  This paper is concerned with the Cauchy-Dirichlet problem for fast diffusion
equations posed in bounded domains, where every energy solution vanishes in
finite time and a suitably rescaled solution converges to an asymptotic
profile. Bonforte and Figalli (CPAM, 2021) first proved an exponential
convergence to nondegenerate positive asymptotic profiles for nonnegative
rescaled solutions in a weighted $L^2$ norm for smooth bounded domains by
developing a nonlinear entropy method. However, the optimality of the rate
remains open to question. In the present paper, their result is fully extended
to possibly sign-changing asymptotic profiles as well as general bounded
domains by improving an energy method along with a quantitative gradient
inequality developed by the first author (ARMA, 2023). Moreover, a
(quantitative) exponential stability result for least-energy asymptotic
profiles follows as a corollary, and it is further employed to prove the
optimality of the exponential rate.
