The Fujita exponent for semilinear heat equations with quadratically decaying potential or in an exterior domain

Consider the equation(0.1)ut=Δu−Vu+aupin Rn×(0,T);u(x,0)=ϕ(x)≩0in Rn, where p>1, n⩾2, T∈(0,∞], V(x)∼ω|x|2 as |x|→∞, for some ω≠0, and a(x) is on the order |x|m as |x|→∞, for some m∈(−∞,∞). A solution to the above equation is called global if T=∞. Under some additional technical conditions, we calculate a critical exponent p∗ such that global solutions exist for p>p∗, while for 1