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Extinction and decay estimates of solutions for a porous medium equation with nonlocal source and strong absorption

Xianghui Xu1, Zhong Bo Fang2* and Su-Cheol Yi3

Author Affiliations

1 Department of Mathematics, Pusan National University, Busan, 609-735, Republic of Korea

2 School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, P.R. China

3 Department of Mathematics, Changwon National University, Changwon, 641-773, Republic of Korea

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Boundary Value Problems 2013, 2013:24  doi:10.1186/1687-2770-2013-24


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/24


Received:25 September 2012
Accepted:21 December 2012
Published:5 March 2013

© 2013 Xu et al.; licensee Springer

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

Abstract

In this paper, we investigate extinction properties of the solutions for the initial Dirichlet boundary value problem of a porous medium equation with nonlocal source and strong absorption terms. We obtain some sufficient conditions for the extinction of nonnegative nontrivial weak solutions and the corresponding decay estimates which depend on the initial data, coefficients, and domains.

1 Introduction

We consider the initial Dirichlet boundary value problem for a class of porous medium equations with nonlocal source and strong absorption terms

(1)

(2)

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M6">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M7">View MathML</a>) is a bounded domain with smooth boundary, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M8">View MathML</a> is a nonnegative function. The symbols <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M10">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M11">View MathML</a>, denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M12">View MathML</a>- and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M13">View MathML</a>-norm, respectively, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M14">View MathML</a> denotes the measure of Ω.

Equation (1) describes the fast diffusion of concentration of some Newtonian fluids through a porous medium or the density of some biological species in many physical phenomena and biological species theories, while nonlocal source and absorption terms cooperate and interact with each other during the diffusion. It has been known that the nonlocal source term presents a more realistic model for population dynamics; see [1-3]. In the nonlinear diffusion theory, obvious differences exist among the situations of slow (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M15">View MathML</a>), fast (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M16">View MathML</a>), and linear (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M17">View MathML</a>) diffusions. For example, there is a finite speed propagation in the slow and linear diffusion situations, whereas an infinite speed propagation exists in the fast diffusion situation.

Recently, many scholars have been devoted to the study of blow-up and extinction properties of solutions for nonlinear parabolic equations with nonlocal terms. The blow-up rates and blow-up sets of solutions to equation (1) have been investigated when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M19">View MathML</a>, and the linear absorption term is replaced with a nonlinear term with exponent (cf.[4-9]). Extinction is the phenomenon whereby there exists a finite time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M20">View MathML</a> such that the solution is nontrivial for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M21">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M22">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M23">View MathML</a>. In this case, T is called the extinction time. It is also an important property of solutions for nonlinear parabolic equations which have been studied by many researchers. For instance, Evans and Knerr [10] investigated the extinction behaviors of solutions for the Cauchy problem of the semilinear parabolic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M24">View MathML</a>

by constructing a suitable comparison function. Ferreira and Vazquez [11] studied the extinction phenomena of solutions for the Cauchy problem of the porous medium equation with an absorption term

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M25">View MathML</a>

by using the analysis of self-similar solutions. Li and Wu [12] considered the problem of the porous medium equation with a source term

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M26">View MathML</a>

(4)

subject to (2) and (3). They obtained some conditions for the extinction and non-extinction of solutions to equation (4) and decay estimates by the upper and lower solutions method. On extinctions of solutions to the p-Laplacian equations or the doubly degenerate equations, we refer readers to [13,14] and the references therein.

Replacing the nonlocal term in equation (1) with a local term, Liu [15]et al. considered the initial Dirichlet boundary value problem for a class of porous medium equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M27">View MathML</a>

and obtained sufficient conditions for the extinction and non-extinction of solutions to that equation. Thereafter, Fang and Li [16] extended their results to the doubly degenerate equation in the whole dimensional space.

For equation (1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M29">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M30">View MathML</a>, Han and Gao [17] showed that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M31">View MathML</a> is the critical exponent for the occurrence of extinction or non-extinction. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M29">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M30">View MathML</a>, the conditions for the extinction and non-extinction of solutions and corresponding decay estimates were obtained (cf.[18]). Recently, Fang and Xu [19] considered equation (1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M35">View MathML</a>, when the diffusion term was replaced with a p-Laplacian operator in the whole dimensional space, and showed that the extinction of the weak solution is determined by the competition of two nonlinear terms. They also obtained the exponential decay estimates which depend on the initial data, coefficients, and domains. The extinctions of solutions to equation (1) with nonlocal source terms do not depend on the first eigenvalue of the corresponding operator, which is different from the case of local source terms. The extinction and decay estimates for solutions to the nonlocal fast diffusion equations with nonzero coefficients and strong absorption terms, like equation (1), are still being investigated.

Motivated by the above works, we investigate whether the existence of strong absorption can change extinction behaviors for solutions to problem (1)-(3) in the whole dimensional space. The main tools we use are the integral estimate method and the Gagliardo-Nirenberg inequality. This technique has a wide application, especially for equations that do not satisfy the maximum principle (cf.[20]). Our goals are to show that the extinction of nonnegative nontrivial weak solutions to problem (1)-(3) occurs when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M36">View MathML</a> and to find the decay estimates depending on the initial data, coefficients, and domains.

Our paper is organized as follows. In Section 2, we give preliminary knowledge including lemmas that are required in the proofs of our results and present the proofs for the results in Section 3.

2 Preliminary knowledge

Due to the singularity of equation (1), problem (1)-(3) has no classical solutions in general. To state the definition of the weak solution, we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M37">View MathML</a> and firstly define the class of nonnegative testing functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M38">View MathML</a>

Definition 1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M39">View MathML</a> is called a weak subsolution (supersolution) of problem (1)-(3) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M40">View MathML</a> if the following conditions hold:

a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M41">View MathML</a> in Ω;

b. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M42">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M43">View MathML</a>;

c. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M44">View MathML</a> and every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M45">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M46">View MathML</a>

A function u is called a locally weak solution of problem (1)-(3) if it is both a subsolution and a supersolution for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M20">View MathML</a>.

Remark 1 The existence and uniqueness of locally nonnegative solutions in time to problem (1)-(3) can be obtained by the standard parabolic regular theory that can be applied to get suitable estimates in the standard limiting process (cf.[2,21,22]). The proof is similar to the ones in the cited references, and so it is omitted here.

Lemma 1Letk, αbe positive constants and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M48">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M49">View MathML</a>is a nonnegative absolutely continuous function on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M50">View MathML</a>satisfying the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M51">View MathML</a>

then we have the decay estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M52">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M53">View MathML</a>.

Proof

By solving the initial problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M54">View MathML</a>

and using the comparison principle, one can easily obtain the result. □

Lemma 2 (The Gagliardo-Nirenberg inequality) [23]

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M57">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M58">View MathML</a>. We then have the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M59">View MathML</a>

whereCis a constant depending onN, m, r, j, k, andqsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M60">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M61">View MathML</a>. While if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M62">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M63">View MathML</a>, and if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M64">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M65">View MathML</a>.

3 Main results

In this section, we give some extinction properties of nonnegative nontrivial weak solutions of problem (1)-(3) stated in the following theorems. The corresponding decay estimates to the solutions will be presented in the proofs of the theorems for brief expressions instead of in the statements.

Theorem 1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M66">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M67">View MathML</a>. Then the nonnegative nontrivial weak solution of problem (1)-(3) vanishes in finite time for any nonnegative initial data provided that either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M14">View MathML</a>orλis sufficiently small.

Proof We first consider the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M69">View MathML</a>. Multiplying both sides of (1) by u and integrating the result over Ω, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M70">View MathML</a>

By Hölder’s inequality, we get the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M71">View MathML</a>

In particular, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M72">View MathML</a>, we get the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M73">View MathML</a>

(5)

from the two expressions above. By using the Sobolev embedding inequality, one can show that there exists an embedding constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M74">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M75">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M76">View MathML</a>

In particular, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M77">View MathML</a>, then the inequality above turns out to be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M78">View MathML</a>

(6)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M16">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M80">View MathML</a>, and hence, inequality (5) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M81">View MathML</a>

(7)

By Lemma 2, we get the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M82">View MathML</a>

(8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M83">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M69">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M66">View MathML</a>, it can be easily seen that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M86">View MathML</a>.

It then follows from (8) and Young’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M87">View MathML</a>

(9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M89">View MathML</a> will be determined later. If we choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M90">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M91">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M92">View MathML</a>. From (9) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M93">View MathML</a>

(10)

By inequalities (7) and (10), we get the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M94">View MathML</a>

Here, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M95">View MathML</a> and λ or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M14">View MathML</a> small enough so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M97">View MathML</a>

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M98">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M99">View MathML</a>.

By Lemma 1, we then obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M100">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M101">View MathML</a>, which give the decay estimates in finite time for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M102">View MathML</a>.

Secondly, we consider the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M103">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M104">View MathML</a>, multiplying both sides of (1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M105">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M106">View MathML</a>) and integrating the result over Ω, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M107">View MathML</a>

By Hölder’s inequality, we have the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M108">View MathML</a>

In particular, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M109">View MathML</a>, we then get the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M110">View MathML</a>

(11)

by the two expressions above. By the Sobolev embedding inequality, one can see that there exists an embedding constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M111">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M112">View MathML</a>

(12)

Using Hölder’s inequality again, we have the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M113">View MathML</a>

(13)

From inequalities (11), (12), and (13), we then obtain the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M114">View MathML</a>

(14)

By Lemma 2, we can also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M115">View MathML</a>

(15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M116">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M117">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M66">View MathML</a>, one can easily see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M119">View MathML</a>. Then it follows from (15) and Young’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M120">View MathML</a>

(16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M121">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M122">View MathML</a> will be determined later. If we choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M123">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M124">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M125">View MathML</a>. We then have the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M126">View MathML</a>

(17)

by (16). From inequalities (14) and (17), we can also obtain the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M127">View MathML</a>

Here, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M128">View MathML</a> and λ or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M14">View MathML</a> small enough so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M130">View MathML</a>. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M131">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M132">View MathML</a> from the inequality above. By Lemma 1, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M133">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M134">View MathML</a>, which give the decay estimates in finite time for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M103">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M136">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M137">View MathML</a>, one can show that there exists an embedding constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M138">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M139">View MathML</a>

by multiplying both sides of (1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M105">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M141">View MathML</a>) and integrating the result over Ω, and the Sobolev embedding inequality. By using the inequality above and a similar argument as above, the following decay estimates can be obtained:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M142">View MathML</a>

provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M143">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M144">View MathML</a>. □

Theorem 2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M66">View MathML</a>, then the nonnegative nontrivial weak solution of problem (1)-(3) vanishes in finite time provided that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M14">View MathML</a>orλis sufficiently small, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M148">View MathML</a>, where if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M69">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M150">View MathML</a>, and if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M30">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M106">View MathML</a>.

Proof Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M153">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M69">View MathML</a>, multiplying both sides of (1) by u and integrating the result over Ω, we have the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M155">View MathML</a>

(18)

By (10) and (18), and using Hölder’s inequality, we get the inequality

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M95">View MathML</a> small enough so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M158">View MathML</a>, we obtain the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M159">View MathML</a>

Hence, we have the inequality

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

provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M161">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M162">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M163">View MathML</a>, from which and a similar argument as the one used in the proof of Theorem 1, the following decay estimates can be obtained:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M164">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M165">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M103">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M16">View MathML</a>, multiplying both sides of (1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M105">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M106">View MathML</a>) and integrating the result over Ω, we get the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M170">View MathML</a>

(19)

By (17) and (19), and using Hölder’s inequality, we obtain the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M171">View MathML</a>

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M128">View MathML</a> small enough so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M173">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M174">View MathML</a>

Therefore, we obtain the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M175">View MathML</a>

provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M176">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M177">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M178">View MathML</a>, which yields the following decay estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M179">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M180">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M106">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M182">View MathML</a>, and hence, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M183">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M184">View MathML</a>.

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M185">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M186">View MathML</a> is the first eigenvalue of the boundary problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M187">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M188">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M189">View MathML</a>, is an eigenfunction corresponding to the eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M186">View MathML</a>, then for sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M191">View MathML</a>, it can be easily shown that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M192">View MathML</a> is an upper solution of problem (1)-(3) provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M193">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M194">View MathML</a>. We then have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M195">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M196">View MathML</a> by the comparison principle. Therefore, from equation (19), we can obtain the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M197">View MathML</a>

from which the following decay estimates can be obtained:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M198">View MathML</a>

provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M199">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M200">View MathML</a>. □

Remark 2 Since the Sobolev embedding inequality cannot be used in the proof of Theorem 2, it is not necessary to consider the cases that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M117">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M137">View MathML</a>, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M30">View MathML</a>. In addition, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M204">View MathML</a>, the conditions in Theorem 2 imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M205">View MathML</a>.

Theorem 3Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M206">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M207">View MathML</a>. Then the nonnegative nontrivial weak solution of problem (1)-(3) vanishes in finite time for any nonnegative initial data provided thatβis sufficiently large.

Proof We first consider the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M69">View MathML</a>. Multiplying both sides of (1) by u and integrating the result over Ω, and using Hölder’s inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M209">View MathML</a>

By Lemma 2, we have the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M210">View MathML</a>

(20)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M211">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M212">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M213">View MathML</a>. It then follows from (20) and Young’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M214">View MathML</a>

(21)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M215">View MathML</a> will be determined later. From (18) and (21), one can see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M216">View MathML</a>

We then obtain the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M217">View MathML</a>

by (6) and the inequality above, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M218">View MathML</a>. We can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M215">View MathML</a> small enough so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M220">View MathML</a>. Once <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M215">View MathML</a> is fixed, we may choose β large enough so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M222">View MathML</a>

Hence, we have the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M223">View MathML</a>

from which the following decay estimates can be obtained by a similar argument as the one used in the proof of Theorem 1:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M224">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M225">View MathML</a>.

Secondly, we consider the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M103">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M227">View MathML</a>, multiplying both sides of (1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M105">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M106">View MathML</a>) and integrating the result over Ω, and then using Hölder’s inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M230">View MathML</a>

By Lemma 2, it can be shown that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M231">View MathML</a>

(22)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M232">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M233">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M234">View MathML</a>. It then follows from (22) and Young’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M235">View MathML</a>

(23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M236">View MathML</a> will be determined later. From (19) and (23), one can see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M237">View MathML</a>

By (12), (13), and the inequality above, we can obtain the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M238">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M239">View MathML</a>. We can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M236">View MathML</a> small enough so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M241">View MathML</a>. Once <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M236">View MathML</a> is fixed, we can choose β large enough so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M243">View MathML</a>

Hence, we can obtain the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M244">View MathML</a>

from which the following decay estimates can be obtained:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M245">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M246">View MathML</a>.

Similarly, one can obtain the following decay estimates for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M137">View MathML</a>:

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M249">View MathML</a>. □

Remark 3 One can see from Theorems 1-3 that the extinction of nonnegative nontrivial weak solutions to problem (1)-(3) occurs when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M250">View MathML</a>.

Remark 4 Theorems 1-3 all require <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M14">View MathML</a>, λ, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/24/mathml/M146">View MathML</a> to be sufficiently small or β to be sufficiently large.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

The second and third authors were supported by the National Science Foundation of Shandong Province of China (ZR2012AM018) and Changwon National University in 2012, respectively. The authors would like to express their sincere gratitude to the anonymous reviewers for their insightful and constructive comments.

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