  Here, cosmology of the late and future universe is obtained from
$f(R)$-gravity with non-linear curvature terms $R^2$ and $R^3$ ($R$ being the
Ricci scalar curvature). It is different from $f(R)$-dark enrgy models, where
non-linear curvature terms are taken as gravitational alternative of dark
energy. In the present model, neither linear nor no-linear curvature terms are
taken as dark energy. Rather, dark energy terms are induced by curvature terms
in the Friedmann equation derived from $f(R)$-gravitational equations. It has
advantage over $f(R)$- dark energy models in the sense that the present model
satisfies WMAP results and expands as $\sim t^{2/3}$ during matter-dominance.
So, it does not have problems due to which $f(R)$-dark energy models are
criticized. Curvature-induced dark energy, obtained here, mimics phantom.
Different phases of this model, including acceleration and deceleration during
phantom phase, are investigated here.It is found that expansion of the universe
will stop at the age $(3.87 t_0 + 694.4 {\rm kyr})$ ($t_0$ being the present
age of the universe) and after this epoch, it will contract and collapse by the
time $(336.87 t_0 + 694.4 {\rm kyr})$. Further,it is shown that universe will
escape predicted collapse (obtained using classical mechanics) on making
quantum gravity corrections relevant near collapse time due to extremely high
energy density and large curvature analogous to the state of very early
universe. Interestingly, cosmological constant is also induced here, which is
very small in classical domain, but very high in quantum domain.
