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 = u −V (x)u+ up in RN ×(0,∞), u(x, 0) = φ(x) 0 in RN, where p > 1 and V behaves like ω|x|−2(1 + o(1)) with ω > 0, as |x| → ∞. In this paper we determine the so-called Fujita exponent p∗ for this Cauchy problem. Furthermore, for the critical case p = p∗, we prove that the Cauchy problem has no global positive solutions. © 2008 Elsevier Inc. All rights reserved