  We consider the parabolic equation $$u_t-\Delta u=F(x,u),\quad
(t,x)\in\R_+\times\R^n\tag{P}$$ and the corresponding semiflow $\pi$ in the
phase space $H^1$. We give conditions on the nonlinearity $F(x,u)$, ensuring
that all bounded sets of $H^1$ are $\pi$-admissibile in the sense of
Rybakowski. If $F(x,u)$ is asymptotically linear, under appropriate
non-resonance conditions, we use Conley's index theory to prove the existence
of nontrivial equilibria of (P) and of heteroclinic trajectories joining some
of these equilibria. The results obtained in this paper extend earlier results
of Rybakowski concerning parabolic equations on {\it bounded} open subsets of
$\R^n$.
