The problem of blow-up in nonlinear parabolic equations

The course aims at presenting an introduction to the subject of singularity formation in nonlinear evolution problems usually known as blowup. In short, we are interested in the situation where, starting from a smooth initial configuration, and after a first period of classical evolution, the solution (or in some cases its derivatives) becomes infinite in finite time due to the cumulative effect of the nonlinearities. We concentrate on problems involving differential equations of parabolic type, or systems of such equations. A first part of the course introduces the subject and discusses the classical questions addressed by the blow-up theory. We propose a list of main questions that extends and hopefully updates on the existing literature. We also introduce extinction problems as a parallel subject. In the main bulk of the paper we describe in some detail the developments in which we have been involved in recent years, like rates of growth and pattern formation before blow-up, the characterization of complete blow-up, the occurrence of instantaneous blow-up (i.e., immediately after the initial moment) and the construction of transient blow-up patterns (peaking solutions), as well as similar questions for extinction. In a final part we have tried to give an idea of interesting lines of current research. The survey concludes with an extensive list of references. Due to the varied and intense activity in the field both aspects are partial, and reflect necessarily the authors' tastes.