Scalar-Tensor Quintessence with a linear potential: Avoiding the Big
  Crunch cosmic doomsday
  All quintessence potentials that are either monotonic with negative interval
or have a minimum at negative values of the potential, generically predict a
future collapse of the scale factor to a "doomsday" singularity. We show that
this doomsday is generically avoided in models with a proper non-minimal
coupling of the quintessence scalar field to the curvature scalar $R$. For
simplicity we consider linear quintessence potential $V=-s\phi$ and linear
non-minimal coupling $F=1-\lambda \phi$. However our result is generic and is
due to the fact that the non-minimal coupling modifies the effective potential
that determines the dynamics of the scalar field. Thus for each positive value
of the parameter $s$ we find a critical value $\lambda_{crit}(s)$ such that for
$\lambda>\lambda_{crit}(s)$ the negative potential energy does not dominate the
universe and the cosmic doomsday Big Crunch singularity is avoided because the
scalar field eventually rolls up its potential. We find that
$\lambda_{crit}(s)$ increases approximately linearly with $s$. For
$\lambda>\lambda_{crit}(s)$ the potential energy of the scalar field becomes
positive and it eventually dominates while the dark energy equation of state
parameter tends to $w=-1$ leading to a deSitter Universe.
