Asymptotic expansion of moment functions of solutions of nonlinear parabolic equations

M. I. Vishik ; A. V. Fursikov

Sbornik. Mathematics, Tome 24 (1974) no. 4, pp. 575-591

Résumé

Let $ v(t,\xi)$ be the Fourier coefficients of the solution of the Cauchy problem for a nonlinear parabolic equation of the form $$ \frac{\partial u}{\partial t}=-A(D)u+f(u,D^\gamma u),\qquad|\gamma|\leqslant m, $$ where $A(D)$ is a linear elliptic operator of order $m$ and $f(u,D^\gamma u)$ is the nonlinear part of the equation. Then $M(t,\xi_1,\dots,\xi_k,\sigma)$ are the moment functions of the equation, i.e. the average of the function $v(t,\xi_1)\cdots v(t,\xi_k)$ with respect to a probability measure $\mu_\sigma$, where $\sigma$ characterizes the degree of concentration of the measure. In this paper we give an asymptotic expansion for the functions $M(t,\xi_1,\dots,\xi_k,\sigma)$ as $\sigma\to0$. Bibliography: 8 titles.