Qs. Zhang, ON A PARABOLIC EQUATION WITH A SINGULAR LOWER ORDER TERM - PART II - THE GAUSSIAN BOUNDS, Indiana University mathematics journal, 46(3), 1997, pp. 989-1020
We establish a lower and an upper Gaussian bound for the fundamental s
olutions of some general parabolic equations with potentials under a m
inimum integrability assumption. These parabolic equations include the
second order uniformly parabolic equations in Euclidean spaces and th
e heat equations corresponding to some subelliptic operators. Even in
the uniformly parabolic case, the existence of a Gaussian upper bound
in the presence of these singular potentials was not known and has bee
n questioned by a number of people. We also use these bounds to prove
the Harnack inequality, the local boundedness and continuity of weak s
olutions. Part II is logically independent of the previous paper [Z].