autonomous and non autonomous attactiors for differential equations with delays

ARTICLE IN PRESS J Differential Equations 208 (2005) 9–41 Autonomous and non-autonomous attractors for differential equations with delays T Caraballo,a,Ã P Marı´ n-Rubio,a and J Valerob a Dpto de Ecuaciones Diferenciales y Ana´lisis Nume´rico, Universidad de Sevilla, Apdo de Correos 1160, 41080 Sevilla, Spain b Universidad Cardenal Herrera CEU, Comissari 3, 03203 Elche, Alicante, Spain Received March 27, 2003 Dedicated to Professor George R Sell on the occasion of his 65th birthday Abstract The asymptotic behaviour of some types of retarded differential equations, with both variable and distributed delays, is analysed In fact, the existence of global attractors is established for different situations: with and without uniqueness, and for both autonomous and non-autonomous cases, using the classical notion of attractor and the recently new concept of pullback one, respectively r 2003 Elsevier Inc All rights reserved MSC: 34D45; 34K20; 34K25; 37L30 Keywords: Autonomous and non-autonomous (pullback) attractors; Delay differential equations; Integro-differential equations; Non-uniqueness of solutions; Multi-valued semiflows; Multi-valued processes Introduction Physical reasons, non-instant transmission phenomena, memory processes, and specially biological motivations (e.g [5,20,31,37]) like species’ growth or incubating time on disease models among many others, make retarded differential equations an important area of applied mathematics Ã Corresponding author E-mail addresses: caraball@us.es (T Caraballo), pmr@us.es (P Marı´ n-Rubio), valer.el@ceu.es (J Valero) 0022-0396/$ - see front matter r 2003 Elsevier Inc All rights reserved doi:10.1016/j.jde.2003.09.008 ARTICLE IN PRESS 10 T Caraballo et al / J Differential Equations 208 (2005) 9–41 Moreover, the asymptotic behaviour of such models has meaningful interpretations like permanence of species on a given domain, with or without competition, their possible extinction, instability and sometimes chaotic developments, being therefore of obvious interest However, most studies use to deal with stability concepts concerning fixed points The study of global attractors and the equations for which the existence of an attractor (and so both stable and unstable regions) can be ensured is therefore an interesting subject The theory of global attractors for autonomous systems as developed by Hale [24] owes much to examples arising in the study of (finite and infinite) retarded functional differential equations [26] (for slightly different approaches see [3,32,41]) Although the classical theory can be extended in a relatively straightforward manner to deal with time-periodic equations, general non-autonomous equations such as x0 ðtÞ ¼ F ðt; xðtÞ; xðt À rðtÞÞÞ; ð1Þ with variable delay, or x0 ðtÞ ¼ Z bðt; s; xðt þ sÞÞ ds; ð2Þ Àh for distributed delay terms, including the possibility of being h ¼ þN; fall outside its scope Recently, a theory of ‘pullback attractors’ has been developed for stochastic and non-autonomous systems in which the trajectories can be unbounded when time increases to infinity, allowing many of the ideas for the autonomous theory to be extended to deal with such examples In this case, the global attractor is defined as a parameterized family of sets AðtÞ depending on the final time, such that attracts solutions of the system ‘from ÀN’, i.e initial time goes to ÀN while the final time remains fixed Moreover, in [7] this theory has been successfully extended to deal with variable delay equations, and some sufficient conditions have been proved to guarantee the existence of pullback attractor for Eq (1) (see also [12,15]) However, as far as we know, there exists a wide variety of situations of great interest from the point of view of applications that still has not been analysed For instance, delay differential equations without uniqueness (in both the autonomous and non-autonomous framework), differential inclusions, integro-differential equations in a non-autonomous context with or without uniqueness, all the previous situations but considering infinite delays, etc Consequently, we are mainly interested in providing some results on two of the previous situations: autonomous functional equations without uniqueness, and nonautonomous functional and/or integro-differential equations with and without uniqueness with finite delay The content of the paper is as follows In Section we include some preliminaries on the existence of solutions to functional differential equations and their properties The construction of the semiflows and processes associated to our delay models is ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 11 carried out in Section Some results ensuring the existence of autonomous and nonautonomous attractors are collected in Section Finally, in Section 5, which is the main one, our theory is applied to some interesting and general situations arising in applications and several examples are exhibited Preliminaries First, let us introduce some notation Let h40 be a given positive number (the delay time) and denote by C the Banach space Cð½Àh; 0; Rn Þ endowed with the norm jjcjj ¼ supsA½Àh;0 j cðsÞj; which is the usual phase space when we deal with delay differential equations However, it is sometimes useful to consider the solutions as mappings from R into Rn (we will consider in Rn its usual Euclidean topology and denote by /Á; ÁS; j Á j its scalar product and norm, respectively) Let us point out that the case of infinite delay needs a more careful choice of the phase space (cf [1,25]), but we will not get into those details here By xt we will denote the element in C given by xt ðsÞ ¼ xðt þ sÞ for all sA½Àh; 0: Also, it will be useful to denote Rd ¼ fðt; sÞAR2 ; tXsg: We will now recall some well-known results for a general functional differential equation with finite delay (cf [23, Chapter 2]): x0 ðtÞ ¼ f ðt; xt Þ; xt0 ¼ cAC: ð3Þ Theorem (Existence of solutions) Suppose O is an open subset in R Â C and f ACðO; Rn Þ: If ðt0 ; cÞAO; then there is a solution of (3), i.e a function x : ½t0 À h; t0 þ aÞ-Rn with a40; which satisfies (3) in a classical sense Remark As in the non-delay case, uniqueness results hold if, for instance, f satisfies a locally Lipschitz condition on compact sets with respect to its second variable (cf [23, Chapter 2, Theorem 2.3]) However, we will be concerned with both situations, i.e with and without uniqueness, establishing a more general theory The existence of global solutions in time of (3) is obviously essential for our purpose We have the following result from [23, Chapter 2]: Theorem (Non-continuable solutions) Suppose O is an open set in R Â C and f ACðO; Rn Þ: If x is a non-continuable solution of Eq (3) on ½t0 À h; bÞ; then, for any compact set W in O; there is a tW such that ðt; xt ÞeW for tW ptob: ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 12 As a straightforward consequence of this result, we have an analogous result to the non-delay case with non-explosion a priori estimates Suppose E is a metric space and denote by PðEÞ; CðEÞ; BðEÞ and KðEÞ the sets of non-empty, non-empty and closed, non-empty and bounded, and non-empty and compact subsets of E: Definition Given two metric spaces X and Y ; a single (or multi-valued resp.) mapping U : X -Y ðPðY Þ resp.) is said to be bounded if for every BABðX Þ; U ðBÞABðY Þ: Remark Observe that, if the map f is only bounded, we cannot in general ensure that the solutions of (3) are defined in the future, even in the case without delays, as the simple example x0 ¼ x2 shows Corollary Let f ACðR Â C; Rn Þ be a bounded map, and assume that the equation in (3) satisfies the property that possible solutions x corresponding to an initial datum c remain in a bounded set of C; in other words, ðt; t0 ÞARd ; 8cAC; (D ¼ Dðt; t0 ; cÞABðCÞ such that solution xðÁÞ of ð3Þ defined in ½t0 À h; tÞ it holds xt0 AD 8t0 A½t0 ; tÞ: ð4Þ Then, all solutions are defined globally in time Proof By a contradiction argument, consider a non-continuable solution x of (3) defined in ½t0 À h; tÞ; with initial datum cAC: Then, from (4) we deduce the existence of a bounded set D ¼ Dðt; t0 ; cÞABðCÞ such that xt0 AD 8t0 A½t0 ; tÞ: Define now the set ( o¼ ) n jAC ð½Àh; 0; R Þ : jjjjjpjjDjj; jjj jjp sup j f ðr; ZÞj ¼ M ; ðr;ZÞA½t0 Àh;tÂD which is compact thanks to the Ascoli–Arzela` Theorem Thus, the set W ¼ ½t0 ; t Â o is compact and we can apply Theorem to obtain the existence of tW such that ðt0 ; xt0 ÞeW for tW pt0 ot: In particular, for tW it holds that either jjxtW jj4jjDjj or jjx0tW jj4M: The first possibility obviously contradicts (4) For the second, observe that jjx0tW jj ¼ sup jx0tW ðyÞj yA½Àh;0 ¼ sup jx0 ðtW þ yÞj yA½Àh;0 ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 13 ¼ sup j f ðtW þ y; xtW þy Þj yA½Àh;0 p sup j f ðr; ZÞj ¼ M; ðr;ZÞA½t0 Àh;tÂD and the proof is complete & Semiflows and processes for retarded differential equations In this section we aim to establish the definitions of (multi-valued) semiflows and processes associated to our two cases under study (autonomous functional equations and non-autonomous integro-differential equations with or without uniqueness) and some useful properties about them In order to avoid unnecessary repetitions, we shall first state the results for the non-autonomous case and will particularize later on for the autonomous framework Hereafter O will denote the set R Â C unless otherwise is specified We also suppose that the assumptions in Corollary hold, what jointly with Theorem 1, guarantees the existence of global solutions to (3) Since in [7] only the case containing variable delays was considered, we will develop here most of the time our theory and applications for a mixed case of both retarded terms, with the following canonical form: x ðtÞ ¼ f ðt; xt Þ ¼ F ðt; xðtÞ; xðt À rðtÞÞÞ þ Z bðt; s; xðt þ sÞÞ ds; ð5Þ Àh xt0 ¼ cAC; ð6Þ where F ACðR2nþ1 ; Rn Þ contains the dependence on the variable delays (for simplicity, we will consider only one delay function r : R-½0; h; although the analysis can be extended to a more general setting in a straightforward way), and with the distributed delay term described by bACðR Â ½Àh; 0 Â Rn ; Rn Þ (which implies that f ACðR Â C; Rn Þ as can be proved by using the Lebesgue Theorem), and such that the solutions to (5) satisfy (4) According to Remark 2, if in addition f is such that uniqueness of solutions holds, then the standard single-valued process can be defined as follows: Rd Â C{ðt; t0 ; cÞ/Uðt; t0 ; cÞ ¼ xt AC; ð7Þ where xðÁÞ is the unique solution of (5)–(6) However, when f is such that the uniqueness of the problem does not hold or cannot be guaranteed, the process will not necessarily be single-valued but multi-valued in general ARTICLE IN PRESS 14 T Caraballo et al / J Differential Equations 208 (2005) 9–41 In this respect, the definition given by (7) becomes [ fxt : xðÁÞ is a solution to ð5Þ2ð6Þ defined globallyg: Uðt; t0 ; cÞ ¼ But, owing to some realistic reasons related to the models under study (e.g., biological, physical, etc.), we may be interested just in solutions which remain in a closed subset X CC; what motivates the construction of a multi-valued semiflow in X instead of in the whole space C: To this end, we assume that for any cAX there exists at least one solution to (5)–(6) defined globally in time and that remains in X for all tXt0 ; and denote by Dðt0 ; cÞ the set of all solutions of (5)–(6) defined for all tXt0 which remain in X for all tXt0 : Then, we can define the multi-valued process generated by (5)–(6) as [ Uðt; t0 ; cÞ ¼ fxt g: ð8Þ xðÁÞADðt0 ;cÞ Let us recall this concept and some of its properties more precisely (cf [6]) Consider a complete metric space X which in our situation will be a closed subset of C: Definition The map U : Rd Â X -PðX Þ is said to be a multi-valued dynamical process (MDP) on X if (1) Uðt; t; ÁÞ ¼ Id (identity map); (2) Uðt; s; xÞCUðt; t; Uðt; s; xÞÞ; for all xAX ; sptpt; S where Uðt; t; Uðt; s; xÞÞ ¼ yAUðt;s;xÞ Uðt; t; yÞ: The MDP U is said to be strict if Uðt; s; xÞ ¼ Uðt; t; Uðt; s; xÞÞ; for all xAX ; sptpt: Lemma Under the previous assumptions the multi-valued mapping U defined by (8) is a strict MDP Proof It is easy to check that U is well defined and satisfies (1) in Definition Let us now prove that (2) also holds Indeed, consider fAUðt; s; cÞ: Then from the definition of U; there exists a solution xðÁÞ to (5) with initial datum xs ¼ c and xt ¼ f: If tXs; then xt AUðt; s; cÞ; and as Uðt; t; xt Þ ¼ fzt : zðÁÞ is solution to ð5Þ with zt ¼ xt g; obviously xt ¼ fAUðt; t; xt ÞCUðt; t; Uðt; s; cÞÞ: To prove that the MDP is strict, let us consider fAUðt; t; Uðt; s; cÞÞ: Then there exists a solution xðÁÞ to (5) such that xt ¼ yt ; where yðÁÞ is another solution to (5) with initial value ys ¼ c: We now define & yðrÞ if s À hprpt; zðrÞ ¼ xðrÞ if tprpt: ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 15 It is clear that zðÁÞ is solution to Eq (5), and it also holds that zs ¼ ys ¼ c; and zt ¼ xt ¼ f; which means that fAUðt; s; cÞ: & Lemma The map Uðt; s; ÁÞ is bounded for all spt if and only if 8ðt; t0 ÞARd ; 8B0 ABðX Þ; (Bðt; t0 ; B0 ÞABðX Þ such that 8xðÁÞADðt0 ; B0 Þ it follows that xt0 ABðt; t0 ; B0 Þ 8t0 A½t0 ; t; S where Dðt0 ; B0 Þ ¼ cAB0 Dðt0 ; cÞ: ð9Þ Proof It is clear that (9) implies that Uðt; s; ÁÞ is bounded The converse is a consequence of the fact that the sets Uðt0 þ kh; t0 ; B0 Þ are bounded in Cð½Àh; 0; Rn Þ for any kAN: Indeed, for any xðÁÞADðt0 ; B0 Þ and any sA½t0 ; t we have jxðsÞjpjjxt0 þks h jjpC0 ; where ks is the minimum integer such that spt0 þ ks h and sup jjyjj: & C0 ¼ max 0pkpkt yAUðt0 þkh;t0 ;B0 Þ For the multi-valued map F : X -2X we shall denote DðF Þ ¼ fxAX j F ðxÞAPðX Þg: The multi-valued map F is said to be upper semicontinuous if for any xADðF Þ and any neighbourhood O of F ðxÞ there exists d40 such that F ðyÞCO; provided that rðx; yÞod: Once again the Ascoli–Arzela` Theorem allows us to prove the following useful result: Proposition 10 Let f ACðR Â C; Rn Þ be a bounded map, and assume that the solutions to (5) satisfy condition (9) Consider the process U : Rd Â X -PðX Þ generated by (5), which is therefore bounded Then the next properties hold: (i) If tXs þ h; the process Uðt; s; ÁÞ : X -PðX Þ is compact, that is, for all DABðX Þ; one has that Uðt; s; DÞAKðX Þ: (ii) Given a time sAR; if xn -xAX ; and xn : ½s À h; NÞ-Rn is a sequence of solutions to (5) with xns ¼ xn ; then there exists a subsequence fxm gm such that xmt -xt in X ; 8tXs; where x : ½s; NÞ-Rn satisfies xs ¼ x and Eq (5) ARTICLE IN PRESS 16 T Caraballo et al / J Differential Equations 208 (2005) 9–41 (iii) For any spt the map Uðt; s; ÁÞ is upper semicontinuous and has compact values and closed graph Proof Let DABðX Þ; and consider any sequence of points jm AUðt; s; DÞ: Thus, there exists a sequence of solutions of (5), xm : ½s À h; t-Rn ; with m xm t ¼j : As t X s þ h; the solutions are differentiable and their derivatives are bounded by C¼ sup j f ðs; ZÞj; ½tÀh;tÂD% where D˜ ¼ [ Uðt þ y; s; DÞABðX Þ: yA½Àh;0 The uniform bound and the equicontinuity allow us to apply the Ascoli–Arzela` Theorem and conclude the compactness of Uðt; s; ÁÞ for all tXs þ h: To prove (ii) we proceed analogously Observe that the condition tXs þ h is not necessary now (in ½s À h; s the convergence is assumed) We have initial data xm at time s (converging to x in X ) and solutions from there xm : ½s À h; NÞ-Rn : The Ascoli–Arzela` Theorem applied in successive steps of length h (and a diagonal Cantor argument) implies the existence of a converging subsequence xm to a function x : ½s; NÞ-Rn (the convergence is uniform on compact intervals of time) Using m m x ðtÞ ¼ x ð0Þ þ Z s t f ðr; xmr Þ dr; by the Lebesgue Theorem, we pass to the limit and obtain that & xðrÞ ˜ ¼ xðrÞ; xðrÞ; rA½s À h; s; rXs; solves (5) with initial data x at time s: Point (iii) is a consequence of (ii) Indeed, if the map Uðt; s; ÁÞ is not upper semicontinuous at some xAX ; then there exist a neighbourhood O of Uðt; s; xÞ and sequences xn -x; yn AUðt; s; xn Þ such that yn eO; for all n: But (ii) implies that for some subsequence ynk -yAUðt; s; xÞ; which is a contradiction The compactness of the values and the graph of Uðt; s; ÁÞ is proved in a similar way & Remark 11 The above result also shows that the MDP U is, in the autonomous multi-valued case, a generalized semigroup in the sense of Ball [4] ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 17 In the autonomous framework, all the previous results hold true but now for the multi-valued semiflow (MSF) G : Rþ Â X -PðX Þ generated by the autonomous functional differential equation x0 ðtÞ ¼ F ðxðtÞ; xðt À hÞÞ þ Z bðs; xðt þ sÞÞ ds ¼ f ðxt Þ; tX0; ð10Þ Àh x0 ¼ cAX ; ð11Þ which is defined, roughly, as Gðt; cÞ ¼ fxt : xðÁÞ is solution of ð10Þ2ð11Þ defined globally in timeg: In general, we have the following definition of MSF Definition 12 The multi-valued map G : Rþ Â X -PðX Þ is called a multi-valued semiflow (MSF or m-semiflow) if the next conditions are satisfied: (1) Gð0; xÞ ¼ fxg; for all xAX ; (2) Gðt1 þ t2 ; xÞCGðt1 ; Gðt2 ; xÞÞ; for all t1 ; t2 ARþ ; xAX ; where Gðt; BÞ ¼ S xAB Gðt; xÞ; BCX : This definition generalizes the concept of semigroup to the case where an equation can admit more than one solution for a fixed initial value This approach has already been used for some differential equations and inclusions (cf [2,13,14,27,28,36,42]) Another definition of generalized semigroup (using trajectories instead of multivalued maps) is given in [4,21], with applications to three-dimensional Navier–Stokes and parabolic degenerate equations We note that this semigroup satisfies in fact the conditions of Definition 12, so that it is a particular case (see [10] for a comparison of both theories) A different method for treating the problem of non-uniqueness is used in [17,40] For our equation, the map G is defined in the following way which is analogous to the non-autonomous case Let X be a closed subset of C such that for any cAX there exists at least one solution xðÁÞ of (10)–(11) such that xðtÞAX ; for all tX0: We denote by DðcÞ the set of all solutions of (10)–(11) defined for all tX0 which remain in X for all tX0: Then Gðt; cÞ ¼ [ xt ðÁÞ: xðÁÞADðcÞ In any case, we can always define the multi-valued process U but for this autonomous situation However, it is easy to check that Uðt; s; cÞ ¼ Gðt À s; cÞ: ARTICLE IN PRESS 18 T Caraballo et al / J Differential Equations 208 (2005) 9–41 Taking into account this fact, one can obtain autonomous versions of the results in this section in a straightforward manner Then we have: Lemma 13 The map G is an m-semiflow, and, moreover, Gðt þ s; xÞ Gðt; Gðs; xÞÞ; for all xAX ; tX0: Proof It is a consequence of Lemma and the previous comment & Autonomous and non-autonomous attractors for MSF and MDP In this section we shall collect the main definitions and results involving multivalued semiflows and processes and their attractors Let us consider a complete metric space X with metric r: 4.1 Autonomous attractor for a MSF 4.1.1 Abstract theory of attractors for multi-valued semiflows Let also denote by distðA; BÞ the Hausdorff semi-metric, i.e., for given subsets A and B we have distðA; BÞ ¼ sup inf rðx; yÞ: xAA yAB Definition 14 It is said that the set RCX is a global attractor of the m-semiflow G if: (1) It is attracting, i.e., distðGðt; BÞ; RÞ-0 as t- þ N; for all BABðX Þ; (2) R is negatively semi-invariant, i.e., RCGðt; RÞ; for all tX0; (3) It is minimal, that is, for any closed attracting set Y ; we have RCY : In applications it is desirable for the global attractor to be compact and invariant (i.e R ¼ Gðt; RÞ; for all tX0) S Let us denote gþ t ðBÞ ¼ tXt Gðt; BÞ: The MSF G is called asymptotically upper semi-compact if for all BABðX Þ such that for some TðBÞARþ ; gþ TðBÞ ðBÞABðX Þ; any sequence xn AGðtn ; BÞ; tn - þ N; is precompact in X : The m-semiflow G is called pointwise dissipative if there exists B0 ABðX Þ such that distðGðt; xÞ; B0 Þ-0; as t- þ N; for all xAX : The following two results can be found in [36] (see also [13,14]) Proposition 15 Let X be a Banach space and let Gðt; ÁÞ ¼ Sðt; ÁÞ þ Kðt; ÁÞ be an msemiflow, where Kðt0 ; ÁÞ : X -PðX Þ is a compact map for some t0 40 and ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 27 Example Consider a model of human respiratory duðtÞ ¼ ÀauðtÞ þ avðtÞ þ c; dt dvðtÞ ¼ buðtÞ À bvðtÞ À pðvðtÞ À gÞjðvðt À hÞÞ; dt where a; b; c; g; p are positive constants, u; v are the CO2 partial pressures in tissues and lungs, respectively, xðtÞ ¼ ðuðtÞ; vðtÞÞ; yðtÞ ¼ ðuðt À hÞ; vðt À hÞÞ; and j is a continuous function such that jðy2 Þ ¼ 0; for y2 px0 (for some x0 X0), and strictly increasing for y2 4x0 : For the physical meaning of the constants see [11] We set L ¼ fxAR2 : uXg; vXgg: Lemma 33 All the solutions starting in X ¼ Cð½Àh; 0; LÞ remain in this space Consequently, condition (H1) holds Proof Let xðtÞ ¼ ðuðtÞ; vðtÞÞ be an arbitrary solution with xt0 ¼ cð0ÞA@L: The case u ¼ v ¼ g is trivial, so we suppose one of the components is not null We shall suppose that xðtÞeL in some interval ðt0 ; t1 Þ: If uðt0 Þ ¼ g and less than g in ðt0 ; t1 Þ; then there exists t0 ot2 pt1 such that ÀauðtÞ þ avðtÞ þ c40; for all tAðt0 ; t2 Þ: Hence, uðtÞXg; for all tAðt0 ; t2 Þ: Then we have to assume that vðtÞog; for all tAðt0 ; t2 Þ: But in such a case buðtÞ À bvðtÞ À pðvðtÞ À gÞjðvðt À hÞÞ40; for all tAðt0 ; t2 Þ: Hence, vðtÞXg; for all tAðt0 ; t2 Þ; and we obtain a contradiction The solution xðtÞ cannot leave the set L: On the other hand, condition (H3) follows directly from /F ðx; yÞ; xSp À au2 þ ða þ bÞuv À bv2 þ cupCð1 þ jxj2 Þ: Thus, by the proof of Proposition 29, we have that each solution is defined globally in time This implies that (H1) holds & In order to check (H2) we need additional assumptions on the constants of the model Lemma 34 Let g4x0 and let gp4ða À bÞ2 =4a; where g ¼ jðgÞ: Then (H2) holds Proof First let aXb: We note that since jðy2 ÞXg; for all y2 Xg; and Àau2 þ ða þ bÞuv À bv2 ¼ Àbðu À vÞðab u À vÞ we have a u À v þ cu À pvðv À gÞjðy2 Þ b a p À bðu À vÞ u À v þ cu À gpvðv À gÞ: b /F ðx; yÞ; xS ¼ À bðu À vÞ ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 28 We shall consider the following three regions: n a o R1 ¼ xAL : vX u ; b & ' b R ¼ xAL : vpupv ; a R3 ¼ fxAL : vpug: À Á Note that the function cðu; vÞ ¼ Àbðu À vÞ ab u À v takes positive values only in R2 : Let xAR1 : Since vXu; jxjXK1 implies 2v2 Xu2 þ v2 XK12 ; using that cðu; vÞp0; we obtain /F ðx; yÞ; xSp cv À gpvðv À gÞ p C1 À gpv2 =2 p À 1; if jxjXK1 ¼ ð4ðC1 þ 1Þ=ðgpÞÞ1=2 : Let now xAR2 : Denote x ¼ gp À ða À bÞ2 =4a: Using that v2 ða À bÞ2 =4a ¼ max b uA½a v;v cðu; vÞ and upv; we have /F ðx; yÞ; xSp v2 ða À bÞ2 =4a þ cv À gpvðv À gÞ p C2 À xv2 =2 p À 1; if jxjXK2 ¼ ð4ðC2 þ 1Þ=xÞ1=2 : Finally, let xAR3 : In this case, we have cðu; vÞp À bðu À vÞ2 : Hence, /F ðx; yÞ; xSp À bðu À vÞ2 þ cu À gpvðv À gÞ p C3 À bðu À vÞ2 =2 À gpv2 =2: Denote Z ¼ ð2ðC3 þ 1Þ=bÞ1=2 : If u À vXZ; then /F ðx; yÞ; xSp À 1: Otherwise, jxjXK3 implies 3v2 þ 2Z2 Xu2 þ v2 XK32 : It follows that /F ðx; yÞ; xSp À 1; if jxjXK3 ¼ ð2Z2 þ 6ðC3 þ 1Þ=ðgpÞÞ1=2 : ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 29 Now let aob and define the regions: R4 ¼ fxAL : vXug; & ' b R5 ¼ xAL : vpup v ; a n a o R ¼ xAL : vp u : b Region R4 is treated in the same way as R1 : For R5 ; since vXab u; jxjXK5 implies ð1 þ b2 =a2 Þv2 Xu2 þ v2 XK52 ; using v2 ða À bÞ2 =4a ¼ max b uA½v;a v cðu; vÞ; we have /F ðx; yÞ; xSp v2 ða À bÞ2 =4a þ cbv=a À gpvðv À gÞ p C5 À xv2 =2 p À 1; if jxjXK5 ¼ ð2ð1 þ ðb=aÞ2 ÞðC5 þ 1Þ=xÞ1=2 : Finally, consider xAR6 : In this case, we have cðu; vÞp À bðau=b À vÞ2 : Hence, /F ðx; yÞ; xSp À bðau=b À vÞ2 þ cu À gpvðv À gÞ p C6 À bðau=b À vÞ2 =2 À gpv2 =2: Denote Z ¼ ð2ðC6 þ 1Þ=bÞ1=2 : If au=b À vXZ; then /F ðx; yÞ; xSp À 1: Otherwise, jxjXK6 implies ð1 þ 2b2 =a2 Þv2 þ 2b2 Z2 =a2 Xu2 þ v2 XK62 : It follows that /F ðx; yÞ; xSp À if jxjXK6 ¼ ð2b2 Z2 =a2 þ 2ð1 þ 2b2 =a2 ÞðC6 þ 1Þ=ðgpÞÞ1=2 : Taking K ¼ maxfK1 ; K2 ; K3 g (or maxfK4 ; K5 ; K6 gÞ condition (H2) holds for dðeÞ ¼ 1: & One of the typical functions used in such syn2 =ðyn þ yn2 Þ; with s; n; y40: In this case it that the condition gp4ða À bÞ2 =4a implies pðvðtÞ À gÞjðvðt À hÞÞ; which controls the air models is the Hill controller jðy2 Þ ¼ is clear that x0 ¼ 0og: We also note a strong enough effect of the term flow in the lungs 5.2 Non-autonomous case 5.2.1 Dissipative and sub-linear terms We are now interested in considering a situation which takes into account the possible appearance of variable and distributed delays together ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 30 Consider the equation x0 ðtÞ ¼ F0 ðt; xðtÞÞ þ F1 ðt; xðt À rðtÞÞÞ þ Z bðt; s; xðt þ sÞÞ ds ¼ f ðt; xt Þ ð19Þ Àh with F0 ; F1 ACðR Â Rn ; Rn Þ; rAC ðR; ½0; hÞ Assume the following conditions: bACðR Â ½Àh; 0 Â Rn ; Rn Þ: and (1) There exist positive scalar functions m0 ; m1 AL1 ð½Àh; 0Þ such that jbðt; s; xÞjpm0 ðsÞ þ m1 ðsÞjxj; 8tAR: ð20Þ (2) There exist positive constants k1 ; k2 ; a and a positive function bðÁÞ such that /x; F0 ðt; xÞSp À ajxj2 þ bðtÞ; jF1 ðt; xÞj2 pk12 þ k22 jxj2 ; jr0 ðtÞjprÃ o1; 8tAR; xARn ; ð21Þ 8tAR; xARn ; ð22Þ 8tAR; ð23Þ where b is such that Z t bðrÞedr droN; 8tAR; 8d40: ð24Þ ÀN Also we denote mi ¼ Z mi ðsÞ ds; i ¼ 0; 1: Àh We take as the phase space X ¼ Cð½Àh; 0; Rn Þ: Then, we have the following result: Theorem 35 Let conditions (20)–(24) hold Also, assume that 2m1 eho1; ð25Þ k22 oeÀ1 ð1 À rÃ Þaða À lÃ Þ; ð26Þ and where lÃ Aðl0 ; l1 Þ; being l0 ol1 the solutions of the equation leÀlh ¼ 2m1 ; and let lÃ oa: ð27Þ Then, Eq (19) generates a MDP which has the global compact non-autonomous attractor fAðtÞgtAR : ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 31 Remark 36 (i) We note that the equation leÀlh ¼ 2m1 has two solutions l0 ðhÞol1 ðhÞ if condition (25) holds, and l0 ðhÞ-0 as h-0 (or m1 -0) Hence (25) holds for h (or m1 ) small, whereas (27) holds if h (or m1 ) is small enough, or if a is large On the other hand, (26) is satisfied for k2 small or a large These conditions can be read as: a combination of strong dissipativity and small effects of the delay (in terms of h; m1 or k2 small) ensure the existence of the attractor (ii) Conditions (26) and (27) are stronger than what is really needed in the proof below, that is, there exist positive values l and e such that l À 2a þ e þ elh k22 eð1ÀrÃ Þo0 and lAðl0 ; l1 Þ: Proof Let xðtÞ be an arbitrary solution with jjxt0 jjpd: First, we fixed two positive parameters l and e to be chosen later on Then, we have d lt ðe jxðtÞj2 Þ dt ¼ lelt jxðtÞj2 þ 2elt /xðtÞ; F0 ðt; xðtÞÞ þ F1 ðt; xðt À rðtÞÞÞS ( Z þ 2elt xðtÞ; ) bðt; r; xðt þ rÞÞ dr Àh plelt jxðtÞj2 þ 2elt ðÀajxðtÞj2 þ bðtÞÞ þ eelt jxðtÞj2 þ ( Z þ 2elt xðtÞ; ) elt jF1 ðxðt À rðtÞÞÞj2 e bðt; r; xðt þ rÞÞ dr Àh elt ðk þ k22 jxðt À rðtÞÞj2 Þ e ( ) Z lt bðt; r; xðt þ rÞÞ dr : þ 2e xðtÞ; pðl À 2a þ eÞelt jxðtÞj2 þ 2elt bðtÞ þ Àh Integrating between t0 and t; we obtain 2 e jxðtÞj p e jxðt0 Þj þ ðl À 2a þ eÞ lt lt0 þ e þ2 Z Z t ls e jxðsÞj þ Z t0 t t0 Z t t0 t els bðsÞ ds t0 els ðk12 þ k22 jxðs À rðsÞÞj2 Þ ds ( Z els xðsÞ; ) bðs; r; xðs þ rÞÞ dr ds: Àh Now we estimate the integral containing the variable delay by using the change of variables s À rðsÞ ¼ u; taking into account that r takes values in ½0; h; ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 32 and 1 1Àr0 ðsÞp1ÀrÃ : Z t els jxðs À rðsÞÞj2 ds t0 p Z t elu t0 Àh lh elh jxðuÞj2 du À rÃ ! Z t0 Z t elu jxðuÞj2 du þ elu jxðuÞj2 du e À rÃ t0 Àh t0 lh Z t0 lh Z t e d e p elu du þ elu jxðuÞj2 du; À rÃ t0 Àh À rÃ t0 ¼ where we have used that jjxt0 jjpd for some d40: Thus, we obtain Z t elh k22 els jxðsÞj2 ds elt jxðtÞj2 p elt0 jxðt0 Þj2 þ l À 2a þ e þ eð1 À rÃ Þ t0 Z t k2 k2 elh d ðelt0 À elðt0 ÀhÞ Þ þ2 els bðsÞ ds þ ðelt À elt0 Þ þ el leð1 À rÃ Þ t0 ) Z Z t ( els xðsÞ; bðs; r; xðs þ rÞÞ dr ds; þ2 Àh t0 and it follows for the last integral in (28) that ) Z t ( Z els xðsÞ; bðs; r; xðs þ rÞÞ dr ds t0 p2m0 pe% Z Z Àh t jxðsÞjels ds þ 2m1 t0 t t0 m2 els jxðsÞj2 ds þ e% Z Z t jxðsÞjjjxs jjels ds t0 t ls e ds þ 2m1 Z t0 t els jjxs jj2 ds; t0 where e% is another positive constant to be determined later on Therefore, Z t elh k22 2 lt lt0 þ e% e jxðtÞj p e jxðt0 Þj þ l À 2a þ e þ els jxðsÞj2 ds eð1 À rÃ Þ t0 Z t k m þ2 els bðsÞ ds þ þ ðelt À elt0 Þ el le% t0 Z t lh k e d ðelt0 À elðt0 ÀhÞ Þ þ 2m1 þ els jjxs jj2 ds: leð1 À rÃ Þ t0 Choosing e ¼ a; l ¼ lÃ and using (26) we obtain l À 2a þ e þ e% þ elh k22 o0; eð1 À rÃ Þ ð28Þ ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 33 for e% small enough Then, it holds Ã Ã k12 m20 e bðsÞ ds þ e jxðtÞj p e jxðt0 Þj þ ðel t À el t0 Þ Ãþ Ã al l e% t0 Z t lÃ h Ã Ã Ã k e d ðel t0 À el ðt0 ÀhÞ Þ þ 2m1 el s jjxs jj2 ds: þ Ã2 l að1 À rÃ Þ t0 lÃ t lÃ t0 Z t lÃ s Ã Setting now t þ y instead of t (where yA½Àh; 0), multiplying by eÀl ðtþyÞ and using standard estimates, it follows jxðt þ yÞj2 peÀl Ã ðtþyÞ lÃ t0 þ e k12 Ã la þ Ã jxðt0 Þj2 þ 2eÀl ðtþyÞ m20 Ã l e% Ã þ 2m1 eÀl ðtþyÞ Z Ã Ã Ã ð1 À el ðt0 ÀtÀyÞ Þ þ eÀl ðtþyÞ Z tþy Ã Ã þ k12 Ã Ã la þ Ã m20 Ã Ã þ 2m1 eÀl t el h Ã h Z t Ã el s bðsÞ ds t0 l e% Ã Ã k22 el h d ðel t0 À el ðt0 ÀhÞ Þ Ã l að1 À rÃ Þ el s jjxs jj2 ds peÀl t el ðt0 þhÞ jxðt0 Þj2 þ 2eÀl t el Ã el s bðsÞ ds t0 t0 Ã tþy Ã ð1 À e Z t lÃ ðt0 ÀtþhÞ Þþe Ã Ã k22 e2l h d ðel t0 À el ðt0 ÀhÞ Þ Ã l að1 À rÃ Þ ÀlÃ t Ã el s jjxs jj2 ds: t0 Neglecting the negative terms we deduce Ã Ã el t jjxt jj2 p el ðt0 þhÞ d þ 2el Ã h Z t Ã el s bðsÞ ds t0 Ã Ã þ C e l t þ C d e l t0 þ L Z t t0 where we have denoted C1 ¼ k12 m20 Ã þ Ã ; l a l e% C2 ¼ k22 e2l h ; l að1 À rÃ Þ Ã Ã Ã L ¼ 2m1 el h : Ã el s jjxs jj2 ds ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 34 Gronwall’s Lemma and the Fubini Theorem yield Ã el t jjxt jj2 Ã Ã pel ðt0 þhÞ d þ 2el h þ LeLt Z Z t Ã Ã t0 t eÀLs el t0 Ã Ã pel ðt0 þhÞ d þ 2el h Ã Z Ã el s bðsÞ ds þ C1 el t þ C2 d el t0 Ã ðt0 þhÞ d þ 2el h Z s ! Ã Ã Ã el r bðrÞ dr þ C1 el s þ C2 d el t0 ds t0 t Ã Ã Ã el s bðsÞ ds þ C1 el t þ C2 d el t0 þ LeLt t0 Ã el ðt0 þhÞ d þ C2 d el Â L ! Ã C1 þ Ã eðl ÀLÞt : l ÀL Ã t0 Ã ðeÀLt0 À eÀLt Þ þ 2el h Z t Ã el r bðrÞ t0 eÀLr À eÀLt dr L Therefore, jjxt jj2 p el Ã ðt0 þhÞ ÀlÃ t d e Ã þ 2el h eÀl Ã t Z t Ã Ã el s bðsÞ ds þ C1 þ C2 d el t0 eÀl Ã t t0 Ã tþLðtÀt0 Þ Ã ðhÀtÞþLt þ eÀl þ 2el Ã Ã ðel ðt0 þhÞ d þ C2 d el t0 Þ Z t t0 Ã el r bðrÞðeÀLr À eÀLt Þ dr þ LC1 : lÃ À L ð29Þ We can see that (29) and Corollary imply that all solutions exist globally in time (so, U is well defined), and also that the maps Uðt; t0 ; ÁÞ are bounded On the other hand, condition (25) implies that lÃ À L40: Then it follows from the previous inequality that < LC1 Ã Ã Rt = lÃ s jjyjj2 p2el h eÀl t ÀN þ Z e bðsÞ ds þ lÃ À L ; BðtÞ ¼ yAC : þ C1 þ 2elÃ ðhÀtÞþLt R t elÃ r bðrÞðeÀLr À eÀLt Þ dr ; ÀN where Z40; is a family of bounded absorbing sets We conclude the proof by applying Theorem 26 & We can consider also the case where F0 ; F1 and b not depend on the time variable, that is, we have an autonomous equation We note that in such ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 35 a situation bðtÞ b; and then neglecting negative terms (29) becomes Ã Ã Ã jjxt jj2 p el ðt0 þhÞ d eÀl t þ þ eÀl Ã tþLðtÀt0 Þ Ã Ã 2bel h þ C1 þ C2 d el t0 eÀl t lÃ Ã Ã ðel ðt0 þhÞ d þ C2 d el t0 Þ Ã 2bel h LC1 þ : þ Ã l À L lÃ À L Now t0 ¼ is fixed and t- þ N: It follows that the set & ' Ã 1 LC1 B0 ¼ yAC : jjyjj2 pC1 þ 2bel h Ã þ Ã þ Ã l l ÀL l ÀL is attracting for the m-semiflow G; which is also bounded We obtain the same result as in Theorem 32 Remark 37 The statement of Theorem 35 remains valid if X ¼ Cð½Àh; 0; LÞ; being L a closed subset of Rn ; supposing that for each cAX and t0 AR there exists at least one solution such that xðtÞAX ; for all tXt0 : Corollary 38 Assume that L ¼ Rnþ and that for each i ¼ 1; y; n one of the following conditions holds: R (1) F i ðt; cð0ÞÞ þ F i ðt; cðÀrðtÞÞÞ þ bi ðt; s; cðsÞÞ ds ¼ 0; for all tAR; and Àh cACð½Àh; 0; LÞ; with ci ð0Þ ¼ 0; R (2) F i ðt; cð0ÞÞ þ F i ðt; cðÀrðtÞÞÞ þ bi ðt; s; cðsÞÞ ds40; for all tAR; and Àh cACð½Àh; 0; Rn Þ; with ci ð0Þo0: Then for each cAX ; t0 AR there exists at least one solution xðtÞ of (19) such that xðtÞAX for all tXt0 : Proof It is similar to the proof of Corollary 30 & Remark 39 We can consider two simpler situations: Assume that b satisfies jbðt; s; xÞjpmðsÞjxj; ð30Þ and that F0 ðt; xÞ ¼ Àax; F1 ðt; xÞ ¼ 0: Then, we can deduce (obtaining an estimate rather similar to (29)) that the pullback attractor exists and is just one point (the null solution in C) Also, this attractor is a global attractor in the usual forward sense (as t- þ N), and implies that the null solution is asymptotically stable (what means extinction in a biological model) On the other hand, if we delete the bound of b on x; we can reproduce the same calculus but extending the dependence of the bound on t: More exactly, given bACðR Â ½Àh; 0 Â Rn ; Rn Þ ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 36 verifying jbðt; s; xÞjpKðt; sÞ 8xARn ; ð31Þ an analogous result is obtained if Z RðtÞ :¼ Kðt; sÞ ds Àh satisfies Z r R2 ðtÞe2*at dtoN 8rAR; for some a* Að0; aÞ: ð32Þ ÀN 5.2.2 Weaker assumptions on the dissipativity In a similar way, we can also weaken the dissipativity assumption on the function F0 in the sense that more general non-autonomous situations can be covered by our results To this end, we start again from Eq (19), but with F ðt; xt Þ ¼ F0 ðt; xðtÞÞ þ F1 ðt; xðt À rðtÞÞÞ where rACðR; ½Àh; 0Þ and /F0 ðt; xÞ; xSpðÀa þ g1 ðtÞÞjxj2 þ g2 ðtÞ; jF1 ðt; xÞjpg3 ðtÞ; ð33Þ 8tAR; 8d40; ð34Þ being gi positive functions satisfying Z t ÀN Â Ã g1 ðsÞ þ eds ðg2 ðsÞ þ g23 ðsÞÞ dsoN; and b verifying conditions (31)–(32) Then ( ) Z d jxðtÞj2 ¼ 2/xðtÞ; x0 ðtÞS ¼ 2/xðtÞ; F ðt; xt ÞS þ xðtÞ; bðt; s; xðt þ sÞÞ ds dt Àh Z p 2ðg1 ðtÞ À aÞjxðtÞj þ 2g2 ðtÞ þ 2jxðtÞj g3 ðtÞ þ Kðt; sÞ ds Àh g2 ðtÞ p ½2ðg1 ðtÞ À aÞ þ ejxðtÞj2 þ þ 2g2 ðtÞ þ R2 ðtÞ: e e By Gronwall Lemma, it holds jxðtÞj2 p e Rt t0 ½2ðg1 ðsÞÀaÞþe ds jxðt0 Þj2 Rt Z t g2 ðsÞ þ R2 ðsÞ ½2ðg ðrÞÀaÞþe dr 2g2 ðsÞ þ ds e s e t0 Rt ð2aÀeÞðt0 ÀtÞ t0 g1 ðsÞ ds e jxðt0 Þj2 ¼e þ ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 þe ðeÀ2aÞt Z t e ð2aÀeÞs e Rt s g1 ðrÞ dr t0 37 g23 ðsÞ þ R2 ðsÞ 2g2 ðsÞ þ ds e p eð2aÀeÞðt0 ÀtÞ eMt jxðt0 Þj2 þ eðeÀ2aÞt eMt Z t t0 g2 ðsÞ þ R2 ðsÞ eð2aÀeÞs 2g2 ðsÞ þ ds; e where we have denoted Mt ¼ Z t ÀN g1 ðrÞ dr: The existence of a family of bounded absorbing sets in C is already standard (choosing some eo2a) Hence, we obtain the same result as in Theorem 35 5.2.3 Examples Let us consider now some examples from real applications In all of these examples we shall consider that L ¼ Rþ ; hence X ¼ Cð½Àh; 0; LÞ: Also, in all the examples Corollary 38 implies that the semiprocess U is well defined Example Mackey–Glass model of production of blood cells [35]: dxðtÞ bðtÞ ¼ À dxðtÞ; dt þ jxðt À hÞjn Rt where n; d40; bðtÞ40 is continuous and ÀN ees b2 ðsÞ dso þ N; for any tAR and e40: Conditions (33)–(34) are fulfilled A generalization of this model is the following [37,38, p 20]: dxðtÞ bðtÞjxðt À hÞjm ¼ À dxðtÞ; dt þ jxðt À hÞjn where we suppose that 0pmpn þ and jbðtÞjpk; for all tAR: Let also d2 k2 o ; e if m ¼ n þ 1: For conditions (25)–(27) note that lÃ ¼ rÃ ¼ m1 ¼ 0; a ¼ d: So, (26) follows from the condition on k if m ¼ n þ and by Young inequality if mon þ 1: Another generalization of this model appears if we consider an integral term: dxðtÞ ¼ dt Z Àh bðt; sÞ ds À dxðtÞ; þ jxðt À hÞjn ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 38 where bðt; sÞ40 satisfies jbðt; sÞjpm0 ðsÞ; m0 AL1 ð½Àh; 0Þ (or jbðt; sÞjpKðt; sÞ with K satisfying (32)) All the conditions of Theorem 35 are satisfied (note that m1 ¼ k2 ¼ lÃ ¼ 0) Example Lasota and Wazewska model of production of blood cells [33]: dxðtÞ ¼ bðtÞ expðÀxðtÞxðt À hÞÞ À dxðtÞ; dt Rt where d40; bðtÞ40; xðtÞX0 are continuous and ÀN ees b2 ðsÞ dso þ N; for any tAR and e40: It is clear again that (33)–(34) are satisfied Consider also the following generalization of the model: dxðtÞ ¼ dt Z bðt; sÞ expðÀxðt þ sÞxðt þ sÞÞ ds À dxðtÞ; Àh where b satisfies the same conditions of the previous example Example Consider the model: dxðtÞ ¼ bðtÞjxðt À hÞjn expðÀxðtÞxðt À hÞÞ À dxðtÞ; dt where d40; 0onp1; bðtÞ40; xðtÞX0 are continuous and jbðtÞj2 pk2 ; for any tAR: If n ¼ we have to assume also that k2 ode : For conditions (25)–(27) note that lÃ ¼ m1 ¼ 0; a ¼ d: So, (26) follows from the condition on k if n ¼ and by Young inequality if no1: In the particular case where n ¼ this is the Nicholson model of blowflies [22] We can consider also the following model: dxðtÞ ¼ dt Z bðtÞjxðt À hÞjn expðÀxðt þ sÞxðt þ sÞÞ ds À dxðtÞ; Àh where d40; 0onp1; bðtÞ40; xðtÞX0 are continuous and jbðtÞj2 pk2 ; for any tAR: If n ¼ we have to assume that m1 ¼ kh is small enough, so that (25) and (27) hold If no1 we can make m1 as small as we want using the Young inequality Hence, (25)–(27) are satisfied with a ¼ d; k2 ¼ rÃ ¼ 0: An analysis of persistence and extinction for these models is given in [34] Conclusions The existence of an attractor (autonomous or pullback) has been proved in several situations arising in real applications, when some hereditary features ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 39 appear in the models The use of multi-valued semiflows and processes allowed us to provide results covering also the cases in which non-uniqueness of solutions can take place However, our analysis has been done by considering only finite delays Therefore, it is an interesting task to study the framework with unbounded (infinite) delays, as well as the situations modelled by differential inclusions rather than differential equations (whose importance comes, for instance, from viability reasons in biological problems) We plan to investigate these points in subsequent papers Acknowledgments The two first authors would like to thank all the people in the Universidad Cardenal Herrera CEU (Elche, Spain) for the kind hospitality while they were visiting it to prepare this work Special thanks goes to Jose´ Valero and his wife P.M.-R also thanks Clotilde Martı´ nez for useful discussions about the subject This work has been partially supported by MCYT (Ministerio de Ciencia y Tecnologı´ a, Spain), grant HA2001-0075, by MCYT (Ministerio de Ciencia y Tecnologı´ a, Spain), and FEDER (Fondo Europeo de Desarrollo Regional), grants BFM2002-03512 and BFM2002-03068, and by Generalitat Valenciana (Spain), grant CTIDIB/2002/342 References [1] F.V Atkinson, J.R Haddock, On determining phase spaces for functional differential equations, Funkcial Ekvac 31 (1988) 331–347 [2] A.V Babin, Attractor of the generalized semigroup generated by an elliptic equation in a cylindrical domain, Russian Acad Sci Izv Math 44 (1995) 207–223 (Transl Izv Ross Akad Nauk Ser Mat 58 (1994)) [3] A.V Babin, M.I Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 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initial condition at least one solution is globally defined in X ; (ii) Uðt; s; ÁÞ : X -X is a bounded map for. .. Attractors for differential equations with variable delays, J Math Anal Appl 260 (2) (2001) 421–438 [8] T Caraballo, J.A Langa, J Valero, Global attractors for multi-valued random dynamical systems generated by random differential inclusions with multiplicative noise, J Math Anal Appl 260 (2001) 602–622 [9] T Caraballo, J.A Langa, J Valero, Global attractors for multi-valued random dynamical systems, Nonlinear... model with delay time, Appl Math Comput 136 (2003) 27–36 [39] B SchmalfuX, Backward cocycles and attractors of stochastic differential equations, in: V Reitmann, T Redrich, N.J Kosch (Eds.), International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behaviour, Technische Universita¨t, Dresden, 1992, pp 185–192 [40] G Sell, Non -autonomous differential equations and. .. Schmalfuss, Global attractors of nonautonomous disperse dynamical systems and differential inclusions, Bull Acad Sci Rep Moldova Mat 29 (1) (1999) 3–22 [16] V.V Chepyzhov, M.I Vishik, A Hausdorff dimension estimate for kernel sections of non -autonomous evolution equations, Indiana Univ Math J 42 (3) (1993) 1057–1076 [17] V.V Chepyzhov, M.I Vishik, Evolution equations and their trajectory attractors,... attractors, J Math Pures Appl 76 (1997) 913–964 [18] H Crauel, A Debussche, F Flandoli, Random attractors, J Dynamics Differential Equations 9 (2) (1997) 307–341 [19] H Crauel, F Flandoli, Attractors for random dynamical systems, Probab Theory Related Fields 100 (3) (1994) 365–393 [20] J.M Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, Vol... 1969–1983 [29] P.E Kloeden, B SchmalfuX, Non -autonomous systems, cocycle attractors and variable time-step discretization, Numer Algorithms 14 (1–3) (1997) 141–152 [30] P.E Kloeden, D.J Stonier, Cocycle attractors in non- autonomously perturbed differential equations, Dynamic Contin Discrete Impuls Systems 4 (2) (1998) 211–226 [31] Y Kuang, Delay Differential Equations with Applications in Population Dynamics,... Þ40; which is a contradiction If Fi ðx; yÞ ¼ 0; for all x; yAL such that xi ¼ 0; and xi ðt1 Þ ¼ 0; for some t1 X0; then we can put xi ðtÞ ¼ 0; for all tXt1 ; and continue solving the system of equations for the rest of the components xj : Hence, with this procedure we obtain the desired solution xðtÞ: & ARTICLE IN PRESS T Caraballo et al / J Differential Equations 208 (2005) 9–41 25 Theorem 31 Let conditions... Caraballo et al / J Differential Equations 208 (2005) 9–41 39 appear in the models The use of multi-valued semiflows and processes allowed us to provide results covering also the cases in which non- uniqueness of solutions can take place However, our analysis has been done by considering only finite delays Therefore, it is an interesting task to study the framework with unbounded (infinite) delays, as well... developed in [18,29,30] As it has already been mentioned, in the case of non -autonomous differential equations the initial time is as important as the final time, and the classical semigroup property of autonomous dynamical systems is no longer suitable Therefore, the notions of the classical theory need to be adapted to deal with MDP Definition 19 Let tAR: The set DðtÞ is said to attract (in the pullback... situations Lemma 17 Let b and F be continuous and let for each initial condition at least one solution to (10)–(11) be globally defined in X : We assume that Gðt; ÁÞ is a bounded map for any tX0: Then the map Gðt; ÁÞ has closed values and is upper semicontinuous Proof It is a consequence of Proposition 10 and Lemma 9 & Theorem 18 Let b and F be continuous and let for each initial condition at least one solution ... space and denote by PðEÞ; CðEÞ; BðEÞ and KðEÞ the sets of non-empty, non-empty and closed, non-empty and bounded, and non-empty and compact subsets of E: Definition Given two metric spaces X and. .. Equations, Studies in Mathematics and its Applications, Vol 25, North-Holland, Amsterdam, 1992 [4] J.M Ball, Continuity properties and global attractors of generalized semiflows and the Navier–Stokes equations,... Then the map Gðt; ÁÞ has closed values and is upper semicontinuous Proof It is a consequence of Proposition 10 and Lemma & Theorem 18 Let b and F be continuous and let for each initial condition