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

Haiyan Li1 and Yuming Qin2*

Author Affiliations

1 College of Information Science and Technology, Donghua University, Shanghai, 201620, P.R. China

2 Department of Applied Mathematics, Donghua University, Shanghai, 201620, P.R. China

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Boundary Value Problems 2013, 2013:191  doi:10.1186/1687-2770-2013-191

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/191


Received:25 April 2013
Accepted:6 August 2013
Published:27 August 2013

© 2013 Li and Qin; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M1">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M2">View MathML</a> 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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M3">View MathML</a>

(1.1)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M4">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M6">View MathML</a> is an adequate given delay function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M7">View MathML</a> is the initial velocity field at the initial time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M8">View MathML</a>, φ is the initial datum on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M9">View MathML</a>.

Equation (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M10">View MathML</a> becomes the classical three-dimensional Navier-Stokes (NS) equation. In the past decades, many authors [1-6] 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 -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 [16-18] 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M12">View MathML</a> when the external forcing term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M13">View MathML</a> and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M14">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M15">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M16">View MathML</a> be a metric space, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M17">View MathML</a> be the class of nonempty subsets of X. As usual, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M18">View MathML</a> the Hausdorff semi-distance in X between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M20">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M21">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M22">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M23">View MathML</a> for any pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M24">View MathML</a> of real numbers such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M25">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M26">View MathML</a> is a family of sets. A multi-valued process U is said to be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27">View MathML</a>-asymptotically compact if for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M28">View MathML</a>, any sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M29">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M31">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M32">View MathML</a>, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M33">View MathML</a> is relatively compact in X.

Lemma 2.1If a multi-valued processUis<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27">View MathML</a>-asymptotically compact, then the sets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M35">View MathML</a>are nonempty compact subsets ofX, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M36">View MathML</a>

Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M35">View MathML</a>attracts in a pullback sense to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27">View MathML</a>at timet, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M39">View MathML</a>

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

Definition 2.3 The family of subsets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M40">View MathML</a> is said to be pullback-absorbing with respect to a multi-valued process U if for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M28">View MathML</a> and all bounded subset B of X, there exists a time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M42">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M43">View MathML</a>

Lemma 2.2Let the family of sets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M44">View MathML</a>be pullback-absorbing for the multi-valued processU, and letUbe<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27">View MathML</a>-asymptotically compact. Then, for any bounded setsBofX, it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M46">View MathML</a>

Definition 2.4 Suppose that U is a multi-valued process. A family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M47">View MathML</a> is said to be a pullback attractor for a multi-valued process U if the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M48">View MathML</a> is compact for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M49">View MathML</a> and attracts at time t to any bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M50">View MathML</a> in a pullback sense, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M51">View MathML</a>

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 thatUis a multi-valued process, andUis<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27">View MathML</a>-asymptotically compact and a family<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27">View MathML</a>is pullback-absorbing forU. Then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M28">View MathML</a>and any bounded subsetBofX, the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M55">View MathML</a>

is a nonempty compact subset contained in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M35">View MathML</a>, which attracts toBin a pullback sense. In fact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M35">View MathML</a>defined above is the minimal closed set with this property.

Furthermore, for any bounded setB, the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M58">View MathML</a>is a pullback attractor. From Definitions 2.3 and 2.4, it is easy to see that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M59">View MathML</a>.

If there exists a time<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M60">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M61">View MathML</a>is bounded, for boundedB, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M62">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M63">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M64">View MathML</a>, the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M65">View MathML</a> is upper semi-continuous from X into , that is to say, given a converging sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M67">View MathML</a>, for some sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M68">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M69">View MathML</a> for all n, there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M68">View MathML</a> converging in X to an element of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M71">View MathML</a>.

Lemma 2.4[22]

Assume that a multi-valued processUand a family<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27">View MathML</a>. If, in addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M65">View MathML</a>is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27">View MathML</a>-asymptotically compact and upper semi-continuous, then the family<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M75">View MathML</a>is negatively invariant, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M76">View MathML</a>

whereBis a bounded setBofX. The family<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M77">View MathML</a>is also negatively invariant and the family<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M48">View MathML</a>defined in Lemma 2.3 is also negatively invariant.

Lemma 2.5[22]

Given a universe, if a multi-valued processUis-asymptotically compact, then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M81">View MathML</a>and for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M82">View MathML</a>, the omega limit set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M83">View MathML</a>is a nonempty compact set ofXthat attracts to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M84">View MathML</a>at timetin a pullback sense. Indeed, it is the minimal closed set with this property. If, in addition, the multi-valued processUis upper semi-continuous, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M85">View MathML</a>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 2.6Suppose thatis a universe and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M87">View MathML</a>is pullback-absorbing for a multi-valued processU, which is also<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27">View MathML</a>-asymptotically compact. Then the results in Lemma 2.5 hold. Furthermore, the family<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M90">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M91">View MathML</a>, and the following results hold:

(1) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M28">View MathML</a>, the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M93">View MathML</a>defined above is compact.

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M94">View MathML</a>attracts pullback to any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M95">View MathML</a>.

(3) SupposeUis upper semi-continuous, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M96">View MathML</a>is negatively invariant.

(4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M97">View MathML</a>.

(5) Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M98">View MathML</a>, the minimal family of closed sets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M94">View MathML</a>attracts pullback to elements of.

(6) Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M98">View MathML</a>, each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M102">View MathML</a>is closed and the universeis inclusion-closed, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M104">View MathML</a>and it is the only family ofwhich satisfies the above properties (1), (2) and (3).

(7) Ifcontains the families of fixed bounded sets, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M107">View MathML</a>defined in Lemma 2.3 is the minimal pullback attractor of bounded sets, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M108">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M28">View MathML</a>. In addition, if there exists some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M110">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M61">View MathML</a>is bounded, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M112">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M113">View MathML</a>.

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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M114">View MathML</a>

The symbols H, V denote the closures of in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M117">View MathML</a>, respectively. In other words, H= the closure of in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M119">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M120">View MathML</a> and the inner product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M121">View MathML</a>, where for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M122">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M123">View MathML</a>

V= the closure of in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M125">View MathML</a> with the norm associated to the inner product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M126">View MathML</a>, where for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M127">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M128">View MathML</a>

We shall use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M129">View MathML</a> to denote the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M130">View MathML</a>. The value of a functional from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M130">View MathML</a> on an element from V is denoted by brackets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M132">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M133">View MathML</a>, and the injections are dense and compact.

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M134">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M135">View MathML</a>, where P is the ortho-projector from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M119">View MathML</a> onto H. Considering the properties of the operator A, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M137">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M138">View MathML</a>

We define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M139">View MathML</a>

for every function u, v, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M140">View MathML</a>, and the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M141">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M142">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M143">View MathML</a> is a continuous trilinear form such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M144">View MathML</a>

which yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M145">View MathML</a>

Moreover, b and B satisfy the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M146">View MathML</a>

Now, we make some assumptions. The given delay function ρ satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M147">View MathML</a>, and there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M148">View MathML</a> independent of t satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M149">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M150">View MathML</a>.

Moreover, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M151">View MathML</a> satisfies the following assumptions:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M152">View MathML</a> is measurable for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M153">View MathML</a>.

(H2) For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M155">View MathML</a> is continuous.

(H3) There exist two functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M156">View MathML</a>. The above <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M157">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M158">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M159">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M160">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M163">View MathML</a>.

As to the initial datum, we assume

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M164">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M165">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M166">View MathML</a>.

Next, we shall consider the solution of (1.1).

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M167">View MathML</a>

(3.2)

Definition 3.1 It is said that u is a weak solution to (1.1) if u belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M168">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M64">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M170">View MathML</a> coincides with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M171">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M9">View MathML</a> and satisfies equation (3.2) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M130">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M64">View MathML</a>.

If u is a solution of (3.2), then it is easy to get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M175">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M176">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M159">View MathML</a>. From the property of the operator A, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M178">View MathML</a>. On the other hand, reasoning as in [12], we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M179">View MathML</a>.

Now, we define a functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M180">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M181">View MathML</a> is a differentiable and nonnegative strictly increasing function given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M182">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M183">View MathML</a>

(3.3)

From the above analysis, taking into account <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M184">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M159">View MathML</a>, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M186">View MathML</a>. Hence, it is clear that u is a weak solution to (1.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M188">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M159">View MathML</a>, and satisfies the energy equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M190">View MathML</a>

in the distributional sense on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M191">View MathML</a>.

Theorem 3.1Suppose 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M192">View MathML</a>. As to different value τ, we only proceed by translation.

Consider the Hilbert basis of H formed by the eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M193">View MathML</a> of the above operator A, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M194">View MathML</a>. In fact, these elements allow to define the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M195">View MathML</a>, which is the orthogonal projection of H and V in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M196">View MathML</a> with their respective norms.

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M197">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M198">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M199">View MathML</a>, are unknown real functions satisfying the finite-dimensional problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M200">View MathML</a>

(3.4)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M201">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M202">View MathML</a>.

By (3.4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M203">View MathML</a>

(3.5)

Multiplying (3.5) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M204">View MathML</a>, summing from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M205">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M206">View MathML</a>, and using the properties of the operator b, we easily get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M207">View MathML</a>

(3.6)

Observing (H3) and using the Young inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M208">View MathML</a>

(3.7)

Considering the above inequality in (3.6) and observing that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M209">View MathML</a>

(3.8)

we easily get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M210">View MathML</a>

(3.9)

From the above inequality and the Gronwall inequality, one has that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M211">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M212">View MathML</a>, also in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M213">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M214">View MathML</a>. Observing (3.5), we know that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M215">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M216">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M214">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M218">View MathML</a> (we relabel the same) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M219">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M214">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M221">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M9">View MathML</a>. Recalling (3.9) and the above analysis, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M223">View MathML</a>

(3.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M224">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M225">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M214">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M227">View MathML</a>

(3.11)

By the properties of operator A and (3.10), we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M228">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M216">View MathML</a>. Reasoning as in [26] on page 76, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M230">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M231">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M214">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M214">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M234">View MathML</a>

(3.12)

and as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M211">View MathML</a> is bounded in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M236">View MathML</a> obtained from (3.9), by (H3), we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M237">View MathML</a>

(3.13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M238">View MathML</a>. Thus we can easily derive from (3.12) and (3.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M239">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M159">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M241">View MathML</a>, then for the solutions u, v,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M242">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M244">View MathML</a> is measurable.

(H2) For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M246">View MathML</a> is continuous.

(H3) There are two nonnegative functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M247">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M248">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M160">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M250">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M153">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M252">View MathML</a>

To construct a multi-valued process, we introduce symbols <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M253">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M254">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M166">View MathML</a> as two phase spaces. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M256">View MathML</a> denote the set of global solutions of (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M257">View MathML</a> and the initial datum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M258">View MathML</a>.

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, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M259">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M260">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M261">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M262">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M263">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M264">View MathML</a>, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M265">View MathML</a>

(4.1)

From (4.1), we can find that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M266">View MathML</a>

(4.2)

and for any σ: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M267">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M268">View MathML</a>

(4.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M269">View MathML</a>.

Lemma 4.1Suppose that (H1′)-(H3′) hold, for any initial datum<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M258">View MathML</a>and any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M271">View MathML</a>, it holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M272">View MathML</a>

(4.4)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M273">View MathML</a>.

Proof Let u be a solution of (3.2), so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M271">View MathML</a>. Multiplying (1.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M275">View MathML</a> and using the energy equality and the Poincaré inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M276">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M277">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M278">View MathML</a>

(4.5)

where we have denoted

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M279">View MathML</a>

Considering

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M280">View MathML</a>

(4.6)

and integrating (4.5) from τ to t, we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M281">View MathML</a>

(4.7)

where we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M282">View MathML</a>

Observing the above estimates, we easily deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M283">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M259">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M261">View MathML</a> defined above are pullback-absorbing. To obtain this, we propose the assumptions

(H4′)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M286">View MathML</a>

and the relation among constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M287">View MathML</a>, d defined above, and α in (1.1) satisfies

(H5′)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M288">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M289">View MathML</a> satisfies

(H6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M290">View MathML</a>

where the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M291">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M292">View MathML</a> is the collection of the sets of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M293">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M294">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M295">View MathML</a> be the class of all families <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M296">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M297">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M298">View MathML</a>. In the same way, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M299">View MathML</a> denote the class of all families <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M300">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M301">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M298">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M303">View MathML</a> be any fixed bounded subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M304">View MathML</a>. Observing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M305">View MathML</a>, which is inclusion-closed, by (H4) and (H5), we deduce that the family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M306">View MathML</a> is contained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M307">View MathML</a>. With regard to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M308">View MathML</a>, we use the same method and obtain a similar conclusion if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M303">View MathML</a> is included in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M310">View MathML</a>.

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

Lemma 4.2Suppose that (H1)-(H6) hold and the constantsα, d, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M311">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M312">View MathML</a>satisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M313">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M314">View MathML</a>.

(1) Then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M28">View MathML</a>and any family<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M316">View MathML</a>, there exits<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M317">View MathML</a>such that any initial datum<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M258">View MathML</a>and any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M271">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M320">View MathML</a>satisfy that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M321">View MathML</a>, where the positive continuous function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M322">View MathML</a>is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M323">View MathML</a>

(2) Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M324">View MathML</a>be included<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M325">View MathML</a>which is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M326">View MathML</a>

Then the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M327">View MathML</a>and is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M307">View MathML</a>-pullback absorbing for the process<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M329">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27">View MathML</a>is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M299">View MathML</a>-pullback absorbing for the process<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M332">View MathML</a>.

Proof Since the proof is a consequence of the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M295">View MathML</a>, we only sketch it here. From Lemma 4.1, (H5) and (H6) , we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M334">View MathML</a>

(4.9)

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

For the two processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M329">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M332">View MathML</a>, 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 4.3Suppose that the assumptions in Lemma 4.2 hold. The two processes<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M332">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M329">View MathML</a>are<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M27">View MathML</a>-asymptotically compact.

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M340">View MathML</a> , a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M341">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M30">View MathML</a> and a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M343">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M344">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M345">View MathML</a>, we shall prove that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M346">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M310">View MathML</a>. By the properties concerning operator b mentioned in Section 3, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M348">View MathML</a>

It is easy to get the above estimate which is independent of n. The sequences of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M343">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M350">View MathML</a> possess their subsequence, relabeled the same in suitable spaces such that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M351">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M352">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M353">View MathML</a>

(4.10)

According to the assumptions on a function g and analogously as in Theorem 3.1, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M354">View MathML</a> strongly in V a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M355">View MathML</a>.

By the Lebesgue theorem and the uniform estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M356">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M357">View MathML</a>, we deduce that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M358">View MathML</a> converges to the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M359">View MathML</a> strongly. Therefore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M360">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M361">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M362">View MathML</a>

(4.11)

The uniform estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M350">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M364">View MathML</a> implies that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M343">View MathML</a> is equicontinuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M130">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M367">View MathML</a>. In addition, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M343">View MathML</a> is bounded, which is independent of n in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M369">View MathML</a>. Using the Ascoli-Arzelà theorem, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M370">View MathML</a>

(4.12)

From the uniform boundedness of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M343">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M369">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M373">View MathML</a>, we can also obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M374">View MathML</a>, weakly in V.

By the analogous argument, for any compact sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M375">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M376">View MathML</a>, we obtain that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M377">View MathML</a> is convergent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M378">View MathML</a> weakly in V. To achieve our result in Lemma 4.3, we only need to prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M379">View MathML</a>

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) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M380">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M381">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M382">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M383">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M384">View MathML</a>. We shall see <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M385">View MathML</a> in V. In order to achieve the last claim, because the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M377">View MathML</a> is weakly convergent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M378">View MathML</a> in V, we only need the convergence of the norms above. In other words, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M388">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M389">View MathML</a>.

From the weak convergence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M390">View MathML</a> in V, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M391">View MathML</a>

Therefore we have to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M392">View MathML</a>

(4.13)

From the energy equality, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M393">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M394">View MathML</a>

(4.14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M395">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M396">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M360">View MathML</a>, define the continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M398">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M399">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M400">View MathML</a>

By (4.14), it is clear that J and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M401">View MathML</a> are non-increasing functions. By the convergence (4.10), for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M402">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M403">View MathML</a>. Using the same analysis method as in [22], we can deduce that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M404">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M405">View MathML</a>, which gives (4.13) as desired. □

We can apply the technical method for any family in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M307">View MathML</a> in Lemma 4.3. Suppose that the assumptions in Lemma 4.3 hold. We can deduce that the processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M332">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M329">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M409">View MathML</a>-asymptotically compact and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M410">View MathML</a>-asymptotically compact.

Lemma 4.4Suppose that (H1′)-(H6′) hold. The two processes<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M332">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M329">View MathML</a>are semi-continuous and that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M413">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M414">View MathML</a>have compact values in their respective topologies.

Proof In fact, the upper semi-continuity of the process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M329">View MathML</a> can be obtained by similar arguments to those used for the Galerkin sequence in Theorem 3.1.

As to the process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M332">View MathML</a>, applying the same energy-procedure in Lemma 4.3, we shall obtain that in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M417">View MathML</a> 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 4.1Suppose that (H1′)-(H6′) hold. For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M28">View MathML</a>, then there exist global pullback attractors<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M419">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M420">View MathML</a>for the process<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M332">View MathML</a>in the universe of fixed bounded sets and in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M299">View MathML</a>, respectively. Moreover, they are unique in the sense of Lemma 2.5 and negatively and strictly invariant forUrespectively, and the following holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M423">View MathML</a>

The above theorem proves that there exist pullback attractors in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M310">View MathML</a> framework, while we shall prove that there exist pullback attractors in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M304">View MathML</a> framework in the following theorem.

Theorem 4.2Suppose that the assumptions in Theorem 4.1 hold. For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M49">View MathML</a>, there exist global pullback attractors<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M427">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M428">View MathML</a>for the process<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M329">View MathML</a>in the universes of fixed bounded sets and in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M307">View MathML</a>. They are unique in the sense of Lemma 2.5 and negatively and strictly invariant forU, respectively, and we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M431">View MathML</a>. Moreover, the relationship between the attractors for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M329">View MathML</a>and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M332">View MathML</a>is as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M434">View MathML</a>

(4.15)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M435">View MathML</a>is the continuous mapping defined by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M436">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M65">View MathML</a> maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M304">View MathML</a> into bounded sets in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M310">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M440">View MathML</a>, and also maps bounded sets from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M304">View MathML</a> into bounded sets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M310">View MathML</a>.

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M443">View MathML</a> is the minimal closed set, and using Lemma 2.5 and the above arguments, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M444">View MathML</a> also attracts bounded sets in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M304">View MathML</a> in a pullback sense. Therefore the inclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M446">View MathML</a> holds.

As to the opposite inclusion of the first identification in (4.15), for any bounded set B, it follows from the continuous injection <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M447">View MathML</a> and the attractor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M448">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/191/mathml/M449">View MathML</a>, whence the opposite inclusion of the first identification in (4.15) holds.

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

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Haiyan carried out the main part of this manuscript. Yuming participated in the discussion and corrected the main theorems. All authors read and approved the final manuscript.

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).

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