{\em Perpetual strategies} in $\lambda$-calculus are analyzed from a {\em semantical}
perspective. This is achieved using suitable denotational models, {\em
computationally adequate} with respect to the {\em observational} ({\em
operational}) {\em equivalence}, $\approx_p$, induced by perpetual strategies. A
necessary and sufficient condition is given for an $\omega$-algebraic lattice,
isomorphic to the space of its {\em strict continuous self-maps}, to be
computationally adequate w.r.t $\approx_p$. While many such models exist, it is
shown, however, that none is {\em fully abstract} w.r.t. $\approx_p$. The
computationally adequate lattice model ${\cal D}^p$ is studied in detail. It is
used to give a semantical proof of the Conservation Theorem for the $\lambda$-calculus,
and to provide {\em coinductive} and {\em mixed inductive-coinductive} characterizations
of $\approx_p$. The coinductive characterization allows to show that the term model of
$\approx_p$ is a denotational model; this yields a new characterization of {\em
perpetual} redexes in $\lambda$-calculus.