{\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.