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 nonsimultaneous blowup occurs. Moreover, we get some conditions on which simultaneous blowup must occur. Furthermore, we also get a result on the coexistence of both simultaneous and nonsimultaneous blowups.
MSC: 35B33, 35K65, 35K55.
Keywords:
simultaneous blowup; nonsimultaneous blowup; parabolic system; nonlinear boundary flux1 Introduction and main results
In this paper, we study the following system of two heat equations coupled by nonlinear boundary conditions,
where
and
The system (1.1) can be used to describe heat propagation of a twocomponent 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 crossboundary 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 blowup, and has been observed
for several scalar equation since the pioneering work of Fujita. Blowup may also
happen for systems, X. F. Song considered the blowup conditions and blowup rates
of system (1.1), when
However, it can only show
whether the blowup is simultaneous or nonsimultaneous is not known yet.
Recently, the simultaneous and nonsimultaneous blowup 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
They got a complete and optimal classification on nonsimultaneous and simultaneous blowups by four sufficient and necessary conditions.
Motivated by the above works, we will focus on the simultaneous and nonsimultaneous blowup 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
In the sequel, we assume the blowup indeed occurs. Then we get the conditions, under which simultaneous or nonsimultaneous blowup occurs.
Theorem 1.2There exist initial data such that nonsimultaneous blowup occurs if and only if
Corollary 1.1Any blowup is simultaneous if and only if
Theorem 1.3If
both nonsimultaneous and simultaneous blowups may occur.
In order to show the conditions more clearly, we graph Figure 1 with the region of nonsimultaneous and simultaneous blowups occur in the parameter space.
Figure 1. Nonsimultaneous and simultaneous blowups.
The rest of this paper is organized as follows: In next section, we consider the blowup conditions of system (1.1), give the proof of Theorem 1.1. In Section 3, we will study the sufficient and necessary conditions of nonsimultaneous blowup, in order to prove Theorem 1.2. In Section 4, we consider the coexistence of both simultaneous and nonsimultaneous blowups; Theorem 1.3 is proved.
2 Blowup
In this section, we prove the blowup 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
Proof Set
From the hypothesis of initial data, we can get
By the comparison principle,
Proof of Theorem 1.1 It is easy to check that
Let
By the results of [17], the solutions of (2.1) blow up with large initial data if
3 Nonsimultaneous blowup
In this section, we prove Theorem 1.2 with four lemmas. Firstly, we define the set
of initial data with a fixed constant
Lemma 3.1For any
Proof Set
By computations, we can check that
By the comparison principle,
Lemma 3.2For any
Proof First, we prove (3.2). From (3.1), we get
then
Integrating (3.4) from t to T,
thus
then
Similarly, we can also prove (3.3) from (3.1),
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
Proof Firstly, we prove the sufficiency.
Let
be the fundamental solution of the heat equation. Assume
Consider the auxiliary problem
For
thus,
Combining the radial symmetry and the monotonicity of the initial data with the estimate (3.2), we have
So, v satisfies that
By the comparison principle,
Secondly, we prove the necessity. Assume u blows up while v remains bounded, say
By Green’s identity, we have
for any
hence,
For some
Similarly to Lemma 3.1, we can prove there must be
Then
Integrating the above inequality from
The boundedness of v requires that
The following lemma proves the sufficient and necessary condition on the existence of v blowing up alone.
Lemma 3.4There exist suitable initial data such thatvblows up whileuremains bounded if and only if
Proof Firstly, we prove the sufficiency. Assume
Consider the auxiliary problem
For
So
From (3.3), we have
By the comparison principle,
Secondly, we prove the necessity. Assume v blows up while u remains bounded, say
By Green’s identity, we have
For any
thus
From (3.5) and (3.6), we have
Integrating (3.7) from
The boundedness of u requires that
4 Coexistence of simultaneous and nonsimultaneous blowup
In this section, we consider the coexistence of both simultaneous and nonsimultaneous blowups. In order to prove Theorem 1.3, we introduce following lemma.
Lemma 4.1The set of
Proof Let
By Lemma 3.4, we know
where radially symmetric
Denote
Since v blows up at time T, there exists small
provided
Consider the auxiliary system,
By Green’s identity,
Meanwhile, from (3.3), we get
So, we have
By the comparison principle,
According to the continuity with respect to initial data for bounded solutions, there
must exist a neighborhood of
Similarly, we can prove the set of
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
Moreover, Lemma 4.1 shows that such sets are open. Clearly, the two open sets are
disjoint. That is to say, there exists
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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