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O.D.E. type behavior of blowup solutions of nonlinear heat equations
The problem Of blowup in nonlinear parabolic equations
1.  Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom 
2.  Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain 
A first part of the course introduces the subject and discusses the classical questions addressed by the blowup 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 blowup, the characterization of complete blowup, the occurrence of instantaneous blowup (i.e., immediately after the initial moment) and the construction of transient blowup 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.
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