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

2023/11/23 by Dewan, Utsav
#35K05 #35R03 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary 43A80 #Secondary 28A78

paper · doi:10.48550/arxiv.2311.14051

Abstract

Let (\mathbbG,∘) be a stratified Lie group. We estimate the Hausdorff dimension (with respect to the Carnot-Carathéodory metric) of the singular sets in \mathbbG, 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 \mathbbG 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 ℝⁿ.

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