We study the existence/nonexistence of global-in-time positive solutions of the Cauchy problem of semilinear heat equations with potential $V$. It is well known that the critical Fujita exponent, which separate that the problem possesses global-in-time solutions or not, depends on the behavior of the potential $V$. In particular, the case where $V$ decays quadratically at the space infinity is on the borderline where the critical Fujita exponent can vary in $(1,\infty]$, and there are several partial results. In this talk we identify the critical Fujita exponent for the case where $V$ is a radially symmetric potential decaying quadratically. The identification of the critical Fujita exponent for this problem is a delicate issue, in particular, when the Schrödinger operator $L_V:=-\Delta+V$ on $L^2$ is critical. Indeed, in the critical case, the critical Fujita exponent is different from previous results and it depends on a new threshold number. This talk is based on a joint work with Kazuhiro Ishige (University of Tokyo).