  We study the long-time behavior of solutions of the $k$-Hessian evolution
equation $u_t=S_{k}(D^2 u)$, posed on a bounded domain of the $n$-dimensional
space with homogeneous boundary conditions. To this end, we construct a
separable solution and we show that the long-time behavior of $u$ is precisely
described by this special solution. Further, we initiate the study of that
dynamic phenomenon on the entire space, providing a new class of explicit and
radially symmetric self-similar solutions that we call $k$-Barenblatt
solutions. These solutions present some common properties as those of
well-known Barenblatt solutions for the porous media equation and the
$p$-Laplacian equation. It is known that self-similar solutions are important
in describing the intermediate asymptotic behavior of general solutions.
