Singular solutions to parabolic equations in nondivergence form

TL;DR

This paper constructs examples of solutions to parabolic equations with isolated singularities, demonstrating limitations on regularity and uniqueness, and explores the existence of nonhomogeneous solutions with singularities.

Contribution

It provides explicit examples of singular solutions in parabolic equations, showing the nonexistence of certain highly regular singular solutions and constructing nonhomogeneous solutions with singularities.

Findings

Existence of solutions with isolated singularities not better than $C^eta$.

No solutions with isolated singularities that are $C^2$ except at the singularity and analytic elsewhere.

Numerical verification of a nonhomogeneous solution with an isolated singularity.

Abstract

For any $\alpha \in (0,1)$, we construct an example of a solution to a parabolic equation with measurable coefficients in two space dimensions which has an isolated singularity and is not better that $C^\alpha$. We prove that there exists no solution to a fully nonlinear uniformly parabolic equation, in any dimension, which has an isolated singularity where it is not $C^2$ while it is analytic elsewhere, and it is homogeneous in $x$ at the time of the singularity. We build an example of a non homogeneous solution to a fully nonlinear uniformly parabolic equation with an isolated singularity, which we verify with the aid of a numerical computation.