Rates of Convergence to Non-degenerate Asymptotic Profiles for Fast Diffusion via Energy Methods

This paper is concerned with a quantitative analysis of asymptotic behaviors of (possibly sign-changing) solutions to the Cauchy-Dirichlet problem for the fast diffusion equation posed on bounded domains with Sobolev subcritical exponents. More precisely, rates of convergence to non-degenerate asymptotic profiles are revealed via an energy method. The sharp rate of convergence to positive ones was recently discussed by Bonforte and Figalli (Commun Pure Appl Math 74:744-789, 2021) based on an entropy method. An alternative proof for their result is also provided. Furthermore, the dynamics of fast diffusion flows with changing signs is discussed more specifically under concrete settings; in particular, exponential stability of some sign-changing asymptotic profiles is proved in dumbbell domains for initial data with certain symmetry.