.e., it belongs 40 Balibrea, Caraballo, Kloeden & Valero to F(see [Hu & Papageorgiou, 1997; Proposition 2.1.4]). Clearly, all this is also valid if we replace P by Pcand Fby F. If we do not specify which probability space we are using (Por Pc), it will mean that the result is valid for both cases. It is also evident that if Dis a random set with respect to P, then it is also random with respect to Pc. We now introduce multi-valued nonautonomous and random dynamical systems. Definition 5.42. A multi-valued map U:R+× Ω×X→Pf(X) is called a multi-valued nonautonomous dynamical system (MNDS) if i) U(0, ω, ·) = idX, ii) U(t+τ, ω, x)⊂U(t, θτω, U(τ, ω, x)) for all t, τ ∈R+, x ∈X, ω ∈Ω. It is called a strict MNDS if iii) U(t+τ, ω, x) = U(t, θτω, U(τ, ω, x)) for all t, τ ∈R+, x ∈X, ω ∈Ω. An MNDS is called a multi-valued random dynamical system (MRDS) if the multi-valued mapping (t, ω, x)→U(t, ω, x) is B(R+)⊗ F ⊗ B(X) measurable, i.e. {(t, ω, x) : U(t, ω, x)∩O6=∅} ∈ B(R+)⊗F ⊗B(X) for every open set Oof the topological space X. For the above composition of multi-valued mappings we use that for any non-empty set V⊂ X,U(t, ω, V ) is defined by U(t, ω, V ) = [ x0∈V U(t, ω, x0). We also note that the above measurability hypothesis is not standard at least for single-valued random dynamical system. However, for MRDS it is more difficult to derive measurability than for single valued systems. We now introduce some topological properties of the MNDS U. Definition 5.43. U(t, ω, ·) is called upper semicontinuous at x0if for every neighborhood Uof the set U(t, ω, x0) there exists δ > 0 such that if dX(x0, y)< δ then U(t, ω, y)∈ U. U(t, ω, ·) is called upper semi-continuous if it is upper semi-continuous at every x0in X. Remark 5.44. (i) Note that if a mapping U(t, ω, ·) is upper semi-continuous at x0, then for all ε > 0 there exists δ(ε)>0 such that distX(U(t, ω, y), U (t, ω, x0)) ≤ε, for any ysatisfying dX(y, x0)≤δ(ε). (ii) The converse is true when U(t, ω, x0) is compact, see [Aubin & Cellina, 1984]. It is not difficult to extend Definition 5.43 if we consider the upper-semi-continuity with respect to all variables assuming that Ω is a Polish space. We now formulate a general condition ensuring that an MNDS defines an MRDS. We need the particular assumption that Ω is a Polish space and F the associated Borel-σ-algebra. Lemma 5.45. Let Ωbe a Polish space and let F be the Borel-σ-algebra. Suppose that (t, ω, x)7→ U(t, ω, x)is upper semi-continuous. Then this mapping is measurable in the sense of Definition 5.42. Proof. Thanks to Proposition 1.2.5 in [Hu & Papageorgiou, 1997], we have that for each closed subset C⊂X, the set U−1(C) = {(t, ω, x) : U(t, ω, x)∩C6=∅} is closed, and thus is a Borel set in B(R+)⊗ F ⊗ B(X) = B(R+×Ω×X). This implies that the inverse of a closed set is measurable and then, by Theorem 2.2.4 in [Hu & Papageorgiou, 1997], the map is measurable. 5.8(A). Non-autonomous and random attractors for MNDS In this section we generalize the concept of pullback and random attractors to the case of an MNDS and prove a general result for the existence and uniqueness of attractors. Three perspectives 41 As a preparation we need the following definitions. A multi-valued mapping Dis said to be negatively, strictly, or positively invariant for the MNDS Uif D(θtω)⊂ = ⊃ U(t, ω, D(ω)) for ω∈Ω, t ∈R+. Let Dbe the family of multi-valued mappings with values in C(X). We say that a family K∈ D is pullback D-attracting if for every D∈ D and all ω∈Ω lim t→+∞distX(U(t, θ−tω, D (θ−tω)), K(ω)) = 0. B∈ D is said to be pullback D-absorbing if for every D∈ D there exists T=T(ω, D)>0 such that U(t, θ−tω, D (θ−tω)) ⊂B(ω),for all t≥T. (65) The following definition provides the main objective of this article. We have to introduce a particular set system (see [Schmalfuß, 2000]): let Dbe a set of multi-valued mappings in C(X) satisfying the inclusion closed property: suppose that D∈ D and let D′be a multi-valued mapping in C(X) such that D′(ω)⊂D(ω) for ω∈Ω, then D′∈ D. Definition 5.46. A family A∈ D is said to be a global pullback D-attractor for the MNDS Uif it satisfies: 1. A(ω) is compact for any ω∈Ω; 2. Ais pullback D-attracting; 3. Ais negatively invariant. Ais said to be a strict global pullback D- attractor if the invariance property in the third item is strict. A natural modification of this definition for MRDS is Definition 5.47. Suppose Uis an MRDS and suppose that the properties of Definition 5.46 are satisfied. In addition, we suppose that Ais a random set, with respect to Pc.Then Ais called a random global pullback D-attractor. Remark 5.48. In contrast to the theory of random attractors for single valued random dynamical systems we have weaker assumptions on the measurability of A. Of course, it is desirable to obtain that Ais a random set with respect to P, but usually we need stronger assumptions in the applications to obtain this property. A consequence of the pullback convergence and invariance of Pis that it reflexes the forward convergence to the attractor P−lim t→+∞distX(U(t, ω, D(ω)), A(θtω)) = 0 for all sets Dsuch that ω→U(t, ω, D(ω)) is measurable for t≥0. Indeed, we have to replace in the formula for the pullback convergence ωby θtω. However, this is only true in the weaker convergence in probability. There exist counterexamples which show that in general the forward convergence does not hold almost surely (see [Arnold, 1998; page 488]). The main tool to prove the existence of an attractor is the pullback-omega-limit set for the MNDS U. For some multi-valued mappings D we define a pullback-omega-limit set as an ωdependent set Λ(D, ω) given by Λ (D, ω) = \ s≥0[ t≥s U(t, θ−tω, D (θ−tω)). This set is obviously closed, but in general it can be empty. It is not difficult to prove that y∈Λ (D, ω) if and only if there exist tn→ +∞and yn∈U(tn, θ−tnω, D (θ−tnω)) such that limn→+∞yn=y. We then have the following lemma, which is a generalization of Theorem 6 and Lemma 8 in [Caraballo et al., 2003b] to the case in which we consider the family Dinstead of the bounded sets of X. Lemma 5.49. Suppose that the MNDS U(t, ω, ·)is upper semi-continuous for t≥0and ω∈Ω. Let B be a multi-valued mapping such that the MNDS is asymptotically compact with respect to B, i.e., for every sequence tn→+∞, ω ∈Ωevery sequence yn∈U(tn, θ−tnω, B(θ−tnω)) is pre-compact. Then for ω∈Ωthe pullback-omega-limit set Λ (B, ω)is non-empty, compact, and lim t→+∞distX(U(t, θ−tω, B (θ−tω)),Λ (B, ω)) = 0, 48 Balibrea, Caraballo, Kloeden & Valero set-valued dynamical systems on parameters,” Set-Valued Anal. 13, 405-416. Kloeden, P.E., & Mar´ın-Rubio, P. [2003] “Weak pullback attractors of nonautonomous difference inclusions,” J. Difference Equ. Appl.,9, 489-502. Kloeden, P.E., & Rasmussen, M. [2010], Nonautonomous Dyamical Systems (to appear). Kloeden, P.E., & Siegmund, S. [2005] “Bifurcations and continuous transitions of attractors in autonomous and non-autonomous systems”, Internat. J. Bifur. Chaos 15 (3), 743–762. Kloeden, P.E. & Schmalfuß, B. [1996] “Lyapunov functions and attractors under variable timestep discretization,” Discrete and Continuous Dynamical Systems 2(2), 163–172. Kloeden, P.E. & Schmalfuß, [1998] “Asymptotic behavior of non-autonomous difference inclusions,” Systems Control Lett. 33 (4), 275–280. Kloeden, P.E. & Valero, J. [2005] “Attractors of weakly asymptotically compact setvalued dynamical systems,” Set-Valued Analysis 13, 381– 404. Kloeden, P.E. & Valero, J. [2007] “The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier-Stokes equations,” Proc. R. Soc. A 463, 1491-1508. Kocan, Z., Korneck´a, V., & M´alek, M. [2010] “On the center of ω-limit points of continuous maps on dendrites,” Topol. Appl. (to appear). Kocic, V. & Ladas, G. [1993] Global behavior of non-linear difference equations of higher order with applications (Kluwer Academic Publishers, Dordrecht. Kolyada, S. & Snoha, L. [1996] “Topological entropy of non-autonomous dynamical systems,” Random Com. Dynamics,4, 205-233. Kolyada, S., Misiurewicz, M. & Snoha, L. [1999] “Topological entropy of non-autonomous piecewise monotone dynamical systems on the interval,” Fund. Math. 160, 161-181. Kulenovic, M. & Ladas, G. [2002] Dynamics of second order rational difference equations with open problems and conjectures (Chapman & Hall, London). Ladyzhenskaya, O.A. [1972] “Dynamical system, generated by Navier-Stokes equations,” Zap. Nauch. Sem. LOMI 27, 91–115. Ladyzhenskaya, O.A. [1982] “On finitedimensionality of bounded invariant sets for Navier-Stokes equations and other dissipative systems,” Zapiski Nauchnih Seminarov LOMI 115, 137-155. Ladyzhenskaya, O.A. [1987] “On finding minimal global attractors for the Navier-Stokes equations and other PDE,” Uspekhi Mat. Nauk. 42, 25-60. Ladyzhenskaya, O.A. [1990] “Some comments to my papers on the theory of attractors for abstract semi-groups,” Zapiski Nauchnikh seminarov LOMI 182, 102-112. English translation in J. Soviet Math.,62 (1992), 1789-1794. Ladyzhenskaya, O.A. [1991] Attractors for semigroups and Evolution Equations (Cambridge University Press, Cambridge). Langa, J.A. & Schmalfuß, B. [2004] “Finite dimensionality of attractors for non-autonomous dynamical systems given by partial differential equations,” Stochastics and Dynamics 4(3), 385-404. Li, D., Wang, Y. & Wang, S. [2008] “On the dynamics of non-autonomous general dynamical systems and differential inclusions,” Set-Valued Anal. 16, 651–671. Lions, J.L. [1969] Quelques m´ethodes de r´esolution des probl`emes aux limites non lin´eaires (Gauthier-Villar, Paris). Lu, K. & Wang, B. [2001] “Global attractors for the Klein-Gordon-Schr¨odinger equation in unbounded domains,” J. Differential Equations 170, 281-316. Lukaszewicz, G. & Sadowski, W. [2004] “Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains,” Z. Angew. Math. Phys. 55, 1-11. Three perspectives 49 Mai, J.H. [2005] “Pointwise-recurrent graphs maps,” Ergod.Th. & Dynam.Sys. 25 (2), 629- 637. Malek, J. & Necas, J. [1996] “A finite-dimensional attractor for three dimensional flow of incompresible fluids,” J. Differential Equations 127, 498-518. Malek, J. & Prazak, D. [2002] “Large time behavior via the method of l-trajectories,” J. Differential Equations 181, 243-279. Marion, M. [1987] “Attractors for reaction-diffusion equations: existence and estimate of their dimension,” Appl. Anal. 25, 101-147. Marotto, F.R. [1978] “ Snap-back repellers imply chaos in Rn,” J. Math. Anal. Appl. 63, 109-223. Melnik, V.S. [1994] “multi-valued dynamics of nonlinear infinite dimensional systems,” Preprint of NAS of Ukraine, 94-17, Institute of Cybernetics, Kyiv. Melnik, V.S. [1998] “Estimates of the fractal and hausdorff dimensions of sets invariant under multimappings,” Math. Notes 63, 190-196. Melnik, V.S. & Valero, J. [1998] “On attractors of multi-valued semi-flows and differential inclusions,” Set Valued Anal. 6, 83–111. Melnik, V.S. & Valero, J. [2008] “Addendum to ”On attractors of multi-valued semi-flows and differential inclusions” [Set-Valued Anal., 6 (1998), 83-111],” Set Valued Anal. 16, 507-509. Miller, R. K. [1965] “Almost periodic differential equations as dynamical systems with applications to the existence of almost periodic solutions”, J. Differential Equations 1, 337-395. Minkevic, M.I. [1948] “A theory of integral funnels for dynamical systems without uniqueness,” Moskov. Gos. Ped. Inst. Uchen. Zap. 135, 134- 151. English translation in Amer. Math. Soc. Transl. 95 (1970), 11-34. Misiurewicz, M. [1989], “Horseshoes for mappings of an interval,” Bull. Acad. Pol. Sci., S´e. Sci. Math. 315, 167-169. Misiurewicz; M. & Szlenk, W.[1980], “Entropy of piecewise monotone mappings,” Studia Math. 67 (1980), 45-63. Moise, I., Rosa, R. & Wang, X. [1998] “Attractors for non-compact semi-groups via energy equations,” Nonlinearity 11, 1369-1393. Morillas, F. & Valero, J. [2005] “Attractors for reaction-diffusion equations in RNwith continuous nonlinearity,” Asymptot. Anal. 44, 111-130. Morillas, F. & Valero J. [2008] “Asymptotic compactness and attractors for phase-field equations in R3,” Set-Valued Anal. 16, 861-897. Morillas, F. & Valero J. [2009] “A Peano’s theorem and attractors for lattice dynamical systems,” Internat J. Bifur. Chaos 19, 557-578. Norman, D.E. [1999] “Chemically reacting fluid flows: weak solutions and global attractors,” J. Differential Equations 152, 75-135. Nadler, S.B. [1992] Continuum Theory: An Introduction, Monogrphs and Textbooks In Pure and Applied Math., Vol 158, Marcel Dekker, New York, Basel, Hong Kong, 1992. Otani, M. [1977] “On existence of strong solutions for du dt +∂ψ1(u(t))−∂ψ2(u(t)) ∋f(t),” J. Fac. Sci. Univ. Tokio. Sect. IA Math 24, 575-605. Otani, M. [1984] “Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, periodic problems problems,” J. Differential Equations 54, 248- 273. Papageorgiuou, N.S. & Papalini, F. [1996] “On the structure of the solution set of evolution inclusions with time-dependent subdifferentials,” Acta Math. Univ. Comenianae 65, 33-51. Papaschinopolous, G. & Schinas, C.J. [2008],“On a non-autonomous kth-order rational equation difference equation” J. Difference Equ. Appl. 14 (6), 645-655 Parrondo, J.M., Harmer, G.P. & Abbott, D. [2000] “New paradoxical games based on brownian ratchets,” Phys. Rev. Lett. 85, 5226-5229. 50 Balibrea, Caraballo, Kloeden & Valero Rosa, R. [1998] “The global attractor for the 2D Navier-Stokes flow on some unbounded domains,” Nonlinear Anal. 32, 71-85. Rosa, R. [2006] “Asymptotic regularity conditions for the strong convergence towards weak limit set and weak attractors of the 3D Navier-Stokes equations,” J. Differential Equations 130, 196- 212. Rossi, R., Segatti, A. & Stefanelli, U. [2008] “Attractors for gradient flows of non convex functionals and appplications,” Arch. Rational Mech. Anal. 187, 91-135. Roxin, E. [1965] “Stability in general control systems,” J. Differential Equations 1, 115-150. Roxin, E. [1965] “On generalized dynamical systems defined by contingent equations,” J. Differential Equations 1, 188-205. Schimperna, G. [2007] “Global attractors for Cahn- Hilliard equations with nonconstant mobility,” Nonlinearity 20, 1987-2006. Schmalfuß, B. [1992] “Backward cocycles and attractors of stochastic differential equations,” in International seminar on Applied Mathematics – Nonlinear Dynamics: Attractor Approximation and Global behavior (Eds. V. Reitmann, T. Riedrich, and N. Koksch) (Technische Universit¨at, Dresden), 185–192. Schmalfuß, B. [2000] “Attractors for nonautonomous dynamical systems,” in Proc. Equadiff 99, Berlin (Eds. B. Fiedler, K. Gr¨oger and J. Sprekels) (World Scientific, 2000), 684-689. Segatti, A. [2006] “Global attractor for a class of doubly nonlinear abstract evolution equations,” Discrete Contin. Dyn. Systems 14, 801-820. Segatti, A. [2007] “On the hyperbolic relaxation of the Cahn-Hilliard equation in 3D: approximation and long time behaviour,” Math. Models Methods Appl. Sci. 17, 411-437. Sell, G.R. [1971] Topological Dynamics and Ordinary Differential Equations, (Van Nostrand Reinhold Mathematical Studies, London). Sell, G.R. [1996] “Global attractors for the threedimensional Navier-Stokes equations,” J. Dynamics Differential Equations 8, 1–33. Sell, G.R. & You, Y. [1995] Dynamics of evolutionary equations (Springer, New-York). Sharkovskii A.N. [1964] “Co-existence of the cycles of a continuous mappings of the line into itself” (in Ukrainian), Ukrain. Math. Zh. 16 (1), 64-71. Sharkovskii, A.N. [1994] “Co-existence of the cycles of a continuous mappings of the line into itself” (in Ukrainian), Proceedings of the Conference ”Thirty Years after Sharkovskii’s theorem: New Perspectives” (Murcia, 1994), Internat. J. Bifur. Chaos 5(1995), 1263-1273. Shi, Y.M. & “Discrete chaos in Banach spaces”, Science in China Series A: Mathematics,34, 595- 609. Shnirelman, A. [1997] “On the nonuniqueness of weak solution of the Euler equation,” Comm. Pure Appl. Math. L, 261-1286. Simsen, J. & Gentile, C. [2008] “On attractors for multi-valued semi-groups defined by generalized semi-flows,” Set-Valued Anal. 16, 105-124. Spitalsk´y, V. [2008] “Omega-limit sets in hereditarily locally connected continua,” Topol. Appl. 155 (11), 1237-1255. Smoller, J. [1983] Shock waves and reactiondiffusion equations (Springer, New-York). Stuart, A.M. & Humphries, A.R. [1996] Dynamical Systems and Numerical Analysis, (Cambridge University Press, Cambridge). Szego, G.P. & Treccani, G. [1969] Semi-grupi di transformazioni multivoche (Springer-Verlag, Berlin). Temam, R. [1979] Navier-Stokes Equations (North- Holland, Amsterdam). Temam, R. [1988] Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer- Verlag, New York. Terman D. [1983] “A free boundary problem arising from a bistable reaction-diffusion equation,” Siam J. Math. Anal. 14, 1107-1129. Three perspectives 51 Terman D. [1085] “A free boundary arising from a model for nerve conduction,” J. Differential Equations 58, 345-363. Tolstonogov, A.A. & Umansky, Ya.I. [1992] “On solutions of evolution inclusions I,” Siberian Math. J. 33, 500–511. Tolstonogov, A.A. & Umansky, Ya.I. [1992] “On solutions of evolution inclusions II,” Siberian Math. J. 33, 693–702. Valero, J. [2000] “Finite and infinite dimensional attractors of multi-valued reaction-diffusion equations”, Acta Math. Hungary. 3, 239–258. Valero, J. [2001] “Attractors of parabolic equations without uniqueness,” J. Dynamics Differential Equations 13, 711–744. Valero, J. [2005] “On the Kneser property for some parabolic problems”, Topology Appl. 153, 975- 989. Vishik, M.I. [1992] “Asymptotic behavior of solutions of evolutionary equations” (Cambridge Univ. Press, Cambridge). Vorotnikov, D. & Zvyagin, V. [2007] “Uniform attractors for non-autonomous motion equations of viscoelastic medium,” J. Math. Anal. Appl. 325, 438-458. Vrabie, I.I. [1997] Compactness Methods for Nonlinear Equations (Pitman Longman, London). Wang, B. [1999] “Attractors for reaction-diffusion equations in unbounded domains,” Physica D 128, 41-52. Wang, Y. & Zhou, Sh [2007] “Kernel sections and uniform attractors of multi-valued semiprocesses,” J. Differential Equations 232, 573- 622. Yamazaki, N. [2004] “Attractors of asymptotically periodic multi-valued dynamical systems governed by time-dependent subdifferentials,” Elec. J. Differential Equations 107, 1-22. Ye, X. [1993] “The center and the depth of the center of a tree map,” Bull. Austral. Math. Soc. 48, 347-350.