The critical Fujita exponent for a diffusion equation with a potential term

摘要

We study the initial-boundary-value problem of the diffusion equation u_t=Δu~m-V (x)u~m+u~p in a conelike domain D=[1,∞)× Ω, where V(x)~ω_2|x|~(-2) with ω_2>0. 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+ω_2. We prove that if m≤m+2/(n+l), then the problem has no global nonnegative solutions for any nonnegative u0 unless u_0=0; if p>m+2/(n+l), then the problem has global solutions for some u_0≥0.