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Global existence and blow-up of solutions for p-Laplacian evolution equation with nonlinear memory term and nonlocal boundary condition

Zhong Bo Fang* and Jianyun Zhang

Author Affiliations

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

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Boundary Value Problems 2014, 2014:8  doi:10.1186/1687-2770-2014-8

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/8


Received:13 September 2013
Accepted:11 December 2013
Published:7 January 2014

© 2014 Fang 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 cited.

Abstract

In this paper, we deal with an initial boundary value problem for a p-Laplacian evolution equation with nonlinear memory term and inner absorption term subject to a weighted linear nonlocal boundary condition. We find the effects of a weighted function as regards determining blow-up of nonnegative solutions or not and establish the precise blow-up estimate for the linear diffusion case under some suitable conditions.

Keywords:
p-Laplacian evolution equation; nonlinear memory; global existence; blow-up; weight function

1 Introduction

In the past decades, there have been many works dealing with global existence and blow-up properties of solutions for nonlinear parabolic equations, especially the initial boundary value problems with nonlocal terms in equations or boundary conditions, we refer to [1-10] and references therein. For the study of an initial boundary value problem for local parabolic equations with nonlocal boundary condition, we refer to [1-4]. For example, Friedman [1] studied the linear parabolic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M1">View MathML</a>

subject to the weighted linear nonlocal Dirichlet boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M2">View MathML</a>

(1.1)

where A is an elliptic operator,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M3">View MathML</a>

and the nonnegative continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M4">View MathML</a> satisfies suitable conditions. He proved that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M5">View MathML</a>, the solution approaches to 0 monotonously and exponentially as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M6">View MathML</a>. As regards more general discussions on an initial boundary value problem for a linear parabolic equation with a weighted linear nonlocal Neumann boundary condition, one can refer to [2] by Pao, where the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M7">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M8">View MathML</a>

was considered. He studied the asymptotic behavior of solutions and found the influence of the weight function on the existence of global and blow-up solutions. Later, Akila [3] adopted the method of an upper-lower solution to consider the semilinear parabolic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M9">View MathML</a>

under a similar weighted linear nonlocal boundary condition. Wang et al.[4] studied a porous medium equation with power form source term,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M10">View MathML</a>

under the weighted linear nonlocal Dirichlet boundary condition (1.1). By virtue of the method of an upper-lower solution, they obtained global existence, blow-up properties, and blow-up rate of solutions.

For the study of initial boundary value problem with nonlocal parabolic equation, especially the nonlocal problem with time-integral, we refer to [5-10]. Under a homogeneous Dirichlet boundary condition, Li and Xie [5] studied the nonlinear diffusion equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M11">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M13">View MathML</a>. They obtained the sufficient conditions of global existence and blow-up of solutions under appropriate critical conditions. Furthermore, under the following assumptions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M14">View MathML</a>

(1.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M15">View MathML</a>

(1.3)

they derived the following blow-up rate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M16">View MathML</a>

(1.4)

for the special case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M18">View MathML</a>. It is necessary to point out that assumption (1.2) seems to be reasonable, but unfortunately, the authors of [5] did not give a relationship between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M19">View MathML</a> and equation (1.2). The characterization of the monotonicity condition (1.2) was given by Souplet in [6], who proved the existence of monotone in time solutions for the above problem and obtained the blow-up rate (1.4) without the assumption of condition (1.3).

Zhou et al.[7] considered the following singular diffusion equation with memory term:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M20">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M23">View MathML</a>. They got similar results by the method of upper-lower solution. We should notice that this kind of equation can be turned into a degenerate porous medium equation by suitable transformation. In addition, for the system of porous medium equations with nonlinear memory terms and a homogeneous Dirichlet boundary condition, one can refer for example to [8,9].

Recently, Liu and Mu [10] considered the following semilinear parabolic equation with memory term:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M24">View MathML</a>

subject to a weighted nonlinear nonlocal boundary,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M25">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M26">View MathML</a>. They gave the conditions of global existence and blow-up of solutions and the blow-up rate of solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M28">View MathML</a> by establishing an auxiliary function.

In view of the works mentioned above, a nonlocal parabolic equation with time-integral term does not seem to be so much investigated as nonlocal equations with space-integral terms. Already at first glance, the problem with a memory term has some difficulties in proving the existence of non-global solutions. First if t is sufficiently small, the nonlinear memory term vanishes, and then it is not clear whether the comparison principle holds in proving the existence of global small solutions. As far as we know, there are a few papers about the blow-up phenomenon for the p-Laplacian evolution equation with nonlinear memory term. Motivated by it, we consider the global existence and blow-up properties of the following p-Laplacian evolution equation with nonlinear memory term and inner absorption term:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M29">View MathML</a>

(1.5)

subject to weighted linear nonlocal boundary and initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M30">View MathML</a>

(1.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M31">View MathML</a>

(1.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M36">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M37">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M38">View MathML</a>) is a bounded domain with smooth boundary. The weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M39">View MathML</a> in the boundary condition is continuous, nonnegative on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M40">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M41">View MathML</a> on Ω, while the nonnegative and nontrivial initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M42">View MathML</a> satisfies the compatibility conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M43">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M45">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M46">View MathML</a>, which is the closed relationship for local solvability of our problem (1.5)-(1.7) (see Section 2).

The nonlocal diffusion model like equation (1.5) arises in many natural phenomena. In some sense, this kind of nonlocal problem is closer to the actual model than the local problem, such as the model of non-Newton flux through a porous medium, the model for compressible reactive gases, the model of population dynamics, and the model of biological species with human-controlled distribution (see [2,11-14] and references therein). From a physics point of view, equation (1.5) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M48">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M49">View MathML</a> appears in the theory of nuclear reactor dynamics in which case the nonlocal term with time-integral is called the memory term [15]. In fact, there are some important phenomena formulated as parabolic equations which are coupled with weighted nonlocal boundary conditions in mathematical models, such as thermoelasticity theory. In this case, the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M50">View MathML</a> describes the entropy per volume of the materia1 (see [16,17]).

Our main goal is to find the effects of weight function on global or non-global existence of solutions for problem (1.5)-(1.7), the suitable range of nonlinear exponent, and to give the blow-up rate estimate under some suitable conditions. In addition, we treat the nonlocal nonlinearity Hölder (non-Lipschitz) cases m or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M51">View MathML</a>, as well as the Lipschitz cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M52">View MathML</a> in this paper. We get our main results by establishing a modified comparison principle, constructing the suitable upper and lower solutions (including the self-similar lower solutions, the eigenfunction argument and the technique of ordinary differential equation and so on) and the auxiliary function. Moreover, our results extend part of or all results in [8-10]. The detailed results are stated as follows.

• For arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M53">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M54">View MathML</a>, then the solution of problem (1.5)-(1.7) blows up in finite time for sufficiently large initial data.

• If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M55">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M57">View MathML</a>, then the solution of problem (1.5)-(1.7) blows up in finite time for all strictly positive initial dates with T sufficiently large.

• If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M58">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M47">View MathML</a>, the initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M19">View MathML</a> satisfies conditions (H1)-(H2) (see Section 3) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M50">View MathML</a> is the blow-up solution of problem (1.2)-(1.4), then the blow-up rate is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M64">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M66">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M67">View MathML</a> is a constant.

• If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M58">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56">View MathML</a>,

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M70">View MathML</a>, then the solution of problem (1.5)-(1.7) exists globally for small initial data.

(ii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M71">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M72">View MathML</a>, then the solution of problem (1.5)-(1.7) exists globally for small initial data.

(iii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M72">View MathML</a>, then the solution of problem (1.5)-(1.7) exists globally for small enough initial data.

The rest of the paper is organized as follows. In Section 2, we give the preliminaries for our research. The proofs of blow-up results and blow-up rate of solutions are given in Section 3. In Section 4, we will deduce the results of global existence.

2 Comparison principle and local existence

Since equation (1.2) is degenerate when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M74">View MathML</a>, there is no classical solution in general. Hence, it is reasonable to find a weak solution. To this end, we first give the following definition of nonnegative weak solution of problem (1.5)-(1.7).

Definition 1 If the nonnegative function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M50">View MathML</a> satisfies the following conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M76">View MathML</a>

(2.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M77">View MathML</a>

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M78">View MathML</a> is nonnegative, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M79">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M80">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M81">View MathML</a>

(2.3)

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M50">View MathML</a> is called the weak solution of problem (1.5)-(1.7).

If the equalities in equations (2.1)-(2.3) are replaced by ‘≤’ and ‘≥’, we can get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M83">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M84">View MathML</a> which are called the lower solution and upper solution of problem (1.5)-(1.7), respectively.

The following modified comparison principle plays a crucial role in our proofs, which can be obtained by establishing a suitable test function and Gronwall’s inequality.

Proposition 1 (Comparison principle)

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M83">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M84">View MathML</a>are the lower and upper solutions of problem (1.2)-(1.4), respectively. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M88">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M89">View MathML</a>, whereεis any positive constant, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M90">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M91">View MathML</a>.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M44">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M83">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M84">View MathML</a> are the lower and upper solutions of problem (1.5)-(1.7), respectively, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M95">View MathML</a>

Choose a test function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M96">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M97">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M98">View MathML</a> is a characteristic function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M99">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M100">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M101">View MathML</a>

By Lemma 4.10 in [18], we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M102">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M32">View MathML</a>. Moreover, it follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M106">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M107">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M108">View MathML</a>) is bounded, and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M109">View MathML</a>, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M110">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M112">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M88">View MathML</a>. Furthermore, because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M83">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M84">View MathML</a> are bounded functions, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M117">View MathML</a>

That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M118">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M89">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M120">View MathML</a>

By Gronwall’s inequality, we can deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M121">View MathML</a>, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M122">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M123">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M125">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M126">View MathML</a>

in the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M127">View MathML</a> in Ω. Therefore, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M90">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M129">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M130">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M131">View MathML</a>. □

Next, we state the theorem of local existence and uniqueness without proof.

Theorem (Local existence and uniqueness)

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M35">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M36">View MathML</a>, the nonnegative initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M137">View MathML</a>satisfies the compatibility conditions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M43">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M44">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M45">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M46">View MathML</a>. Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M142">View MathML</a>such that problem (1.5)-(1.7) admits a nonnegative solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M143">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M144">View MathML</a>. Furthermore, either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M145">View MathML</a>or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M146">View MathML</a>

Remark 1 The existence of local nonnegative solutions in time to problem (1.5)-(1.7) can be obtained by combining Theorem 1.2 in [4] with Theorem A4′ in [19]. By the comparison principle above, we can get the uniqueness of the solutions to problem (1.5)-(1.7) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M104">View MathML</a>.

3 Blow-up solutions and blow-up rate

Comparing with the problem under a general homogeneous Dirichlet boundary condition, the existence of weight function in the boundary condition has a great influence on the global and non-global existence of solutions.

Theorem 1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M149">View MathML</a>, then the solution of problem (1.5)-(1.7) blows up in finite time for arbitrary<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M53">View MathML</a>and sufficiently large initial data.

Proof In order to prove the blow-up result, we need to establish a self-similar blow-up solution. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M151">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M153">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M154">View MathML</a>

It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M155">View MathML</a> blows up at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M156">View MathML</a> as t approaches T. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M157">View MathML</a>

An explicit calculation yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M158">View MathML</a>

Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M159">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M160">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M161">View MathML</a>

Therefore we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M162">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M163">View MathML</a>

In view of the above, this gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M164">View MathML</a>

We will discuss the problem for two cases.

Case 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M166">View MathML</a>. We need to show that for sufficiently small T,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M167">View MathML</a>

That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M168">View MathML</a>

Let δ be sufficiently large, satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M169">View MathML</a>. By the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M166">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M171">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M172">View MathML</a>, we just have to make the following equality hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M173">View MathML</a>

It is obvious that it holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M174">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M175">View MathML</a>, choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M176">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M177">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M178">View MathML</a>

Case 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M179">View MathML</a>, choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M180">View MathML</a> and we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M181">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M182">View MathML</a>

(3.1)

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M183">View MathML</a>; we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M184">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M185">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M149">View MathML</a>, choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M187">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M188">View MathML</a> to satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M189">View MathML</a>. However,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M190">View MathML</a>

thus we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M191">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M192">View MathML</a>

in the sense of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M171">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M194">View MathML</a>.

In order to get the result, we have to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M195">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M171">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M188">View MathML</a>, we choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M198">View MathML</a>

in which case we can get the result.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M175">View MathML</a>; we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M200">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M201">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M171">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M203">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M204">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M205">View MathML</a>

(3.2)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M207">View MathML</a>, it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M208">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M209">View MathML</a>. Substituting equation (3.2) into equation (3.1) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M210">View MathML</a>

However, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M211">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M208">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M213">View MathML</a>

Finally, we need to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M214">View MathML</a>

Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M215">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M216">View MathML</a>, we have to show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M217">View MathML</a>

In other words, we just need the following inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M218">View MathML</a>

to hold. So choose T to be small enough, and we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M219">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M221">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M222">View MathML</a>

choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M223">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M155">View MathML</a> is the lower solution of problem (1.5)-(1.7). This implies that the solution blows up in finite time for large enough initial data. □

Theorem 2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M225">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M57">View MathML</a>, then the solution of problem (1.5)-(1.7) blows up in finite time for all strictly positive initial dates withTsufficiently large.

Proof Consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M228">View MathML</a>

(3.3)

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M72">View MathML</a>, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M230">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M231">View MathML</a>. Therefore, the solution of equation (3.3) is an upper solution of the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M232">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M233">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M60">View MathML</a>, the solution of this problem blows up in finite time if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M235">View MathML</a>.

It is obvious that the solution of problem (3.3) is a lower solution of problem (1.5)-(1.7) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M236">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M237">View MathML</a>. By Proposition 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M50">View MathML</a> is a blow-up solution. □

Suppose that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M50">View MathML</a> of problem (1.5)-(1.7) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M47">View MathML</a> blows up in finite time, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M241">View MathML</a>. We suppose that the initial data satisfies the following assumptions:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M242">View MathML</a>.

(H2) There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M67">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M244">View MathML</a>.

Theorem 3Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M245">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M19">View MathML</a>satisfies condition (H1)-(H2), and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M50">View MathML</a>is the blow-up solution of problem (1.5)-(1.7) in finite timeTwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M47">View MathML</a>, then the blow-up rate is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M251">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M65">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M253">View MathML</a>.

Remark 2 Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M255">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M256">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M257">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M258">View MathML</a>, one can easily verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M259">View MathML</a> satisfies (C1)-(C2), conditions in Theorem 3 are thus valid.

Lemma 1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M19">View MathML</a>satisfies condition (H1)-(H2), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M47">View MathML</a>, then there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M65">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M263">View MathML</a>.

Proof It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M264">View MathML</a> is Lipschitz continuous and differentiable almost everywhere. By equation (1.5) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M266">View MathML</a>, it yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M267">View MathML</a>

and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M268">View MathML</a>

Integrating the result above over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M269">View MathML</a>, we can obtain the conclusion. □

Proof of Theorem 3 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M270">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M67">View MathML</a> is sufficiently small. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M60">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M273">View MathML</a>

so we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M67">View MathML</a> to be small enough and thus obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M275">View MathML</a>

On the other hand, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M276">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M277">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M278">View MathML</a>, By Jensen’s inequality, this gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M279">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M280">View MathML</a>

It follows from the assumptions of (H1)-(H2) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M281">View MathML</a>. Therefore, it is easy to deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M282">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M283">View MathML</a>. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M284">View MathML</a> and integrating this over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M269">View MathML</a> yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M286">View MathML</a>. Combining the results with Lemma 1, we obtain the desired result. □

4 Global existence of solutions

In this section, we give sufficient conditions of the global existence of solutions.

Theorem 4Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M287">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M70">View MathML</a>, then the solution of problem (1.5)-(1.7) exists globally for small initial data.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M290">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M291">View MathML</a> are determined later, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M292">View MathML</a> solves the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M293">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M294">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M295">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M296">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M297">View MathML</a>

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M298">View MathML</a>, choosing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M299">View MathML</a>

we can infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M300">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M294">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M303">View MathML</a>

Selecting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M304">View MathML</a>, we can deduce that the result holds. □

Theorem 5Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M5">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M71">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M72">View MathML</a>, then the solution of problem (1.5)-(1.7) exists globally for small initial data.

Proof Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M309">View MathML</a> solves the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M310">View MathML</a>

in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M311">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M312">View MathML</a>.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M313">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M314">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M315">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M316">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M317">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M71">View MathML</a>, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M315">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M320">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M321">View MathML</a>. It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M322">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M323">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M314">View MathML</a>, choosing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M325">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M326">View MathML</a>

we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M327">View MathML</a>

On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56">View MathML</a> and sufficiently large A, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M329">View MathML</a>

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M330">View MathML</a> to be sufficiently small such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M331">View MathML</a>, we can conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M332">View MathML</a> is an upper solution of problem (1.5)-(1.7). The proof is completed. □

Theorem 6Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M287">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M56">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M72">View MathML</a>, then the solution of problem (1.5)-(1.7) exists globally for small enough initial data.

Proof Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M336">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/8/mathml/M337">View MathML</a>, it is easy to see that the result holds. □

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

This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2012AM018) and the Fundamental Research Funds for the Central Universities (No. 201362032). The authors would like to warmly thank all the reviewers for their insightful and constructive comments.

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