On the evolution equation with a dynamic Hardy-type potential

Abstract

Motivated by the celebrated paper of Baras and Goldstein (1984), we study the heat equation with a dynamic Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, non-existence and uniqueness of solutions, and obtain a sharp lower and upper bound near the singular point. Proofs are given by using solutions of the radial heat equation, some precise estimates for an equivalent integral equation and the comparison principle.