On the Fujita exponent for a semilinear heat equation with a potential term

We consider the existence and nonexistence of positive global solutions for the Cauchy problem,{∂tu=Δu−V(x)u+upinRN×(0,∞),u(x,0)=ϕ(x)⩾0inRN, where p>1p>1 and V   behaves like ω|x|−2(1+o(1))ω|x|−2(1+o(1)) with ω>0ω>0, as |x|→∞|x|→∞. In this paper we determine the so-called Fujita exponent p∗p∗ for this Cauchy problem. Furthermore, for the critical case p=p∗p=p∗, we prove that the Cauchy problem has no global positive solutions.