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Large time behavior of solutions for the porous medium equation with a nonlinear gradient source

Nan Li1, Pan Zheng23*, Chunlai Mu2 and Iftikhar Ahmed2

Author Affiliations

1 Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, P.R. China

2 College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.R. China

3 College of Mathematics and Physics, Chongqing University of Posts and Telecommunications, Chongqing, 400065, P.R. China

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Boundary Value Problems 2014, 2014:98  doi:10.1186/1687-2770-2014-98


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/98


Received:23 December 2013
Accepted:3 April 2014
Published:6 May 2014

© 2014 Li et al.; licensee Springer.

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

Abstract

This paper deals with the large time behavior of non-negative solutions for the porous medium equation with a nonlinear gradient source <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M2">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5">View MathML</a>, we prove that the global solution converges to the separate variable solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M6">View MathML</a>. While <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M7">View MathML</a>, we note that the global solution converges to the separate variable solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M8">View MathML</a>. Moreover, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M9">View MathML</a>, we show that the global solution also converges to the separate variable solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M8">View MathML</a> for the small initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11">View MathML</a>, and we find that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12">View MathML</a> blows up in finite time for the large initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11">View MathML</a>.

MSC: 35K55, 35K65, 35B40.

Keywords:
large time behavior; separate variable solution; porous medium equation; gradient source; blow-up

1 Introduction

In this paper, we investigate the large time behavior of non-negative solutions for the following initial-boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M14">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4">View MathML</a>, Ω is a bounded domain of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M17">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M18">View MathML</a>) with smooth boundary Ω, and the initial function is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M19">View MathML</a>

(1.2)

Equation (1.1) arises in the study of the growth of surfaces and it has been considered as a mathematical model for a variety of physical problems (see [1,2]). For instance, in ballistic deposition processes, the evolution of the profile of a growing interface is described by the diffusive Hamilton-Jacobi type equation (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M20">View MathML</a> (see [3]).

One of the particular feature of problem (1.1) is that the equation is a slow diffusion equation with nonlinear source term depending on the gradient of a power of the solution. In general, there is no classical solution. Therefore, it turns out that a suitable framework for the well-posedness of the initial-boundary value problem (1.1) is the theory of viscosity solutions (see [4-6]), so we first define the notion of solutions.

Definition 1.1 A non-negative function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M21">View MathML</a> is called a solution of (1.1), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12">View MathML</a> is a viscosity solution to (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M23">View MathML</a> and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M24">View MathML</a>

(1.3)

Under some assumptions, the global (local) existence in time, uniqueness and regularity of solutions to reaction-diffusion equations with gradient terms have been extensively investigated by many authors (see [7-12] and the references therein). In particular, in [1], Andreucci proved the existence of solutions for the following degenerate parabolic equation with initial data measures:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M25">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M28">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M29">View MathML</a>.

The main purpose of this paper is to further study the large time behavior of non-negative solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12">View MathML</a> to (1.4) with homogeneous Dirichlet boundary conditions. In recent years, many authors have investigated the asymptotic behavior of solutions to the viscous Hamilton-Jacobi equations (see [3,4,7,8,10,13-21] and the references therein). For example, for the special case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M20">View MathML</a>, Gilding [22] studied the large time behavior of solutions to the following Cauchy problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M32">View MathML</a>

(1.5)

and he gave the temporal decay estimates.

In [23], Stinner investigated the asymptotic behavior of solutions for the following one space dimensional viscous Hamilton-Jacobi equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M33">View MathML</a>

(1.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M35">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M36">View MathML</a>, and he proved that these solutions converge to the steady states by Lyapunov functional.

In higher dimensional case, Barles et al.[24] studied the large time behavior of solutions for the following initial-boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M37">View MathML</a>

(1.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M39">View MathML</a>, and they showed that the non-negative radially symmetric solutions converge to the stationary solution.

Recently, in [25], Laurencot et al. extended the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M39">View MathML</a> of the problem (1.7) to the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M41">View MathML</a>, and derived that these solutions converge to two different separate variables solutions according to the cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M43">View MathML</a> in the general bounded domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M44">View MathML</a>, respectively.

Motivated by the above work, by using the modified comparison argument, the self-similar transformation method, and the half-relaxed limits technique used in [6,25,26], we investigate the asymptotic behavior of non-negative solutions to (1.1). Our main results in this paper are stated as follows.

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M48">View MathML</a>satisfies (1.2). Then there exists a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12">View MathML</a>to (1.1) in the sense of Definition 1.1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M50">View MathML</a>

(1.8)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M51">View MathML</a>is the unique positive solution to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M52">View MathML</a>

(1.9)

Moreover, we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M53">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M54">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M55">View MathML</a>

(1.10)

Theorem 1.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M7">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M48">View MathML</a>satisfies (1.2). Then there exists a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12">View MathML</a>to (1.1) in the sense of Definition 1.1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M61">View MathML</a>

(1.11)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M62">View MathML</a>is the unique positive solution to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M63">View MathML</a>

(1.12)

Theorem 1.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M9">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M48">View MathML</a>satisfies (1.2) and suppose further that there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M68">View MathML</a>satisfying (1.2) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M69">View MathML</a>

(1.13)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M70">View MathML</a>is defined in (1.10). Then there exists a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12">View MathML</a>to (1.1) in the sense of Definition 1.1 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M72">View MathML</a>

(1.14)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M62">View MathML</a>is defined in (1.12).

Theorem 1.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M9">View MathML</a>. Assume that there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M77">View MathML</a>depending only onm, l, q, s, andεsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M78">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M80">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M81">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M82">View MathML</a>. Then the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12">View MathML</a>of the problem (1.1) blows up in finite time in the sense of weak solution. Moreover, the upper bound of blow-up time is given as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M84">View MathML</a>

(1.15)

Remark 1.1 Compared to the results in [25], we extend the results of p-Laplacian equation to the porous medium equation (1.1) with a nonlinear gradient source.

Remark 1.2 In Theorem 1.4, we only give an upper estimate of the blow-up time. But the lower estimate of the blow-up time is an open problem.

This paper is organized as follows. In Section 2, we establish the comparison lemmas to prove the uniqueness of the positive solution to (1.9) and the identification of the half-relaxed limits. In Section 3, using the comparison principle, we construct the global solutions to obtain the upper bound and Hölder estimate of solutions to (1.1), and we prove Theorems 1.1 and 1.2 by the half-relaxed limits method. Moreover, we give the large time behavior of solutions to (1.1) with the small initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M9">View MathML</a>, and we prove Theorem 1.3 in Section 4. Finally, we obtain the blow-up case, and we prove Theorem 1.4 in Section 5.

2 Comparison lemmas

In this section, we establish the following comparison lemma between positive supersolutions and non-negative subsolutions to the elliptic equation in (1.9): which is an important tool for the uniqueness of the positive solution to (1.9) and the identification of the half-relaxed limits later.

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M90">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M91">View MathML</a>are respectively a bounded upper semicontinuous (usc) viscosity subsolution and a bounded lower semicontinuous (lsc) viscosity supersolution to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M92">View MathML</a>

(2.1)

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M93">View MathML</a>

(2.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M94">View MathML</a>

(2.3)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M95">View MathML</a>

(2.4)

Proof The proof is based on the idea as in [25,27], but with different auxiliary functions.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M96">View MathML</a> large enough, it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M97">View MathML</a> is a non-empty open subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M98">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M99">View MathML</a> is compact and W is lower semicontinuous, the function W has a minimum in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M99">View MathML</a>. By the positivity (2.3) of W in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M99">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M102">View MathML</a>

(2.5)

Similarly, the compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M103">View MathML</a>, the upper semicontinuity and boundedness of w imply that w has at least one point of maximum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M104">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M105">View MathML</a> and we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M106">View MathML</a>

(2.6)

It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M107">View MathML</a> and w vanishes on Ω by (2.2) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M108">View MathML</a>.

Next, we claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M109">View MathML</a>

(2.7)

Indeed, owing to the compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M110">View MathML</a> and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M111">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M112">View MathML</a> and a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M113">View MathML</a> (not relabeled) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M114">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M115">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M116">View MathML</a>, we deduce from the upper semicontinuity of w that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M117">View MathML</a>

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M118">View MathML</a> small enough, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M119">View MathML</a> large enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M120">View MathML</a>

Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M121">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M122">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M123">View MathML</a>, we conclude that zero is a cluster point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M124">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M115">View MathML</a>. The claim (2.7) follows from the monotonicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M124">View MathML</a>.

Now, fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M127">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M128">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M96">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M130">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M131">View MathML</a>

It follows from the assumptions on w and W that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M132">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M133">View MathML</a> are, respectively, a bounded usc viscosity subsolution and a bounded lsc viscosity supersolution to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M134">View MathML</a>

and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M135">View MathML</a>

Moreover, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M136">View MathML</a>

(2.8)

then it follows from (2.5) and (2.8) that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M137">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M138">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M139">View MathML</a>, we deduce from (2.6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M140">View MathML</a>

By the comparison principle [6], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M141">View MathML</a>

(2.9)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M96">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M128">View MathML</a> satisfying (2.8).

According to (2.8), the parameter δ can be taken to be arbitrarily small in (2.9). Therefore, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M144">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M96">View MathML</a>.

Passing to the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M115">View MathML</a>, it follows from (2.7) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M147">View MathML</a>

Finally, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M148">View MathML</a> and take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M149">View MathML</a>; then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M150">View MathML</a>

The proof of Lemma 2.1 is complete. □

A straightforward consequence of Lemma 2.1 is the uniqueness of the solution to (1.9).

Corollary 2.1There is at most one positive viscosity solution to (1.9).

By the similar argument, we have the following result to (1.12).

Lemma 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M90">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M152">View MathML</a>be respectively a bounded upper semicontinuous (usc) viscosity subsolution and a bounded lower semicontinuous (lsc) viscosity supersolution to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M153">View MathML</a>

(2.10)

satisfying (2.2) and (2.3). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M154">View MathML</a>

(2.11)

Proof The proof is similar as in Lemma 2.1, so we omit it here. □

3 Proofs of Theorems 1.1 and 1.2

In this section, we obtain the well-posedness and large time behavior of solutions to (1.1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M155">View MathML</a>, and we prove Theorems 1.1-1.2. To do this, we first obtain the well-posedness to (1.1) by the following proposition.

Proposition 3.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M155">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M159">View MathML</a>satisfies (1.2). Then there exists a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12">View MathML</a>to (1.1) in the sense of Definition 1.1.

Proof The idea of the proof is same as in [25,28], so we omit here. □

Next, in order to prove the large time behavior of the solution to (1.1), we shall need several steps: Step 1, we will find that the temporal decay rate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M161">View MathML</a> is indeed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M162">View MathML</a>. Step 2, we prove the boundary estimates for the large time which guarantee that no loss of boundary condition occurs throughout the time evolution. Step 3, the half-relaxed limits technique is applied to show the expected convergence after introducing self-similar variables. The approach is developed by Laurençot and Stinner in [25,27]. To do this, we need the following lemmas.

Lemma 3.1 (Upper bound)

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5">View MathML</a>, and the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M48">View MathML</a>satisfies (1.2). Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M167">View MathML</a>depending only onm, l, q, Ω, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M168">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M169">View MathML</a>

(3.1)

Proof Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M170">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M171">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M172">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M173">View MathML</a>, we define the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M174">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M175">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M176">View MathML</a> satisfies the following condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M177">View MathML</a>

(3.2)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M170">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M179">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M180">View MathML</a>-smooth in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M181">View MathML</a>. Moreover, according the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M185">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M186">View MathML</a>. Therefore, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M187">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M188">View MathML</a>, it follows from (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M189">View MathML</a>

(3.3)

Hence, the condition (3.2) guarantees that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M179">View MathML</a> is a supersolution to (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M191">View MathML</a>. In addition, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M188">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M193">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M194">View MathML</a>, we deduce from (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M195">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M196">View MathML</a>

By the comparison principle, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M197">View MathML</a>

The proof of Lemma 3.1 is complete. □

Lemma 3.2 (Upper bound)

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M200">View MathML</a>, and the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M201">View MathML</a>satisfies (1.2). Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M202">View MathML</a>depending only onm, l, q, Ω, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M168">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M204">View MathML</a>

(3.4)

Proof Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M170">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M171">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M172">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M173">View MathML</a>, we define the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M209">View MathML</a>

where the positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M210">View MathML</a>, R, and δ satisfy the following condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M211">View MathML</a>

(3.5)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M170">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M213">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M180">View MathML</a>-smooth in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M181">View MathML</a>. Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M187">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M188">View MathML</a>, it follows from (3.5) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M218">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M219">View MathML</a>

(3.6)

Therefore, the condition (3.5) guarantees that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M213">View MathML</a> is a supersolution to (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M191">View MathML</a>. In addition, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M188">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M193">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M194">View MathML</a>, we deduce from (3.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M225">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M226">View MathML</a>

By the comparison principle, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M227">View MathML</a>

The proof of Lemma 3.2 is complete. □

Lemma 3.3 (Hölder estimate)

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M155">View MathML</a>, and the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M48">View MathML</a>satisfies (1.2). Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M232">View MathML</a>depending only onm, l, q, Ω, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M233">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M234">View MathML</a>

(3.7)

Proof Since the boundary Ω of Ω is smooth, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M235">View MathML</a> such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M236">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M237">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M238">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M239">View MathML</a>. It follows from the initial data condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M201">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11">View MathML</a> is Lipschitz continuous, i.e., there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M242">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M243">View MathML</a>

(3.8)

Next, we define the open subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M244">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M98">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M246">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M247">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M248">View MathML</a>, and denote the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M249">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M250">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M251">View MathML</a>.

Moreover, we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M252">View MathML</a>

(3.9)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M253">View MathML</a>

(3.10)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M255">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M256">View MathML</a> are defined in Lemmas 3.1 and 3.2, respectively.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M257">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M258">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M180">View MathML</a>-smooth in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M260">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M261">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M262">View MathML</a>. By a direct computation, we infer from (3.9)-(3.10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M263">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M258">View MathML</a> is a supersolution to (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M265">View MathML</a>. In addition, it follows from (3.8)-(3.10) and the mean value theorem that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M266">View MathML</a>

Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M267">View MathML</a>, we have either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M268">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M269">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M268">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M271">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M269">View MathML</a>, by Lemmas 3.1-3.2 and (3.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M273">View MathML</a>

By the comparison principle [6] we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M274">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M275">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M276">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M277">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M278">View MathML</a>

(3.11)

Finally, we consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M193">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M280">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M281">View MathML</a>, it follows from Lemmas 3.1-3.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M282">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M254">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M255">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M256">View MathML</a> are defined in Lemmas 3.1 and 3.2, respectively.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M286">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M237">View MathML</a> satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M238">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M239">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M193">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M291">View MathML</a>

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M292">View MathML</a>. Therefore, we deduce from (3.11) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M293">View MathML</a>

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M294">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M295">View MathML</a>

The proof of Lemma 3.3 is complete. □

We next proceed as in [29] to deduce the Hölder continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12">View MathML</a> from Lemma 3.3. Therefore, we obtain the following corollary.

Corollary 3.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M155">View MathML</a>, and the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M48">View MathML</a>satisfies (1.2). Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M301">View MathML</a>depending only onm, l, q, Ω, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M302">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M303">View MathML</a>

(3.12)

Proof The proof is similar to the argument in [25,29], so we omit here. □

Proofs of Theorems 1.1 and 1.2 The proofs are based on the ideas in [25], but we give the details of the argument for the reader’s convenience. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M304">View MathML</a> be the solution to the porous medium equation with homogeneous Dirichlet boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M305">View MathML</a>

(3.13)

According to the nonnegativity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M306">View MathML</a>, it follows from the comparison principle [6] that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M307">View MathML</a>

(3.14)

We introduce the scaling variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M308">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M309">View MathML</a> and denote the new unknown function v and V by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M310">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M311">View MathML</a>

Then v is a viscosity solution to the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M312">View MathML</a>

(3.15)

In addition, owing to Lemmas 3.1-3.3 and (3.14), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M313">View MathML</a>

(3.16)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M314">View MathML</a>

(3.17)

Next, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M80">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M316">View MathML</a>

and the half-relaxed limits

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M317">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M318">View MathML</a>.

By (3.16), it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M319">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M320">View MathML</a> are well-defined and do not depend on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M321">View MathML</a>. Moreover, it readily follows from (3.15) and (3.17) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M322">View MathML</a>

(3.18)

By a direct computation, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M323">View MathML</a> is a solution to the following initial-boundary problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M324">View MathML</a>

(3.19)

Next, we shall give proofs of Theorems 1.1 and 1.2. To do this, we distinguish the two cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M200">View MathML</a>.

Case 1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5">View MathML</a>. We use the stability of semicontinuous viscosity solutions [6] to deduce from (3.19) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M328">View MathML</a>

(3.20)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M329">View MathML</a>

(3.21)

In addition, by [30], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M330">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M331">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M332">View MathML</a>. Moreover, it follows from (3.16) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M319">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M320">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M335">View MathML</a>

(3.22)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M336">View MathML</a> in Ω by [30], we deduce from (3.22) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M319">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M320">View MathML</a> are positive and bounded in Ω and vanish on Ω by (3.18). Owing to (3.20) and (3.21), we infer from Lemma 2.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M339">View MathML</a>

By (3.22), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M340">View MathML</a>

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M341">View MathML</a>, we deduce from (3.18), (3.20), (3.21), and (3.22) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M51">View MathML</a> is a positive viscosity solution to (2.1) and solves (1.9). Therefore, the existence of a positive solution to (1.9) is proved. Moreover, by Corollary 2.1, we obtain the uniqueness of solution to (1.9).

Furthermore, it follows from the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M343">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M344">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M345">View MathML</a>

(3.23)

Therefore, we infer from the scaling transformation that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M346">View MathML</a>

(3.24)

Finally, Corollary 3.1 gives the last statement of Theorem 1.1. The proof of Theorem 1.1 is complete.

Case 2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M200">View MathML</a>. We use once more the stability of semicontinuous viscosity solutions [6] to deduce from (3.19) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M348">View MathML</a>

(3.25)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M349">View MathML</a>

(3.26)

In addition, by [30], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M330">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M331">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M332">View MathML</a>. Moreover, it follows from (3.16) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M319">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M320">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M355">View MathML</a>

(3.27)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M336">View MathML</a> in Ω by [30] and is a solution to (2.10), we can apply Lemma 2.2 to conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M357">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M110">View MathML</a>. Owing to (3.27), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M359">View MathML</a>

Therefore, we deduce from (3.19) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M360">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M361">View MathML</a>

(3.28)

Thus, we infer from the scaling transformation that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M362">View MathML</a>

(3.29)

The proof of Theorem 1.2 is complete. □

4 Proof of Theorem 1.3

In this section, we shall consider the well-posedness and the large time behavior of solutions to (1.1) with the small initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M9">View MathML</a> by the method used in [25]. To do this, we need the following lemma.

Lemma 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4">View MathML</a>. Assume that G is the corresponding solution to (1.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M68">View MathML</a>satisfying (1.2) for the case<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5">View MathML</a>, and denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M369">View MathML</a>

(4.1)

where the parameter<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M70">View MathML</a>is defined in (1.10). Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M371">View MathML</a>is a solution to (1.1) with the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M372">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M374">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M375">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M376">View MathML</a>is a supersolution to (1.1) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M218">View MathML</a>.

Proof According to the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M70">View MathML</a> in (1.10), it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M374">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M375">View MathML</a>.

Next, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M381">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M382">View MathML</a> has a local minimum at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M383">View MathML</a>. Since ℱ is smooth with respect to the time variable and Hölder continuous with respect to the space variable, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M384">View MathML</a>

(4.2)

Moreover, introducing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M385">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M386">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M387">View MathML</a> has a local minimum at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M388">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M389">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M390">View MathML</a>

(4.3)

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M376">View MathML</a> is a supersolution to (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5">View MathML</a>. In a similar way, it can be shown that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M376">View MathML</a> is also a subsolution. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M376">View MathML</a> is a solution to (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M5">View MathML</a>.

Furthermore, we deduce from (4.2), (4.3), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M218">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M397">View MathML</a>

(4.4)

The proof of Lemma 4.1 is complete. □

Proposition 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M4">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M9">View MathML</a>. Assume that the initial data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M401">View MathML</a>satisfies (1.2), moreover, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M402">View MathML</a>satisfying (1.2) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M403">View MathML</a>

(4.5)

where the parameter<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M70">View MathML</a>is defined in (1.10). Then there exists a unique solution u to (1.1) in the sense of Definition 1.1 and it satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M405">View MathML</a>

(4.6)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M376">View MathML</a>is defined in (4.1).

Proof On the one hand, the solution U to the porous medium equation (3.13) is clearly a subsolution to (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M191">View MathML</a>.

On the other hand, it follows from Lemma 4.1 that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M376">View MathML</a> is a supersolution to (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M191">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M376">View MathML</a> is a supersolution to (3.13).

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M411">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M412">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M413">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M414">View MathML</a> by (4.5), we infer from the comparison principle [6] that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M415">View MathML</a>

This property and the simultaneous vanishing of U and ℱ on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M412">View MathML</a> allow us to use the classical Perron method to establish the existence of a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12">View MathML</a> to (1.1) in the sense of Definition 1.1 which satisfies (4.6). The uniqueness next follows from the comparison principle [6]. The proof of Proposition 4.1 is complete. □

Proof of Theorem 1.3 We notice that Lemma 3.2 is also valid in that case. It readily follows from Lemma 3.3 and Proposition 4.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M418">View MathML</a>

The convergence proof is similar to that performed in the proof of Theorem 1.2 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M200">View MathML</a>. The proof of Theorem 1.3 is complete. □

5 Proof of Theorem 1.4

In this section, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M9">View MathML</a>, we shall prove that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12">View MathML</a> of (1.1) blows up in finite time for the large initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11">View MathML</a> in the sense of weak solution by the method used in [26].

In order to obtain a blow-up condition corresponding to (1.1), we have to modify the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M423">View MathML</a> used in [31-33], and introduce a test function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M424">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M425">View MathML</a>

Proof of Theorem 1.4 Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12">View MathML</a> is the solution of the problem (1.1) and T is the blow-up time of the solution. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M79">View MathML</a>, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M428">View MathML</a>

(5.1)

By a direct calculation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M429">View MathML</a>

(5.2)

By Young’s inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M430">View MathML</a>

(5.3)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M431">View MathML</a>, it follows from (5.2), (5.3), and Poincaré’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M432">View MathML</a>

(5.4)

According to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M433">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M431">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M435">View MathML</a> and Hölder’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M436">View MathML</a>

(5.5)

Thus, by (5.4) and (5.5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M437">View MathML</a>

(5.6)

Owing to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M433">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M431">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M435">View MathML</a>, and Jensen’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M441">View MathML</a>

(5.7)

Therefore, it follows from (5.6) and (5.7) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M442">View MathML</a>

(5.8)

as long as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M443">View MathML</a>

(5.9)

Taking

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M444">View MathML</a>

Since the initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M11">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M446">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M447">View MathML</a>

(5.10)

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M448">View MathML</a> increases and remains above <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M449">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M450">View MathML</a>.

By (5.8), integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M451">View MathML</a> yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M452">View MathML</a>

(5.11)

Hence, it follows from (5.10) and (5.11) that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12">View MathML</a> of (1.1) blows up in finite time, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M454">View MathML</a>.

Moreover, by (5.10), we obtain the upper estimate on the blow-up time T of the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M12">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/98/mathml/M456">View MathML</a>

(5.12)

The proof of Theorem 1.4 is complete. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The authors would like to express sincere gratitude to the referees for their valuable suggestions and comments on the original manuscript. The first author is supported by Scientific Research Fund of Sichuan Provincial Science and Technology Department (2014JY0098); the second author is supported in part by the Fundamental Research Funds for the Central Universities, Project No. CDJXS 12 10 00 14; the third author is supported in part by NSF of China (11371384).

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