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

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.

In this paper, we study the following system of two heat equations coupled by nonlinear boundary conditions,

where , parameters , . Assume the non-zero, non-negative initial data , are radially symmetric non-increasing continuous functions, vanishing on ∂Ω, as well as satisfy the compatibility conditions,

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 [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 and , in [16].

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 [8] 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

Theorem 1.3If

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.

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 2.1Let (u, v) be a solution of system (1.1), then, for all.

Proof Set

From the hypothesis of initial data, we can get

By the comparison principle, , for . □

Proof of Theorem 1.1 It is easy to check that

Let be a solution of the following system:

By the results of [17], 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. □

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, there must be

Proof Set

By computations, we can check that

By the comparison principle, , for . □

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 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 [5], we have

thus, , for any . So satisfies

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, , 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 .

By Green’s identity, we have

for any , take t such that , then

hence,

For some , we can find a suitable , such that

Similarly to Lemma 3.1, we can prove there must be

Then

Integrating the above inequality from to t, we have

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 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

So satisfies

From (3.3), we have

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 .

By Green’s identity, we have

For any , take t such that , then

thus

From (3.5) and (3.6), we have

Integrating (3.7) from to t, we obtain that

The boundedness of u requires that . □

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 ofinsuch thatvblows up whileuremains 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.

By Lemma 3.4, we know . Take , let be the solution of the following problem:

where radially symmetric is to be determined and is the maximal existence time.

Denote

Since v blows up at time T, there exists small , such that blows up and is small, satisfying

provided .

Consider the auxiliary system,

By Green’s identity, . Hence,

Meanwhile, from (3.3), we get

So, we have

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. □

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.

1. Bebernes, J, Eberly, D: Mathematical Problem from Combustion Theory, Springer, Berlin (1989)

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.. 4(3), 523–536 (2005)

3. 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

4. Deng, K: Blow-up rates for parabolic systems. Z. Angew. Math. Phys.. 47, 132–143 (1996). Publisher Full Text

5. Friedman, A: Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood Cliffs (1964)

6. Lieberman, GM: Second Order Parabolic Differential Equations, World Scientific, Singapore (1996)

7. 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

8. 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

9. 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

10. 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

11. Pinasco, JP, Rossi, JD: Simultaneous versus non-simultaneous blow-up. N.Z. J. Math.. 29, 55–59 (2000)

12. 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

13. Quirós, F, Rossi, JD: Non-simultaneous blow-up in a nonlinear parabolic system. Adv. Nonlinear Stud.. 3(3), 397–418 (2003)

14. 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

15. Rossi, JD, Souplet, P: Coexistence of simultaneous and non-simultaneous blow-up in a semilinear parabolic system. Differ. Integral Equ.. 18, 405–418 (2005)

16. 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

17. Wang, MX: Parabolic systems with a nonlinear boundary conditions. Chin. Sci. Bull.. 40(17), 1412–1414 (1995)

18. 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

19. Zheng, SN, Qiao, L: Non-simultaneous blow-up in a reaction-diffusion system. Appl. Math. Comput.. 180, 309–317 (2006). Publisher Full Text

20. 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