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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Study of solutions to an initial and boundary value problem for certain systems with variable exponents

Yunzhu Gao1 and Wenjie Gao2

Author Affiliations

1 Department of Mathematics and Statistics, Beihua University, Jilin City, P.R. China

2 Institute of Mathematics, Jilin University, Changchun, 130012, P.R. China

Boundary Value Problems 2013, 2013:76  doi:10.1186/1687-2770-2013-76


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/76


Received:15 February 2013
Accepted:18 March 2013
Published:5 April 2013

© 2013 Gao and Gao; licensee Springer

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

Abstract

In this paper, the existence and blow-up property of solutions to an initial and boundary value problem for a nonlinear parabolic system with variable exponents is studied. Meanwhile, the blow-up property of solutions for a nonlinear hyperbolic system is also obtained.

Keywords:
existence; blow-up; parabolic system; hyperbolic system; variable exponent

1 Introduction

In this paper, we first consider the initial and boundary value problem to the following nonlinear parabolic system with variable exponents:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M2">View MathML</a> is a bounded domain with smooth boundary Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M5">View MathML</a> denotes the lateral boundary of the cylinder <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M6">View MathML</a>, and the source terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M8">View MathML</a> are in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M9">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M10">View MathML</a>

respectively, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M14">View MathML</a> are functions satisfying conditions (2.1) below.

In the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M12">View MathML</a> are constants, system (1.1) provides a simple example of a reaction-diffusion system. It can be used as a model to describe heat propagation in a two-component combustible mixture. There have been many results about the existence, boundedness and blow-up properties of the solutions; we refer the readers to the bibliography given in [1-7].

The motivation of this work is due to [2], where the following system of equations is studied.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M17">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M18">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M19">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M20">View MathML</a>, and p, q are positive numbers. The authors investigated the boundedness and blow-up of solutions to problem (1.2). Furthermore, the authors also studied the uniqueness and global existence of solutions (see [3]).

Besides, this work is also motivated by [8] in which the following problem is considered:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M21">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M22">View MathML</a> is a bounded domain with smooth boundary Ω, and the source term is of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M23">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M24">View MathML</a>. The author studied the blow-up property of solutions for parabolic and hyperbolic problems. Parabolic problems with sources like the ones in (1.3) appear in several branches of applied mathematics, which can be used to model chemical reactions, heat transfer or population dynamics etc. We also refer the interested reader to [9-23] and the references therein.

We also study the following nonlinear hyperbolic system of equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M25">View MathML</a>

(1.4)

The aim of this paper is to extend the results in [2,8] to the case of parabolic system (1.1) and hyperbolic system (1.4). As far as we know, this seems to be the first paper, where the blow-up phenomenon is studied with variable exponents for the initial and boundary value problem to some parabolic and hyperbolic systems. The main method of the proof is similar to that in [3,8].

We conclude this introduction by describing the outline of this paper. Some preliminary results, including existence of solutions to problem (1.1), are gathered in Section 2. The blow-up property of solutions are stated and proved in Section 3. Finally, in Section 4, we prove the blow-up property of solutions for hyperbolic problem (1.4).

2 Existence of solutions

In this section, we first state some assumptions and definitions needed in the proof of our main result and then prove the existence of solutions.

Throughout the paper, we assume that the exponents <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M26">View MathML</a> and the continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M27">View MathML</a> satisfy the following conditions:

(2.1)

Definition 2.1 We say that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M29">View MathML</a> for problem (1.1) blows up in finite time if there exists an instant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M30">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M31">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M32">View MathML</a>

Our first result here is the following.

Theorem 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M33">View MathML</a>be a bounded smooth domain, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M37">View MathML</a>satisfy the conditions in (2.1), and assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M38">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M39">View MathML</a>are nonnegative, continuous and bounded. Then there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M41">View MathML</a>, such that problem (1.1) has a nonnegative and bounded solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M42">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M43">View MathML</a>.

Proof We only prove the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M45">View MathML</a>, and the proofs to the cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M46">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M47">View MathML</a> are similar.

Let us consider the equivalent systems of (1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M48">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M49">View MathML</a> is the corresponding Green function. Then the existence and uniqueness of solutions for a given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M50">View MathML</a> could be obtained by a fixed point argument.

We introduce the following iteration scheme:

and the convergence of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M52">View MathML</a> follows by showing that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M53">View MathML</a>

is a contraction in the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M54">View MathML</a> to be defined below.

Now, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M55">View MathML</a>

where

We denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M57">View MathML</a>

and for arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M58">View MathML</a>, define the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M59">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M61">View MathML</a> is a fixed positive constant.

We claim that Ψ is a contraction on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M54">View MathML</a>. In fact, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M63">View MathML</a> fixed, we have

and we always have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M65">View MathML</a>

(2.2)

Now, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M66">View MathML</a>

It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M67">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M68">View MathML</a>.

Then, by using inequality (2.2), we get

Hence, for sufficiently small t, we have

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M71">View MathML</a> is a constant. Then Ψ is a strict contraction. □

3 Blow-up of solutions

In this section, we study the blow-up property of the solutions to problem (1.1). We need the following lemma.

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M72">View MathML</a>be a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M73">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M76">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M77">View MathML</a>are given constants. Then, there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M78">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M79">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M72">View MathML</a>cannot be globally defined; in fact,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M81">View MathML</a>

(3.1)

Proof It is sufficient to take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M78">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M83">View MathML</a>

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M84">View MathML</a>

(3.2)

By a direct integration to (3.2), then we get immediately (3.1), which gives an upper bound for the blow-up time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M85">View MathML</a>. □

The next theorem gives the main result of this section.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M33">View MathML</a>be a bounded smooth domain, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M42">View MathML</a>be a positive solution of problem (1.1), with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M37">View MathML</a>satisfying conditions in (2.1). Then any solutions of problem (1.1) will blow up at finite time<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M92">View MathML</a>if the initial datum<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M93">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M94">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M95">View MathML</a>is the first eigenfunction of the homogeneous Dirichlet Laplacian on Ω and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M77">View MathML</a>is a constant depending only on the domain Ω and the bounds<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M98">View MathML</a>given in condition (2.1).

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M99">View MathML</a> be the first eigenvalue of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M100">View MathML</a>

with the homogeneous Dirichlet boundary condition, and let φ be a positive function satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M101">View MathML</a>

We introduce the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M102">View MathML</a>. First of all, we consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M45">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M105">View MathML</a>

We now deal with the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M106">View MathML</a>. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M20">View MathML</a>, we divide Ω into the following four sets:

Then we have

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M110">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M112">View MathML</a>.

From the convex property of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M76">View MathML</a> and Jensen’s inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M115">View MathML</a>

Then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M116">View MathML</a>

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M117">View MathML</a>

Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M118">View MathML</a> big enough, the result follows from Lemma 3.1.

Next, we state briefly the proof to the theorem in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M46">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M47">View MathML</a>. We repeat the previous argument under defining <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M102">View MathML</a>, and we obtain in much the same way

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M122">View MathML</a>

In view of the property of φ, we get

According to the convex property of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M76">View MathML</a>, and by using Jensen’s inequality, by considering again <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M129">View MathML</a> as before, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M130">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M131">View MathML</a> depends only on γ, p and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M133">View MathML</a> denotes the measure of Ω. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M134">View MathML</a>

By Lemma 3.1, the proof is complete. □

4 Blow-up of solutions for a hyperbolic system

Lemma 4.1[15]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M135">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M136">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M138">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M139">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M140">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M141">View MathML</a>wheneveryexists; and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M142">View MathML</a>

(4.1)

Now, let us study the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M143">View MathML</a>

(4.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M144">View MathML</a> and they are not identically zero, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M146">View MathML</a> as above respectively.

Theorem 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M147">View MathML</a>be a solution of problem (4.2), and let the conditions in (2.1) hold. Then there exist sufficiently large initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M151">View MathML</a>such that any solutions of problem (4.1) blew up at finite time<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M92">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M153">View MathML</a> be the first eigenvalue and eigenfunction of Laplacian in Ω with homogeneous Dirichlet boundary conditions as before. We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M45">View MathML</a>, the other is similar. We also define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M102">View MathML</a>, so we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M157">View MathML</a>

The term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M158">View MathML</a> is dealt with as before, then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M159">View MathML</a>

By virtue of the convex property of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M76">View MathML</a>, and Jensen’s inequality, we still obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M162">View MathML</a>

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M163">View MathML</a>

Now, we can apply Lemma (4.1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M165">View MathML</a> large enough such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M166">View MathML</a>, and note that

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M168">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/76/mathml/M42">View MathML</a> blows up before the maximal time of existence defined in inequality (4.1) is reached. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

YG performed the calculations and drafted the manuscript. WG supervised and participated in the design of the study and modified the draft versions. All authors read and approved the final manuscript.

Acknowledgements

Supported by NSFC (11271154) and by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education and by the 985 program of Jilin University.

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