Open Access

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

Boundary Value Problems20122012:85

DOI: 10.1186/1687-2770-2012-85

Received: 16 April 2012

Accepted: 25 July 2012

Published: 6 August 2012

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,
{ u t = Δ u , v t = Δ v , ( x , t ) Ω × ( 0 , T ) , u η = e p v u α , v η = u q e β v , ( x , t ) Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x Ω ,
(1.1)
where Ω = B R = { | x | < R } R N , parameters α , q 1 , p , β 0 . Assume the non-zero, non-negative initial data u 0 , v 0 are radially symmetric non-increasing continuous functions, vanishing on Ω, as well as satisfy the compatibility conditions,
{ u 0 η = e p v 0 u 0 α , v 0 η = u 0 q e β v 0 , x Ω
(1.2)

and Δ u 0 , Δ v 0 0 , for x Ω .

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 p , q > 0 0 α < 1 and 0 β < p , in [16].

However, it can only show
lim t T sup { u ( , t ) + v ( , t ) } = ,

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, 1113, 15, 19, 20]. For example, B. C. Liu and F. J. Li [8] considered the nonlinear parabolic system
{ u t = Δ u + u m e p v , v t = Δ v + u q e n v , ( x , t ) Ω × ( 0 , T ) , u ( x , t ) = v ( x , t ) = 0 , ( x , t ) Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x Ω .

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 1.1 There exist initial data such that the solutions of (1.1) blow up, if
α > 1 , or β > 0 , or p q > β ( α 1 ) .

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 1.2 There exist initial data such that non-simultaneous blow-up occurs if and only if
α > q + 1 , or β > p .
Corollary 1.1 Any blow-up is simultaneous if and only if
{ α q + 1 , β p .
Theorem 1.3 If
{ α > q + 1 , β > p

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.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-85/MediaObjects/13661_2012_Article_188_fig1.jpg
Figure 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 2.1 Let (u, v) be a solution of system (1.1), then u t , v t 0 , for all ( x , t ) B R × ( 0 , T ) .

Proof Set
M = u t , N = v t , ( x , t ) B R × ( 0 , T ) .
From the hypothesis of initial data, we can get
{ M t = Δ M , N t = Δ N , ( x , t ) B R × ( 0 , T ) , M η = p u α e p v N + α u α 1 e p v M , N η = q u q 1 e β v M + β u q e β v N , ( x , t ) B R × ( 0 , T ) , M ( x , 0 ) = Δ u 0 0 , N ( x , 0 ) = Δ v 0 0 , x B R .

By the comparison principle, M ( x , t ) , N ( x , t ) 0 , for ( x , t ) B R × ( 0 , T ) . □

Proof of Theorem 1.1 It is easy to check that
{ u η = e p v u α v p u α , v η = u q e β v u q ( β β + 1 ) β + 1 v β + 1 .
Let ( u ̲ , v ̲ ) be a solution of the following system:
{ u ̲ t = Δ u ̲ , v ̲ t = Δ v ̲ , ( x , t ) Ω × ( 0 , T ) , u ̲ η = u ̲ α v ̲ β , v ̲ η = ( β β + 1 ) β + 1 u ̲ q v ̲ β + 1 , ( x , t ) Ω × ( 0 , T ) , u ̲ ( x , 0 ) = u 0 ( x ) , v ̲ ( x , 0 ) = v 0 ( x ) , x Ω .
(2.1)

By the results of [17], the solutions of (2.1) blow up with large initial data if α > 1 , or β > 0 , or p q > β ( α 1 ) . By the comparison principle, ( u ̲ , v ̲ ) 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 ε ( 0 , 1 ) ,
V 0 = { ( u 0 , v 0 ) | Δ u 0 ε u 0 α e p v 0 0 , Δ v 0 ε u 0 q e β v 0 0 , x B R } .
Lemma 3.1 For any ( u 0 , v 0 ) V 0 , there must be
u t ( x , t ) ε ( u α e p v ) ( x , t ) v t ( x , t ) ε ( u q e β v ) ( x , t ) ( x , t ) B R × [ 0 , T ) .
(3.1)
Proof Set
J = u t ε u α e p v , K = v t ε u q e β v , ( x , t ) B R × [ 0 , T ) .
By computations, we can check that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-85/MediaObjects/13661_2012_Article_188_Equl_HTML.gif

By the comparison principle, J ( x , t ) , K ( x , t ) 0 , for ( x , t ) B R × [ 0 , T ) . □

Lemma 3.2 For any t [ 0 , T )
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-85/MediaObjects/13661_2012_Article_188_Equ5_HTML.gif
(3.2)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-85/MediaObjects/13661_2012_Article_188_Equ6_HTML.gif
(3.3)
Proof First, we prove (3.2). From (3.1), we get
u t ( 0 , t ) ε u α ( 0 , t ) e p v ( 0 , t ) ,
then
u t ( 0 , t ) ε u α ( 0 , t ) e p v 0 ( 0 , t ) .
(3.4)
Integrating (3.4) from t to T,
t T u t ( 0 , t ) d t u α ( 0 , t ) ε e p v 0 ( 0 ) ( T t ) ,
thus
u α + 1 ( 0 , t ) ε e p v 0 ( 0 ) ( T t ) ( α + 1 ) ,
then
u ( 0 , t ) [ ( α 1 ) ε e p v 0 ( 0 ) ] 1 α 1 ( T t ) 1 α 1 .
Similarly, we can also prove (3.3) from (3.1),
v t ( 0 , t ) ε u q ( 0 , t ) e β v ( 0 , t ) ε u 0 q ( 0 ) e β v ( 0 , t ) .
Integrating the above inequality from t to T, then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-85/MediaObjects/13661_2012_Article_188_Equr_HTML.gif

 □

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

Lemma 3.3 There exist suitable initial data such that u blows up while v remains bounded if and only if α > q + 1 .

Proof Firstly, we prove the sufficiency.

Let
Γ ( x , y , t , τ ) = 1 [ 4 π ( t τ ) ] N / 2 exp { | x y | 2 4 ( t τ ) }
be the fundamental solution of the heat equation. Assume ( u ˜ 0 , v ˜ 0 ) is a pair of initial data such that the solution of (1.1) blows up. Fix radially symmetric v 0 ( v ˜ 0 ) in B R and take M 0 > v 0 ( 0 ) . Let the minimum of u 0 ( u ˜ 0 ) be large such that T is small and satisfies
M 0 v 0 ( 0 ) + α 1 α q 1 [ ( α 1 ) ε e p v 0 ( 0 ) ] q α 1 e β M 0 T α q 1 α 1 .
Consider the auxiliary problem
{ v ¯ t = Δ v ¯ , ( x , t ) B R × ( 0 , T ) , v ¯ η = [ ( α 1 ) ε e p v 0 ( 0 ) ] q α 1 e β M 0 ( T t ) q α 1 , ( x , t ) B R × ( 0 , T ) , v ¯ ( x , 0 ) = v 0 ( x ) , x B R .
For α > q + 1 and by Green’s identity [5], we have
v ¯ ( x , t ) = B R Γ ( x , y , t , 0 ) v 0 ( y ) d y + 0 t B R Γ ( x , y , t , τ ) v ¯ η d S y d τ = B R Γ ( x , y , t , 0 ) v 0 ( y ) d y + 0 t B R Γ ( x , y , t , τ ) [ ( α 1 ) ε e p v 0 ( 0 ) ] q α 1 e β M 0 ( T τ ) q α 1 d S y d τ v 0 ( 0 ) + α 1 α q 1 [ ( α 1 ) ε e p v 0 ( 0 ) ] q α 1 e β M 0 T α q 1 α 1 M 0 ,
thus, M 0 v ¯ ( x , t ) , for any ( x , t ) B R × ( 0 , T ) . So v ¯ satisfies
{ v ¯ t = Δ v ¯ , ( x , t ) B R × ( 0 , T ) , v ¯ η [ ( α 1 ) ε e p v 0 ( 0 ) ] q α 1 e β v ¯ ( T t ) q α 1 , ( x , t ) B R × ( 0 , T ) , v ¯ ( x , 0 ) = v 0 ( x ) , x B R .
Combining the radial symmetry and the monotonicity of the initial data with the estimate (3.2), we have
u q ( | x | , t ) u q ( 0 , t ) [ ( α 1 ) ε e p v 0 ( 0 ) ] q α 1 ( T t ) q α 1 ( x , t ) B R × ( 0 , T ) .
So, v satisfies that
{ v t = Δ v , ( x , t ) B R × ( 0 , T ) , v η [ ( α 1 ) ε e p v 0 ( 0 ) ] q α 1 e β v ( T t ) q α 1 , ( x , t ) B R × ( 0 , T ) , v ( x , 0 ) = v 0 ( x ) , x B R .

By the comparison principle, v v ¯ M 0 , so v remains bounded up to time T. Since ( u 0 , v 0 ) ( u ˜ 0 , v ˜ 0 ) , ( u , v ) blows up, hence only u blows up at time T.

Secondly, we prove the necessity. Assume u blows up while v remains bounded, say v C .

By Green’s identity, we have
u ( 0 , t ) u ( 0 , z ) + C u α ( 0 , t ) ( T z ) ,
for any z ( 0 , T ) , take t such that u ( 0 , t ) = 2 u ( 0 , z ) , then
u ( 0 , z ) C u α ( 0 , z ) ( T z ) ,
hence,
u ( 0 , t ) C ( T t ) 1 α 1 t ( 0 , T ) .
For some t 1 ( 0 , T ) , we can find a suitable ε 1 ( 0 , 1 ) , such that
u t ( x , t 1 ) ε 1 ( u 0 α e p v 0 ) ( x , t 1 ) 0 .
Similarly to Lemma 3.1, we can prove there must be
u t ( x , t ) ε 1 ( u α e p v ) ( x , t ) v t ( x , t ) ε 1 ( u q e β v ) ( x , t ) ( x , t ) B R × [ t 1 , T ) .
(3.5)
Then
v t ( 0 , t ) ε 1 e β v 0 ( 0 ) C q ( T t ) q α 1 , t [ t 1 , T ) .
Integrating the above inequality from t 1 to t, we have
v ( 0 , t ) ε 1 e β v 0 ( 0 ) C q t 1 t ( T τ ) q α 1 d τ + v ( 0 , t 1 ) .

The boundedness of v requires that α > q + 1 . □

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

Lemma 3.4 There exist suitable initial data such that v blows up while u remains bounded if and only if β > p .

Proof Firstly, we prove the sufficiency. Assume ( u ˜ 0 , v ˜ 0 ) is a pair of initial data such that the solution of (1.1) blows up. Fix radially symmetric u 0 ( u ˜ 0 ) in B R and take M 1 > u 0 ( 0 ) . Let the minimum of v 0 ( v ˜ 0 ) be large such that T is small and satisfies
M 1 u 0 ( 0 ) + β β p [ β ε u 0 q ( 0 ) ] p β M 1 α T β p β .
Consider the auxiliary problem
{ u ¯ t = Δ u ¯ , ( x , t ) B R × ( 0 , T ) , u ¯ η = [ β ε u 0 q ( 0 ) ] p β M 1 α ( T t ) p β , ( x , t ) B R × ( 0 , T ) , u ¯ ( x , 0 ) = u 0 ( x ) , x B R .
For β > p , and by Green’s identity, we have
u ¯ ( x , t ) u 0 ( 0 ) + β β p [ β ε u 0 q ( 0 ) ] p β M 1 α T β p β M 1 .
So u ¯ satisfies
u ¯ η [ β ε u 0 q ( 0 ) ] p β u ¯ α ( T t ) p β , ( x , t ) B R × ( 0 , T ) .
From (3.3), we have
u η [ β ε u 0 q ( 0 ) ] p β u α ( T t ) p β , ( x , t ) B R × ( 0 , T ) .

By the comparison principle, u u ¯ M 1 . Since ( u 0 , v 0 ) ( u ˜ 0 , v ˜ 0 ) , ( u , v ) blows up, hence only v blows up at time T.

Secondly, we prove the necessity. Assume v blows up while u remains bounded, say u C .

By Green’s identity, we have
v ( 0 , t ) v ( 0 , z ) + C e β v ( 0 , t ) ( T z ) .
For any z ( 0 , T ) , take t such that v ( 0 , t ) = v ( 0 , z ) + 1 , then
C e β v ( 0 , z ) ( T z ) 1 ,
thus
v ( 0 , t ) ln [ C ( T t ) ] 1 β , t ( 0 , T ) .
(3.6)
From (3.5) and (3.6), we have
u t ( 0 , t ) ε 1 u 0 α ( 0 ) C p β ( T t ) p β , t ( t 1 , T ) .
(3.7)
Integrating (3.7) from t 1 to t, we obtain that
u ( 0 , t ) u ( 0 , t 1 ) + ε 1 u 0 α ( 0 ) C p β t 1 t ( T τ ) p β d τ .

The boundedness of u requires that β > p . □

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 4.1 The set of ( u 0 , v 0 ) in V 0 such that v blows up while u remains bounded is open in L -topology.

Proof Let ( u , v ) be a solution of (1.1) with initial data ( u 0 , v 0 ) V 0 such that v blows up at T while u remains bounded, that is 0 < u ( 0 , t ) M . We only need to find a L -neighborhood of ( u 0 , v 0 ) in V 0 , such that any solution ( u ˆ , v ˆ ) of (1.1) coming from this neighborhood maintains the property that v ˆ blows up while u ˆ remains bounded.

By Lemma 3.4, we know β > p . Take M 2 > M + u 0 ( 0 ) 2 , let ( u ˜ , v ˜ ) be the solution of the following problem:
{ u ˜ t = Δ u ˜ , v ˜ t = Δ v ˜ , ( x , t ) B R × ( 0 , T 0 ) , u ˜ η = e p v ˜ u ˜ α , v ˜ η = u ˜ q e β v ˜ , ( x , t ) B R × ( 0 , T 0 ) , u ˜ ( x , 0 ) = u ˜ 0 ( x ) , v ˜ ( x , 0 ) = v ˜ 0 ( x ) , x B R ,

where radially symmetric ( u ˜ 0 , v ˜ 0 ) is to be determined and T 0 is the maximal existence time.

Denote
N ( u 0 , v 0 ) = { ( u ˜ 0 , v ˜ 0 ) V 0 | u ˜ 0 ( 0 ) u ( 0 , T ε 0 ) , v ˜ 0 ( 0 ) v ( 0 , T ε 0 ) < u 0 ( 0 ) 2 } .
Since v blows up at time T, there exists small ε 0 > 0 , such that ( u ˜ , v ˜ ) blows up and T 0 is small, satisfying
M 2 > M + u 0 ( 0 ) 2 + β β p [ β ε ( u 0 ( 0 ) 2 ) q ] p β T 0 β p β M 2 α ,

provided ( u ˜ 0 , v ˜ 0 ) N ( u 0 , v 0 ) .

Consider the auxiliary system,
{ u ¯ t = Δ u ¯ , ( x , t ) B R × ( 0 , T 0 ) , u ¯ η = [ β ε u ˜ 0 q ( 0 ) ] p β M 2 α ( T 0 t ) p β , ( x , t ) B R × ( 0 , T 0 ) , u ¯ ( x , 0 ) = u ˜ 0 ( x ) , x B R .
By Green’s identity, u ¯ M 2 . Hence,
u ¯ η [ β ε u ˜ 0 q ( 0 ) ] p β u ¯ α ( T 0 t ) p β , ( x , t ) B R × ( 0 , T 0 ) .
Meanwhile, from (3.3), we get
v ˜ ( 0 , t ) ln { [ β ε u ˜ 0 q ( 0 ) ] 1 β ( T 0 t ) 1 β } .
So, we have
u ˜ η [ β ε u ˜ 0 q ( 0 ) ] p β u ˜ α ( T 0 t ) p β , ( x , t ) B R × ( 0 , T 0 ) .

By the comparison principle, u ˜ u ¯ M 2 , then v ˜ must blow up.

According to the continuity with respect to initial data for bounded solutions, there must exist a neighborhood of ( u 0 , v 0 ) in V 0 such that every solution ( u ˆ , v ˆ ) starting from the neighborhood, will enter N ( u 0 , v 0 ) at time T ε 0 , and keeps the property that v ˆ blows up while u ˆ remains bounded. □

Similarly, we can prove the set of ( u 0 , v 0 ) in V 0 such that u blows up while v remains bounded is open in L -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 ( u 0 , v 0 ) in V 0 such that u blows up and v remains bounded is nonempty. And from Lemma 3.4, we also know the set of ( u 0 , v 0 ) in V 0 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 ( u 0 , v 0 ) such that u and v blow up simultaneously at a finite time T. □

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematics, Jiangxi Vocational College of Finance and Economics
(2)
Department of Information Engineering, Jiangxi Vocational College of Finance and Economics

References

  1. Bebernes J, Eberly D: Mathematical Problem from Combustion Theory. Springer, Berlin; 1989.View Article
  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. 2005, 4(3):523-536.MATHMathSciNetView Article
  3. Brändle C, Quirós F, Rossi JD: The role of non-linear diffusion in non-simultaneous blow-up. J. Math. Anal. Appl. 2005, 308: 92-104. 10.1016/j.jmaa.2004.11.004MATHMathSciNetView Article
  4. Deng K: Blow-up rates for parabolic systems. Z. Angew. Math. Phys. 1996, 47: 132-143. 10.1007/BF00917578MATHMathSciNetView Article
  5. Friedman A: Partial Differential Equations of Parabolic Type. Prentice Hall, Englewood Cliffs; 1964.MATH
  6. Lieberman GM: Second Order Parabolic Differential Equations. World Scientific, Singapore; 1996.MATHView Article
  7. Lin ZG, Xie CH: The blow-up rate for a system of heat equations with nonlinear boundary conditions. Nonlinear Anal. 1998, 34: 767-778. 10.1016/S0362-546X(97)00573-7MATHMathSciNetView Article
  8. Liu BC, Li FJ: Blow-up properties for heat equations coupled via different nonlinearities. J. Math. Anal. Appl. 2008, 347: 294-303. 10.1016/j.jmaa.2008.06.004MATHMathSciNetView Article
  9. Liu BC, Li FJ: Non-simultaneous blow-up and blow-up rates for reaction-diffusion equations. Nonlinear Anal. 2012, 13: 764-778. 10.1016/j.nonrwa.2011.08.015MATHView Article
  10. Pedersen M, Lin ZG: Blow-up estimates of the positive solution of a parabolic system. J. Math. Anal. Appl. 2001, 255: 551-563. 10.1006/jmaa.2000.7261MATHMathSciNetView Article
  11. Pinasco JP, Rossi JD: Simultaneous versus non-simultaneous blow-up. N.Z. J. Math. 2000, 29: 55-59.MATHMathSciNet
  12. Quirós F, Rossi JD: Non-simultaneous blow-up in a semilinear parabolic system. Z. Angew. Math. Phys. 2001, 52(2):342-346. 10.1007/PL00001549MATHMathSciNetView Article
  13. Quirós F, Rossi JD: Non-simultaneous blow-up in a nonlinear parabolic system. Adv. Nonlinear Stud. 2003, 3(3):397-418.MATHMathSciNet
  14. Rossi JD: The blow-up rate for a system of heat equations with non-trivial coupling at the boundary. Math. Methods Appl. Sci. 1997, 20: 1-11. 10.1002/(SICI)1099-1476(19970110)20:1<1::AID-MMA843>3.0.CO;2-EMATHMathSciNetView Article
  15. Rossi JD, Souplet P: Coexistence of simultaneous and non-simultaneous blow-up in a semilinear parabolic system. Differ. Integral Equ. 2005, 18: 405-418.MATHMathSciNet
  16. Song XF: Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws. Nonlinear Anal. 2008, 69: 1971-1980. 10.1016/j.na.2007.07.040MATHMathSciNetView Article
  17. Wang MX: Parabolic systems with a nonlinear boundary conditions. Chin. Sci. Bull. 1995, 40(17):1412-1414.MATH
  18. Zhao LZ, Zheng SN: Blow-up estimates for system of heat equations coupled via nonlinear boundary flux. Nonlinear Anal. 2003, 54: 251-259. 10.1016/S0362-546X(03)00060-9MATHMathSciNetView Article
  19. Zheng SN, Qiao L: Non-simultaneous blow-up in a reaction-diffusion system. Appl. Math. Comput. 2006, 180: 309-317. 10.1016/j.amc.2005.12.019MATHMathSciNetView Article
  20. Zheng SN, Liu BC, Li FJ: Non-simultaneous blow-up for a multi-coupled reaction-diffusion system. Nonlinear Anal. 2006, 64: 1189-1202. 10.1016/j.na.2005.05.061MATHMathSciNetView Article

Copyright

© Xu and Zeng; licensee Springer 2012

This article is published under license to BioMed Central Ltd. 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.