Profiles of blow-up solution of a weighted diffusion system
© Zeng et al.; licensee Springer 2014
Received: 8 May 2014
Accepted: 27 May 2014
Published: 11 September 2014
In this paper, we study the blow-up profiles for a coupled diffusion system with a weighted source term involved in a product with local term. We prove that the solutions have a global blow-up and the profile of the blow-up is precisely determined in all compact subsets of the domain.
where B is an open ball of , with radius R; α, β, p, q are nonnegative constants and satisfy and .
System (1.2) is usually used as a model to describe heat propagation in a two-component combustible mixture . In this case u and v represent the temperatures of the interacting components, thermal conductivity is supposed constant and equal for both substances, a volume energy release given by some powers of u and v is assumed.
was studied by Pao and Zheng  and they obtained the blow-up rates and boundary layer profiles of the solutions.
As for problem (1.2), it is well known that problem (1.2) has a classical, maximal in time solution and that the comparison principle is true (using the methods of ). A number of papers have studied problem (1.2) from the point of view of blow-up and global existence (see , ).
assuming , or , or , he proved that the solution blows up in finite time if the initial data and are large enough.
If , this system possesses uniform blow-up profiles.
If , this system presents single point blow-up patterns.
Recently, Zhang and Yang  studied the problem of (1.1), but they only obtained the estimation of the blow-up rate, which is not precisely determined. In , the authors proved there are initial data such that simultaneous and non-simultaneous blow-up occur for a diffusion system with weighted localized sources, but they did not study the profile of the blow-up solution. There are many known results concerning blow-up properties for parabolic system equations, of which the reaction terms are of a nonlinear localized type. For more details as regards a parabolic system with localized sources, see –.
Our present work is partially motivated by –. The purpose of this paper is to determine the blow-up rate of solutions for a nonlinear parabolic equation system with a weighted localized source. That is, we prove that the solutions u and v blow up simultaneously and that the blow-up rate is uniform in all compact subsets of the domain. Moreover, the blow-up profiles of the solutions are precisely determined.
In the following section, we will build the profile of the blow-up solution of (1.1).
2 Blow-up profile
Throughout this paper, we assume that the functions , , and satisfy the following three conditions:
(A1) ; in B and on ∂B.
(A2) , , and are radially symmetric; , , and are non-increasing for ().
(A3) and satisfy and in B, respectively.
- (i)If , and , then
- (ii)If and , then
- (iii)If and , then
- (iv)If and , then
From , it follows that . Applying similar arguments as above to the equation of v in system (1.2), it is reasonable that . □
The following lemma will play a key role in proving Theorem 2.1, which will give the relationships among u, v, , and .
- (i)and , then
- (ii)and , then
- (iii)and , then
- (iv)and , then
- (i)When and . A simple computation shows that(2.1)
Denote , the first eigenvalue of −Δ in and by and the corresponding eigenfunction, normalized by and .
and using , we deduce that , so that (2.3) implies . By a process analogous to above, we arrive at .
uniformly in all compact subsets of B.
The rest of the proof of case (i) is similar to Lemma 2.2(i). Cases (ii), (iii), and (iv) can be treated similarly. Now we prove Theorem 2.1 by using Lemma 2.2. □
Proof of Theorem 2.1
- (i)If and . By Lemma 2.2(i), we know that for choosing positive constants , there exists such that
The arguments of cases (ii), (iii), and (iv) are very similar to the above, we omit the details. Therefore, we have completed the proof of Theorem 2.1. □
This work was supported by the National Natural Science Foundation of China under Grant 61374194, the China Postdoctoral Science Foundation Founded Project under Grant 2013M540405, the National Key Technologies R&D Program of China under Grant 2009BAG13A06, the National High-tech R&D Program of China (863 Program) under Grant 2008AA040202, and the Natural Science Foundation of Jiangsu Province under grant BK20140638.
- Bebernes J, Eberly D: Mathematical Problems from Combustion Theorem. Springer, New York; 1989.View ArticleGoogle Scholar
- Pao L, Zheng S: Critical exponents and asymptotic estimates of solutions to parabolic systems with localized nonlinear sources. J. Math. Anal. Appl. 2004, 292: 621-635. 10.1016/j.jmaa.2003.12.011MathSciNetView ArticleGoogle Scholar
- Deng W: Global existence and finite time blow up for a degenerate reaction-diffusion system. Nonlinear Anal. 2005, 60: 977-991. 10.1016/j.na.2004.10.016MathSciNetView ArticleGoogle Scholar
- Wang L, Chen Q: The asymptotic behavior of blow-up solution of localized nonlinear equation. J. Math. Anal. Appl. 1996, 200: 315-321. 10.1006/jmaa.1996.0207MathSciNetView ArticleGoogle Scholar
- Wang MX: Global existence and finite time blow up for a reaction-diffusion system. Z. Angew. Math. Phys. 2000, 51: 160-167. 10.1007/PL00001504MathSciNetView ArticleGoogle Scholar
- Chen H: Global existence and blow-up for a nonlinear reaction-diffusion system. J. Math. Anal. Appl. 1997, 212: 481-492. 10.1006/jmaa.1997.5522MathSciNetView ArticleGoogle Scholar
- Li H, Wang M: Properties of blow-up solution to a parabolic system with nonlinear localized terms. Discrete Contin. Dyn. Syst. 2005, 13: 683-700. 10.3934/dcds.2005.13.683MathSciNetView ArticleGoogle Scholar
- Zhang R, Yang Z: Uniform blow-up rates and asymptotic estimates of solutions for diffusion systems with weighted localized sources. J. Appl. Math. Comput. 2010, 32: 429-441. 10.1007/s12190-009-0261-6MathSciNetView ArticleGoogle Scholar
- Ling Z, Wang Z: Simultaneous and non-simultaneous blow-up criteria of solutions for a diffusion system with weighted localized sources. J. Appl. Math. Comput. 2012, 40: 183-194. 10.1007/s12190-012-0570-zMathSciNetView ArticleGoogle Scholar
- Bimpong KB, Ross PO: Far-from-equilibrium phenomena at local sites of reaction. J. Chem. Phys. 1974, 80: 3124-3133. 10.1063/1.1681498View ArticleGoogle Scholar
- Pedersen M, Lin ZG: Coupled diffusion systems with localized nonlinear reactions. Comput. Math. Appl. 2001, 42: 807-816. 10.1016/S0898-1221(01)00200-0MathSciNetView ArticleGoogle Scholar
- Souplet P: Blow up in nonlocal reaction-diffusion equations. SIAM J. Math. Anal. 1998, 29(6):1301-1334. 10.1137/S0036141097318900MathSciNetView ArticleGoogle Scholar
- Escobedo M, Herrero M: A semi-linear parabolic system in a bounded domain. Ann. Mat. Pura Appl. (4) 1993, CLXV: 315-336. 10.1007/BF01765854MathSciNetView ArticleGoogle Scholar
- Li F, Liu B: Non-simultaneous blow-up in parabolic equations coupled via localized sources. Appl. Math. Lett. 2010, 23: 871-874. 10.1016/j.aml.2010.03.025MathSciNetView ArticleGoogle Scholar
- Liu QL, Li YX, Gao HJ: Uniform blow-up rate for diffusion equations with nonlocal nonlinear source. Nonlinear Anal. 2007, 67: 1947-1957. 10.1016/j.na.2006.08.030MathSciNetView ArticleGoogle Scholar
- Kong L, Wang J, Zheng S: Asymptotic analysis to a parabolic equation with a weighted localized source. Appl. Math. Comput. 2008, 197: 819-827. 10.1016/j.amc.2007.08.016MathSciNetView ArticleGoogle Scholar
- Liu QL, Li YX, Gao HJ: Uniform blow-up rate for diffusion equations with localized nonlinear source. J. Math. Anal. Appl. 2006, 320: 771-778. 10.1016/j.jmaa.2005.07.058MathSciNetView ArticleGoogle Scholar
- Zeng WL, Lu XB, Tan XH: Uniform blow-up rate for a porous medium equation with a weighted localized source. Bound. Value Probl. 2011., 2011: 10.1186/1687-2770-2011-57Google Scholar
This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.