The critical Fujita exponent for one-dimensional semilinear heat equations with potentials and space-dependent nonlinearities

This paper is concerned with the existence/nonexistence of nontrivial global-in-time solutions to the Cauchy problem {equation} {cases}{P}∂_tu-∂_x^2u+Vu=(1+x^2)-m/2u^p,&x,\ t>0,\\ u(x,0)=u_0(x)≥0,&x, {cases} {equation} where $p>1$, m≥0, u₀∈ BC(R) and the potential V=V(x)∈ BC(R) satisfies a certain property. More precisely, we determine the critical Fujita exponent for (P), that is, the threshold for the global existence/nonexistence of (P).