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 ut=Δum−V(x)um+uput=Δum−V(x)um+up in Rn×(0,T)Rn×(0,T), where 1−2mα+n1p>1, n≥2n≥2, V(x)∼ω|x|2 with ω≥0ω≥0 as |x|→∞|x|→∞, and αα is the positive root of mα(mα+n−2)−ω=0mα(mα+n−2)−ω=0. We obtain the critical Fujita exponent pc=m+2mα+n to the problem in the sense that every nontrivial solution blows up in finite time when 1pcp>pc.