Analytic first integrals of nonlinear parabolic equations and their applications

First integrals of the nonlinear parabolic equation
\begin{equation} \frac{\partial u(t,x)}{\partial t}=\mathfrak U(u),\qquad x\in R^n,\quad t>t_0, \end{equation}
are considered, i.e. functionals $G(t,u)$ that are constant on solutions $u(t,x)$ of (1): $G(t,u(t,x))=\mathrm{const}$. Every first integral satisfies a first-order variational differential equation. A solution of the Cauchy problem is constructed for this equation. The method of constructing these solutions, i.e. first integrals, affords a number of corollaries concerning statistical characteristics of solutions of (1).
Bibliography: 4 titles.