Archive for Rational Mechanics and Analysis

, Volume 109, Issue 1, pp 73–80

The value of the critical exponent for reaction-diffusion equations in cones


  • Howard A. Levine
    • Department of MathematicsIowa State University
  • Peter Meier
    • Department of MathematicsIowa State University

DOI: 10.1007/BF00377980

Cite this article as:
Levine, H.A. & Meier, P. Arch. Rational Mech. Anal. (1990) 109: 73. doi:10.1007/BF00377980


Let DRN be a cone with vertex at the origin i.e., D = (0, ∞)xΩ where Ω ⊂ SN−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: ut = Δu+up in D×(0, T), u=0 on ∂Dx(0, T) with u(x, 0)=u0(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) = ω1. 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 ut=Δ+¦x¦σup.

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© Springer-Verlag 1990