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 equation 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. (C) 2019 Elsevier Inc. All rights reserved.