Extinction and non-extinction for a polytropic filtration equation with a nonlocal source
Extinction and non-extinction for a polytropic filtration equation with a nonlocal source
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DOI：
10.1080/00036811.2011.632766
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发表时间：
2013-02
影响因子：
1.1
通讯作者：
Yuzhu Han;Wenjie Gao
中科院分区：
文献类型：
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作者：
Yuzhu Han;Wenjie Gao
在本文中，作者建立了快速扩散多方过滤方程 u t = div(|∇u m | p−2∇u m ) + a∫Ω u q (y, t)dy 的有限时间内解消失的条件，其中 a, q, m > 0, p > 1, m(p − 1) 2)。更准确地说，如果 q > m(p − 1)，任何具有小初始数据的非负解都会在有限时间内消失，如果 0 0。对于临界情况 q = m(p − 1)，解是否会在有限时间内消失取决于 a 和 μ 之间的比较，其中 μ = ∫ Ωφ p−1(x)dx 且 φ 是椭圆问题的唯一正解 −div(|∇φ| p−2∇φ) = 1, x ∈ Ω; φ(x) = 0, x ∈ ∂Ω。
In this article, the authors establish the conditions for the extinction of solutions, in finite time, of the fast diffusive polytropic filtration equation u t = div(|∇u m | p−2∇u m ) + a∫Ω u q (y, t)dy with a, q, m > 0, p > 1, m(p − 1) 2). More precisely speaking, it is shown that if q > m(p − 1), any non-negative solution with small initial data vanishes in finite time, and if 0 0. For the critical case q = m(p − 1), whether the solutions vanish in finite time or not depends on the comparison between a and μ, where μ = ∫ Ωφ p−1(x)dx and φ is the unique positive solution of the elliptic problem −div(|∇φ| p−2∇φ) = 1, x ∈ Ω; φ(x) = 0, x ∈ ∂Ω.