Kolmogorov nonlinear diffusion equation
1982
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Kolmogorov nonlinear diffusion equation
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1982
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The Kolmogorov nonlinear diffusion equation, u_t=\frac{1}{2}u_{xx}+f(u), was first investigated by Kolmogorov, Petrovsky, and Piscounov in their celebrated paper in 1937. After a long pause, there has been renewed interest in this equation over the past decade. In extending this equation to more general settings, various tools from both analysis and probability theory have been applied, including phase plane analysis, the maximum principle, Brownian motion estimates, and the Feynman-Kac formula. It is the intent of these notes to summarize much of the work done on the Kolmogorov equation to date, while emphasizing the interaction between analytic and probabilistic techniques. These notes are to a large extent based on a course given by Don Aronson and myself last spring at the University of Minnesota in Minneapolis. (Laboratoire de Probabilités, Université de Paris VI, Spring 1982)
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This file is a reproduction of notes on the KPP equation that I handed out for a course in spring, 1982 at the University of Paris VI. I have sporadically received requests for copies of the notes and, following a suggestion by Jay Rosen, it seems reasonable to put them on the web. Over the past ten years there have been many new developments in the area and some parts of these notes are now obsolete, but other parts may be of use to individuals interested in background in the subject or in a summary of my 1983 Memoirs paper. I have left these notes as they were, except for a few typos. The reader is warned that the summary of material to be covered that is given on pages 9-10 of the introductory chapter is inaccurate, since the content was modified as the course proceeded. In particular, Chapters 8-10 and 13 were reordered and the material intended for Chapters 11 and 12 was omitted.
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Bramson, Maury. (1982). Kolmogorov nonlinear diffusion equation. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/151582.
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