This talk is concerned with the heat equation with a time-dependent Hardy-type singular potential. In the subcritical case, it is shown that there exist two types of positive solutions if the motion of the singularity is not so quick (at least $1/2$-H\"older continuous). On the other hand, when the singular point moves more quickly like a fractional Brownian motion with the Hurst index smaller than $1/2$, it is shown that a positive solution exists for a wider range of parameters.