  In this paper we analyse the stability of black hole Cauchy horizons arising
in a class of 2d dilaton gravity models. It is shown that due to the
characteristic asymptotic Rindler form of the metric of these models, time
dependent gravitational perturbations generated in the external region do not
necessarily blow-up when propagated along the Cauchy horizon. There exists, in
fact, a region of nonzero measure in the space of the parameters characterizing
the solutions such that both instability and mass inflation are avoided. This
is a new result concerning asymptotically flat space-times, not shared by the
well-known solutions of General Relativity. Despite this fact, however, quantum
back-reaction seems to produce a scalar curvature singularity there.
