Flows generated by symmetric functions of the eigenvalues of the Hessian

The global unique solvability of the first initial-boundary value problem for fully nonlinear equations of the form
$$ -u_t+f(\lambda_1[u],\dots,\lambda_n[u])=g $$
is proved. Here, $\lambda_i[u]$, $i=1,\dots,n$, are eigenvalues of the Hessian $u_{xx}$ and $f$ is a symmetric function satisfying some conditions. Bibliography: 7 titles.