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 \begin{equation} \begin{cases}\tag{P}\partial_tu-\partial_x^2u+Vu=(1+x^2)^{-\frac{m}{2}}u^p,&x\in$\mathbb{R}$,\ t>0,\ u(x,0)=u_0(x)\ge0,&x\in$\mathbb{R}$, \end{cases} \end{equation} where $p>1$, $m\ge 0$, $u_0\in BC(\mathbb{R})$ and the potential $V=V(x)\in BC(\mathbb{R})$ 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).