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

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.