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Blow-up profile for a degenerate parabolic equation with a weighted localized term

Weili Zeng1*, Xiaobo Lu1, Shumin Fei1 and Miaochao Chen2

Author Affiliations

1 School of Automation, Southeast University, Nanjing, 210096, China

2 Department of Mathematics, Southeast University, Nanjing, 210096, China

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Boundary Value Problems 2013, 2013:269  doi:10.1186/1687-2770-2013-269


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/269


Received:23 June 2013
Accepted:21 November 2013
Published:12 December 2013

© 2013 Zeng et al.; licensee Springer.

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

Abstract

In this paper, we investigate the Dirichlet problem for a degenerate parabolic equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M1">View MathML</a>. We prove that under certain conditions the solutions have global blow-up, and the rate of blow-up is uniform in all compact subsets of the domain. Moreover, the blow-up profile is precisely determined.

Keywords:
degenerate parabolic equation; localized source; uniform blow-up rate

1 Introduction

In this paper, we consider the following parabolic equation with nonlocal and localized reaction:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M2">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M3">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M4">View MathML</a>

(1.3)

where Ω is an open ball of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M6">View MathML</a> with radius R, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M7">View MathML</a>.

Many of localized problems arise in applications and have been widely studied. Equations (1.1)-(1.3), as a kind of porous medium equation, can be used to describe some physical phenomena such as chemical reactions due to catalysis and an ignition model for a reaction gas (see [1-3]).

As for our problem (1.1)-(1.3), to our best knowledge, many works have been devoted to the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M8">View MathML</a> (see [4-7]). Let us mention, for instance, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M9">View MathML</a>, blow-up properties have been investigated by Okada and Fukuda [7]. Moreover, they proved that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M11">View MathML</a> is sufficiently large, every radial symmetric solution (maximal solution) has a global blow-up and the solution satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M12">View MathML</a>

(1.4)

in all compact subsets of Ω as t is near the blow-up time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M13">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M15">View MathML</a> are two positive constants. Souplet [4,8] investigated that global blow-up solutions have uniform blow-up estimates in all compact subsets of the domain.

The work of this paper is motivated by the localized semi-linear problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M16">View MathML</a>

(1.5)

with Dirichlet boundary condition (1.2) and initial condition (1.3). In the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M18">View MathML</a>, the uniform blow-up profiles were studied in [5,9] and [10], respectively.

It seems that the result of [5,9,10] can be extended to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M20">View MathML</a> are two functions. Motivated by this, in this paper, we extend and improve the results of [5,9,10]. Our approach is different from those previously used in blow-up rate studies.

In the following section, we establish the blow-up rate and profile to (1.1)-(1.3).

2 Blow-up rate and profile

Throughout this paper, we assume that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M11">View MathML</a> satisfy the following two conditions:

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M26">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M11">View MathML</a> are positive in Ω.

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M11">View MathML</a> are radially symmetric; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M34">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M35">View MathML</a> are non-increasing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M36">View MathML</a>.

Theorem 2.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M37">View MathML</a>satisfies (A1) and (A2). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M38">View MathML</a>, then the solutions of (1.1)-(1.3) blow up in finite time for large initial data.

The proof of this theorem bears much resemblance to the result in [7,11,12] and is, therefore, omitted here.

Next we will show that in the situation of localized source dominating (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M39">View MathML</a>), problem (1.1)-(1.3) admits some uniform blow-up profile.

Theorem 2.2Assume (A1) and (A2). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M40">View MathML</a>be the blow-up solution of (1.1)-(1.3) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M40">View MathML</a>is non-decreasing in time. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M42">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M43">View MathML</a>

(2.1)

uniformly in all compact subsets of Ω.

Throughout this paper, we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M44">View MathML</a>

In our consideration, a crucial role is played by the Dirichlet eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M45">View MathML</a>

Denote by λ the first eigenvalue and by φ the corresponding eigenfunction with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M46">View MathML</a> in Ω, normalized by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M47">View MathML</a>. In the following, C is different from line to line. Also, we will sometimes use the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M48">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M49">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M13">View MathML</a> the blow-up time for (1.1)-(1.3).

In order to prove the results of Section 2, first we derive a fact of the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M51">View MathML</a>

(2.2)

Lemma 2.1Assume (A1), (A2) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M39">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M40">View MathML</a>be the blow-up solution of (2.2) and assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M40">View MathML</a>is non-decreasing in time, we then have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M55">View MathML</a>

(2.3)

uniformly in all compact subsets of Ω.

Proof Assumption (A2) implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M56">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M57">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M58">View MathML</a>. From (2.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M59">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M60">View MathML</a>

(2.4)

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M62">View MathML</a>.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M66">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M67">View MathML</a>, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M68">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M69">View MathML</a>.

Introducing a function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M70">View MathML</a>

In the following, we only consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M71">View MathML</a>. For the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M72">View MathML</a>, the proof is similar.

A series calculation yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M73">View MathML</a>

(2.5)

In addition, note

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M74">View MathML</a>

(2.6)

Now, according to (2.5) and (2.6), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M75">View MathML</a>

(2.7)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M77">View MathML</a>.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M80">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M81">View MathML</a> and note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M82">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M83">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M84">View MathML</a>.

Therefore, in view of (2.7), we observe

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M85">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M87">View MathML</a>. We then obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M88">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M89">View MathML</a> is a sup-solution of the following equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M90">View MathML</a>

(2.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M91">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M92">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M93">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M94">View MathML</a>. Here we also assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M95">View MathML</a> is a symmetric and non-increasing function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M96">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M97">View MathML</a>).

By the maximum principle, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M99">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M92">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M101">View MathML</a>.

Similar to the proof of Theorem 3.1 in [9] that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M102">View MathML</a>

uniformly in all compact subsets of Ω.

By the arbitrariness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M92">View MathML</a>, we obtain that the following limit converges uniformly in all compact subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M92">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M105">View MathML</a>

(2.9)

In particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M106">View MathML</a>

This inequality and (2.4) infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M107">View MathML</a>

Multiplying both sides of (2.2) by φ and integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M108">View MathML</a>, we have, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M77">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M110">View MathML</a>

(2.10)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M111">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M112">View MathML</a>, it then follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M113">View MathML</a>

(2.11)

Next we prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M114">View MathML</a>

uniformly in any compact subsets of Ω.

Assume on the contrary that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M115">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M116">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M117">View MathML</a>

Then there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M119">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M120">View MathML</a>

Using the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M21">View MathML</a>, we see that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M122">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M123">View MathML</a>) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M124">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M125">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M126">View MathML</a> and (2.9), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M127">View MathML</a>

This contradicts (2.11) and we then complete the proof of Lemma 2.1. □

Lemma 2.2Under the assumption of Theorem 2.2, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M40">View MathML</a>be the blow-up solution of (1.1)-(1.3), then it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M129">View MathML</a>

(2.12)

uniformly in all compact subsets in Ω.

Proof Proceeding as in (2.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M130">View MathML</a>

(2.13)

which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M62">View MathML</a>.

Now, according to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M133">View MathML</a>, it then follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M40">View MathML</a> is a sub-solution of the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M135">View MathML</a>

By the maximum principle, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M136">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M99">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M138">View MathML</a>. Using Lemma 2.1, it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M139">View MathML</a>

uniformly in all compact subsets of Ω.

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M140">View MathML</a>

(2.14)

uniformly in any compact subsets of Ω, which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M141">View MathML</a>

(2.15)

Combining (2.13) with (2.15), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M142">View MathML</a>

(2.16)

Multiplying both sides of (1.1) by φ and integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M143">View MathML</a>, we find, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M77">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M145">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M146">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M39">View MathML</a>, it then follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M148">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M149">View MathML</a>

By analogy with the argument taken in Lemma 2.1, we complete the proof of this lemma. □

Proof of Theorem 2.2 By Lemma 2.2, we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M150">View MathML</a>

hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M151">View MathML</a>

Integrating this equivalence between t and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M13">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M153">View MathML</a>

(2.17)

The result finally follows by returning (2.17) to (2.12). □

Remark 2.1 It seems that in the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/269/mathml/M154">View MathML</a>, the blow-up rate remains valid in all compact subsets, but we do not know how to treat it. (It is an open problem in this case.)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All the authors typed, read and approved the final manuscript.

Acknowledgements

This work was supported by the China Postdoctoral Science Foundation Founded Project (2013M540405), the National Natural Science Foundation of China (61374194), and the Scientific Innovation Research of College Graduate in Jiangsu Province (CXZZ_0163).

References

  1. Bebernes, J, Bressan, A: Total blow-up versus single point blow-up. J. Differ. Equ.. 73(1), 30–44 (1988). Publisher Full Text OpenURL

  2. Bimpong-Bota, K, Ortoleva, P, Ross, J: Far-from-equilibrium phenomena at local sites of reaction. J. Chem. Phys.. 60(8), 3124–3133 (1974). Publisher Full Text OpenURL

  3. Ortoleva, P, Ross, J: Local structures in chemical reactions with heterogeneous catalysis. J. Chem. Phys.. 56(9), 4397–4400 (1972). Publisher Full Text OpenURL

  4. Souplet, P: Uniform blow-up profiles and boundary for diffusion equations with nonlocal nonlinear source. J. Differ. Equ.. 153(2), 374–406 (1999). Publisher Full Text OpenURL

  5. Liu, QL, Li, YX, Gao, HJ: Uniform blow-up rate for diffusion equations with localized nonlinear source. J. Math. Anal. Appl.. 320(2), 771–778 (2006). Publisher Full Text OpenURL

  6. Zheng, SN, Wang, JH: Total versus single point blow-up in heat equations with coupled localized sources. Asymptot. Anal.. 51(2), 133–156 (2007)

  7. Okada, A, Fukuda, I: Total versus single point blow-up of solution of a semilinear parabolic equation with localized reaction. J. Math. Anal. Appl.. 281(2), 485–500 (2003). Publisher Full Text OpenURL

  8. Souplet, P: Blow-up in non-local reaction-diffusion equations. SIAM J. Math. Anal.. 29(6), 1301–1334 (1998). Publisher Full Text OpenURL

  9. Liu, QL, Li, YX, Gao, HJ: Uniform blow-up rate for diffusion equations with nonlocal nonlinear source. Nonlinear Anal.. 67(6), 1947–1957 (2007). Publisher Full Text OpenURL

  10. Wang, JH, Kong, LH, Zheng, SN: Asymptotic analysis for a localized nonlinear diffusion equation. Comput. Math. Appl.. 56(9), 2294–2304 (2008). Publisher Full Text OpenURL

  11. Fukuda, I, Suzuki, R: Blow-up behavior for a nonlinear heat equation with a localized source in a ball. J. Differ. Equ.. 218(2), 217–291 (2005)

  12. Du, LL, Xiang, ZY: A further blow-up analysis for a localized porous medium equation. Appl. Math. Comput.. 179(1), 200–208 (2006). Publisher Full Text OpenURL