This talk addresses some quantitative properties of nonnegative solutions to the fast diffusion equations and porous medium equations on smooth bounded domains with the homogeneous Dirichlet boundary condition. The presentation consists of two main parts. The first part surveys the complete characterization of solution behavior in the Sobolev subcritical regime. The second part explores recent developments in the Sobolev critical case, where we establish a dichotomy result: under a two-bubble energy threshold condition on the initial data, solutions either converge uniformly to a steady state or develop a blow-up profile with precise bubble structure, with convergence measured in relative error.