Wiman–Valiron theory and entire solutions of differential equations

The Wiman–Valiron theory is more than one hundred years old. It enables one to estimate the growth of derviatives of an entire function $f(z)$ at the point $z_0$ of the circle $|z|=R$ where $|f(z)|$ attains its maximum value on this cicrle, in terms of $|f(z_0)|$ and quantities equivalent to the order and type of $f$. The talk is about various ways of generalization of this theory for functions of several complex variables and applications to two problems: the proof of absence of non-constant entire solutions of the Korteweg–de Vries equation and a description of solutions of complexity one for the heat equation.