Open Access Research

Averaging of the 3D non-autonomous Benjamin-Bona-Mahony equation with singularly oscillating forces

Mingxia Zhao1, Xinguang Yang2* and Lingrui Zhang3

Author Affiliations

1 College of Mathematics and Information Science, Pingdingshan University, Pingdingshan, 467009, P.R. China

2 College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, P.R. China

3 College of Education and Teacher Development, Henan Normal University, Xinxiang, 453007, P.R. China

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Boundary Value Problems 2013, 2013:111  doi:10.1186/1687-2770-2013-111


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/111


Received:29 January 2013
Accepted:16 April 2013
Published:30 April 2013

© 2013 Zhao 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

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M1">View MathML</a>, we investigate the convergence of corresponding uniform attractors of the 3D non-autonomous Benjamin-Bona-Mahony equation with singularly oscillating force contrast with the averaged Benjamin-Bona-Mahony equation (corresponding to the limiting case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M2">View MathML</a>). Under suitable assumptions on the external force, we shall obtain the uniform boundedness and convergence of the related uniform attractors as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M3">View MathML</a>.

MSC: 35B40, 35Q99, 80A22.

Keywords:
Benjamin-Bona-Mahony equation; singularly oscillating forces; uniform attractors; translational bounded functions

1 Introduction

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M4">View MathML</a> be a fixed parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M5">View MathML</a> be a bounded domain with sufficiently smooth boundary Ω. We investigate the long-time behavior for the non-autonomous 3D Benjamin-Bona-Mahony (BBM) equation with singularly oscillating forces:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M6">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M7">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M8">View MathML</a>

(1.3)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M10">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M11">View MathML</a> is the velocity vector field, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M12">View MathML</a> is the kinematic viscosity, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M13">View MathML</a> is a nonlinear vector function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M14">View MathML</a> is the singularly oscillating force.

Along with (1.1)-(1.3), we consider the averaged Benjamin-Bona-Mahony equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M15">View MathML</a>

(1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M16">View MathML</a>

(1.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M17">View MathML</a>

(1.6)

formally corresponding to the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M2">View MathML</a> in (1.1).

The function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M19">View MathML</a>

(1.7)

represents the external forces of problem (1.1)-(1.3) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M20">View MathML</a> and of problem (1.4)-(1.6) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M2">View MathML</a>, respectively.

The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M23">View MathML</a> are taken from the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M24">View MathML</a> of translational bounded functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M25">View MathML</a>, namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M26">View MathML</a>

(1.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M27">View MathML</a>

(1.9)

for some constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M28">View MathML</a>.

Defining

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M29">View MathML</a>

as a straightforward consequence of (1.7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M30">View MathML</a>

(1.10)

note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M31">View MathML</a> is of the order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M32">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M33">View MathML</a>.

The BBM equation is a well-known model for long waves in shallow water which was introduced by Benjamin, Bona, and Mahony ([1], 1972) as an improvement of the Korteweg-de Vries equation (KdV equation) for modeling long waves of small amplitude in two dimensions. Contrasting with the KdV equation, the BBM equation is unstable in high wavenumber components. Further, while the KdV equation has an infinite number of integrals of motion, the BBM equation only has three. For more results on the wellposedness and infinite dimensional dynamical systems for BBM equations, we can refer to [2-7].

In this paper, firstly, we shall study the asymptotic behavior of the non-autonomous BBM equation depending on the small parameter ε, which reflects the rate of fast time oscillations in the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M34">View MathML</a> with amplitude of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M32">View MathML</a>, then we shall consider the boundedness and convergence of corresponding uniform attractors of (1.1)-(1.3) in contrast to (1.4)-(1.6).

2 Preliminaries

Throughout this paper, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M36">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M37">View MathML</a>) is the generic Lebesgue space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M38">View MathML</a> is the Sobolev space. We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M39">View MathML</a>, H, V, W is the closure of the set E in the topology of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M42">View MathML</a> respectively. ‘⇀’ stands for the weak convergence of sequences.

Lemma 2.1For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M43">View MathML</a>, every nonnegative locally summable functionϕon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M44">View MathML</a>and every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M45">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M46">View MathML</a>

(2.1)

holds for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M47">View MathML</a>.

Proof See, e.g., [8]. □

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M48">View MathML</a>fulfill that for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M47">View MathML</a>, the differential inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M50">View MathML</a>

(2.2)

where, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M47">View MathML</a>, the scalar functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M52">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M53">View MathML</a>satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M54">View MathML</a>

(2.3)

for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M56">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M57">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M58">View MathML</a>

(2.4)

Proof See, e.g., [8]. □

For the non-autonomous general Benjamin-Bona-Mahony (BBM) equation,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M59">View MathML</a>

(2.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M60">View MathML</a>

(2.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M61">View MathML</a>

(2.7)

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M62">View MathML</a>, the nonlinear vector function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M64">View MathML</a>, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M65">View MathML</a>

(2.8)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M66">View MathML</a>

(2.9)

In addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M67">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M68">View MathML</a>) is a smooth function satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M69">View MathML</a>

(2.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M70">View MathML</a>

(2.11)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M71">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M73">View MathML</a> are positive constants.

Similar to [5], by the Galerkin method and a priori estimate, we easily derive the existence of a global weak solution and a uniform attractor which shall be stated in the following theorems.

Theorem 2.3Assume that (2.8)-(2.11) hold, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M62">View MathML</a> (orV) , then there exists a unique global weak solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M76">View MathML</a>of the problem (2.5)-(2.7) which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M77">View MathML</a>

(2.12)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M43">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M79">View MathML</a>.

Theorem 2.4Assume that the external force<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M74">View MathML</a>and (2.8)-(2.11) hold, then the processes<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M81">View MathML</a>generated by the global solution possess uniform attractors<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M82">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M83">View MathML</a>for the non-autonomous system (2.5)-(2.7).

3 Some lemmas

Lemma 3.1The functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M22">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M23">View MathML</a>are taken from the space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M24">View MathML</a>of translational bounded functions in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M25">View MathML</a>, then the processes<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M88">View MathML</a>generated by system (1.1)-(1.3) have a uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M89">View MathML</a>) compact attractor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90">View MathML</a>for any fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M1">View MathML</a>.

Proof As a similar argument in Section 2, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M92">View MathML</a> in Theorem 2.4, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M94">View MathML</a> are translational bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M25">View MathML</a>, then for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M97">View MathML</a> is translational bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M25">View MathML</a> and we can easily deduce the existence of uniformly compact attractors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90">View MathML</a>. □

We can briefly describe the structure of the uniform attractor as follows: if the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M101">View MathML</a> are translational bounded, problem (1.1)-(1.3) generates the dynamical processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M102">View MathML</a> acting on V which is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M47">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M105">View MathML</a> is the solution to (1.1)-(1.3). The processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M102">View MathML</a> have a uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M107">View MathML</a>) absorbing set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M108">View MathML</a>

(3.1)

which is bounded in V for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M1">View MathML</a>.

On the other hand, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90">View MathML</a> is also bounded in V for each fixed ε since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M111">View MathML</a>. Assuming <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M112">View MathML</a>, the external force <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M113">View MathML</a> appearing in equation (1.1) belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M114">View MathML</a> also. Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M116">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M117">View MathML</a>

(3.2)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M118">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M119">View MathML</a>. In this case, to describe the structure of the uniform attractor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90">View MathML</a>, we consider the family of equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M121">View MathML</a>

(3.3)

For every external force <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M122">View MathML</a>, equation (3.3) generates a class of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M123">View MathML</a> on V, which shares similar properties to those of the processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M124">View MathML</a>, corresponding to the original equation (1.1) with the external force <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M113">View MathML</a>. Moreover, the map

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M126">View MathML</a>

(3.4)

is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M127">View MathML</a>-continuous.

Lemma 3.2If the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M128">View MathML</a>in (1.4) is taken from the space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M24">View MathML</a>of translational bounded functions in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M25">View MathML</a>, then the processes<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M131">View MathML</a>generated by system (1.4)-(1.6) have a uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M132">View MathML</a>) compact attractor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M133">View MathML</a>.

Proof Use a similar technique as that in Theorem 2.4, we can easily deduce the existence of a uniformly compact attractor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M133">View MathML</a> if we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M135">View MathML</a>. □

4 Uniform boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90">View MathML</a>

Firstly, we shall consider the auxiliary linear equation with a non-autonomous external force and give some useful lemmas, and then we shall prove the uniform boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90">View MathML</a>.

Considering the linear equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M138">View MathML</a>

(4.1)

we get the following lemma.

Lemma 4.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M139">View MathML</a>, then problem (4.1) has a unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M140">View MathML</a>

(4.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M141">View MathML</a>

(4.3)

Moreover, the following inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M142">View MathML</a>

(4.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M143">View MathML</a>

(4.5)

hold for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M47">View MathML</a>and some constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M145">View MathML</a>, independent of the initial time<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M43">View MathML</a>.

Proof Firstly, using the Galerkin approximation method, we can deduce the existence of a global solution for (4.1), here we omit the details.

Then multiplying (4.1) by Y and AY respectively, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M147">View MathML</a>

(4.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M148">View MathML</a>

(4.7)

By the Gronwall inequality and Poincaré inequality, we can easily prove the lemma. □

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M43">View MathML</a>, we have the following lemma.

Lemma 4.2Assume that the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M152">View MathML</a>

(4.8)

holds for some constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M153">View MathML</a>, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M154">View MathML</a>. Then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M155">View MathML</a>yields the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M156">View MathML</a>

(4.9)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M1">View MathML</a>satisfying the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M158">View MathML</a>

(4.10)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M145">View MathML</a>is constant independent ofK.

Moreover, we also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M160">View MathML</a>

(4.11)

Proof Noting that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M161">View MathML</a>

(4.12)

we can derive the following estimates from (4.8):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M162">View MathML</a>

From Lemma 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M163">View MathML</a>

(4.13)

Hence, from the Poincaré inequality, combining (4.12) and (4.4)-(4.5), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M164">View MathML</a>

(4.14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M165">View MathML</a>

(4.15)

Setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M166">View MathML</a>

(4.16)

we deduce that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M47">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M168">View MathML</a>

(4.17)

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M169">View MathML</a>.

Integrating (4.9) with respect to time variable from τ to t, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M170">View MathML</a> is a solution to the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M171">View MathML</a>

(4.18)

such that from (4.13) and (4.14), we can derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M172">View MathML</a>

(4.19)

By virtue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M175">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M176">View MathML</a>

(4.20)

Hence, we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M177">View MathML</a>

(4.21)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M178">View MathML</a>

(4.22)

The proof is finished. □

Now, we shall use the auxiliary linear equation and some estimates to prove the uniform boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90">View MathML</a> in V. For convenience, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M180">View MathML</a>

(4.23)

and assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M181">View MathML</a>

(4.24)

holds for some constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M153">View MathML</a>.

Theorem 4.3The attractors<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90">View MathML</a>of problem (1.1)-(1.3) (or (1.4)-(1.6)) are uniformly (w.r.t. ε) bounded inV, namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M184">View MathML</a>

(4.25)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M185">View MathML</a> be the solution to (1.1)-(1.3) with the initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M186">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M20">View MathML</a>, we consider the auxiliary linear equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M188">View MathML</a>

(4.26)

From Lemma 4.2, we have the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M189">View MathML</a>

(4.27)

Setting the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M190">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M191">View MathML</a>

(4.28)

which satisfies the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M192">View MathML</a>

(4.29)

Taking the scalar product of (4.28) with w, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M193">View MathML</a>

(4.30)

Using the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M194">View MathML</a>

(4.31)

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M195">View MathML</a>

(4.32)

where λ is the first eigenvalue of −Δ.

Moreover, notice that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M196">View MathML</a>

(4.33)

and inserting (4.29)-(4.30) into (4.28), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M197">View MathML</a>

(4.34)

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M198">View MathML</a>

(4.35)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M199">View MathML</a>

(4.36)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M200">View MathML</a>

(4.37)

Therefore using (1.8), we derive from (4.33)-(4.36) that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M47">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M202">View MathML</a>

(4.38)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M203">View MathML</a>

(4.39)

Applying Lemma 2.2 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M204">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M205">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M207">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M208">View MathML</a>

(4.40)

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M209">View MathML</a>

(4.41)

Recalling that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M210">View MathML</a>, and using (4.25) and (4.37), we end up with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M211">View MathML</a>

(4.42)

Thus, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M212">View MathML</a>, the processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M213">View MathML</a> have an absorbing set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M214">View MathML</a>

(4.43)

On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M215">View MathML</a>, the processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M213">View MathML</a> also possess an absorbing set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M217">View MathML</a>

(4.44)

In conclusion, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M218">View MathML</a>, the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M219">View MathML</a>

(4.45)

is an absorbing set for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M213">View MathML</a> which is independent of ε. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M221">View MathML</a>, (4.24) follows and hence the proof is complete. □

5 Convergence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M223">View MathML</a>

The main result of the paper reads as follows.

Theorem 5.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M224">View MathML</a>and (4.23) holds. Then the uniform attractor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M90">View MathML</a> (for problem (1.1)-(1.3)) converges to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M133">View MathML</a> (for problem (1.4)-(1.6)) as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M227">View MathML</a>in the following sense:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M228">View MathML</a>

(5.1)

Next, we shall study the difference of two solutions for (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M20">View MathML</a> and (1.4) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M2">View MathML</a> which share the same initial data. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M231">View MathML</a>

(5.2)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M232">View MathML</a> belonging to the absorbing set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M233">View MathML</a> which can be found in Section 4. In particular, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M234">View MathML</a>, the formula corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M2">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M236">View MathML</a>

(5.3)

holds for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M237">View MathML</a>, as the size of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M238">View MathML</a> depends on ρ.

Lemma 5.2For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M234">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M242">View MathML</a>, the difference

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M243">View MathML</a>

(5.4)

satisfies the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M244">View MathML</a>

(5.5)

for some positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M245">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M246">View MathML</a>, both independent of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M20">View MathML</a>.

Proof Since the difference <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M190">View MathML</a> solves the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M249">View MathML</a>

(5.6)

the difference

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M250">View MathML</a>

(5.7)

fulfills the Cauchy problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M251">View MathML</a>

(5.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M252">View MathML</a> is the solution to (4.25).

Taking an inner product of equation (5.8) with q in H, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M253">View MathML</a>

(5.9)

Noting that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M254">View MathML</a>

(5.10)

where λ is the first eigenvalue of −Δ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M255">View MathML</a> is the upper bound for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M256">View MathML</a> (by Lemma 3.1) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M257">View MathML</a>

thus, it follows from (5.9) and (5.10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M258">View MathML</a>

(5.11)

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M259">View MathML</a>, by the Gronwall inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M260">View MathML</a>

(5.12)

Moreover, we can derive the following formulas:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M261">View MathML</a>

(5.13)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M262">View MathML</a>

(5.14)

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M263">View MathML</a>

(5.15)

holds for some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M264">View MathML</a>. Finally, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M265">View MathML</a>, using (4.26) to control <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M266">View MathML</a>, we may obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M267">View MathML</a>

(5.16)

where R is a positive constant. The proof is finished. □

Next, we want to generalize Lemma 5.2 to derive the convergence of corresponding uniform attractors. Let the external force in equation (3.3) as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M268">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M269">View MathML</a> satisfies inequality (5.22).

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M270">View MathML</a>

(5.17)

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M271">View MathML</a>

(5.18)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M272">View MathML</a>, we observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M273">View MathML</a> is a solution to (3.3) with the external force <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M274">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M275">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M20">View MathML</a>, we investigate the property of the difference

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M277">View MathML</a>

(5.19)

Lemma 5.3The inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M278">View MathML</a>

(5.20)

holds, hereDandRare defined in Lemma 5.2.

Proof As the similar discussion in the proof of Lemma 5.2, replacing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M279">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M280">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M281">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M256">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M94">View MathML</a>, respectively, noting that (5.1) still holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M285">View MathML</a>, and the family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M123">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M122">View MathML</a>), is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M288">View MathML</a>-continuous, using (5.18) in place of (4.23), we can finally complete the proof of the lemma. □

Proof of Theorem 5.1 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M290">View MathML</a>, we obtain that there exists a complete bounded trajectory <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M291">View MathML</a> of equation (3.3), with some external force

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M292">View MathML</a>

(5.21)

such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M293">View MathML</a>.

We choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M294">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M295">View MathML</a>

(5.22)

From the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M296">View MathML</a>

(5.23)

applying Lemma 5.3 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M297">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M298">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M299">View MathML</a>

(5.24)

On the other hand, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M133">View MathML</a> attracts all sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M301">View MathML</a> uniformly when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M302">View MathML</a>. Then, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M303">View MathML</a>, there exists some time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M304">View MathML</a> which is independent of L such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M305">View MathML</a>

(5.25)

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M306">View MathML</a> and collecting (5.15)-(5.16), we readily get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M307">View MathML</a>

(5.26)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M290">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M303">View MathML</a> is arbitrary, taking the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M3">View MathML</a>, we can prove the theorem. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

All authors give their thanks to the reviewer’s suggestions, XY was in part supported by the Innovational Scientists and Technicians Troop Construction Projects of Henan Province (No. 114200510011) and the Young Teacher Research Fund of Henan Normal University (qd12104).

References

  1. Benjamin, TB, Bona, JL, Mahony, JJ: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A. 272, 47–48 (1972). Publisher Full Text OpenURL

  2. Avrin, J, Goldstein, JA: Global existence for the Benjamin-Bona-Mahony equation in arbitrary dimensions. Nonlinear Anal. TMA. 9(8), 861–865 (1985). Publisher Full Text OpenURL

  3. Goldstein, JA, Wichnoski, BJ: On the Benjamin-Bona-Mahony equation in higher dimensions. Nonlinear Anal. TMA. 4, 665–675 (1980). Publisher Full Text OpenURL

  4. Park, JY, Park, SH: Pullback attractors for the non-autonomous Benjamin-Bona-Mahony equation in bounded domains. Sci. China Math.. 54(4), 741–752 (2011). Publisher Full Text OpenURL

  5. Qin, Y, Yang, X, Liu, X: Pullback attractors for the non-autonomous Benjamin-Bona-Mahony equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M311">View MathML</a>. Acta. Math. Sci.. 32B(4), 1338–1348 (2012)

  6. Stanislavova, M, Stefanow, A, Wang, B: Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/111/mathml/M313">View MathML</a>. J. Differ. Equ.. 219, 451–483 (2005). Publisher Full Text OpenURL

  7. Wang, B: Strong attractors for the Benjamin-Bona-Mahony equation. Appl. Math. Lett.. 10, 23–28 (1997)

  8. Chepyzhov, VV, Vishik, MI: Attractors for Equations of Mathematical Physics, Am. Math. Soc., Providence (2001)