Pullback Attractors for Globally Modified Navier-Stokes Equations with Infinite Delays

DISCRETE AND CONTINUOUS doi: DYNAMICAL SYSTEMS Volume , Number 0,pp. 1–XX PULLBACK ATTRACTORS FOR GLOBALLY MODIFIED NAVIER-STOKES EQUATIONS WITH INFINITE DELAYS Pedro Mar´ ın-Rubio1, Antonio M. M´ arquez-Dur´ an1,2& Jos´ e Real1 1Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080–Sevilla, SPAIN 2Departamento de Econom´ıa, M´etodos Cuantitativos e Historia Econ´omica, Universidad Pablo de Olavide, Ctra. de Utrera, Km. 1, 41013–Sevilla, SPAIN (Communicated by Hans-Otto Walther) Abstract. We establish the existence of pullback attractors for the dynamical system associated to a globally modified model of the Navier-Stokes equations containing delay operators with infinite delay in a suitable weighted space. Actually, we are able to prove the existence of attractors in different classes of universes, one is the classical of fixed bounded sets, and the other is given by a tempered condition. Relationship between these two kind of objects is also analyzed. 1. Introduction. Let Ω ⊂R3be an open bounded set with regular boundary Γ, and let N∈(0,+∞) be fixed. Let us define FN: [0,+∞)→(0,1] by FN(r) := min 1,N r, r ∈[0,+∞), and consider the following system of globally modified Navier-Stokes equations (GMNSE for short) on Ω, with infinite delays and homogeneous Dirichlet boundary condition              ∂u ∂t −ν∆u+FN(kuk) [(u· ∇)u] + ∇p=f(t) + g(t, ut) in (τ, +∞)×Ω, ∇ · u= 0 in (τ, +∞)×Ω, u= 0 on (τ, +∞)×Γ, u(τ+s, x) = φ(s, x), s ∈(−∞,0], x ∈Ω, (1) where ν > 0 is the kinematic viscosity, uthe velocity field of the fluid, pthe pressure, τ∈Ran initial time, f(t) a given external force field, gis another external force field containing some hereditary characteristic, φis a given function defined in the interval (−∞,0],and we denote by utthe function defined on (−∞,0] by the relation ut(s) = u(t+s), s ∈(−∞,0]. 2010 Mathematics Subject Classification. Primary: 35K55; 35Q30; 34D45. Key words and phrases. Globally Modified Navier-Stokes Equations; pullback attractors; infinite delays. Partially supported by Ministerio de Ciencia e Innovaci´on (Spain), project MTM2008-00088 and Junta de Andaluc´ıa, grant P07-FQM-02468. 1 2 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN AND J. REAL The GMNSE (1), with or without delays, are indeed global modifications of the Navier-Stokes equations – the modifying factor FN(kuk) depends on the norm kuk=k∇uk(L2(Ω))3×3, which in turn depends on ∇uover the whole domain Ω and not just at or near the point x∈Ω under consideration. Essentially, it prevents large gradients dominating the dynamics and leading to explosions. It violates the basic laws of mechanics, but mathematically the GMNSE (1) are a well defined system of equations, just like the modified versions of the NSE of Leray and others with other mollifications of the nonlinear term, see the review paper [12]. It is worth mentioning that a global cut off function involving the D(A1/4) norm for the two dimensional stochastic Navier-Stokes equations is used in [13], and a cut-off function similar to the one we will use here was considered in [32]. The globally modified Navier-Stokes equations, in the case without delays, were introduced and studied in [1] (see also [3,2,19,20,21,7,24] and the review paper [18]). This modified model of the three-dimensional Navier-Stokes equations has some good properties: global existence, uniqueness, regularity, contrary to the original Navier-Sokes model, where the analysis of the asymptotic behaviour of solutions needs to be carried out in some non-standard way, e.g. cf. [28] and the references therein. These results are interesting in their own right, but also GMNSE are useful in obtaining new results about the three-dimensional Navier-Stokes equations, e.g., they were used in [1] to establish the existence of bounded entire weak solutions for them. Also, in [21], GMNSE were used to show that the attainability set of the weak solutions of the three-dimensional Navier-Stokes equations satisfying an energy inequality are weakly compact and weakly connected. For convergence results of solutions of GMNSE to solutions of the three-dimensional Navier-Stokes equations, see [1,24]. However, there are situations in which the model is better described if some terms containing delays appear in the equations. These delays may appear, for instance, when one wants to control the system by applying a force which takes into account not only the present state but the complete history of the solutions. To our knowledge, the references [8,9,10] are the first papers devoted to consider existence of solutions for the Navier-Stokes equations with delays and to study their asymptotic behaviour (see also [15,25] for the same task in some unbounded domains). However, all these papers deal with finite delays, while the case of infinite delays has been treated more recently for autonomous and non-autonomous dynamical systems (e.g. cf. [6,27]). In this paper we are interested in the case of a GMNSE model in which terms containing infinite delays appear (see [7] for the case with finite delays). The problem (1) was studied in [23], where existence and uniqueness of solution, and convergence to stationary solutions were obtained. Our goal in this paper is to prove more general results on the asymptotic behaviour of problem (1) than those shown in [23]. Namely, we will establish for a suitable process related to problem (1) that we can assure the existence of minimal pullback attractors under less restrictive assumptions than those in Theorem 3 and Theorem 4 in [23]. In fact, we will obtain two minimal pullback attractors for the PULLBACK ATTRACTORS FOR GMNSE WITH INFINITE DELAYS 3 process associated to problem (1). The first one is the minimal pullback attractor of fixed bounded sets of Cγ(H), which is the most usual in the literature. The second one, is the pullback attractor in the framework of a universe of families of time dependent sets with a tempered growth condition, following the ideas of [4,26]. The structure of the paper is the following. In Section 2we recall the basic results on existence of solution for the GMNSE problem with infinite delays. Indeed, an improvement on the conditions imposed for the existence is done. In Section 3we state some well-known results on the theory for the existence of minimal pullback attractors, in a unified approach for an abstract given universe. This will be applied to two cases, one the classical case of fixed bounded sets, and the other is a universe defined by a tempered condition. Finally, in Section 4we apply the above results to problem (1) obtaining two different kind of families of minimal pullback attractors. The main key is an asymptotic compactness result, whose proof relies on an energy method that makes the most of the continuity properties of the solutions and the corresponding non-increasing energy functions. Relationship between these objects is also analyzed. 2. Existence of solutions. To set our problem in the abstract framework, we consider the following usual abstract spaces (e.g. cf. [22] and [30,31]): V=nu∈(C∞ 0(Ω))3: div u= 0o, H= the closure of Vin (L2(Ω))3with inner product (·,·) and associate norm |·| , where for u, v ∈(L2(Ω))3, (u, v) = 3 X j=1 ZΩ uj(x)vj(x)dx, V= the closure of Vin (H1 0(Ω))3with scalar product ((·,·)) and associate norm k·k ,where for u, v ∈(H1 0(Ω))3, ((u, v)) = 3 X i,j=1 ZΩ ∂uj ∂xi ∂vj ∂xi dx. We will use k·k∗for the norm in V0and h·,·i for the duality pairing between V0and V. Finally, we will identify every u∈Hwith the element fu∈V0given by hfu, vi= (u, v) for all v∈V. It follows that V⊂H⊂V0,where the injections are dense and compact. We consider the linear continuous operator A:V→V0defined by hAu, vi= ((u, v)) for all u, v ∈V. Denoting D(A) = {u∈V:Au ∈H},with inner product (u, v)D(A)= (Au, Av), then, by the regularity of Γ, D(A) = (H2(Ω))3∩V, and Au =−P∆u, for all u∈D(A),is the Stokes operator (Pis the ortho-projector from (L2(Ω))3onto H). Let us denote λ1= inf v∈V\{0} kvk2 |v|2>0, the first eigenvalue of the Stokes operator. 4 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN AND J. REAL Now we define b(u, v, w) = 3 X i,j=1 ZΩ ui ∂vj ∂xi wjdx, for all measurable functions u, v, w defined on Ω with values in R3for which the integrals in the right-hand member of the above equality are finite. In particular, bis a trilinear continuous form on V×V×V, i.e., there exists a constant C1>0 only dependent on Ω (namely, C1= (2λ1/4 1)−1) such that |b(u, v, w)| ≤ C1kukkvkkwk,for all u, v, w ∈V, and b(u, v, v)=0,for all u, v ∈V. We denote bN(u, v, w) = FN(kvk)b(u, v, w). The form bNis linear in uand w, but it is nonlinear in v. By the definition of FN,if we denote hBN(u, v), wi=bN(u, v, w),for all u, v, w ∈V, we have kBN(u, v)k∗≤NC1kuk,for all u, v ∈V. (2) We recall (cf. [30]) that there exists a constant C2>0 depending only on Ω such that |b(u, v, w)| ≤ C2kuk1/2|Au|1/2kvk|w|,(3) for all u∈D(A), v ∈V, w ∈H, and |b(u, v, w)| ≤ C2kukkvk|w|1/2kwk1/2,(4) for all u, v, w ∈V. (See [29] for the proof of (4)). Let γ > 0 be fixed. One possibility to deal with infinite delays, and which we will use here (cf. [27,16,17]), is to consider the space Cγ(H) = ϕ∈C((−∞,0]; H) : ∃lim s→−∞ eγsϕ(s)∈H, which is a Banach space with the norm kϕkγ:= sup s∈(−∞,0] eγs|ϕ(s)|. We will assume that f∈L2 loc(R; (L2(Ω))3).For the term g, in which the delay is present, we assume that g:R×Cγ(H)→(L2(Ω))3satisfies (g1) For any ξ∈Cγ(H) the mapping R3t7→ g(t, ξ) is measurable, (g2) g(t, 0) = 0 for all t∈R, (g3) there exists a constant Lg>0 such that for any t∈Rand all ξ, η ∈Cγ(H), |g(t, ξ)−g(t, η)| ≤ Lgkξ−ηkγ. An example of an operator satisfying assumptions (g1)-(g3) is given in [23]. PULLBACK ATTRACTORS FOR GMNSE WITH INFINITE DELAYS 5 Definition 1. A weak solution of (1) is a function u∈C((−∞, T]; H)∩L2(τ, T ;V) for all T > τ, such that for all v∈V, d dt(u(t), v) + ν((u(t), v)) + bN(u(t), u(t), v)=(f(t), v)+(g(t, ut), v), in the sense of D0(τ, +∞),and uτ=φ. Remark 1. If uis a weak solution of (1), then usatisfies the energy equality, |u(t)|2+ 2νZt s ku(r)k2dr =|u(s)|2+ 2 Zt s [(f(r), u(r)) + (g(r, ur), u(r))] dr for all s, t ∈[τ, +∞). We have the following existence and uniqueness result: Theorem 1. Suppose that f∈L2 loc(R; (L2(Ω))3), γ > 0, and g:R×Cγ(H)→ (L2(Ω))3satisfying the assumptions (g1)–(g3), are given. Then, for any τ∈Rand φ∈Cγ(H),there exists a unique weak solution u=u(·;τ, φ)of (1), which in fact is a strong solution in the sense that u∈C((τ, T ]; V)∩L2(τ+ε, T;D(A)), for all ε > 0and any T > τ +ε. Moreover, if φ(0) ∈V, then usatisfies u∈C([τ, T ]; V)∩L2(τ, T;D(A)), for all T > τ. Proof. The proof can be seen in [23]. There, the additional assumption 2γ > νλ1 was made. That this assumption is unnecessary can be seen as follows. For the Galerkin approximations umdefined by (8) on page 661 of [23], one has d dt|um(t)|2+ 2νkum(t)k2= 2(f(t), um(t)) + 2(g(t, um t), um(t)) ≤νkum(t)k2+1 νλ1 |f(t)|2+ 2Lgkum tk2 γ, and therefore, |um(t)|2+νZt τ kum(s)k2ds ≤ |u(τ)|2+Zt τ (|f(s)|2/(νλ1)+2Lgkum sk2 γ)ds, (5) for all t≥τ. Using (5) instead of inequality (9) of [23], one obtains kum tk2 γ≤max sup θ∈(−∞,τ−t] e2γθ|φ(θ+t−τ)|2, sup θ∈[τ−t,0] (e2γθ|u(τ)|2+e2γθ Zt+θ τ (|f(s)|2/(νλ1)+2Lgkum sk2 γ)ds) ≤max sup θ∈(−∞,τ−t] e2γθ|φ(θ+t−τ)|2, |u(τ)|2+Zt τ (|f(s)|2/(νλ1)+2Lgkum sk2 γ)ds,(6) 6 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN AND J. REAL and therefore, observing that sup θ∈(−∞,τ−t] eγθ|φ(θ+t−τ)|= sup θ≤0 eγ(θ−(t−τ))|φ(θ)| =e−γ(t−τ)kφkγ ≤ kφkγ, and |u(τ)|=|φ(0)| ≤ kφkγ,we deduce from (6) that kum tk2 γ≤ kφk2 γ+Zt τ (|f(s)|2/(νλ1)+2Lgkum sk2 γ)ds for all t≥τ. Thus, by the Gronwall lemma, we have kum tk2 γ≤e2Lg(t−τ)kφk2 γ+ (νλ1)−1Zt τ e2Lg(t−s)|f(s)|2ds, for all t≥τ. Using this inequality and (5), one also obtains (10) and (12) in [23]. Now, the proof of the theorem follows as in that paper. The following continuous dependence result was also proved in [23, Prop.1]. Proposition 1 (Continuity of solutions with respect to initial data).Under the assumptions of Theorem 1, for any τ∈Rthe solutions obtained for (1) are continuous with respect to the initial condition φ, and more exactly, there exists a constant C3>0,only dependent on νand the constant C2appearing in (4), such that if ui=ui(·;τ, φi),for i= 1,2,are the corresponding solutions to initial data φi∈Cγ(H), i = 1,2,the following estimate holds: max r∈[τ,t]|u1(r)−u2(r)|2≤|φ1(0) −φ2(0)|2+Lg 2γkφ1−φ2k2 γ ×e(3Lg+2C3N4)(t−τ), for all t≥τ. 3. Abstract results on attractors theory. Existence of minimal pullback attractors. In this section we recall some abstract results on pullback attractors theory. We present a summary of some results on the existence of minimal pullback attractors obtained in [14] (see also [26,4,5]). In particular, we consider the process Ubeing closed (see Definition 2below). Consider given a metric space (X, dX),and let us denote R2 d={(t, τ)∈R2:τ≤ t}. A process on Xis a mapping Usuch that R2 d×X3(t, τ, x)7→ U(t, τ)x∈X with 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 2. Let Ube a process on X. a) Uis said to be continuous if for any pair τ≤t, the mapping U(t, τ) : X→X is continuous. b) Uis said to be closed if for any τ≤t, and any sequence {xn} ⊂ X, if xn→x∈Xand U(t, τ)xn→y∈X, then U(t, τ)x=y. Remark 2. It is clear that every continuous process is closed. More generally, every strong-weak continuous process (see [26] for the definition) is a closed process. PULLBACK ATTRACTORS FOR GMNSE WITH INFINITE DELAYS 7 Let us denote 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) [observe that we do not require any additional condition on these sets as compactness or boundedness]. Definition 3. 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. 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 4. 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)≤t such that U(t, τ)D(τ)⊂D0(t) for all τ≤τ0(t, b D). Observe that in the definition above b D0does not belong necessarily to the class D. Definition 5. Given a family parameterized in time, b D={D(t) : t∈R} ⊂ P(X), it is said that a process Uon Xis b D−asymptotically compact if for any t∈Rand any sequences {τn} ⊂ (−∞, t] and {xn} ⊂ Xsatisfying τn→ −∞ and xn∈D(τn) for all n, the sequence {U(t, τn)xn}is relatively compact in X. Definition 6. A process Uon Xis said to be pullback D−asymptotically compact if it is b D-asymptotically compact for any b D∈ D. Denote Λ( b D0, t) := \ s≤t[ τ≤s U(t, τ)D0(τ) X for all t∈R, where {· · · }Xis the closure in X. Finally, we denote by distX(O1,O2) the Hausdorff semi-distance in Xbetween two sets O1and O2, defined as distX(O1,O2) = sup x∈O1 inf y∈O2 dX(x, y) for O1,O2⊂X. We have the following result on existence of minimal pullback attractors (cf. [14]). Theorem 2. 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. Then, the family AD={AD(t) : t∈R}defined by AD(t) = [ b D∈D Λ( b D, t) X t∈R, has the following properties: 8 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN AND J. REAL (a) for any t∈R,the set AD(t)is a nonempty compact subset of X, and AD(t)⊂Λ( b D0, t), (b) ADis pullback D−attracting, i.e., lim τ→−∞ distX(U(t, τ)D(τ),AD(t)) = 0 for all b D∈ D, t ∈R, (c) ADis invariant, i.e., U(t, τ)AD(τ) = AD(t)for all τ≤t, (d) if b D0∈ D,then AD(t) = Λ( b D0, t)⊂D0(t)X,for all t∈R. The family ADis minimal in the sense that if b C={C(t) : t∈R}⊂P(X)is a family of closed sets such that for any b D={D(t) : t∈R} ∈ D, lim τ→−∞ distX(U(t, τ)D(τ), C(t)) = 0, then AD(t)⊂C(t). Remark 3. Under the assumptions of Theorem 2, the family ADis called the minimal pullback D−attractor for the process U. If AD∈ D,then it is the unique family of closed subsets in Dthat satisfies (b)–(c). A sufficient condition for AD∈ D is to have that b D0∈ D, the set D0(t) is closed for all t∈R,and the family Dis inclusion-closed (i.e., if b D∈ D,and b D0={D0(t) : t∈R}⊂P(X) with D0(t)⊂D(t) for all t, then b D0∈ D). We will denote by DF(X) the universe of fixed nonempty bounded subsets of X, i.e., the class of all families b Dof the form b D={D(t) = D:t∈R}with Da fixed nonempty bounded subset of X. In the particular case of the universe DF(X), the corresponding minimal pullback DF(X)−attractor for the process Uis the pullback attractor defined by Crauel, Debussche, and Flandoli, [11, Th.1.1, p.311], and will be denoted by ADF(X). Now, it is easy to conclude the following result. Corollary 1. [cf. [25, Cor.20]] Under the assumptions of Theorem 2, if the universe Dcontains the universe DF(X),then both attractors, ADF(X)and AD,exist, and the following relation holds: ADF(X)(t)⊂ AD(t)for all t∈R. Remark 4. It can be proved (see [26]) that, under the assumptions of the preceding corollary, if, moreover, b D0∈ D,and for some T∈Rthe set ∪t≤TD0(t) is a bounded subset of X, then ADF(X)(t) = AD(t) for all t≤T. 4. Existence of pullback attractors for the process associated to (1).Now, by the previous results, we are able to define correctly a process Uon Cγ(H) associated to (1), and to obtain the existence of minimal pullback attractors. Proposition 2. Assume that f∈L2 loc(R; (L2(Ω))3), γ > 0, and g:R×Cγ(H)→ (L2(Ω))3satisfying the assumptions (g1)–(g3), are given. Then, the bi-parametric family of maps U(t, τ) : Cγ(H)→Cγ(H),with τ≤t, given by U(t, τ)φ=ut,(7) PULLBACK ATTRACTORS FOR GMNSE WITH INFINITE DELAYS 9 where u=u(·;τ, φ)is the unique weak solution of (1), defines a continuous process on Cγ(H). Proof. It is a consequence of Theorem 1and Proposition 1. Lemma 1. Under the assumptions of Proposition 2, let µbe such that 0< µ < ν and (ν−µ)λ1≤γ. (8) Then, the following estimates hold for the solution to (1) for all t≥τ: kutk2 γ≤e−2((ν−µ)λ1−Lg)(t−τ)kφk2 γ +(µλ1)−1Zt τ e−2((ν−µ)λ1−Lg)(t−s)|f(s)|2ds, (9) µZt τ ku(s)k2ds ≤e2(ν−µ)λ1(t−τ)|u(τ)|2+e2Lg(t−τ)kφk2 γ +(µλ1)−1e−2(ν−µ)λ1τZt τ e2(ν−µ)λ1s|f(s)|2ds +(µλ1)−1e2Lgt−2(ν−µ)λ1τZt τ e2((ν−µ)λ1−Lg)s|f(s)|2ds.(10) Proof. Take a µsuch that 0 <µ<ν. By the energy equality (see Remark 1), one has d dt|u(t)|2+ 2νku(t)k2= 2(f(t), u(t)) + 2(g(t, ut), u(t)) ≤2λ−1/2 1|f(t)|ku(t)k+ 2Lgkutkγ|u(t)| ≤µku(t)k2+1 µλ1 |f(t)|2+ 2Lgkutk2 γ,a.e. t > τ. Thus, d dt|u(t)|2+ 2(ν−µ)λ1|u(t)|2+µku(t)k2≤d dt|u(t)|2+ 2(ν−µ)ku(t)k2+µku(t)k2 ≤1 µλ1 |f(t)|2+ 2Lgkutk2 γ,a.e. t > τ, and therefore, |u(t)|2+µZt τ e−2(ν−µ)λ1(t−s)ku(s)k2ds (11) ≤e−2(ν−µ)λ1(t−τ)|u(τ)|2+Zt τ e−2(ν−µ)λ1(t−s)(|f(s)|2/(µλ1)+2Lgkusk2 γ)ds, for all t≥τ. Consequently kutk2 γ≤max sup θ∈(−∞,τ−t] e2γθ|φ(θ+t−τ)|2, sup θ∈[τ−t,0] (e2γθ−2(ν−µ)λ1(t−τ+θ)|u(τ)|2 +e2γθ Zt+θ τ e−2(ν−µ)λ1(t+θ−s)(|f(s)|2/(µλ1)+2Lgkusk2 γ)ds). 16 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN AND J. REAL But observe that sup s≤τn−t0 e2γs|φn(s+t0−τn)|2= sup s≤τn−t0 e2γ(s+t0−τn)e2γ(τn−t0)|φn(s+t0−τn)|2 =e2γ(τn−t0)kφnk2 γ ≤e2γ(τn−t0)ρ2(τn) ≤e2γ(τn−t0)+Me(2γ−σ)(τn−t0)≤ε/4, thanks to the choice of nε. Finally, by (17) with T=Tε,we also have sup s∈[τn−t0,−Tε] e2γs|un(s+t0)|2= sup θ∈[τn−t0+Tε,0] e2γ(θ−Tε)|un(t0−Tε+θ)|2 ≤e−2γTεkun t0−Tεk2 γ ≤e−2γTεR(t0, Tε) =e−2γTε+Me(σ−2γ)Tε ≤ε/4. The proof is completed. Joining all the above statements we obtain the existence of minimal pullback attractors for the process Uon Cγ(H) associated to problem (1). Theorem 3. Assume that f∈L2 loc(R; (L2(Ω))3), γ > 0, and g:R×Cγ(H)→ (L2(Ω))3satisfying the assumptions (g1)–(g3), (12)and (13), are given. Then, there exist the minimal pullback DF(Cγ(H))-attractor ADF(Cγ(H)) ={ADF(Cγ(H))(t) : t∈R} and the minimal pullback Dσ((Cγ(H)))-attractor ADσ(Cγ(H)) ={ADσ(Cγ(H))(t) : t∈R}, for the process Udefined by (7). The family ADσ(Cγ(H)) belongs to Dσ(Cγ(H)), and the following relation holds: ADF(Cγ(H))(t)⊂ ADσ(Cγ(H))(t)⊂BCγ(H)(0, ρ(t)) for all t∈R, where ρ(t)is the expression given in (16). Proof. The result is a direct consequence of Theorem 2, Remark 3, Corollary 1, Proposition 2, Corollary 2, and Proposition 3. As a consequence of Theorem 3, we have the following result, which just discusses about the existence of a bigger tempered universe and bigger corresponding pullback attractor, if condition (12) is not optimized. Corollary 3. Under the assumptions of Theorem 3, if µ∈(0, ν)satisfies (ν− µ)λ1< γ, then there exists ˜µ∈(0, µ)such that it still fulfills (ν−˜µ)λ1≤γ, and therefore there exists the minimal pullback D˜σ((Cγ(H)))-attractor AD˜σ(Cγ(H)) ={AD˜σ(Cγ(H))(t) : t∈R}, for the process Udefined by (7), where ˜σ= 2((ν−˜µ)λ1−Lg),and the universe D˜σ(Cγ(H)) is defined analogously as in Definition 7, but with parameter ˜σ. The family AD˜σ(Cγ(H)) belongs to D˜σ(Cγ(H)), and the following relation holds: ADσ(Cγ(H))(t)⊂ AD˜σ(Cγ(H))(t)⊂BCγ(H)(0,˜ρ(t)) for all t∈R,(33) PULLBACK ATTRACTORS FOR GMNSE WITH INFINITE DELAYS 17 where ˜ρ2(t) = 1 + (˜µλ1)−1Zt −∞ e−˜σ(t−s)|f(s)|2ds. Proof. The result is a consequence of Theorem 3, provided that fand µsatisfying (12) and (13) imply that fand ˜µalso satisfy the analogous conditions. Moreover, observe that in this case any family of Dσ(Cγ(H)) also belongs to D˜σ(Cγ(H)),whence the first inclusion in (33) follows. The second inclusion is again a consequence of Theorem 3with parameter ˜µ. Remark 8. i) Under the assumptions of Corollary 3, i.e., with µsatisfying (ν−µ)λ1< γ, there are two possibilities. a) If νλ1> γ, the optimal value (cf. Remark 5) in order to obtain the maximal family of minimal pullback attractor in the biggest universe is ˜µ=ν−γλ−1 1∈(0, ν). b) If νλ1≤γ, then for any arbitrarily small value ˜µthe conditions (12)and (13) with such a value are satisfied. 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