Our purpose in this paper is to study solitons of the spacelike mean curvature flow in a generalized Robertson-Walker (GRW) spacetime -I x_f Mⁿ. Under suitable constraints on the warping function f and on the curvatures of the Riemannian fiber Mⁿ, we apply suitable maximum principles in order to obtain nonexistence and uniqueness results concerning these solitons. Applications to standard models of GRW spacetimes, namely, the Einstein-de Sitter spacetime, steady state type spacetimes, de Sitter and anti-de Sitter spaces, are given. Furthermore, we establish new Calabi-Bernstein type results related to entire spacelike mean curvature flow graphs constructed over the Riemannian fiber of the ambient spacetime.

The first author is partially supported by CNPq [Grant: 308440/2021-8 and 405468/2021-0], and FAPEAL [Grant: E:60030.0000001758/2022], and the third author is partially supported by CNPq [Grant: 301970/2019-0]. The second author is partially supported by INdAM-GNAMPA Research Project 2020 titled \textit{Equazioni alle derivate parziali: problemi e modelli}. The fourth author is partially supported by CAPES [Finance Code 001], Brazil.