Critical Fujita Exponents for Semilinear Heat Equations with Quadratically Decaying Potential

We study the existence/nonexistence of global-in-time positive solutions of the Cauchy problem (P) {partial derivative(t)u = Delta u - V(x)u + u(p), x is an element of R-N, t > 0, u(x,0) = phi(x) >= 0, x is an element of R-,(N) where p > 1 and V is a radially symmetric function decaying quadratically at the space infinity. We identify the so-called critical Fujita exponent for problem (P).