Some questions in the theory of nonlinear elliptic and parabolic equations

For the nonlinear parabolic equation of order $m$
\begin{equation} \frac{\partial u}{\partial t}=-A(D)u+f(u,D^\gamma u),\qquad|\gamma|\leqslant m, \end{equation}
where the nonlinear part $f$ depends analytically on its arguments, in the case of periodic boundary conditions we prove a theorem about the unique solvability in a certain space of generalized functions if the initial condition is a eneralized function from the same class. We prove an analogous theorem for nonlinear elliptic equations.
We construct an asymptotic expansion (as $t\to\infty$) for the $\xi$th Fourier coefficient $v(t,\xi)$ of the solution $u(t,x)$ of a parabolic equation of the form (1).
Bibliography: 3 titles.