Non-Autonomous Attractor for Integro-Differential Evolution Equations

DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2009.2.xx DYNAMICAL SYSTEMS SERIES S Volume 2, Number 1, March 2009 pp. 1–XXX NON-AUTONOMOUS ATTRACTORS FOR INTEGRO-DIFFERENTIAL EVOLUTION EQUATIONS T. Caraballo Departamento de Ecuaciones Diferenciales y An´alisis Num´erico Facultad de Matem´aticas, Universidad de Sevilla Apartado de Correos 1160, 41080-Sevilla, Spain P. E. Kloeden J.W. Goethe-Universit¨at , FB Mathematik, Postfach 11 19 32 D-60054 Frankfurt a.M., Germany Abstract. We show that infinite-dimensional integro-differential equations which involve an integral of the solution over the time interval since starting can be formulated as non-autonomous delay differential equations with an infinite delay. Moreover, when conditions guaranteeing uniqueness of solutions do not hold, they generate a non-autonomous (possibly) multi-valued dynamical system (MNDS). The pullback attractors here are defined with respect to a universe of subsets of the state space with sub-exponetial growth, rather than restricted to bounded sets. The theory of non-autonomous pullback attractors is extended to such MNDS in a general setting and then applied to the original integro-differential equations. Examples based on the logistic equations with and without a diffusion term are considered. 1. Introduction. The main aim of this paper is to show that a wide class of integro-differential partial differential equations can be analyzed within the framework of non-autonomous dynamical systems, and the long-time behaviour of their solutions can be investigated with the help of the theory of pullback attractors. This theory is now well established as has been extensively developed over the last one and a half decades. Pullback attractors have proven to be appropriate concepts to describe the long–time behaviour of many dynamical systems arising in science, especially those exhibiting non-autonomity (see, e.g. Caraballo et al. [15], Cheban et al. [20], Chepyzhov and Vishik [21]), Chueshov [22], Crauel and Flandoli [23], Flandoli and Schmalfuß [25], Kloeden [28], Kloeden and Schmalfuß [29], Robinson [33], Schmalfuß [34], amongst many others). Integro-differential equations appear in various branches of science (e.g. in modelling the growth of parasite population, in Lotka-Volterra predator-prey systems, in reaction-diffusion models with memory, and their relevance is without doubt. In general, the models containing in their equations some kind of delay terms are now being studied extensively, since it is assumed that in many phenomena from reality, 2000 Mathematics Subject Classification. Primary: 34G25, 34K25, 35R10; Secondary: 34D45, 37C70, 47H20. Key words and phrases. Integro-differential equation, differential equation with infinite delay, set-valued process, set-valued non-autonomous dynamical system, pullback attractor. Partially supported by Ministerio de Educaci´on y Ciencia (Spain), FEDER (European Community) under grant MTM2005-01412, and Consejer´ıa de Innovaci´on Ciencia y Empresa de la Junta de Andaluc´ıa (Spain) under grant P07-FQM-02468. 1 2 T. CARABALLO AND P. E. KLOEDEN the principle of causality does not seem appropriate, and it is assumed that the past history of the phenomena have a decisive influence in the future evolution of the systems. There are now many papers dealing with the asymptotic behaviour of ordinary or partial differential equations within the framework of the pullback theory for nonautonomous dynamical systems. In principle, as soon as we have an equation with a non-constant delay term, the problem becomes non-autonomous. However, we have not found in the literature any papers concerning its applications to integrodifferential equations of the form du dt=Au +F(u) + Zt 0 G(t, s, u(s))ds, (1) where, for instance, Ais a linear operator and F,Gare nonlinear, in an infinite dimensional Banach state space H. On the other hand, for a finite dimensional state space, we are aware only of some indirectly related papers (see below). The integral term, essentially a memory term, in equation (1) means that it is in effect a differential equation with unbounded (infinite) delay. Caraballo et al. [10,16] used such a formulation when the space His finite dimensional for logistic– like equations involving an integral over the entire negative time axis of a function of the solution. The following observation shows that equation (1) can also be formulated as a differential equation with infinite delay. Considering only the last term in the equation and denoting by utthe segment solution defined for s≤0 as ut(s) = u(t+s), a change of variables gives Zt 0 G(t, s, u(s))ds =Z0 −t G(t, t +s, u(t+s))ds =Z0 −t G(t, t +s, ut(s))ds =G(t, ut), where Gis defined in a suitable phase space Cγ, a Banach subspace of C(−∞,0; H) satisfying appropriate additional assumptions (e.g. the limt→−∞ u(t)eγt exists for a suitable weight γ). In other words, G:R×R×Cγ→His defined as G(t, φ) = Z0 −t G(t, t +s, φ(s))ds. Then, equation (1) can be written as du dt=Au +F(u) + G(t, ut) and even the term F(u) can be included in the delay term by setting F(t, ut) = F(ut(0)) + G(t, ut) and then our model becomes du dt=Au +F(t, ut).(2) An analogous situation holds when the Banach space His infinite dimensional, but leads to an abstract functional partial differential equation. Although we could carry out our investigation working directly with equation (1), we prefer to develop a general abstract theory for equation (2), and then analyze our motivating model as a particular case, since our results then also apply to many other situations. ATTRACTORS FOR INTEGRO-DIFFERENTIAL PDE 3 We will assume very weak assumptions on the operators in our model so that uniqueness of solutions will be not guaranteed from the very beginning. For this reason we need the theory of multi-valued (or set-valued) non-autonomous dynamical systems. The structure of the paper is the following. We recall the definition of a nonautonomous set-valued dynamical system in the next section and, then, in section 3 we present the definition and properties of pullback attractors of such systems along with statements of theorems, which will be proved in the appendix, for their existence. The pullback attractors here are defined with respect to a universe of subsets of the state space with sub-exponential growth, rather than restricted to bounded sets. In section 4 we show that general class of infinite-dimensional non-autonomous differential equations with infinite delay generate such a non-autonomous set-valued dynamical system which establish the existence of a pullback attractor under certain structural conditions. Two examples are given in section 5, both with a logistic structure with an integral term, one with and one without an additional diffusion term. An appendix contains proofs of theorems presented earlier in the paper as well as some results that were used earlier. 2. Non-autonomous set-valued dynamical systems. First we recall some basic definitions for set–valued non–autonomous dynamical systems and establish a sufficient condition for the existence of a pullback attractor for these systems. For a more general random context the reader is referred to [8] Let X= (X, dX) denote a Polish space, let 2Xbe the set of all subsets of Xand let Pc(X) be the set of all non-empty closed subsets of the space X. A mapping D :t∈R→D(t)∈2Xis called a multi–function or set-valued mapping. We denote by C(X) the set of all multi–functions D:t∈R→D(t)∈2Xwith closed and non–empty images and use the notation b D={D(t) : t∈R}for any element in C(X). A multi–valued map U:R2 d×X→ Pc(X), where R2 d:= {(t, s)∈R2:t≥s}, is called a multi–valued non–autonomous dynamical system (MNDS) [12,13,14] if i) U(s, s, ·) = idX(·) for all s∈R, ii) U(t, τ, x)⊂U(t, s, U(s, τ, x)) for all τ≤s≤t,x∈X(process property), where U(t, τ, V ) := ∪x0∈VU(t, τ, x0) for any non–empty set V⊂X. Moreover, an MNDS is said to be strict if iii) U(t, τ, x) = U(t, s, U(s, τ, x)) for all τ≤s≤tand x∈X, and to be upper–semicontinuous at x0if iv) for every neighborhood Uin Xof the set U(t, τ, x0)there exists δ > 0such that U(t, τ, y)∈ U whenever dX(x0, y)< δ. Finally, U(t, τ, ·) is said to be upper–semicontinuous, if it is upper–semicontinuous at every x0in X. We note that, if the mapping U(t, τ, ·) is upper–semicontinuous at x0, then for all ε > 0 there exists δ(ε)>0 such that distX(U(t, τ, y), U (t, τ, x0)) ≤ε, for any ysatisfying dX(y, x0)≤δ(ε), where distXdenotes the Hausdorff semidistance which is defined for two non-empty sets A, B as distX(A, B) = sup x∈A inf y∈BdX(x, y). 4 T. CARABALLO AND P. E. KLOEDEN The converse is true when U(t, τ, x0) is compact, see Aubin and Cellina [2]. 3. Pullback attractors for MNDS. We will now establish a sufficient condition ensuring the existence of pullback attractors with respect to a universe of sets (as in [15]). When this universe consists of bounded sets, the results have already been proved in [12]. A multi–valued mapping b D={D(t) : t∈R}is said to be negatively, strictly, or positively invariant (resp.) for the MNDS Uif D(t)⊂,=,⊃(resp.) U(t, τ, D(τ)) for (t, τ)∈R2 d. Let Dbe the family of multi–valued mappings with values in C(X). We say that a family b K∈ D is pullback D-attracting if for every b D∈ D lim τ→+∞distX(U(t, t −τ, D (t−τ)), K(t)) = 0,for all t∈R. b B∈ D is said to be pullback D-absorbing if for every b D∈ D and every t∈R, there exists T:= T(t, b D)>0 such that U(t, t −τ, D (t−τ)) ⊂B(t) for all τ≥T. (3) The following definition provides the main objective of this article. For this we need a particular set system called a universe (see Schmalfuß [34]): Let Dbe a set of multi–valued mappings in C(X)satisfying the inclusion closure property: suppose that b D∈ D and let c D′be a multi–valued mapping in C(X)such that D′(t)⊂D(t) for t∈R, then c D′∈ D. Definition 3.1. A family b A∈ D is said to be a global pullback D-attractor for the MNDS Uif it satisfies: i): A(t) is compact for any t∈R; ii): b Ais pullback D-attracting; iii): b Ais negatively invariant. b Ais said to be a strict global pullback D-attractor if the invariance property in the third item is strict. As usual, the main tool to prove the existence of an attractor is the concept of pullback-omega-limit set. For a multi–valued mappings b Dwe define the pullbackomega-limit set as the t–dependent set Λ( b D, t) given by Λ( b D, t) = \ τ≥0[ s≥τ U(t, t −s, D (t−s)). This set is closed, but it may be empty. It can be proved that y∈Λ( b D, t) if and only if there exist tn→+∞and yn∈U(t, t −tn, D (t−tn)) such that lim n→+∞yn=y. We then have the following lemma, which is a generalization of Theorem 6 and Lemma 8 in Caraballo et al. [12] to the case in which we consider a general universe Dinstead of just the bounded sets of X. The proof is given in the appendix. ATTRACTORS FOR INTEGRO-DIFFERENTIAL PDE 5 Lemma 3.2. Assume the MNDS U(t, τ, ·)is upper–semicontinuous for (t, τ)∈R2 d,. Let b Bbe a multi–valued mapping such that the MNDS is asymptotically compact with respect to b B, i.e., for every sequence tn→+∞, t ∈R, every sequence yn∈ U(t, t −tn, B(t−tn)) is pre–compact. Then, for t∈R, the pullback–omega–limit set Λb B, tis non–empty, compact, and lim τ→+∞distX(U(t, t −τ, B (t−τ)),Λb B, t) = 0,(4) Λb B, t⊂Ut, τ, Λb B, τ, for all (t, τ)∈R2 d.(5) We can now present a sufficient condition ensuring the existence of pullback attractor. The proof of this theorem is also given in the appendix. Theorem 3.3. Assume the hypotheses in Lemma 3.2. In addition, suppose that b B∈ D is pullback D–absorbing. Then, the set b Agiven by A(t) := Λ b B, t is a pullback D-attractor. Moreover, b Ais the unique element from Dwith these properties. In addition, if Uis a strict MNDS then b Ais strictly invariant. 4. MNDS generated by infinite-delay partial differential equations. In this section we consider the following evolution equation dy dt =Ay +f(t, yt),(6) which includes, in particular, our integro-differential model (1). Here we suppose that Ais the generator of a C0contraction semigroup (eAt)t≥0 on a separable Banach space (H, k · k) such that keAtxk ≤ kxke−αt,for some α > 0 and every t≥0, and assume the operators eAt are compact for t > 0. The non–linear term fdepends on tand on a delay term yt,which is defined as follows: For a function y(·) : R→H, and any t∈R, we define yt: (−∞,0] →Has yt(s) = y(t+s), s ∈(−∞,0]. When we equip (6) with an initial value in order to have a an initial value problem, we need to set y(t) = φ(t),for t≤0,(7) where φ: (−∞,0] →His a suitable function. Thus, if y(·) denotes a solution to (6) such that (7) holds, then ytdenotes yt(s) = y(t+s) for s∈[−t, 0] φ(s+t)s < −t where t≥0. Before describing the assumptions on f, we first introduce the function space Cγ={u∈C((−∞,0]; H) : lim τ→−∞ u(τ)eγτ exists}, where γ > α, and set kukγ:= supτ∈(−∞,0] eγτ ku(τ)k<∞.This is a separable Banach space [26, p.15]. 6 T. CARABALLO AND P. E. KLOEDEN Our objective now is to show the existence of a pullback attractor for the dynamical system generated by (6). In what follows we assume that there exist two non–negative functions ci:R→ R,i= 1,2, such that t→ci(t) is integrable with respect to every finite interval (a, b) and sub–exponentially growing for t→ ±∞. We also suppose that c2satisfies lim t→±∞ 1 tZt 0 c2(τ+s)dτ = ¯c2(s)∈R,for all s∈R. Finally, we suppose that φ∈Cγand that f:R×Cγ→H is a continuous function for which kf(t, φ)k ≤ c1(t) + c2(t)kφkγfor t∈Rand φ∈Cγ.(8) Notice that we do not assume that fis Lipschitz continuous. We now prove that for every φ∈Cγ(6) possesses at least one mild solution. Definition 4.1. For a given φ∈Cγ,a function [t0, T ]∋t→yt(·) := yt(·;t0, φ)∈ Cγis said to be a mild solution of (6) with initial function φat time t0(< T ), if yt(s) = eA(t−t0+s)φ(0) + Rs+t t0eA(t+s−τ)f(τ, yτ)dτ :s∈[−(t−t0),0] φ(s+t−t0) : s < −(t−t0),(9) for all t∈[t0, T ]. Note that in the last definition we express that the mild solution has the state space Cγ, not H. Alternatively, we can define a mild solution to (6) with state space H, setting s= 0, by y(t) = eA(t−t0)φ(0) + Rt t0eA(t−τ)f(τ, yτ)dτ :t≥t0 φ(t−t0) : t < t0. Briefly we can write that y(·;t0, φ) is a mild solution to the IVP (IVP)t0,φ (dy dt=Ay +f(t, yt),for t≥t0, yt0=φ. We now introduce the following notation. Let y∈C([t0, T ]; H) with y(t0) = φ(0) and φ∈Cγ. Then, for τ∈[t0, T ],we denote by y∨t0,τ φthe mapping from R−to Hdefined by y∨t0,τφ(s) := y(t0+τ+s) : s∈(−τ, 0] φ(τ+s) : s≤ −τ. We observe that for such function ythe integral in (9) is well defined. Theorem 4.2. Suppose that the above assumptions on eAt and fare satisfied. Then, for every interval [t0, T ],and any φ∈Cγ,the initial value problem (IVP)t0,φ possesses a mild solution in Cγ. The global existence follows by the a priori estimates established in Theorem 4.3 below, assuming that one has the local existence of solutions. The proof of local existence, given in the Appendix, follows Pazy [32] Theorem 6.2.1. ATTRACTORS FOR INTEGRO-DIFFERENTIAL PDE 7 Theorem 4.3. Let ytbe any mild solution of (6) on [t0, T ), T ∈R+∪ {+∞}with a initial function φ∈Cγ. Then ytsatisfies the inequality kytkγ≤e−α(t−t0)+Rt t0c2(τ)dτ kφkγ+Zt t0 e−α(t−τ)+Rt τc2(q)dqc1(τ)dτ. (10) Proof. We have kytkγ≤max sup s≤−(t−t0) kφ(s+t−t0)keγs,sup s∈[−(t−t0),0] keA(t−t0+s)φ(0)keγs + sup s∈[−(t−t0),0] kZs+t t0 eA(t+s−τ)f(τ, yτ)dτkeγs. The first term on the right hand side of the last inequality is equal to sup s≤0 kφ(s)keγ(s−t+t0)=e−γ(t−t0)kφkγ. For the second term we have the estimate sup s∈[−(t−t0),0] keA(t−t0+s)φ(0)keγs ≤sup s∈[−(t−t0),0] e−α(s+t−t0)kφ(0)keγs ≤e−α(t−t0)sup s∈[−(t−t0),0] e(−α+γ)skφ(0)k ≤e−α(t−t0)kφ(0)k. The third term can be estimated as follows sup s∈[−(t−t0),0] kZs+t t0 eA(t+s−τ)f(τ, yτ)dτkeγs ≤sup s∈[−(t−t0),0] Zs+t t0 eα(−t−s+τ)(c1(τ) + c2(τ)kyτkγ)dτeγs ≤Zt t0 e−α(t−τ)c1(τ)dτ +Zt t0 e−α(t−τ)c2(τ)kyτkγdτ. Collecting all these estimates we have that kytkγ≤max e−γ(t−t0)kφkγ, e−α(t−t0)kφ(0)k+Zt t0 e−α(t−τ)c1(τ)dτ +Zt t0 e−α(t−τ)c2(τ)kyτkγdτ ≤e−α(t−t0)kφkγ+Zt t0 e−α(t−τ)(c1(τ) + c2(τ)kyτkγ)dτ. We obtain the desired inequality by the Gronwall lemma. Remark 1. A consequence of this theorem is that, in case of a finite maximal interval of existence [t0, tmax) of a solution, no explosions are allowed, i.e. lim sup t↑tmax kytkγ<∞. But the case of such a finite interval carrying a bounded solution can be excluded similar to Pazy [32] Theorem 6.2.2 applying (8). Hence for every φ∈Cγ, t0∈R every mild solution of (6) is global. 8 T. CARABALLO AND P. E. KLOEDEN 4.1. Pullback attractors for the equation with infinite delay. Throughout this subsection we assume the same conditions on Aand fgiven at the beginning of Section 4. We define the multi–valued mapping U(t, τ, φ) to be the set of mild solutions (9) in the sense of Definition 4.1 at time t∈R, that is, U(t, τ, φ) = ∪yt, where the union is taken within the set of entire mild solutions (−∞,+∞)∋t→yt(·;τ, φ)∈Cγ such that yτ(·;τ, φ) = φ. We stress here that we know from above that every local solution can be extended to a global solution. Lemma 4.4. The map Uis a strict MNDS. In particular, for any fixed t∈Rwe have U(t, τ, D (τ)) ∈ C(Cγ)if ˆ D∈ C(Cγ). Proof. Let us first prove that U(t, τ, φ)⊂U(t, s, U(s, τ, φ)) for all t≥s≥τ, and all φ∈Cγ. Let z∈U(t, τ, φ). Then there exists a solution yof (9) such that z=yt(= yt(·;τ, φ)). Denote, for short, ut=yt(·;τ, φ). Hence us=ys(·;τ, φ) (for τ≤s≤t),and therefore usolves Eq. (9) with the initial value ys(·;τ, φ) at t0=s. Indeed, for θ∈[−(t−s),0] we have ut(θ) = yt(θ;τ, φ)) = eA(t−τ+θ)φ(0) + Zt+θ τ eA(t+θ−σ)f(σ, yσ)dσ =eA(t−s+θ)eA(s−τ)φ(0) + Zs τ eA(t−s+θ)eA(s−σ)f(σ, yσ)dσ +Zt+θ s eA(t+θ−σ)f(σ, yσ)dσ =eA(t−s+θ)eA(s−τ)φ(0) + Zs τ eA(s−σ)f(σ, yσ)dσ +Zt+θ s eA(t+θ−σ)f(σ, yσ)dσ =eA(t−s+θ)ys(0; τ, φ) + Zt+θ s eA(t+θ−σ)f(σ, yσ(·;τ, φ))dσ, and hence ut(θ) = yt(θ;s, ys(·;τ, φ)) for θ∈[−(t−s),0]. On the other hand, if θ∈(−∞,−(t−s)),then ut(θ) = ys(t+θ−s;τ, φ) = yt(θ;τ, φ) and hence ut(·) = yt(·;s, ys(·;τ, φ)). This implies that z=ut∈U(t, s, ys(·;τ, φ)) ⊂U(t, s, U(s, τ, φ)), and, consequently, U(t, τ, φ)⊂U(t, s, U (s, τ, φ)). As for the other inclusion, let us consider z∈U(t, s, U(s, τ, φ)).Then, there exist z1∈U(s, τ, φ) (i.e. z1=y1 s(·;τ, φ)) and z2∈U(t, s, z1) = U(t, s, y1 s(·;τ, φ)) (i.e. z2=y2 t(·;s, y1 s(·;τ, φ))) such that z=y2 t(·;s, y1 s(·;τ, φ)). Now, by concatenating these solutions, we construct uσ=y1 σ(·;τ, φ) if σ < s, y2 σ(·;s, y1 s(·;τ, φ)) if s≤σ. It is not difficult to check that utis a mild solution of (6), so z=ut=yt(·;τ, φ) and, therefore, z∈U(t, τ, φ). We also note that U(t, τ, D(τ)) belongs to C(Cγ) if b D∈ C(Cγ) where the proof follows by the continuity of Cγ∋φ→f(τ, φ), (8) and the Lebesgue domination theorem. ATTRACTORS FOR INTEGRO-DIFFERENTIAL PDE 9 For our analysis below, we will consider the system Dgiven by the multi–valued mapping Din C(Cγ) with D(t)⊂BCγ(0, (t)),the closed ball with center zero and radius , with sub–exponential growth: lim t→±∞ log+(t) t= 0. This universe Dis called the family of sub–exponentially growing multi–functions in C(Cγ). Of course, the properties of Dgiven in Definition 3.1 hold. Lemma 4.5. In addition to the previous assumptions, suppose that c2(t)< α for all t∈R. Then, the balls B(t) = BCγ(0, R(t)) with R(t) := 2 Z0 −∞ eατ+R0 τc2(t+s)dsc1(t+τ)dτ = 2 Zt −∞ eα(τ−t)+Rt τc2(s)dsc1(τ)dτ (11) form a family b B∈ D. In addition, ˆ Bis pullback D–absorbing in the sense of (3) which is forward invariant, i.e., U(t, τ, B(τ)) ⊂B(t) for (t, τ)∈R2 d. Remark 2. We note that R∋t7→ R(t) is continuous because this function solves the linear ordinary differential equation initial value problem dr dt = (−α+c2(t))r+ 2c1(t), r(0) = R(0) = 2 Z0 −∞ eατ+R0 τc2(s)dsc1(τ)dτ. Moreover, by a similar analysis to the one in Caraballo et al. [11] for a random situation, it follows that t→R(t) is sub–exponentially growing. Proof. The first part of the Lemma holds thanks to the previous remark. Let us now prove that B(t) is pullback D-absorbing. Consider b D∈ D, and pick φ∈b D. Then, if we replace in the formula in Theorem 4.3 the parameter t0by t−s, we obtain an estimate for yt:= yt(·;t−s, φ)∈U(t, t −s, D(t−s)), ||yt||γ≤e−αs+Rt t−sc2(τ)dτ ||φ||γ+Zt t−s e−α(t−τ)+Rt τc2(q)dqc1(τ)dτ. (12) We prove now that the right-hand side of Eq. (12) tends to R(t) as sgoes to +∞, what means that there exists T(b D, t)>0 such that U(t, t −s, D(t−s)) ⊂B(t) for all s > T (b D, t). First, since 1 sZt t−s c2(τ)dτ →c2(t)< α, then there exists s0(t)>0 such that α−1 sZt t−s c2(τ)dτ ≥ε > 0,for all s≥s0,for a certain ε > 0, and, hence e−αs+Rt t−sc2(τ)dτ →0 as s→+∞. Therefore, for the first term on the right-hand side of Eq. (12) we have e−αs+Rt t−sc2(τ)dτ ||φ||γ≤1 2R(t). 16 T. CARABALLO AND P. E. KLOEDEN on a bounded domain Ω in Rdwith smooth boundary ∂Ω and the Dirichlet boundary condition u|∂Ω= 0 and the non-negative initial condition u(x, 0) = u0(x)≥0. The crucial step here is to show there exists an absorbing set in L2(Ω). For this we use the following estimates for any existing positive solution. d dt|u(t)|2 2=−2|∇u(t)|+ 2rZΩ u(x, t)2dx −2r KZΩ u(x, t)3dx ≤ −2λ1|u(t)|2 2+ 2r|u(t)|2 2−2r K|u(t)|3 3 =−2(λ1−r)|u(t)|2 2−2r K|u(t)|3 3 by the Poincar´e inequality, where λ1>0 is the first eigenvalue of the Laplace operator on Ω with the Dirichlet boundary condition. Then we use the inequality |u(t)|2 2=ZΩ u(x, t)2dx ≤ZΩ u(x, t)3dx2/3 |Ω|1/3=|u(t)|2/3 3|Ω|1/3 to obtain |Ω|−3/2|u(t)|9 2≤ |u(t)|3 3. Finally, this leads to the inequality d dt|u(t)|2 2≤ −2(λ1−r)|u(t)|2 2−2r K|Ω|3/2|u(t)|9 2 or d dt|u(t)|2≤ −(λ1−r)|u(t)|2−r K|Ω|3/2|u(t)|8 2 If r≤λ1the zero solution is globally asymptotically stable, but if r > λ1then it follows that |u|7 2≤1 + K|Ω|3/2(r−λ1) r. Again, by a similar argument as the one in Lemma 26 from [16], we can obtain the existence of an absorbing set. 6. Appendix. Proof of Lemma 3.2 Consider a sequence yn∈U(t, t −tn, B(t−tn)) with tn→+∞. As Uis pullback–asymptotically compact with respect to b B, there exists a converging subsequence and its limit ybelongs to Λ b B, t, so that Λ b B, tis non–empty. We now prove that Λ b B, tis compact. For any sequence {yn} ⊂ Λb B, t there exist tn→+∞and zn∈U(t, t −tn, B(t−tn)), such that dX(yn, zn)<1 n. Using again the pullback asymptotic compactness of Uthe existence of a converging subsequence znk→z∈Λb B, tfollows. Then, ynk→z, so that Λ b B, tis compact. We prove (4) by contradiction. If (4) does not hold, then there exist ε > 0 and yn∈U(t, t −tn, B (t−tn)) with tn→+∞, such that distXyn,Λb B, t> ε. ATTRACTORS FOR INTEGRO-DIFFERENTIAL PDE 17 As Uis pullback–asymptotically compact with respect to b B, it follows that there exists a subsequence (relabelled again the same) yn→y∈Λb B, t, which is not possible. Let us now prove that (5) holds. Fix (t, τ)∈R2 d.Then, if y∈Λb B, t, there exist sequences yn∈U(t, t −(tn−τ), xn), xn∈B(t−(tn−τ)) with tn→+∞, such that yn→y. For tn≥t, the process property implies U(t, t −tn+τ, xn)⊂U(t, τ, U (τ, t −tn+τ, xn)) , and then yn∈U(t, τ, zn), where zn∈U(τ, t −tn+τ, xn). As before, up to a subsequence, zn→z∈Λb B, τ. Since x7→ U(t, τ, x) is upper–semicontinuous with closed values, we have y∈U(t, τ, z)⊂U(t, τ, Λb B, τ).  Proof of Theorem 3.3 First we need to prove that lim τ→+∞distX(U(t, t −τ, D (t−τ)), A(t)) = 0 for every b D∈ D.(25) Indeed, thanks to (4), for every ε > 0 and t∈R, there exists T(t, ε) such that for τ≥T(t, ε) distX(U(t, t −τ, B (t−τ)), A(t)< ε. But, for every b D∈ D, U(t−τ, t −τ−T(t−τ, b D), D(t−τ−T(t−τ, b D))) ⊂B(t−τ) so that distX(U(t, t −τ, D (t−τ)), A(t)) < ε for τlarge. The third property in Definition 3.1 follows from (5). Since U(t, t −τ, B(t−τ)) ⊂B(t) for τ≥T(t, b B), we have the relation A(t)⊂B(t) for each t∈R,so that b A∈ D. But this shows that Ais unique. Indeed suppose we have another pullback D–attractor c A′,then as A′(t)⊂U(t, t −τ, A′(t−τ)) and lim τ→+∞distX(U(t, t −τ, A′(t−τ)), A(t)) = 0, we have that A′(t)⊂A(t). Exchanging b Aand c A′it follows that b A=c A′. Finally, assume that Uis a strict MNDS. Then, U(t, r, A (r)) ⊂U(t, τ, U (τ, r −τ, A(r−τ))) =U(t, r −τ, A (r−τ)) ,for all τ≥0. As b Apullback attracts itself, it follows that lim τ→+∞distX(U(t, r −τ, A(r−τ)), A(t)) = 0, and, consequently, given ε > 0,there exists T(ε, t, r)>0 such that, for τ≥T(ε, t, r) distX(U(t, r −τ, A(r−τ)), A(t)) < ε, 18 T. CARABALLO AND P. E. KLOEDEN and as U(t, r, A (r)) ⊂U(t, r −τ, A (r−τ)) ,we have distX(U(t, r, A(r)), A(t)) < ε, for all ε > 0, so U(t, r, A(r)) ⊂A(t), as required.  Proof of local existence in Theorem 4.2 The proof follows Pazy [32] Theorem 6.2.1. Let us fix some φ∈Cγand t,t0∈R. Consider B(R) = {y∈C([t0, T ]; H) : y(t0) = φ(0),sup s∈[t0,T ] kφ(0) −y(s)k ≤ R}. B(R) is a convex and bounded set in C([t0, T ]; H). For any T > t0(with T−t0 small enough) we define the mapping TT:B(R)→C([t0, T ]; H) by TT(y)[t] := eA(t−t0)φ(0) + Zt t0 eA(t−t0−τ)f(τ, y ∨t0,τ x0)dτ, t ∈[t0, T ]. We note that TT(y)∈4C([t0, T ]; H) because τ→ kf(τ, y ∨t0,τ φ)k ∈ L1([t0, T ]). To see that the operator TTmaps B(R) into itself, for appropriate Rand T, we note that kf(r, y ∨t0,rφ)k ≤ c1(r) + c2(r) sup ∈[0,r] eγ(−r)ky(t0+)k +c2(r) sup ≤−r eγkφ(r+)k ≤c1(r) + c2(r) sup ∈[t0,T ] ky()k +c2(r)e−γr sup ≤−r eγ(+r)kφ(r+)k, so kf(r, y ∨t0,rφ)k ≤ c1(r) + c2(r) sup ∈[t0,T ] ky()k+c2(r)kφkγ.(26) The term sup∈[t0,T ]ky()kis bounded by kφkγ+R. In addition, keA(t−t0−τ)xk ≤ e−α(t−t0−τ)kxkso that, for small T−t0>0 (depending on r), we have TT(B(R)) ⊂B(R). In view of the continuity of Cγ∋ξ→f(t, ξ) and (8) we obtain by the Lebesgue domination theorem that TTis continuous on B(R) with the topology of C([t0, T ]; H). To see that TTis compact we first note that the sets Zt:= {z=TT(y)[t], y ∈B(R)}, t ∈[t0, T ] are pre–compact. This is trivially true for t=t0. For t > t0we introduce for sufficiently small ε > 0 Tε T(y)[t] = eA(t−t0)φ(0) + Zt−ε t0 eA(t−t0−τ)f(τ, y ∨t0,τ φ)dτ = eA(t−t0)φ(0) + eAε Zt−ε t0 eA(t−t0−τ−ε)f(τ, y ∨t0,τ φ)dτ. By (26) and the integrability conditions on c1and c2 sup y∈B(R) kZt−ε t0 S(t−t0−τ−ε)f(τ, y ∨t0,τ φ)dτk ATTRACTORS FOR INTEGRO-DIFFERENTIAL PDE 19 is finite for appropriate ε > 0 so that Tε T(B(R))[t] is pre-compact by the compactness of eAε. Then for every ε′>0 we have an ε > 0 such that kT ε T(y)[t]− TT(y)[t]k ≤ Zt t−ε e−α(t−t0−τ)kf(τ, y ∨t0,τ φ)kdτ ≤ε′ uniformly for y∈B(R) so that Ztis totally bounded, hence pre–compact. To apply the Arzel`a–Ascoli theorem we show that TT(y) with y∈B(R),is equicontinuous. Notice that, for t2> t1> t0, kTT(y)[t2]− TT(y)[t1]k ≤k(eAt2−eAt1)φ(0)k +Zt1 0 keA(t2−τ)−eA(t1−τ)kkf(τ, y ∨t0,τ φ)kdτ +Zt2 t1 keA(t2−τ)kkf(τ, y ∨t0,τ φ)kdτ. Since S(t) = eAt is a compact operator for t > 0 we know that the mapping t→ S(t) is norm-continuous for t > 0. The Lebesgue domination theorem together with (26) imply the equicontinuity for t > t0. Similar arguments hold for t1=t=t0. Indeed, in the above formula the second term on the right hand side disappears for t1=t0. The Schauder theorem gives the existence of a fixed point of TTwhich is a local solution for (6).  Acknowledgments. We would like to thank the referee for his several interesting and helpful remarks and suggestions. REFERENCES [1] L. Arnold, “Random Dynamical Systems,” Springer-Verlag, Berlin, 1998. [2] J. P. Aubin and A. Cellina, “Differential Inclusions,” Springer-Verlag, Berlin, 1984. [3] J. P. Aubin and H. Frankowska, “Set-Valued Analysis,” Birkh¨auser, Boston, 1990. [4] P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6(2006), 1–21. [5] H. Brezis, “Functional Analysis. Theory and Applications,” Masson, Paris, 1993. [6] T. Caraballo, I. D. Chueshov, P. Mar´ın-Rubio and J. Real, Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory, Discrete and Continuous Dynamical Systems A, 18 (2007), 253–270. [7] T. Caraballo, M. J. Garrido-Atienza and B. Schmalfuß, Exponential stability of stationary solutions for semilinear stochastic evolution equations with delays, Discrete and Continuous Dynamical Systems A, 18 (2007), 271–293. [8] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete and Continous Dynamical Systems, Series A, 21 (2008), 415–443. [9] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Asymptotic behaviour of of a stochastic semilinear dissipative functional equation without uniqueness of solutions, in preparation. [10] T. Caraballo, P. Kloeden and P. Mar´ın-Rubio, Numerical and finite delay approximations of attractors for logistic differential-integral equations with infinite delay, Discrete and Continuous Dynamical Systems A, 19 (2007), 177–196. [11] T. Caraballo, P. E. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183– 207. [12] T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set Valued Analysis, 11 (2003), 153–201. [13] T. Caraballo, J. A. Langa and J. Valero, Global attractors for multivalued random dynamical systems, Nonlinear Anal., 48 (2002), 805–829. 20 T. CARABALLO AND P. E. KLOEDEN [14] T. Caraballo, J. A. Langa and J. Valero, Global attractors for multivalued random dynamical systems generated by random differential inclusions with multiplicative noise, J. Math. Anal. Appl., 260 (2001), 602–622. [15] T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Analysis (TMA), 64 (2006), 484–498. [16] T. Caraballo, P. Mar´ın-Rubio and J. Valero, Attractors of differential equations with undounded delays, J. Differential Eqns., 239 (2007), 311–342. [17] T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Diff. Eqns., 205 (2004), 271–297. [18] T. Caraballo, J. Real and T. Taniguchi, The exponential stability of neutral stochastic delay partial differential equations, Discrete and Continuous Dynamical Systems A, 18 (2007), 295–313. [19] C. Castaing and M. Valadier, “Convex Analysis and Measurable Multifunctions,” Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin, 1977. [20] D. Cheban, P. E. Kloeden and B. Schmalfuss, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems, Nonlinear Dynamics & Systems Theory, 2(2002), 9–28. [21] V. V. Chepyzhov and M. I Vishik, “Attractors for Equations of Mathematical Physics,” American Mathematical Society Colloquium Publications, Vol. 49, American Mathematical Society, Providence, RI, 2002. [22] I. Chueshov,“Monotone Random Systems – Theory and Applications,” Springer Lecture Notes in Mathematics, vol. 1779, Springer–Verlag, Heidelberg, 2002. [23] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields, 100 (1994), 365–393. [24] Qixiang Dong, Gang Li and Jin Zhang, Quasilinear nonlocal integrodifferential equations in Banach spaces, Elec. J. Diff. Eqns., 2008 (2008), 1–8. [25] F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise. Stochastics & Stochastics Reports, 59 (1996), 21–45. [26] Y. Hino, S. Murakami and T. Naito, “Functional Differential Equations with Infinite Delay,” Lecture Notes in Mathematics, Vol. 1473, Springer-Verlag, Berlin, 1991. [27] S. Hu and N. S. Papageorgiou, “Handbook of Multivalued Analysis,” Vol. I, volume 419 of “Mathematics and its Applications,” Kluwer Academic Publishers, Dordrecht, 1997. [28] P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stochastics & Dynamics, 3(2003), 101–112. [29] P. E. Kloeden and B. Schmalfuß, Nonautonomous systems, cocycle attractors and variable time-step discretization, Numer. Algorithms, 14 (1997), 141–152. [30] R. K. Miller and G. R. Sell, “Volterra Integral Equations and Topological Dynamics,” Amer. Math. Soc. Memoirs 1970, 67 pp., [31] S. Novo, R. Obaya and A. M. Sanz, Exponential stability in non-autonomous delayed equations with applications to neural networks, Discrete and Continuous Dynamical Systems A, 18 (2007), 517–536. [32] A. Pazy, “Semigroups of Linear Operators and Applications to Partial Differential Equations,” Vol. 44 of “Applied Mathematical Sciences,” Springer-Verlag, New York, 1983. [33] J. C. Robinson, “Infinite-Dimensional Dynamical Systems,” Cambridge University Press, Cambridge, 2001. [34] B. Schmalfuß, Attractors for the non-autonomous dynamical systems, in “International Conference on Differential Equations,” Vol. 1,2(Berlin, 1999), 684–689, World Sci. Publishing, River Edge, NJ, 2000. Received May 2008; revised July 2008. E-mail address, T. Caraballo: [email protected] E-mail address, P.E. Kloeden: [email protected]