Fujita-type conditions for fast diffusion equation with variable source

Abstract

This article studies Fujita-type theorems to the fast diffusion equation with variable source u t =Δu m + u p(x), x ∈ Ω ⊆ ℝ N , t ∈ (0, T), where m is a constant and p(x) is a continuous bounded function 0 < p − = inf p ≤ p(x) ≤ sup p = p +. First, all solutions are global if and only if p + ≤ p 0 = 1. Furthermore, when Ω = ℝ N , there are nontrivial global solutions when , while any nontrivial nonnegative solutions blow up in finite time if . Especially, in the case of , there are functions p(x) such that any nontrivial nonnegative solutions blow up in finite time and functions p(x) such that there exist nontrivial global solutions. In addition, for bounded Ω, some Fujita-type conditions are obtained as well: there are functions p(x) and domain Ω such that any nontrivial nonnegative solutions blow up in finite time, and the problem admits nontrivial global solutions provided Ω small enough, independent of the size of p(x).