Reaction-Diffusion Equations with Non-Autonomous Force In H-1 and Delays Under Measurability Conditions on The Driving Delay Term

Reaction-diffusion equations with non-autonomous force in H−1and delays under measurability conditions on the driving delay term J. Garc´ıa-Luengo and P. Mar´ın-Rubio∗ Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla Apdo. de Correos 1160, 41080-Sevilla (Spain) e-mails: [email protected] [email protected] February 6, 2014 Abstract In this paper we analyze the existence of solutions for a reaction-diffusion problem with hereditary effects and a time-dependent force term with values in H−1.The main novelty is that the delay term may be driven by a function under very minimal assumptions, namely, just measurability. This is due to the fact that we only deal with a phase-space of functions continuous in time, allowing this general setting, which might be more useful when less regularity is known in the hereditary mechanism. After that, we obtain uniform estimates and asymptotic compactness properties (via an energy method) that allow us to ensure the existence of pullback attractors for the associated process to the problem. Actually, we obtain two different families of minimal pullback attractors, namely, those of fixed bounded sets but also for a class of time-dependent families (universe) given by a tempered condition. Finally, from comparison results, we establish relations among them, and under suitable additional assumptions we conclude that these families of attractors are in fact the same object. Keywords: reaction-diffusion equations; delay; pullback attractors. 1 Introduction and statement of the problem Reaction-diffusion equations have been intensively developed during the last decades because of their applications in Chemistry, Biology, etcetera. Even just in the last few years it is possible to find many related results and features of such models in the literature, as epidemic systems, cellular neural networks, or problems within random environments (e.g. cf. [5, 15, 22, 25] and the references therein among many others). Apart of the analysis in finite-time intervals, it has also been deeply studied the asymptotic behaviour of solutions for such kind of problems. Jointly with stability questions, the theory of attractors for the associated ∗Corresponding author. Phone number: (+34) 95 455 99 09. Fax number: (+34) 95 455 28 98. 1 dynamical systems has also experimented significant improvements. Namely, examples of such advances are the internal structure of the global attractor for Chaffee-Infante equations (for autonomous systems), finitedimensionality reduction, and trajectory and pullback attractors for non-autonomous dynamical systems (e.g. cf. some results in the monographs [21, 13, 6], the recent study [1], or the papers [12, 20, 26, 18, 23, 19, 24, 17, 2, 16, 3, 4, 14]). Our aim in this paper is to present an improvement on the conditions for dealing with time-dependent delayed reaction-diffusion problems. Namely, we may consider a reaction-diffusion problem only under measurability conditions on the driving delay terms appearing in the equation and we are able to establish existence results and to study the long-time behaviour of such solutions as well. Before continuing with the description of our results, let us state our problem properly. Let Ω ⊂RNbe a bounded domain. We consider the following non-autonomous reaction-diffusion problem with delay effects and homogeneous Dirichlet boundary condition          ∂u ∂t −∆u=f(u) + g(t, ut) + k(t) in Ω ×(τ, ∞), u= 0 on ∂Ω×(τ, ∞), u(x, τ +s) = φ(x, s)x∈Ω, s ∈[−h, 0], (1) where τ∈R, f ∈C(R), g is an operator acting on the solution containing some hereditary characteristics (assumptions on gare given below), the time-dependent force term is k∈L2 loc(R;H−1(Ω)),and φ∈ C([−h, 0]; L2(Ω)) is the initial datum, being h > 0 the length of the delay effect, and where for each t≥τ, we denote by utthe function defined in [−h, 0] by ut(s) = u(t+s), s ∈[−h, 0]. Before describing the assumptions on the function fand the operator g, we introduce some notation that will be used in the paper. We will denote by (·,·) and |·| the scalar product and norm in L2(Ω) respectively; the norm in Lp(Ω) will be written as k · kLp(Ω); in H1 0(Ω) we will use as (equivalent) scalar product ((·,·)) = (∇·,∇·),with corresponding norm k · k.We will denote by CL2the Banach space C([−h, 0]; L2(Ω)),equipped with the sup-norm. For an element u∈CL2,its norm will be written as |u|CL2= maxt∈[−h,0] |u(t)|.The duality between Lp(Ω) and its dual Lq(Ω),where q=p/(p−1) (when 1 <p<∞), will be also denoted by (·,·); and the duality between H1 0(Ω) and its dual H−1(Ω) will be written as h·,·i.Finally, we will use k·k∗for the norm in H−1(Ω). Concerning the function fand the operator g, we make the following assumptions. There exist positive constants α1, α2, κ, l, and p > 2 such that −κ−α1|s|p≤f(s)s≤κ−α2|s|p∀s∈R,(2) (f(s)−f(r))(s−r)≤l(s−r)2∀r, s ∈R.(3) From (2) we deduce that there exists c > 0 such that |f(s)| ≤ c(|s|p−1+ 1) ∀s∈R.(4) It is well-known that any polynomial of odd degree f(s) = P2m+1 j=0 cjsjwith c2m+1 <0 satisfies the above assumptions (a typical example is the function f(s) = −s3+λs, that in reaction-diffusion leads to Chaffee- 2 Infante equations, and more in particular, when λ= 1,gives rise to the double-potential, for instance, in phase-field models). Since it will be used below, let us denote F(s) := Zs 0 f(r)dr. From (2), there exist positive constants eα1,eα2,and eκsuch that −eκ−eα1|s|p≤ F(s)≤eκ−eα2|s|p∀s∈R.(5) For the operator gwe will assume that it is well-defined as g:R×CL2→L2(Ω),and it satisfies (I) for all ξ∈CL2,the function R3t7→ g(t, ξ)∈L2(Ω) is measurable, (II) g(t, 0) = 0 for all t∈R, (III) there exists Lg>0 such that for all t∈Rand ξ, η ∈CL2,it holds |g(t, ξ)−g(t, η)| ≤ Lg|ξ−η|CL2. Observe that the assumption (II) does not represent any restriction, since we may consider new functions ˜g and ˜ kdefined for each t∈Rand ξ∈CL2by ˜g(t, ξ) = g(t, ξ)−g(t, 0) and ˜ k(t) = k(t) + g(t, 0).Now, ˜gsatisfies (I)–(III) provided that gsatisfied (I) and (III); and ˜ k∈L2 loc(R;H−1(Ω)) if g(·,0) ∈L2 loc(R;L2(Ω)). When comparing with similar non-autonomous delayed problems in the literature (for instance, with variable time delay), one may check that the assumptions on the delay operator in order to deal with phase-spaces square integrable in time include more restrictions than our hypotheses, as C1regularity for the function driving the delay time and control of square norms in time of the delay operators by the original arguments in an augmented time interval. We do not require such additional conditions, since our phase-space is composed by functions continuous in time, which in this framework does not really seem a restriction. Indeed, we emphasize that the assumptions on g(introduced in [8]) and the fact that we are dealing just with CL2terms allow us to include driving delay terms under the only assumption of measurability. Namely, suppose that ρ:R→[0, h] measurable and G:L2(Ω) →L2(Ω) globally Lipschitz with G(0) = 0 are given. Then, g:R×CL2→L2(Ω) defined by R×CL23(t, ξ)7→ g(t, ξ) := G(ξ(−ρ(t))) satisfies the assumptions (I)–(III) given above. Moreover, examples with multiple delays can also be included (cf. [8, Remark 2.1]). The structure of the paper is the following. In Section 2 we establish existence and uniqueness of solution for the problem under the above conditions. Continuity with respect to initial data of the solution operator is also studied. Then, in Section 3, after a very brief survey of some abstract results of pullback attractors theory, we consider the natural (non-autonomous) dynamical system associated to the problem through the previous result and we analyze conditions in order to obtain pullback attractors for it. An energy method that relies strongly on the continuity of the solutions is involved. Actually, we obtain two different families of minimal pullback attractors, namely, those of fixed bounded sets but also for a class of time-dependent families (universe) given by a tempered condition. Finally, from comparison results, we establish relations among them, and under suitable additional assumptions we conclude that these families of attractors are in fact the same object. 3 2 Existence and uniqueness of solution In this paragraph we analyze the existence and uniqueness of solution to problem (1). Definition 1 A weak solution to (1) is a function u∈C([τ−h, ∞); L2(Ω)) such that u∈L2(τ, T ;H1 0(Ω)) ∩ Lp(τ, T ;Lp(Ω)) for all T > τ, with u(t) = φ(t−τ)for all t∈[τ−h, τ],and such that for all v∈H1 0(Ω)∩Lp(Ω), it satisfies d dt(u(t), v) + ((u(t), v)) = (f(u(t)), v)+(g(t, ut), v) + hk(t), viin D0(τ, ∞). Remark 2 Observe that if uis a weak solution to (1), then it satisfies the energy equality |u(t)|2+ 2 Zt s ku(r)k2dr =|u(s)|2+ 2 Zt s [(f(u(r)), u(r)) + (g(r, ur), u(r)) + hk(r), u(r)i]dr for all τ≤s≤t. The main result of this section is the following Theorem 3 Consider fand gsatisfying (2) and (3), and (I)–(III) respectively, k∈L2 loc(R;H−1(Ω)), τ ∈R, and φ∈CL2given. Then, there exists a unique weak solution u(·) = u(·;τ, φ)to (1). Proof. Uniqueness. Suppose that uand vare two weak solutions to (1), and denote w=u−v, which solves the equation ∂w ∂t −∆w=f(u)−f(v) + g(t, ut)−g(t, vt) with wτ= 0 as initial datum, and homogeneous Dirichlet boundary condition. Then, from the energy equality (cf. Remark 2) we have |w(t)|2+ 2 Zt τ kw(s)k2ds = 2 Zt τ [(f(u(s)) −f(v(s)), w(s)) + (g(s, us)−g(s, vs), w(s))]ds for all t≥τ. Using (3), (III), and Cauchy-Schwarz inequality, we deduce that |w(t)|2+ 2 Zt τ kw(s)k2ds ≤2lZt τ |w(s)|2ds + 2LgZt τ |ws|2 CL2ds for all t≥τ. In particular, and since wτ= 0,we conclude that |wt|2 CL2≤2(l+Lg)Zt τ |ws|2 CL2ds ∀t≥τ, whence uniqueness follows by Gronwall’s inequality. Existence. We split the proof in three steps, proving an intermediate result. Consider {wj}j≥1⊂H1 0(Ω)∩ Lp(Ω) a Hilbert basis of L2(Ω) such that span{wj}j≥1is dense in H1 0(Ω) ∩Lp(Ω). Claim. Consider k∈L2 loc(R;L2(Ω)), n ∈Nfixed, and φ∈C([−h, 0]; span{wj}n j=1) (in particular, observe that φ(0) ∈H1 0(Ω) ∩Lp(Ω)). Then, there exists a (unique) weak solution ˜uto (1). The uniqueness has been proved in the previous argument. In order to establish the existence, we state the following approximate system for any m≥n. We seek ˜um(t, x) = Pm j=1 γmj(t)wj(x) (observe that we use 4 an upper script to avoid misleading with the delay terminology, but no brackets are used since no possible confusion arises in this sense) that solves    d dt(˜um(t), wj) + ((˜um(t), wj)) = (f(˜um(t)), wj)+(g(t, ˜um t), wj)+(k(t), wj), a.e. t > τ, 1≤j≤m, ˜um τ=φ. It is well-known that the above finite-dimensional delayed system is well-posed (e.g. cf. [9]), at least locally. We will provide a priori estimates that show that these solutions are well-defined in every interval [τ−h, T ] for any T > τ. Step 1: First a priori estimates. Multiplying each equation in the above system by γmj(t) respectively and summing from j= 1 to m, we obtain 1 2 d dt|˜um(t)|2+k˜um(t)k2= (f(˜um(t)),˜um(t)) + (g(t, ˜um t),˜um(t)) + (k(t),˜um(t)) ≤κ|Ω| − α2k˜um(t)kp Lp(Ω) +Lg|˜um t|CL2|˜um(t)|+1 2kk(t)k2 ∗+1 2k˜um(t)k2, a.e. t > τ, where we have used (2), (II), (III), and Cauchy-Schwarz and Young inequalities. Therefore, we arrive at d dt|˜um(t)|2+k˜um(t)k2+ 2α2k˜um(t)kp Lp(Ω) ≤2κ|Ω|+ 2Lg|˜um t|2 CL2+kk(t)k2 ∗, a.e. t > τ. Integrating in [τ, t] we obtain |˜um(t)|2+Zt τ k˜um(s)k2ds + 2α2Zt τ k˜um(s)kp Lp(Ω)ds ≤ |˜um(τ)|2+ 2LgZt τ |˜um s|2 CL2ds +Zt τ kk(s)k2 ∗ds + 2κ|Ω|(t−τ) (6) for all t≥τ. In particular, putting t+θinstead of t, with θ∈[−h, 0],we deduce that |˜um t|2 CL2≤ |φ|2 CL2+ 2LgZt τ |˜um s|2 CL2ds +Zt τ kk(s)k2 ∗ds + 2κ|Ω|(t−τ)∀t≥τ, and by Gronwall’s inequality, it yields |˜um t|2 CL2≤|φ|2 CL2+Zt τ kk(s)k2 ∗ds + 2κ|Ω|(t−τ)e2Lg(t−τ)∀t≥τ, m ≥n. (7) So, from this and (6), we obtain that {˜um}m≥nis bounded in L2(τ, T;H1 0(Ω)) ∩Lp(τ, T ;Lp(Ω)) ∩C([τ, T ]; L2(Ω)) for all T > τ. Now, using (4), we have that {f(˜um)}m≥nis bounded in Lq(τ, T;Lq(Ω)) for all T > τ. So, there exist functions ˜u∈L∞(τ, T ;L2(Ω)) ∩Lp(τ, T;Lp(Ω)) ∩L2(τ, T;H1 0(Ω)) and eχ∈Lq(τ, T ;Lq(Ω)) for all T > τ, and a subsequence (relabelled the same) such that              ˜um∗ *˜uweakly-star in L∞(τ, T;L2(Ω)), ˜um*˜uweakly in Lp(τ, T;Lp(Ω)), ˜um*˜uweakly in L2(τ, T;H1 0(Ω)), f(˜um)*eχweakly in Lq(τ, T ;Lq(Ω)) (8) 5 for all T > τ. Step 2: Uniform estimates for the time-derivatives. Now, we proceed to obtain a second set of estimates that will provide the convergence of a subsequence in CL2. Multiplying each equation of the approximate system by γ0 mj(t) and summing from j= 1 to m, we obtain |(˜um)0(t)|2+1 2 d dtk˜um(t)k2= (f(˜um(t)),(˜um)0(t)) + (g(t, ˜um t) + k(t),(˜um)0(t)) ≤d dt ZΩ F(˜um(t, x))dx +|g(t, ˜um t)|2+|k(t)|2+1 2|(˜um)0(t)|2, a.e. t > τ. Integrating between τand t, from (II), (III), and by (5), we have Zt τ |(˜um)0(s)|2ds +k˜um(t)k2+ 2eα2k˜um(t)kp Lp(Ω) ≤4˜κ|Ω|+ 2eα1k˜um(τ)kp Lp(Ω) +k˜um(τ)k2+ 2L2 gZt τ |˜um s|2 CL2ds + 2 Zt τ |k(s)|2ds for all t≥τand any m≥n. Since ˜um τ=φfor all m≥nand in particular ˜um(τ) = φ(0) ∈H1 0(Ω) ∩Lp(Ω),by (7), we deduce that {˜um}m≥nis bounded in L∞(τ, T;H1 0(Ω) ∩Lp(Ω)) for all T > τ and {(˜um)0}m≥nis bounded in L2(τ, T;L2(Ω)) (9) for all T > τ. Then, we improve the regularity of ˜uobtained in Step 1. Actually, ˜u∈L∞(τ, T ;H1 0(Ω) ∩Lp(Ω)) with ˜u0∈L2(τ, T;L2(Ω)) for all T > τ. Fixing (an arbitrary value) T > τ, since |˜um(t2)−˜um(t1)|2=Zt2 t1 (˜um)0(s)ds 2 ≤ k(˜um)0k2 L2(τ,T ;L2(Ω))|t2−t1| ∀ t1, t2∈[τ, T ], from (9), the compactness of the embedding of H1 0(Ω) into L2(Ω), by the Ascoli-Arzel`a Theorem, and taking into account the initial data for all the sequence, we deduce that there exists a subsequence (relabelled the same) such that ˜um→˜uin C([τ−h, T]; L2(Ω)) (10) for all T > τ and a.e. in Ω ×(τ, ∞). Since f∈C(R),we conclude that f(˜um)→f(˜u) a.e. in Ω ×(τ, ∞).So, from (8) and [10, Lemma 1.3, page 12] we may identify eχ=f(˜u). Thus, from (8) and (10) we may pass to the limit in the equations satisfied by {˜um}and, thanks to the fact that span{wj}j≥1is dense in H1 0(Ω) ∩Lp(Ω),we conclude that ˜uis a weak solution to (1). The claim is proved. Step 3: Proof of the general statement by density. For each n∈N,define φn(s) = Pn j=1(φ(s), wj)wj, s∈[−h, 0].[Due to the fact that {wj}j≥1is a Hilbert basis of L2(Ω),it is not difficult to check by contradiction that φn→φin CL2.] Let also consider a sequence {kn} ⊂ L2 loc(R;L2(Ω)) converging to kin L2 loc(R;H−1(Ω)). 6 Denote by unthe corresponding solution to (1) with kreplaced by knand initial data un τ=φn(according to the claim proved in steps 1 and 2). From the energy equality for each un,we have |un(t)|2+ 2 Zt τ kun(s)k2ds =|un(τ)|2+ 2 Zt τ (f(un(s)), un(s))ds + 2 Zt τ (g(s, un s) + kn(s), un(s))ds ∀t≥τ. Reasoning analogously to Step 1, we conclude that {un}is bounded in L2(τ, T ;H1 0(Ω)) ∩Lp(τ, T ;Lp(Ω)) ∩C([τ−h, T]; L2(Ω)) for all T > τ. Now, using (4), we have that {f(un)}is bounded in Lq(τ, T ;Lq(Ω)) for all T > τ. So, there exist functions u∈L∞(τ−h, T;L2(Ω))∩Lp(τ, T ;Lp(Ω))∩L2(τ, T;H1 0(Ω)) and χ∈Lq(τ, T ;Lq(Ω)) for all T > τ, and a subsequence (relabelled the same) such that              un∗ * u weakly-star in L∞(τ−h, T;L2(Ω)), un* u weakly in Lp(τ, T ;Lp(Ω)), un* u weakly in L2(τ, T;H1 0(Ω)), f(un)* χ weakly in Lq(τ, T;Lq(Ω)) (11) for all T > τ. Actually, we may improve some of the above convergences. Reasoning as in the proof of the uniqueness of solution, i.e. taking into account the difference of the equations satisfied by unand um,and writing the energy equality for un−um,we have |um(t)−un(t)|2+Zt τ kum(s)−un(s)k2ds ≤ |um(τ)−un(τ)|2+2lZt τ |um(s)−un(s)|2ds+2LgZt τ |um s−un s|2 CL2ds+Zt τ kkm(s)−kn(s)k2 ∗ds (12) for all t≥τ, and so, in particular, |um t−un t|2 CL2≤ |φm−φn|2 CL2+ 2(l+Lg)Zt τ |um s−un s|2 CL2ds +Zt τ kkm(s)−kn(s)k2 ∗ds ∀t≥τ. From Gronwall’s inequality and (12), we have {un}is a Cauchy sequence in L2(τ, T;H1 0(Ω)) ∩C([τ−h, T]; L2(Ω)) for all T > τ. So, up to a subsequence (still relabelled the same), we have un→ua.e. in Ω ×(τ, ∞). Thus, as before, from (11) and [10, Lemma 1.3, page 12] we may identify χ=f(u); and from (11) we may pass to the limit in the equations satisfied by {un}and we conclude that uis a weak solution to (1). Although the following regularity result is not important for the goal of this paper (but of a forthcoming one), it is worth to present it here since is a simple generalization of the intermediate claim proved above. 7 Corollary 4 Consider fand gsatisfying (2) and (3), and (I)–(III) respectively, k∈L2 loc(R;L2(Ω)), φ ∈ CL2with φ(0) ∈H1 0(Ω) ∩Lp(Ω),and suppose that there exist a family {wj}j≥1⊂H1 0(Ω) ∩Lp(Ω) dense in H1 0(Ω) ∩Lp(Ω) and {φn} ⊂ CL2with each φn∈C([−h, 0]; span{wi}n i=1), φn→φin CL2and φn(0) →φ(0) in H1 0(Ω) ∩Lp(Ω).Then, there exists a unique weak solution uto (1), which also satisfies u∈L∞(τ, T;H1 0(Ω) ∩ Lp(Ω)) and u0∈L2(τ, T ;L2(Ω)) for all T > τ. Proof. It is analogous to steps 1 and 2 in the proof of Theorem 3, with the natural modification in the initial data in the Galerkin scheme, taking into account that, without loss of generality, we may assume that the functions {wj}j≥1are linearly independent. Remark 5 Given a family {wj}j≥1⊂H1 0(Ω)∩Lp(Ω) dense in H1 0(Ω)∩Lp(Ω) (and therefore dense in L2(Ω)), it is always possible to assume that {wj}j≥1is a Hilbert basis in L2(Ω) –by a Gram-Schmidt ortho-normalization process–, and then Pnφ:= Pn j=1(φ, wj)wj→φin CL2.However, it might not occur at the same time that Pnφ(0) →φ(0) in H1 0(Ω) ∩Lp(Ω),which motivates this requirement in the statement given above. Nevertheless, it is really the case in some situations, as for instance if N≤2p/(p−2) (without extra conditions on Ω) or with Ω⊂RNa bounded Ckdomain with k≥2and k≥N(p−2)/(2p),since then, a special basis formed by eigenfunctions of −∆with homogeneous Dirichlet boundary conditions satisfies all the requirements (namely in the first case H1 0(Ω) is continuously embedded in Lp(Ω); and in the second case span{wj}j≥1is dense in Hk(Ω) ∩H1 0(Ω),which is continuously embedded in Lp(Ω)). Concerning the solutions to (1), we have the following result, which shows continuity with respect to initial data. Proposition 6 Consider fand gsatisfying (2) and (3), and (I)–(III) respectively, and k∈L2 loc(R;H−1(Ω)) given. Then, for any τ∈R,and φ, ψ ∈CL2,the solutions u=u(·;τ, φ)and v=v(·;τ, ψ)to (1) with respective initial data φand ψsatisfy |ut−vt|2 CL2≤ |φ−ψ|2 CL2e2(l+Lg)(t−τ)∀t≥τ. (13) Proof. Denote by w=u−v, which solves the problem ∂w/∂t −∆w=f(u)−f(v) + g(t, ut)−g(t, vt),with wτ=φ−ψand homogeneous Dirichlet boundary condition. Then, from the energy equality we have 1 2 d dt|w(t)|2+kw(t)k2= (f(u(t)) −f(v(t)), w(t)) + (g(t, ut)−g(t, vt), w(t)), a.e. t > τ. Using (3) and (III), and integrating in time, we may deduce that |w(t)|2+ 2 Zt τ kw(s)k2ds ≤ |w(τ)|2+ 2(l+Lg)Zt τ |ws|2 CL2ds ∀t≥τ, whence in particular |wt|2 CL2≤ |wτ|2 CL2+ 2(l+Lg)Zt τ |ws|2 CL2ds ∀t≥τ. Now, by applying Gronwall’s inequality, it yields to (13). 8 3 Pullback attractors The goal of this section is to ensure, under certain conditions, the existence of pullback attractors for a suitable dynamical system associated to problem (1). In order to proceed, let us first introduce some definitions and abstract results that will be applied later. Consider given a metric space (X, dX),and let us denote R2 d={(t, τ)∈R2:τ≤t}. A process Uon Xis a mapping R2 d×X3(t, τ, x)7→ U(t, τ)x∈Xsuch that U(τ, τ)x=xfor any (τ, x)∈R×X, and U(t, r)(U(r, τ)x) = U(t, τ)xfor any τ≤r≤tand all x∈X. Definition 7 A process Uon Xis said to be continuous if for any τ≤t, the map U(t, τ) : X→Xis continuous. The process Uis said to be closed if for any τ≤t, and any sequence {xn} ⊂ Xwith xn→x∈Xand U(t, τ)xn→y∈X, then U(t, τ)x=y. If a process is continuous, then it is closed. Therefore, it is more general (and can be more useful for applications) to establish a theory within the concept of closed process. Let us denote by P(X) the family of all nonempty subsets of X, and consider a family of nonempty sets b D0={D0(t) : t∈R} ⊂ P(X). Definition 8 We say that a process Uon Xis pullback b D0-asymptotically compact if for any t∈Rand any sequences {τn} ⊂ (−∞, t] and {xn} ⊂ Xsatisfying τn→ −∞ and xn∈D0(τn) for all n, the sequence {U(t, τn)xn}is relatively compact in X. Denote ΛX(b D0, t) = \ s≤t[ τ≤s U(t, τ)D0(τ) X ∀t∈R, where {· · · }Xis the closure in X. Given two subsets of X, O1and O2,we denote by distX(O1,O2) the Hausdorff semi-distance in Xbetween them, defined as distX(O1,O2) = sup x∈O1 inf y∈O2 dX(x, y). Let be given Da nonempty class of families parameterized in time b D={D(t) : t∈R}⊂P(X).The class Dwill be called a universe in P(X). Definition 9 A process Uon Xis called pullback D-asymptotically compact if it is pullback b D-asymptotically compact for any b D∈ D. It is said that b D0={D0(t) : t∈R}⊂P(X)is pullback D-absorbing for the process Uon Xif for any t∈Rand any b D∈ D, there exists a τ0(t, b D)≤tsuch that U(t, τ)D(τ)⊂D0(t)for all τ≤τ0(t, b D). With the above definitions, we may establish the following result (cf. [7, Theorem 3.11]). Theorem 10 Consider a closed process U:R2 d×X→X, a universe Din P(X),and a family b D0={D0(t) : t∈R}⊂P(X)which is pullback D-absorbing for U, and assume also that Uis pullback b D0-asymptotically compact. 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