Self-similar asymptotics in evolution equations

A function is called a self-similar solution of an evolution equation if its value at a given time t0 (e.g. at t0 = 1 ) it sufficient to calculate this function for all other values of time t > 0 via so-called self-similar transformation. Self-similar solutions have always played in important role in the study of properties of other solutions to linear and nonlinear evolution equations. Very often, one can find explicit self-similar solutions which describe typical properties of other solutions. For example, the Gauss-Weierstrass kernel G(x, t) = (4πt)−n/2e−|x| 2/(4t) is the most famous solution of the heat equation ut = ∆u. This explicit solution appears in the asymptotic expansions as t→∞ of other solutions to the initial value problem for the heat equation. In this series of lecture, participants will learn methods how to find for self-similar solutions of different nonlinear problem and how to use them to understand properties of other solutions.