Blow-up profile for a degenerate parabolic equation with a weighted localized term
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Blow-up profile for a degenerate parabolic equation with a weighted localized term Weili Zeng1* , Xiaobo Lu1 , Shumin Fei1 and Miaochao Chen2 * Correspondence: [email protected] 1 School of Automation, Southeast University, Nanjing, 210096, China Full list of author information is available at the end of the article
Abstract In this paper, we investigate the Dirichlet problem for a degenerate parabolic equation ut – um = a(x)up (0, t) + b(x)uq (x, t). We prove that under certain conditions the solutions have global blow-up, and the rate of blow-up is uniform in all compact subsets of the domain. Moreover, the blow-up profile is precisely determined. Keywords: degenerate parabolic equation; localized source; uniform blow-up rate
1 Introduction In this paper, we consider the following parabolic equation with nonlocal and localized reaction: ut – um = a(x)up (, t) + b(x)uq (x, t), u(x, τ ) = ,
x ∈ , < t < T ∗ ,
(.)
x ∈ ∂, t > ,
(.)
x ∈ ,
(.)
u(x, ) = u (x),
where is an open ball of RN , N ≥ with radius R, and p ≥ q > m > . Many of localized problems arise in applications and have been widely studied. Equations (.)-(.), as a kind of porous medium equation, can be used to describe some physical phenomena such as chemical reactions due to catalysis and an ignition model for a reaction gas (see [–]). As for our problem (.)-(.), to our best knowledge, many works have been devoted to the case m = (see [–]). Let us mention, for instance, when a(x) = b(x) = , blow-up properties have been investigated by Okada and Fukuda []. Moreover, they proved that if p ≥ q > and u (x) is sufficiently large, every radial symmetric solution (maximal solution) has a global blow-up and the solution satisfies –/(p–) –/(p–) ≤ u(x, t) ≤ C T ∗ – t , C T ∗ – t
(.)
in all compact subsets of as t is near the blow-up time T ∗ , where C and C are two positive constants. Souplet [, ] investigated that global blow-up solutions have uniform blow-up estimates in all compact subsets of the domain. The work of this paper is motivated by the localized semi-linear problem ut – um = λ up (, t) + λ uq (x, t),
x ∈ , < t < T ∗ ,
(.)
©2013 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 2013, 2013:269 http://www.boundaryvalueproblems.com/content/2013/1/269
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with Dirichlet boundary condition (.) and initial condition (.). In the case of m = and m > , the uniform blow-up profiles were studied in [, ] and [], respectively. It seems that the result of [, , ] can be extended to λ and λ are two functions. Motivated by this, in this paper, we extend and improve the results of [, , ]. Our approach is different from those previously used in blow-up rate
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