Let u be a solution of the heat equation which can be written as the difference of two non-negative solutions, and let v be a non-negative solution. A study is made of the behaviour of u(x, t)/v(x, t) as t → 0+. The methods are based on the Gauss-Weierstrass integral representation of solutions on Rn × ]0, a[ and results on the relative differentiation of measures, which are employed in a novel way to obtain several domination, non-negativity, uniqueness and representation theorems.

The aim of this article is to review the progress made in the last few years in the representation theory of solutions of parabolic systems in the sense of Petrowskii.

We establish a method of constructing kernels of Bergman operators for second-order linear partial differential equations in two independent variables, and use the method for obtaining a new class of Bergman kernels, which we call modified class E kernels since they include certain class E kernals. They also include other kernels which are suitable for global representations of solutions (whereas Bergman operators generally yield only local representations).

Both S. Bergman [1] and I. N. Vekua [13] have constructed integral operators which map ordered pairs of analytic functions of one complex variable onto solutions of fourth order elliptic equations in two independent variables. Such operators play an important role in the investigation of the analytic properties of solutions to higher order elliptic equations and in the approximation of solutions to boundary value problems associated with these equations. Unfortunately, little progress has been made in developing an analogous theory for elliptic equations in more than two independent variables. Recently, however, Colton and Gilbert [7] constructed integral operators for a class of fourth order elliptic equations with spherically symmetric coefficients in p + 2 (p ≥ 0) independent variables, and at present Dean Kukral [11], a student of R. P. Gilbert, is in the process of trying to extend some recent results of Colton [3, 4, 5] for second order equations in three independent variables to the fourth order case.