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

Abstract

The Cauchy--Dirichlet problem for the fast diffusion equation on a smooth bounded domain admits a natural bound in the weighted space $L_{{\Phi }_1}^p$, 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.