Another proof for the removable singularities of the heat equation

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 $n=2$.

Singularities of solutions of partial differential equations appear in many problems.
For example singularities appears in the study of the solutions of the harmonic map [13] and the harmonic map heat flow [3]. In [14] S. Sato and E. Yanagida studied the solutions for a semilinear parabolic equation with moving singularities. Singularities of solutions also appears in the study of hyperbolic partial differential equations [15] and in the study of the touchdown behavior of the micro-electromechanical systems equation [4], [5], [6].
It is interesting to find the necessary and sufficient condition for the solutions of the equations to have removable singularities. In [8] S.Y. Hsu proved the following theorem.
The proof in [8] is based on the Green function estimates of [9] and a careful analysis of the behavior of the solution near the singularities using Dehamel principle. In this paper we will use the Schauder estimates for heat equation [2], [12], and the technique of [1] and [7] to give two different simple proofs of the above result. We also obtain the following result for the solution of the heat equation in 2-dimension.
Theorem 2. Let 0 ∈ Ω ⊂ R 2 be a domain. Suppose u is a solution of the heat equation in (Ω \ {0}) × (0, T ). Then u has removable singularities at {0} × (0, T ) if and only if for any 0 < t 1 < t 2 < T and δ ∈ (0, 1) there exists B R 0 (0) ⊂ Ω depending on t 1 , t 2 and δ, such that for any 0 < |x| ≤ R 0 and t 1 ≤ t ≤ t 2 . We start with some definitions. For any set A we let χ A be the characteristic function of the set A. Let 0 ∈ Ω ⊂ R n be a bounded domain. We say that a solution u of the heat equation (1) Proof of Theorem 1: Suppose u has removable singularities at {0} × (0, T ). By the same argument as the proof in section 3 of [8] for any 0 < t 1 < t 2 < T and δ ∈ (0, 1) there exists B R 0 ⊂ Ω depending on t 1 , t 2 and δ, such that (2) holds.
Proof of Theorem 2: Theorem 2 follows by an argument very similar to the proof of Theorem 1 but with (3) replacing (2) in the argument.