Non-simultaneous blow-up for a parabolic system with nonlinear boundary flux which obey different laws
© Xu and Zeng; licensee Springer 2012
Received: 16 April 2012
Accepted: 25 July 2012
Published: 6 August 2012
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.
1 Introduction and main results
and , for .
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 . 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 and , in .
whether the blow-up is simultaneous or non-simultaneous is not known yet.
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.
In the sequel, we assume the blow-up indeed occurs. Then we get the conditions, under which simultaneous or non-simultaneous blow-up occurs.
both non-simultaneous and simultaneous blow-ups may occur.
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.
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, for all.
By the comparison principle, , for . □
By the results of , the solutions of (2.1) blow up with large initial data if , or , or . By the comparison principle, is a sub-solution of (1.1), thus the solutions of (1.1) also blow up. □
3 Non-simultaneous blow-up
By the comparison principle, , for . □
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.
Proof Firstly, we prove the sufficiency.
By the comparison principle, , so v remains bounded up to time T. Since , blows up, hence only u blows up at time T.
Secondly, we prove the necessity. Assume u blows up while v remains bounded, say .
The boundedness of v requires that . □
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.
By the comparison principle, . Since , blows up, hence only v blows up at time T.
Secondly, we prove the necessity. Assume v blows up while u remains bounded, say .
The boundedness of u requires that . □
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 ofinsuch that v blows up while u remains bounded is open in-topology.
Proof Let be a solution of (1.1) with initial data such that v blows up at T while u remains bounded, that is . We only need to find a -neighborhood of in , such that any solution of (1.1) coming from this neighborhood maintains the property that blows up while remains bounded.
where radially symmetric is to be determined and is the maximal existence time.
By the comparison principle, , then must blow up.
According to the continuity with respect to initial data for bounded solutions, there must exist a neighborhood of in such that every solution starting from the neighborhood, will enter at time , and keeps the property that blows up while remains bounded. □
Similarly, we can prove the set of in such that u blows up while v remains bounded is open in -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 in such that u blows up and v remains bounded is nonempty. And from Lemma 3.4, we also know the set of in 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 such that u and v blow up simultaneously at a finite time T. □
We would like to thank the referees for their valuable comments and suggestions.
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