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,
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 .
However, it can only show
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  considered the nonlinear parabolic system
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
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
Corollary 1.1Any blow-up is simultaneous if and only if
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.
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.
In this section, we prove the blow-up criterion of system (1.1). First, we check the monotonicity of the solution.
From the hypothesis of initial data, we can get
Proof of Theorem 1.1 It is easy to check that
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 computations, we can check that
Proof First, we prove (3.2). From (3.1), we get
Integrating (3.4) from t to T,
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.
Proof Firstly, we prove the sufficiency.
be the fundamental solution of the heat equation. Assume is a pair of initial data such that the solution of (1.1) blows up. Fix radially symmetric () in and take . Let the minimum of () be large such that T is small and satisfies
Consider the auxiliary problem
For and by Green’s identity , we have
Combining the radial symmetry and the monotonicity of the initial data with the estimate (3.2), we have
So, v satisfies that
By Green’s identity, we have
Similarly to Lemma 3.1, we can prove there must be
The following lemma proves the sufficient and necessary condition on the existence of v blowing up alone.
Proof Firstly, we prove the sufficiency. Assume is a pair of initial data such that the solution of (1.1) blows up. Fix radially symmetric () in and take . Let the minimum of () be large such that T is small and satisfies
Consider the auxiliary problem
From (3.3), we have
By Green’s identity, we have
From (3.5) and (3.6), we have
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.
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.
Consider the auxiliary system,
Meanwhile, from (3.3), we get
So, we have
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. □
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.
The authors declare that they have no competing interests.
The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
We would like to thank the referees for their valuable comments and suggestions.
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
Deng, K: Blow-up rates for parabolic systems. Z. Angew. Math. Phys.. 47, 132–143 (1996). Publisher Full Text
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
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
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
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
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
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
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
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
Zheng, SN, Qiao, L: Non-simultaneous blow-up in a reaction-diffusion system. Appl. Math. Comput.. 180, 309–317 (2006). Publisher Full Text
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