Pullback attractors for three-dimensional Navier-Stokes-Voigt equations with delays

  • Haiyan Li1 and

    Affiliated with

    • Yuming Qin2Email author

      Affiliated with

      Boundary Value Problems20132013:191

      DOI: 10.1186/1687-2770-2013-191

      Received: 25 April 2013

      Accepted: 6 August 2013

      Published: 27 August 2013

      Abstract

      Our aim in this paper is to study the existence of pullback attractors for the 3D Navier-Stokes-Voigt equations with delays. The forcing term g ( t , u ( t ρ ( t ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq1_HTML.gif containing the delay is sub-linear and continuous with respect to u. Since the solution of the model is not unique, which is caused by the continuity assumption, we establish the existence of pullback attractors for our problem by using the theory of multi-valued dynamical system.

      Keywords

      3D Navier-Stokes-Voigt equations pullback attractors delay terms multi-valued process

      1 Introduction

      Let Ω R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq2_HTML.gif be an open, bounded and connected set. We consider the following problem for three-dimensional Navier-Stokes-Voigt (NSV) equations with delays in continuous and sub-linear operators:
      { u t ν u α 2 u t + ( u ) u + p = g ( t , u ( t ρ ( t ) ) ) , in  ( τ , + ) × Ω , div u = 0 , in  ( τ , + ) × Ω , u ( x , t ) = 0 , on  ( τ , + ) × Ω , u ( τ , x ) = u 0 ( x ) , x Ω , u ( τ + t , x ) = φ ( t , x ) , t ( h , 0 ) , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ1_HTML.gif
      (1.1)

      Here u = ( u 1 ( t , x ) , u 2 ( t , x ) , u 3 ( t , x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq3_HTML.gif is the velocity vector field, ν is a positive constant, α is a characterizing parameter of the elasticity of the fluid, p is the pressure, g is the external force term which contains memory effects during a fixed interval of time of length h > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq4_HTML.gif, ρ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq5_HTML.gif is an adequate given delay function, u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq6_HTML.gif is the initial velocity field at the initial time τ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq7_HTML.gif, φ is the initial datum on the interval ( h , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq8_HTML.gif.

      Equation (1.1) with α = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq9_HTML.gif becomes the classical three-dimensional Navier-Stokes (NS) equation. In the past decades, many authors [16] investigated intensively the classical three-dimensional incompressible NS equation. For the sake of direct numerical simulations for NS equations, the NSV model of viscoelastic incompressible fluid has been proposed as a regularization of NS equations.

      Equation (1.1) governs the motion of a Klein-Voigt viscoelastic incompressible fluid. Oskolkov [7] was the first to introduce the system which gives an approximate description of the Kelvin-Voigt fluid (see, e.g., [8, 9]). In 2010, Levant et al. [10] investigated numerically the statistical properties of the Navier-Stokes-Voigt model. Kalantarov and Titi [11] studied a global attractor of a semigroup generated by equation (1.1) for the autonomous case. Recently, Luengo et al. [12] obtained asymptotic compactness by using the energy method, and they further got the existence of pullback attractors for three-dimensional non-autonomous NSV equations.

      Let us recall some related results in the literature. Yue and Zhong [13] studied the long time behavior of the three-dimensional NSV model of viscoelastic incompressible fluid for system (1.1) by using a useful decomposition method. The authors in [12, 14] deduced the existence of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq10_HTML.gif -pullback attractors for 3D non-autonomous NSV equations using the energy method. As we know from [15], delay terms appear naturally. In recent years, Caraballo and Real [1618] developed a fruitful theory of existence, uniqueness, stability of solutions and global attractors for Navier-Stokes models including some hereditary characteristics such as constant, variable delay, distributed delay, etc. However, our present problem has no uniqueness of solutions. To overcome the difficulty, we may cite the results by Ball [19] and by Marín-Rubio and Real [20]. Gal and Medjo [21] proved the existence of uniform global attractors for a Navier-Stokes-Voigt model with memory. As commented before, in comparison with three-dimensional Navier-Stokes equations, there is no regularizing effect. Our result of this paper is to establish the existence of pullback attractors for three-dimensional NSV equations in H 0 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq11_HTML.gif when the external forcing term g ( t , u ) H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq12_HTML.gif and the function g ( t , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq13_HTML.gif is continuous with respect to u. Another difficulty is to obtain that the multi-valued processes are asymptotically compact. In 2007, Kapustyan and Valero [22] presented a method suitable for verifying the asymptotic compactness. The authors [20] applied this method to 2D Navier-Stokes equations with delays in continuous and sub-linear operators. We shall apply the energy method to prove that our multi-valued processes are asymptotically compact by making some minor modifications caused by the term α 2 Δ u t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq14_HTML.gif in (1.1).

      This paper is organized as follows. In Section 2, we recall briefly some results on the abstract theory of pullback attractors. In Section 3, we introduce some abstract spaces necessary for the variational statement of the problem and give the proof of the global existence of solutions. In Section 4, we consider the asymptotic behavior of problem (1.1).

      2 Basic theory of pullback attractors

      By using the framework of evolution processes, thanks to [20, 23, 24], we now briefly recall some theories of pullback attractors and the related results. On the one hand, we have to overcome some difficulties caused by multi-valued processes. On the other hand, since our model is non-autonomous, we should use the related results for classical multi-valued processes in [23, 24], but which are not completely adapted to our model.

      Let ( X , d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq15_HTML.gif be a metric space, and let P ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq16_HTML.gif be the class of nonempty subsets of X. As usual, we denote by dist X ( B 1 , B 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq17_HTML.gif the Hausdorff semi-distance in X between B 1 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq18_HTML.gif and B 2 X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq19_HTML.gif, i.e.,
      dist X ( B 1 , B 2 ) = sup x B 1 inf y B 2 d X ( x , y ) for  B 1 , B 2 X , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equa_HTML.gif

      where d X ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq20_HTML.gif denotes the distance between two points x and y in X.

      We now formulate an abstract result in order to establish the existence of pullback attractors for the multi-valued dynamical system associated with (1.1).

      Definition 2.1 A multi-valued process U is a family of mappings U ( t , τ ) : X P ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq21_HTML.gif for any pair τ t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq22_HTML.gif of real numbers such that
      U ( t , τ ) x U ( t , r ) U ( r , τ ) x , x X , τ r t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equb_HTML.gif

      If the above relation is not only an inclusion but also an equality, we say that the multi-valued process is strict. For example, the relation generalized by 3D non-autonomous NSV equation (see, e.g., [12]) is an equality, while the relation generalized by 3D NS equations (see, e.g., [22]) is strict.

      Definition 2.2 Suppose that D ˆ 0 = { D 0 ( t ) } t R P ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq23_HTML.gif is a family of sets. A multi-valued process U is said to be D ˆ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq24_HTML.gif-asymptotically compact if for any t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq25_HTML.gif, any sequences { τ n } n = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq26_HTML.gif with τ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq27_HTML.gif, x n D 0 ( τ n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq28_HTML.gif, and ξ n U ( t , τ n ) x n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq29_HTML.gif, the sequence { ξ n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq30_HTML.gif is relatively compact in X.

      Lemma 2.1 If a multi-valued process U is D ˆ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq24_HTML.gif-asymptotically compact, then the sets Λ ( D ˆ 0 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq31_HTML.gif are nonempty compact subsets of X, where
      Λ ( D ˆ 0 , t ) = s t τ s U ( t , τ ) D 0 ( τ ) ¯ X , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equc_HTML.gif
      Furthermore, Λ ( D ˆ 0 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq31_HTML.gif attracts in a pullback sense to D ˆ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq24_HTML.gif at time t, i.e.,
      lim τ dist ( U ( t , τ ) D 0 ( τ ) , Λ ( D ˆ 0 , t ) ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equd_HTML.gif

      Indeed, it is the minimal closed set with this property.

      Definition 2.3 The family of subsets D ˆ 0 = { D 0 ( t ) } t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq32_HTML.gif is said to be pullback-absorbing with respect to a multi-valued process U if for every t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq25_HTML.gif and all bounded subset B of X, there exists a time τ ( t , B ) t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq33_HTML.gif such that
      U ( t , τ ) B D 0 ( t ) , τ τ ( t , B ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Eque_HTML.gif
      Lemma 2.2 Let the family of sets D ˆ 0 = { D 0 ( t ) } t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq34_HTML.gif be pullback-absorbing for the multi-valued process U, and let U be D ˆ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq24_HTML.gif-asymptotically compact. Then, for any bounded sets B of X, it holds that
      lim τ dist ( U ( t , τ ) B , Λ ( D ˆ 0 , t ) ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equf_HTML.gif
      Definition 2.4 Suppose that U is a multi-valued process. A family A = { A ( t ) } t R P ( X ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq35_HTML.gif is said to be a pullback attractor for a multi-valued process U if the set A ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq36_HTML.gif is compact for any t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq37_HTML.gif and attracts at time t to any bounded set B X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq38_HTML.gif in a pullback sense, i.e.,
      lim τ dist ( U ( t , τ ) B , A ( t ) ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equg_HTML.gif

      We can see obviously that a pullback attractor does not need to be unique. However, it can be considered unique in the sense of minimal, that is, the minimal closed family with such a property. In this sense, we obtain the following property and the existence of pullback attractors.

      Lemma 2.3 [25]

      Assume that U is a multi-valued process, and U is D ˆ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq24_HTML.gif-asymptotically compact and a family D ˆ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq24_HTML.gif is pullback-absorbing for U. Then, for any t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq25_HTML.gif and any bounded subset B of X, the set
      Λ ( B , t ) = s t τ s U ( t , τ ) B ¯ X , t R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equh_HTML.gif

      is a nonempty compact subset contained in Λ ( D ˆ 0 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq31_HTML.gif, which attracts to B in a pullback sense. In fact, Λ ( D ˆ 0 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq31_HTML.gif defined above is the minimal closed set with this property.

      Furthermore, for any bounded set B, the set A ( t ) = B Λ ( B , t ) ¯ X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq39_HTML.gif is a pullback attractor. From Definitions 2.3 and 2.4, it is easy to see that A ( t ) Λ ( D ˆ 0 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq40_HTML.gif.

      If there exists a time T R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq41_HTML.gif such that t T D 0 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq42_HTML.gif is bounded, for bounded B, then
      A ( t ) = B Λ ( B , t ) ¯ X = Λ ( D ˆ 0 , t ) , t T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equi_HTML.gif

      As we know, for the single-valued processes, the continuity of processes provides invariance; while in the multi-valued processes, the upper semi-continuity (defined below) provides negatively invariance of the omega limit sets Λ ( B , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq43_HTML.gif and the attractor. The following is the definition of the upper semi-continuity of the multi-valued processes (see in [22]).

      Definition 2.5 Let U be a multi-valued process on X. It is said to be upper semi-continuous if for all t τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq44_HTML.gif, the mapping U ( t , τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq45_HTML.gif is upper semi-continuous from X into http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq46_HTML.gif , that is to say, given a converging sequence x n x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq47_HTML.gif, for some sequence { y n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq48_HTML.gif such that y n U ( t , τ ) x n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq49_HTML.gif for all n, there exists a subsequence of { y n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq48_HTML.gif converging in X to an element of U ( t , τ ) x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq50_HTML.gif.

      Lemma 2.4 [22]

      Assume that a multi-valued process U and a family D ˆ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq24_HTML.gif. If, in addition, U ( t , τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq45_HTML.gif is D ˆ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq24_HTML.gif-asymptotically compact and upper semi-continuous, then the family { Λ ( D ˆ 0 , t ) } t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq51_HTML.gif is negatively invariant, i.e.,
      Λ ( D ˆ 0 , t ) U ( t , τ ) Λ ( D ˆ 0 , τ ) , t τ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equj_HTML.gif

      where B is a bounded set B of X. The family { Λ ( B , t ) } t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq52_HTML.gif is also negatively invariant and the family A ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq36_HTML.gif defined in Lemma  2.3 is also negatively invariant.

      Lemma 2.5 [22]

      Given a universe http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq10_HTML.gif , if a multi-valued process U is http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq53_HTML.gif -asymptotically compact, then, for any t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq54_HTML.gif and for any D ˆ D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq55_HTML.gif, the omega limit set Λ ( D ˆ , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq56_HTML.gif is a nonempty compact set of X that attracts to D ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq57_HTML.gif at time t in a pullback sense. Indeed, it is the minimal closed set with this property. If, in addition, the multi-valued process U is upper semi-continuous, then { Λ ( D ˆ , t ) } t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq58_HTML.gif is negatively invariant.

      From the above lemmas, we obtain the following results which are rather similar to Theorem 3 in [20]. We only sketch it here.

      Lemma 2.6 Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq53_HTML.gif is a universe and D ˆ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq59_HTML.gif is pullback http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq53_HTML.gif -absorbing for a multi-valued process U, which is also D ˆ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq24_HTML.gif-asymptotically compact. Then the results in Lemma  2.5 hold. Furthermore, the family A D = { A D ( t ) } t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq60_HTML.gif, where A D ( t ) = Λ ( D ˆ 0 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq61_HTML.gif, and the following results hold:
      1. (1)

        For each t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq25_HTML.gif, the set A D ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq62_HTML.gif defined above is compact.

         
      2. (2)

        A D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq63_HTML.gif attracts pullback to any D ˆ D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq64_HTML.gif.

         
      3. (3)

        Suppose U is upper semi-continuous, then A D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq65_HTML.gif is negatively invariant.

         
      4. (4)

        A D ( t ) = Λ ( D ˆ 0 , t ) = D ˆ D Λ ( D ˆ , t ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq66_HTML.gif.

         
      5. (5)

        Assume D ˆ 0 D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq67_HTML.gif, the minimal family of closed sets A D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq63_HTML.gif attracts pullback to elements of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq10_HTML.gif .

         
      6. (6)

        Assume D ˆ 0 D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq67_HTML.gif, each D 0 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq68_HTML.gif is closed and the universe http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq10_HTML.gif is inclusion-closed, then A D D http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq69_HTML.gif and it is the only family of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq53_HTML.gif which satisfies the above properties (1), (2) and (3).

         
      7. (7)

        If http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq53_HTML.gif contains the families of fixed bounded sets, then A = { A ( t ) } t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq70_HTML.gif defined in Lemma  2.3 is the minimal pullback attractor of bounded sets, and A ( t ) A D ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq71_HTML.gif for each t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq25_HTML.gif. In addition, if there exists some T R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq72_HTML.gif such that t T D 0 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq42_HTML.gif is bounded, then A ( t ) = A D ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq73_HTML.gif for all t T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq74_HTML.gif.

         

      3 Introduction to some abstract spaces and the existence of solutions

      We first recall some notations about the function spaces which will be used later to discuss the regularity of pullback attracting sets. Let us consider the following abstract space:
      V = { u ( C 0 ( Ω ) ) 3 : div u = 0 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equk_HTML.gif
      The symbols H, V denote the closures of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq75_HTML.gif in L 2 ( Ω ) 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq76_HTML.gif, H 0 1 ( Ω ) 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq77_HTML.gif, respectively. In other words, H= the closure of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq75_HTML.gif in ( L 2 ( Ω ) ) 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq78_HTML.gif with the norm | | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq79_HTML.gif and the inner product ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq80_HTML.gif, where for u , v ( L 2 ( Ω ) ) 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq81_HTML.gif,
      ( u , v ) = i = 1 2 Ω u i ( x ) v i ( x ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equl_HTML.gif
      V= the closure of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq75_HTML.gif in ( H 0 1 ( Ω ) ) 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq82_HTML.gif with the norm associated to the inner product ( ( , ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq83_HTML.gif, where for u , v ( H 0 1 ( Ω ) ) 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq84_HTML.gif,
      ( ( u , v ) ) = i , j = 1 2 Ω u j x i v j x i d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equm_HTML.gif

      We shall use v V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq85_HTML.gif to denote the norm of V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq86_HTML.gif. The value of a functional from V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq86_HTML.gif on an element from V is denoted by brackets , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq87_HTML.gif. It follows that V H H V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq88_HTML.gif, and the injections are dense and compact.

      Define Au = P u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq89_HTML.gif for all u D ( A ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq90_HTML.gif, where P is the ortho-projector from ( L 2 ( Ω ) ) 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq78_HTML.gif onto H. Considering the properties of the operator A, we have A : V V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq91_HTML.gif as
      Au , v : = ( ( u , v ) ) , u , v V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equn_HTML.gif
      We define
      b ( u , v , w ) = i , j = 1 3 Ω u i v j x i w j d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equo_HTML.gif
      for every function u, v, w : Ω R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq92_HTML.gif, and the operator B : V × V V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq93_HTML.gif as
      B ( u , v ) , w = b ( u , v , w ) , u , v , w V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equp_HTML.gif
      Obviously, b ( u , v , w ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq94_HTML.gif is a continuous trilinear form such that
      | b ( u , v , w ) | C 1 u v w , u , v , w V , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equq_HTML.gif
      which yields
      B ( u , v ) V C 1 u v , u , v V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equr_HTML.gif
      Moreover, b and B satisfy the following:
      b ( u , v , w ) = b ( u , w , v ) , u , v , w V , b ( u , v , v ) = 0 , u , v , w V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equs_HTML.gif
      Now, we make some assumptions. The given delay function ρ satisfies ρ C 1 ( [ 0 , + ) ; [ 0 , h ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq95_HTML.gif, and there is a constant ρ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq96_HTML.gif independent of t satisfying
      ρ ( t ) ρ 0 < 1 t 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ2_HTML.gif
      (3.1)

      where ρ = d ρ d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq97_HTML.gif.

      Moreover, we assume that g : [ τ , + ) × H H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq98_HTML.gif satisfies the following assumptions:

      (H1) g ( , u ) : [ τ , + ) H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq99_HTML.gif is measurable for all u V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq100_HTML.gif.

      (H2) For all t τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq44_HTML.gif, g ( t , ) : H H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq101_HTML.gif is continuous.

      (H3) There exist two functions γ , β : [ τ , + ) [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq102_HTML.gif. The above γ L p ( τ , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq103_HTML.gif and β L 1 ( τ , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq104_HTML.gif for all T > τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq105_HTML.gif, for 1 p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq106_HTML.gif, such that g ( t , u ) V 2 γ ( t ) u 2 + β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq107_HTML.gif, t τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq108_HTML.gif, u V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq109_HTML.gif.

      As to the initial datum, we assume

      (H4) u 0 V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq110_HTML.gif, and φ L 2 q ( h , 0 ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq111_HTML.gif, where 1 p + 1 q = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq112_HTML.gif.

      Next, we shall consider the solution of (1.1).
      { u ( t ) + α 2 Au ( t ) + τ t ( ν Au ( s ) + B ( u ( s ) ) ) d s = u 0 + α 2 Au 0 + τ t g ( s , u ( s ρ ( s ) ) ) d s , t τ , u ( τ + t ) = φ ( t ) , a.e.  t ( h , 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ3_HTML.gif
      (3.2)

      Definition 3.1 It is said that u is a weak solution to (1.1) if u belongs to u L 2 q ( τ h , T ; V ) L ( τ , T ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq113_HTML.gif for all t τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq44_HTML.gif such that u ( τ + t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq114_HTML.gif coincides with φ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq115_HTML.gif in ( h , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq8_HTML.gif and satisfies equation (3.2) in V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq86_HTML.gif for all t τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq44_HTML.gif.

      If u is a solution of (3.2), then it is easy to get u ( t ) + α 2 Au ( t ) L 2 ( τ , T ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq116_HTML.gif, and d d t ( u ( t ) + α 2 Au ( t ) ) L 1 ( τ , T ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq117_HTML.gif for all T > τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq105_HTML.gif. From the property of the operator A, we have u ( t ) + α 2 Au ( t ) C ( τ , + ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq118_HTML.gif. On the other hand, reasoning as in [12], we have u C ( [ τ , + ) ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq119_HTML.gif.

      Now, we define a functional γ ˜ ( t ) = γ ζ 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq120_HTML.gif, where ζ : [ τ , + ) [ ρ ( τ ) , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq121_HTML.gif is a differentiable and nonnegative strictly increasing function given by ζ ( s ) = s ρ ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq122_HTML.gif. We have
      τ t g ( t , u ( t ρ ( t ) ) ) V 2 d t τ T γ ( t ) u ( t ρ ( t ) ) 2 d t + τ T β ( t ) d t 1 1 ρ 0 τ ρ ( τ ) T ρ ( T ) γ ˜ ( t ) u ( t ) 2 d t + τ T β ( t ) d t 1 1 ρ 0 ( ρ ( τ ) 0 γ ˜ ( t + τ ) φ 2 d t + τ T γ ˜ ( t ) u ( t ) 2 d t ) + τ T β ( t ) d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ4_HTML.gif
      (3.3)
      From the above analysis, taking into account γ ˜ ( t ) L P ( ρ ( τ ) , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq123_HTML.gif for all T > τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq105_HTML.gif, we obtain that g ( t , u ( t ρ ( t ) ) ) L 2 ( τ , T ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq124_HTML.gif. Hence, it is clear that u is a weak solution to (1.1) if u C ( [ τ , + ) ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq125_HTML.gif, u L 2 ( τ , T ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq126_HTML.gif for all T > τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq105_HTML.gif, and satisfies the energy equality
      1 2 d d t ( | u ( t ) | 2 + α 2 u ( t ) 2 ) + ν u ( t ) 2 = g ( t , u ( t ρ ( t ) ) ) , u ( t ) , a.e.  t > τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equt_HTML.gif

      in the distributional sense on ( τ , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq127_HTML.gif.

      Theorem 3.1 Suppose that (H1)-(H4) hold. Then there exists a global solution u to (3.2).

      Proof We shall prove the result by the Faedo-Galerkin scheme and compactness method. For convenience and without loss of generality, we set the initial time τ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq128_HTML.gif. As to different value τ, we only proceed by translation.

      Consider the Hilbert basis of H formed by the eigenfunctions { v k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq129_HTML.gif of the above operator A, i.e., A v k = λ k v k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq130_HTML.gif. In fact, these elements allow to define the operator P m v = k = 1 m ( v k , v ) v k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq131_HTML.gif, which is the orthogonal projection of H and V in V m : = span [ v 1 , , v m ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq132_HTML.gif with their respective norms.

      Denote u m ( t ) = k = 1 m η m k ( t ) v k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq133_HTML.gif, where η m k ( t ) = ( u m ( t ) , v k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq134_HTML.gif, k = 1 , 2 , , m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq135_HTML.gif, are unknown real functions satisfying the finite-dimensional problem
      { ( u m , v k ) + α 2 ( Au m ( t ) , v k ) + ν 0 t Au m ( s ) , v k d s + 0 t B ( u m ( s ) , u m ( s ) ) , v k d s = ( u 0 , v k ) + α 2 ( Au 0 , v k ) + 0 t ( g ( s , u m ( s ρ ( s ) ) ) , v k ) d s , t > 0 , 1 k m , u m ( t ) = φ m ( t ) , a.e.  t ( h , 0 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ5_HTML.gif
      (3.4)

      with φ m ( t ) = P m φ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq136_HTML.gif. From [12], we can obtain the local well-posedness of this finite-dimensional delay problem. The following provides estimates which imply that the solutions are well defined in the whole [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq137_HTML.gif.

      By (3.4), we obtain
      d d t ( ( u m , v k ) + α 2 ( Au m ( t ) , v k ) ) + ν Au m ( t ) , v k + B ( u m ( t ) , u m ( t ) ) , v k = ( g ( t , u m ( t ρ ( t ) ) ) , v k ) , t > 0 , 1 k m . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ6_HTML.gif
      (3.5)
      Multiplying (3.5) by ( u m ( t ) , v k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq138_HTML.gif, summing from k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq139_HTML.gif to k = m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq140_HTML.gif, and using the properties of the operator b, we easily get
      d d t ( | u m ( t ) | 2 + α 2 u m ( t ) 2 ) + 2 ν u m ( t ) 2 = 2 g ( t , u m ( t ρ ( t ) ) ) , u m ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ7_HTML.gif
      (3.6)
      Observing (H3) and using the Young inequality, we have
      2 g ( t , u m ( t ρ ( t ) ) ) , u m ( t ) 2 g ( t , u ( t ρ ( t ) ) ) V u m ( t ) 2 ( γ 1 2 ( t ) u m ( t ρ ( t ) ) + β 1 2 ( t ) ) u m ( t ) = 2 γ 1 2 ( t ) u m ( t ρ ( t ) ) u m ( t ) + 2 β 1 2 ( t ) u m ( t ) ν ( 1 ρ 0 ) u m ( t ρ ( t ) ) 2 + γ ( t ) ν ( 1 ρ 0 ) u m ( t ) 2 + β ( t ) + u m ( t ) 2 = ν ( 1 ρ 0 ) u m ( t ρ ( t ) ) 2 + ( 1 + γ ( t ) ν ( 1 ρ 0 ) ) u m ( t ) 2 + β ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ8_HTML.gif
      (3.7)
      Considering the above inequality in (3.6) and observing that
      0 t u m ( s ρ ( s ) ) 2 d s 1 1 ρ 0 ρ ( 0 ) t ρ ( t ) u m ( s ) 2 d s 1 1 ρ 0 ( h 0 φ ( s ) 2 d s + 0 t u m ( s ) 2 d s ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ9_HTML.gif
      (3.8)
      we easily get
      | u m ( t ) | 2 + α 2 u m ( t ) 2 + 2 ν 0 t u m ( s ) 2 d s | u 0 | 2 + α 2 ( t ) u 0 2 + 0 T β ( s ) d s + ν h 0 φ ( s ) 2 d s + 0 t ( 1 + ν + γ ( t ) ν ( 1 ρ 0 ) ) u m ( s ) 2 d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ10_HTML.gif
      (3.9)

      From the above inequality and the Gronwall inequality, one has that { u m } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq141_HTML.gif is bounded in L 2 ( 0 , T ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq142_HTML.gif, also in L ( 0 , T ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq143_HTML.gif, for any T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq144_HTML.gif. Observing (3.5), we know that the sequence { d u m d t } m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq145_HTML.gif is bounded in L 2 ( 0 , T ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq146_HTML.gif for all T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq144_HTML.gif. The reason is the same as in [12].

      By the compactness of the injection of V into H, using the above estimates and the Ascoli-Arzelà theorem, we deduce that there exist a subsequence { u m } m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq147_HTML.gif (we relabel the same) and u W 1 , 2 ( 0 , T ; V ) L 2 q ( h , T ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq148_HTML.gif for any T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq144_HTML.gif with u = φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq149_HTML.gif in ( h , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq8_HTML.gif. Recalling (3.9) and the above analysis, we have
      u m u weakly-star in  L ( 0 , T ; V ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ11_HTML.gif
      (3.10)
      u m u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq150_HTML.gif strongly in C ( 0 , T ; H ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq151_HTML.gif for all T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq144_HTML.gif, and
      u m ( t ) u ( t ) , a.e.  t > 0  in  V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ12_HTML.gif
      (3.11)

      By the properties of operator A and (3.10), we deduce that Au m Au http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq152_HTML.gif weakly in L 2 ( 0 , T ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq146_HTML.gif. Reasoning as in [26] on page 76, we deduce that B ( u m ) B ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq153_HTML.gif weakly in L 2 ( 0 , T ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq154_HTML.gif for any T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq144_HTML.gif.

      On the other hand, observing that by (3.11), (H2) and the hypothesis on ρ in (3.1), for any time t, we conclude for any T > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq144_HTML.gif
      g ( t , u m ( t ρ ( t ) ) ) g ( t , u ( t ρ ( t ) ) ) in  V  a.e.  0 < t < T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ13_HTML.gif
      (3.12)
      and as { u m } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq141_HTML.gif is bounded in the space L ( 0 , T ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq155_HTML.gif obtained from (3.9), by (H3), we deduce
      g ( t , u m ( t ρ ( t ) ) ) V 2 { β ( t ) + γ ( t ) , t > ρ ( t ) , β ( t ) + γ ( t ) φ ( t ρ ( t ) ) 2 , t < ρ ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ14_HTML.gif
      (3.13)
      where C = sup m 1 u m L ( 0 , T ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq156_HTML.gif. Thus we can easily derive from (3.12) and (3.13)
      g ( t , u m ( t ρ ( t ) ) ) g ( t , u ( t ρ ( t ) ) ) in  L 2 ( 0 , T ; V ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ15_HTML.gif
      (3.14)

      From the above discussion, passing to the limit, we prove that u is a global solution of (3.2) in the sense of Definition 3.1. □

      Remark 3.1 We can obtain the uniqueness if there are additional assumptions on the forcing term g. For example, if we suppose that g satisfies (H1), (H3), for an arbitrary T > τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq105_HTML.gif and C T = C 1 max t [ τ , T ] ( u ( t ) , v ( t ) ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq157_HTML.gif, then for the solutions u, v,
      g ( t , u ) g ( t , v ) V ( ν + C T ) u v , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equu_HTML.gif

      then we obtain the uniqueness of solutions.

      4 Existence of pullback attractors

      In this section, we discuss the existence of pullback attractors for the 3D Navier-Stokes-Voight equations with delays in continuous and sub-linear operators. At first, we propose the assumptions for g given in Section 3:

      (H1) For all u V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq100_HTML.gif, g ( , u ) : R H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq158_HTML.gif is measurable.

      (H2) For all t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq25_HTML.gif, g ( t , ) : H H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq159_HTML.gif is continuous.

      (H3) There are two nonnegative functions γ ( t ) , β ( t ) : R [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq160_HTML.gif with γ ( t ) L l o c p ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq161_HTML.gif for some 1 p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq106_HTML.gif and β ( t ) L l o c 1 ( R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq162_HTML.gif such that for any u V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq100_HTML.gif,
      g ( t , u ) V 2 γ ( t ) u 2 + β ( t ) for any  t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equv_HTML.gif

      To construct a multi-valued process, we introduce symbols C V = C ( [ h , 0 ] ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq163_HTML.gif and S V 2 q = V × L 2 q ( h , 0 ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq164_HTML.gif, where 1 p + 1 q = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq112_HTML.gif as two phase spaces. Let D ( τ , u 0 , φ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq165_HTML.gif denote the set of global solutions of (1.1) in [ τ , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq166_HTML.gif and the initial datum ( u 0 , φ ) S V 2 q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq167_HTML.gif.

      By Theorem 3.1, we know there exists a solution to problem (3.2) although we have no discussion on the uniqueness of solutions to problem (3.2). We may define two strict processes, ( C V , { U ( , ) } ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq168_HTML.gif as
      U ( t , τ ) φ = { u ( t ) : u D ( τ , φ ( 0 ) , φ ) } for any  φ C V , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equw_HTML.gif
      and ( S V 2 q , { U ( , ) } ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq169_HTML.gif as
      U ( t , τ ) ( u 0 , φ ) = { u ( t ) : u D ( τ , u 0 , φ ) } for any  ( u 0 , φ ) S V 2 q . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equx_HTML.gif

      Considering the regularity of the problem, the asymptotic behavior of the two processes shall be the same, as we shall see in what follows.

      In order to simplify the calculation form, we introduce a function κ σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq170_HTML.gif. For any σ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq171_HTML.gif, we set
      κ σ ( t , s ) = ( d σ + d α 2 ) ( t s ) e d h α 2 d ( 1 ρ 0 ) s t γ ( r ) d r , t , s R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ16_HTML.gif
      (4.1)
      From (4.1), we can find that
      κ σ ( t , s ) = κ σ ( 0 , t ) κ σ ( 0 , s ) , t , s R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ17_HTML.gif
      (4.2)
      and for any σ: 0 < σ < α 2 d d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq172_HTML.gif, then
      κ σ ( 0 , r ) κ σ ( 0 , t ) + ( d σ + d α 2 ) h , r [ t h , t ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ18_HTML.gif
      (4.3)

      where d = ν min { λ 1 , 1 α 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq173_HTML.gif.

      Lemma 4.1 Suppose that (H1′)-(H3′) hold, for any initial datum ( u 0 , φ ) S V 2 q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq167_HTML.gif and any u D ( τ , u 0 , φ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq174_HTML.gif, it holds
      u ( t ) 2 1 α 2 C τ e κ σ ( t , τ ) + 1 α 2 σ 1 τ t e κ σ ( t , s ) β ( s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ19_HTML.gif
      (4.4)

      where C τ = d h 0 e d r φ ( r ) 2 d r + | u 0 | 2 + α 2 u 0 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq175_HTML.gif.

      Proof Let u be a solution of (3.2), so u D ( τ , u 0 , φ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq174_HTML.gif. Multiplying (1.1) by u ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq176_HTML.gif and using the energy equality and the Poincaré inequality, we have
      d d t ( | u ( t ) | 2 + α 2 u ( t ) 2 ) + d ( | u ( t ) | 2 + α 2 u ( t ) 2 ) 2 g ( t , u ( t ρ ( t ) ) ) V u ( t ) V , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equy_HTML.gif
      where d = ν min { λ 1 , 1 α 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq177_HTML.gif. Thus
      d d t [ e d t ( | u ( t ) | 2 + α 2 u ( t ) 2 ) ] 2 e d t g ( t , u ( t ρ ( t ) ) ) V u ( t ) V 2 e d t ( γ 1 2 ( t ) u ( t ρ ( t ) ) + β 1 2 ( t ) ) u ( t ) 2 e d t γ 1 2 ( t ) u ( t ) u ( t ρ ( t ) ) + 2 e d t β 1 2 ( t ) u ( t ) C 0 1 e d t u ( t ρ ( t ) ) 2 + ( C 0 γ ( t ) + σ ) e d t u ( t ) 2 + σ 1 e d t β ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ20_HTML.gif
      (4.5)
      where we have denoted
      C 0 = e d h d ( 1 ρ 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equz_HTML.gif
      Considering
      τ t e d s u ( s ρ ( s ) ) 2 d s e d h 1 ρ 0 ( τ h t e d r u ( r ) d r ) = e d h 1 ρ 0 ( e d τ h 0 e d r φ ( r ) 2 d r + τ t e d r u ( r ) 2 d r ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ21_HTML.gif
      (4.6)
      and integrating (4.5) from τ to t, we deduce
      e d t | u ( t ) | 2 + α 2 e d t u ( t ) 2 C 0 1 τ t e d s u ( s ρ ( s ) ) 2 d s + τ t ( C 0 γ ( s ) + σ ) e d s u ( s ) 2 d s + σ 1 τ t β ( s ) d s + e d τ | u 0 | 2 + α 2 e d τ u 0 2 C 0 1 × e d h 1 ρ 0 × e d τ h 0 e d r φ ( r ) 2 d r + C 0 1 × e d h 1 ρ 0 τ t e d r u ( r ) 2 d r + τ t ( C 0 γ ( s ) + σ ) e d s u ( s ) 2 d s + σ 1 τ t e d s β ( s ) d s + e d τ | u 0 | 2 + α 2 e d τ u 0 2 = d e d τ h 0 e d r φ ( r ) 2 d r + τ t e d s ( C 0 γ ( s ) + σ + d ) u ( s ) 2 d s + σ 1 τ t e d r β ( r ) d r + e d τ | u 0 | 2 + α 2 e d τ u 0 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ22_HTML.gif
      (4.7)
      where we set
      C τ = d h 0 e d s φ ( s ) 2 d s + | u 0 | 2 + α 2 u 0 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equaa_HTML.gif
      Observing the above estimates, we easily deduce
      e d t | u ( t ) | 2 + α 2 e d t u ( t ) 2 e d τ C τ + τ t ( C 0 γ ( s ) + σ + d ) e d r u ( r ) 2 d r + σ 1 τ t e d r β ( r ) d r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ23_HTML.gif
      (4.8)

      Applying the Poincaré inequality and the Gronwall inequality to (4.8), we deduce that (4.4) holds. This finishes the proof of this lemma. □

      Next, we shall prove that the processes ( C V , { U ( , ) } ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq168_HTML.gif and ( S V 2 q , { U ( , ) } ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq169_HTML.gif defined above are pullback-absorbing. To obtain this, we propose the assumptions (H4′)
      lim sup t 1 t 0 t γ ( s ) d s = γ ¯ [ 0 , + ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equab_HTML.gif
      and the relation among constants σ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq178_HTML.gif, d defined above, and α in (1.1) satisfies(H5′)
      d σ + d α 2 C 0 α 2 > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equac_HTML.gif

      and β ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq179_HTML.gif satisfies

      (H6)
      0 e κ σ ( 0 , r ) β ( r ) d r < + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equad_HTML.gif

      where the function κ σ ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq180_HTML.gif is given by (4.1).

      Before proving that the two multi-valued processes possess pullback-absorbing sets, we introduce the definition of the two natural tempered universes which shall play the key role for our main purpose.

      Definition 4.1 Suppose that R σ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq181_HTML.gif is the collection of the sets of all functions r : R [ 0 , + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq182_HTML.gif such that
      lim t e κ σ ( 0 , t ) r 2 ( t ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equae_HTML.gif

      Let D S V 2 q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq183_HTML.gif be the class of all families D ˆ = { D ( t ) : t R } P ( S V 2 q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq184_HTML.gif such that D ( t ) B ¯ S V 2 q ( 0 , r D ˆ ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq185_HTML.gif for some r D ˆ R σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq186_HTML.gif. In the same way, let D C V σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq187_HTML.gif denote the class of all families D ˆ = { D ( t ) : t R } P ( C V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq188_HTML.gif satisfying D ( t ) B ¯ C V ( 0 , r D ˆ ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq189_HTML.gif for some r D ˆ R σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq186_HTML.gif.

      Let B 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq190_HTML.gif be any fixed bounded subset of S V 2 q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq191_HTML.gif. Observing that D C V σ D S V 2 q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq192_HTML.gif, which is inclusion-closed, by (H4) and (H5), we deduce that the family B ˆ = { B ( t ) B 0 , t R } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq193_HTML.gif is contained in D S V 2 q σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq194_HTML.gif. With regard to D C V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq195_HTML.gif, we use the same method and obtain a similar conclusion if B 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq190_HTML.gif is included in C V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq196_HTML.gif.

      The following lemma provides that there exist pullback-absorbing sets for the two processes mentioned above.

      Lemma 4.2 Suppose that (H1)-(H6) hold and the constants α, d, C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq197_HTML.gif, γ ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq198_HTML.gif satisfy d σ + d α 2 C 0 α 2 γ ¯ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq199_HTML.gif and γ ¯ < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq200_HTML.gif.
      1. (1)
        Then, for any t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq25_HTML.gif and any family B ˆ = { B ( t ) : t R } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq201_HTML.gif, there exits τ ( B ˆ , t ) t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq202_HTML.gif such that any initial datum ( u 0 , φ ) S V 2 q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq167_HTML.gif and any u D ( τ , u 0 , φ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq174_HTML.gif for any τ τ ( B ˆ , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq203_HTML.gif satisfy that u ( t ) R V ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq204_HTML.gif, where the positive continuous function R V ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq205_HTML.gif is given by
        R V 2 ( t ) = 1 + 1 α 2 σ 1 τ t e κ σ ( t , s ) β ( s ) d s , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equaf_HTML.gif
         
      2. (2)
        Let D ˆ 0 = { D 0 ( t ) : t R } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq206_HTML.gif be included P ( C V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq207_HTML.gif which is given by
        D 0 ( t ) = B ¯ C V ( 0 , R ˜ V ( t ) ) and R ˜ V ( t ) = max t h r t R V ( r ) , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equag_HTML.gif
         

      Then the set D ˆ 0 D C V σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq208_HTML.gif and is D S V 2 q σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq194_HTML.gif-pullback absorbing for the process ( S V 2 q , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq209_HTML.gif. Therefore, D ˆ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq24_HTML.gif is D C V σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq187_HTML.gif-pullback absorbing for the process ( C V , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq210_HTML.gif.

      Proof Since the proof is a consequence of the definition of D S V 2 q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq183_HTML.gif, we only sketch it here. From Lemma 4.1, (H5) and (H6) , we have
      u ( t ) 2 1 α 2 C τ e κ σ ( t , τ ) + 1 α 2 σ 1 τ t e κ σ ( t , s ) β ( s ) d s = 1 α 2 C τ e κ σ ( 0 , t ) e κ ( 0 , τ ) + 1 α 2 σ 1 τ t e κ σ ( t , s ) β ( s ) d s = 1 α 2 C τ e κ σ ( 0 , t ) e ( d σ + d α 2 ) τ C 0 α 2 0 τ γ ( r ) d r + 1 α 2 σ 1 τ t e κ σ ( t , s ) β ( s ) d s C τ α 2 e κ σ ( 0 , t ) e ( d σ + d α 2 C 0 α 2 ) τ + 1 α 2 σ 1 τ t e κ σ ( t , s ) β ( s ) d s 1 + 1 α 2 σ 1 t e κ σ ( t , s ) β ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ24_HTML.gif
      (4.9)

      We complete the proof of (1). Estimate (2) is a consequence of (1). □

      For the two processes ( S V 2 q , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq209_HTML.gif and ( C V , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq210_HTML.gif, they possess pullback-absorbing sets. In order to apply Lemma 2.3 to obtain the existence of pullback attractors, it is necessary to prove that the two multi-valued processes are asymptotically compact. This will be done in the following lemma.

      Lemma 4.3 Suppose that the assumptions in Lemma  4.2 hold. The two processes ( C V , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq210_HTML.gif and ( S V 2 q , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq209_HTML.gif are D ˆ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq24_HTML.gif-asymptotically compact.

      Proof For any t 0 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq211_HTML.gif , a sequence { τ n } ( , t 0 2 h ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq212_HTML.gif with τ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq27_HTML.gif and a sequence { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq213_HTML.gif with u n D ( τ n , φ n ( 0 ) , φ n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq214_HTML.gif with φ n D 0 ( τ n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq215_HTML.gif, we shall prove that the sequence { u t 0 n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq216_HTML.gif is relatively compact in C V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq196_HTML.gif. By the properties concerning operator b mentioned in Section 3, we deduce that
      ( u n ) V + α 2 ( u n ) ν u n + b ( u n , u n , ) V + g ( t , u n ( t ρ ( t ) ) ) V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equah_HTML.gif
      It is easy to get the above estimate which is independent of n. The sequences of { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq213_HTML.gif and { ( u n ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq217_HTML.gif possess their subsequence, relabeled the same in suitable spaces such that there exist u L ( t 0 2 h ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq218_HTML.gif and u L 2 ( t 0 h , t 0 ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq219_HTML.gif satisfying
      { u n u weakly-star in  L ( t 0 2 h , t 0 ; V ) , u n u weakly in  L 2 ( t 0 2 h , t 0 ; V ) , d d t u n d d t u weakly in  L 2 ( t 0 h , t 0 ; V ) , u n ( r ) u ( r ) strongly in  V  a.e.  r ( t 0 2 h , t 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ25_HTML.gif
      (4.10)

      According to the assumptions on a function g and analogously as in Theorem 3.1, we deduce that g ( r , u n ( r ρ ( r ) ) ) g ( r , u ( r ρ ( r ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq220_HTML.gif strongly in V a.e. r ( t 0 h , t 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq221_HTML.gif.

      By the Lebesgue theorem and the uniform estimate of u n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq222_HTML.gif in L ( t 0 2 h , t 0 ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq223_HTML.gif, we deduce that the function g ( r , u n ( r ρ ( r ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq224_HTML.gif converges to the function g ( r , u ( r ρ ( r ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq225_HTML.gif strongly. Therefore, for any t [ t 0 h , t 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq226_HTML.gif, we have u C ( [ t 0 h , t 0 ] ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq227_HTML.gif and
      u ( t ) + α 2 Au ( t ) + t 0 h t ( ν Au ( r ) + B ( u ( r ) ) ) d r = u ( t 0 h ) + α 2 Au ( t 0 h ) + t 0 h t g ( r , u ( r ρ ( r ) ) ) d r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ26_HTML.gif
      (4.11)
      The uniform estimate of { ( u n ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq217_HTML.gif in L 2 ( t 0 h , t 0 ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq228_HTML.gif implies that the sequence { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq213_HTML.gif is equicontinuous in V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq86_HTML.gif for any t 0 h t t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq229_HTML.gif. In addition, the sequence { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq213_HTML.gif is bounded, which is independent of n in C ( [ t 0 h , t 0 ] ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq230_HTML.gif. Using the Ascoli-Arzelà theorem, we can obtain
      u n u strongly in  C ( [ t 0 h , t 0 ] ; V ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ27_HTML.gif
      (4.12)

      From the uniform boundedness of the sequence { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq213_HTML.gif in C ( [ t 0 h , t 0 ] ; V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq230_HTML.gif, for any r [ t 0 h , t 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq231_HTML.gif, we can also obtain u n ( r ) u ( r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq232_HTML.gif, weakly in V.

      By the analogous argument, for any compact sequence { r n } [ t 0 h , t 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq233_HTML.gif and { r n } r [ t 0 h , t 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq234_HTML.gif, we obtain that the sequence { u n ( r n ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq235_HTML.gif is convergent to u ( r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq236_HTML.gif weakly in V. To achieve our result in Lemma 4.3, we only need to prove
      u n u strongly in  C ( [ t 0 h , t 0 ] ; V ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equai_HTML.gif

      The proof is slightly different from Proposition 6 in [20] or in [22]. We only sketch it here. We use a contradiction argument. Suppose that it is not true, then there would exist a value ε, a sequence (relabeled the same) { r n } [ r n h , t 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq237_HTML.gif, and r [ t 0 h , t 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq238_HTML.gif with r n r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq239_HTML.gif satisfying u n ( r n ) u ( r ) ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq240_HTML.gif for all n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq241_HTML.gif. We shall see u n ( r n ) u ( r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq242_HTML.gif in V. In order to achieve the last claim, because the sequence { u n ( r n ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq235_HTML.gif is weakly convergent to u ( r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq236_HTML.gif in V, we only need the convergence of the norms above. In other words, u n ( r n ) u ( r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq243_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq244_HTML.gif.

      From the weak convergence of u n ( r n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq245_HTML.gif in V, we get
      u ( r ) lim n + inf u n ( r ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equaj_HTML.gif
      Therefore we have to check that
      lim n + sup u n ( r n ) u ( r ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ28_HTML.gif
      (4.13)
      From the energy equality, for any t 0 h r t t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq246_HTML.gif, we obtain
      1 2 | z ( t ) | 2 + 1 2 z ( t ) 2 + ν r t z ( s ) 2 d s = 1 2 | z ( r ) | 2 + 1 2 z ( r ) 2 + r t ( g ( s , z ( s ρ ( s ) ) ) ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ29_HTML.gif
      (4.14)
      where z = u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq247_HTML.gif or z = u n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq248_HTML.gif. For any t [ t 0 h , t 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq226_HTML.gif, define the continuous functions J ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq249_HTML.gif and J n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq250_HTML.gif as
      J ( t ) = 1 2 | u ( t ) | 2 + 1 2 u ( t ) 2 t 0 h t ( g ( s , u ( s ρ ( s ) ) ) , u ( s ) ) d s , J n ( t ) = 1 2 | u n ( t ) | 2 + 1 2 u n ( t ) 2 t 0 h t ( g ( s , u n ( s ρ ( s ) ) ) , u n ( s ) ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equak_HTML.gif

      By (4.14), it is clear that J and J n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq251_HTML.gif are non-increasing functions. By the convergence (4.10), for any t ( t 0 h , t 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq252_HTML.gif, we have that J n ( t ) J ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq253_HTML.gif. Using the same analysis method as in [22], we can deduce that for n n ( κ ε ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq254_HTML.gif, J n ( r n ) J ( r ) ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq255_HTML.gif, which gives (4.13) as desired. □

      We can apply the technical method for any family in D S V 2 q σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq194_HTML.gif in Lemma 4.3. Suppose that the assumptions in Lemma 4.3 hold. We can deduce that the processes ( C V , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq210_HTML.gif and ( S V 2 q , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq209_HTML.gif are D C V σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq256_HTML.gif-asymptotically compact and D S V 2 q σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq257_HTML.gif-asymptotically compact.

      Lemma 4.4 Suppose that (H1′)-(H6′) hold. The two processes ( C V , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq210_HTML.gif and ( S V 2 q , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq209_HTML.gif are semi-continuous and that U ( t , τ ) : S V 2 q P ( S V 2 q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq258_HTML.gif and U ( t , τ ) : C V P ( C V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq259_HTML.gif have compact values in their respective topologies.

      Proof In fact, the upper semi-continuity of the process ( S V 2 q , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq209_HTML.gif can be obtained by similar arguments to those used for the Galerkin sequence in Theorem 3.1.

      As to the process ( C V , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq210_HTML.gif, applying the same energy-procedure in Lemma 4.3, we shall obtain that in [ τ , t ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq260_HTML.gif any set of solutions possesses a converging subsequence in this process, whence the assertion in Lemma 4.4 follows. □

      According to the results in Section 2, the following two theorems shall be obtained, which are our result in this paper. Observing Lemmas 4.3 and 4.4, and applying Lemma 2.6, we obtain the following theorem.

      Theorem 4.1 Suppose that (H1′)-(H6′) hold. For any t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq25_HTML.gif, then there exist global pullback attractors A C V = { A C V ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq261_HTML.gif and A D C V σ = { A D C V σ ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq262_HTML.gif for the process ( C V , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq210_HTML.gif in the universe of fixed bounded sets and in D C V σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq187_HTML.gif, respectively. Moreover, they are unique in the sense of Lemma  2.5 and negatively and strictly invariant for U respectively, and the following holds:
      A C V ( t ) A D C V σ ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equal_HTML.gif

      The above theorem proves that there exist pullback attractors in the C V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq196_HTML.gif framework, while we shall prove that there exist pullback attractors in the S V 2 q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq191_HTML.gif framework in the following theorem.

      Theorem 4.2 Suppose that the assumptions in Theorem  4.1 hold. For any t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq37_HTML.gif, there exist global pullback attractors A S V 2 q = { A S V 2 q ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq263_HTML.gif and A D S V 2 q σ = { A D S V 2 q σ ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq264_HTML.gif for the process ( S V 2 q , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq209_HTML.gif in the universes of fixed bounded sets and in D S V 2 q σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq194_HTML.gif. They are unique in the sense of Lemma  2.5 and negatively and strictly invariant for U, respectively, and we have A S V 2 q ( t ) A D S V 2 q σ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq265_HTML.gif. Moreover, the relationship between the attractors for ( S V 2 q , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq209_HTML.gif and for ( C V , U ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq210_HTML.gif is as follows:
      A S V 2 q ( t ) = f ( A C V ( t ) ) and A D S V 2 q σ ( t ) = f ( A D C V σ ( t ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_Equ30_HTML.gif
      (4.15)

      where f : C V S V 2 q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq266_HTML.gif is the continuous mapping defined by f ( φ ) = ( φ ( 0 ) , φ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq267_HTML.gif.

      Proof The proof is rather similar to that of Theorem 5 in [20]. Since the regularity is different from [20], we only sketch the proof of (4.15) here.

      By Theorem 3.1, we can conclude that U ( t , τ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq45_HTML.gif maps S V 2 q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq191_HTML.gif into bounded sets in C V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq196_HTML.gif if t τ + h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq268_HTML.gif, and also maps bounded sets from S V 2 q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq191_HTML.gif into bounded sets of C V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq196_HTML.gif.

      Noting that A S V 2 q ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq269_HTML.gif is the minimal closed set, and using Lemma 2.5 and the above arguments, we deduce that f ( A C V ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq270_HTML.gif also attracts bounded sets in S V 2 q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq191_HTML.gif in a pullback sense. Therefore the inclusion A S V 2 q ( t ) f ( A C V ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq271_HTML.gif holds.

      As to the opposite inclusion of the first identification in (4.15), for any bounded set B, it follows from the continuous injection f ( C V ) S V 2 q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq272_HTML.gif and the attractor A C V ( t ) = B Λ C V ( B , t ) ¯ C V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq273_HTML.gif. Thus f ( A C V ( t ) ) = B f ( Λ C V ( B , t ) ) ¯ C V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-191/MediaObjects/13661_2013_Article_447_IEq274_HTML.gif, whence the opposite inclusion of the first identification in (4.15) holds.

      Analogously, it is obvious that the second relation in (4.15) holds. □

      Declarations

      Acknowledgements

      This work was supported in part by the NNSF of China (No. 11271066, No. 11031003), by the grant of Shanghai Education Commission (No. 13ZZ048), by Chinese Universities Scientific Fund (No. CUSF-DH-D-2013068).

      Authors’ Affiliations

      (1)
      College of Information Science and Technology, Donghua University
      (2)
      Department of Applied Mathematics, Donghua University

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      Copyright

      © Li and Qin; licensee Springer 2013

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