THE VALUE OF THE CRITICAL EXPONENT FOR REACTION-DIFFUSION EQUATIONS IN CONES

Let D ⊂ RN 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. © 1990 Springer-Verlag.