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Uniform attractors for the non-autonomous p-Laplacian equations with dynamic flux boundary conditions

Kun Li1 and Bo You2*

Author Affiliations

1 Department of Basic, Henan Mechanical and Electrical Engineering College, Xinxiang, 453003, P.R. China

2 School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, P.R. China

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Boundary Value Problems 2013, 2013:128  doi:10.1186/1687-2770-2013-128


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/128


Received:27 December 2012
Accepted:1 May 2013
Published:17 May 2013

© 2013 Li and You; 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

This paper studies the long-time asymptotic behavior of solutions for the non-autonomous p-Laplacian equations with dynamic flux boundary conditions in n-dimensional bounded smooth domains. We have proved the existence of the uniform attractor in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a> for the non-autonomous p-Laplacian evolution equations subject to dynamic nonlinear boundary conditions by using the Sobolev compactness embedding theory, and the existence of the uniform attractor in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M2">View MathML</a> by asymptotic a priori estimate.

1 Introduction

We are concerned with the existence of uniform attractors for the process associated with the solutions of the following non-autonomous p-Laplacian equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M3">View MathML</a>

(1)

Equation (1) is subject to the dynamic flux boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M4">View MathML</a>

(2)

and the initial condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M5">View MathML</a>

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M6">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M7">View MathML</a>) is a bounded domain with smooth boundary Γ, ν denotes the outer unit normal on Γ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M8">View MathML</a>, the nonlinearity f and the external force g satisfy some conditions specified later.

Non-autonomous equations appear in many applications in the natural sciences, so they are of great importance and interest. The long-time behavior of solutions of such equations has been studied extensively in recent years (e.g., see [1-4]). The first attempt was to extend the notion of a global attractor to the non-autonomous case, leading to the concept of the so-called uniform attractor (see [5]). It is remarkable that the conditions ensuring the existence of a uniform attractor are parallel with those for the autonomous case. A uniform attractor need not be ‘invariant’, unlike a global attractor for autonomous systems. Moreover, it is well known that the trajectories may be unbounded for many non-autonomous systems when the time tends to infinity, and there does not exist a uniform attractor for these systems.

Dynamic boundary conditions are very natural in many mathematical models such as heat transfer in a solid in contact with a moving fluid, thermoelasticity, diffusion phenomena, heat transfer in two mediums, problems in fluid dynamics (see [1-4,6-11]).

In recent years, many authors have studied p-Laplacian equations (see [12-17]) and the problem (1)-(3) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M9">View MathML</a> (see [3,7,9,10]) by discussing the existence and uniqueness of local solutions, the blow-up of solutions, the global existence of solutions, the global attractors of solutions and the eigenvalue problems, etc. In [18], the authors have proved the global existence of solutions for quasi-linear elliptic equations with dynamic boundary conditions. Due to the complications inherent to nonlinear dynamic boundary conditions, these problems (1)-(3) still need to be investigated. In [15-17,19], the authors have considered the eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M10">View MathML</a>

and obtained some results, and some p-Laplacian elliptic equations with nonlinear boundary condition have been studied by using these results mentioned in [15-17,19]. In [14,20], the authors have proved the existence of uniform attractors for the non-autonomous p-Laplacian equations with Dirichlet boundary conditions in a bounded and an unbounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M11">View MathML</a>. The authors have proved the existence of global attractors for the autonomous p-Laplacian equations with dynamic flux boundary conditions in [21]. In [11], the authors have used a new type of uniformly Gronwall inequality and proved the existence of a pullback attractor in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M12">View MathML</a> of the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M13">View MathML</a>

under the assumptions that f, g satisfy the polynomial growth condition with order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M16">View MathML</a> satisfies some weak assumption

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M17">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M18">View MathML</a>, where θ is some positive constant.

Moreover, the existence of uniform attractors for the non-autonomous p-Laplacian equations with dynamical boundary conditions remains unsolvable.

To study problem (1)-(3), we assume the following conditions.

(H1) The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M19">View MathML</a> and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M20">View MathML</a>

(4)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M21">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M22">View MathML</a>

(5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M23">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M24">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M26">View MathML</a>.

(H2) The external force <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M27">View MathML</a> is locally Lipschitz continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M29">View MathML</a> and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M30">View MathML</a>

(6)

(H3) Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M31">View MathML</a> is uniformly bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M32">View MathML</a> with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M18">View MathML</a>, i.e., there exists a positive constant K such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M34">View MathML</a>

The main purpose of this paper is to study the long-time dynamical behavior for the non-autonomous p-Laplacian evolutionary equations (1)-(3) under quite general assumptions (4)-(6). We first prove the existence and the uniqueness of solutions for (1)-(5), and then the existence of uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>) absorbing sets for the process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36">View MathML</a> corresponding to (1)-(5) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M2">View MathML</a>, respectively, is obtained. Finally, the existence of the uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>) attractor for the process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36">View MathML</a> corresponding to (1)-(5) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M41">View MathML</a> is obtained by the Sobolev compactness embedding theory and the existence of the uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>) attractor for the process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36">View MathML</a> corresponding to (1)-(5) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M44">View MathML</a> is obtained by asymptotic a priori estimate.

This paper is organized as follows. In Section 2, we give some notations and lemmas used in the sequel. The existence and the uniqueness of solutions for the problem (1)-(5) have been proved in Section 3. Section 4 is devoted to proving the existence of the uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>) absorbing sets in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M48">View MathML</a>, respectively, for the process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36">View MathML</a> corresponding to (1)-(5) and the existence of the uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>) attractors in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M48">View MathML</a>, respectively, for the process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36">View MathML</a> corresponding to (1)-(5).

Throughout this paper, we denote the inner product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M32">View MathML</a> (or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M56">View MathML</a>) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M57">View MathML</a>, and let C be a positive constant, which may be different from line to line (and even in the same line); we denote the trace operator by γ.

2 Preliminaries

In order to study the problem (1)-(5), we recall the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M58">View MathML</a> defined as the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M59">View MathML</a> in the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M60">View MathML</a>

and denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M61">View MathML</a> the dual space of X. We also define the Lebesgue spaces as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M62">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M63">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M64">View MathML</a>. Moreover, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M65">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M66">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M67">View MathML</a>, where the measure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M68">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M69">View MathML</a> is defined for any measurable set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M70">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M71">View MathML</a>. In general, any vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M72">View MathML</a> will be of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M73">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M75">View MathML</a>, and there need not be any connection between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M77">View MathML</a>.

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M78">View MathML</a>

and let the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M79">View MathML</a> be defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M80">View MathML</a>

(7)

Next, we recall briefly some lemmas used to prove the well-posedness of the solutions and the existence of the uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) attractors for (1)-(3) under some assumptions on f.

Lemma 2.1[22]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M82">View MathML</a>be a bounded domain in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M11">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M84">View MathML</a>, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M85">View MathML</a>be given. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M86">View MathML</a>, whereCis independent ofn, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M87">View MathML</a>, as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M88">View MathML</a>, almost everywhere in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M82">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M90">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M87">View MathML</a>, as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M88">View MathML</a>weakly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M93">View MathML</a>.

Lemma 2.2[13]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M94">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M95">View MathML</a>be the standard scalar product in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M11">View MathML</a>. Then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M8">View MathML</a>, there exist two positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M98">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M99">View MathML</a>, which depend onp, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M100">View MathML</a>

Lemma 2.3[23]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M101">View MathML</a>and Ω be a bounded subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M11">View MathML</a>with smooth boundary Γ. Then the inclusion

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M103">View MathML</a>

is compact for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M104">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M105">View MathML</a>

Lemma 2.4[24]

LetAbe defined in (7) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M106">View MathML</a>. Then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M107">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M108">View MathML</a>

Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M109">View MathML</a>if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M110">View MathML</a>a.e. in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M111">View MathML</a>.

Lemma 2.5[25]

LetXbe a given Banach space with dual<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M112">View MathML</a>, and letuandgbe two functions belonging to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M113">View MathML</a>. Then the following three conditions are equivalent:

(i) uis almost everywhere equal to a primitive function ofg, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M114">View MathML</a>

for almost every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M115">View MathML</a>;

(ii) For each test function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M116">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M117">View MathML</a>

(iii) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M118">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M119">View MathML</a>

in the scalar distribution sense on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M120">View MathML</a>.

If (i)-(iii) are satisfied, uis almost everywhere equal to a continuous function from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M121">View MathML</a>intoX.

3 The well-posedness of solutions

In what follows, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M122">View MathML</a> is given.

Definition 3.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M123">View MathML</a> is called a weak solution of (1)-(3) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M124">View MathML</a> if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M125">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M126">View MathML</a>

for all test functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M127">View MathML</a>.

Theorem 3.1Let Ω be a bounded domain in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M11">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M7">View MathML</a>). Assume thatfsatisfies (H1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M27">View MathML</a>is locally Lipschitz continuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M29">View MathML</a>. Then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132">View MathML</a>, any initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M122">View MathML</a>and any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M134">View MathML</a>, there exists a unique weak solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M123">View MathML</a>of (1)-(3), and the mapping

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M136">View MathML</a>

is continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a>.

Proof We first prove the existence of solutions for (1)-(5) by the Faedo-Galerkin method (see [25]).

Consider the approximating solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M138">View MathML</a> in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M139">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M140">View MathML</a> is an orthogonal basis of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a>, which is included in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M44">View MathML</a>. We get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M143">View MathML</a> from solving the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M144">View MathML</a>

(8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M145">View MathML</a>

(9)

Since f is continuous and g is locally Lipschitz continuous, using the Peano theorem, we get the local existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M146">View MathML</a>. Next, we establish some a priori estimates for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M146">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M148">View MathML</a>

Thanks to (5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M149">View MathML</a>

(10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M150">View MathML</a>

(11)

by virtue of the following inequality (see Theorem 2.3.1 in [26]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M151">View MathML</a>

(12)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M152">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M153">View MathML</a>, we deduce from (10) and (12) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M154">View MathML</a>

(13)

Integrating (13) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M155">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M156">View MathML</a>

(14)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M157">View MathML</a>.

Due to (14), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M158">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M159">View MathML</a> is uniformly bounded in n in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M161">View MathML</a>, respectively, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M162">View MathML</a> is uniformly bounded in n in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M163">View MathML</a>, and one can extract a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M164">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M159">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M166">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M167">View MathML</a> be a projection. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M168">View MathML</a>, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M169">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M170">View MathML</a>

(15)

We perform the following estimate deduced from the Hölder inequality and the Young inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M171">View MathML</a>

Using the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M159">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M160">View MathML</a> again, we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M174">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M176">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M177">View MathML</a>, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M178">View MathML</a>

Therefore we can extract a subsequence such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M179">View MathML</a>

By virtue of the Aubin compactness theorem, we can extract a further subsequence (still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M164">View MathML</a>) such that additionally

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M181">View MathML</a>

(16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M182">View MathML</a>

(17)

Due to the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M159">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M161">View MathML</a> and (5), we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M185">View MathML</a> is uniformly bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M186">View MathML</a> and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M187">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M186">View MathML</a>, similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M189">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M190">View MathML</a>. By virtue of (16)-(17), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M191">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M111">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M193">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M194">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M195">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M111">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M197">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M194">View MathML</a>. Thanks to Lemma 2.1, we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M199">View MathML</a>

Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M200">View MathML</a>

(18)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M168">View MathML</a>.

In order to prove that u is a weak solution of (1)-(3), it remains to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M202">View MathML</a>. Noticing that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M203">View MathML</a>

(19)

it follows from the formulation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M205">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M206">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M208">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M56">View MathML</a>. Moreover, by the lower semi-continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M210">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M211">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M212">View MathML</a>

(20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M213">View MathML</a>

(21)

Meanwhile, by the Lebesgue dominated theorem, one can check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M214">View MathML</a>

This fact and (20)-(21) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M215">View MathML</a>

(22)

In view of (18), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M216">View MathML</a>

This and (22) deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M217">View MathML</a>

(23)

To this end, we first observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M218">View MathML</a>

On the other hand, it follows from Lemma 2.4 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M219">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M220">View MathML</a>

(24)

Combining (24) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M221">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M160">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M223">View MathML</a>

Therefore, from Lemma 2.2, the Hölder inequality and the Young inequality, we deduce that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M224">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M225">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M226">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M227">View MathML</a>, hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M202">View MathML</a>.

Finally, we prove the uniqueness and continuous dependence of the initial data of the solutions. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M229">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M230">View MathML</a> be two solutions of (1)-(5) with the initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M232">View MathML</a>, respectively. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M233">View MathML</a>. Taking the inner product of the equation with w, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M234">View MathML</a>

(25)

By virtue of (4) and Lemma 2.2, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M235">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M236">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M237">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M238">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M239">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M69">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M123">View MathML</a> is continuously dependent on the initial data.

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M242">View MathML</a>

by use of Lemma 2.5, we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M243">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M244">View MathML</a> is meaningful. □

By Theorem 3.1, we can define a family of continuous processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M245">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a> as follows: For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M247">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M248">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M249">View MathML</a> is the solution of (1)-(5) with initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M250">View MathML</a>. That is, a family of mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M251">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M252">View MathML</a>

4 Existence of uniform attractors

In this section, we prove the existence of uniform attractors for (1)-(3).

4.1 Abstract results

In this subsection, let Σ be a parameter set, let X, Y be two Banach spaces, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M253">View MathML</a> continuously. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a> is a family of processes in a Banach space X. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M255">View MathML</a> the set of all bounded subsets of X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M256">View MathML</a>. In the following, we give some basic definitions and some abstract results about the existence of bi-space uniform (with respect to (w.r.t.) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) attractors.

Definition 4.1[5,27]

A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M258">View MathML</a> is called to be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) absorbing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M261">View MathML</a> if for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132">View MathML</a> and any bounded subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M263">View MathML</a>, there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M264">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M265">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M266">View MathML</a>.

A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M267">View MathML</a> is said to be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) attracting for the family of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a>, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M271">View MathML</a>

for an arbitrary fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132">View MathML</a> and any bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M263">View MathML</a>.

Definition 4.2[5]

A closed set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M274">View MathML</a> is said to be an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259">View MathML</a>-uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) attractor for the family of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a> if it is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) attracting and it is contained in any closed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) attracting set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M282">View MathML</a> for the family of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a>: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M284">View MathML</a>.

Definition 4.3[5]

Define the uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) ω-limit set of B by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M286">View MathML</a>. This can be characterized by the following: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M287">View MathML</a> if and only if there are sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M288">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M289">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M290">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M291">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M292">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M88">View MathML</a>).

Definition 4.4[5]

A family of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a> possessing a compact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) absorbing set is called <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259">View MathML</a>-uniformly compact. A family of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a> is called <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259">View MathML</a>-uniformly asymptotically compact if it possesses a compact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) attracting set, i.e., for any bounded subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M263">View MathML</a> and any sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M303">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M304">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M305">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M288">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M307">View MathML</a> is precompact in Y.

Lemma 4.1[20]

If a family of processes<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a>is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259">View MathML</a>-uniformly asymptotically compact, then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M311">View MathML</a>,

(i) for any sequences<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M288">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M289">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M290">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M291">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M88">View MathML</a>, there is a convergent subsequence of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M317">View MathML</a>inY,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M318">View MathML</a>is nonempty and compact inY,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M319">View MathML</a>,

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M320">View MathML</a>,

(v) ifAis a closed set and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) attractingB, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M323">View MathML</a>.

Assumption 1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M324">View MathML</a> be a family of operators acting on Σ and satisfying:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M325">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M326">View MathML</a>,

(ii) translation identity:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M327">View MathML</a>

Definition 4.5[5]

The kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M328">View MathML</a> of the process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M329">View MathML</a> acting on X consists of all bounded complete trajectories of the process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M329">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M331">View MathML</a>

The set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M332">View MathML</a> is said to be kernel section at time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M333">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M334">View MathML</a>.

Definition 4.6[5]

A family of processes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a> is said to be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M336">View MathML</a>-weakly continuous if for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132">View MathML</a>, the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M339">View MathML</a> is weakly continuous from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M340">View MathML</a> to Y.

Assumption 2 Let Σ be a weakly compact set and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a> be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M336">View MathML</a>-weakly continuous.

Lemma 4.2[20]

Under Assumptions 1 and 2 with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M343">View MathML</a>, which is a weakly continuous semigroup, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a>acting onXis<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) asymptotically compact, then it possesses an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259">View MathML</a>-uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) attractor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M349">View MathML</a>, which is compact inYand attracts all the bounded subsets ofXin the topology ofY.

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M350">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M351">View MathML</a>is a bounded neighborhood of the compact<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259">View MathML</a>-uniformly attracting set inY; i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M351">View MathML</a>is a bounded<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M259">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) absorbing set of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M357">View MathML</a>is the section at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M333">View MathML</a>of kernel<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M359">View MathML</a>of the process<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M329">View MathML</a>with symbol<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M359">View MathML</a>is nonempty for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>.

From the ideas of [4,20,28], we give the following results, which are very useful for the existence of a uniform attractor in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M364">View MathML</a>.

Lemma 4.3[20]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a>be a family of processes on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M366">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M367">View MathML</a>) and suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a>has a bounded<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M369">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) absorbing set in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M366">View MathML</a>. Then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M372">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132">View MathML</a>and any bounded subset<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M374">View MathML</a>, there exist two positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M375">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M376">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M377">View MathML</a>

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M378">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M379">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>.

Lemma 4.4[4,28]

Let a family of processes<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a>be<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M382">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) asymptotically compact, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a>is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M385">View MathML</a>-uniformly asymptotically compact for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M386">View MathML</a>, if

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a>has a bounded<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M385">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) absorbing set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M351">View MathML</a>,

(ii) for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M391">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132">View MathML</a>and any bounded subset<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M393">View MathML</a>, there exist two positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M394">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M395">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M396">View MathML</a>

From Theorem 3.1, we know that the problem (1)-(5) generates a process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a> acting in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a> and the time symbol is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M399">View MathML</a>. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M400">View MathML</a> the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M401">View MathML</a> endowed with a locally weak convergence topology. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M402">View MathML</a> be the hull of g in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M400">View MathML</a>, i.e., the closure of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M404">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M400">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M406">View MathML</a>.

Lemma 4.5[5]

Ifis reflective separable and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M407">View MathML</a>, then

(i) for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M408">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M409">View MathML</a>,

(ii) the translation group<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M410">View MathML</a>is weakly continuous on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M411">View MathML</a>,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M412">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M413">View MathML</a>,

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M411">View MathML</a>is weakly compact.

Due to Lemma 4.5, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M402">View MathML</a> is weakly compact and the translation semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M416">View MathML</a> satisfies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M417">View MathML</a> and is weakly continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M402">View MathML</a>. Because of the uniqueness of solution, the following translation identity holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M419">View MathML</a>

Theorem 4.1The family of processes<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36">View MathML</a>corresponding to problem (1)-(5) is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M421">View MathML</a>-weakly continuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M422">View MathML</a>-weakly continuous.

Proof For any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M423">View MathML</a> and τ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M424">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M426">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M88">View MathML</a>) weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M429">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M402">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M88">View MathML</a>, denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M432">View MathML</a>. The same estimates for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M433">View MathML</a> given in the Galerkin approximations (in Section 3) are valid for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M138">View MathML</a> here. Therefore, for some subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M435">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M249">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M423">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M438">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M439">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M441">View MathML</a>. And the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M442">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M443">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M444">View MathML</a>. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M445">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M446">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M447">View MathML</a> the weak limits of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M448">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M449">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M450">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M451">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M452">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M453">View MathML</a>, respectively. So, we get the following equation for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M454">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M455">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M168">View MathML</a>.

By the same method as the proof of Theorem 3.1, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M457">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M458">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M459">View MathML</a>, which means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M460">View MathML</a> in V is the weak solution of (1)-(5) with the initial condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M461">View MathML</a>. Due to the uniqueness of the solution, we state that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M462">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M441">View MathML</a>. For any other subsequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M465">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M466">View MathML</a> satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M467">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M469">View MathML</a>, by the same process, we obtain the analogous relation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M470">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M441">View MathML</a> holds. Then it can be easily seen that for any weakly convergent initial sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M473">View MathML</a> and weakly convergent sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M474">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M475">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M441">View MathML</a>. □

Lemma 4.6[25]

(The uniform Gronwall lemma) Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M478">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M479">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M480">View MathML</a>be three positive locally integrable functions on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M481">View MathML</a>, and for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M482">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M266">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M478">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M479">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M486">View MathML</a>satisfy the following inequalities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M487">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M488">View MathML</a>

whereR, A, Bare three positive constants. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M489">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M490">View MathML</a>.

4.2 The existence of uniformly absorbing sets

In this subsection, we prove the existence of uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M81">View MathML</a>) absorbing sets for the process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M254">View MathML</a> corresponding to (1)-(5).

Theorem 4.2Assume thatfandgsatisfy (H1)-(H2). Then the family of processes<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36">View MathML</a>corresponding to problem (1)-(5) has a bounded<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M494">View MathML</a>- and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M495">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>) absorbing set. That is, for any bounded subsetBof<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a>and any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M499">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M500">View MathML</a>and two positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M501">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M502">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M503">View MathML</a>

(26)

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M504">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M505">View MathML</a>

(27)

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M506">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M507">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M508">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M501">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M502">View MathML</a>are specified in (33), (41), (32) and (40), respectively.

Proof Taking the inner product of (1) with u, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M511">View MathML</a>

(28)

By virtue of (5), the Hölder inequality and the Young inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M512">View MathML</a>

(29)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M152">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M514">View MathML</a>, we deduce from (12) and (29) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M515">View MathML</a>

(30)

It follows from the classical Gronwall inequality and Lemma 4.5 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M516">View MathML</a>

(31)

where we have used the following inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M517">View MathML</a>

From (31), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M518">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M519">View MathML</a>

(32)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M520">View MathML</a>

(33)

Integrating (30) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M521">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M522">View MathML</a>

(34)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M523">View MathML</a>, we deduce from (5) that there exist three positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M524">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M525">View MathML</a>, β such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M526">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M527">View MathML</a>

(35)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M528">View MathML</a>

(36)

Thanks to (34), we deduce from (35)-(36) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M529">View MathML</a>

(37)

On the other hand, taking the inner product of (1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M530">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M531">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M532">View MathML</a>

(38)

Combining (37) with (38), by virtue of the uniform Gronwall Lemma 4.6, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M533">View MathML</a>

(39)

which implies that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M534">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132">View MathML</a>, there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M502">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M537">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M538">View MathML</a>

(40)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M539">View MathML</a>

(41)

 □

From Theorem 4.2, the compactness of the Sobolev embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M540">View MathML</a>, the compactness of the Sobolev trace embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M541">View MathML</a> and Lemma 4.2, we have the following result.

Corollary 4.1The family of processes<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36">View MathML</a>generated by (1)-(5) with initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M122">View MathML</a>has an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M544">View MathML</a>-uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>) attractor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M546">View MathML</a>, which is compact in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a>and attracts every bounded subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a>in the topology of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M550">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M351">View MathML</a>is the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M552">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>) absorbing set in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M357">View MathML</a>is the section at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M333">View MathML</a>of kernel<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M359">View MathML</a>of the process<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36">View MathML</a>with symbol<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>.

4.3 The existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M560">View MathML</a>-uniform attractor

The main purpose of this subsection is to give an asymptotic a priori estimate for the unbounded part of the modular <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M561">View MathML</a> for the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M562">View MathML</a> of problem (1)-(5) in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M47">View MathML</a>-norm.

Theorem 4.3The family of processes<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36">View MathML</a>corresponding to problem (1)-(5) with initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M122">View MathML</a>has an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M560">View MathML</a>-uniform (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>) attractor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M568">View MathML</a>, which is compact in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M47">View MathML</a>and attracts every bounded subsetBof<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a>in the topology of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M47">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M572">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M351">View MathML</a>is the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M560">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>) absorbing set and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M357">View MathML</a>is the section at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M333">View MathML</a>of kernel<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M359">View MathML</a>of the process<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M329">View MathML</a>with symbol<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>.

Proof We need only prove that the process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M36">View MathML</a> satisfies the assumption (ii) of Lemma 4.4.

From (H3), we deduce that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M583">View MathML</a>

Moreover, from Lemma 4.3 and Theorem 4.2, we know that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M584">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M585">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M378">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M587">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M589">View MathML</a>

Multiplying (1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M590">View MathML</a> and integrating over Ω, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M591">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M592">View MathML</a> denotes the positive part of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M593">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M594">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M595">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M596">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M597">View MathML</a>

Due to (5), we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M598">View MathML</a> large enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M599','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M599">View MathML</a>

for some positive constant c. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M600','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M600">View MathML</a>

(42)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M601">View MathML</a>

(43)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M602">View MathML</a>

(44)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M603">View MathML</a>

(45)

From (42)-(45), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M604">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M605">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M606','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M606">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M607">View MathML</a>

It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M608">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M610">View MathML</a> and the classical Gronwall inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M611">View MathML</a>

which implies that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M612">View MathML</a>, there exist two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M613">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M614">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M615">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M616">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M617">View MathML</a>

Repeating the same steps as above, just taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M618">View MathML</a> instead of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M619">View MathML</a>, we deduce that there exist two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M620">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M621">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M622">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M623">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M624">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M625">View MathML</a>

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M626">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M627">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M622">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M629">View MathML</a>.

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M630">View MathML</a>

 □

4.4 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M631">View MathML</a>-uniform attractor

In this subsection, we prove the existence of an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M631">View MathML</a>-uniform attractor. For this purpose, we first give some a priori estimates about <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M633','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M633">View MathML</a> endowed with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a>-norm.

Theorem 4.4Under assumptions (H1)-(H3), for any bounded subset<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M635">View MathML</a>, any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>, there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M638">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M639">View MathML</a>

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M378">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M641">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M643">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M644">View MathML</a>is a positive constant which is independent ofBandσ.

Proof First, we differentiate (1) and (2) in time, and denoting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M645">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M646">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M647">View MathML</a>

(46)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M648">View MathML</a>

(47)

where ‘⋅’ denotes the dot product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M11">View MathML</a>.

Multiplying (46) by ζ and integrating over Ω, and combining (4) with (47), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M650">View MathML</a>

On the other hand, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M651">View MathML</a>, integrating (38) from r to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M652">View MathML</a> and using (39), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M653">View MathML</a>

Therefore, we deduce from the uniformly Gronwall inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M654">View MathML</a>

which implies that there exist two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M655','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M655">View MathML</a> and a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M644">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M657">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M534">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M132">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M660">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M661">View MathML</a>

 □

Next, we prove the process <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M662">View MathML</a> is uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>) asymptotically compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M44">View MathML</a>.

Theorem 4.5Assume thatfandgsatisfy (H1)-(H3). Then the family of processes<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M662">View MathML</a>corresponding to problem (1)-(5) with initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M122">View MathML</a>is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M667','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M667">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>) asymptotically compact, i.e., there exists a compact uniformly attracting set in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M44">View MathML</a>, which attracts any bounded subset<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M635">View MathML</a>in the topology of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M44">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M351">View MathML</a> be an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M631">View MathML</a>-uniformly (w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M35">View MathML</a>) absorbing set obtained in Theorem 4.2, then we need only to show that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M675">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M676">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M291">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M678">View MathML</a> is pre-compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M44">View MathML</a>.

Thanks to Lemma 4.2, it is sufficient to verify that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M675">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M676">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M291">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M678">View MathML</a> is pre-compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M58">View MathML</a>.

In fact, from Corollary 4.1 and Theorem 4.3, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M678">View MathML</a> is pre-compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M47">View MathML</a>.

Without loss of generality, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M678">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M47">View MathML</a>.

Now, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M678">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M58">View MathML</a>.

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M693','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M693">View MathML</a>, we deduce from Lemma 2.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M694','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M694">View MathML</a>

We now estimate separately the two terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M695','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M695">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M696','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M696">View MathML</a>. By simple calculations and the Hölder inequality, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M697">View MathML</a>

(48)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M698','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/128/mathml/M698">View MathML</a>

(49)

which combining with Corollary 4.1, Theorem 4.3 and Theorem 4.4 yields Theorem 4.5 immediately. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors typed, read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referees for their valuable suggestions.

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