Uniform blow-up rate for a porous medium equation with a weighted localized source
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RESEARCH
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Uniform blow-up rate for a porous medium equation with a weighted localized source Weili Zeng1, Xiaobo Lu2* and Qilin Liu3 * Correspondence: [email protected] 2 School of Automation, Southeast University, Nanjing 210096, China Full list of author information is available at the end of the article
Abstract In this article, we investigate the Dirichlet problem for a porous medium equation with a more complicated source term. In some cases, we prove that the solutions have global blow-up and the rate of blow-up is uniform in all compact subsets of the domain. Moreover, in each case, the blow-up rate of |u(t)|∞ is precisely determined. Keywords: porous medium equation, localized source, blow-up, uniform blow-up rate
1 Introduction Let Ω be a bounded domain in ℝN (N ≥ 1) with smooth boundary ∂Ω. We consider the following parabolic equation with a localized reaction term vτ − vm = a(x)vq1 (x, τ )vs1 (x0 , τ ), v(x, τ ) = 0,
x ∈ , τ > 0,
(1:1)
x ∈ ∂, τ > 0,
(1:2)
x ∈ ,
(1:3)
v(x, 0) = v0 (x),
where m ≥ 1, q1 ≥ 0, s1 >0 and x0 Î Ω is a fixed point. Throughout this article, we assume the functions a(x) and v0(x) satisfy the following conditions: (A1) a(x) and v0(x) Î C2(Ω); a(x), v0(x) >0 in Ω and a(x) = v0(x) = 0 on ∂Ω. When Ω = B = {x Î ℝN; |x| < R}, we sometimes assume (A2) a(x) and v0(x) are radially symmetric; a(r) and v0(r) are non-increasing for r Î [0, R]. Problems (1.1)-(1.3) arise in the study of the flow of a fluid through a porous medium with an internal localized source and in the study of population dynamics (see [1-3]). Porous medium equations (m >1) with or without local sources have been studied by many authors [4-6]. Concerning (1.1)-(1.3), to the best of authors knowledge, a number of articles have studied it from the point of the view of blow-up and global existence [7-10]. Many studies have been devoted to the case m = 1 [10-13]. The case m = 1, a(x) = 1, q1 = 0, s1 ≥ 1 and m = 1, a(x) = 1, q1, s1 >1 were studied by Souple [10,11]. Souple [10] demonstrated that the positive solution blows up in finite time if the initial value v0 is large enough. In the case a(x) = 1, q1 = 0, and s1 >1, Souple [11] showed that the solution v(x, τ) blows up © 2011 Zeng et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Zeng et al. Boundary Value Problems 2011, 2011:57 http://www.boundaryvalueproblems.com/content/2011/1/57
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globally and the blow-up rate is precisely determined. The case q1 = 0 and s1 >0 was studied by Cannon and Yin [12] and Chandam et al. [13]. Cannon and Yin [12] studied its local solvability and Chandam et al. [13] investigated its blow-up properties. The study of this article is motivated by some recent results of related problems (see [14][15][16]. In the case of a(x)(= constant), the global existence and
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