We consider the parabolic equation
(P) $u_t - Δu = F(x,u)$, (t,x) ∈ ℝ₊ × ℝⁿ,
and the corresponding semiflow π in the phase space H¹. We give conditions on the nonlinearity F(x,u), ensuring that all bounded sets of H¹ are π-admissible 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 extend earlier results of Rybakowski concerning parabolic equations on bounded open subsets of ℝⁿ.