Initial singularities of positive solutions of the Heat equation on Stratified Lie groups
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916930327150592 |
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| author | Dewan, Utsav |
| author_facet | Dewan, Utsav |
| contents | 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 $\mathbb{R}^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_14051 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Initial singularities of positive solutions of the Heat equation on Stratified Lie groups Dewan, Utsav Classical Analysis and ODEs Analysis of PDEs Primary 43A80, 35R03, Secondary 28A78, 35K05 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 $\mathbb{R}^n$. |
| title | Initial singularities of positive solutions of the Heat equation on Stratified Lie groups |
| topic | Classical Analysis and ODEs Analysis of PDEs Primary 43A80, 35R03, Secondary 28A78, 35K05 |
| url | https://arxiv.org/abs/2311.14051 |