  We define the hyperbolic Yamabe flow and we obtain some properties of its
stationary solutions, namely, of hyperbolic Yamabe solitons. We also consider
immersed submanifolds as hyperbolic Yamabe solitons and we prove that, under
certain assumptions, a hyperbolic Yamabe soliton hypersurface is
pseudosymmetric or metallic shaped. Further we characterize the hyperbolic
Yamabe soliton factor manifolds of a multiply twisted, multiply warped, doubly
warped, and warped product manifold, and we provide a classification for a
complete gradient hyperbolic Yamabe soliton factor manifold. Finally, we
determine the conditions when the factor manifolds are hyperbolic Yamabe
solitons if the manifold is a hyperbolic Yamabe soliton and we illustrate this
result for the Robertson--Walker spacetime.
