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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The work was supported by the 985 program of Jilin University.
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Lian, S., Liu, C. On the existence and nonexistence of global solutions for the porous medium equation with strongly nonlinear sources in a cone. Arch. Math. 94, 245–253 (2010). https://doi.org/10.1007/s00013-009-0081-9
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DOI: https://doi.org/10.1007/s00013-009-0081-9