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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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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,

{ 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 [2,3,8,9,11-13,15,19,20]. 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 1.1There 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 1.2There exist initial data such that non-simultaneous blow-up occurs if and only if

α > q + 1 , or β > p .

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

{ α ≤ q + 1 , β ≤ p .

Theorem 1.3If

{ 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.

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 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 3.1For 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

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

Lemma 3.2For any t ∈ [ 0 , T )

(3.2)
(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

 □

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 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 3.4There exist suitable initial data such thatvblows up whileuremains 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 4.1The set of ( u 0 , v 0 ) in V 0 such thatvblows up whileuremains 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. □

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.

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