Attractors for 2D-Navier-Stokes models with delays

Attractors for 2D-Navier-Stokes models with delays T. Caraballo ∗and J. Real Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain Abstract The existence of an attractor for a 2D-Navier-Stokes system with delay is proved. The theory of pullback attractors is successfully applied to obtain the results since the abstract functional framework considered turns out to be nonautonomous. However, on some occasions, the attractors may attract not only in the pullback sense but in the forward one as well. Also, this formulation allows to treat, in a unified way, terms containing various classes of delay features (constant, variable, distributed delays, etc.). As a consequence, some results for the autonomous model are deduced as particular cases of our general formulation. Key words: 2D-Navier-Stokes equations, pullback attractor, forward attractor, variable delay, distributed delay 1991 MSC: 35R10, 35B40, 47H20, 58F39, 73K70 1 Introduction Navier-Stokes equations have received very much attention over the last decades due to their importance in the understanding of fluids motion and turbulence (see [1], [10], [12], [15], [18], [26], amongst others). Very recently, in [7],[8] we started an investigation involving Navier-Stokes models in which the forcing term contains some hereditary features. These situations may appear, for instance, when we want to control the system by applying a force which takes ?Partly supported by Ministerio de Ciencia y Tecnolog´ıa (Spain) and FEDER (European Community), projects HA2001-0075 and BFM2002-03068. ∗Corresponding author Email addresses: [email protected] (T. Caraballo), [email protected] (J. Real). Article to be published in J. Differential Equations 28 October 2003 into account not only the present state of the system but the history of the solutions. No doubt at all, the asymptotic behaviour of dynamical systems is an interesting and challenging problem, since it can provide useful information on the future evolution of the system. This will be the main aim of this paper. To this respect, some sufficient conditions ensuring the exponential behaviour of solutions to a 2D-Navier-Stokes delay model were proved in [8]. Roughly speaking, when the viscosity is large, there exists a unique stationary solution to some models and this solution is exponentially stable (which means that the global attractor for these situations becomes the unique stationary solution). However, when the viscosity is small it is expected something similar to what happens in the non-delay framework, i.e., the existence of a compact invariant attracting set (a global attractor for the associated semigroup). But on this occasion, we need to be careful with our analysis since we have to consider the semigroup in a different phase space. In fact, the dynamical system needs to be defined in a phase space of trajectories (for a similar approach for nondelay models see [20]). To be more precise, our intention is to consider an abstract functional model for the delay so that a wide range of hereditary characteristics (constant or variable delay, distributed delay, etc) can be treated in a unified way. Although for some particular cases, the resulting abstract equation becomes autonomous (e.g. for constant delays) and the standard technique for autonomous dynamical systems can be adapted to solve the problem, most cases need of a nonautonomous model to describe the system and, consequently, a nonautonomous technique is necessary to handle the problem. Being possible various options to deal with the problem of attractors for nonautonomous systems (kernel sections [10], skew-product formalism [24], etc.), for our particular situation we have preferred to choose that of pullback attractor (see [9], [16], [17], [23]) which has also proved extremely fruitful, particularly in the case of random dynamical systems (see [13], [14], [23]). The main reason is that, although when one knows the explicit dependence of the delay (e.g. as in the cases of variable or distributed delays) it could be possible to construct the parameters set which is needed to have a skew-product flow (or the symbols set in the theory of kernel sections), it is not known how to construct them when one is trying to develop a general theory concerning abstract delay terms, i.e. under a general functional formulation (see [6] for more details). It is also worth pointing out that, after proving our theory for the nonautonomous delay model, we will obtain similar results for an autonomous version in a straightforward way. As far as we know, not many papers have been published dealing with the existence of attractors for partial differential equations with delay. We would like to mention that, for instance, a linear partial differential equation containing a nonlinear autonomous term with finite delay is considered in [11], and a class of retarded partial differential equations of second order with respect to 2 the time variable is analyzed in [3]. However, we do not know any work concerning nonautonomous delay terms. Some results in the finite dimensional context can be found in [6], [5] (see also Mallet-Paret and Sell [21], [22] for some preliminary and interesting results on the structure of the attractors for ordinary differential delay systems). In Section 2, we will recall some preliminary results on the existence, uniqueness and regularity of solutions of our model as well as some results on the theory of pullback attractors. Section 3 is devoted to prove the existence of the attractor of our nonautonomous delay models. In fact, under suitable uniform assumptions we prove the existence of a pullback attractor. In addition, some applications are exhibited (variable and distributed delays), and we also point out how can be obtained corresponding results for the autonomous framework as a particular case of our general model. 2 Preliminaries In this section we will include some preliminaries on the existence and uniqueness of solutions to our problem and recall some facts from the theory of pullback attractors. 2.1 Existence and uniqueness of solutions The general formulation for our model is the following. Let Ω ⊂R2be an open bounded set with regular boundary Γ, and consider the following functional 2D−Navier-Stokes problem (for further details and notations see Lions [19] and Temam [25]):                                ∂u ∂t −ν∆u+ 2 X i=1 ui ∂u ∂xi =f− ∇p+g(t, ut) in (τ, +∞)×Ω, div u= 0 in (τ, +∞)×Ω, u= 0 on (τ, +∞)×Γ, u(τ, x) = u0(x), x ∈Ω, u(t, x) = φ(t−τ, x), t ∈(τ−h, τ)x∈Ω, where ν > 0 is the kinematic viscosity, uis the velocity field of the fluid, pthe pressure, τ∈Rthe initial time, u0the initial velocity field, fa nondelayed external force field, ganother external force with some hereditary characteris3 tics and φthe initial datum in the interval of time (−h, 0),where his a fixed positive number. To set our problem in the abstract framework, we consider the following usual abstract spaces: V=nu∈(C∞ 0(Ω))2: div u= 0o, H= the closure of Vin (L2(Ω))2with norm |·| ,and inner product (·,·) where for u, v ∈(L2(Ω))2, (u, v) = 2 X j=1 ZΩuj(x)vj(x)dx, V= the closure of Vin (H1 0(Ω))2with norm k·k ,and associated scalar product ((·,·)),where for u, v ∈(H1 0(Ω))2, ((u, v)) = 2 X i,j=1 ZΩ ∂uj ∂xi ∂vj ∂xi dx. It follows that V⊂H≡H0⊂V0,where the injections are dense and compact. Finally, we will use k·k∗for the norm in V0and h·,·i for the duality pairing between Vand V0. Now we define the trilinear form bon V×V×Vby b(u, v, w) = 2 X i,j=1 ZΩui ∂vj ∂xi wjdx∀u, v, w ∈V. Given T > τ and u: (τ−h, T)→(L2(Ω))2, for each t∈(τ, T) we denote by ut the function defined on (−h, 0) by the relation ut(s) = u(t+s), s ∈(−h, 0).We also denote CH=C0([−h, 0]; H), CV=C0([−h, 0]; V), L2 H=L2(−h, 0; H) and L2 V=L2(−h, 0; V). Now, we establish suitable hypotheses on the term containing the delay. Let g:R×CH→(L2(Ω))2satisfy the following assumptions: (I) ∀ξ∈CH,t∈R→g(t, ξ)∈(L2(Ω))2is measurable, (II) ∀t∈R,g(t, 0) = 0, (III) ∃Lg>0 s.t.∀t∈R,∀ξ, η ∈CH |g(t, ξ)−g(t, η)| ≤ Lgkξ−ηkCH, 4 (IV) ∃m0≥0, Cg>0: ∀m∈[0, m0], τ ≤t, u, v ∈C0([τ−h, t]; H) Zt τems |g(s, us)−g(s, vs)|2ds≤C2 gZt τ−hems |u(s)−v(s)|2ds. Observe that (I)-(III) imply that given u∈C0([τ−h, T]; H), the function gu:t∈[τ, T]→(L2(Ω))2defined by gu(t) = g(t, ut)∀t∈[τ, T], is measurable (see Bensoussan et al. [2]) and, in fact, belongs to L∞(τ, T; (L2(Ω))2). Then, thanks to (IV), the mapping G:u∈C0([τ−h, T]; H)→gu∈L2(τ, T; (L2(Ω))2) has a unique extension to a mapping e Gwhich is uniformly continuous from L2(τ−h, T;H) into L2(τ, T; (L2(Ω))2). From now on, we will denote g(t, ut) = e G(u)(t) for each u∈L2(τ−h, T;H), and thus, ∀t∈[τ, T],∀u, v ∈L2(τ− h, T;H),we will have Zt τ|g(s, us)−g(s, vs)|2 (L2(Ω))2ds≤C2 gZt τ−h|u(s)−v(s)|2ds. Assume now that u0∈H,φ∈L2 H,f∈L2 loc(R;V0), and g:R×CH→ (L2(Ω))2satisfies hypotheses (I)-(IV). For example, when the function gis defined by g(t, φ) = G(φ(−ρ(t)) for a suitable differentiable delay function ρand a Lipschitz continuous mapping G:R2→R2, the assumptions above hold (see Caraballo & Real [7] for more details and examples). Set A:V→ V0as hAu, vi= ((u, v)), B :V×V→V0by hB(u, v), wi=b(u, v, w), ∀u, v, w ∈V, and B(u) = B(u, u).Denoting D(A)=(H2(Ω))2∩V, then Au =−P∆u, ∀u∈D(A),(Pthe ortho-projector from (L2(Ω))2onto H). For each τ∈Rwe consider the problem:              To find u∈L2(τ−h, T;H)∩L2(τ, T;V)∩L∞(τ, T;H)∀T > τ, d dtu(t) + νAu(t) + B(u(t)) = f(t) + g(t, ut) in D0(τ, +∞;V0), u(τ) = u0, u(t) = φ(t−τ), t ∈(τ−h, τ), (1) The following result can be proved as Theorem 2.3 in Caraballo & Real [8]. Theorem 1 Let us consider u0∈H,φ∈L2 H,f∈L2 loc(R;V0), and assume that g:R×CH→(L2(Ω))2satisfies hypotheses (I)-(IV). Then, for each τ∈R, a) There exists a unique solution to (1) which, in addition, belongs to the space C0([τ, +∞); H). b) If f∈L2 loc(R; (L2(Ω))2)and u0∈V, then the solution uto (1) is a strong 5 solution, that is, u∈L2(τ, T;D(A)) ∩C0([τ, T]; V)and u0∈L2(τ, T;H)∀T > τ. (2) In particular, if φ∈CVand u0=φ(0),then u∈C0([τ−h, +∞); V). 2.2 Preliminaries on pullback attractors We now discuss the theory of pullback attractors, as developed in Kloeden and Stonier [16], Kloeden and Schmalfuss [17], and Crauel et al. [14]. As it is well known, in the case of nonautonomous differential equations the initial time is just as important as the final time, and the classical semigroup property of autonomous dynamical systems is no longer available. Instead of a family of one time-dependent maps S(t) we need to use a twoparameter process U(t, τ) on the complete metric space X(which in our case will be CHor H×L2 H) (cf. Sell [24]); U(t, τ)ψuses to denote the value of the solution at time twhich was equal to the initial value ψat time τ. The semigroup property is replaced by the process composition property U(t, τ)U(τ, r) = U(t, r) for all t≥τ≥r, and, obviously, the initial condition implies U(τ, τ) =Id. As with the semigroup composition S(t)S(τ) = S(t+τ), this just expresses the uniqueness of solutions. It is also possible to present the theory within the more general framework of cocycle dynamical systems. In this case the second component of Uis viewed as an element of some parameter space J, so that the solution can be written as U(t, p)φ, and a shift map θt:J→Jis defined so that the process composition becomes the cocycle property, U(t+τ, p) = U(t, θτp)U(τ, p). However, when one tries to develop a theory which can include several kinds of hereditary characteristics under a unified abstract formulation, what means that we do not know a priori the explicit expression of the delay appearing in the problem, the context of cocycle (or skew-product flows) may not be the most appropriate to deal with the problem, since it is not known how to construct the set J(the same happens with the construction of the symbols set if one wishes to apply the theory of kernel sections as developed by Chepyzhov and Vishik [10]). For this reason, we do not pursue this approach here, but note that it has proved extremely fruitful, particularly in the case of random 6 dynamical systems. For various examples using this general setting, see Kloeden and Schmalfuss [17], or Sell [24]. For this reason, pullback attractors are often referred to as ‘cocycle attractors’. As in the standard theory of attractors, we seek an invariant attracting set. However, since the equation is nonautonomous this set also depends on time. Definition 2 Let Ube a process on a complete metric space X. A family of compact sets {A(t)}t∈Ris said to be a (global) pullback attractor for Uif, for all τ∈R, it satisfies i) U(t, τ)A(τ) = A(t)for all t≥τ, and ii) lims→∞ dist(U(t, t −s)D, A(t)) = 0, for all bounded subsets Dof X. The pullback attractor is said to be uniform if the attraction property is uniform in time, i.e. lim s→∞ sup t∈R dist(U(t, t −s)D, A(t)) = 0,for all bounded subsets D⊂X. Definition 3 A family of compact sets {A(t)}t∈Ris said to be a (global) forward attractor for Uif, for all τ∈R, it satisfies i) U(t, τ)A(τ) = A(t)for all t≥τ, and ii) limt→∞ dist(U(t, τ)D, A(t)) = 0, for all bounded subsets Dof X. The forward attractor is said to be uniform if the attraction property is uniform in time, i.e. lim t→∞ sup τ∈R dist(U(t+τ, τ)D, A(t+τ)) = 0,for all bounded subsets D⊂X. The reader is referred to Cheban et al. [9] for a detailed analysis on the relationship between these concepts. We emphasize that the property of uniform pullback attraction is equivalent to that of uniform forward attraction. In the definition, dist(A, B) is the Hausdorff semidistance between Aand B, defined as dist(A, B) = sup a∈A inf b∈Bd(a, b),for A, B ⊆X. Property i) is a generalization of the invariance property for autonomous dynamical systems. The pullback attracting property ii) considers the state of the system at time twhen the initial time t−sgoes to −∞ (see also Chepyzhov and Vishik [10]) The notion of an attractor is closely related to that of an absorbing set. Definition 4 The family {B(t)}t∈Ris said to be (pullback) absorbing with respect to the process Uif, for all t∈Rand all D⊂Xbounded, there exists 7 TD(t)>0such that for all s≥TD(t) U(t, t −s)D⊂B(t). The absorption is said to be uniform if TD(t)does not depend on the time variable t. Indeed, just as in the autonomous case, the existence of compact absorbing sets is the crucial property in order to obtain pullback attractors. For the following result see Crauel and Flandoli [13] or Schmalfuss [23]. Theorem 5 Let U(t, τ)be a two-parameter process, and suppose U(t, τ) : X→Xis continuous for all t≥τ. If there exists a family of compact (pullback) absorbing sets {B(t)}t∈R, then there exists a pullback attractor {A(t)}t∈R, and A(t)⊂B(t)for all t∈R. Furthermore, A(t) = [ D⊂X bounded ΛD(t), where ΛD(t) = \ n∈N[ s≥n U(t, t −s)D. Remark 6 It is worth mentioning that the uniqueness of the pullback attractor, as defined above, does not hold in general (see Caraballo and Langa [4]). However, the one given in the preceding theorem is minimal with respect to set inclusion (see Crauel and Flandoli [13]). But, if we impose in the definition of pullback attractor that the family {A(t)}t∈Ris uniformly bounded (i.e. there exists a bounded set B⊂Xsuch that A(t)⊂Bfor all t∈R) or we are interested in finding uniformly bounded attractors, then the uniqueness of this attractor follows immediately. A sufficient condition ensuring this is that the family of compact absorbing sets in Theorem 5 is also uniformly bounded. Finally, there exists another possibility to ensure the uniqueness of the pullback attractor which is related to the fact that the attractor is asked to belong to a certain class of set valued functions which are attracted by the attractor (see [9]). 3 Existence of the attractor We denote by λ1the first eigenvalue of the operator A. 8 3.1 Construction of the associated process Now we will apply the theory in the previous section to prove the existence of an attractor for our nonautonomous Navier-Stokes model with delay. To this end, we consider g:R×CH→(L2(Ω))2satisfying (I)-(IV) and assume that u0∈H,φ∈L2 Hand f∈L2 loc(R;V0). Then, for each initial time τ∈R, Theorem 1 ensures that problem (1) possesses a unique solution u(·;τ, (u0, φ)) which belongs to the space L2(τ, T;V)∩L2(τ−h, τ;H)∩C0([τ, T]; H) for all T > τ. We can now proceed in two different forms to construct the evolution process which can help us in the analysis of the long-time behaviour of our model. On the one hand, we can define a process in the phase space CHas the family of mappings U(t, τ) : CH→CHgiven by U(t, τ)φ=ut(·;τ, (φ(0), φ)),for any φ∈CH,and any τ≤t. (3) However, it may seem that the product space M2 H=H×L2 Hcan be more convenient since this is the usual space where the initial data are taken. This space is a Hilbert space with associated norm k(u0, φ)k2 M2 H=|u0|2+Z0 −h|φ(s)|2ds, for (u0, φ)∈M2 H. In this way, we can define the corresponding process as S(t, τ)(u0, φ) = (u(t;τ, (u0, φ)), ut(·;τ, (u0, φ))),for (u0, φ)∈M2 H,τ≤t. (4) Although, due to the continuity of trajectories, it seems sensible to consider only the first case, with a little more of additional work we will be able to handle both situations at the same time. Of course, it is sensible to expect that the attractors for both situations should be related. We will prove that this is indeed the case. Remark 7 Associated to the processes U(·,·)and S(·,·)we will consider the family of mappings ˜ U(·,·) : M2 H→L2 Hdefined as ˜ U(t, τ)(u0, φ) = ut(·;τ, (u0, φ)),for (u0, φ)∈M2 H,and τ≤t. (5) Observe that U(t, τ)φ=˜ U(t, τ)(φ(0), φ)for any t≥τ, and any φ∈CH.(6) In this way, we then have that the process S(t, τ)can be rewritten as S(t, τ)(u0, φ) = (u(t;τ, (u0, φ)),˜ U(t, τ)(u0, φ)).(7) These facts will allow us to prove the estimates for the processes Uand Sin a straightforward way by using the previously obtained ones for the process ˜ U. 9 If we now consider the time t−sinstead of τ(i.e. u(·) denotes now u(·;t−s, (u0, φ)) , so that we can use more easily the definition of absorbing sets) we have ° ° °˜ U(t, t −s)(u0, φ)° ° °CH =kutk2 CH≤|f|2 mσ +˜ d2emh (1 + Cg)e−ms for all t, and s≥h. and denoting by ˜ρ2=|f|2 mσ and ˜ρ2 H= 2˜ρ2,it easily follows that there exists e T˜ D(t)(= e T˜ D)≥hsuch that for all s≥e T˜ D(t) and all (u0, φ)∈M2 H, it holds ° ° °˜ U(t, t −s)(u0, φ)° ° °CH ≤˜ρH,which means that the balls B(t) = BCH(0,˜ρH) form an absorbing family of bounded sets for the mappings ˜ U(t, τ). Corollary 13 Under the assumptions in Theorem 12, there exists a family {B(t)}t∈Rof bounded absorbing sets in CHfor the process U, which is given by B(t) = B1=BCH(0,˜ρH)for all t∈R. Moreover, the family {B(t)}t∈Rgiven by B(t)=BH(0,˜ρH)×BL2 H(0, h1/2˜ρH)⊂M2 Hfor all t∈Ris absorbing for the process S. PROOF. The first part follows from the previous Theorem 12 and Lemma 11. As for the second, observe that {j(B(t))}t∈Ris a family of bounded absorbing sets for S(·,·).On the other hand, as kφk2 L2 H≤hkφk2 CHand j(B(t)) = {(φ(0), φ) : φ∈BCH(0,˜ρH)}, it follows that j(B(t)) ⊂BH(0,˜ρH)×BL2 H(0, h1/2˜ρH) = B(t), what implies that the family {B(t)}t∈Ris absorbing for the process S(·,·). Remark 14 If we assume that f∈V0, the previous results also hold true by modifying slightly the proofs and substituting |f|by kfk∗. 3.3 Existence of an absorbing family of sets in CV We now prove the existence of an absorbing family of sets in CVand a necessary bound on the term Rt+θ2 t+θ1|Au(s)|2ds. We proceed in a similar way as we have already done in the previous subsection. Theorem 15 Under the assumptions in Theorem 12, there exist positive constants ˜ρV,e β1,e β2such that for any bounded set ˜ D⊂M2 Hand for e T˜ Dthe ab16 sorbing time corresponding to the set B1in Theorem 12, it follows ° ° °˜ U(t, t −s)(u0, φ)° ° °2 CV = max θ∈[−h,0] ku(t+θ;t−s, (u0, φ))k2≤˜ρ2 V, Zt+θ2 t+θ1 |Au(σ;t−s, (u0, φ))|2dσ ≤e β1|θ2−θ1|+e β2, for all s≥e T˜ D+1+h, t ∈R,(u0, φ)∈˜ D, and θ1, θ2∈[−h, 0]. PROOF. As in the proof of Theorem 12, let ˜ D⊂M2 Hbe a bounded set, i.e. there exists ˜ d > 0 such that k(u0, φ)kM2 H≤˜ dfor all (u0, φ)∈˜ D. Denote u(·) = u(·;t0−s, (u0, φ)) for (u0, φ)∈˜ D, where t0∈Ris a fixed number, and let us take s≥e T˜ D,where we have chosen the same σand mthan in that proof. We can then integrate in (11) between tand t+1 for t≥t0and s≥e T˜ D. We obtain |u(t+ 1)|2− |u(t)|2+³2ν−(σ+Cg)λ−1 1´Zt+1 tku(r)k2dr ≤|f|2 σ+1 CgZt+1 t|g(r, ur)|2dr ≤|f|2 σ+1 Cg·C2 gZt+1 t−h|u(r)|2dr¸ ≤|f|2 σ+CgZt t−h|u(r)|2dr+CgZt+1 t|u(r)|2dr ≤|f|2 σ+CgZt t−h|u(r)|2dr+Cgλ−1 1Zt+1 tku(r)k2dr, and ³2ν−(σ+ 2Cg)λ−1 1´Zt+1 tku(r)k2dr≤|f|2 σ+CgZt t−h|u(r)|2dr+|u(t)|2 ≤|f|2 σ+CgZt t−hkurk2 CHdr+ ˜ρ2 H ≤|f|2 σ+ (1 + hCg) ˜ρ2 H. Therefore, Zt+1 tku(r)k2dr≤e IV,∀t≥t0,(15) where e IV=1 2ν−(σ+ 2Cg)λ−1 1Ã|f|2 σ+ (1 + hCg) ˜ρ2 H!. On the other hand, we take the inner product with Au and obtain for r≥t0 1 2 d drkuk2+ν|Au|2+b(u, u, Au)≤(f, Au)+(g(r, ur), Au).(16) 17 Now we evaluate the terms. First, notice that |(f, Au)|+|(g(r, ur), Au)| ≤ |Au|(|f|+|g(r, ur)|) ≤ν 4|Au|2+2 ν³|f|2+|g(r, ur)|2´.(17) Next, |b(u, u, Au)| ≤ c1|u|1/2kuk |Au|3/2(18) ≤ν 4|Au|2+c0 1 ν3|u|2kuk4. Thanks to (17)-(18), and the fact that kϕk ≤ λ−1 1|Aϕ|for ϕ∈D(A), we can deduce from Eq. (16) d drkuk2+ν|Au|2≤ν 4³|f|2+L2 gkurk2 CH´+2c0 1 ν3|u|2kuk4,(19) and d drkuk2+νλ1kuk2≤ν 4³|f|2+L2 gkurk2 CH´+2c0 1 ν3|u|2kuk4 ≤ν 4³|f|2+L2 g˜ρ2 H´+2c0 1 ν3|u|2kuk4. Now, we can apply the uniform Gronwall lemma for s≥e T˜ D(see Temam [26]). Then, ku(r)k2≤(a3+a2)ea1,for all r≥t0+ 1, provided s≥e T˜ D, where a3=e IV a2=ν 4³|f|2+L2 g˜ρ2 H´ a1=2c0 1 ν3˜ρ2 He IV, and, consequently, if we take s≥e T˜ D+1+h, sup θ∈[−h,0] ku(t0+θ)k2≤(a3+a2)ea1= ˜ρ2 V, (20) where the constants appearing in (20) are independent of the fixed time t0∈R. So, (20) holds true for all t0∈R. Denoting from now on u(·) = u(·;t−s, (u0, φ)), and, taking into account that part b) in Theorem 1 ensures that ut(·)∈CV for s > h, we indeed have kutkCV≤˜ρV,for all t∈R, provided s≥e T˜ D+1+h. 18 Finally we will obtain the bound on the term Rt+θ2 t+θ1|Au(r)|2dr. Indeed, from (19) it follows |Au|2≤α1+α2|u|2kuk4−1 ν d dr kuk2. If we choose s≥e T˜ D+1+hand θ1, θ2∈[−h, 0] with e.g. θ2> θ1,we have Zt+θ2 t+θ1 |Au(r)|2dr≤α1|θ2−θ1|+α2Zt+θ2 t+θ1 |u(r)|2ku(r)k4dr −1 νku(t+θ2)k2+1 νku(t+θ1)k2 ≤³α1+α2˜ρ2 H˜ρ4 V´|θ2−θ1|+1 ν˜ρ2 V, as desired. Corollary 16 Under the assumptions in Theorem 12, there exist positive constants ρV, β1, β2such that for any bounded set D⊂CHand for TD=˜ Tj(D) with ˜ Tj(D)the absorbing time corresponding to the set B1in Theorem 12, it follows kU(t, t −s)φk2 CV=kut(·;t−s, j(φ))k2 CV= max θ∈[−h,0] ku(t+θ;t−s, j(φ))k2≤ρ2 V, Zt+θ2 t+θ1 |Au(σ;t−s, j(φ))|2dσ≤β1|θ2−θ1|+β2, for all s≥TD+1+h, t ∈R, φ ∈D, and θ1, θ2∈[−h, 0].In particular, the family {B2(t)}t∈R,where B2(t) = B2=BCV(0, ρV),is absorbing for the process U(·,·). Moreover, the family {BS(t)}t∈R, where BS(t) = BCV(0, ρV)×BL2 V(0, h1/2ρV), is absorbing for S(·,·). PROOF. The proof follows the same lines as those of Corollary 13. 3.4 Existence of the pullback attractors Now we can prove the following result. Theorem 17 Under the assumptions in Theorem 12, there exist a unique uniformly bounded pullback attractor {ACH(t)}t∈Rfor the process U(·,·)in CH, and a unique uniformly bounded pullback attractor {AM2 H(t)}t∈Rfor S(·,·)in M2 H. Futhermore, AM2 H(t)⊂H×CHfor all t∈Rand both attractors are related by means of AM2 H(t) = j(ACH(t)) ,for all t∈R. 19 PROOF. Let us consider the family {B2(t)}t∈R,where B2(t) = B2=BCV(0; ρV) for all t∈R. This is a family of bounded sets in CV, which is also (uniformly) absorbing for ˜ U(·,·).Take now ˜ B2=j(B2).Then, using the previous notation, there exists ˜ T0 ˜ B2=TB2+1+h > 0 such that ˜ U(t, t −s)˜ B2⊂B2,for all t∈R, and all s≥˜ T0 ˜ B2. Now, for each t∈R, consider the set B3(t) = [ s≥˜ T0 ˜ B2 ˜ U(t, t −s)˜ B2⊂B2⊂CV. Thus, {B3(t)}t∈Ris a family of uniformly bounded sets in CVwhich is (uniformly) absorbing for ˜ U(·,·). If we prove that each B3(t) is relatively compact in CH,then {B3(t)}t∈R(where the closure is taken in CH) is a family of compact absorbing set in CHfor ˜ U(·,·).Consequently, it is also a family of compact (uniform) absorbing sets for the process U(·,·) in CH,and {j³B3(t)´}t∈Ris another family of compact (uniform) absorbing sets for S(·,·) in M2 H,what ensures the existence of the pullback attractors for the processes. The uniqueness of these attractors holds since they are uniformly bounded (see Remark 6). Let us now prove this compactness property. To this end, we will use the Ascoli-Arzel`a theorem, in other words, we have to check (A) The set [ s≥˜ T0 ˜ B2 ˜ U(t, t−s)˜ B2is equicontinuous (i.e. ∀ε > 0,∃δ > 0 such that if |θ1−θ2| ≤ δ, then ¯¯¯˜ U(t, t −s) (j(φ)) (θ1)−˜ U(t, t −s) (j(φ)) (θ2)¯¯¯≤ε, ∀t∈ R,s≥˜ T0 ˜ B2,∀φ∈B2.) (B) For each θ∈[−h, 0], [ s≥˜ T0 ˜ B2[ φ∈B2 ˜ U(t, t −s) (j(φ)) (θ) is a compact set in H. To prove (B) we need to check that, for any fixed θ∈[−h, 0] and t∈R, the set nu(t+θ;t−s, j (φ)) : s≥˜ T0 ˜ B2, φ ∈B2o is relatively compact. But this holds since this set is bounded in V(see Theorem 15) and the injection V⊂His compact. 20 Finally, in order to prove (A) we proceed by estimating ¯¯¯˜ U(t, t −s) (j(φ)) (θ1)−˜ U(t, t −s) (j(φ)) (θ2)¯¯¯ =|u(t+θ1;t−s, j(φ)) −u(t+θ2;t−s, j(φ))| for t∈R,θ1, θ2∈[−h, 0], s ≥˜ T0 ˜ B2and φ∈B2.Then we obtain (denoting for simplicity u(·;t−s, j(φ)) by u(·) and assuming θ2> θ1) |u(t+θ1)−u(t+θ2)|=¯¯¯¯¯Zt+θ2 t+θ1 u0(r)dr¯¯¯¯¯ ≤Zt+θ2 t+θ1 |u0(r)|dr ≤Zt+θ2 t+θ1 (ν|Au(r)|+|B(u(r))|+|f|+|g(r, ur)|) dr ≤ |f| |θ1−θ2| +Zt+θ2 t+θ1³ν|Au(r)|+c1|Au(r)| ku(r)k+LgkurkCH´dr ≤ |f| |θ1−θ2| +Zt+θ2 t+θ1³(ν+c1ku(r)k)|Au(r)|+LgkurkCH´dr, (21) and, consequently, for t∈R,s≥˜ T0 ˜ B2 |u(t+θ1)−u(t+θ2)| ≤ |f| |θ1−θ2| +Zt+θ2 t+θ1³(ν+c1ku(r)k)|Au(r)|+LgkurkCH´dr ≤(|f|+ρHLg)|θ1−θ2|+Zt+θ2 t+θ1 (ν+c1ρV)|Au(r)|dr ≤(|f|+ρHLg)|θ1−θ2| + (ν+c1ρV)|θ1−θ2|1/2Zt+θ2 t+θ1 |Au(r)|2dr ≤(|f|+ρHLg)|θ1−θ2| + (ν+c1ρV) (β1|θ1−θ2|+β2)|θ1−θ2|1/2, which implies the needed equicontinuity. Finally, we will prove the interesting relationship that there exists between the attractors {ACH(t)}t∈Rand {AM2 H(t)}t∈R. Observe that from the properties of the mapping j(·), the results in Lemma 11 and Corollary 16, it is straightforward to check that {j(ACH(t))}t∈Ris a uniformly bounded family of compact sets in M2 Hwhich is pullback attracting for the process S(·,·), and it is also invariant. Taking into account the uniqueness of the uniformly 21 bounded attractors, it follows immediately that AM2 H(t) = j(ACH(t)) for all t∈R. The proof is now complete. Remark 18 As we have already mentioned, our analysis can be extended to deal with more general nonautonomous fand g. The technique we have used in the previous subsections can be performed to treat this case in a straightforward way, although with additional difficulties in the computations. For instance, if we assume that f∈L2 loc(R;L2(Ω)2),and satisfies Zt −∞ ems|f(s)|2ds < +∞,for all t∈R, and m > 0, then, under assumptions (I)-(IV) with m0>0,and νλ1> Cg, it is not difficult to check that there exists a family {B(t)}t∈Rof bounded absorbing sets in CHfor ˜ U(·,·). To be more precise, B(t) = BCH(0, ρH(t)) where ρ2 H(t) = 2emhe−mt Rt −∞ ems|f(s)|2ds, for a positive but small enough m. Under these assumptions, we can then prove similarly the existence of the nonautonomous absorbing family in CV,and conclude with the existence of the pullback attractor. We leave the details to the reader. 3.5 An application: a forcing term with variable delay Consider that operator gis given by g(t, ut) = G(u(t−ρ(t))), with G:R2→R2a function satisfying G(0) = 0 and such that there exists L1>0 for which |G(u)−G(v)|R2≤L1|u−v|R2,∀u, v ∈R2, and ρ∈C1(R), ρ(t)≥0 for all t∈R,h= supt∈Rρ(t)∈(0,+∞) and ρ∗= supt∈Rρ0(t)<1. This situation is within our framework and satisfies our assumptions (Conditions (I)-(IV)) ensuring the existence and uniqueness of solutions (see Caraballo & Real [7]). Moreover, (IV) is fulfilled by setting 22 C2 g=L2 1em0h/(1 −ρ∗) for any m0>0.Indeed, it follows for t≥τ Zt τems|g(s, us)−g(s, vs)|2ds=Zt τems|G(u(s−ρ(s))) −G(v(s−ρ(s)))|2ds ≤L2 1Zt τems|u(s−ρ(s)) −v(s−ρ(s))|2ds ≤L2 1emh 1−ρ∗Zt−ρ(t) τ−ρ(τ)emσ|u(σ)−v(σ)|2dσ ≤L2 1em0h 1−ρ∗Zt τ−hems|u(s)−v(s)|2ds, m ∈[0, m0). Observe that if νλ1> L1/(1 −ρ∗)1/2,our result ensures the existence of a pullback attractor ACH(t)⊂CHfor the process U(·,·) (and also another pullback attractor AM2 H(t) for S(·,·)).Indeed, we only need to check that νλ1> Cg=L1em0h/2/(1 −ρ∗)1/2.But, if νλ1> L1/(1 −ρ∗)1/2,then for a sufficiently small but positive m0, we have that νλ1> L1em0h/2/(1 −ρ∗)1/2. Notice that the analysis done in Caraballo and Real [8] ensures that if the viscosity νis larger, i.e., if for instance, for certain positive constants k1and k2(depending only on Ω), it holds that 2νλ1>(2 −ρ∗)L1 (1 −ρ∗)+k1|f| ν−λ−1 1L1 +k2|f|3 ν2(ν−λ−1 1L1)3, then, there exists a unique stationary solution u∞∈Vto our problem and every solution approaches this stationary solution exponentially fast. In other words, ACH(t) consists of this unique stationary solution. Notice that in the particular case ρ∗= 0 (which means that the delay function ρis not increasing) we obtain an attractor for our model if νλ1> L1,and this attractor becomes a unique point if 2νλ1>2L1+k1|f| ν−λ−1 1L1 +k2|f|3 ν2(ν−λ−1 1L1)3. 3.6 Remarks on the autonomous case We are now interested in the following autonomous version of our problem              To find u∈L2(0, T;H)∩L2(−h, T;V)∩L∞(0, T;H)∀T > 0, s.t. d dtu(t) + νAu(t) + B(u(t)) = f+g(ut) in D0(0,+∞;V0), u(0) = u0, u(t) = φ(t), t ∈(−h, 0), (22) where f∈(L2(Ω))2,and g:CH→(L2(Ω))2satisfies (II), (III) and (IV) in Section 2. Owing to the fact that gdoes not explicitly depend on the time 23 variable t, these conditions can be rewritten as follows: (g1) g(0) = 0 (g2) there exists Lg>0 such that ∀ξ, η ∈CH |g(ξ)−g(η)| ≤ Lgkξ−ηkCH, (g3) ∃m0≥0, Cg>0: ∀m∈[0, m0],0≤t, u, v ∈C0([−h, t]; H) Zt 0ems|g(us)−g(vs)|2ds≤C2 gZt −hems|u(s)−v(s)|2ds. For each initial function φ∈CHand taking as initial value u0=φ(0),there exists a unique solution u(·;φ) to problem (22) such that u∈C0([−h, +∞); H). Then, for any t≥0 we can define an operator U0(t) : CH→CHas U0(t)φ=ut(·;φ). Bearing in mind the analysis done in the previous section, we can proceed only on the phase space CHsince the existence of an attractor in CHenables us to obtain another one in M2 H.In this sense, it is not difficult to prove, in a similar fashion as we have done in the preceding subsections, that this dynamical system U0(·) possesses a global attractor in CH.But, we note that, considering this problem as a nonautonomous one and setting U(t, τ) for its associated process, it holds that U(t, τ) = U(t−τ, 0),for all t≥τ, and, consequently, U0(t) = U(t, 0),for all t≥0, is a semigroup of nonlinear continuous operators. Now, our previously developed theory allows the reader to prove as an easy exercise the following result. Theorem 19 (Existence of global attractor) Assume that (g1),(g2) and (g3) hold with m0>0. If, in addition, νλ1> Cg,then there exists the global attractor ACH⊂CHfor the semigroup U0(t). Remark 20 Needless to say that a similar result can also be proved if we consider the semigroup S0(t) = S(t, 0). As an application, we will now consider an example in which the forcing term contains a distributed delay. Let G: [−h, 0] ×R2→R2be a measurable function satisfying G(s, 0) = 0 for all s∈[−h, 0] and assume that there exists a function γ∈L2(−h, 0) such 24 that |G(s, u)−G(s, v)|RN≤γ(s)|u−v|RN,∀u, v ∈RN∀s∈[−h, 0]. Then, we define g(ξ)(x) = R0 −hG(s, ξ(s)(x)) dsfor each ξ∈C0([0, T]; H) and x∈Ω. In this case, the delayed term gin our problem becomes g(ut) = Z0 −hG(s, u(t+s)) ds. It holds that gsatisfies the hypotheses in Theorem 19. Indeed, (g1) is evident. As for (g2), notice that, if ξ, η ∈CH, we obtain |g(ξ)−g(η)|2≤RΩ³R0 −h|G(s, ξ(s)(x)) −G(s, η(s)(x))|RNds´2dx ≤RΩ³R0 −hγ(s)|ξ(s)(x)−η(s)(x)|RNds´2dx ≤RΩkγk2 L2(−h,0) ³R0 −h|ξ(s)(x)−η(s)(x)|2 RNds´dx ≤hkγk2 L2(−h,0)kξ−ηk2 CH. Finally, if u, v ∈C0([−h, T]; H) then, for each t > 0, m0>0 and all m∈ [0, m0],it follows Zt 0emτ |g(uτ)−g(vτ)|2dτ ≤ kγk2 L2(−h,0) Zt 0emτ µZ0 −h|u(s+τ)−v(s+τ)|2ds¶dτ ≤ kγk2 L2(−h,0) Z0 −hµZt 0emτ |u(s+τ)−v(s+τ)|2dτ¶ds ≤ kγk2 L2(−h,0) Z0 −hµZt+s sem(r−s)|u(r)−v(r)|2dr¶ds ≤ kγk2 L2(−h,0) Z0 −he−ms µZt −hemr|u(r)−v(r)|2dr¶ds ≤ kγk2 L2(−h,0)hem0hZt −hemr|u(r)−v(r)|2dr. Consequently, Theorem 19 ensures the existence of the global attractor in CH provided νλ1>kγkL2(−h,0)h1/2em0h/2.But, we note that if νλ1>kγkL2(−h,0)h1/2, we can choose m0small enough such that νλ1>kγkL2(−h,0)h1/2em0h/2.It is also remarkable that when h→0 the sufficient condition ensuring the existence of the global attractor becomes νλ1>0, which is the usual one in the case without delays (and trivially fulfilled). 25