On the existence and nonexistence of global solutions for the porous medium equation with strongly nonlinear sources in a cone

In this paper, we study the initial-boundary value problem of the porous medium equation ut = Δum + V(x)up in a cone D = (0, ∞) × Ω, where V(x) ~ (1 + |x|)σ. Let ω1 denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω and let l denote the positive root of l2 + (n − 2)l = ω1. We prove that if m ≤ p ≤ m + (2 + σ)/(n + l), then the problem has no global nonnegative solutions for any nonnegative u0 unless u0 = 0; if p > m + (2 + σ)/n, then the problem has global solutions for some u0 ≥ 0.