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Singularity Formation for the Semi-Linear Heat Equation
A seminar to be conducted by Dr. Charles Collot, New York University in Abu Dhabi, United Arab Emirates.
Abstract: This talk is about singularity formation (also called blow-up) for evolution partial differential equations. Solutions to such equations, sometimes, may only be well defined up to a finite maximal time of existence T>0. Some phenomenon then happens and prevents the solution to be extended beyond that time. A general goal is to understand if a singularity formation is possible, and then to describe it, or if it is not. The semi-linear heat equation (will be given during the talk) is a model equation for this issue. Despite its simple form, the dynamics of the solutions are indeed very rich. The blow-up problem was initiated by Fujita in the 1960s and still receives attention today with new techniques coming from elliptic and dispersive equations.
The first part of the presentation will be an introduction to the different types of singularities, the key property being the so-called "self-similarity."
The second part will focus on the results recently obtained by the author, in partial collaboration with F. Merle, P. Raphael and J. Szeftel.
The major part of the talk is aimed at a general audience with background in dynamical systems, modelling, functional analysis or PDEs. References to singularity formation in fluid mechanics and dispersive equations will be given.
Dr. Collot's research area is PDEs and their applications.