Some questions in the theory of nonlinear elliptic and parabolic equations

M. I. Vishik ; A. V. Fursikov

Sbornik. Mathematics, Tome 23 (1974) no. 2, pp. 287-318

Résumé

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.