$f(R)$ Gravity $k$-Essence Late-time Phenomenology
  In this work we shall study the late-time behavior of $k$-Essence $f(R)$
gravity without scalar potential, in the presence of matter and radiation
perfect fluids. We quantify the late-time study by using the statefinder
function $Y_H(z)=\frac{\rho_{DE}}{\rho_m^{(0)}}$, which is a function of the
redshift and of the Hubble rate. By appropriately rewriting the Friedmann
equation in terms of the redshift and of the function $Y_H(z)$, we numerically
solve it using appropriate initial conditions, and we critically examine the
effects of the $k$-Essence higher order kinetic terms. As we demonstrate, the
effect of the higher order scalar field kinetic terms on the late-time dynamics
is radical, since the dark energy oscillations are absent, and in addition, the
cosmological physical quantities are compatible with the latest Planck data and
also the model is almost indistinguishable from the $\Lambda$ Cold Dark Matter
model. This is in contrast to the standard $f(R)$ gravity case, where the
oscillations are present. Furthermore, by choosing a different set of values of
two of the free parameters of the model, and specifically the coefficient of
the higher order kinetic term and of the exponent of $R^{\delta}$ appearing in
the $f(R)$ action, we demonstrate that it is possible to obtain $\rho_{DE}<0$
for redshifts $z\sim 2-3.8$, which complies phenomenologically with, and seems
to explain, the observational data for the same redshifts, and also to obtain a
viable cosmological evolution at $z\sim 0$, at least when the dark energy
equation of state parameter and the dark energy density parameters are
considered.
