Green’s function for second order parabolic equations with singular lower order coefficients

We construct Green’s functions for second order parabolic operators of the form Pu = ∂tu−div(A∇u+bu)+c·∇u+du in (−∞, ∞)×Ω, where Ω is an open connected set in Rⁿ. It is not necessary that Ω to be bounded and Ω = Rⁿ is not excluded. We assume that the leading coefficients A are bounded and measurable and the lower order coefficients b, c, and d belong to critical mixed norm Lebesgue spaces and satisfy the conditions d − divb ≥ 0 and div(b−c) ≥ 0. We show that the Green’s function has the Gaussian bound in the entire (−∞, ∞) × Ω.