Extinction and non-extinction for a polytropic filtration equation with a nonlocal source |
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Authors: | Yuzhu Han Wenjie Gao |
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Institution: | 1. Institute of Mathematics , Jilin University , Changchun 130012 , P.R. China hanyuzhu2003@yahoo.cn;3. Institute of Mathematics , Jilin University , Changchun 130012 , P.R. China |
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Abstract: | 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)?1, in a bounded domain Ω???R N (N?>?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?q?m(p???1), there exists a solution which is positive in Ω for all t?>?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. |
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Keywords: | polytropic filtration equation nonlocal source extinction |
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