  The stability of asymptotic profiles of solutions to the Cauchy-Dirichlet
problem for Fast Diffusion Equation (FDE, for short) is discussed. The main
result of the present paper is the stability of any asymptotic profiles of
least energy. It is noteworthy that this result can cover non-isolated
profiles, e.g., those for thin annular domain cases. The method of proof is
based on the Lojasiewicz-Simon inequality, which is usually used to prove the
convergence of solutions to prescribed limits, as well as a uniform extinction
estimate for solutions to FDE. Besides, local minimizers of an energy
functional associated with this issue are characterized. Furthermore, the
instability of positive radial asymptotic profiles in thin annular domains is
also proved by applying the Lojasiewicz-Simon inequality in a different way.
