SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Extinction and asymptotic behavior of solutions for nonlinear parabolic equations with variable exponent of nonlinearity

Yanchao Gao* and Wenjie Gao

Author Affiliations

School of Mathematics, Jilin University, Changchun, 130012, P.R. China

For all author emails, please log on.

Boundary Value Problems 2013, 2013:164  doi:10.1186/1687-2770-2013-164


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/164


Received:11 March 2013
Accepted:26 June 2013
Published:10 July 2013

© 2013 Gao and Gao; 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

The aim of this paper is to study the existence and extinction of weak solutions of the initial and boundary value problem for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M1">View MathML</a>. First, the authors apply the method of parabolic regularization and Galerkin’s method to prove the existence of solutions to the problem mentioned and then obtain the comparison principle by arguing by contradiction. Furthermore, the authors prove that the solution vanishes in finite time and approaches 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M2">View MathML</a> norm as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M3">View MathML</a>.

Keywords:
nonlinear parabolic equation; nonstandard growth condition; extinction; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M4">View MathML</a>-Laplace operator

1 Introduction

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M6">View MathML</a>) be a bounded simply connected domain and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M7">View MathML</a>. Consider the following quasilinear degenerate parabolic problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M8">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M10">View MathML</a> denotes the lateral boundary of the cylinder <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M12">View MathML</a> with the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M13">View MathML</a> is a positive constant and the nonlinear source <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M14">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M15">View MathML</a>

(1.2)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M18">View MathML</a>. It will be assumed throughout the paper that the exponents <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M20">View MathML</a> are continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M21">View MathML</a> with the logarithmic module of continuity:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M22">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M23">View MathML</a>

(1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M24">View MathML</a>

(1.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M25">View MathML</a>

Model (1.1) may describe some properties of image restoration in space and time. Especially when the nonlinear source <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M26">View MathML</a>, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M28">View MathML</a> represent a recovering image and its observed noisy image, respectively. In the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M20">View MathML</a> are fixed constants, there have been many results about the existence, uniqueness, blowing-up and so on; we refer to the bibliography [1-3]. When p, σ are functions with respect to the space variable and time variable, this problem arises from elastic mechanics, electro-rheological fluids dynamics and image processing, etc.; see [4-9].

To the best of our knowledge, there are only a few works about parabolic equations with variable exponents of nonlinearity. In [6], Chen, Levine and Rao obtained the existence and uniqueness of weak solutions with the assumption that the exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M32">View MathML</a>. In [10], we applied the method of parabolic regularization and Galerkin’s method to prove the existence of weak solutions to problem (1.1) with the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M34">View MathML</a>. In this paper, we generalize the results in [10]. Especially, unlike [10], we obtain the existence and uniqueness of weak solution not only in the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M36">View MathML</a>, but also in the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M38">View MathML</a>. Furthermore, we apply energy estimates and Gronwall’s inequality to obtain the extinction of solutions when the exponents <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M39">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M40">View MathML</a> belong to different intervals; as we know such results are seldom seen for the problems with variable exponents. At the end of this paper, we prove that the solution approaches 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M2">View MathML</a> norm as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M3">View MathML</a> by some techniques in convex analysis.

The outline of this paper is the following. In Section 2, we introduce the function spaces of Orlicz-Sobolev type, give the definition of weak solution to the problem and prove the existence of weak solutions with a method of regularization and the uniqueness of solutions by arguing by contradiction. Section 3 is devoted to the proof of the extinction of the solution obtained in Section 2. In Section 4, we get the long time asymptotic behavior of the solution.

2 Existence and uniqueness of weak solutions

We study the existence of weak solutions in this section. Let us introduce the Banach spaces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M43">View MathML</a>

and denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M44">View MathML</a> the dual of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M45">View MathML</a> with respect to the inner product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M46">View MathML</a>.

Definition 2.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M47">View MathML</a> is called a weak solution of problem (1.1) if for every test-function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M48">View MathML</a>

and every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M49">View MathML</a>, the following identity holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M50">View MathML</a>

(2.1)

Following the line of the proof of Theorem 2.1 in [10,11], we have the following theorem about the existence of weak solutions.

Theorem 2.1Let the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M14">View MathML</a>and the exponents<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M20">View MathML</a>satisfy Conditions (1.2)-(1.5). If the following conditions hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M54">View MathML</a>

then problem (1.1) has at least one weak solutionusatisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M55">View MathML</a>.

The theorems about the uniqueness of weak solutions are as follows.

Theorem 2.2Suppose that the conditions in Theorem 2.1 are fulfilled and the following condition is satisfied:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M56">View MathML</a>

Then the nonnegative bounded solution of problem (1.1) is unique within the class of all nonnegative bounded weak solutions.

Proof We argue by contradiction. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M58">View MathML</a> are two nonnegative weak solutions of problem (1.1) and there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M59">View MathML</a> such that for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M61">View MathML</a> on the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M62">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M63">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M64">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M65">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M66">View MathML</a>.

By the definition of weak solutions, pick a test-function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M67">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M68">View MathML</a>

(2.2)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M69">View MathML</a>.

Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M70">View MathML</a>, then we estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M74">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M75">View MathML</a>

(2.3)

Let us consider first the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M76">View MathML</a>. By virtue of the first inequality of Lemma 4.4 in [2], we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M77">View MathML</a>

(2.4)

According to the condition (H3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M78">View MathML</a>

and then applying Young’s inequality, we may estimate the integrand of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M73">View MathML</a> in the following way:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M80">View MathML</a>

(2.5)

Substituting (2.5) into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M73">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M82">View MathML</a>

(2.6)

Secondly, we consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M84">View MathML</a>. According to the second inequality of Lemma 4.4 in [2], it is easily seen that the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M85">View MathML</a>

(2.7)

Using Young’s inequality, we may evaluate integrand of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M73">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M87">View MathML</a>

(2.8)

Plugging (2.8) into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M73">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M89">View MathML</a>

(2.9)

Plugging the above estimates (2.3), (2.4), (2.6) and (2.3), (2.7), (2.9) into (2.2) and dropping the nonnegative terms, we arrive at the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M90">View MathML</a>

(2.10)

with a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M91">View MathML</a> independent of ε.

Noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M92">View MathML</a>, we obtain a contradiction. This means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M94">View MathML</a>, a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M95">View MathML</a>. □

In the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M96">View MathML</a>, following the lines of the proof of Theorem 2.2, we have the following theorem.

Theorem 2.3Suppose that the conditions in Theorem 2.1 are fulfilled and the following condition is satisfied:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M97">View MathML</a>

Then the nonnegative solution of problem (1.1) is unique within the class of all nonnegative weak solutions.

3 Localization of weak solutions

In this section, we study the localization of the weak solution to problem (1.1). Namely, we study the extinction of the solution. We discuss the extinction of weak solutions in the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M100">View MathML</a>, respectively. Our main results are the following.

Theorem 3.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M98">View MathML</a>, then any bounded nonnegative solution of problem (1.1) vanishes in finite time for any nonnegative initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M103">View MathML</a>and satisfies the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M104">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M107">View MathML</a>are two positive constants.

Proof In Definition 2.1, we choose u as a test-function to show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M108">View MathML</a>

(3.1)

Applying the conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M109">View MathML</a>, the inequalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M110">View MathML</a> and the imbedding theorem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M111">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M112">View MathML</a>

(3.2)

By (3.1), (3.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M113">View MathML</a>

(3.3)

In (3.3), let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M115">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M116">View MathML</a>), multiply (3.3) by h and apply Lebesgue’s dominated convergence theorem to show that as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M117">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M118">View MathML</a>

(3.4)

By Gronwall’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M119">View MathML</a>

 □

Theorem 3.2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M100">View MathML</a>, then any bounded nonnegative solution of problem (1.1) vanishes in finite time for any nonnegative initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M103">View MathML</a>and satisfies the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M124">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M107">View MathML</a>are two positive constants.

Proof In Definition 2.1, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M129">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M130">View MathML</a>) as a test-function to show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M131">View MathML</a>

(3.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M133">View MathML</a>. The conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M135">View MathML</a>, the inequalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M136">View MathML</a> and the imbedding theorem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M137">View MathML</a> show that the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M138">View MathML</a>

(3.6)

A similar argument as above gives that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M139">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M140">View MathML</a> satisfies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M141">View MathML</a>

 □

Remark 3.1 In the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M143">View MathML</a>, it is not clear whether any bounded nonnegative solution of problem (1.1) vanishes in finite time.

4 Asymptotic behavior of weak solutions

In this section, we study the asymptotic properties of the weak solution to problem (1.1). Namely, we study the long time asymptotic behavior of the solution, our main result is as follows.

Theorem 4.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M101">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M145">View MathML</a>. If the following condition is satisfied

(H4) there exists a positive continuous function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M146">View MathML</a>such that the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M147">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M146">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M149">View MathML</a>.

Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M150">View MathML</a>, the solution to problem (1.1) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M151">View MathML</a>

Proof Step 1. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M152">View MathML</a>

then it is easy to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M153">View MathML</a> is a convex functional on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M2">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M155">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M156">View MathML</a>, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M157">View MathML</a> (δ representing Gâteaux differential) and the convexity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M153">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M159">View MathML</a>

(4.1)

for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M161">View MathML</a>. Integrating inequality (4.1) with respect to τ over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M162">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M163">View MathML</a>

(4.2)

Multiplying both sides of (4.2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M164">View MathML</a>, and letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M165">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M166">View MathML</a>

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M167">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M168">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M169">View MathML</a>

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M114">View MathML</a>, we get from the definition of solutions that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M172">View MathML</a>

(4.3)

Step 2. We prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M173">View MathML</a>. According to Theorem 2.1 and (H4), we have the following conclusions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M174">View MathML</a>

Applying Theorem 5 in [12], we get that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M175">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M176">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M173">View MathML</a>.

Step 3. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M178">View MathML</a>. Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M173">View MathML</a>, then it is easy to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M180">View MathML</a> is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M181">View MathML</a>, so we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M182">View MathML</a>.

By the imbedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M183">View MathML</a>, Lemmas (2.1)-(2.3) in [5] and (H4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M184">View MathML</a>

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M185">View MathML</a>.

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M187">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M188">View MathML</a>, and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M189">View MathML</a>

This completes the proof of Theorem 4.1. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors collaborated in all the steps concerning the research and achievements presented in the final manuscript.

Acknowledgements

The work was supported by the Natural Science Foundation of China (11271154) and by the 985 program of Jilin University. We are very grateful to the anonymous referees for their valuable suggestions that improved the article.

References

  1. Wei, DM: Existence, uniqueness and numerical analysis of solutions of a quasilinear parabolic problem. SIAM J. Numer. Anal.. 29(2), 484–497 (1992). Publisher Full Text OpenURL

  2. Dibenedetto, E: Degenerate Parabolic Equations, Springer, New York (1993)

  3. Zhao, JN: On the Cauchy problem and initial traces for the evolution p-Laplacian equations with strongly nonlinear sources. J. Differ. Equ.. 121, 329–383 (1995). Publisher Full Text OpenURL

  4. Ruzicka, M: Electrorheological Fluids: Modelling and Mathematical Theory, Springer, Berlin (2000)

  5. Antontsev, SN, Shmarev, SI: Anisotropic parabolic equations with variable nonlinearity. Publ. Math.. 53, 355–399 (2009)

  6. Chen, Y, Levine, S, Rao, M: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math.. 66, 1383–1406 (2006). Publisher Full Text OpenURL

  7. Acerbi, E, Mingione, G, Seregin, GA: Regularity results for parabolic systems related to a class of non newtonian fluids. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 21, 25–60 (2004)

  8. Acerbi, E, Mingione, G: Regularity results for a class of functionals with nonstandard growth. Arch. Ration. Mech. Anal.. 156(1), 121–140 (2001)

  9. Diening, L, Harjulehto, P, Hästö, P, Rûžička, M: Lebesgue and Sobolev Spaces with Variable Exponents, Springer, Heidelberg (2011)

  10. Guo, B, Gao, WJ: Study of weak solutions for parabolic equations with nonstandard growth conditions. J. Math. Anal. Appl.. 374(2), 374–384 (2011). Publisher Full Text OpenURL

  11. Guo, B, Gao, WJ: Existence and asymptotic behavior of solutions for nonlinear parabolic equations with variable exponent of nonlinearity. Acta Math. Sci.. 32(3), 1053–1062 (2012)

  12. Simon, J: Compact sets in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M190">View MathML</a>. Ann. Math. Pures Appl.. 4(146), 65–96 (1987)