  In this paper we deal with a reaction-diffusion equation in a bounded
interval of the real line with a nonlinear diffusion of Perona-Malik's type and
a balanced bistable reaction term. Under very general assumptions, we study the
persistence of layered solutions, showing that it strongly depends on the
behavior of the reaction term close to the stable equilibria $\pm1$, described
by a parameter $\theta>1$. If $\theta\in(1,2)$, we prove existence of steady
states oscillating (and touching) $\pm1$, called $compactons$, while in the
case $\theta=2$ we prove the presence of $metastable$ $solutions$, namely
solutions with a transition layer structure which is maintained for an
exponentially long time. Finally, for $\theta>2$, solutions with an unstable
transition layer structure persist only for an algebraically long time.
