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Analytic first integrals of nonlinear parabolic equations and their applications
Sbornik. Mathematics, Tome 21 (1973) no. 3, pp. 339-369
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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. 
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M. I. Vishik; A. V. Fursikov. Analytic first integrals of nonlinear parabolic equations and their applications. Sbornik. Mathematics, Tome 21 (1973) no. 3, pp. 339-369. http://geodesic.mathdoc.fr/item/SM_1973_21_3_a0/

[1] M. I. Vishik, A. V. Fursikov, “Analiticheskie pervye integraly kvazilineinykh parabolicheskikh uravnenii”, Vestnik MGU, seriya matem., mekh., 1974, no. 1

[2] M. D. Donsker, J. L. Lions, “Volterra variational equations, boundary value problems and function space integrals”, Acta Math., 108 (1962), 147–228 | DOI | MR | Zbl

[3] E. Hopf, “Statistical hydromechanics and functional calculus”, J. Rational Mech., Anal., 1 (1952), 87–123 | MR | Zbl

[4] Ch. Foyash, “Statisticheskie resheniya nelineinykh evolyutsionnykh uravnenii”, Matematika, 17:3 (1973), 90–113