I this talk I will discuss some of the mathematical theories on nonlinear partial differential equations motivated by the desire of providing better models for various propagation phenomena. The talk will start by looking at the classical works of Fisher, Kolmogorov-Petrovskii-Piskunov and Aronson-Weinberger, and compare them with some recent results on models with free boundaries. Some new trends of research in this direction will also be mentioned.
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