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arXiv · 2605.14622

Optimal Weighted Smoothing and Asymptotics of Ancient Solutions for Fast Diffusion Equations

Beomjun Choi·Xiqin Jiang·Hua-Yang Wang·Jingang Xiong
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Abstract

The Cauchy--Dirichlet problem for the fast diffusion equation on a smooth bounded domain admits a natural bound in the weighted space $L^p_{\Phi_1}$, where $\Phi_1$ is the first Dirichlet eigenfunction. A key regularization question is whether this implies the stronger $L^\infty$ bound. We provide a complete resolution, showing that the critical exponent coincides with the classical Brezis--Turner exponent from semilinear elliptic theory. As a primary application, we derive improved global Harnack inequalities and describe asymptotic behavior of positive ancient solutions.

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Beomjun Choi, Xiqin Jiang, Hua-Yang Wang, Jingang Xiong. 2026-05-14. Optimal Weighted Smoothing and Asymptotics of Ancient Solutions for Fast Diffusion Equations. https://arxiv.org/abs/2605.14622

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