Another proof for the removable singularities of the heat equation

TL;DR

This paper provides two simple proofs for the conditions under which singularities are removable in the heat equation for dimensions three and higher, and establishes a necessary and sufficient condition for two-dimensional cases.

Contribution

It introduces new straightforward proofs for the removability of singularities in the heat equation and characterizes the conditions for the two-dimensional case.

Findings

Two simple proofs for $n \\ge 3$ cases.

Necessary and sufficient condition for $n=2$.

Clarification of singularity removability criteria.

Abstract

We give two different simple proofs for the removable singularities of the heat equation in $(\Omega\setminus\{x_0\})\times (0,T)$ with $n\ge 3$. We also give a necessary and sufficient condition for removable singularities of the heat equation in $(\Omega\setminus\{x_0\})\times (0,T)$ for the case $n=2$.