We consider two types of equations on a cylindrical domain
$\Omega\times(0,\infty)$, where $\Omega$ is a bounded domain
in $R^N$, $N\ge 2$. The first type is a semilinear damped
wave equation, in which the unbounded direction of
$\Omega\times(0,\infty)$ is reserved for time $t$.
The second type is an elliptic equation with a singled out
unbounded variable $t$. In both cases, we consider solutions
which are defined and bounded on  $\Omega\times(0,\infty)$
and satisfy Dirichlet boundary condition on
$\partial \Omega\times(0,\infty)$. We show that for some
nonlinearities, the equations have bounded solutions that
do not stabilize to any single function $\phi:\Omega\to R$,
as $t\to \infty$; rather they approach a continuum of such
functions. This happens despite the presence of damping
in the equation which forces the $t$-derivative of bounded
solutions to converge to $0$ as $t\to\infty$. Our results
contrast with known stabilization properties of solutions
of such equations in the case $N=1$.