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A lecture exploring quantitative properties of nonnegative solutions to fast diffusion equations and porous medium equations on smooth bounded domains with homogeneous Dirichlet boundary conditions. Delivered as part of the Workshop on "Degenerate and Singular PDEs" at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI), this 48-minute presentation is structured in two main parts. First, survey the complete characterization of solution behavior in the Sobolev subcritical regime. Then, examine recent developments in the Sobolev critical case, establishing a dichotomy result where solutions either converge uniformly to a steady state or develop a blow-up profile with precise bubble structure under a two-bubble energy threshold condition on the initial data, with convergence measured in relative error.
