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Energy decay and nonexistence of solution for a reaction-diffusion equation with exponential nonlinearity

Huiya Dai12* and Hongwei Zhang2

Author Affiliations

1 School of Mathematics and Statistics, Xi’an Jiaotong University, Xianning Road, Xi’an, P.R. China

2 Department of Mathematics, Henan University of Technology, Lianhua Street, Zhengzhou, P.R. China

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Boundary Value Problems 2014, 2014:70  doi:10.1186/1687-2770-2014-70


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/70


Received:2 December 2013
Accepted:13 March 2014
Published:25 March 2014

© 2014 Dai and Zhang; 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 credited.

Abstract

In this work we consider the energy decay result and nonexistence of global solution for a reaction-diffusion equation with generalized Lewis function and nonlinear exponential growth. There are very few works on the reaction-diffusion equation with exponential growth f as a reaction term by potential well theory. The ingredients used are essentially the Trudinger-Moser inequality.

Keywords:
reaction-diffusion equation; stable and unstable set; exponential reaction term; decay rate; global nonexistence

1 Introduction

In this paper, we study the following initial boundary value problem with generalized Lewis function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M1">View MathML</a> which depends on both spacial variable and time:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M2">View MathML</a>

(1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M3">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M4">View MathML</a>

(3)

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M5">View MathML</a> is a reaction term with exponential growth at infinity to be specified later, Ω is a bounded domain with smooth boundary Ω in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M6">View MathML</a>.

For the reaction-diffusion equation with polynomial growth reaction terms (that is, equation (1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M8">View MathML</a>), there have been many works in the literature; one can find a review of previous results in [1,2] and references therein, which are not listed in this paper just for concision. Problem (1)-(3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M9">View MathML</a> describes the chemical reaction processes accompanied by diffusion [2]. The author of work [1] proved the existence and asymptotic estimates of global solutions and finite time blow-up of problem (1)-(3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M9">View MathML</a> and the critical Sobolev exponent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M11">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M12">View MathML</a>.

In this paper we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M5">View MathML</a> is a reaction term with exponential growth like <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M14">View MathML</a> at infinity. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M16">View MathML</a>, model (1)-(3) was proposed by [3] and [4]. In this case, Fujita [5] studied the asymptotic stability of the solution. Peral and Vazquez [6] and Pulkkinen [7] considered the stability and blow-up of the solution. Tello [8] and Ioku [9] considered the Cauchy problem of heat equation with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M17">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M18">View MathML</a>.

Recently, Alves and Cavalcanti [10] were concerned with the nonlinear damped wave equation with exponential source. They proved global existence as well as blow-up of solutions in finite time by taking the initial data inside the potential well [11]. Moreover, they also got the optimal and uniform decay rates of the energy for global solutions.

Motivated by the ideas of [1,10], we concentrate on studying the uniform decay estimate of the energy and finite time blow-up property of problem (1)-(3) with generalized Lewis function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M1">View MathML</a> and exponential growth f as a reaction term. To the authors’ best knowledge, there are very few works in the literature that take into account the reaction-diffusion equation with exponential growth f as a reaction term by potential well theory. The majority of works in the literature make use of the potential well theory when f possesses polynomial growth. See, for instance, the works [12-16] and a long list of references therein. The ingredients used in our proof are essentially the Trudinger-Moser inequality (see [17,18]). We establish decay rates of the energy by considering ideas from the work of Messaoudi [15]. The case of nonexistence results is also treated, where a finite time blow-up phenomenon is exhibited for finite energy solutions by the standard concavity method adapted for our context.

The remainder of our paper is organized as follows. In Section 2 we present the main assumptions and results, Section 3 and Section 4 are devoted to the proof of the main results.

Throughout this study, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M22">View MathML</a> the usual norms in spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M25">View MathML</a>, respectively.

2 Assumptions and preliminaries

In this section, we present the main assumptions and results. We always assume that:

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M1">View MathML</a> is a positive differentiable function and is bounded for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M28">View MathML</a>.

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M29">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M30">View MathML</a> function. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M31">View MathML</a> is increasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M32">View MathML</a>, and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M33">View MathML</a>, there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M34">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M35">View MathML</a>

(4)

(A3) For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M33">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M38">View MathML</a> fixed, there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M39">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M40">View MathML</a>

(5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M41">View MathML</a>

(6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M42">View MathML</a>.

(A4) There exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M43">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M44">View MathML</a>

(7)

A typical example of functions satisfies (A2)-(A4) is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M45">View MathML</a>, with given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M48">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M49">View MathML</a>.

Now we define some functional as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M50">View MathML</a>

(8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M51">View MathML</a>

(9)

then the ‘potential depth’ given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M52">View MathML</a>

is a positive constant [10]. Hence, we are able to define stable and unstable sets respectively as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M53">View MathML</a>

We also need the following lemmas.

Lemma 2.1[17,18]

Let Ω be a bounded domain in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M6">View MathML</a>. For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M55">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M56">View MathML</a>

(10)

and there exist positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M57">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M58">View MathML</a>

(11)

Lemma 2.2[19]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M59">View MathML</a>be a nonincreasing and nonnegative function on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M60">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M61">View MathML</a>

(12)

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M62">View MathML</a>

whereC, ωare positive constants depending on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M63">View MathML</a>and other known qualities.

Lemma 2.3[20]

Suppose that a positive, twice-differentiable function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M64">View MathML</a>satisfies on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M65">View MathML</a>the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M66">View MathML</a>

(13)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M67">View MathML</a>, then there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M68">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M69">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M70">View MathML</a>.

In order to state and prove our main results, we remind that by the embedding theorem there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M71">View MathML</a> depending on p and Ω only such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M72">View MathML</a>

(14)

By multiplying equation (1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M73">View MathML</a>, integrating over Ω, using integration by parts and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M74">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M75">View MathML</a>

(15)

Our main results read as follows.

Theorem 2.1Let (A1)-(A4) hold. Assume further that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M76">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M77">View MathML</a>

(16)

for some sufficiently small<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M78">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M79">View MathML</a>. Then there exist positive constantsKandksuch that the energy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M80">View MathML</a>satisfies the decay estimates for larget

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M81">View MathML</a>

(17)

Theorem 2.2Let (A1)-(A4) hold. Assume further that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M83">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M84">View MathML</a>, then the solutions of (1)-(3) blow up in finite time.

3 Proof of decay of the energy

In this section we prove Theorem 2.1. We divide the proof into two lemmas.

Lemma 3.1Under the assumptions of Theorem 2.1, we have, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M86">View MathML</a>.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M87">View MathML</a>, then there exists (by continuity) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M88">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M89">View MathML</a>

This and (A4) give

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M90">View MathML</a>

(18)

So, by (15) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M91">View MathML</a>

(19)

We then use (5), the Holder inequality and the embedding theorem to obtain, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M92">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M93">View MathML</a>

(20)

Once <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M94">View MathML</a>, we choose β such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M95">View MathML</a>, then, from Trudinger-Moser inequality (11),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M96">View MathML</a>

and therefore, by (16) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M98">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M99">View MathML</a>

(21)

By virtue of (21) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M100">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M101">View MathML</a>

This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M86">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M92">View MathML</a>. By repeating this procedure and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M104">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M105">View MathML</a>

This is extended to T. □

Lemma 3.2Under the assumptions of Theorem 2.1, we have, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M106">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M107">View MathML</a>

(22)

Proof It suffices to rewrite (21) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M108">View MathML</a>

(23)

Thus (22) follows from (23). □

Proof of Theorem 2.1 We integrate (15) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M109">View MathML</a> to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M110">View MathML</a>

(24)

Now we multiply (1) by u and integrate over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M111">View MathML</a> to arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M112">View MathML</a>

(25)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M113">View MathML</a>. Exploiting (14) and (19), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M114">View MathML</a>

(26)

Using (7), (23) and (22), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M115">View MathML</a>

(27)

Integrating both sides of (27) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M116">View MathML</a> and using (26), one can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M117">View MathML</a>

(28)

By using (15) again, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M119">View MathML</a>, hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M120">View MathML</a>

(29)

Inserting (29) in (24) and using (27), we easily have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M121">View MathML</a>

(30)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M122">View MathML</a> a constant depending on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M71">View MathML</a>, A, θ, η only. We then use Young’s inequality to get from (30) and (24)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M124">View MathML</a>

(31)

By (12) in Lemma 2.2 we then get the results. □

4 Proof of the blow-up result

In this section, we shall prove Theorem 2.2 by adapting the concavity method (see Levine [20]). We recall the following lemma in [10].

Lemma 4.1[10]

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M83">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M126">View MathML</a>, then it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M127">View MathML</a>

(32)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M128">View MathML</a>

(33)

Proof of Theorem 2.2 Assume by contradiction that the solution is global. Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M129">View MathML</a>, we consider the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M130">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M131">View MathML</a>

(34)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M132">View MathML</a>, T, ρ are positive constants which will be fixed later. Direct computations show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M133">View MathML</a>

(35)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M134">View MathML</a>

(36)

Then, due to equations (1), (7) and (33), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M135">View MathML</a>

(37)

Now we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M136">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M137">View MathML</a> (this ρ can be chosen since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M138">View MathML</a>), and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M139">View MathML</a>

(38)

We also note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M140">View MathML</a>

Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M142">View MathML</a> are both positive. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M143">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M28">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M85">View MathML</a>, by the construction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M64">View MathML</a>, it is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M147">View MathML</a>

(39)

Thus, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M148">View MathML</a>, from (35), (38) and (39) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M149">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M150">View MathML</a>

That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M151">View MathML</a>

Then we complete the proof by the standard concavity method (Lemma 2.3) since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/70/mathml/M43">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

We thank the referees for their valuable suggestions which helped us improve the paper so much. This work was supported by the National Natural Science Foundation of China (11171311).

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