Note on the strong hyperbolicity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> gravity with dynamical shifts

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