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

TL;DR

This paper derives sharp weighted smoothing estimates for fast diffusion equations, linking critical exponents to classical elliptic theory, and applies these results to improve inequalities and understand ancient solutions.

Contribution

It introduces new weighted smoothing estimates for fast diffusion equations and connects critical exponents to classical elliptic results, enhancing understanding of solution behavior.

Findings

Critical exponent matches the Brezis–Turner exponent.

Established sharp weighted smoothing estimates for solutions.

Derived improved global Harnack inequalities and asymptotic descriptions.

Abstract

We establish sharp weighted smoothing estimates for limit solutions to the Cauchy-Dirichlet problem for the fast diffusion equation on smooth bounded domains. We demonstrate that the critical exponent governing these estimates coincides with the classical Brezis--Turner exponent known in the theory of semilinear elliptic equations. As a primary application, we derive improved global Harnack inequalities and describe asymptotic behavior of positive ancient solutions.