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

2026/05/31 by Beomjun Choi, Xiqin Jiang, Hua-Yang Wang +1
Mathematics · #math.AP

paper · pdf

arxiv created 2026/08/01 · arxiv updated 2026/08/04

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_Φ1, where Φ₁ is the first Dirichlet eigenfunction. A key regularization question is whether this implies the stronger L^∞ 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.

Citations

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