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Critical exponents for a non‐Newtonian polytropic filtration equation with weighted nonlocal inner sources
Author(s) -
Zheng Yadong,
Bo Fang Zhong
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6051
Subject(s) - polytropic process , mathematics , exponent , critical exponent , initial value problem , mathematical analysis , function (biology) , newtonian fluid , cauchy problem , mathematical physics , physics , classical mechanics , geometry , linguistics , philosophy , evolutionary biology , biology , scaling
This article considers the Cauchy problem for a non‐Newtonian polytropic filtration equation with weighted nonlocal inner sourcesu t = div ( | ∇ u m | p − 2 ∇ u m ) +∫ R NK ( x ) u q ( x , t ) d xr − 1 qu s + 1 , ( x , t ) ∈ R N × ( 0 , T ) , where N ≥ 1, 1 − 1 + m 1 + m N < m ( p − 1 ) < 1 , 0< m ≤ 1, q >1, r ≥ 1, 0 ≤ s < p N + m ( p − 1 ) − 1 , and r + s >1. We first obtain a new critical Fujita exponent by virtue of the auxiliary function method and the forward self‐similar solution and then determine the second critical exponent to classify global and nonglobal solutions of the problem in the coexistence region via the decay rates of an initial data at spatial infinity. Moreover, the large time behavior of global solution and the life span of nonglobal solution are derived.