Open Access Research

Non-simultaneous blow-up for a parabolic system with nonlinear boundary flux which obey different laws

Si Xu1* and Jinfa Zeng2

Author affiliations

1 Department of Mathematics, Jiangxi Vocational College of Finance and Economics, Jiujiang, Jiangxi, 332000, P.R. China

2 Department of Information Engineering, Jiangxi Vocational College of Finance and Economics, Jiujiang, Jiangxi, 332000, P.R. China

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Citation and License

Boundary Value Problems 2012, 2012:85  doi:10.1186/1687-2770-2012-85


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/85


Received:16 April 2012
Accepted:25 July 2012
Published:6 August 2012

© 2012 Xu and Zeng; 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 consider a system of two heat equations with nonlinear boundary flux which obey different laws, one is exponential nonlinearity and another is power nonlinearity. Under certain hypotheses on the initial data, we get the sufficient and necessary conditions, on which there exist initial data such that non-simultaneous blow-up occurs. Moreover, we get some conditions on which simultaneous blow-up must occur. Furthermore, we also get a result on the coexistence of both simultaneous and non-simultaneous blow-ups.

MSC: 35B33, 35K65, 35K55.

Keywords:
simultaneous blow-up; non-simultaneous blow-up; parabolic system; nonlinear boundary flux

1 Introduction and main results

In this paper, we study the following system of two heat equations coupled by nonlinear boundary conditions,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M2">View MathML</a>, parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M4">View MathML</a>. Assume the non-zero, non-negative initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M6">View MathML</a> are radially symmetric non-increasing continuous functions, vanishing on Ω, as well as satisfy the compatibility conditions,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M7">View MathML</a>

(1.2)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M8">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M9">View MathML</a>.

The system (1.1) can be used to describe heat propagation of a two-component combustible mixture in a bounded region. In this case, u and v represent the temperatures of the interacting components, thermal conductivity is supposed constant and equal for both substances, and a volume energy release given by powers of u and v is assumed; see [1,6]. The nonlinear Neumann boundary conditions can be physically interpreted as the cross-boundary fluxes, which obey different laws; some may obey power laws [4,7,10,14], some may follow exponential laws [18]. It is interesting when the two types of boundary fluxes meet. In system (1.1), the coupled boundary flux obey a mixed type of power terms and exponential terms.

As we shall see, under certain conditions the solutions of this problem can become unbounded in a finite time. This phenomenon is known as blow-up, and has been observed for several scalar equation since the pioneering work of Fujita. Blow-up may also happen for systems, X. F. Song considered the blow-up conditions and blow-up rates of system (1.1), when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M10">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M12">View MathML</a>, in [16].

However, it can only show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M13">View MathML</a>

whether the blow-up is simultaneous or non-simultaneous is not known yet.

Recently, the simultaneous and non-simultaneous blow-up problems of parabolic systems have been widely considered by many authors [2,3,8,9,11-13,15,19,20]. For example, B. C. Liu and F. J. Li [8] considered the nonlinear parabolic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M14">View MathML</a>

They got a complete and optimal classification on non-simultaneous and simultaneous blow-ups by four sufficient and necessary conditions.

Motivated by the above works, we will focus on the simultaneous and non-simultaneous blow-up problems to system (1.1), and get our main results as follows.

Theorem 1.1There exist initial data such that the solutions of (1.1) blow up, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M15">View MathML</a>

In the sequel, we assume the blow-up indeed occurs. Then we get the conditions, under which simultaneous or non-simultaneous blow-up occurs.

Theorem 1.2There exist initial data such that non-simultaneous blow-up occurs if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M16">View MathML</a>

Corollary 1.1Any blow-up is simultaneous if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M17">View MathML</a>

Theorem 1.3If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M18">View MathML</a>

both non-simultaneous and simultaneous blow-ups may occur.

In order to show the conditions more clearly, we graph Figure 1 with the region of non-simultaneous and simultaneous blow-ups occur in the parameter space.

thumbnailFigure 1. Non-simultaneous and simultaneous blow-ups.

The rest of this paper is organized as follows: In next section, we consider the blow-up conditions of system (1.1), give the proof of Theorem 1.1. In Section 3, we will study the sufficient and necessary conditions of non-simultaneous blow-up, in order to prove Theorem 1.2. In Section 4, we consider the coexistence of both simultaneous and non-simultaneous blow-ups; Theorem 1.3 is proved.

2 Blow-up

In this section, we prove the blow-up criterion of system (1.1). First, we check the monotonicity of the solution.

Lemma 2.1Let (u, v) be a solution of system (1.1), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M19">View MathML</a>, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M20">View MathML</a>.

Proof Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M21">View MathML</a>

From the hypothesis of initial data, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M22">View MathML</a>

By the comparison principle, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M23">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M20">View MathML</a>. □

Proof of Theorem 1.1 It is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M25">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M26">View MathML</a> be a solution of the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M27">View MathML</a>

(2.1)

By the results of [17], the solutions of (2.1) blow up with large initial data if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M28">View MathML</a>, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M29">View MathML</a>, or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M30">View MathML</a>. By the comparison principle, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M31">View MathML</a> is a sub-solution of (1.1), thus the solutions of (1.1) also blow up. □

3 Non-simultaneous blow-up

In this section, we prove Theorem 1.2 with four lemmas. Firstly, we define the set of initial data with a fixed constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M32">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M33">View MathML</a>

Lemma 3.1For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M34">View MathML</a>, there must be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M35">View MathML</a>

(3.1)

Proof Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M36">View MathML</a>

By computations, we can check that

By the comparison principle, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M38">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M39">View MathML</a>. □

Lemma 3.2For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M40">View MathML</a>

(3.2)
(3.3)

Proof First, we prove (3.2). From (3.1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M43">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M44">View MathML</a>

(3.4)

Integrating (3.4) from t to T,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M45">View MathML</a>

thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M46">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M47">View MathML</a>

Similarly, we can also prove (3.3) from (3.1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M48">View MathML</a>

Integrating the above inequality from t to T, then

 □

The following lemma proves the sufficient and necessary condition on the existence of u blowing up alone.

Lemma 3.3There exist suitable initial data such thatublows up whilevremains bounded if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M50">View MathML</a>.

Proof Firstly, we prove the sufficiency.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M51">View MathML</a>

be the fundamental solution of the heat equation. Assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M52">View MathML</a> is a pair of initial data such that the solution of (1.1) blows up. Fix radially symmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M6">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M54">View MathML</a>) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M55">View MathML</a> and take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M56">View MathML</a>. Let the minimum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M58">View MathML</a>) be large such that T is small and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M59">View MathML</a>

Consider the auxiliary problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M60">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M50">View MathML</a> and by Green’s identity [5], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M62">View MathML</a>

thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M63">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M20">View MathML</a>. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M65">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M66">View MathML</a>

Combining the radial symmetry and the monotonicity of the initial data with the estimate (3.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M67">View MathML</a>

So, v satisfies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M68">View MathML</a>

By the comparison principle, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M69">View MathML</a>, so v remains bounded up to time T. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M71">View MathML</a> blows up, hence only u blows up at time T.

Secondly, we prove the necessity. Assume u blows up while v remains bounded, say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M72">View MathML</a>.

By Green’s identity, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M73">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M74">View MathML</a>, take t such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M75">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M76">View MathML</a>

hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M77">View MathML</a>

For some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M78">View MathML</a>, we can find a suitable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M79">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M80">View MathML</a>

Similarly to Lemma 3.1, we can prove there must be

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M81">View MathML</a>

(3.5)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M82">View MathML</a>

Integrating the above inequality from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M83">View MathML</a> to t, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M84">View MathML</a>

The boundedness of v requires that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M50">View MathML</a>. □

The following lemma proves the sufficient and necessary condition on the existence of v blowing up alone.

Lemma 3.4There exist suitable initial data such thatvblows up whileuremains bounded if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M86">View MathML</a>.

Proof Firstly, we prove the sufficiency. Assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M52">View MathML</a> is a pair of initial data such that the solution of (1.1) blows up. Fix radially symmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M5">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M58">View MathML</a>) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M55">View MathML</a> and take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M91">View MathML</a>. Let the minimum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M6">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M54">View MathML</a>) be large such that T is small and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M94">View MathML</a>

Consider the auxiliary problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M95">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M86">View MathML</a>, and by Green’s identity, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M97">View MathML</a>

So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M98">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M99">View MathML</a>

From (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M100">View MathML</a>

By the comparison principle, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M101">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M71">View MathML</a> blows up, hence only v blows up at time T.

Secondly, we prove the necessity. Assume v blows up while u remains bounded, say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M104">View MathML</a>.

By Green’s identity, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M105">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M74">View MathML</a>, take t such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M107">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M108">View MathML</a>

thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M109">View MathML</a>

(3.6)

From (3.5) and (3.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M110">View MathML</a>

(3.7)

Integrating (3.7) from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M83">View MathML</a> to t, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M112">View MathML</a>

The boundedness of u requires that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M86">View MathML</a>. □

4 Coexistence of simultaneous and non-simultaneous blow-up

In this section, we consider the coexistence of both simultaneous and non-simultaneous blow-ups. In order to prove Theorem 1.3, we introduce following lemma.

Lemma 4.1The set of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M114">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M115">View MathML</a>such thatvblows up whileuremains bounded is open in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M116">View MathML</a>-topology.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M71">View MathML</a> be a solution of (1.1) with initial data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M34">View MathML</a> such that v blows up at T while u remains bounded, that is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M119">View MathML</a>. We only need to find a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M116">View MathML</a>-neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M114">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M115">View MathML</a>, such that any solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M123">View MathML</a> of (1.1) coming from this neighborhood maintains the property that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M124">View MathML</a> blows up while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M125">View MathML</a> remains bounded.

By Lemma 3.4, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M86">View MathML</a>. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M127">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M128">View MathML</a> be the solution of the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M129">View MathML</a>

where radially symmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M52">View MathML</a> is to be determined and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M131">View MathML</a> is the maximal existence time.

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M132">View MathML</a>

Since v blows up at time T, there exists small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M133">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M128">View MathML</a> blows up and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M131">View MathML</a> is small, satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M136">View MathML</a>

provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M137">View MathML</a>.

Consider the auxiliary system,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M138">View MathML</a>

By Green’s identity, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M139">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M140">View MathML</a>

Meanwhile, from (3.3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M141">View MathML</a>

So, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M142">View MathML</a>

By the comparison principle, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M143">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M144">View MathML</a> must blow up.

According to the continuity with respect to initial data for bounded solutions, there must exist a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M114">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M115">View MathML</a> such that every solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M123">View MathML</a> starting from the neighborhood, will enter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M148">View MathML</a> at time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M149">View MathML</a>, and keeps the property that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M124">View MathML</a> blows up while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M125">View MathML</a> remains bounded. □

Similarly, we can prove the set of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M114">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M115">View MathML</a> such that u blows up while v remains bounded is open in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M116">View MathML</a>-topology, we omit the proof here.

Now, we give the proof of Theorem 1.3.

Proof of Theorem 1.3 Under our assumptions, from Lemma 3.3, we know that the set of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M114">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M115">View MathML</a> such that u blows up and v remains bounded is nonempty. And from Lemma 3.4, we also know the set of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M114">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M115">View MathML</a> such that v blows up and u is bounded is nonempty.

Moreover, Lemma 4.1 shows that such sets are open. Clearly, the two open sets are disjoint. That is to say, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/85/mathml/M114">View MathML</a> such that u and v blow up simultaneously at a finite time T. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

We would like to thank the referees for their valuable comments and suggestions.

References

  1. Bebernes, J, Eberly, D: Mathematical Problem from Combustion Theory, Springer, Berlin (1989)

  2. Brändle, C, Quirós, F, Rossi, JD: Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Commun. Pure Appl. Anal.. 4(3), 523–536 (2005)

  3. Brändle, C, Quirós, F, Rossi, JD: The role of non-linear diffusion in non-simultaneous blow-up. J. Math. Anal. Appl.. 308, 92–104 (2005). Publisher Full Text OpenURL

  4. Deng, K: Blow-up rates for parabolic systems. Z. Angew. Math. Phys.. 47, 132–143 (1996). Publisher Full Text OpenURL

  5. Friedman, A: Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs (1964)

  6. Lieberman, GM: Second Order Parabolic Differential Equations, World Scientific, Singapore (1996)

  7. Lin, ZG, Xie, CH: The blow-up rate for a system of heat equations with nonlinear boundary conditions. Nonlinear Anal.. 34, 767–778 (1998). Publisher Full Text OpenURL

  8. Liu, BC, Li, FJ: Blow-up properties for heat equations coupled via different nonlinearities. J. Math. Anal. Appl.. 347, 294–303 (2008). Publisher Full Text OpenURL

  9. Liu, BC, Li, FJ: Non-simultaneous blow-up and blow-up rates for reaction-diffusion equations. Nonlinear Anal.. 13, 764–778 (2012). Publisher Full Text OpenURL

  10. Pedersen, M, Lin, ZG: Blow-up estimates of the positive solution of a parabolic system. J. Math. Anal. Appl.. 255, 551–563 (2001). Publisher Full Text OpenURL

  11. Pinasco, JP, Rossi, JD: Simultaneous versus non-simultaneous blow-up. N.Z. J. Math.. 29, 55–59 (2000)

  12. Quirós, F, Rossi, JD: Non-simultaneous blow-up in a semilinear parabolic system. Z. Angew. Math. Phys.. 52(2), 342–346 (2001). Publisher Full Text OpenURL

  13. Quirós, F, Rossi, JD: Non-simultaneous blow-up in a nonlinear parabolic system. Adv. Nonlinear Stud.. 3(3), 397–418 (2003)

  14. Rossi, JD: The blow-up rate for a system of heat equations with non-trivial coupling at the boundary. Math. Methods Appl. Sci.. 20, 1–11 (1997). Publisher Full Text OpenURL

  15. Rossi, JD, Souplet, P: Coexistence of simultaneous and non-simultaneous blow-up in a semilinear parabolic system. Differ. Integral Equ.. 18, 405–418 (2005)

  16. Song, XF: Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws. Nonlinear Anal.. 69, 1971–1980 (2008). Publisher Full Text OpenURL

  17. Wang, MX: Parabolic systems with a nonlinear boundary conditions. Chin. Sci. Bull.. 40(17), 1412–1414 (1995)

  18. Zhao, LZ, Zheng, SN: Blow-up estimates for system of heat equations coupled via nonlinear boundary flux. Nonlinear Anal.. 54, 251–259 (2003). Publisher Full Text OpenURL

  19. Zheng, SN, Qiao, L: Non-simultaneous blow-up in a reaction-diffusion system. Appl. Math. Comput.. 180, 309–317 (2006). Publisher Full Text OpenURL

  20. Zheng, SN, Liu, BC, Li, FJ: Non-simultaneous blow-up for a multi-coupled reaction-diffusion system. Nonlinear Anal.. 64, 1189–1202 (2006). Publisher Full Text OpenURL