We consider three types of semilinear second order
PDEs on a cylindrical domain $\Omega\times(0,\infty)$,
where $\Omega$ is a bounded domain in $R^N$, $N\ge 2$.
Among these, two are evolution problems of parabolic and
hyperbolic types, in which the unbounded direction of
$\Omega\times(0,\infty)$ is reserved for time $t$,
the third type is an elliptic equation with a singled
out unbounded variable $t$. We discuss the asymptotic
behavior, as $t\to\infty$, of solutions which are
defined and bounded on $\Omega\times(0,\infty)$.

This paper is based on the author's lecture at the conference
EQUADIFF 10 held in Prague in 2001. Detailed proofs are
contained in papers with F. Simondon and M.-A. Jendoubi.