in some domain D C Rn+l that vanishes locally on some distinguished part of OD. The term refers to the fact that in some sense, along that part of OD, u resembles the fundamental solution of the heat equation in D. This is particularly the case when the domain under consideration, D, is the intersection of some n+1-dimensional cube Q with Q, one side of a Lipschitz graph; i.e., Q = {fx > f(x', t)} (that is D = Q n Q) and the distinguished part of OD is precisely (&Q) n Q. This is the main first area of study of this paper, relying on work of Fabes, Garofalo, Salsa [FGS] on backward Harnack type inequalities for such domains. Lipschitz regularity in time, versus Lipschitz regularity in space, is not, of course, the natural homogeneity balance for the study of parabolic equations, but it is so for the study of phase transition relations of the form