  We study deterministic escape dynamics in the framework of the discrete
Klein-Gordon modelwith a repulsive quartic on-site potential. Using a
combination of analytical techniques, based on differential and algebraic
inequalities and selected numerical illustrations, we first derive conditions
for collapse of an initially excited single-site unit, for both the Hamiltonian
and the linearly damped versions of the system and showcase different potential
fates of the single-site excitation, such as the possibility to be "pulled
back" from outside the well or to "drive over" the barrier some of its
neighbors. Next, we study the evolution of a uniform (small) segment of the
chain and, in turn, consider the conditions that support its escape and
collapse of the chain. Finally, our path from one to the few and finally to the
many excited sites is completed by a modulational stability analysis and the
exploration of its connection to the escape process for plane wave initial
data. This reveals the existence of three distinct regimes, namely modulational
stability, modulational instability without escape and, finally, modulational
instability accompanied by escape. These are corroborated by direct numerical
simulations. In each of the above cases, the variations of the relevant model
parameters enable a consideration of the interplay of discreteness and
nonlinearity within the observed phenomenology.
