This talk is concerned with classification of extinction profiles for the Sobolev-critical fast diffusion equation, subject to the zero Dirichlet boundary condition within bounded domains. We show that, upon appropriate rescaling, each solution either converges strongly to a steady-state or undergoes blow-up at a finite set of points, with a trivial weak limit. Fine blow up behavior is obtained. This is based on joint work with Zheng-Chao Han and Tianling Jin. 

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