On a parabolic equation with a singular lower order term

Abstract

We obtain the existence of the weak Green’s functions of parabolic equations with lower order coefficients in the so called parabolic Kato class which is being proposed as a natural generalization of the Kato class in the study of elliptic equations. As a consequence we are able to prove the existence of solutions of some initial boundary value problems. Moreover, based on a lower and an upper bound of the Green’s function, we prove a Harnack inequality for the non-negative weak solutions.