Blow-up problems for a compressible reactive gas model
© Ling and Wang; licensee Springer 2012
Received: 19 May 2012
Accepted: 29 August 2012
Published: 11 September 2012
This paper investigates a compressible reactive gas model with homogeneous Dirichlet boundary conditions. Under the parameters and the initial data satisfying some conditions, we prove that the solutions have global blow-up, and the blow-up rate is uniform in all compact subsets of the domain. Moreover, the blow-up rates of and are precisely determined.
MSC:35K05, 35K55, 35D55.
Keywordsdegenerate parabolic system nonlocal sources finite time blow-up blow-up rate
1 Introduction and main results
where () is a ball centered at the origin with the radius , , exponents , , and is the maximal existence time of a solution, .
The system (1.1) models such as heat propagations in a two-components combustible mixture gases . This problem is worth studying because of the applications to heat and mass transport processes (see [2, 3]). In addition, there exist interesting interactions among the multi-nonlinearities described by these exponents in the problem (1.1).
In the past decades, many physical phenomena have been formulated into nonlocal mathematical models and studied by many authors. Here, we will recall some of those results concerning the first initial boundary value problem.
we consider a more general degenerate parabolic system (1.1) which includes the problems considered in [7, 8] and  as special cases. Employing the ideas in [7, 8], we describe the blow-up rate of the radially symmetric solutions to (1.1). Here we discuss the blow-up of radially symmetric solutions as well as derive their blow-up rate. Moreover, we get the accurate coefficient of the blow-up rate. For the related discussion on a radially symmetric solution, we refer the readers to  and references therein.
In this paper, we always assume that the initial data (V is defined by (1.8)) and satisfies the following (H1)-(H3) or (H4):
(H1) , .
(H2) in B, , on ∂B.
(H3) , are radially symmetric, for , .
It is noted that the set V is not empty. For example, for the simplest case and , for any constant exponents m, n and , , , , there exist positive constants , such that with , , .
here , and , , are defined by (2.9) and (2.6).
Then, our main results read as follows in detail.
Theorem 1 Assume that and satisfies (H 1)-(H 3). If , then the positive solution of (1.1) blows up in finite time, where ρ is defined by (2.12).
where and are defined by (3.1).
This paper is organized as follows. The result pertaining to blow-up of a solution in finite time is presented in Section 2, while results regarding the blow-up rates are established in Section 3. Some discussions are given in Section 4.
2 Proof of Theorem 1
Set , then we have:
Lemma 1 If , then the positive solution of (2.11) blows up in finite time.
That is . In view of , it follows that there exists such that , and hence blows up in finite time. □
for all .
Thus it follows from (2.13) and (2.15) that is a sub-solution of (2.8). Hence, by the comparison principle. □
Lemma 3 The solution of (2.8) blows up in finite time if and , satisfy (H 1)-(H 3).
Therefore, it suffices to show that blows up in finite time, because if so, its upper bound does exist up to a finite time T.
with the corresponding initial and boundary conditions and .
It follows from (2.20), (2.23) and the comparison principle that . Hence blows up in finite time, and so does the solution of (2.8) from (2.19). The proof now is completed. □
Considering Lemma 3 and (2.10), we directly obtain the results of Theorem 1.
3 Proofs of Theorems 2 and 3
In this section, we assume that the solution of (1.1) blows up in finite time T and will prove Theorems 2 and 3. We use c or C to denote the generic constant depending only on the structural data of the problem, and it may be different even in the same formula.
Then we have
where by (1.7).
here , .
By the comparison principle of Lemma 1 in , we have . This completes the proof. □
Integrating (3.15) from t to T, we end the proof. □
uniformly on compact subsets of B.
This completes the proof of the theorem. □
Combining with Lemma 7, we obtain the results of Theorem 3 immediately. □
The authors are supported by National Natural Science Foundation of China and they would like to express their many thanks to the editor and reviewers for their constructive suggestions to improve the previous version of this paper. This work is supported by the NNSF of China (11071100).
- Escobedo M, Herrero M: Boundedness and blow-up for a semilinear reaction-diffusion system. J. Differ. Equ. 1991, 89(1):176-202. doi:10.1016/0022-0396(91)90118-S 10.1016/0022-0396(91)90118-SMATHMathSciNetView ArticleGoogle Scholar
- Galaktionov V, Kurdyumov S, Samarskii A: A parabolic system of quasilinear equations I. Differ. Equ. 1984, 19: 1558-1574.Google Scholar
- Souplet P: Blow up in nonlocal reaction diffusion equations. SIAM J. Math. Anal. 1998, 29: 1301-1334. 10.1137/S0036141097318900MATHMathSciNetView ArticleGoogle Scholar
- Souplet P: Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear sources. J. Differ. Equ. 1999, 153: 374-406. doi:10.1006/jdeq.1998.3535 10.1006/jdeq.1998.3535MATHMathSciNetView ArticleGoogle Scholar
- Pao C: Nonexistence of global solutions for an integrodifferential system in reactor dynamics. SIAM J. Math. Anal. 1980, 11(3):559-564. 10.1137/0511053MATHMathSciNetView ArticleGoogle Scholar
- Guo JS, Su HW: The blow-up behaviour of the solution of an integrodifferential equation. Differ. Integral Equ. 1992, 5(6):1237-1245.MATHMathSciNetGoogle Scholar
- Li F, Xie CH: Global existence and blow-up for a nonlinear porous medium equation. Appl. Math. Lett. 2003, 16: 185-192. doi:10.1016/S0893-9659(03)80030-7 10.1016/S0893-9659(03)80030-7MATHMathSciNetView ArticleGoogle Scholar
- Liu QL, Li YX, Gao HG: Uniform blow-up rate for a nonlocal degenerate parabolic equations. Nonlinear Anal. TMA 2007, 66: 881-889. doi:10.1016/j.na.2005.12.029 10.1016/j.na.2005.12.029MATHMathSciNetView ArticleGoogle Scholar
- Deng WB, Li YX, Xie CH: Existence and nonexistence of global solutions of some nonlocal degenerate parabolic equations. Appl. Math. Lett. 2003, 16: 803-808. doi:10.1016/S0893-9659(03)80118-0 10.1016/S0893-9659(03)80118-0MATHMathSciNetView ArticleGoogle Scholar
- Liu QL, Li YX, Gao HG: Uniform blow-up rate for diffusion equations with nonlocal nonlinear source. Nonlinear Anal. TMA 2007, 67: 1947-1957. doi:10.1016/j.na.2006.08.030 10.1016/j.na.2006.08.030MATHMathSciNetView ArticleGoogle Scholar
- Quirós F, Rossi JD: Non-simultaneous blow-up in a semilinear parabolic system. Z. Angew. Math. Phys. 2001, 52(2):342-346. doi:10.1007/PL00001549 10.1007/PL00001549MATHMathSciNetView ArticleGoogle Scholar
- Li FC, Huang SX, Xie CH: Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete Contin. Dyn. Syst. 2003, 9(6):1519-1532. doi:10.3934/dcds.2003.9.1519MATHMathSciNetView ArticleGoogle Scholar
- Ling ZQ, Wang ZJ: Simultaneous and non-simultaneous blow-up criteria of solutions for a diffusion system with weighted localized sources. J. Appl. Math. Comput. 2012. doi:10.1007/s12190-012-0570-zGoogle Scholar
- Li HL, Wang MX: Critical exponents and lower bounds of blow-up rate for a reaction-diffusion system. Nonlinear Anal. TMA 2005, 63(8):1083-1093. doi:10.1016/j.na.2005.05.037 10.1016/j.na.2005.05.037MATHView ArticleGoogle Scholar
- Li HL, Wang MX: Blow-up behaviors for semilinear parabolic systems coupled in equations and boundary conditions. J. Math. Anal. Appl. 2005, 304(1):96-114. doi:10.1016/j.jmaa.2004.09.020 10.1016/j.jmaa.2004.09.020MATHMathSciNetView ArticleGoogle Scholar
- Song XF, Zheng SN, Jiang ZX: Blow-up analysis for a nonlinear diffusion system. Z. Angew. Math. Phys. 2005, 56: 1-10. 10.1007/s00033-004-1152-1MATHMathSciNetView ArticleGoogle Scholar
- Deng WB, Li YX, Xie CH: Blow-up and global existence for a nonlocal degenerate parabolic system. J. Math. Anal. Appl. 2003, 277: 199-217. doi:10.1016/S0022-247X(02)00533-4 10.1016/S0022-247X(02)00533-4MATHMathSciNetView ArticleGoogle Scholar
- Ling ZQ, Wang ZJ: Blow-up and global existence for a degenerate parabolic system with nonlocal sources. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 956564. doi:10.1155/2012/956564Google Scholar
- Gidas B, Ni WM, Nirenberg L:Symmetry of positive solutions of nonlinear elliptic equations in . Advances in Math. Suppl. Studies 7A. In Math. Anal. and Applications, Part A. Edited by: Nachbin L. Academic Press, San Diego; 1981:369-402.Google Scholar
- Li HL, Wang MX: Global solutions and blow-up problems for a nonlinear degenerate parabolic system coupled via nonlocal sources. J. Math. Anal. Appl. 2007, 333(2):984-1007. doi:10.1016/j.jmaa.2006.11.023 10.1016/j.jmaa.2006.11.023MathSciNetView ArticleGoogle Scholar
- Friedman A, Mcleod B: Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J. 1985, 34(2):425-447. doi:10.1512/iumj.1985.34.34025 10.1512/iumj.1985.34.34025MATHMathSciNetView ArticleGoogle Scholar