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Uniform attractors for non-autonomous suspension bridge-type equations

Xuan Wang12*, Lu Yang2 and Qiaozhen Ma1

Author Affiliations

1 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, P.R. China

2 School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, P.R. China

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Boundary Value Problems 2014, 2014:75  doi:10.1186/1687-2770-2014-75

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/75


Received:5 July 2013
Accepted:18 March 2014
Published:28 March 2014

© 2014 Wang et al.; 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

We discuss the long-time dynamical behavior of the non-autonomous suspension bridge-type equation, where the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M1">View MathML</a> is translation compact and the time-dependent external forces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M2">View MathML</a> only satisfy Condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3">View MathML</a>) instead of being translation compact. By applying some new results and the energy estimate technique, the existence of uniform attractors is obtained. The result improves and extends some known results.

MSC: 34Q35, 35B40, 35B41.

Keywords:
non-autonomous suspension bridge equation; uniform Condition (C); uniform attractor

1 Introduction

Consider the following equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M4">View MathML</a>

(1.1)

Suspension bridge equations (1.1) have been posed as a new problem in the field of nonlinear analysis [1] by Lazer and McKenna in 1990. This model has been derived as follows. In the suspension bridge system, the suspension bridge can be considered as an elastic and unloaded beam with hinged ends. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M5">View MathML</a> denotes the deflection in the downward direction; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M6">View MathML</a> represents the viscous damping. The restoring force can be modeled owing to the cable with one-sided Hooke’s law so that it strongly resists expansion but does not resist compression. The simplest function to model the restoring force of the stays in the suspension bridge can be denoted by a constant k times u, the expansion, if u is positive, but zero if u is negative, corresponding to compression; that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M7">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M8">View MathML</a>

Besides, the right-hand side of (1.1) also contains two terms: the large positive term l corresponding to gravity, and a small oscillatory forcing term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M9">View MathML</a>, possibly aerodynamic in origin, where ϵ is small.

There are many results for the problem (1.1) (cf.[1-7]), for instance, the existence, multiplicity and properties of the traveling wave solutions, etc.

In the study of equations of mathematical physics, the attractor is a proper mathematical concept as regards the depiction of the behavior of the solutions of these equations when time is large or tends to infinity, which describes all the possible limits of solutions. In the past two decades, many authors have proved the existence of an attractor and discussed its properties for various mathematical physics models (e.g., see [8-10] and the references therein). For the long-time behavior of suspension bridge-type equations, for the autonomous case, in [11,12] the authors have discussed long-time behavior of the solutions of the problem on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M10">View MathML</a> and obtained the existence of global attractors in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M12">View MathML</a>.

It is well known that, for a model to describe the real world which is affected by many kinds of factors, the corresponding non-autonomous model is more natural and precise than the autonomous one, moreover, it always presents a nonlinear equation but not just a linear one. Therefore, in this paper, we will discuss the following non-autonomous suspension bridge-type equation: Let Ω be an open bounded subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M13">View MathML</a> with smooth boundary, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M14">View MathML</a>, and we add the nonlinear forcing term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M1">View MathML</a> (which is dependent on the deflection u and time t) to (1.1) and neglect gravity, then we can obtain the following initial-boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M16">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M5">View MathML</a> is an unknown function, which could represent the deflection of the road bed in the vertical plane; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M1">View MathML</a> are time-dependent external forces; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M7">View MathML</a> represents the restoring force, k denotes the spring constant; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M21">View MathML</a> represents the viscous damping, α is a given positive constant.

To our knowledge, this is the first time for one to consider the non-autonomous dynamics of equation (1.2). At the same time, in mathematics, we only assume that the force term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M2">View MathML</a> satisfies the so-called Condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3">View MathML</a>) (introduced in [13]), which is weaker than the assumption of being translation compact (see [8] or Section 2 below).

This paper is organized as follows. At first, in Section 2, we give (recall) some preliminaries, including the notation we will use, the assumption on nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M24">View MathML</a> and some general abstract results for a non-autonomous dynamical system. In Section 3 we prove our main result about the existence of a uniform attractor for the non-autonomous dynamical system generated by the solution of (1.2).

2 Notation and preliminaries

With the usual notation, we introduce the spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M27">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M28">View MathML</a>. We equip these spaces with an inner product and a norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M34">View MathML</a>, respectively,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M35">View MathML</a>

Obviously, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M36">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M38">View MathML</a> is the dual space of H, V, respectively, the injections are continuous and each space is dense in the following one.

In the following, the assumption on the nonlinearity g is given. Let g be a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M39">View MathML</a> function from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M40">View MathML</a> to ℝ and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M41">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M42">View MathML</a>, and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M43">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M44">View MathML</a>

(2.2)

Suppose that γ is an arbitrary positive constant, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M45">View MathML</a>

(2.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M46">View MathML</a>

(2.4)

where δ is a sufficiently small constant.

As a consequence of (2.1)-(2.2), if we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M47">View MathML</a>, then there exist two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M49">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M50">View MathML</a>

(2.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M51">View MathML</a>

(2.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M52">View MathML</a>, and we can take m sufficiently small.

By virtue of (2.3), we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M53">View MathML</a>

(2.7)

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M28">View MathML</a>, the problem (1.2) is equivalent to the following equations in H:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M55">View MathML</a>

(2.8)

From the Poincaré inequality, there exists a proper constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M56">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M57">View MathML</a>

(2.9)

We introduce the Hilbert spaces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M58">View MathML</a>

and endow this space with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M59">View MathML</a>

To prove the existence of uniform attractors corresponding to (2.8), we also need the following abstract results (e.g., see [8]).

Let E be a Banach space, and let a two-parameter family of mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M60">View MathML</a> on E:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M61">View MathML</a>

Definition 2.1 ([8])

Let Σ be a parameter set. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M63">View MathML</a> is said to be a family of processes in Banach space E, if for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65">View MathML</a> is a process; that is, the two-parameter family of mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65">View MathML</a> from E to E satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M67">View MathML</a>

(2.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M68">View MathML</a>

(2.11)

where Σ is called the symbol space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64">View MathML</a> is the symbol.

Note that the following translation identity is valid for a general family of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64">View MathML</a>, if a problem has unique solvability and for the translation semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M72">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M73">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M74">View MathML</a>

A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M75">View MathML</a> is said to be a uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M76">View MathML</a>) absorbing set for the family of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64">View MathML</a> if for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M79">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M80">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M81">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M82">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M83">View MathML</a>. A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M84">View MathML</a> is said to be uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64">View MathML</a>) attracting for the family of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64">View MathML</a>, if for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M88">View MathML</a> and every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M80">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M90">View MathML</a>

(2.12)

Definition 2.2 ([8])

A closed set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M91">View MathML</a> is said to be the uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64">View MathML</a>) attractor of the family of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64">View MathML</a> if it is uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64">View MathML</a>) attracting (attracting property) and contained in any closed uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M76">View MathML</a>) attracting set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M97">View MathML</a> of the family of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64">View MathML</a>: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M100">View MathML</a> (minimality property).

Now we recall the results in [14].

Definition 2.3 ([14])

A family of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64">View MathML</a>, is said to be satisfying the uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64">View MathML</a>) Condition (C) if for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M106">View MathML</a>, there exist a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M107">View MathML</a> and a finite dimensional subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M108">View MathML</a> of E such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M109">View MathML</a> is bounded; and

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M111">View MathML</a>,

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M113">View MathML</a> is abounded projector.

Theorem 2.4 ([14])

Let Σ be a complete metric space, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M114">View MathML</a>be a continuous invariant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M115">View MathML</a>semigroup on Σ satisfying the translation identity. A family of processes<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64">View MathML</a>, possess a compact uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64">View MathML</a>) attractor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M119">View MathML</a>inEsatisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M120">View MathML</a>

(2.13)

if it

(i) has a bounded uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64">View MathML</a>) absorbing set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M122">View MathML</a>; and

(ii) satisfies uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M64">View MathML</a>) Condition (C),

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M124">View MathML</a>. Moreover, ifEis a uniformly convex Banach space, then the converse is true.

Let X be a Banach space. Consider the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M125">View MathML</a> of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M127">View MathML</a> with values in X that are 2-power integrable in the Bochner sense. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M128">View MathML</a> is a set of all translation compact functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M130">View MathML</a> is the set of all translation bound functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M129">View MathML</a>.

In [13], the authors have introduced a new class of functions which are translation bounded but not translation compact. In Section 3, let the forcing term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M2">View MathML</a> satisfy Condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3">View MathML</a>); we can prove the existence of compact uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M135">View MathML</a>) attractor for a non-autonomous suspension bridge equation in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>.

Definition 2.5 ([13])

Let X be a Banach space. A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M137">View MathML</a> is said to satisfy Condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3">View MathML</a>) if, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M106">View MathML</a>, there exists a finite dimensional subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M140">View MathML</a> of X such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M141">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M142">View MathML</a> is the canonical projector.

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M143">View MathML</a> the set of all functions satisfying Condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3">View MathML</a>). From [13], we can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M145">View MathML</a>.

Remark 2.6 In fact, the function satisfying Condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3">View MathML</a>) implies the dissipative property in some sense, and Condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3">View MathML</a>) is very natural in view of the compact condition, and the uniform Condition (C).

Lemma 2.7 ([13])

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M148">View MathML</a>, then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M106">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M88">View MathML</a>we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M151">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M142">View MathML</a>is the canonical projector andδis a positive constant.

In order to define the family of processes of the equations (2.8), we also need the following results:

Proposition 2.8 ([8])

IfXis reflexive separable, then

(i) for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M154">View MathML</a>;

(ii) the translation group<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M114">View MathML</a>is weakly continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M156">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M157">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M158">View MathML</a>.

Proposition 2.9 ([8])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M159">View MathML</a>, then

(i) for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M161">View MathML</a>, and the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M162">View MathML</a>is bound in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M130">View MathML</a>;

(ii) the translation group<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M114">View MathML</a>is continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M162">View MathML</a>with the topology of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M166">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M167">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M158">View MathML</a>.

3 Uniform attractors in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>

To describe the asymptotic behavior of the solutions of our system, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M170">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M171">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M172">View MathML</a> denotes the closure of a set in topological space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M173">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M174">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M175">View MathML</a>; this is to be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M176">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M177">View MathML</a> denotes the norm in H.

3.1 Existence and uniqueness of solutions

At first, we give the concept of solutions for the initial-boundary value problem (2.8).

Definition 3.1 Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M178">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M179">View MathML</a>. We suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M182">View MathML</a> satisfying (2.1)-(2.4) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M183">View MathML</a>. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M184">View MathML</a> is said to be a weak solution to the problem (2.8) in the time interval I, with initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M185">View MathML</a>, provided

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M186">View MathML</a>

(3.1)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M187">View MathML</a> and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M188">View MathML</a>.

Then, by using of the methods in [15] (Galerkin approximation method), we get the following result as regards the existence and uniqueness of solutions:

Theorem 3.2 (Existence and uniqueness of solutions)

Define<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M190">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M180">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M182">View MathML</a>satisfying (2.1)-(2.4). Then for any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M194">View MathML</a>, there is a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M195">View MathML</a>for the problem (2.8) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>. Furthermore, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M197">View MathML</a>, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M198">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M199">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M200">View MathML</a>) be two initial conditions, and denote by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M201">View MathML</a>the corresponding solutions to the problem (2.8). Then the estimates hold as follows: for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M202">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M203">View MathML</a>

(3.2)

Thus, (2.8) will be written as an evolutionary system, introduced as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M205">View MathML</a> for brevity, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M206">View MathML</a>, the system (2.8) can be written in the operator form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M207">View MathML</a>

(3.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M208">View MathML</a> is the symbol of (3.3). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M194">View MathML</a>, then the problem (3.3) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M210">View MathML</a>. This implies that the process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65">View MathML</a> given by the formula <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M212">View MathML</a> is defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>.

Now we define the symbol space. A fixed symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M214">View MathML</a> can be given, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M215">View MathML</a> is in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M216">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M217">View MathML</a> satisfying (2.1)-(2.4), and ℳ is a Banach space,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M218">View MathML</a>

endowed with the following norm:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M219">View MathML</a>

Obviously, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M214">View MathML</a> is in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M221">View MathML</a>. We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M222">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M223">View MathML</a> denotes the closure of a set in topological space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M224">View MathML</a> (or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M225">View MathML</a>). So, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M226">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M2">View MathML</a> all satisfy Condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3">View MathML</a>).

Applying Proposition 2.8, Proposition 2.9, and Theorem 3.2, we can easily know that the family of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M230">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M232">View MathML</a>, are defined. Furthermore, the translation semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M233">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M234">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M235">View MathML</a>, and the following translation identity:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M236">View MathML</a>

holds.

Then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134">View MathML</a>, the problem (3.3) with σ instead of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M238">View MathML</a> possesses a corresponding process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65">View MathML</a> acting on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>.

Consequently, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M214">View MathML</a> (here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M244">View MathML</a> satisfying (2.1)-(2.4)), we can define a process

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M245">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134">View MathML</a>, is a family of processes on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>.

3.2 Bounded uniformly absorbing set

Before we show the existence of bounded uniformly absorbing set, we firstly make a prior estimate of solutions for equations (2.8) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>.

Lemma 3.3Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M250">View MathML</a>is a solution of (2.8) with initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M251">View MathML</a>. If the nonlinearity<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M1">View MathML</a>satisfies (2.1)-(2.4), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M253">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M180">View MathML</a>, then there is a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M256">View MathML</a>such that for any bounded (in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>) subsetB, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M258">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M259">View MathML</a>

(3.4)

Proof Now we will prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M195">View MathML</a> to be bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M261">View MathML</a>.

We assume that ϱ is positive and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M262">View MathML</a>

(3.5)

Multiplying (2.8) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M263">View MathML</a> and integrating over Ω, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M264">View MathML</a>

(3.6)

We easily see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M265">View MathML</a>

(3.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M266">View MathML</a>

(3.8)

Then, substituting (3.7)-(3.8) into (3.6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M267">View MathML</a>

(3.9)

In view of (2.4) and (2.6), we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M268">View MathML</a>

(3.10)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M269">View MathML</a>

(3.11)

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M270">View MathML</a>

(3.12)

We introduce the functional as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M271">View MathML</a>

(3.13)

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M272">View MathML</a>, we choose proper positive constants m and δ, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M273">View MathML</a>

(3.14)

hold, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M274">View MathML</a>.

We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M275">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M276">View MathML</a>

(3.15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M277">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M278">View MathML</a>.

Analogous to the proof of Lemma 2.1.3 in [8], we can estimate the integral and obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M279">View MathML</a>

(3.16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M280">View MathML</a>.

By virtue of (2.3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M281">View MathML</a>

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M282">View MathML</a>, we obtain from (3.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M283">View MathML</a>

(3.17)

In consideration of (2.7) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M284">View MathML</a>, we see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M285">View MathML</a>

(3.18)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M286">View MathML</a>

(3.19)

Combining (3.16), (3.17), and (3.19), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M287">View MathML</a>

Assuming that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M288">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M289">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M290">View MathML</a>

(3.20)

We thus complete the proof. □

And then, combining Theorem 3.2 with Lemma 3.3, we get the result as follows.

Theorem 3.4 (Bounded uniformly absorbing set)

Presume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M291">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M253">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M293">View MathML</a>satisfy (2.1)-(2.4), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M174">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M296">View MathML</a>be the family of processes corresponding to (2.8) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65">View MathML</a>has a uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134">View MathML</a>) absorbing set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M300">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>. That is, for any bounded subset<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M302">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M258">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M304">View MathML</a>

3.3 The existence of uniform attractor

We will show the existence of uniform attractor to the problem (2.8) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>.

Theorem 3.5 (Uniform attractor)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M306">View MathML</a>be the family of processes corresponding to the problem (2.8). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M307">View MathML</a>satisfying (2.1), (2.2), (2.5), and (2.6), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M253">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M309">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65">View MathML</a>possesses a compact uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134">View MathML</a>) attractor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M312">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>, which attracts any bounded set in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M315">View MathML</a>, satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M316">View MathML</a>

(3.21)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M122">View MathML</a>is the uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134">View MathML</a>) absorbing set in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>.

Proof From Theorem 2.4 and Theorem 3.4, we merely need to prove that the family of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134">View MathML</a> satisfy the uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134">View MathML</a>) Condition (C) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>. We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M324">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M325">View MathML</a> are eigenvalue of operator A in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M326">View MathML</a>, satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M327">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M328">View MathML</a> denotes eigenvector corresponding to eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M324">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M330">View MathML</a> , which forms an orthogonal basis in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M326">View MathML</a>, at the same time they are also a group of canonical basis in V or H, and they satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M332">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M333">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M334">View MathML</a> is an orthogonal projector. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M335">View MathML</a>, we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M336">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M337">View MathML</a>.

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M338">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M339">View MathML</a>. Taking the scalar product with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M340">View MathML</a> for (2.8) in H, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M341">View MathML</a>

(3.22)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M342">View MathML</a>

(3.23)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M343">View MathML</a>

(3.24)

Clearly, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M344">View MathML</a>

(3.25)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M345">View MathML</a>

(3.26)

Combining (3.23)-(3.26), we obtain from (3.22)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M346">View MathML</a>

(3.27)

We define the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M347">View MathML</a>

and we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M348">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M349">View MathML</a>

(3.28)

By Gronwall’s lemma, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M350">View MathML</a>

(3.29)

Obviously, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M351">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M352">View MathML</a>

so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M353">View MathML</a>

(3.30)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M354">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M355">View MathML</a>, from Lemma 2.7, we can know for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M356">View MathML</a>, there exists a constant m large enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M357">View MathML</a>

(3.31)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M358">View MathML</a>

(3.32)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M359">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M360">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M361">View MathML</a>

So for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M363">View MathML</a>

(3.33)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M364">View MathML</a>.

Therefore, the family of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M365">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134">View MathML</a> satisfy uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134">View MathML</a>) Condition (C) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>. Applying Theorem 2.4, we can obtain the existence of a uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134">View MathML</a>) attractor of the family of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M365">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M134">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M136">View MathML</a>, which satisfies (3.21).

We thus complete the proof. □

So we can draw the conclusion: when the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M1">View MathML</a> is translation compact and the time-dependent external forces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M2">View MathML</a> only satisfies Condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M3">View MathML</a>) instead of translation compact, the uniform attractors in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/75/mathml/M376">View MathML</a> exist.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors read and approved the final manuscript.

Acknowledgements

This work is partly supported by NSFC (11361053, 11201204, 11101134, 11261053, 11101404) of China and the Young Teachers Scientific Research Ability Promotion Plan of Northwest Normal University (NWNU-LKQN-11-5).

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