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On the Heat Equation with a Time-Dependent Singular Potential☆

Abstract We introduce various classes of potentials on [0, T ]× R n and study the corresponding nonautonomous heat equation given by ∂u ∂t = 1 2 Δu−V(t,x)u. We prove the existence of the Feynman–Kac propagators for the nonautonomous heat equation and the ( L p − L q )-smoothing theorem for the propagators.