We consider the prescribed mean curvature problem of spacelike graphs in Robertson-Walker spacetimes of flat fiber with homogeneous Dirichlet conditions on an Euclidean ball. Under reasonable assumptions, it is shown that every possible solution must be radially symmetric. Besides, an existence result for a singular nonlinear equation is proved by making use of the classical Schauder fixed point Theorem. The results (which assume a sufficiently small radius of the ball) are applied to relevant examples of Robertson-Walker spacetimes.

In the second part, we improve the result deleting the size radius assumption. As a consequence, we provide sufficient conditions for the existence of entire spacelike graphs with prescribed mean curvature in a Robertson-Walker spacetime with flat fiber. The proof is based on the analysis of the associated homogeneous Dirichlet problem on an Euclidean ball together with suitable bounds for the gradient which permit the prolongability of the solution to the whole space.