Singular solutions to parabolic equations in nondivergence form

For any alpha is an element of (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.

Automated summary

This article presents some properties of solutions to parabolic equations and fully nonlinear uniformly parabolic equations. By constructing examples and using numerical computations, it proves certain properties of the solutions under specific conditions. These studies are of great importance for a deeper understanding and solving problems related to parabolic equations and nonlinear uniformly parabolic equations.