On the Heat Equation with a Time-Dependent Singular Potential

We introduce various classes of potentials on [0, T]×Rn and study the corresponding nonautonomous heat equation given by ∂u∂t=12Δu−V(t,x)u. We prove the existence of the Feynman–Kac propagators for the nonautonomous heat equation and the (Lp−Lq)-smoothing theorem for the propagators.