On the existence and nonexistence of global solutions for the porous medium equation with strongly nonlinear sources in a cone |
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Authors: | Songzhe Lian Changchun Liu |
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Institution: | 1. Department of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China
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Abstract: | In this paper, we study the initial-boundary value problem of the porous medium equation u
t
= Δu
m
+ V(x)u
p
in a cone D = (0, ∞) × Ω, where V(x) ~ (1 + |x|)
σ
. Let ω
1 denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω and let l denote the positive root of l
2 + (n − 2)l = ω
1. We prove that if m ≤ p ≤ m + (2 + σ)/(n + l), then the problem has no global nonnegative solutions for any nonnegative u
0 unless u
0 = 0; if p > m + (2 + σ)/n, then the problem has global solutions for some u
0 ≥ 0. |
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Keywords: | |
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