Abstract
Let D ⊂ R N be a cone with vertex at the origin i.e., D = (0, ∞)xΩ where Ω ⊂ S N−1 and x ε D if and only if x = (r, θ) with r=¦x¦, θ ε Ω. We consider the initial boundary value problem: u t = Δu+u p in D×(0, T), u=0 on ∂Dx(0, T) with u(x, 0)=u 0(x) ≧ 0. Let ω1 denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let γ + denote the positive root of γ(γ+N−2) = ω 1. Let p * = 1 + 2/(N + γ+). If 1 < p < p *, no positive global solution exists. If p>p *, positive global solutions do exist. Extensions are given to the same problem for u t=Δ+¦x¦ σ u p.
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Communicated by J. Serrin
This research was supported in part by the Air Force Office of Scientific Research under Grant # AFOSR 88-0031 and in part by NSF Grant DMS-8 822 788. The United States Government is authorized to reproduce and distribute reprints for governmental purposes not withstanding any copyright notation therein.
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Levine, H.A., Meier, P. The value of the critical exponent for reaction-diffusion equations in cones. Arch. Rational Mech. Anal. 109, 73–80 (1990). https://doi.org/10.1007/BF00377980
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DOI: https://doi.org/10.1007/BF00377980