Critical exponent for a quasilinear parabolic equation with inhomogeneous density in a cone

In this paper, we study the initial-boundary value problem of porous medium equation ρ(x)u _t  = Δu ^m  + V(x)h(t)u ^p in a cone D = (0, ∞) × Ω, where ({V(x),{sim}, |x|^sigma, h(t),{sim}, t^s}). 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. We prove that if ({m < p leq 1+(m-1)(1+s)+frac{2(s+1)+sigma}{n+l}}), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if ({p >1 +(m-1)(1+s)+frac{2(s+1)+sigma}{n+l}}), then the problem has global solutions for some u ₀ ≥ 0.