Bubbling blow - up in critical parabolic problems

These lecture notes are devoted to the analysis of blow-up of solutions for some parabolic equations that involve bubbling phenomena. The term bubbling refers to the presence of families of solutions which at main order look like scalings of a single stationary solution which in the limit become singular but at the same time have an approximately constant energy level. This arise in various problems where critical loss of compactness for the underlying energy appears. Three main equations are studied, namely: the Sobolev critical semilinear heat equation in \(\mathbb{R}^{n}\), the harmonic map flow from \(\mathbb{R}^{2}\) into S2, the Patlak-Keller-Segel system in \(\mathbb{R}^{2}\).