We consider the Dirichlet problem for the semilinear heat  equation
\begin{equation}\label{eq1} \tag{A}
  u_t=\Delta u+g(x,u),   \quad  x\in \Omega,
\end{equation}
where  $\Omega$ is an arbitrary bounded domain in $\mathbb R^N$, $N\ge 2$, with $C^2$ boundary. We find a $C^\infty$-function $g(x,u)$ such that \eqref{eq1} has a bounded solution whose $\omega$-limit set is a  continuum of equilibria. This extends and improves an earlier result   of the first author with Rybakowski,   in which $\Omega$ is a disk in $\mathbb R^2$ and $g$ is of finite differentiability class. We also show that  \eqref{eq1} can have an infinite-dimensional manifold of nonconvergent bounded trajectories.