On parabolic convergence of positive solutions of the heat equation

TL;DR

This paper investigates the boundary behavior of positive solutions to the heat equation in the upper half-space, establishing the equivalence between parabolic limits and strong derivatives of boundary measures.

Contribution

It proves that the existence of a parabolic limit at a boundary point is equivalent to the existence of a strong derivative of the boundary measure, linking boundary limits to measure derivatives.

Findings

Parabolic limits exist if and only if strong derivatives of boundary measures exist.

Parabolic limits and strong derivatives are equal at boundary points.

Provides a characterization of boundary behavior for positive heat equation solutions.

Abstract

In this article, we study certain type of boundary behaviour of positive solutions of the heat equation on the upper half-space of $\R^{n+1}$. We prove that the existence of the parabolic limit of a positive solution of the heat equation at a point in the boundary is equivalent to the existence of the strong derivative of the boundary measure of the solution at that point. Moreover, the parabolic limit and the strong derivative are equal.