The critical Fujita exponent for the fast diffusion equation with potential

This paper studies the Cauchy problem for positive solutions of the fast diffusion equation with source and quadratically decaying potential u t = Δ u m − V ( x ) u m + u p in R n × ( 0 , T ) , where 1 − 2 m α + n < m < 1 , p > 1 , n ≥ 2 , V ( x ) ∼ ω | x | 2 with ω ≥ 0 as | x | → ∞ , and α is the positive root of m α ( m α + n − 2 ) − ω = 0 . We obtain the critical Fujita exponent p c = m + 2 m α + n to the problem in the sense that every nontrivial solution blows up in finite time when 1 < p ≤ p c , and there are both global and non-global solutions if p > p c .