  The well-posedness of the gravitational equations of $f(R)$ gravity are
studied in this paper. Three formulations of the $f(R)$ gravity with dynamical
shifts (which are all based on the Arnowitt-Deser-Misner (ADM) formalism of the
equations) are investigated. These three formulations are all proved to be
strongly hyperbolic by pseudodifferential reduction. The first one is the
Baumagarte-Shapiro-Shibata-Nakamura (BSSN) formulation with the so-called
"hyperbolic $K$-driver" condition and the "hyperbolic Gamma driver" condition.
The second one is the ADM formulation with modified harmonic gauge conditions.
We find that the equations are not strong hyperbolic in traditional Z4
formulation for $f(R)$ gravity. So, in the third formulation, we improve the Z4
formulation, and show these equations are strong hyperbolic with modified
harmonic gauge conditions.
