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arXiv · 2311.14051

Initial singularities of positive solutions of the Heat equation on Stratified Lie groups

Utsav Dewan
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Abstract

Let $(\mathbb{G},\circ)$ be a stratified Lie group. We estimate the Hausdorff dimension (with respect to the Carnot-Carath\'eodory 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 $\mathbb{R}^n$.

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Utsav Dewan. 2023-11-23. Initial singularities of positive solutions of the Heat equation on Stratified Lie groups. https://arxiv.org/abs/2311.14051

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