Initial value problems for diffusion equations with singular potential

Let V be a nonnegative locally bounded function defined inQ∞ := R × (0,∞). We study under what conditions on V and on a Radon measure μ in R does it exist a function which satisfies ∂tu−∆u+ V u = 0 in Q∞ and u(., 0) = μ. We prove the existence of a subcritical case in which any measure is admissible and a supercritical case where capacitary conditions are needed. We obtain a general representation theorem of positive solutions when tV (x, t) is bounded and we prove the existence of an initial trace in the class of outer regular Borel measures.