Non-Newtonian polytropic filtration systems with nonlinear boundary conditions
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Non-Newtonian polytropic filtration systems with nonlinear boundary conditions Wanjuan Du* and Zhongping Li * Correspondence: [email protected] College of Mathematic and Information, China West Normal University, Nanchong 637002, PR China
Abstract This article deals with the global existence and the blow-up of non-Newtonian polytropic filtration systems with nonlinear boundary conditions. Necessary and sufficient conditions on the global existence of all positive (weak) solutions are obtained by constructing various upper and lower solutions. Mathematics Subject Classification (2000) 35K50, 35K55, 35K65 Keywords: Polytropic filtration systems, Nonlinear boundary conditions, Global existence, Blow-up
Introduction In this article, we study the global existence and the blow-up of non-Newtonian polytropic filtration systems with nonlinear boundary conditions (uki i )t = mi ui (i = 1, . . . , n), n m ∇mi ui · ν = uj ij (i = 1, . . . , n),
x ∈ , t > 0, x ∈ ∂, t > 0,
(1:1)
j=1
¯ ui (x, 0) = ui0 (x) > 0 (i = 1, . . . , n), x ∈ ,
where mi ui = div(|∇ui |mi −1 ∇ui ) =
N
(|∇ui |mi −1 uixj )x , ∇mi ui = (|∇ui |mi −1 uix1 , . . . , |∇ui |mi −1 uixN ), j
j=1
Ω ⊂ ℝN is a bounded domain with smooth boundary ∂Ω, ν is the outward normal vector on the boundary ∂Ω, and the constants ki, mi > 0, mij ≥ 0, i, j = 1,..., n; ui0(x) (i = 1,..., n) are positive C1 functions, satisfying the compatibility conditions. The particular feature of the equations in (1.1) is their power- and gradient-dependent diffusibility. Such equations arise in some physical models, such as population dynamics, chemical reactions, heat transfer, and so on. In particular, equations in (1.1) may be used to describe the nonstationary flows in a porous medium of fluids with a power dependence of the tangential stress on the velocity of displacement under polytropic conditions. In this case, the equations in (1.1) are called the non-Newtonian polytropic filtration equations which have been intensively studied (see [1-4] and the references therein). For the Neuman problem (1.1), the local existence of solutions in time have been established; see the monograph [4]. © 2011 Du and Li; 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.
Du and Li Boundary Value Problems 2011, 2011:2 http://www.boundaryvalueproblems.com/content/2011/1/2
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We note that most previous works deal with special cases of (1.1) (see [5-13]). For example, Sun and Wang [7] studied system (1.1) with n = 1 (the single-equation case) and showed that all positive (weak) solutions of (1.1) exist globally if and only if m11 ≤ k1 when k1 ≤ m1; and exist globally if and only if m11 ≤
m1 (k1 +1) m1 +1
when k1 >m1. In [13],
Wang studied the case n = 2 of (1.1) in one dimension. Recently, Li et al. [5] extended th
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