Initial singularities of positive solutions of the Heat equation on Stratified Lie groups
Utsav Dewan · arXiv (Cornell University) · 2023
Let $(\mathbb{G},\circ)$ be a stratified Lie group. We estimate the Hausdorff dimension (with respect to the Carnot-Carathéodory metric) of the singular sets in $\mathbb{G}$, where a positive solution of the Heat equation corresponding to a sub-Laplacian, blows up faster than a prescribed rate along normal limits, in terms of the homogeneous dimension of $\mathbb{G}$ and the rate of the blowup parameter. This generalizes a classical result of Watson for the Euclidean Heat. We also obtain the corresponding sharpness result, which is new even for $\mathbb{R}^n$.