  Stationary solutions to the problem of particle acceleration at shock waves
in the non-linear regime, when the dynamical reaction of the accelerated
particles on the shock cannot be neglected, are known to show a prominent
energy flux escaping from the shock towards upstream infinity. On physical
grounds, the escape of particles from the upstream region of a shock has to be
expected in all those situations in which the maximum momentum of accelerated
particles, $p_{max}$, decreases with time, as is the case for the Sedov-Taylor
phase of expansion of a shell Supernova Remnant, when both the shock velocity
and the cosmic ray induced magnetization decrease. In this situation, at each
time $t$, particles with momenta larger than $p_{max}(t)$ leave the system from
upstream, carrying away a large fraction of the energy if the shock is strongly
modified by the presence of cosmic rays. This phenomenon is of crucial
importance for explaining the cosmic ray spectrum detected at Earth. In this
paper we discuss how this escape flux appears in the different approaches to
non-linear diffusive shock acceleration, and especially in the quasi-stationary
semi-analytical kinetic ones. We apply our calculations to the Sedov-Taylor
phase of a typical supernova remnant, including in a self-consistent way
particle acceleration, magnetic field amplification and the dynamical reaction
on the shock structure of both particles and fields. Within this framework we
calculate the temporal evolution of the maximum energy reached by the
accelerated particles and of the escape flux towards upstream infinity. The
latter quantity is directly related to the cosmic ray spectrum detected at
Earth.
