Blow-up of positive solutions for the semilinear heat equation with a potential

We study nonnegative solutions of the equation u(t) - Delta u + V(x)u = vertical bar u vertical bar(p-1)u in R-n, t>0, under the assumption that V(x) is an element of C-1(R-n) satisfies vertical bar V(x)vertical bar <= c. We establish a new blow-up criterion that depends only on V(x) and p. In addition, as the supreme of V(x) decreases, we find an interesting phenomenon that the Fujita exponent goes to infinity, in the sense that every nonnegative solution blows up in finite time whenever p>1. Furthermore, we obtain the blow-up rate estimate in the subcritical case. In the end, under the assumption of V(x) >= 0, we give the refined blow-up estimate of the blow-up solutions near the blow-up time.